On Analogue-Gravity Models As Seen by In-Universe Observers a Thesis

On analogue-gravity models as seen by in-universe observers

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

Scott Lockerbie Todd

BAppSc(Nano)/BAppSc(Phys)(Hons) RMIT

School of Science

College of Science, Engineering and Health

RMIT University

October 2020

Declaration

I certify that except where due acknowledgement has been made, the work is that of the author alone; the work has not been submitted previously, in whole or in part, to qualify for any other academic award; the content of the thesis is the result of work which has been carried out since the oﬃcial commencement date of the approved research program; any editorial work, paid or unpaid, carried out by a third party is acknowledged; and, ethics procedures and guidelines have been followed. I acknowledge the support I have received for my research through the provision of an Australian Government Research Training Program Scholarship.

Scott L. Todd October 16, 2020

Acknowledgements

Attempting to acknowledge every specific person who was of importance in my journey as a PhD candidate—important in the production of this thesis—seems to be at best a daunting task, and at worst a near-impossible task. If you, the reader, are somebody in particular to whom I owe a debt of gratitude, and if your name does not appear in the acknowledgements to come, then I sincerely apologize for the oversight; the omission is, I promise, unintentional. So where to begin? I guess I’ll try to do this somewhat chronologically. While it should go without saying, first and foremost: thank you, Mum and Dad. It’s difficult for me to properly articulate the multitude of ways in which I am thankful to you, but at the very least—and of direct relevance to this thesis—I think I can safely say that I likely would not be the inquisitive person that I am today (and I likely would not be writing an acknowledgement section for a doctoral thesis) had you not put so much effort into attempting to sate my curiosity at a young age. Much love and thanks to the both of you. I think it only fair to thank all of the education staff—from those earliest of teachers who taught me in primary school to those academics still teaching me in the present—whose collective hard work played no small role in my path through life leading to this point. Despite not knowing any of them personally, and despite basically all of my interactions with them having been remote, I feel like it would be unfair to not thank the staff at the RMIT library: so to the RMIT library staff, thank you! I definitely appreciate all of the effort that must have gone into acquiring all of the documents that I requested over the course of my PhD once my own attempts had failed to bear fruit. I am deeply thankful to the RMIT physics department collectively. Thank you to the academic staff and my fellow students for all of the guidance—both academic and personal—that has individually and collectively been imparted to me in my time as a student at RMIT. In particular of the RMIT physics department academics, thank you to Professor Andrew Greentree, Professor Jared Cole, and Professor Salvy Russo: the three of you have certainly put up with more unpleasantness from me than most of the academic staff at RMIT, in spite of which you have all remained absolutely phenomenal pillars of support both academically and personally. For that, I thank the three of you deeply. To Daniel Stavrevski and Giannis Thalassinos: you are two of the best friends that I have ever had the pleasure of making. The pair of you have ensured that my time at RMIT has been enjoyable since day one of our undergraduate (actually, day zero in Daniel’s case), and in complete honesty, I genuinely do not think that I could have gotten this far without the two of you. Your friendship and comradery means more to me than I’ll probably ever be able to express. I thank you both deeply for the everything. Furthermore on the student-front, I would like to specifically acknowledge Akib Karim, David Ing, Josef Worboys, Sam Wilkinson, and Tommy Bartolo. I could write paragraphs of thanks to each of you... but I won’t. Suffice it to say, your collective support, guidance, and friendship has been of immense importance to me through my time as a PhD student, and I am thankful towards the lot of you! Along with Daniel and Giannis, you guys have provided much entertainment, many fun times, and a whole lot of food for thought. Sam, you get bonus props for being an awesome DM during our D&D sessions (and I use the word “awesome” both in the original sense of the word and its more modern-day usage here)! Thank you to Dr Jamie Booth for all of the interesting discussions, guidance, and endless supply of entertaining anecdotes and stories! It should go without saying (though I shall of course say it anyhow) that I owe an immense debt of gratitude to the supervisory team that has seen me through my PhD candidature. Of my supervisors, I owe a special debt of gratitude to my primary supervisor Professor Nicolas C. Menicucci who has been a phenomenal mentor in more ways than I can adequately articulate: thank you very much for your endless patience, your guidance, and your feedback; I am—without doubt—a much better physicist for it. Thank you also to Doctor Valentina Baccetti, who joined as a secondary supervisor relatively late on during the course of my PhD candidature, but whose input as both a supervisor and a co-author has been invaluable. Salvy, you already got a thanks earlier on, so don’t get greedy! Penultimately, thank you to Past Scott, who in a cruel twist of fate can never read this acknowledgement. Past Scott was far from perfect—Present Scott isn’t either, and I very much doubt Future Scott will be for that matter—however he did put in a good amount of leg work at the start of the PhD that made putting this thesis together much easier than it otherwise might have been (and even then, it was a struggle at times). In spite of his desire to procrastinate, Past Scott made a decent effort to write explicit derivations down for Future (now Present) Scott’s reference/use, and I thank him for that. Finally, though definitely not least, thank you to my wonderful partner Thamale! Your endless love, support, and fantastic sense of humour has kept me sane and has made this whole thing possible. My love for you is nothing short of Jupiter-sized! rrr

iv Summary

The contents of this thesis include several investigations into analogue-gravity models from an internal or in-universe perspective. The philosophy driving these studies is that operational notions of observation and measurement have been of crucial importance in the construction of our theories of physics to date, and this was especially true in the construction of Einstein’s theories of special and general relativity: despite this, so far very little work appears to have been done on investigating analogue-gravity models from the perspective of such observers. Speciﬁcally, within this thesis it is demonstrated that it is possible to construct devices whose measurements behave in an equivalent manner to the reference frames within Einstein’s theory of special relativity. In particular, the collective measurements of these devices would lead observers who use them (in-universe observers) to view their analogue-gravity universe as containing a sonic analogue to special relativity (where the speed of sound 2s plays the role of the speed of light 2 from special relativity). Of course, the sonic analogue to special relativity (sonic relativity) is not fundamental: to this end, this thesis also provides a demonstration of how certain experiments can be performed by in-universe observers to reveal the fact that their universe has a preferred frame of reference (the frame deﬁned by their analogue-gravity medium). The existence of a preferred frame of reference reveals to in-universe observers that there exists a bigger universe (the actual universe) outside of their analogue-gravity universe, and that sonic relativity is not a fundamental description of reality in this bigger universe. Towards the end of this thesis, it is speculated that it may be possible to map a certain type of experiment that violates sonic relativity to a type of experiment that does not violate sonic relativity. Should this be true, studying analogue-gravity models that are not fully sonically relativistic may still provide us with insights into real relativistic systems.

CONTENTS

List of Figures xi

I Background

1 Introduction1 1.1 Structure of this thesis ...... 2 1.2 The grand scheme of things...... 3 1.2.1 Gravity: fundamental or emergent? ...... 4 1.2.2 A small detour: Semiclassical gravity ...... 7 1.2.3 The case for quantizing gravity...... 8 1.2.3.1 A glance at the ﬁeld of quantum gravity . . . . . 10 1.2.3.2 One ﬁnal and curious quantum gravity result . . 13 1.2.4 The case for not quantizing gravity...... 14 1.2.5 Other possibilities: questioning our postulates? ...... 16 1.3 Analogue gravity...... 17 1.3.1 Origins...... 17 1.3.1.1 Gordon’s optical-metric model...... 17 1.3.1.2 Unruh’s acoustic black-hole model ...... 18 1.3.2 Developments...... 20 1.4 Making sense of it all: the role of analogue-gravity models in the hunt for a microscopic theory of gravity and the aim of this thesis 22

2 Theory 25 2.1 Relativity...... 25 2.1.1 Tensors and Einstein summation convention ...... 26 2.1.2 Special relativity ...... 33 2.1.3 General relativity...... 39 2.2 Quantum physics...... 42 2.2.1 Basics of quantum physics...... 42 2.2.2 The quantum harmonic oscillator ...... 45 2.2.3 Scattering theory...... 49 2.3 Field theory ...... 51 2.3.1 Quantum ﬁeld theory ...... 53 2.4 Analogue gravity...... 57 2.4.1 Eﬀective spacetime metrics ...... 57 2.4.1.1 Geometrical acoustic metrics...... 57 2.4.1.2 Physical acoustic metrics...... 59 2.5 An interesting toy model...... 65 2.5.1 The Lagrangian...... 67 2.5.1.1 The continuum limit...... 68 2.5.2 Equations of motion ...... 70 2.5.3 Bringing it all together...... 72 2.5.4 Including curvature (and thus some analogy to gravity) . 74

vii II Analogue-gravity universes and in-universe observers: Pub- lished/Submitted for publication

3 Sound clocks and sonic relativity 79 3.1 Introduction...... 79 3.2 Approach...... 83 3.3 Simple sound clocks ...... 84 3.4 Sound clock chains...... 88 3.4.1 Calibrating clock separation...... 88 3.4.2 Synchronisation of clocks...... 91 3.5 Relativistic eﬀects observed by stationary sound clocks ...... 94 3.5.1 Time dilation as seen by stationary observers ...... 95 3.5.2 Length contraction as seen by stationary observers . . . . 97 3.6 Relativistic eﬀects observed by moving sound clocks ...... 100 3.6.1 Time dilation as seen by moving observers ...... 100 3.6.2 Length contraction as seen by moving observers ...... 102 3.7 Sonic relativity...... 104 3.8 Discussion...... 107 3.9 Conclusion...... 109

4 Particle scattering in analogue-gravity models 111 4.1 Introduction...... 111 4.2 In-universe observers...... 114 4.3 Aim and Approach...... 116 4.3.1 Aim...... 116 4.3.2 Sketch of our approach...... 117 4.3.3 Schematic of our scattering experiment ...... 119 4.4 Phonon scattering kinematics...... 121 4.4.1 Phonon scattering from internal particles ...... 121 4.4.1.1 The laboratory frame kinematics of phonon scattering from internal particles ...... 121 4.4.1.2 The co-moving in-universe-observer frame kinematics of phonon scattering from internal particles123 4.4.2 Phonon scattering from external particles ...... 125 4.4.2.1 The laboratory frame kinematics of phonon scattering from external particles ...... 125 4.4.2.2 Physical meaning of the two solutions ...... 127 4.4.2.3 The co-moving in-universe-observer frame kinematics of phonon scattering from external particles128 4.5 Scattering cross sections ...... 130 4.5.1 General cross section deﬁnition ...... 131 4.5.2 Cross section in the in-universe observers’ reference frame 133 4.5.2.1 Reaction rate...... 133 4.5.2.2 Flux...... 134 4.5.2.3 Total cross section...... 136 4.5.2.4 Diﬀerential cross section ...... 136

viii 4.5.3 Phonons, quantized external particles, and the interaction Hamiltonian in the laboratory frame ...... 138 4.5.3.1 Interaction Hamiltonian ...... 139 4.5.4 Internal particle cross section ...... 139 4.5.5 External particle cross section ...... 141 4.5.5.1 Initial and ﬁnal states ...... 141 4.5.5.2 Cross section derivation ...... 141 4.5.5.3 Scattering cross sections in the laboratory frame . 144 4.5.5.4 Scattering cross sections in the co-moving in- universe observer frame ...... 146 4.5.5.5 Using Lorentz-violating sonic Compton scattering to determine absolute motion ...... 146 4.6 Discussion...... 148 4.7 Conclusion...... 149

III Work-in-progress and additional unpublished work

5 From sound clocks to the Lorentz transformation 155 5.1 Deﬁning four-vectors using sound clocks ...... 155 5.1.1 Clock-readings ...... 156 5.1.2 In-universe observer measurement of velocity ...... 156 5.1.3 Parametrising coordinates in one frame in terms of coordinates in another, or how I learned to stop worrying and love the Lorentz transformation ...... 158

6 Lorentz violating scattering 165 6.1 Notation and terminology...... 165 6.2 The Lorentz symmetry...... 165 6.3 Scattering from ﬁelds with diﬀerent relativities ...... 166 6.3.1 In-universe observers and external particles ...... 166 6.3.2 External observers and internal particles ...... 172 6.3.3 Sonic relativity, photonic relativity, and the standard-model extension...... 174 6.3.3.1 Analogue gravity and the standard model extension in the laboratory ...... 180

7 Towards sonic atoms 183 7.1 Sonically-relativistic quantum detectors ...... 183 7.2 A toy model for a sonic atom ...... 184 7.2.1 General requirements for sonic atoms ...... 188 7.2.2 An approximately one to one analogue of an actual atom . 190

IV Retrospective and outlook

8 Conclusion 197 Main references...... 200

ix Other references (etymology, interesting facts, etc..) ...... 220

V Appendix

A General transition probability derivation 223

x LISTOFFIGURES

2.1 Coordinate systems are a useful abstract mathematical tool, but unlike the things that they describe, coordinates are not inherently physically meaningful...... 28 2.2 A one-dimensional lattice of coupled pendula...... 66 2.3 The path of a single pendulim...... 67

3.1 The operation of a sound clock...... 85 3.2 The time taken for a sound clock to tick...... 86 3.3 Sound clock chains...... 87 3.4 Calibrating the seperation of sound clocks in a chain...... 90 3.5 Comparison of moving and stationary sound clocks: synchronisation of clocks and calibration of sound clock seperation...... 94 3.6 Measurement of the believed time taken for a moving sound clock to tick as measured from the reference frame of a stationary chain of sound clocks...... 96 3.7 Measurement of the believed seperation of pairs of sound clocks in a moving sound clock chain from the reference frame of a stationary chain of sound clocks...... 98 3.8 Measurement of the believed time taken for a stationary sound clock to tick as measured from the reference frame of a moving chain of sound clocks...... 101 3.9 Measurement of the believed seperation of pairs of sound clocks in a stationary sound clock chain from the reference frame of a moving chain of sound clocks...... 103

4.1 A scattering event as seen in (a) the laboratory frame and (b) the co-moving in-universe-observer frame...... 120 4.2 Phonon scattering from an external particle as viewed in the laboratory frame...... 132 4.3 Differential scattering cross sections for phonon scattering from external particles for fixed 0 and varying i of the external particle...... 147 4.4 Differential scattering cross sections for phonon scattering from external particles for fixed i of the external particle and varying 0...... 148

5.1 Two sound clock chains pass by one another in the laboratory, with the clocks at the origin of each chain being instantaneously synchronous.159

7.1 A diagrammatic representation of a Lorentz-obeying sonic atom. . . . . 187 7.2 A diagrammatic representation of a Lorentz-violating sonic atom . . . . 187

Part I

Background

INTRODUCTION

The title of this thesis is, “On analogue-gravity models as seen by in-universe observers”; the term “analogue gravity” may or may-not be familiar to the reader, and so, for clarity, analogue-gravity models will be deﬁned for the purposes of this thesis as such: 1 Analogue-gravity models are models of non-gravitational physical systems whose behaviour, in certain regimes, can be expressed in the mathematical language of general relativity.

An example might be illustrative here. Perhaps the most conceptually easy-to- understand and most well-known analogue-gravity model is that of the acoustic black-hole as proposed by Unruh in 1981 [1]. At the level of a verbal analogy, the idea is quite simple: a purely radially inflowing fluid that is allowed to accelerate as it flows will eventually flow faster than the speed of sound within the fluid itself; the region of the fluid at which the fluid-flow velocity is equal to that of the speed of sound in the fluid can be treated as an acoustic analogue to the event horizon of a black hole. Sound waves that are generated downstream of this acoustic-horizon will be swept downstream by the fluid flow quicker than they can travel upstream, never crossing to the region of the fluid on the other side of the horizon. This is—at least at the level of this verbal analogy—analogous to light’s inability to climb out of the gravitational well of a black hole. It turns out, however, that this analogy goes much further than merely a verbal one: as shall be discussed more heavily in Section 1.3, the mathematical formulation of this problem can be cast into a form that mirrors that of a massless field (such as light) propagating through a Schwarzschild spacetime in general relativity (GR), thus completing the promise made above when defining analogue-gravity models. While hydrodynamical analogue-gravity models are possibly the most well known—probably owed partly to their conceptual simplicity as can be seen from the acoustic black-hole example described above, and partly to the ease of their experimental realization (at least in comparison to some of the more exotic models)—plenty of other non-hydrodynamical analogue-gravity models do exist, and some subset of these others models will be discussed in Section 1.3 and beyond. However, before we can move on to these more interesting discussions, some housekeeping is in order: to this end, Section 1.1 provides the structure of this thesis.

1 2 Introduction |

1.1 Structure of this thesis

PartI –the current part of the thesis—contains two background chapters (detailed below) that are intended to be a relatively pedagogical introduction to analogue gravity and its related fields. If these chapters serve their intended purpose, then they should be of most use to students who are unfamiliar with analogue gravity and the fields from which it stems. While the chapters contained within this part of the thesis are far from comprehensive, they should ideally motivate analogue gravity as a research discipline, motivate the specific research topics of this thesis, and also serve as a relatively self-contained reference for future parts of this thesis.

Chapter1 —the chapter that you are currently reading—is a literature review that reviews, at a mostly non-technical level, the research endeavour into unifying general relativity and quantum theory, analogue gravity, and its possible role in aiding this endeavour. Chapter2 provides the necessary mathematical machinery to make sense of everything that follows. The reader unfamiliar with the mathematical machinery of, in particular, general relativity and quantum field theory may benefit by reading Chapter2 before returning here to read the current one in full, as specific calculations in the literature will be displayed and discussed in Subsection 1.2.3 and Subsection 1.2.4. Alternatively, the reader can continue on bravely, consulting Chapter2 as necessary.

PartII contains two chapters, the content of which is work that is either published (Chapter3 ) or currently under review for publication (Chapter4 ).

Chapter3 is a minorly modified version of the first paper [2] produced during the undertaking of this PhD dissertation in which one method to construct operationally defined classical in-universe observers in analogue-gravity models is discussed. This paper was co-authored with Nicolas C. Menicucci. Chapter4 is a minorly modified version of the second paper produced during the undertaking of this PhD in which a toy model of particle scattering in analogue-gravity models is investigated.1 This paper was co-authored with Giacomo Pantaleoni, Valentina Baccetti, and Nicolas C. Menicucci, all of whom provided major contributions to Chapter 4.5.

1This paper is currently under review for publication with Physical Review D. A preprint version of this paper is accessible on arXiv [3]. The grand scheme of things 3 |

The appendix present in the paper was produced by Valentina Baccetti and is attached as AppendixA in this thesis.

Part III contains three chapters of unpublished work.

Chapter5 is an extension to the published work that makes up Chapter3 in which it is shown how to develop the full four-vector description of special relativity from the operational measurement procedure described in Chapter3 .

Chapter6 contains a rough sketch of where the work from Chapter4 may be taken in the future. In particular, it is demonstrated that it should be possible to leverage the Standard Model Extension (SME)—an extension to the standard model that allows for spontaneous Lorentz symmetry breaking—to further investigate such scenarios.

Chapter7 contains discussions and some preliminary back-of-the-envelope calculations on a particular type of sonically relativistic quantum detector.

PartIV contains as its sole chapter Chapter8 , which itself contains concluding remarks of this thesis and suggestions for where future research eﬀorts may be best focussed.

PartV is the ﬁnal part of this thesis and contains Chapter 4.5 as its sole entry, which as mentioned above is the appendix that was prepared by Valentina Baccetti for a paper that is currently under review with Physical Review D (the arXiv preprint can be found in reference [3]).

1.2 The grand scheme of things

To best understand the role of analogue-gravity models in the current landscape of physics, it is instructive to take a short wander through the history of the effort to unify general relativity with quantum theory. To that end, Subsection 1.2.1 will give a brief and high-level overview on the different conceptual avenues that are being considered to tackle this particular problem. The specifics of each avenue of research will be mostly delayed until Subsection 1.2.3 and Subsection 1.2.4, after which the reader should have a sufficient enough understanding of the research landscape to be able to understand the role of analogue-gravity models in the research endeavour, which shall be discussed in more detail in Section 1.3. 4 Introduction |

1.2.1 Gravity: fundamental or emergent?

Before moving on, it is important to explicitly explain what is meant by a fundamental interaction (alternatively, fundamental force, fundamental phenomena, or any other such terms formed by using the word fundamental as an adjective).

• Fundamental interactions are those physical interactions that cannot themselves be explained in terms of other physical interactions. They are, in a sense, axiomatic (in that they do not follow from something prior). These interactions are intrinsic to certain physical objects, and so if the associated physical objects are present, then their corresponding interactions are also, by necessity, present.

As of the publication of this thesis in 2020, it is now widely believed that the physical laws that govern the universe are fundamentally quantum mechanical in nature. 2 The standard model of particle physics (SM)—whose mathematical framework is that of quantum field theory (QFT)—has had extraordinary success in explaining the observed properties of the material world3 to a high precision, specifically by providing a theoretical basis for the electromagnetic interaction, the strong interaction, and the weak interaction: three of the so-called fundamental interactions—or fundamental forces—of nature. Gravity is typically considered another fundamental interaction of nature bringing the total to four. While we have had an excellent classical theory of gravity—general relativity—for over 100 years, as of 2020, we have yet to establish an accepted theory of quantum gravity (QG). There have, of course, been many attempts to formulate such a theory , though as of present no candidate theory has been experimentally verified. It is entirely possible that future work may reveal one (or more) of these proposed theories to be a valid description of nature, but this remains to be seen.

2There are, of course, views to the contrary. Of historical note is the famous EPR-paradox [4], the suggested resolution of which was to accept that the wavefunction in quantum mechanics is an incomplete descriptions of reality and that there must exist additional elements of (physical) reality (or hidden variables), the observation of which would remove the indeterminate nature of quantum theories. The EPR-paradox, in-part, formed the basis for Bell’s famous paper [5] in which it was proven—under a reasonable set of assumptions about reality—that quantum mechanics cannot possibly be a local hidden-variable theory. It has since been suggested that our reality may in fact be consistent with a relaxing of Bell’s original assumptions, in which case Bell’s conclusions may be avoidable [6–10]. In recent years, ’t Hooft has investigated models of cellular automata (see [11] and references therein) as a means by which to describe quantum mechanics from a deterministic model: such models can be used to reproduce a 1 1 -dimensional quantum ﬁeld ( + ) theory of non-interacting bosons [12]. 3By “material world” what is meant is matter and its associated non-gravitational interactions/behaviours. Gravity: fundamental or emergent? 5 |

Must there exist a theory of quantum gravity, though? Not necessarily. The strongly held view that such a theory must exist is based—as all non-axiomatic things are—on assumptions. There are certain fundamental assumptions that we make about the nature of the universe that, if invalid, would remove the supposed-need for a quantum theory of gravity. Two major assumptions that we typically make about our universe are the following:

1. Gravity is a fundamental phenomenon; 2. All fundamental phenomena are quantum mechanical in nature.

If one abandons the first assumption—that gravity is a fundamental phenomenon— then it must be the case that gravity is some kind of emergent phenomena (a definition of this term will be offered shortly). While certain emergent phenomena admit a meaningful quantum mechanical description (e.g., phonons and other condensed matter phenomena), others do not (e.g., thermodynamic properties like temperature). If gravity is not fundamental, and if it only exist as some kind of thermodynamic property of large ensembles of particles, then there may not exist any meaningful theory of quantum gravity. With that said, abandoning the first assumption does not necessarily lead to this conclusion: an emergent theory of gravity might still be meaningful before taking the thermodynamic limit (i.e., before all of the quantum mechanical degrees of freedom have been averaged out). It is therefore conceivable that we could abandon the first assumption and yet still be faced with the necessity of requiring a theory of quantum gravity. Abandoning the second assumption—that all fundamental phenomena are quantum mechanical in nature—is somewhat more controversial [13], though in principle this approach may actually be reconcilable with our current understanding of the universe [14–18]; this shall be discussed in more detail below. Are there any reasons to actually believe that either (or both) of these assumptions might be invalid? In the case of the first assumption—that gravity is a fundamental phenomenon—the answer is both yes and no. Currently, there is no direct experimental evidence to support the notion that gravity is not a fundamental phenomenon, however, there exists both theoretical indications and a large body of indirect experimental evidence—by analogy—that gravity might be an emergent phenomenon. The specific arguments about the nature of gravity—fundamental or emergent—and the evidence to support these arguments will be relegated to the discussions in Subsection 1.2.3 and Subsection 1.2.4; before moving on to these discussions, however, we should discuss the notion of emergent phenomena in general in order to elucidate their possible role in explaining gravity. 6 Introduction |

The term emergent (and similarly related terms that are constructed by using “emergent” as an adjective) are often used fairly loosely, and so an attempt shall be made to make the idea somewhat more concrete for the purposes of this thesis. The notion of emergent phenomena/behaviour/physics/etc.. can be encapsulated by the following observation:

• The phenomenology of certain physical systems is not scale-invariant, with some phenomena often only appearing to exist for large values of certain physical parameters (usually the total number of particles in the system, or the physical size of the system).

That is to say, certain phenomena that did not appear to exist for small parameter values appear to emerge at larger parameter values as some ensemble—or collective—property of the microscopic subcomponents (or degrees of freedom) of the system. Perhaps the two most familiar examples of emergent phenomena are that of thermodynamics—which emerges from statistical mechanics—and fluid dynamics—which emerges from molecular dynamics. A slightly more formal definition of emergent phenomena might be the following: emergent phenomena are those phenomena in a physical system that can be described in terms of macroscopic variables that are not inherent in the constituent components of the system. This then turns the question of, “what is meant by emergent phenomena?” into, “how does one define ‘macroscopic variables’?” One such answer to this question is the following: a macroscopic variable describes the average or net behaviour of some aspect of a physical system. A more specific definition might be the following: we choose to define a macroscopic variable to be any variable whose definition can be used to substantially reduce the number of degrees of freedom required to describe the behaviour of a physical system by—in essence—absorbing many individual microscopic degrees of freedom into a single variable whose value is insensitive to the specific values of any sufficiently small subset of those microscopic degrees of freedom from which it is defined. The prototypical example of a macroscopic variable is the temperature of some physical system in thermodynamic equilibrium: in this case, the definition of temperature absorbs a huge number of degrees of freedom—for example, the velocity of every individual molecule in an ideal gas—into a single scalar quantity. In keeping with the definition that was previously offered, the particular value of a system’s temperature is insensitive to a change in the kinetic energy of any sufficiently small subset of the constituent particles of that system (where in this case, by “any sufficiently small subset of the constituent particles”, we mean any number of particles that is small relative to the 1023 particles that one typically ∼ finds in a macroscopic sample of gas). A small detour: Semiclassical gravity 7 |

If it turns out that gravity—and perhaps our entire notion of spacetime—is some emergent phenomenon, then one should not necessarily expect to be able to naïvely quantize the objects that we use to describe gravity in general relativity (specifically, the metric tensor) because these objects might not exist in any meaningful way at the level of a microscopic theory. Whether or not quantization of the gravitational field would be meaningful within a theory of emergent gravity would depend on the specific nature of the emergent gravitational field itself. Before talking more about the methods that have been taken towards describing the underlying nature of gravity (be it fundamental or emergent), let us first take a small detour to talk about semiclassical gravity.

1.2.2 A small detour: Semiclassical gravity

While the ultimate nature of gravity—fundamental or emergent—still eludes us, there does exist a semiclassical theory of gravity [14] whose validity we are relatively confident of in certain regimes. The semiclassical theory of gravity should exist as an effective mean-field theory in the correspondence—or classical— limit of any true quantum theory of gravity, and furthermore it should also produce valid predictions in certain limits of emergent gravity models. In the theory of semiclassical gravity, spacetime and its associated properties are treated classically, while the contents of the spacetime (e.g., matter fields) are treated quantum mechanically. Specifically, the gravitational field is treated as a classical field whose value is given as a function of the expectation values of quantum fields in spacetime. This is embodied mathematically in the semiclassical Einstein field equations [19]:

8 = ) #. (1.2.1) 24 h ˆ i

The spacetime structure (embedded in the Einstein tensor ) is given in its usual classical form on the left-hand-side, while on the right-hand-side we have the expectation value of the quantum energy-momentum tensor operator ()ˆ). The quantum energy-momentum tensor operator is sourced from some spatial distribution of matter described by the wavefunction # (note that # is a subscript on the right-hand-side, and so the wavefunction is not explicitly present in this form of the semiclassical Einstein ﬁeld equations). While the physical regimes that would likely be required to test any full theory of quantum gravity or emergent gravity are currently far beyond our reach, the theory of semiclassical gravity should apply to physical scenarios that will be realizable in near-future experiments. For example, tangentially related to the 8 Introduction | semiclassical Einstein ﬁeld equations is the Schrödinger–Newton equation [20–25]. The Schrödinger–Newton equation can be written as follows

2 %# 2 ¹ # C, A0 © ~ 2 2 3 ì ª 8~ C, A = < 3 A0 ( ) ® # C, A . (1.2.2) %C ( ì) −2< ∇ − ì A A0 ( ì) « ì − ì ¬ This is a modified version of the Schrödinger equation that contains, in addition to all of the usual terms, a potential term that takes into account the self-gravitation of some mass distribution that is represented by the wavefunction of the system. The gravitational field is described by a classical Newtonian potential, hence the name Schrödinger–Newton equation. Research on the implications of the Schrödinger–Newton equation constitute an example of research that is tangentially related to semiclassical gravity. This particular research avenue appears to be a promising one due to the fact that modern optical and optomechanical experiments should—in principle—suffice to determine whether or not the Schrödinger–Newton equation is valid within its regimes of supposed applicability. [26–28] The exact nature of the link between the Schrödinger–Newton equation and the theory of semiclassical gravity is ultimately reliant on whether or not gravity admits a quantum description. If the gravitational field is quantum mechanical in nature, then the Schrödinger–Newton equation is only an approximate description of the behaviour of large ensembles of gravitating particles; that is to say, in the limit of small number of particles, the Schrödinger–Newton equation fails to make valid predictions.4 If, however, the gravitational field only admits a classical description, and if it is also true that the semiclassical Einstein equations are indeed the correct way to describe the gravitational field of quantum mechanical systems, then the Schrödinger–Newton equation is valid even for single particles [25].

1.2.3 The case for quantizing gravity

Throughout this subsection, it will be assumed—unless otherwise explicitly stated—that gravity is a fundamental phenomena. Without experimental guidance, the current arguments in favour of quantizing gravity stem from demands of logical consistency; for example, by arguing that all fundamental phenomena must be quantum mechanical in nature, or conversely, by arguing that fundamental phenomena cannot possibly be classical in nature.

4This is demonstrated in reference [20] which appears to be the ﬁrst instance of the Schrödinger– Newton equation in the literature (though the name was not coined until later, with reference [29] appearing to be its origin). The case for quantizing gravity 9 |

Perhaps the most obvious question that naturally leads one to realize that there should be a quantum description of gravity is the following: what happens when a quantum mechanical object is placed into a spatial superposition [30]? Taking our current understanding of gravity seriously, we observe that any object with mass/energy gravitates, which then leads very obviously to the conclusion that a physical object placed into spatial superposition should itself result in a superposition of gravitational ﬁelds (i.e., a superposition of spacetimes). This seemingly simple consideration appears to immediately necessitate the existence of a theory of quantum gravity.

The apparent necessity for a quantum theory of gravity can also be demonstrated by showing that physical theories involving both quantum mechanics and fundamentally classical gravitational fields lead either to internal inconsistencies or to predictions that cannot be reconciled with already understood physics. A particularly noteworthy example of this type of approach is the 1977 paper by Eppley and Hannah [13]: they argued that a fundamentally classical theory of gravity that is valid at arbitrarily small length scales results in violations of currently accepted and well tested laws of physics. In particular, Eppley and Hannah argued that if gravitational waves of sufficiently small wavelength are used to make measurements of quantum mechanical systems, then a violation of one of the following principles is a logical consequence: momentum conservation, the uncertainty principle, or locality. Eppley and Hannah’s proposal would be (by their own calculations) an engineering nightmare: the particular hypothetical experimental apparatus as envisioned in their paper would have an enormous mass, so much so that entire galaxies would be consumed in its construction. With that said, engineering nightmares are only problems of practicality: Eppley and Hannah’s calculations showed that the mass of their hypothetical experimental apparatus is less than the mass of the observable universe, and thus they con- cluded that their hypothetical experiment was at least possible in principle. The results of Eppley and Hannah would then seem to necessitate the abandonment of either momentum conservation, the uncertainty principle, or locality in any internally-consistent model of physics incorporating both quantum mechanics and a fundamentally classical gravitational field. Given how well these assumptions have stood the test of time so far, the notion of abandoning any of them is difficult to accept.

It should be noted that Eppley and Hannah’s conclusions have subsequently been called into question [15–18]. Huggett and Callender [15] argued that Eppley and Hannah’s result is a consequence of choosing the standard (Copenhagen) interpretation of quantum mechanics and that alternative interpretations of 10 Introduction | quantum mechanics can—in an internally consistent manner—support a fundamentally classical gravitational field. Even within the Copenhagen interpretation of quantum mechanics, the supposed problems that Eppley and Hannah predicted might actually be completely unobservable, and hence, unphysical. To this end, Mattingly [16] disputes the conclusions of Eppley and Hannah by demonstrating that their hypothetical experimental apparatus is, in fact, fundamentally impossible to construct in our universe even in principle. Through a more careful and thorough consideration of the necessary experimental considerations than was originally undertaken by Eppley and Hannah, Mattingly concludes that Eppley and Hannah’s hypothetical experimental apparatus would—by necessity of its design—sit within its own Schwarzschild radius. As a result, the apparatus would collapse into a black hole thus preventing its use as a measurement apparatus as intended. While Mattingly’s analysis rules out Eppley and Hannah’s particular thought experiment it does not necessarily rule out all possible thought experiments that could be designed to demonstrate the same thing. If it were possible to show that no experiment could ever be constructed (even in principle) to reveal the apparent physical inconsistencies that arise from coupling a classical gravitational field to quantum mechanical objects then, troubling though those inconsistencies may be, one could safely ignore them on the grounds that they have no measurable—and thus no actual—effect on reality. Mattingly briefly entertains such a scenario as being analogous to the cosmic protection hypothesis5 [31]6, dubbing it the semiclassical protection hypothesis. As of present, there does not—to the author’s knowledge—exist any no-go theorems that prevent some other hypothetical experiment from demonstrating what Eppley and Hannah’s specific experiment fails to. In the absence of any such no-go theorem, Eppley and Hannah’s result is certainly troubling. If the gravitational field is fundamental in nature then without abandoning or modifying quantum mechanics or other well tested principles of physics, quantization of the gravitational field may well prove necessary.

1.2.3.1 A glance at the ﬁeld of quantum gravity

Many research programmes and theoretical frameworks have come into existence in the eﬀort to ﬁnd a theory of quantum gravity. Of those, the two largest and most noteworthy research programmes to date are that of Loop Quantum Gravity

5More commonly referred to as the “cosmic censorship hypothesis”. 6The original 1969 paper in which the cosmic censor was ﬁrst proposed has been reprinted as [32]. The case for quantizing gravity 11 |

(LQG) [33, 34]—a theory of quantum gravity7—and String Theory8 (ST) [37]—a candidate theory-of-everything that itself contains theories of quantum gravity. Below is a short summary of the history of both loop quantum gravity and string theory. References to comprehensive review articles and textbooks on the subjects will be provided at the end of each summary. Loop Quantum Gravity is a direct continuation of the earliest efforts undertaken in the attempt to build a theory of quantum gravity. These first efforts took the obvious approach of trying to apply canonical quantization techniques to general relativity [38–40]. While these techniques had successfully allowed for the development of quantum mechanics and had paved the way for quantum electrodynamics (QED), attempts at canonical quantum gravity resulted in serious mathematical problems that stalled the research effort for some time. Eventually, work by Ashtekar [41, 42] allowed for the mathematical problems of canonical quantum gravity to be overcome and loop quantum gravity (LQG) emerged as a continuation of the canonical quantum gravity research programme. One particularly noteworthy aspect of loop quantum gravity is that it has been formulated in such a way that it is manifestly background independent, a feature that it shares in common with general relativity. Detailed discussions on the formalism and history of LQG can be found in Rovelli’s book [43] and the review articles [33, 34, 44], while references [45, 46] provide review articles for Loop Quantum Cosmology (LQC), which results from the application of LQG to cosmological scenarios. String theory has a rather more storied history than that of loop quantum gravity. Despite being a quantum theory, the path that leads to string theory in some sense begins with an attempt to form a classical unified theory of physics. Specifically, some of the major conceptual ideas underpinning string theory lie in Kaluza–Klein theory [47, 48], a theory that itself began as an attempt to “geometrize”9 electromagnetism. Originally, Kaluza demonstrated that by extending general relativity to include an additional fourth spatial dimension, both gravity and

7While loop quantum gravity itself is only a theory of quantum gravity, there do exist related theories sharing similar philosophical and mathematical bases that may prove to be candidate theories-of-everything. For example, Bilson-Thompson et al. [35] have shown that topological degrees of freedom within a loop-quantum-gravity like theory can be associated with the quantum numbers that describe particles. In particular, braided graphs within this theory can naturally be thought of as containing the first generation of standard model particles. 8Due to the vast number of theories that are contained under the umbrella term of “string theory”, it is somewhat difficult to point to single reviews on string theory as a whole. The resource letter in reference [36] contains a list of references that may be of use to the reader interested in specific aspects of string theory (e.g., bosonic string theory, superstring theory, M-theory, the holographic principle, AdS/CFT duality, etc..). 9Where by “geometrize” we mean “make geometric”, in the same spirit as to how “quantize” is understood to mean “make quantum”. 12 Introduction |

classical electromagnetism could be realized as a single geometric theory in a five-dimensional spacetime [49]10. A constraint referred to as the cylinder condition was imposed by Kaluza in order to suppresses the fourth spatial dimension such that the resulting theory more accurately agreed with our understanding of reality. Klein later demonstrated that the cylinder condition allowed for the electromagnetic field in Kaluza’s theory to be quantized if the fourth spatial dimensional was curled up—or “compactified”—into a cylinder with a radius of the order 10 30 cm [51, 52]. The notions of extra dimensions and dimensional ∼ − compactification from Kaluza–Klein theory eventually became central to modern string theory [47, 48] which originally emerged as its own theory during the period of time in which the standard model of particle physics was being constructed [53]. String theory was initially an attempt at describing hadrons [53], though ultimately quantum chromodynamics (QCD) succeeded in this endeavour by providing the mathematical formalism describing hadrons, their constituent components (quarks and gluons), and their interactions. Despite QCD’s success, string theory proved to have great theoretical utility in other ways: while not originally conceived of as a theory of quantum gravity, it was eventually realized that string theory naturally contains a quantized massless spin-2 field—the graviton—whose behaviour in the low energy limit is described by general relativity [54, 55]. In fact, not only does string theory allow for the existence of gravitons: it demands it. The existence of gravitons within string theory is, in fact, an unavoidable and necessary consequence on insisting that string theory is a unitary quantum theory, and for this reason—among many others whose detailing would make this introduction unreasonably long—string theory has received considerable attention by those in the quantum gravity community. See the 25th anniversary edition of Green, Schwarz, and Witten’s book for a foundation on string theory and for comments on how the field has progressed since its inception [56, 57]. Review articles on string theory (or particular aspects of it) can be found in references [58, 59], while reference [60] is a review article on String Cosmology, which results from the application of string theory to cosmological scenarios. Both of these research programmes have produced theoretical results that agree with those from the semiclassical gravity literature. In particular, both LQG and string theory admit derivations that correctly obtain results from black-hole thermodynamics, namely expressions for the Hawking temperature of black holes, and the black-hole entropy-area law (LQG references [61–63]; string theory references [64, 65]) . The fact that these theories make predictions that are in

10An English translation of the original German paper is available in reference [50] The case for quantizing gravity 13 | agreement with the predictions made within an existing theoretical framework whose validity we are reasonably certain of is a good reason to suspect that the research programme of quantum gravity is well founded. Loop quantum gravity and string theory are not—as has already been mentioned—the only quantum gravity research programmes. Other examples of research programmes and theoretical frameworks within the quantum gravity research endeavour include causal dynamical triangulation [66], twistor theory [67], and causal sets [68]. Some of the older and more established research programmes within the field of quantum gravity are touched upon by Smolin in reference [69] and Rovelli in reference [70]. This is not a complete list of all aspects of the quantum gravity literature by any means, and indeed such an endeavour is outside of the scope of this thesis. The literature does indeed appear to contain many compelling arguments and results that lend support to the idea that gravity should indeed be quantized. With that said, experiment is ultimately the arbiter of scientific validity, and to date, no experiments have been able to probe the physical regimes within which we expect the effects of quantum gravity to present themselves. While experiments that hope to probe semiclassical gravitational phenomena may provide us with some new experimental insight in the near future (as mentioned in Subsection 1.2.2), verification of the predictions made by semiclassical gravity will not necessarily allow us to uniquely identify the parent theory of which semiclassical gravity is an approximation. No theoretical framework for quantum gravity has yet been blessed with experimental validation, and so for the time being it is an open question as to which—if any—approach best describes the fundamental nature of gravity within our universe.

1.2.3.2 One ﬁnal and curious quantum gravity result

As a ﬁnal note on the quantum gravity literature, it is worth drawing attention to a curious result obtained by Dyson [71]. Assuming that there does indeed exist a theory of quantum gravity, Dyson asked the question “Is a graviton detectable?” in a paper of the same name [71]. While Dyson does not prove that gravitons are undetectable by any and all conceivable experiments, he does show that if one were to attempt to construct a device that follows the same design principles as LIGO (Laser Interferometer Gravitational-Wave Observatory), then in order for it to possess the sensitivity to detect single gravitons its pairs of mirrors would sit within their combined Schwarzschild radius. As a result, the device suﬀers the same fate as Eppley and Hannah’s device [16]: by necessity of its design, it collapse into a black-hole, preventing its intended use. Eppley and 14 Introduction |

Hannah’s hypothetical device was conceived of under the assumption that gravity is fundamentally classical, while Dyson’s hypothetical device was conceived of under the assumption that gravity is fundamentally quantum: in either case, the laws of physics—whether by coincidence or for some deeper reason—seem to conspire in just the right way as to prevent probing the fundamental nature of gravity. If the apparent inability to probe the nature of gravity turns out to be some fundamental limit that extends to any and all such experiments that might conceivably be used to do so, one may pause to wonder if perhaps Mattingly’s “semiclassical protection hypothesis” [16] is actually part of some more fundamental gravitational censorship hypothesis11 that makes it impossible to determine whether or not gravity is quantized. Such a prospect seems quite disturbing from the point of view of the theoretical physicist.

1.2.4 The case for not quantizing gravity

As has been discussed in Section 1.2, there exists the possibility that gravity is an emergent phenomenon rather than a fundamental one. There are many curious coincidences in the scientific literature that are somewhat suggestive of this idea, and some of the more noteworthy examples shall be discussed in this subsection. In 1967, Sakharov [72]12 published a short paper in which he demonstrated that a term corresponding to the Einstein–Hilbert action—which leads to Einstein’s field equations by the principle of least action—can be shown to arise from the one-loop interactions of a quantum field theory of interacting fields. In his paper13, Sakharov assumes that there is some background spacetime described by a Pseudo-Riemannian (specifically, Lorentzian) manifold, and that the Lagrangian density describing a quantum field theory atop this manifold can be expressed in the form of a series of powers of the manifold’s scalar curvature ' (the invariant of

the Ricci tensor '). Sakharov allows the background spacetime to be a dynamic object but does not specify any rules governing its dynamical behaviour (that is to say, the Einstein field equations are not explicitly assumed to apply to the spacetime14). Under the aforementioned considerations, the effective action of the fields at the level of one-loop quantum field theory contains terms that are proportional to the cosmological constant ( '0), the Einstein–Hilbert action ∼ ( '), and functions of higher-powers of the scalar curvature ( '2 ). As a result, ∼ O( ) the fields behave as though they exist within a spacetime that obeys the Einstein field

11A term being coined here. This is not, to the author’s knowledge, a term already in use in the literature. 12See reference [73] for an English translation of the original Russian manuscript 13Though not explicitly spelled out—at least, not in the English translation. 14As Visser points out in reference [74], the Lorentzian manifold is “free to flap in the breeze”. The case for not quantizing gravity 15 | equations, despite this particular dynamical description of the manifold never being explicitly assumed. This idea has become known as induced gravity due to the fact that the fields atop the background manifold induce changes to their own action that are mathematically consistent with the terms from general relativity that are responsible for the dynamic nature of spacetime. For a more thorough and informative read, see Visser [74] in which he elaborates on Sakharov’s relatively short paper. Sakharov’s work is not the only example of (apparently) non-gravitational phenomena appearing to mimic—or perhaps lead to—gravitational phenomena. Perhaps the best known example of such a result is contained within a short letter by Jacobson [75], titled “Thermodynamics of Spacetime: The Einstein Equation of State”. Jacobson’s derivation considers the observer-dependent causal horizons that are seen by uniformly accelerating observers (Rindler horizons). As demonstrated by Davies [76] and Unruh [77], these horizons are thermal (this is commonly referred to as the Unruh effect). Under the assumption that Rindler horizons obey an entropy-area15 law analogous to that obeyed by black holes [78, 79], Jacobson demonstrates that heat flux across the horizon leads, via the second law of thermodynamics, to a thermodynamic equation of state that is term-by-term equivalent to the Einstein field equations. The degrees of freedom that constitute the thermodynamic system are the quantum fields close to either side of the causal horizon, and the correlations between vacuum fluctuations in the quantum fields just-inside of and just-outside of the horizon constitute the entropy of the system16. Jacobson’s work, like Sakharov’s, implies that a dynamical description of spacetime and gravitation that is consistent with general relativity can arise from considerations of phenomena that are not explicitly gravitational. In more recent years, Verlinde [80] and Padmanabhan [81–83] have separately shown that gravitational phenomena can emerge from holographic considerations17. Padmanabhan has demonstrated that the dynamical equations governing the evolution of some bulk18 region of a spacetime manifold—that is, Einstein’s field equations—can emerge as a result of thermodynamic considerations at the boundary of that same region of spacetime. In particular, Padmanabhan

15Formally, the area of a black hole is the area of the event horizon. 16The state of the field behind a causal horizon cannot be measured by those observers for which the horizon exists, and so the correlations between fields just inside-of and just-outside of the horizon are not observable to these observers. It is in this sense that the correlations between the fields just-inside of and just-outside of the horizon constitute a measure of entropy to these observers. 17The holographic principle is one such example of these ideas. The holographic principle is a con- jecture born from string theory and one that has links with ideas from black-hole thermodynamics (such as the entropy-area law mentioned above in Jacobson’s work). Padmanabhan stresses in his work that his use of the word is to be understood to be more general than this [83]. 18The term bulk is typically understood to refer to volumes/voluminous regions. 16 Introduction |

demonstrates that regions of spacetime that are in holographic equipartition are static, whereas regions of spacetime that deviate from holographic equipartition undergo time evolution [83] . Verlinde’s work utilizes holographic ideas to demonstrate that one can recover Newton’s gravitational force law as an entropic force [80] by associating the positions of particles in space with information/entropy on an abstract surface (or holographic screen) that bounds a gravitational source. Furthermore, a sketch of a derivation of Einstein’s field equations within the context of the holographic principle is also offered by Verlinde [80]; in this context, general relativity emerges from entropic considerations in the thermodynamic limit. While the field of emergent gravity possesses many curious results, there are serious difficulties faced by the research endeavour as a whole. To this end, see Carlip [84] and the references therein for a fairly comprehensive review of the challenges faced by the emergent gravity research endeavour, with a specific focus on the problems faced by two major classes of emergent gravity model. In addition to the general problems faced by the field as a whole, there are problems with and criticisms of specific models of emergent gravity. In particular, Kobakhidze [85, 86] has pointed out that Verlinde’s particular entropic gravity approach necessarily leads to predictions that disagree with experiments involving ultra-cold neutrons in the Earth’s gravitational field. Additionally, Visser [87] clearly and concisely lists many problems that are inherent to Verlinde’s entropic gravity model, though makes note of the fact that such concerns do not necessarily apply to Jacobson’s and Padmanabhan’s models of emergent gravity on account of the fact that their models differ substantially from Verlinde’s model. More recently, Dai and Stojkovic [88] have pointed out that Verlinde’s model contains errors leading to internal inconsistencies.

1.2.5 Other possibilities: questioning our postulates?

As a closing thought on this subsection, recall from Subsection 1.2.3 the result of Eppley and Hannah [13]. If Eppley and Hannah’s result is indeed correct then the gravitational field being fundamentally classical would lead to violations of so far well-tested physical laws such as, for example, the uncertainty principle [13]. Must such violations be avoided though? The quantum gravity literature itself contains suggestions that the uncertainty principle may require modification close to the Planck length in the form of generalised uncertainty principles (see both [89] and [90] and references therein). If it is indeed the case that quantizing gravity comes at the cost of needing to modify our understanding of the uncertainty principle then it should not be any more troubling for a similar prospect to arise Analogue gravity 17 | from a theory in which gravity is classical (provided, of course, that any such modified uncertainty relation would limit to our current uncertainty principle for the relevant parameter regimes for which we know it should hold true). In attempting to unify our understanding of both gravity and quantum mechanics, we may well find ourselves having to question our most foundational assumptions of physics.

1.3 Analogue gravity

We ﬁnally arrive back to where we started at the beginning of this chapter: analogue-gravity models. As was stated at the very beginning, analogue-gravity models are models of non-gravitational physical systems whose behaviour, in certain regimes, can be expressed in the mathematical language of general relativity. Given that models of emergent gravity in particular possess precisely this type of feature, analogue gravity models might prove to be an experimentally accessible method of indirectly testing some of the predictions made by such theories.

1.3.1 Origins

1.3.1.1 Gordon’s optical-metric model

Perhaps the earliest example of an analogue-gravity model—or a potential analogue-gravity model, even if not recognised at the time as such—is the optical- metric model developed by Gordon (of Klein–Gordon fame) in 1923 [91] . The optical-metric model describes an eﬀective metric, , ,19 whose geodesics describe ˆ the path of electromagnetic waves through a curved spacetime—characterized by the physical metric tensor , —ﬁlled with a dielectric material whose four-velocity is denoted D . In terms of , , D , and also the dielectric permittivity of free space & and the magnetic permeability of free space , the optical metric tensor is as follows: 1 , = , 1 D D , (1.3.1) ˆ ± − & where Greek indices run over the four spacetime dimensions of general relativity. The choice of plus-or-minus on the right-hand-side is dictated by the metric signature convention of choice: for the , , , signature one picks the positive (− + + +) sign on the right-hand-side, and for the , , , signature one picks the (+ − − −) negative sign on the right-hand-side.

19Note that the circumflex (or hat) is merely used to differentiate the optical metric tensor from the physical metric tensor. The circumflex is not meant to imply that the optical metric tensor is a quantum mechanical operator; the optical-metric model is entirely classical. 18 Introduction |

The physical metric tensor , describes the actual curvature of spacetime, and its corresponding geodesics describe the paths through spacetime followed by non-electromagnetic objects (or objects that do not appreciably interact elec- tromagnetically); for example, the orbits of planets will be, for all intents and

purposes, entirely described by the metric , . However, as mentioned above, it is the geodesics of the optical metric tensor , that dictates the path of electro- ˆ magnetic waves through spacetime. In the particular case of flat spacetime—i.e., when there is no actual, physical spacetime curvature present—the dielectric medium filling space becomes the only source of curvature for the optical metric tensor. If one identifies curvature with gravity, then we find that electromagnetic waves propagating through a flat spacetime that is filled with a dielectric material behave as though they are propagating through a curved spacetime that is not filled with any such material. The optical-metric model in the case of flat spacetime highlights the general feature that characterizes all analogue-gravity models: non-gravitational phenomena can be used to simulate gravitational phenomena through the identification of curvature with gravity. In the optical-metric model, this is manifest in the fact that electromagnetic waves (e.g., light) can be made to follow curved paths20 through otherwise flat spacetime via an interaction with a dielectric medium. In all other models of analogue gravity, there will be some analogous physical behaviour: the trajectories of some appropriate object will be expressible in the language of differential geometry (specifically, Lorentzian geometry), and these trajectories will, in general, behave as though they follow curved paths on some background manifold, where the curvature of the manifold is dictated by some non-gravitational physical property of the system.

1.3.1.2 Unruh’s acoustic black-hole model

Gordon’s optical-metric model may technically be—in retrospect—the first example of an analogue-gravity model, but it is Unruh’s model of an acoustic black hole [1] (or dumb hole21) that seems largely responsible for much of the research in the field of analogue gravity. The importance of Unruh’s model to the analogue gravity community is likely a result of the fact that it provided the first real hope of being able to gain insight—by analogy—into systems for which the phenomenology of quantum gravity is thought to be important, a point that

20Where the notion of a curved path is formalized within the framework of diﬀerential geometry. 21Where the word “dumb” here is to be understood to be synonymous with

mute—its original etymology [a]—rather than being a synonym for stupid. Analogue gravity 19 |

Unruh himself makes towards the end of his letter.22 The specific physical system that Unruh considered in formulating the notion of an acoustic black hole was that of a spherically symmetric convergent fluid flow (i.e., a fluid that flows radially inwards towards some singular point), where the fluid in question is barotropic, inviscid, and irrotational (respectively, these mean: the density of the fluid is a function of pressure only, the viscosity of the fluid is equal to zero, and there are no vortices in the fluid flow). If the local fluid flow velocity at some radius ' (defined from the central point that the fluid flow is convergent on) is instantaneously equal to the speed of sound within the fluid itself (taken to be constant), then sound waves originating at or downstream of this characteristic radius will be unable to cross into the region of the fluid upstream of ', as the propagation of sound upstream will be exactly met by the local fluid’s displacement downstream. At the level of a verbal analogy this sounds somewhat reminiscent of a black hole, in which nothing—not even light—can escape from the black hole’s gravitational influence once it has passed a characteristic radius known as the Schwarzschild radius. As it turns out, this analogy goes much further than a verbal one, and this is precisely what Unruh demonstrated: sound waves propagating through this hydrodynamical system can be mathematically modelled as though they are a massless scalar field propagating within the curved spacetime surrounding a black-hole. In this specific hydrodynamical model, the equations of motion that govern the propagation of sound through the fluid in regions close to the characteristic radius ' can be approximated by the null-trajectories (3B2 = 0) of a metric-equation describing a curved geometry. Ignoring the angular components, the metric equation in question is23 2 0 ' 2 2s 2 3B = ( ) 22s A ' 3 3A , (1.3.2) 2s ( − ) − 2 A ' ( − ) where 0 A is the (radius-dependent) background density of the fluid about ( ) which the hydrodynamical equations of motion are linearized, 2s is the speed of sound in the fluid (taken to be constant everywhere), is a small constant with dimensions of inverse-time that is used to linearize the fluid flow velocity in regions close to the characteristic radius ', and is a modified time coordinate (see reference [1] for details). This metric equation description of this system is noteworthy due to the fact that it is term-wise equivalent to the metric equation

22Speciﬁcally, Unruh makes the claim, “This system forms an excellent theoretical laboratory where many of the unknown eﬀects that quantum gravity could exert on black-hole evaporation can be modelled.” 23 2 Note that an algebraic error in Unruh’s paper [1] results in a missing factor of 2s in the 3A term. 20 Introduction |

describing the spacetime just outside of the event horizon of a Schwarzschild black-hole. Ignoring the angular components of the metric, the metric-equation of Schwarzschild spacetime is (with the , , , metric signature) given by (+ − − −) 1 ' ' − 3B2 = 1 s 223C2 1 s 3A2, (1.3.3) − A − − A where 2 is the speed of light and 's is the Schwarzschild radius. For radial values

close to 's—e.g., close to the event horizon of a Schwarzschild black-hole—the Schwarzschild metric is approximated by24:

A 's 's 3B2 ( − ) 223C2 3A2. (1.3.4) ≈ 's − A 's ( − ) A direct comparison of Equation 1.3.2 and Equation 1.3.4 reveals that the two metric-equations have the exact same mathematical form! That is to say, the mathematical description of sound propagation in a specific region of this specific hydrodynamical system is governed by an effective metric that is formally equivalent to the metric description of Schwarzschild spacetime in regions close to the Schwarzschild radius, which for a black hole is the region of space close to the event horizon. In the spirit of historical completeness, it should be noted that Visser independently discovered that certain fluid mechanical systems could be described by the Lorentzian geometry of general relativity [92]. Until Visser’s rediscovery of this fact, Unruh’s work appears to have gone largely unknown within the physics community.

1.3.2 Developments

Gordon’s optical-metric model and Unruh’s acoustic black-hole model are only two specific examples of what is now a whole catalogue of models that are both many and varied. While a full review of the field of analogue gravity and the many models that it contains is outside of the scope of this thesis, a brief overview of some select results from the analogue gravity research endeavour will be included here. For a significantly more detailed overview of the state of the subject as of the year 2011, see the comprehensive review article by Barceló, Visser, and Liberati [93]. Experiments within the field of analogue-gravity have, in some instances, provided us with indirect evidence for the existence of certain physical phenomena

24Unruh expresses the Schwarzschild metric in units where = 2 = 1 in his paper [1]: in these units, the Schwarzschild radius is given by 's = 2". Analogue gravity 21 | that are predicted to arise by general relativity or semiclassical gravity. Of note, the following phenomena have found their ﬁrst experimental evidence (again, indirectly) within the analogue gravity research endeavour:

1. Hawking radiation has been experimentally observed/inferred in a number of diﬀerent analogue-gravity models in recent years [94–97]. 2. Cosmological particle production has been experimentally observed/inferred in ion-trap experiments in which the trapping potential is rapidly switched [98, 99]. 3. Superradiance has been experimentally observed/inferred in a hydrodynamical analogue of a black-hole [100].

It is important to re-stress that these experimental results only indirectly provide evidence for the existence of the above physical phenomena: they do not (and, in fact, cannot) directly confirm the existence of the actual phenomena that they emulate.25 The field of analogue gravity has also—be it directly or indirectly—resulted in interesting theoretical developments too. In particular, and as Unruh pointed out in his original publication on acoustic black holes, the greatest utility of analogue-gravity models may be to provide us with a source of insight in regards to understanding problems that arise within the realms of semiclassical gravity and quantum gravity. This optimistic prospect afforded to us by analogue-gravity models is likely the main reason that people are drawn to the field, and a fairly considerable research effort has indeed emerged to this end. Some of the noteworthy theoretical results from (or related to26) the analogue gravity literature include:

1. Hawking radiation and gravity while linked, are not fundamentally linked. Hawking radiation exists in non-gravitational settings, with the fundamental requirements being the presence of a horizon on some Lorentzian manifold (hence the appearance of Hawking radiation in analogue-gravity models). [104] 2. Planck scale physics might ultimately prove to be an unimportant consideration in regards to the process of Hawking emission. This is a fact supported by analogue gravity and analogue-gravity inspired numerical [105–107] and analytical [108–111] studies. These studies imply that the existence of (and the speciﬁc mathematical form describing) Hawking radiation is insensitive

25Not only is the evidence indirect, but in some cases it appears to be contentious. See the comment by Leonhardt [101] on the experimental interpretation of reference [96], and also see Steinhauer’s response to Leonhardt’s comment [102]. 26Unruh claims ([103]) that work by Jacobson was inspired by analogue models. 22 Introduction |

to modiﬁcations of physics at the Planck scale, easing theoretical concerns that arise over considerations of trans-Planckian modes within the Hawking emission process. In the case of analogue gravity models, the transition from continuum physics to microscopic physics at length scales close to the medium’s discretization length is treated analogously to the expected breakdown of current physical laws at the Planck scale.

The experimental and theoretical results from the analogue gravity research endeavour provide us with some evidence for the idea that it may be possible to view some subset of the phenomenology of general relativity as emergent. Given that modern analogue-gravity models are conceived of to be experimentally accessible in principle, the ﬁeld appears to be of great utility in probing the phenomenology of emergent gravity theories.

1.4 Making sense of it all: the role of analogue-gravity models in the hunt for a microscopic theory of gravity and the aim of this thesis

Ultimately, the goal of physics is to describe the universe around us. Historically, this has meant that people have observed natural phenomena and have then developed models and theories to explain these phenomena. Often the models and theories that we have constructed have then, in turn, allowed for us to infer the existence of further phenomena. It is by the latter part of this process that we first inferred that their ought to exist a theory of quantum gravity: it appears to be a logical consequence of a universe in which both quantum mechanics and general relativity exist. Subsequently, further results from theory indicated to us that gravity might, in fact, be an emergent phenomena. Whatever the case may be, we have so far found ourselves at a dead-end when it comes to experimentation: owed to the extreme and remote physical circumstances in which the effects of both quantum mechanics and gravity are expected to be important, we have as of yet had no means in obtaining experimental evidence to direct our research efforts. In regards to theory development, there are two major concerns that arise from our lack of ability to perform experiments. First, even if we happened to come across a valid microscopic theory of gravity, an inability to experimentally validate it would prevent us from determining that it is indeed valid. Second, without experimental guidance it is very difficult to put bounds on which parts of theory-space we should explore. Making sense of it all: the role of analogue-gravity models in the hunt for a microscopic theory of gravity and the aim of this thesis 23 | So how does all of this tie together? We want to unify our understanding of gravity with quantum mechanics, and to date we have had no experimental guidance in theory development. We do, however, know of analogue-gravity models: models that mimic some of the phenomenology of general relativity, and importantly, models that are based on physical systems that can actually be (and have actually been) constructed within laboratories. In the absence of any direct tests for our microscopic theories of gravity, indirect tests are the next best thing. Provided that we can identify appropriate candidate systems for a given type of gravitational phenomena, analogue gravity models provide a compelling avenue by which we can indirectly test such phenomena. How best do we go about utilizing analogue-gravity systems to this end then? In some sense, the human experimenters who would deal with an analogue- gravity experiment have a “god’s eye view” on any such analogue gravity universe: they know the rest frame of the medium and they can signal faster (via light) than should be allowed (via sound) in the analogue universe. The actual experimenter’s view—or the laboratory view—is therefore not appropriate if one wishes to use analogue-gravity models as an operational substitute for an actual astrophysical or cosmological experiment. It seems important then to devise some operational procedure by which we can investigate analogue gravity models internally: that is, from the point of view of observers or devices for whom the analogy to relativistic physics appears to be identity, rather than just an analogy. That brings us to the aim of this thesis. At a broad level, the aim of this thesis is conceptually quite straightforward: the aim is to develop and characterize notions of observers/devices/detectors that are natural to analogue gravity universes. By “natural to analogue gravity universes” it is meant that these observers/devices/detectors are constructed and operated in such a way that they obey the symmetries that are inherent to the analogue gravity universe. To this end, the following restriction is placed on the observers/devices/detectors that will be herein considered: any measurements or communication of information can only be made using sound. In the acoustic models of analogue gravity, sound is the object whose trajectories are the null- geodesics of some curved Lorentzian metric, and so by restricting measurements and communication in this way, we can—in the appropriate regimes—consider our observers/devices/detectors to be one-to-one operational substitutes for the observers/devices/detectors that we would ideally have access to in a real (i.e., non-analogue) gravitational experiment.

THEORY

This chapter is intended to provide a somewhat pedagogical overview of necessary mathematical machinery required to understand the field of analogue gravity at the broadest of levels. In essence, this chapter should provide the reader with two things: first, a self-contained reference point to make sense of the notation utilized throughout this thesis; second, the basic conceptual understanding required to make sense of the content of this thesis. 2 The reader who is unfamiliar with—or who feels slightly rusty in regards to—quantum theory and relativistic physics will hopefully benefit from taking the time to read through Sections 2.1 and 2.2; the reader who is reasonably comfortable with these areas of physics can safely skip the corresponding section (or sections) without fear of missing anything of vital importance. Section 2.3 provides some of the basics of field theory—both classical and quantum—and will, as per the preceding sections, be of use to the reader who is either unfamiliar with the field, or the reader who feels that they are in need of a brief recap on the subject matter. Sections 2.4 and 2.5 can be seen as complimentary to one another: both sections detail the construction of specific analogue gravity models, however, while Section 2.4 takes a top-down approach to describe an analogue gravity model based on a particular physical system, Section 2.5 takes a bottom-up approach to describe an analogue gravity system using a toy model. In particular, the former of these two sections—Section 2.4—details the construction of perhaps the most famous class of analogue gravity models: this section is a vital necessity to the reader who is unfamiliar with the details of analogue gravity models; conversely, this section can safely be skipped by the reader with a familiarity on the subject matter. If the reader is interested in specific models of analogue gravity—or the investigation of specific phenomena within analogue gravity models—then a more technical and detailed understanding of the topics discussed within this chapter (and topics not discussed herein) will likely be required. References will be provided throughout for the reader who wishes to read more deeply on the relevant subject matters.

2.1 Relativity

The path that leads to special relativity can be appreciated with a basic observation: from Maxwell’s equations, one can derive a wave equation for electromagnetic waves—of which light is an example—that makes reference to some characteristic

25 26 Theory |

speed: the speed of light. A speed, one would think, should be made with respect to some rest frame; for example, the speed of sound of some given material is defined with respect to the rest frame of the material itself. This leads to the question: with respect to which frame is the speed of light defined? For a not-inconsiderable period of time it was assumed that space—even though it appeared to be a vacuum, totally devoid of anything material—was filled with some light-bearing medium: the luminiferous aether. The conventional wisdom prior to the early 20th century was that it was with respect to the luminiferous aether that the speed of light was defined, though eventually the nature of the luminiferous aether was called into question on both theoretical and experimental grounds: despite repeated attempts to detect the presence—or rather, the effect of the presence—of this hypothetical material, it could not be observed. It is usually considered (at least in retrospect) that the final nail in the coffin for aether theory came after the null result of the famous Michelson–Morley experiment in 1887 [112]: by this point in time, significant developments in optics had permitted for the construction of an interferometer sensitive enough to allow for the detection of the difference in the speed of light that should occur in each arm of the interferometer over the course of a year if the Earth moved through a medium that carried light. No such evidence could be found. There were efforts to save aether theory after the null-result of the Michelson– Morley experiment, and this is discussed somewhat more in Section 3.1, however, this is not our focus here. Our focus here is to give a brief overview of the mathematical machinery of Einstein’s theories of special relativity and general relativity, both of which are constructed under the assumption that there is no preferred universal reference frame.

2.1.1 Tensors and Einstein summation convention

The mathematical foundation of special relativity and general relativity is that of differential geometry, and within the scope of differential geometry, the particular mathematical stage for special relativity and general relativity is that of a pseduo- Riemannian geometry [113, 114]. While a proper review of differential geometry is outside of the scope of this thesis, this subsection will endeavour to provide the reader with the necessary notational rules to understand the content of this thesis (where differential geometry is concerned), and in the process of presenting this some of the basic concepts of differential geometry will be discussed. Before proceeding any further, let us take a moment to make clear the conventions and notational choices that will apply herein:

• We will make use of the coordinate free notation of tensors to refer to Relativity 27 |

tensors in the abstract. In this notation, tensors are denoted with bold font and no indices, e.g., A. When confronted with a symbol like this, one is to understand that we are talking about the tensor A, rather than its individual components or any speciﬁc representation thereof.

• In referring to tensors in regards to their components (be it explicitly or implicitly) we will use abstract index notation. Unless otherwise speciﬁed, indices do not refer to any particular coordinate system. For example, on its own the object is to be understood to refer to the collection of components that constitute the mixed rank-2 tensor A.

• Primed and non-primed symbols will be utilized to distinguish between representations of the same tensor in different coordinate systems. Given the rank-2 tensor A, the non-primed symbol will be used to denote the components of A with respect to one arbitrary coordinate system (G = G1,G2,G3,G4 ), whereas the primed symbol will be used to denote the ( ) 0 components of A with respect to some other arbitrary coordinate system (G = G 1,G 2,G 3,G 4 ). 0 ( 0 0 0 0 ) In general these conventions and notational choices will apply throughout the rest of this chapter unless otherwise stated. The mathematical equations that govern special relativity and general relativity are expressed in terms of tensorial equations. Tensorial equations are expressed in terms of tensors1, which are mathematical objects that, by construction, possess coordinate-independent properties (that is, each tensor is associated with invariant properties under coordinate transformations). The coordinate independent properties of tensors are inherited by tensorial equations, and as a result, these equations are useful for describing physical laws: coordinates are not physical, and therefore, coordinates should play no explicit role in the laws of physics. The non-physicality of coordinates is highlighted in Figure 2.1: two different observers come across some vectorial quantity in space whose existence is entirely separate of the coordinate systems used by those observers to describe it. The vector simply is. The coordinates are an artificial choice made in the minds of the observers. A tensor of rank-# in a -dimensional space is specified by # components or elements. In abstract index notation the rank of a tensor is captured by how many indices must be used to denote its components fully—one index per rank—and the indices themselves take values as many as there are dimensions. For example, a specific rank-2 tensor M in 3-dimensional space can be referred to in terms of its components "8 , where 8, 9 1, 2, 3 : we have two indices for two ranks 9 ∈ { } 1Or, more correctly, tensor fields. 28 Theory |

Figure 2.1: Two observers find a vector floating in space. One observer thinks that a polar coordinate system should be used to describe the vector, but the other observer thinks that a Cartesian coordinate system is the more obvious choice. In reality, the vector’s existence is not contingent on the existence of some coordinate system: it simply is. There is no coordinate system floating around in space with the vector, and the coordinate systems that are imagined by the two observers are merely that: imagined. Coordinate systems are a useful abstract mathematical tool, but unlike the things that they describe, coordinates are not inherently physically meaningful.

(# = 2), each index runs over three values because our space is three-dimensional ( = 3), and so in total we have # = 32 unique combinations of the ordered pair 8, 9 , where a given 8, 9 denotes an element/component of M. ( ) ( ) Tensors of rank-1 and rank-2 can be conveniently represented on paper as arrays. In relativistic physics, we work with = 4 and we usually start our indexing with 0 rather than 1. As a result, in what follows we will most often see tensors denoted in the following way when an explicit matrix representation is given:

0 * ,00 ,01 ,02 ,03 © 1ª © ª * ® ,10 ,11 ,12 ,13® * = ® , = ® . (2.1.1) 2® ® * ® ,20 ,21 ,22 ,23® 3® ® * ,30 ,31 ,32 ,33 « ¬ « ¬ The position of the indices of a tensor—superscript (upstairs) or subscript (downstairs)—serve two main purposes: it provides both a notational way to keep track of the mathematical structure of a given tensor, as well as a notational shorthand for performing mathematical operations on the components of tensors. The mathematical structure that is indicated by the index positions is the following: a tensor Relativity 29 | with only superscript (upstairs) indices is referred to as contravariant, whereas a tensor with only subscript (downstairs) indices is referred to as covariant. Tensors do not have to be completely contravariant or completely covariant: they can be mixed and have both contravariant and covariant properties, e.g., the Riemann curvature tensor often appears in mixed form as ' . The notational shorthand (which is itself a consequence of the mathematical structure of tensors) aﬀorded to us by the index placement is that of Einstein summation convention. Einstein summation convention is the following: when an index is repeated once upstairs and once downstairs, it is to be summed over. Two examples to highlight this are:

1 Õ− 0 1 1 B = 0 1 − 1, (2.1.2) + + · · · + − =0 1 Õ− 0 1 1 B = − . (2.1.3) 0 + 1 + · · · + 1 =0 −

In terms of Einstein summation convention, Equation 2.1.2 is an example of the dot product between two vectors, whereas Equation 2.1.3 is an example of matrix multiplication of a vector. Using Einstein summation convention, we can write the components of a vector in some coordinate system (primed) in terms of the components of the same vector in some other coordinate system (non-primed). The transformation rule for contravariant indices is:

%G0 0 = ; (2.1.4) %G whereas covariant indices transform as follows:

%G 0 = . (2.1.5) %G0 A point of notation: note that when an index appears upstairs in the denominator, it functions as though it were downstairs overall (hence the summation over in Equation 2.1.4, leaving only left). The opposite is also true: an index that is downstairs in the denominator functions as though it were upstairs overall. Also note that these transformation rules apply to a given index (rather than “to the tensor”, and so the transformation rule generalizes in a straight-forward manner to tensors of higher rank. For example, the components of the Riemann curvature tensor, ' , will transform like:

%G %G %G %G ' = 0 ' (2.1.6) 0 $ $ %G %G0 %G0 %G0 30 Theory |

Contravariant rank-1 tensors are just vectors—which we shall refer to as contravariant vectors—whereas covariant rank-1 tensors are one-forms—which we shall refer to as covariant vectors.

The rank-2 tensor of partial derivatives that transforms the components of covariant vectors between coordinate systems in Equation 2.1.5 is the Jacobian matrix relating those two coordinate systems (denoted J), whereas the rank-2 tensor of partial derivatives that transforms the components of contravariant vectors between coordinate systems in Equation 2.1.4 is the inverse Jacobian matrix 1 (denoted J− ). The explicit matrix representation of the Jacobian matrix and the inverse Jacobian matrix take the following forms:

0 0 %G0 %G0 %G %G © 0 0 ª © 0 0 ª %G ··· %G ® %G0 ··· %G0 ® %G0 . . . ® 1 %G . . . ® = = . . . . ® ; − = = . . . . ® ; %G ® %G ® 0 ® 0 0 ® %G0 %G0 ® %G %G ® ··· « %G ··· %G ¬ «%G0 %G0 ¬ (2.1.7) where as previously established, the indices from 0 to , where is the dimensionality of the space. The Jacobian matrix and the inverse Jacobian matrix obey the following identity:

 1 %G0 %G  1, = , − = = = (2.1.8) %G %G 0  0, ≠ ,  where the collection of components constitute the Kronecker delta tensor.

A rank-2 tensor of supreme importance to differential geometry and tensor algebra is the metric tensor g. For our purposes, it is sufficient to say that the metric tensor is defined such that it maps pairs of contravariant vectors to the reals, R in the following way: , = , (2.1.9) where R is a scalar (or a rank-0 tensor). Notationally, this means that the ∈ metric tensor acts on contravariant vectors in such a way as to lower their index (i.e. the metric tensor turns upstairs indices into downstairs indices, turning contravariant vectors into covariant vectors) and swap the label:

, = , (2.1.10) , = , (2.1.11) Relativity 31 | which can be used to induce the summation implied by Einstein summation convention in Equation 2.1.9:

, = = = (2.1.12)

Note that while we may think of deﬁning the metric tensor such that it maps pairs of vectors to the reals, we are not restricted to only use it in this way. The index-lowering property of the metric tensor as seen in Equations 2.1.10 and 2.1.11 is valid on its own, which is to say we can generically lower indices without inducing summation if we so please:

, = . (2.1.13)

The metric tensor can equally well be used to lower the contravariant indices on tensors of arbitrary rank, e.g.,

, , = , . (2.1.14)

From the metric tensor, we can deﬁne the inverse metric tensor whose components , satisfy the following propety:

  1, = , ,, = = . (2.1.15)  0, ≠ .  The inverse metric tensor acts on covariant vectors to raise their index (i.e. it turns covariant vectors into contravariant vectors), in analogy to how the metric tensor acts on contravariant vectors:

, = , (2.1.16) , = . (2.1.17)

By analogy with the metric tensor, the inverse metric tensor can equally well be used to raise the covariant indices of higher order tensors, e.g.,

, , = , . (2.1.18)

A process called tensor contraction—a generalization to the trace of a matrix—can be performed on mixed type tensors with rank-2 or higher. Contraction of a tensor occurs when a contravariant index and a covariant index of that tensor are summed over: the result of this process is to reduce a tensor of rank-# to a 32 Theory |

new tensor of rank- # 2 . For example, the Ricci curvature tensor is formed by ( − ) contracting the Riemann curvature tensor over its contravariant index and its second covariant index:

' B ' = '0 '1 '2 '3 . (2.1.19) 0 + 1 + 2 + 3

When a tensor of rank-2 or greater has more than one contravariant index, the metric tensor can be used to lower one of the contravariant indices and induce contraction. Likewise, when a tensor of rank-2 or greater has more than one covariant index, the inverse metric tensor can be used to raise one of the covariant indices and induce contraction. For example, the inverse metric tensor can be used to contract the Ricci curvature tensor to form the Ricci scalar, a measure of the scalar curvature of a manifold:

' B ,' = ' = ' = ' 0 ' 1 ' 2 ' 3. (2.1.20) 0 + 1 + 2 + 3

The metric tensor—or objects that can be formed from it—captures all of the relevant geometric information of the space that it describes. In a sense, the metric tensor is a full-specification of geometry. One of the most obvious geometric quantities that one often needs to compute in a given geometric space is the length of some path connecting two points. The total length B of some path connecting two points can be computed by integrating the infinitesimal line-element 3B along the path. The infinitesimal line-element, 3B, is given in terms of the coordinate differentials 3G in the following way:

2 3B = , 3G 3G . (2.1.21)

In ordinary Euclidean space described with Cartesian coordinates we have the following

G 1 0 0 © ª © ª 8 ® ® G = H® , ,89 = 0 1 0® , (2.1.22) ® ® I 0 0 1 « ¬ « ¬ from which we can show that Equation 2.1.21 reduces to Pythagoras’ theorem in three dimensions:

3B2 = 3G2 3H2 3I2 (2.1.23) + + Relativity 33 |

2.1.2 Special relativity

Before proceeding, we make some additional notational choices for this subsection:

• Unless otherwise speciﬁed, when the matrix representation of a tensor is given, the Cartesian spatial coordinates G,H,I will be assumed. That is to say, ( ) when the symbol appears on one side of an equals sign, and a matrix corresponding to appears on the other side of that same equals sign, the spatial coordinates referred to by and are assumed to be G,H,I by ( ) default. Note that the abstract index notation convention that we established in Subsection 2.1.1 is overruled in this particular case but not in general (i.e., when the symbol appears on its own with no reference to a matrix, abstract index notation still holds). • Any reference frames under consideration are taken to have aligned spatial coordinate axes. That is to say, if G,H,I and G ,H ,I are the spatial ( ) ( 0 0 0) coordinate systems of two distinct reference frames, then G and G0 are

parallel, H and H0 are parallel, and I and I0 are parallel. • When a specific pair of reference frames are being considered we will assume that there exists some point in time at which their spatial origins coincided. Furthermore, we will assume that clocks positioned at the spatial origins of both reference frames read the same time when both spatial origins are coincident. We will define this time to be zero. In other words, both reference frames share a common origin with respect to both spatial and temporal coordinates. • Any relative motion between pairs of reference frames is restricted to the G-direction (as is conventional), and we take the primed frame to be moving with a positive velocity with respect to the non-primed frame. The historical path—that taken by Einstein—to arrive at the theory of special relativity (alternatively: the special theory of relativity or just special relativity) is based on two physical postulates/principles:2 1. The principle of special relativity: The laws of physics should be the same in any inertial reference frame.3 2. The principle of invariant speed of light: The speed of light takes the same value to all observers. 2There are unspoken mathematical assumptions being made here too. For example, it is assumed that the transformation laws that relate the coordinate systems of different reference frames are linear in nature. 3In other words, the equations that govern the fundamental physics of the universe should not be velocity dependent. That is not to say that velocity should not appear in the equations that govern physical phenomena, but rather that the mathematial structure of the equations themselves should not be velocity depenent. 34 Theory |

Special relativity is often described as “putting space and time on equal footing”, in that time becomes a dynamical variable of the theory rather than just a parameter: physical objects are treated as existing within a four-dimensional geometric space referred to as spacetime—a combination of the three spatial dimensions that we see in reality and the one time dimension that we experience—rather than merely existing in a three-dimensional geometric space with time as a parameter. Mathematically, spacetime is treated as a four-dimensional pseudo-Riemannian manifold characterized by a metric tensor4 that is everywhere nondegenerate5, but that—in contrast to the metric tensor describing a Riemannian manifold—allows for proper-lengths that are negative or zero. The spacetime of special relativity is characterized by the Minkowski metric, whose components are most commonly

labelled . In Cartesian coordinates, the Minkowski metric can be given in one of two ways: 1 0 0 0 ©∓ ª ® 0 1 0 0 ® = ± ® . (2.1.24) 0 0 1 0 ® ± ® 0 0 0 1 « ± ¬ In this subsection we will choose to work with the mostly-positive metric signature, , , , 6. That is to say, we will be working with the Metric: (− + + +)

1 0 0 0 ©− ª ® 0 1 0 0® = ® . (2.1.25) 0 0 1 0® ® 0 0 0 1 « ¬ Let’s make our lives easier by pre-emptively deﬁning the fractional velocity (with respect to light) () and the Lorentz factor () as follows:

E B , (2.1.26) 2 1 B p ; (2.1.27) 1 2 −

4We shall often simply refer to the metric describing a spacetime as “the metric tensor” or “the metric”. 5The requirement that the metric tensor be nondegenerate can be given by the following statement: given a non-zero tangent vector U (i.e., the length of U is non-zero) at any point of the manifold, one can always ﬁnd a tangent vector V at the same point of the manifold such that ,+ * ≠ 0. 6By contrast, the mostly negative metric signature is , , , . The mostly-negative metric (+ − − −) signature will be utilized in other parts of this thesis, and so to prevent confusion the reader will always be made aware of the metric signature of choice in a given scenario. Relativity 35 | where E B v and v is the velocity of an object as is commonly understood. | | Equipped with these deﬁnitions, we can now delve in to the formalism of special relativity.

The four-position X can be represented in ﬂat spacetime using Cartesian spatial-coordinates as follows:

2C © ª ! ® G ® 2C - = ® = , (2.1.28) H ® x ® I « ¬ where we have defined the three-position vector x B G,H,I . In special relativity ( ) the time C is merely another coordinate, and in general different observers will associate different values of C to a given event. The coordinates of one reference frame can be related to coordinates of another reference frame in special relativity using the Lorentz transformation, which is given by the rank-2 tensor Λ, and from 1 which we will denote the inverse Lorentz-transformation as either Λ− or Λ0 (the suggestive notation, as we will see, is intentional). The action of these tensors on the four-position is given as follows:

-0 = Λ - , (2.1.29) 1 - = Λ− -0 (2.1.30)

In its most general form, the matrix representation of Λ is quite cumbersome to work with. Thankfully with our particular choices (Cartesian coordinates, both reference frames have aligned coordinate axes and share a common origin, and the primed frame has positive velocity with respect to the non-primed frame), the Lorentz transformation and its inverse can be represented simply as follows:

0 0 0 0 © − ª © ª ® ® 0 0® 1 0 0® Λ = − ® ; Λ− = Λ0 = ® . (2.1.31) 0 0 1 0® 0 0 1 0® ® ® 0 0 0 1 0 0 0 1 « ¬ « ¬

Written out in matrix form this way, the alternate notation of Λ0 is slightly more clear. The only diﬀerence between the representations of the Lorentz transformation and its inverse under our assumptions is that the sign on diﬀers. If the non-primed frame measures the primed frame to have fractional velocity , then the primed frame measures the non-primed frame to have fractional velocity 36 Theory |

: in this sense, the deﬁnition of the Lorentz transformation matrix in one frame − corresponds to the deﬁnition of the inverse Lorentz-transformation matrix in the 1 other frame, hence the notational choice to have Λ− = Λ0. Let us explicitly use Equation 2.1.30 to write the components - in terms of the components -0 :

2C 0 0 2C 2C G © ª © ª © 0ª © 0 + 0 ª ® ® ® ® G ® 0 0® G0 ® G0 2C0 ® ® = ® ® = + ® (2.1.32) H ® 0 0 1 0® H0 ® H0 ® ® ® ® ® I 0 0 0 1 I0 I0 « ¬ « ¬ « ¬ « ¬

While the times C and C0 are coordinates, there does exist a coordinate independent notion of time within special relativity: the proper time between two events in spacetime is the time elapsed by any inertial clock that is coincident with both events. The proper time is unambiguous and is agreed on by all observers: an inertial clock that is coincident with both events will have a definite, observer-independent time on it when it is coincident with the first event, and it will also have a definite, observer-independent time on it when it is coincident with the second event. If, for the sake of illustration, both events are tied to a single physical object (e.g., a point-like light source turning on at one moment in time and then turning off again at some later point in time), then the proper time can simply be understood to be the time ellapsed between those two events by a clock that is attached to that object. Said another way, the proper time elapsed between two events is the difference in the time coordinate in the frame of those events. The mathematical relation between some observer’s coordinate time and the proper time associated to a pair of events can be obtained easily by considering the components of the four-position X in two separate frames. Take a pair of events to define some primed frame in the following sense: the first event occurs at the coordinates C ,G ,H ,I = 0, 0, 0, 0 , and the second event occurs at the ( 0 0 0 0) ( ) coordinates C ,G ,H ,I = C0 , 0, 0, 0 , where the subscript in C0 is used to remind ( 0 0 0 0) ( 2 ) 2 us that this is the value of the coordinate C0 for the second event. Because both events occur at the same spatial coordinates in the primed frame, any clock that is both stationary with respect to the primed frame and coincident with the first event is then also coincident with the second event. As a result, the proper time between both events is then simply given by the difference in the coordinate

times C0 at the location of those events in the primed frame: = C20 . The Lorentz transformation relating the non-primed frame to the primed frame can then be used to show the following: 3C = . (2.1.33) 3 Relativity 37 |

This is one way to express the notion of time dilation. In the non-primed frame time increments forwards in amounts larger than in the primed frame by a factor of .

From here we are properly equipped to deﬁne the four-velocity U. The components of the four velocity can be deﬁned in the following way: ! 3- 3C 3- 3- 2 * B = = = (2.1.34) 3 3 3C 3C v

One can quickly verify that the four-velocity is indeed a proper tensor using the Minkowski metric. The dot product of the four-velocity with itself yields 22, where 2 is the speed of light: in relativistic physics the speed of light is a − constant to all observers, and so the four-velocity has a characteristic invariant to all observers:

** = ** = 2 22 E2 = 222 1 2 = 22, (2.1.35) (− + ) − ( − ) − where the ﬁnal equality follows from the deﬁnition of the Lorentz factor.

Using the definition of the infinitesimal line element from differential geometry (as defined in Equation 2.1.21), one can express the differential spacetime line-element in terms of the differentials of either the four-position or the four-velocity:

2 2 3B = 3 * * = 3- 3- (2.1.36)

3- 3- is straight-forward to evaluate, and * * is given by Equa- tion 2.1.35. Explicitly evaluating the right-most term with Cartesian spatial coordinates gives the following:

3B2 = 223 2 = 223C2 3G2 3H2 3I2. (2.1.37) − − + + +

To actually compute the length of the spacetime line-element B we can write the following: s ¹ B B0 ¹ 0 ¹ C C0 2 2 2 + + + 1 3G 1 3H 1 3I 3B = 82 3 = 82 1 3C (2.1.38) ˜ ˜ − 2 2 − 2 2 − 2 2 ˜ B0 0 C0 2 3C 2 3C 2 3C ˜ ˜ ˜ C C0 ¹ + q = 82 1 C,G,H,I 23C, (2.1.39) ˜ ˜ C0 − ( ) ¹ C C0 + 3C = 82 ˜. (2.1.40) C0 38 Theory |

Note that we have used tildes above symbols here (e.g., C) to denote when a ˜ variable is an integration variable. The square of the spacetime line-element7 can then be expressed like:

!2 ¹ C C0 + 3C B2 = 22 2 = 22 ˜ . (2.1.41) − − C0

The right-most expression is quite useful as it provides observers in any generic frame an explicit method by which to parametrize (and thus compute) the spacetime line-element between two events in terms of the time that they measure between those two events. The sign of B2 between any two events has an important physical meaning within relativistic theories, dividing spacetime into three distinct regions in the following way:

• s2 < 0 characterizes trajectories through spacetime that are traced out by particles with positive rest-mass. These are called time-like trajectories or time-like curves, and pairs of events along such trajectories are referred to as being time-like separated.

• s2 = 0 characterizes trajectories through spacetime that are traced out by massless particles. These are called null trajectories or null curves, and pairs of events along such trajectories are called light-like separated.

• s2 = 0 > 0 characterizes trajectories through spacetime that cannot be traced out by any known particle as, in order for a particle to trace out such a trajectory, it would have to travel faster than light.8 We refer to these paths through spacetime as space-like curves, and pairs of events along such curves are referred to as being space-like separated.

The regions of spacetime corresponding to the three distinct signs of B2 provide us with a notion of causal structure within relativity: events can only be causally related to one another if they are connected by light-like or time-like trajectories, i.e., for two events to be causally connected they must be characterized by B2 0; ≤ if two events are characterized by B2 > 0 then they cannot possibly have causally inﬂuenced one another (because no real particle can possibly have travelled such a path).

7This is merely convention; we could equally as well write out an equation for B itself, however it is customary in relativistic physics to write down the square on account of the fact that B2 is always a real number (B2 R). ∈ 8Hypothetical particles called tachyons would be able to trace out such trajectories if they existed, though there is no evidence for the existence of such particles. Relativity 39 |

2.1.3 General relativity

For a comprehensive guide to Einstein’s general theory of relativity, consult Gravitation9 by Misner, Thorne and Wheeler [113]. Once special relativity has been established, the principle that leads to general relativity is the the equivalence principle. Conceptually, the principle takes the following form:

• The equivalence principle: the inertial mass of an object is equivalent to its gravitational mass. Equipped with the equivalence principle, one can proceed to construct a theory of gravity within the mathematical framework of diﬀerential geometry. The spacetime of special relativity is described by a metric tensor ﬁeld that has— to inertial observers—the same description everywhere and at all times: the

Minkowski metric, whose components are denoted . On the contrary, the metric tensor field in general relativity—whose components are denoted ,—is much more generic: in general, it is a function of both space and time. In general relativity, we identify the curvature of spacetime—how the spacetime manifold (described by the metric tensor) varies from point-to-point in space and time—to be what we refer to as gravity. Furthermore, unlike in special relativity in which the metric tensor field forms a static background upon which events simply happen to unfold, the metric tensor field of general relativity is a dynamical variable whose structure is determined by the matter and energy content of spacetime itself. The dynamical equations that determines the spacetime metric are called the Einstein Field Equations (EFEs):

1 8 ' ', Λ, = ) (2.1.42) − 2 + 24 where ' are the components of the Ricci curvature tensor, ' is the Ricci scalar

(or the scalar curvature) obtained by contracting the Ricci curvature tensor, , are the components of the metric tensor itself, Λ is the so-called cosmological constant10 (usually identiﬁed to be the intrinsic energy of empty space), is

Newton’s gravitational constant, 2 is the speed of light in vacuum, and ) are the components of the stress-energy tensor (or the energy-momentum tensor), which captures the distribution of matter and energy in spacetime. The only non-scalar quantity in this equation that is not inherently related to the metric tensor is the stress-energy tensor. The Ricci curvature tensor (and, by extension, the Ricci 9Be warned though: owed to its truly comprehensive nature on the subject matter, this book is immense (both in terms of scope of content and physical size). 10Note that the cosmological constant Λ is a scalar that is in no way associated to the rank-2 Lorentz transformation tensor Λ. 40 Theory |

curvature ') is given in terms of Christoffel symbols, and Christoffel symbols are themselves given in terms of the metric tensor: as a result, the left-hand-side of the Einstein field equations can be written entirely in terms of the metric tensor and its derivatives. The Einstein field equations therefore describe a dynamic interplay of two conceptually distinct objects: spacetime (represented by the metric tensor), and the matter and energy content of spacetime (represented by the stress-energy tensor). This dynamical interplay is perhaps most famously summed up by John Archibald Wheeler, who said [115]: “Spacetime tells matter how to move; matter tells spacetime how to curve.”

The components of the Ricci curvature tensor, ', are obtained by contracting

the first and third indices of the Rienmann curvature tensor, ' . The Rienmann curvature tensor is defined in terms of the Levi-Civita connection , which itself ∇ can be defined in terms of Christoffel symbols (of the second kind), the components of which are denoted Γ . Using index notation, the Rienmann curvature tensor has components given by

' = % Γ % Γ Γ Γ Γ Γ , (2.1.43) − + −

and so the Ricci curvature tensor’s components can be computed in the following way: ' = ' = , , ' = ' . (2.1.44) where we have relabelled the dummy-index between the left-hand-side and the right-hand-side of the second equal sign. From here, we can further contract the Ricci curvature tensor to obtain the Ricci scalar (or the scalar curvature):

' = ' = , ' = , ' , (2.1.45) where, again, we have relabelled the dummy-index between the left-hand-side and the right-hand-side of the second equal sign. Christoffel symbols are tensor-like objects (Equation 2.1.51 and the preceding discussion elaborate on this point) that it is useful to define in differential geometry, specifically when considering the Levi-Civita connection on a manifold. We define Christoffel symbols of the second kind to be the array of coefficients Γ that can be used to express the Levi-Civita connection in the coordinate direction e like: ∇

e = Γ e . (2.1.46) ∇

Specifically e defines, at every point of a manifold, a local coordinate basis, i.e., it defines the basis of the tangent space at every point on a manifold. The elements Relativity 41 |

of e are themselves rank-1 tensors, rather than scalars, and the form of e can be given by its action on D, which are the components of the tangent vector u:

u = D e. (2.1.47)

Again it is important to stress that the elements of e are not scalars, which can be appreciated by the fact that the action of e on the components of u (that is, the action of e on D ) is to return the vector u itself. In a sense e provides at any point of a manifold the most natural choice of local coordinate basis.

Explicitly, Christoﬀel symbols of the second kind are given by the equation ! 1 %, %, %, Γ = , . (2.1.48) 2 %G + %G − %G

Perhaps somewhat confusingly, Christoffel symbols of the first kind, Γ, are typically defined in terms of those of the second kind by the equation

Γ = ,Γ . (2.1.49)

Explicitly, Christoﬀel symbols of the ﬁrst kind, Γ, are given by the equation: ! 1 %, %, %, Γ = . (2.1.50) 2 %G + %G − %G

While Christoffel symbols are related to tensorial quantities and are expressed in terms of tensorial index notation, it is important to note that Christoffel symbols do not transform like tensors under a change of coordinates. To highlight the point, considering changing from one coordinate system in which the coordinates are labelled with the indices , , and to the coordinate system in which the coordinates are labelled with the indices , , and . A tensorial object should be described by components that transform as per the rules laid out in Subsection 2.1.1—specifically contravariant indices should transform as per Equation 2.1.4, and covariant indices should transform as per Equation 2.1.5— however, a Christoffel symbol obeys the following transformation rule instead:

2 %G0 %G %G % G %G0 Γ0 = Γ (2.1.51) %G %G0 %G0 + %G0 %G0 %G Despite not being tensors themselves, Christoffel symbols play an important role in general relativity. For example, Christoffel symbols are used in the geodesic equation. Geodesics are a notion from differential geometry that are of great physical 42 Theory | importance to general relativity. In differential geometry, a geodesic between two points on a curved manifold is the path of shortest distance between those two points. In general relativity, geodesics are the trajectories that objects naturally follow through spacetime when free of any non-gravitational forces: we call such trajectories the world-line of the particle. The geodesic equation can be given in terms of the proper time as follows:

32G 3G 3G Γ = 0. (2.1.52) 3 2 + 3 3 The world-line of an object through spacetime is the path through spacetime described by the four-position G such that the geodesic equation is obeyed, and the time measured by an object travelling such a path is the proper time .

2.2 Quantum physics

For comprehensive modern treatments of modern quantum mechanics, consult the well known textbooks of Sakurai [116] and Townsend [117]. Griﬃths undergraduate level quantum mechanics textbook [118] provides an accessible introduction to the topic.

2.2.1 Basics of quantum physics

The mathematical language of quantum mechanics is that of linear algebra. Just as much of classical mechanics can be formulated in terms of vectors (e.g. position and momenta), in quantum mechanics we also use vectors to describe physical systems. In quantum mechanics the vectors that we deal with do not belong to the three-dimensional geometric space of Newtonian mechanics nor the four-dimensional geometric space of relativistic theories—they belong to a more general vector space called a Hilbert space11.A Hilbert space is a type of abstract vector space of arbitrary dimension, specifically a real or complex inner product space that is also a complete metric space. In quantum theories, we use Hilbert spaces that are complex inner product spaces, and the dimensionality of the Hilbert spaces—and thus the vectors—that we encounter range from finite and rather small (two-dimensional Hilbert spaces, for example) to uncountably infinite. The dimensionality of the Hilbert space that one considers when investigating some physical phenomenon is determined by the system itself; exactly how will be elucidated in due course.

11Note, however, that Euclidean geometry is itself an example of a Hilbert space. Euclidean geometry is an example of a ﬁnite dimensional ( = 3) Hilbert space that is a real inner product space rather than a complex inner product space Quantum physics 43 |

The state of a physical system in quantum mechanics is encoded in the state vector12; these vectors are elements of the Hilbert space and are, in general, complex-valued (hence the complex part of complex inner product space). In the case that a system is described by a ﬁnite dimensional Hilbert space, state vectors can be represented on paper in the intuitive way—as a list of numbers13—however, when the system that is being investigated is described by a Hilbert space whose dimension is uncountably inﬁnite, this notational convenience no longer applies. State vectors are represented by some symbol such as #14 within a ket, i.e., like # . While kets represent vectors that are elements of the Hilbert space, | i bras represent elements of a dual vector space (dual to the Hilbert space) and are represented like # . bra–ket notation shall be used throughout this thesis. h | Physical processes—not only interactions, but also the natural evolution of a free system—are encoded into operators in quantum mechanics: we use a “hat”15 to denote such objects, for example ˆ . Operators act on states to produce other states: when acting on kets, operators act from the left, and when acting on bras, operators act from the right, i.e.,

#new = $ # , (2.2.1) | i ˆ | oldi #new = # $ (2.2.2) h | h old| ˆ

Observables are self-adjoint linear operators corresponding to physical processes that can actually be observed (or measured). Consider some physical system that is associated to some observable described by the operator ˆ. If the system is in the state # then the expectation value of the observable is denoted and is | i ˆ h ˆi given by: = # # . (2.2.3) h ˆi h | ˆ| i States are expressed as a linear combination of basis elements, and a basis is nothing more than a choice of coordinates. Because coordinates are not physically meaningful, we have some freedom in picking a basis to describe our Hilbert space, the corresponding states, and the operators that act on those states (provided,

12Really, there is an equivalence class of vectors that define the physical state: any state vector multiplied by any complex number, 2 C, is another valid state vector. As a result, we should ∈ really say that states are defined by rays rather than vectors, however the convention is to refer to state vectors and we shall not break from this convention. 13Of course, when the Hilbert space’s dimensionality is countably infinite, one can also represent parts of the state vector on paper as a list of numbers too. 14The symbol # is usually reserved specifically for the position-space wavefunction in quantum mechanics and spinor fields in quantum field theory. We will use it as a generic label for any type of state, however if the state that we pick happens to coincides with one of these typical uses, we will explicitly make note of this. 15A circumflex. 44 Theory |

of course, that the basis of choice meaningfully describes the physical system of interest). A particularly useful basis when investigating a particular problem in quantum physics is the eigenbasis corresponding to some relevant observable.

The eigenbasis is the set of states #8 (where 8 is just an index—possibly { | i } continuously valued—to indicate which state we are talking about in the set) such that when the observable ˆ acts on a state from this set we obtain

#8 = 08 #8 , (2.2.4) ˆ | i | i where 08 R. The individual elements of the eigenbasis, #8 , are the eigenvectors ∈ | i of the operator ˆ, while the corresponding real numbers 08 are the eigenvalues of the eigenvector. When the system is in an eigenstate then the expectation value of the observable described by ˆ is simply:

= #8 #8 , (2.2.5) h ˆi h | ˆ| i = #8 08 #8 (2.2.6) h | | i = 08 #8 #8 (2.2.7) h | i = 08 . (2.2.8)

In quantum mechanics, the Hamiltonian operator (or simply, the Hamiltonian) is the generator of time translation. This is encoded in the time-dependent Schrödinger equation: 3 8~ # = # . (2.2.9) 3C | i ˆ | i For certain states called stationary states the so-called time-independent Schrödinger equation applies, # = # , (2.2.10) ˆ | i | i where is the energy of the system when its state is the stationary state # . | i Comparing Equation 2.2.10 with Equation 2.2.4 shows that stationary states are eigenvectors of the Hamiltonian. In reality, stationary states actually do have a time-dependent part however the time-dependency is only present in a complex phase factor, and this does not manifest in any observable way. After some time C, an initial stationary state # 0 will have evolved under the Hamiltonian into the | ( )i state # C given by: | ( )i 8C ~ # C = 4− / # 0 , (2.2.11) | ( )i | ( )i where the complex phase factor is the exponential term in front of the state. We mentioned earlier that quantum systems can be described by Hilbert spaces that range from ﬁnite to uncountably inﬁnite in dimensionality. How do we know Quantum physics 45 | the dimensionality of the Hilbert space for a generic system? The answer to this question is the same as the answer to the question, “how many values do I need to specify in order to fully describe the state of the quantum system of interest?” Perhaps the best way to highlight this is by considering one of the most useful models in quantum mechanics: that of the quantum harmonic oscillator.

2.2.2 The quantum harmonic oscillator

The harmonic oscillator is a model of fundamental importance to classical physics, and so it should probably come as no surprise that the quantum analogue to this—the quantum harmonic oscillator—is a system of fundamental importance in quantum physics. Perhaps most obviously, such a model can be used to model the motion of an obviously oscillating quantum system: for example, the motion of diatomic molecules. The usefulness of this model, as we shall see, far exceeds this rather humble application of it. The quantum harmonic oscillator is described, as all quantum systems are, by a Hamiltonian. The Hamiltonian for the one-dimensional quantum harmonic oscillator can be given by ?2 1 = ˆ <$2G, (2.2.12) ˆ 2< + 2 ˆ where ? is the momentum operator and G is the position operator. The position ˆ ˆ operator and the momentum operator obey the following commutation relation:

G, ? B G? ?G, (2.2.13) [ ˆ ˆ] ˆ ˆ − ˆ ˆ = 8~. (2.2.14)

In the position basis the momentum operator has the following representation in one-dimension: % ? = 8~ . (2.2.15) ˆ − %G Using the position basis representation of ?, we can rewrite the Hamiltonian for ˆ the quantum hamonic oscillator in the following way:

~2 32 1 = <$2G (2.2.16) ˆ −2< 3G2 + 2 Recall that the answer to the question, “what is the dimension of our Hilbert space?”, is the same as the answer to the question, “how many values do I need to specify in order to fully describe the state of the quantum system of interest?”. In the position basis, the state of our system can be given by its wavefunction, denoted with the bare symbol #: this represents the quantum amplitude of our system as a function of 46 Theory |

spatial location. In quantum mechanics, we treat space as being continuous, i.e., there are an uncountable infinity of points that make up space, and so to fully specify our wavefunction # we must specify the amplitude for the system at an uncountable infinity of points. The dimensionality of the Hilbert space is given by the dimensionality of the state vectors describing our system and so the Hilbert space of interest is uncountably infinite in dimension. How do we represent our state vector when their dimensionality is uncountably infinite? We simply parametrize them in terms of position, just as we would a continuous function of space: in our case, the wavefunction # has some value for every position. Consider now the eigenstates of the Hamiltonian for the quantum harmonic oscillator as described above. The eigenstates solve the time-independent Schrödinger equation, so we can write: ! ~2 32 1 <$2G # = #. (2.2.17) −2< 3G2 + 2

The family of eigenvectors of this Hamiltonian can be given in terms of Hermite

polynomials, = H . The eigenvectors are represented as continuous functions of ( ) space: 1 4 r ! / 1 <$ <$G2 2~ <$ #= G = 4− / = G , (2.2.18) ( ) √2= =! ~ ~ where = = 0, 1, 2,... , i.e., = N0, and the physicists’ Hermite polynomials are given ∈ by: = = H2 3 H2 = H = 1 4 4− . (2.2.19) ( ) (− ) 3H= Note that here we have deﬁned the physicists’ Hermite polynomials in terms of a continuous variable H and that in our Hamiltonian we have H p <$ G. The = ~ eigenvalues are also given in terms of the variable = and are:

1 = = ~$ = . (2.2.20) + 2

Despite not appearing to exist in the original description of the quantum harmonic oscillator’s Hamiltonian, the quantity = appears to be necessary to completely characterize the system as it must be introduced to solve for the eigenvectors and the eigenvalues of the Hamiltonian. The presence of this variable = means that for every frequency of oscillator we don’t just have a single eigenvector and eigenvalue, but rather an entire family of eigenvectors and eigenvalues—one

eigenvector and corresponding eigenvalue for every = N0. For a given oscillator ∈ (i.e., for some ﬁxed value of $) there is a constant energy gap between adjacent Quantum physics 47 | solutions in its family of solutions (i.e., for solutions described by = and = 1): +

Δ B = 1 = , (2.2.21) + − 1 1 = ~$ = 1 ~$ = , (2.2.22) ( + ) + 2 − + 2 = ~$. (2.2.23)

As a result of this behaviour, a given pair of orbitals in an atom can be modelled as a restricted quantum harmonic oscillator (one for which = only has two values): an electron must absorb or emit a ﬁxed quanta of energy—a single photon of energy ~$—in order to transition between the two orbitals.

Let us now deﬁne the following new operators in terms of the position operator, G, and the momentum operator, ?: r <$ 8? 0 = G ˆ , (2.2.24) ˆ 2~ ˆ + <$ r <$ 8? 0† = G ˆ . (2.2.25) ˆ 2~ ˆ − <$

We can show that these operators obey the following commutation relation,

8 0, 0† = − G, ? = 1. (2.2.26) [ˆ ˆ ] ~ [ ˆ ˆ] With these deﬁnitions, we can rewrite the Hamiltonian for the quantum harmonic oscillator like: 1 = ~$ 0†0 . (2.2.27) ˆ ˆ ˆ + 2 This particular form of the Hamiltonian is very suggestive; it is reminiscent of the expression for the eigenvalues given in Equation 2.2.20! Taking some inspiration from the eigenvalue equation, deﬁne the following operator:

# B 0†0. (2.2.28) ˆ ˆ ˆ

We can also show that this new operator, #ˆ , obeys the following commutation relationships:

#, 0† = 0†, (2.2.29) [ ˆ ˆ ] ˆ #, 0 = 0. (2.2.30) [ ˆ ˆ] −ˆ 48 Theory |

Rewriting the Hamiltonian in terms of the operator # we obtain: | i 1 = ~$ # . (2.2.31) ˆ ˆ + 2

This form of the Hamiltonian is linear in the operator #ˆ and has no other operator dependency, thus the eigenvectors of #ˆ are also eigenvectors of the Hamiltonian. Recall that the eigenvectors of the Hamiltonian obey the time-independent Schrödinger equation: denoting the eigenvectors of # (and thus ) as = , the ˆ ˆ | i time-independent Schrödinger equation can be written as

= = = = , (2.2.32) ˆ | i | i 1 1 ~$ # = = ~$ = = , (2.2.33) ˆ + 2 | i + 2 | i which leads to # = = = = . (2.2.34) ˆ | i | i Because the eigenvalues of the operator #ˆ correspond to how many quanta of energy are in our system, we refer to #ˆ as the number operator, and we call the basis formed by the set of eigenvectors = the number basis (or alternatively the | i Fock basis). We call the individual eigenvectors forming the number basis number states. Note that the number basis is the eigenbasis of the quantum hamrmonic oscillator.

If we act the operator # on the states 0 = and 0 = , we can utilize the ˆ ˆ† | i ˆ | i deﬁnition of #ˆ from Equation 2.2.28 and the commutation relations given by Equations 2.2.26, 2.2.29, and 2.2.30 to show the following:

#0† = = = 1 0† = , (2.2.35) ˆ ˆ | i ( + )ˆ | i #0 = = = 1 0 = . (2.2.36) ˆ ˆ | i ( − )ˆ | i

Recall that the operator # tells us that the eigenvalues of the state = are = quanta ˆ | i of energy. We can then determine that the eigenvalues of the state 0 = are = 1 ˆ† | i + quanta of energy, whereas the eigenvalues of the state 0 = are = 1 quanta of ˆ | i − energy. The action of 0 and 0 on = is then to raise and lower the energy of our ˆ† ˆ | i system by a single quanta of energy, respectively. For this reasons, we call the operator 0 the raising operator and we call the operator 0 the lowering operator. ˆ† ˆ Quantum physics 49 |

2.2.3 Scattering theory

A subject that we will need to consider in due course is that of scattering theory. Scattering experiments are one of the best ways to understand a quantum mechanical system, and we will make explicit use of the formalism of scattering theory in Chapter4 . For everything that follows we work in the interaction picture in which both states and operators are time-dependent. Consider a scattering experiment for which we have an incoming quantum system, e.g., some incoming particle or particles, and this incoming quantum system is incident on some target system (possibly quantum, possibly not).

We describe the incoming state of the particle with an in-state, #in , and the | i outgoing state of the particle with an out-state, #out . Assume that the duration | i of interaction16 between the incoming system and the target system is very small compared with both the time of approach (the time between measuring the in-state and the interaction) and the time of flight afterwards (the time between the interaction and measuring the out-state). In reality the interaction may be infinite in range (for example, the electromagnetic interaction) and thus infinite in duration however, when the incoming system and the target system are sufficiently separated, the interaction will be negligibly small. If the incoming quantum system can be described with first quantization (quantum mechanics), and if we insist on modelling the interaction between the target system and the particle in the way that we understand, i.e., with an operator acting on a state, then we can say the following:

#out = ( #in . (2.2.37) | i ˆ | i

That is, the output state is a function of evolving the input state under the action of some operator. What is the operator (ˆ describing a scattering experiment that takes the input state #in and produces the output state #out ? The speciﬁcs | i | i depend on the kind of system that we are considering, however a particular useful case to consider is that of weak scattering, i.e., when the scattering can be treated as a perturbation to the system. In this case of weak scattering, we can write the Hamiltonian that governs the time-evolution of the system as a perturbation series in the coupling parameter , which if we truncate to ﬁst order in the coupling parameter is:

C = 0 int C . (2.2.38) ˆ ( ) ˆ + ˆ ( )

Written this way, the time dependency of the Hamiltonian in the interaction

16Somewhat of an ill-deﬁned notion. 50 Theory |

picture has shunted into the interaction term; when the interaction is suﬃciently weak, the Hamiltonian is time-independent.

Under the action of this Hamiltonian, the solution to the Schrödinger equation

for time evolution of some initial state at time C0 to some other state at time C 5 can be given by the Dyson series:

= 8 = ¹ C = Õ∞ − Ö Ö 8 8 ~ 0C: 0C: # C = © 3C ª 4 ~ ˆ int4− ~ ˆ # C0 , (2.2.39) | ( 5 )i =! :® T{ ˆ } | ( )i ==0 :=1 C0 :=1 « ¬ where = the desired order of perturbation, and the time-ordering operator17. T If one collects the operator parts of Equation 2.2.39 up into an operator (ˆ – the S-matrix – then Equation 2.2.39 can be simpliﬁed to

# C = ( # C0 , (2.2.40) | ( 5 )i ˆ | ( )i which is of the form desired as per Equation 2.2.37. If we express ( as ( = I 8) ˆ ˆ + ˆ then all of the dynamics relevant to the actual scattering process (i.e. those parts of the dynamics that have nothing to do with the initial conditions of the incoming particle) are put into the T-matrix, )ˆ, while the identity accounts for the cases in which the incoming particle fails to scatter from the particle.

Denote our initial state at time C0 to be #in and denote the state that we | i wish to detect after scattering as #out . To characterize scattering, our job is | i then to determine the amplitude associated with the overlap of the actual ﬁnal

state # C with the state that we desire #out , or equivalently, the amplitude | ( 5 )i | i associated with the initial state #in evolving into the desired state #out : | i | i

#out # C = #out ( #in . (2.2.41) h | ( 5 )i h | ˆ| i

We deﬁne a new object denoted to describe the dynamical (rather than ℳ kinematical) aspects of scattering processes: the amplitude for a given state 0 to

transition to another given state 1 deﬁnes the matrix element 0 1. In reality ℳ → ℳ can only really be considered a matrix if the dimensionality of the Hilbert space under consideration is ﬁnite in size, though the nomenclature sticks regardless of the dimensionality of the Hilbert space. With the relevant elements of the

17The time-ordering operator isn’t a quantum mechanical operator: it’s purpose is to tell you to organize your operators so that everything is ordered in time, i.e. earlier time operators apply to a state before later time operators. Field theory 51 | scattering matrix at our disposal we can then write the following:

4 3 #out 8) #in = 8 in out 2 ( ) ?in ?out in out , (2.2.42) h | ˆ | i ℳ → ( ) ì − ì ( − ) where in out accounts for all of the dynamics of our system (i.e., what interaction ℳ → is actually happening) while the delta functions impose kinematic constraints on the system (i.e., restrict us to the relevant area of phase space).

To obtain a particular description of in out one must have a particular ℳ → system in mind. We will forgo discussing scattering theory in any more depth at this point: additional discussions regarding scattering theory will appear in Section4 when we consider scattering within a particular system.

2.3 Field theory

Field theory is of immense importance in physics in general, and especially so in modern physics: quantum field theory and general relativity—our two best theories of nature to date—are both built atop of field theoretic considerations. Here we shall describe some of the important mathematical waypoints in field theory. At a very basic linguistic level, a physical field is represented by a mathematical object that takes as inputs spatial location and time, and gives in return a single output. The type of output depends on the physical system being modelled: in classical non-relativistic physics scalar fields and vector fields are common; in relativistic physics tensor fields of higher rank are common; in quantum theory one even encounters more exotic fields in the form of spinor fields. We can describe a whole host of spatially dependent quantities in terms of fields: familiar examples are temperature, which takes as input spatial location and time and outputs a real number, and the classical gravitational field from Newtonian mechanics, which takes as input spatial location and time and outputs a vector. While outside of field theory our aim is often to determine trajectories—for example, the position as a function of time, G C , of a ball that is thrown through ì( ) the air—in a field theory, our goal is more ambitious: we want to know the state of the entire field ) G,C (often referred to as the field configuration), which (ì ) usually means the value of the field at every location for a given time or times. In classical or semi-classical physics, very often the state of the field is then itself used to compute the trajectory of some non-field object that interacts with the field: for example, the trajectory of a charged particle through a magnetic field is computed using the state of the field, while the trajectory of an asteroid through 52 Theory |

the solar system can be computed from the state of the gravitational ﬁelds of the sun, nearby planets, and other massive bodies.

How do we determine the entire state of a field at some time? In certain cases, the state of the field is easy to compute (or at least easy compared with the general case): if we can compute the state of the field for one of these specific cases then we can determine the state at some future time using the equations of motion of the field. Generally, the equations of motion of fields are some form of wave equation. For example, the three-dimensional linear wave-equation governing some scalar field variable ) is 1 %2) 0 = 2), (2.3.1) −E2 %C2 + ∇ where E is the propagation speed of the waves. There are, of course, more complicated wave equations: this is just the most basic. Plane waves form a basis of solutions to this equation. In general a plane wave is given by

8 $C k x ) x,C = 4− ( − · ) (2.3.2) ( ) where $ is the (angular) frequency of the wave, : is the (angular) wave-vector, and the relation between $ and : is related by the propagation speed E:

E = $: (2.3.3)

That Equation 2.3.2 is a solution to Equation 2.3.1 can be quickly demonstrated:

1 %2) 1 % %) = = :2), (2.3.4) −E2 %C2 −E2 %C %C % %) % %) % %) 2) = = :2 :2 :2 ) = :2), ∇ %G %G + %H %H + %I %I − G + H + I − (2.3.5) 1 %2) 2) = :2) :2) = 0 (2.3.6) −E2 %C2 + ∇ + (− ) As it turns out, the three-dimensional linear wave-equation—Equation 2.3.1— has the same mathematical form as the relativistic wave-equation for a massless particle (where in the relativistic case we have E = 2). Consider, by comparison, the equations of motion for the electric ﬁeld E and the magnetic ﬁeld B:

1 %2E 0 = 2E, (2.3.7) − 22 %C2 + ∇ 1 %2B 0 = 2B, (2.3.8) − 22 %C2 + ∇ Field theory 53 | where 2 is the speed of light18. While the fields E and B are not themselves Lorentz covariant objects (though they do originate from a Lorentz-covariant tensor), the overall structure of the wave equation that describes them is invariant under a Lorentz transformation. Any other field whose equations of motion are also given by the linear wave-equation, Equation 2.3.1, with waves travelling at the speed of light, E = 2, will also obey a Lorentz transformation and thus be a relativistic field.

2.3.1 Quantum ﬁeld theory

A successful path to building quantum models—both first quantized (non- relativistic quantum mechanics) and second19 quantized (quantum field theory)— is by the process of quantization.20 The guide to building a second quantized theory using the process of quantization can be found in many text books, with the process as given here being an adaptation of that from [119]. At the most general level, the path that takes to construct a quantum field theory is the following:

1. Write down a relativistically invariant Lagrangian density for some classical field theory.21 The Lagrangian could describe a single field or several; it could describe a free theory, or a coupled theory. 2. From the Lagrangian density do the following: a) Utilize the Euler–Lagrange equations to find the momentum density of each field in your Lagrangian density; b) Obtain the Hamiltonian density by performing a Legendre transformation on the Lagrangian density. 3. Declare the fields and their associated momentum-densities to be operators that obey commutation/anti-commutation relations for bosons/fermions respectively. 4. Express the field operators in terms of creation and annihilation operators that act on particle number states.

18 The prefactor in front of the time-derivatives is 00—where 0 is the magnetic permeability of free-space, and 0 is the electric permittivity of free-space—when one derives these equations from Maxwell’s equations: this prefactor has units of inverse velocity, and if we identify this as being the square of the speed of light, we obtain the equations in terms of 2. 19The term “second quantization” is an unfortunate vestige that has been carried into the present from the early history of quantum theory. 20The process of first quantization is somewhat mathematically dubious, however. 21Note that the classical field theory that informs your Lagrangian at this step does not need to correspond to some physical system that is actually manifest in the classical world: the only requirement is that you have a Lagrangian that obeys the mathematical structure of a classical field theory. 54 Theory |

To lead by example, let us demonstrate how one can build the most simple kind of quantum field theory: that being the field theory of a free quantum scalar field in Minkowski (flat) spacetime.

1. Write down a classical relativistically-invariant ﬁeld-theoretic Lagrangian density for the system of your choice.

The form of the simplest relativistic Lagrangian density describing a free scalar- ﬁeld in Minkowski spacetime is often given as follows:

2 2 1 1 < 2 2 = %)% ) ) . (2.3.9) ℒ 2 − 2 ~2 Note that, written this way, the ﬁeld ) must have units of square-root-force in order for the Lagrangian density to have the correct units (energy density, or alternatively, pressure). In order to allow ) to have more sane units, we can write the Lagrangian density with a prefactor like:

2 2 ! 1 < 2 2 = %)% ) ) . (2.3.10) ℒ 2 − ~2

If ) has units of distance (for example, if ) describes the displacement of some object away from some equilibrium position) then —as the suggestive notation may imply—has units of pressure. The prefactor of 1 2 become useful for our / next step.

2a. From the Lagrangian density: Utilize the Euler–Lagrange equations to ﬁnd the momentum density of the ﬁeld.

The Euler–Lagrange equation for a relativistic ﬁeld is:

% © % ª ℒ % ℒ ® = 0, (2.3.11) %) − ® % %) « ¬ the momentum density, Π G , of the ﬁeld ) at some location G is the parenthetical ( ) term in the Euler–Lagrange equation:

% Π G B ℒ . (2.3.12) ( ) % %) G ( ) Field theory 55 |

In our case, the momentum density for the Lagrangian density speciﬁed in Equation 2.3.10 is

% 1 % % )% ) Π = ℒ = , (2.3.13) 2 % %) % %)    %% ) %% )  1   =  % ) % ) , (2.3.14) 2  +  % %) % %)    1 h i = % ) % ) , (2.3.15) 2 + = %), (2.3.16) where we have suppressed the dependency on G for notational brevity. Now that we have the momentum density, we can move on to calculate the Hamiltonian density.

2b. From the Lagrangian density: Obtain the Hamiltonian density by performing a Legendre transformation on the Lagrangian density.

The Hamiltonian density, , is given in terms of the momentum density and the ℋ Lagrangian density: 0 = Π %0) . (2.3.17) ℋ − ℒ Which we can rewrite as 00 = Π0%0) , (2.3.18) ℋ − ℒ which allows us to simplify the algebraic manipulations in the next step. With our choice of metric we have that 00 = 1, so the Hamiltonian density for our − speciﬁc Lagrangian can be written as follows:

2 2 ! 1 1 < 2 2 = %0)%0) %)% ) ) , ℋ − − 2 − 2 ~2

2 2 ! 1 1 1 < 2 2 = %0)%0) %0)%0) %8 )%8 ) ) , − − −2 + 2 − 2 ~2 2 2 1 2 1 2 1 < 2 2 = %0) ) ) . (2.3.19) −2 − 2 ∇ + 2 ~2 Before proceeding, note that while our theory is Lorentz covariant by virtue of the Lagrangian’s construction, this step of our quantization procedure has resulted in our mathematical represention of the theory losing its manifestly Lorentz covariant nature. The Hamiltonian density is deﬁned with respect to a 56 Theory |

given time-coordinate and thus it is not a Lorentz-covariant object. As a result, care must be taken in viewing our quantum field theory from different reference frames; to do so, one must return back to this step and proceed onward with the quantization procedure again. There are however methods by which one can construct quantum field theories in such a way that the mathematical formalism remains explicitly Lorentz covariant: for example, the path integral formulation of quantum field theory achieves this.

3. Declare the ﬁelds and their associated momentum-densities to be operators that obey commutation relations.

Notationally we append hats (circumﬂexes) to quantities to denote that they are operators. In our particular case, that means we perform the following change of notation:

) ), Π0 Π0 (2.3.20) → ˆ → ˆ

We then insist that the following equal-time commutation relations are obeyed:

h 0 i 3 ) C, x , Π C, y = 8( ) x y , (2.3.21) ˆ ( ) ˆ − h i ) C,G , ) C,H = 0, (2.3.22) ˆ ( ) ˆ h i Π0 C,G , Π0 C,H = 0. (2.3.23) ˆ ( ) ˆ

4. Express the operators in terms of creation and annhilation operators that act on number states.

Finally, we express our now operator-valued fields in terms of creation and annihilation operators acting on the vacuum-state of the number basis. Our 0 operator )ˆ—from which Πˆ can be defined—at a specific point in space G can be given in terms of its mode expansion:

¹ 3 3 ? 1 8 p x 8 p x ) G = 0p4− ~ · 0† 4 ~ · , (2.3.24) ˆ 3 2 1 2 p ( ) 2 / / ˆ + ˆ ( ) 2p where is some constant that ensures that the dimensions of the right-hand-side are equal to the dimensions that we have picked for ). Analogue gravity 57 |

2.4 Analogue gravity

In this section, derivations laid out in the review article by Barceló, Liberati, and Visser [93] will be followed very closely as a means to demonstrate explicitly how metric equation descriptions can arise in the description of non-gravitational physical media. Eﬀorts will be made to elaborate on particular points of the derivation that are skipped in [93], and to elucidate the connection between physical ideas and mathematical ideas (for example, an endeavour will be made to explain the connection between physical restrictions and the mathematical approximations that they lead to, and vice-versa).

2.4.1 Eﬀective spacetime metrics

For an analogue gravity model to be an appropriate mathematical analogue for (or of) some subset of features from general relativity, we need to be able to associate some analogue of the spacetime metric to the system. These metrics are not real spacetime metrics in the sense that they don’t characterise the geometric and causal structure of spacetime, rather they are eﬀective metrics describing something to the eﬀect of the apparent geometry experienced by undulations to the underlying medium itself (for example, the apparent geometry experienced by sound in a hydrodynamical system).

2.4.1.1 Geometrical acoustic metrics

Geometrical acoustic metrics are obtained in a very mathematical fashion, without worrying too much about the complicating details that inevitably crop up when one wants to investigate a real-life physical system. We assume, ﬁrst and foremost, that we have some kind of sound-carrying medium. Second, we assume, in analogy with geometric optics, that we can eﬀectively model sound as a set of geometric rays instead of worrying too much about the details of the undulation of some underlying medium. With this kind of picture in mind, we need only take two things for granted in order to obtain a geometrical acoustic metric22; they are:

1. The speed of sound, 2s, relative to to the medium itself is well deﬁned. 2. The velocity of the medium, E, relative to the laboratory is well deﬁned. With these assumptions we can write, in terms of laboratory coordinates, the velocity of rays of sound propagating through the medium as

3x = 2sn v. (2.4.1) 3C + 22More speciﬁcally, an equivalence class of metrics up to some conformal factor. 58 Theory | where n is a unit vector in the direction of the ray being considered. From this, and recalling that the inner product of a unit vector is equal to one, one can obtain an acoustic analogue to the spacetime interval from general relativity,

2 2 2 0 = 2s 3C 3x v3C , (2.4.2) − + ( − ) = 22 E2 3C2 2v 3x3C 3x 3x. (2.4.3) − s − − · + ·

Deﬁne the following (note the uppercase - on the left-hand-side): ! 3C 3X B , (2.4.4) 3x 22 E2 v| 2 © ª , B Ω − − − ® . (2.4.5) v I « − ¬ Where I is the 3 3 identity matrix. With these deﬁnitions, one can write: × ! 22 E2 v| 3C | | 2 © ª 3X ,3X = 3C 3x Ω − − − ® , (2.4.6) v I 3x « − ¬ = Ω2 22 E2 3C2 2v 3x3C 3x 3x , (2.4.7) − − − · + · which is, modulo a conformal factor (Ω2), the same expression as the left-hand-side of Equation 2.4.3:

1 3X|,3X = 22 E2 3C2 2v 3x3C 3x 3x, (2.4.8) Ω2 − − − · + · = 0. (2.4.9)

In actual general relativity we would make the identiﬁcation:

3X|,3X , 3G3G. (2.4.10) →

And from this we can see that the object , (Equation 2.4.5) acts to rays of sound as a conformal class23 of Lorentzian metrics.

23A conformal class of metrics is the set of metrics that diﬀer only by a constant multiplicative function. The reason for the non-uniqueness of the metric in our particular case results from the fact that the metric is constructed to describe the trajectories of rays of sound, which in our sonic analogue to spacetime are null trajectories. As a result, the line element corresponding to these trajectories is zero and so we can freely multiply both sides by an arbitrary function (in this case, Ω2) that can then be absorbed into the metric. Analogue gravity 59 |

2.4.1.2 Physical acoustic metrics

Geometrical acoustics is highly idealized and only allows us to phenomenologi- cally model the trajectory of sound rays in a medium, rather than modelling the entirety of some acoustic field. Physical acoustics is, ideally, where the practicing analogue gravitationalist wants to focus their attention, should their desire be to utilize non-gravitational physical systems as a tool with which to attempt to study gravitational phenomena. Commonly, hydrodynamical systems are the medium of choice as sound in such a system can accurately be modelled as a scalar field, which greatly simplifies calculations. In the general case, a scalar field is not sufficient to model sound in solid systems: only in the case of a 1-dimensional lattice is it appropriate to make this simplification—in general, sound in solid systems must be described by a vector field [120]. Arriving at a useful mathematical model for an analogue gravity system— that is, one that can actually be used to make computations so that theory and experiment can be compared—is a path that is filled with many physical restrictions, mathematical approximations, and technicalities. We shall give a rough sketch of the process that can be used to demonstrate that, in certain limiting cases, sound propagation in a hydrodynamical system can be modelled as a scalar field coupled to some effective metric: as mentioned before, this material follows very closely a derivation in [93]. Before we get stuck into the details of the derivation, recall that the dynamical variables of interest in fluid dynamics are the pressure of the fluid, ?, the density of the fluid, , and the velocity of the fluid flow, v. The dynamical equations of motion within classical hydrodynamics are the equation of continuity (denoted “Continuity Eq.” in margins below) and Euler’s equation (denoted “Euler’s Eq.” in margins below), and these dictate the behaviour of ?, , and v as follows:

%C v = 0, (2.4.11) Continuity Eq. + ∇ · 3v = %Cv v v . (2.4.12) Euler’s Eq. 3C + ( · ∇) To begin down our path to a model of analogue gravity, we begin with a physical restriction: let us restrict our considerations to a very specific type of hydrodynamical system. We will consider a fluid that is barotropic (the density of the fluid is a function of pressure only)24, inviscid (the viscosity of the fluid is equal to zero), and irrotational (there are no vortices in the fluid flow: mathematically, the

24What is meant by this is that the density has explicit functional dependency on the pressure alone. The density of a fluid can of course differ at different locations in space, and the density can of course change with time, but any such spatial or temporal variations in the density are merely inherited implicitly through the density’s functional dependency on the pressure of the fluid. 60 Theory |

curl of the fluid flow velocity is zero everywhere).25 Next, we make another physical restriction that allows us to appropriately use some convenient mathematical approximations: this physical restriction is to assume that the fluid has some background state about which we will consider small perturbations in the perturbative parameter . It will also prove useful moving forwards to define the velocity potential, which can be represented by a scalar field ) such that v = ).26 With this definition, we can—under our physical restrictions—express −∇ the variables ?, , and ) as perturbative series in (where < 1) like:

2 ? = ?0 ?1 ?2 ... ; (2.4.13) + + + 2 = 0 1 2 ... ; (2.4.14) + + + 2 ) = )0 )1 )2 ... ; (2.4.15) + + + where the triplet ?0, 0, )0 deﬁnes the background solution of our system about ( ) which we consider small perturbations. For suﬃciently small perturbations away from the background solution (i.e., when 1) we can ignore all terms that are second-order or higher in , which is to say we can linearize the variables

?, , ) around ?0, 0, )0 . Doing so, we can write the following: ( ) ( ) ? ?0 − ; (2.4.16) ' ?1 0 − (2.4.17) ' 1

The assumption that our fluid is barotropic means that the density is a function of pressure, which is to say changes in pressure result in changes in density, and to first order in those changes are constrained to satisfy Equations 2.4.16 and 2.4.17. Therefore, we can equate Equations 2.4.16 and 2.4.17 to obtain a relation for in terms of ? that is valid to first order in . Doing so and performing some simple algebraic manipulations yields the following linear relation:

1 ? = ? ?0 0, (2.4.18) ( ) ?1 − +

from which we can obtain the following:

1 % %? − 1 = = . (2.4.19) %? % ?1

25One may recall from Section 1.3.1.2 that these are the same conditions that Unruh made use of to derive his model of an acoustic black-hole. 26For an irrotational fluid the definition of the velocity potential is in analogy to the classical physics definitions of the gravitational potential and the electrostatic potential, the respective gradients of which give the gravitational acceleration and the electric field. Analogue gravity 61 |

The aforementioned physical restrictions and mathematical approximations form the bulk of an argument that, through some intermediary steps (such as introducing the speciﬁc enthalpy, ℎ ? ), allows us to recast the equation of ( ) continuity and Euler’s equation into a more useable form:

( %C 0 0v0 = 0 (2.4.20) + ∇ · %C v = 0 Continuity Eq. + ∇ · → %C 1 1v0 0 )1 = 0 (2.4.21) + ∇ · − ∇

 1 2  %C )0 ℎ0 )0 = 0 (2.4.22) 3v  − + + 2 ∇ = %Cv v v Euler’s Eq. 3C + ( · ∇) → ?1  %C )1 v0 )1 = 0 (2.4.23)  − + 0 − · ∇  Equations 2.4.20 and 2.4.22 govern the behaviour of the background (given here by the triplet ?0, 0, v0 ), whereas Equations 2.4.21 and 2.4.23 describe ( ) the behaviour of the ﬂuctuations around this background (given by the triplet

?1, 1, )1 ). The small ﬂuctuations about the background described by the triplet ( ) ?1, 1, )1 is precisely what constitutes sound within the ﬂuid. ( )

Equation 2.4.23 can be inverted to ﬁnd an expression for ?1, which in conjunction with Equation 2.4.19 can be used to write an expression for 1. This expression for 1 can then be substituted into Equation 2.4.21 to obtain a wave equation for )1: % % %C 0 %C )1 v0 )1 0 )1 0v0 %C )1 v0 )1 = 0. − %? + · ∇ + ∇ · ∇ − %? + · ∇ (2.4.24) We can rewrite our equations in a more suggestive form by identifying the speed of sound as follows: s %? 2s = , (2.4.25) % B where the derivative is the partial derivative of pressure with respect to density at constant entropy, B. From Equation 2.4.19 we note that we can replace the partial derivative in Equation 2.4.25 with its inverse, which allows us to obtain the following: % 1 = . (2.4.26) 2 %? 2s Utilising this identity, and recognising that Equation 2.4.24 can be written as a 62 Theory |

matrix equation, we can rewrite Equation 2.4.24 in the following form: " # " # 0 0 0 = % % ) v ) ) v % ) v ) , C 2 C 1 0 1 0 1 2 0 C 1 0 1 − 2s + · ∇ + ∇ · ∇ − 2s + · ∇ (2.4.27) ! 0 %C v0 = | − − · ∇ ) (2.4.28) %C 2 2 1 ∇ 2 2 v0 %C v0 s s ∇ − + · ∇

Note that if we wanted to bring the coeﬃcient 0 out to the left-hand-side as a prefactor for the entire expression then we would need to place further physical restrictions on the ﬂuid: in particular, we would need to assume that the gradient

of 0 is zero (because the gradient operator to the left of 0 acts on both it and the

matrix to the right), i.e. 0 would have to take a constant value across the entire

ﬂuid. We will therefore leave 0 sandwiched in between the two matrices that currently ﬂank it on the grounds that we wish to allow the background solution

?0, 0, v0 to be completely general (or at least completely general within the ( ) scope of generality that is permitted by the physical restrictions that allowed us to reach Equation 2.4.28).

The right-most matrix in Equation 2.4.28 can itself be represented by another matrix equation:

! | ! ! %C v0 1 v0 %C 2 − − · ∇ = − 2 − | )1. (2.4.29) 2 v0 %C v0 v0 2 I v0v s ∇ − + · ∇ − s − 0 ∇ With this, the continuity equation governing sound waves in our ﬂuid can be expressed in the following suggestive form:

| ! ! 0 1 v %C 0 = | − − 0 ) . (2.4.30) %C 2 2 | 1 ∇ 2 v0 2 I v0v s − s − 0 ∇ This is reminiscent of the inner-product of the four-derivative in general relativity, where the matrix in the middle acts as a metric tensor. However, the units of the vectors and the matrix in this expression are slightly strange: they do not have consistent units across all of their components. We can however rewrite the vectors and the matrix from Equation 2.4.30 in such a way that each object has Analogue gravity 63 | consistent units across all of its components by deﬁning the following:

v0 0 B , (2.4.31) 2s 1 %0 B %C , (2.4.32) 2s we can then rewrite Equation 2.4.30 like:

| ! ! 1 %0 | 0 0 = %0 0 − − | )1. (2.4.33) ∇ 0 I 0 − − 0 ∇ Written this way, all components of our vectors have consistent units (inverse distance), and all components of our matrix have consistent units (dimensionless). The vectors have the same structure as the four-derivative from relativity, with the exception that the speed-of-sound replaces the speed-of-light, prompting the following deﬁnitions: ! %0 % B (2.4.34) ∇ | ! 1 0 5 B 0 − − | , (2.4.35) 0 I 0 − − 0 we can utilize the Einstein summation convention to write the continuity equation for sound waves in our ﬂuid in the following way:

0 = % 5 % )1 . (2.4.36)

In general relativity, the action of the d’Alembert operator27 (which we shall denote by the empty-box symbol ) on a scalar ﬁeld is given by:

1 ) B % √ ,, % ) , (2.4.37) √ , − − 28 where , is the determinant of the metric tensor , :

, = , . (2.4.38) | |

27Also referred to as the “the d’Almebertian”, or the “wave operator”. 28 In ﬂat spacetime, i.e. when the metric tensor is just the usual Minkowski metric, , = , with the spatial part described by Cartesian coordinates, we have , = 1, and from this we obtain − the familiar expression ) = %% ). 64 Theory |

By inspection of Equation 2.4.36 we might propose the deﬁnition:

5 B √ ,, , (2.4.39) − with which we can rewrite Equation 2.4.36 in the following way:

0 = % √ ,, % )1 . (2.4.40) −

As long as the determinant of the metric tensor is non-zero we can freely divide both sides of Equation 2.4.40 by √ , to obtain: −

1 0 = % √ ,, % )1 . (2.4.41) √ , − − Recalling from actual relativistic physics that massless scalar fields obey the wave equation ) = 0, we see that we have arrived at an equation of motion for sound waves in a fluid, Equation 2.4.41, that has precisely the same mathematical form as the wave equation for a massless, relativistic scalar-field!

We are now at the point at which we are able to extract an eﬀective metric describing the behaviour of sound waves within the ﬂuid! Recall the following rules for the determinants of square matrices of dimension = =, where uppercase × letters represent matrices, and lowercase letters represent constants:

*+ = * + , (2.4.42) | | | || | 2+ = 2= + . (2.4.43) | | | |

We can use Equation 2.4.42 with the deﬁnition of , (Equation 2.4.38) to obtain the following equalities:

, , = , , = , , = = 1. (2.4.44) | | | || | | | | |

We can use these relations along with Equation 2.4.43 and the deﬁnition of 5 (Equation 2.4.35) to obtain the following (recalling that, in full, our matrices have dimension 4 4): ×

4 4 1 5 = √ ,, = √ , , = √ , = ,. (2.4.45) | | | − | − | | − ,

From the deﬁnition of 5 (Equation 2.4.39) we can explicitly compute 5 . | | Making note of the fact that we must expand in minors, that our matrix has size An interesting toy model 65 |

29 4 4 in full, and that 0 acts as a constant , we obtain the following: ×

5 = 4. (2.4.46) | | − 0

With Equation 2.4.45 and Equation 2.4.46 at hand, , can be obtained by inverting Equation 2.4.39. Once we have , , obtaining , is straightforward. After all is said and done, we arrive at the following:

| ! 1 1 0 , = − − | , (2.4.47) 0 0 I 0 − − 0 1 | | © 0 0 0ª , = 0 − − − ® . (2.4.48) 0 I « − ¬

2.5 An interesting toy model

Here we are going to consider a simple toy model of a one-dimensional lattice of coupled pendula. This model possesses features from all of the previous sections of this chapter, with some of these features being built-in (i.e., exist by construction) and others being emergent properties of the system. Our toy model is that of a one-dimensional lattice (chain) of identical pendula that are subject to gravity and coupled to their nearest neighbours by identical springs. A schematic of this can be seen in Figure 2.2. We imagine the pendula arms to be completely rigid and to only allow motion in the plane of Figure 2.2: this constrains each individual pendulum bob to live on the boundary of a circle. We denote the mass of each pendula and their arm lengths A. The equilibrium positions of adjacent pendula are separated by a distance ΔG, and the springs connecting adjacent pendula have spring constant :. While the microscopic model that we consider is a rather simple one that is encountered in many undergraduate physics courses, we choose to view the continuum description in a diﬀerent manner than is standard. The contents of this section are far from an explicit derivation, though we will note some of the important restrictions/approximations that are used along the way. Denote the displacement away from equilibrium for the 8th pendulum as )ì8. Recalling that all motion is constrained to the plane of Figure 2.2, we explicitly write the displacement of the 8th pendulum in Cartesian coordinates like G H )8 = ) G ) H, (2.5.1) ì 8 ˆ + 8 ˆ 29 As a physical value, the value of 0 is not constant. However it is everywhere described by a single real number, and so for the purposes of the rule given in Equation 2.4.43 0 acts per the constant 2. 66 Theory |

: ΔG

Figure 2.2: A one-dimensional lattice of coupled pendula. The image is for descrip- tive purposes only: ideally the pendulum bobs would be point masses, rather than extended objects. Each pendulum has a mass , and neighbouring pendula are connected via springs of spring constant :. At equilibrium, the separation between adjacent pendula is ΔG.

where symbols with a hat (circumﬂex) represent unit vectors, and we place the component label as a superscript next to ) as so not to clutter the subscript area. Deﬁne the following notational shorthand for brevity:

: : : Δ)8 B )8 1 )8 , (2.5.2) + − where : G,H . ∈ { }

In what follows, it is assumed that we are restricting our considerations to scenarios in which small angle approximation holds. Mathematically, this corresponds to the following:

): 8 1, 8 N, : G,H ; (2.5.3) A ∀ ∈ ∀ ∈ { } ): 8 1, 8 N, : G,H . (2.5.4) ΔG ∀ ∈ ∀ ∈ { }

We shall drop the 8 subscript when referring to these inequalities due to the fact that they are true for all 8, though we shall retain the : superscript to remind ourselves that there are two components to )ì8. With all of this in mind, we now proceed to write down the Lagrangian describing such a system. An interesting toy model 67 |

H A ) − A A H )

Figure 2.3: On the left is an example of the path that may be traced out by the swinging of a pendulum with a ﬁxed arm length. The pendulum’s path is associated to a triangle (right) from which the lengths A, )G and )H can be related by the Pythagorean theorem. In the limit of small angular displacement (not shown here) )H can be parameterized in terms of )G in an analytically useful way.

2.5.1 The Lagrangian

The Lagrangian is given as follows:

Õ ! = )8 + 8 +B 8 , (2.5.5) − ( ,) + ( ) 8

th where )8 is the kinetic energy of the 8 pendulum, +, 8 is the gravitational th ( ) potential experienced by the 8 pendulum, and +B 8 is the spring potential ( ) experienced by the 8th pendulum due to a change in length of the spring by an amount Δ-8. Using the standard deﬁnitions of these energies from classical physics, we ﬁrst express our energy terms in all generality as follow:

H + 8 = ,) , (2.5.6) ( ,) 8 1 2 +B 8 = :Δ-8 , (2.5.7) ( ) 2 !2 1 3)8 ) = ì . (2.5.8) 8 2 3C

Note that we have chosen to denote the 8th pendulum’s spring potential to only result from its coupling to the 8 1 th pendulum: we have ignored the 8 1 th ( + ) ( − ) pendulum as to not double count the spring potentials when summing over all pendula to construct the Lagrangian. In the limit of small angular displacement (): A 1 and ): ΔG 1) the / / H-components of displacement can be parametrized entirely in terms of their 68 Theory |

G-components. Collectively these parametrizations are:

H !2 ) 1 )G 8 8 , (2.5.9) A ≈ 2 A Δ- Δ)G 8 8 , (2.5.10) ΔG ≈ ΔG G 3)8 3) ì 8 . (2.5.11) 3C ≈ 3C

These parametrizations are valid to leading order in ): A and ): ΔG. For ex- / / ample, from Figure 2.3 one can use the binomial/small-angle approximation to H G parametrize )8 entirely in terms of )8 by considering only the leading order terms in the ratios ): A. To obtain Equation 2.5.11, one can utilize Equation 2.5.9 to H / express ) in terms of )G, and then truncate to ﬁrst order in ): A after performing 8 8 / the derivatives. To leading order in the ratios ): A and ): ΔG, the Lagrangian takes the / / following familiar form:

!2 Õ 1 %)G 1 2 1 2 ! = 8 $2 )G : Δ)G ; (2.5.12) 2 %C − 2 0 8 + 2 8 8 where r, $ B . (2.5.13) 0 A

The quantity $0 is the natural (angular) frequency of our pendula. This expression for the angular frequency of a pendulum applies to any pendulum that is set in motion through displacement by a small angle, which is exactly the scenario that we are considering.

2.5.1.1 The continuum limit

The Lagrangian in the continuum limit can be obtained from the deﬁnition of the Riemann integral: ¹ Õ 5 G 3G B lim 5 G8 ΔG8 . (2.5.14) ( ) ΔG8 0 ( ) → 8 Using the fact that ΔG ΔG = 1, Equation 2.5.12 can be written in the following / way:

!2 !2  G G 2 Õ 1 3)8 1 Δ)8 1  ! =  :ΔG $2 )G  ΔG. (2.5.15) 2 ΔG 3C − 2 ΔG − 2 ΔG 0 8  8     An interesting toy model 69 |

: : : Recall that we deﬁned Δ)8 to be shorthand for )8 1 )8 , and further recall that + − the equilibrium separation of adjacent pendula is ΔG. Denoting the equilibrium position of the 8th pendulum G we can write:

: : : Δ)8 )8 1 )8 lim = lim + − , (2.5.16) ΔG 0 ΔG ΔG 0 ΔG → → ): G ΔG ): G = lim ( + ) − ( ) , (2.5.17) ΔG 0 ΔG → %): C (2.5.18) %G

What was merely a discrete set of pendula in our microscopic model has become, in the continuum limit, a field. At any given time C, pick any point G R and you ∈ shall receive the outputs ): G,C for : G,H (though our parametrization of ( ) ∈ { } )H in terms of )G means that only the G-component of the field appears in the dynamical description). What physical object does the field )ì correspond to in the continuum limit? Compression waves in a taught string or a stiff rod would seem a reasonable guess. Indeed, it is in simplified descriptions of such systems in which one is likely to have come across this Lagrangian in the past.

We can also identify the ΔG dependent coeﬃcients of each term in the Lagrangian with some useful physical quantities in the continuum limit. In going from the discrete description of our system to the continuous description, we make the following identiﬁcations: , (2.5.19) ΔG → :ΔG . (2.5.20) → T

These quantities are, respectively, a linear mass-density and a tension. In the microscopic model, the quantity :ΔG is the force that would be required—by Hooke’s force law—to compress the spring between two adjacent pendula by a distance ΔG: the equilibrium separation of adjacent pendula in our model is ΔG and so this is precisely the force required to compress a spring down to zero length. We also have the quantity ΔG in the microscopic description: this is the / cumulative mass of the unit interval of our lattice, which we identify with the linear mass-density in the continuum limit.

With all of these identiﬁcations, our Lagrangian in the continuum limit is:

2 2 ! ¹ 1 %) %) ! = $2)2 3G. (2.5.21) 2T %C − %G − 0 T T 70 Theory |

2.5.2 Equations of motion

The equations of motion of this Lagrangian are well known: via the Euler– Lagrange equation, the Lagrangian leads to the Klein–Gordon equation describing the scalar ﬁeld ) G,C . For the particular Lagrangian that is being considered, the ( ) relevant Euler–Lagrange equation is ! %! 3 %! = 0, (2.5.22) %) G,C − 3C %) G,C ( ) ¤( ) where ) G,C (note the dot above )) is the partial time-derivative of ) G,C . As ¤( ) ( ) was mentioned just above, the solution to the Euler–Lagrange equation is the Klein–Gordon equation: ! %2 %2 T $2 ) G,C = 0. (2.5.23) %C2 − %G2 + 0 ( )

The Klein–Gordon equation is obeyed by scalar ﬁelds ) G,C whose bases of ( ) solutions are plane waves. For a plane wave of angular frequency $ and angular wavenumber :, we can write:

8$C :G 5 G,C = 4 − . (2.5.24) ( )

Take ) G,C = 5 G,C as an ansatz to the Klein–Gordon equation. Then by substi- ( ) ( ) tution of ) G,C = 5 G,C into Equation 2.5.23 we obtain the dispersion relation ( ) ( ) for plane waves in our system:

$2 = T :2 :2 ; (2.5.25) + 0 where we have deﬁned the following:

2 2 :0 B $0. (2.5.26) T Dimensional analysis dictates that the prefactor of :2 must have units of velocity- squared. Note that there are two distinct types of velocity which can be attributed

to a wave: the group velocity (E,) and the phase velocity (E?). These velocities are An interesting toy model 71 | given, respectively, by the following: v tu 2 ! $ :0 E? B = T 1 , (2.5.27) : + :2 %$ 1 E, B = T (2.5.28) %: E?

Where the right-most equalities come from evaluating the deﬁnitions of the phase and group velocities using the dispersion relation given in Equation 2.5.25. Note that from (Equation 2.5.28) we have the following relation:

E E = T . (2.5.29) , ?

In the particular case that :0 : we have, to ﬁrst order in the ratio :0 :: / s E = E = T . (2.5.30) , ?

That is to say, if we restrict ourselves to only consider waves characterized by

:0 :, we have that there is a single characteristic speed governing waves within our system. In this particular case, the speed of the waves can quite unambiguously be called the speed of sound. We therefore deﬁne the speed of sound 2s to be the following: s 2 B T . (2.5.31) s

What is the physical interpretation of the condition :0 :? From our deﬁnition of :0 (Equation 2.5.26) the dispersion relation Equation 2.5.25 can be rewritten as follows: " 2# :0 $2 = T :2 1 (2.5.32) + :

Truncating to ﬁrst order in the ratio :0 : (and noting that, in this limit, we can / replace the square-root term with 2s) we ﬁnd:

$ = 2s :. (2.5.33)

In the limit that :0 : 1 our sound waves obey a linear dispersion relation, which / is precisely the dispersion relation for light in vacuum.

It is important to note here that while the speed of sound 2s is the single characteristic speed of waves within our system for which the limit :0 : 1 is / 72 Theory |

true, even waves that do not obey this limit still, in some sense, know about the

value 2s (through its presence in the Lagrangian).

2.5.3 Bringing it all together

So far, everything within this section has followed a very conventional path. We started with a simple toy model that will be familiar to most people as a simplified microscopic model of something like a taught string. We restricted our considerations in particular ways such that the anticipated dynamical behaviour of our system in the continuum limit was that of longitudinal waves propagating through this string. We then obtained the Klein–Gordon equation as our equation of motion in the continuum limit, and from this we were able to identify the restrictions required for sound to behave in our system like light does in vacuum. Much of the preceding may be familiar to the reader from classical mechanics or solid state physics. Of course, in classical mechanics and solid state physics, one rarely gives much consideration to the theory of relativity. We now take the slightly less conventional route of taking this well known model and framing it in a relativistic way, highlighting its conceptual use as a simplified analogue gravity model. In field theory, the natural quantity to consider is not the Lagrangian itself, but the Lagrangian density. In -dimensional space (not spacetime, just space), the Lagragian density can be defined implicitly through the following equation: ℒ ¹ ! = 3 G. (2.5.34) ℒ

While our toy model explicitly considers two spatial dimensions, our restrictions have removed all reference to the spatial component H in the dynamical description of the system: at the level of dynamics, our system is parametrized in terms of a single spatial component and so we have = 1. The Lagrangian density for our toy model can be obtained from Equation 2.5.21 and is given by:

2 2 ! 1 1 %) %) $2 = 0 )2 . (2.5.35) 2 2 ℒ 2T 2s %C − %G − 2s

Note that we have made use of deﬁnition of 2s here. Make the following deﬁnitions:

1 % % % B %0, %G , %0 B , %G B , (2.5.36) ( ) 2s %C %G ! 1 0, B (2.5.37) 0 1 − An interesting toy model 73 |

With these, we can express the Lagrangian density in a way that appears to be

manifestly Lorentz covariant (except in our case we have the speed of sound 2s as our characteristic speed, rather than the speed of light 2): ! 1 $2 = % )% ) 0 )2 . (2.5.38) 2 ℒ 2T − 2s

In our relativistic notation, the Klein–Gordon equation is then given by: ! $2 % % 0 ) = 0. (2.5.39) 2 − 2s

What initially started as simple toy model of coupled pendula can be viewed—in the appropriate limits—as an analogue gravity model describing the propagation of scalar fields in a 1 1 Minkowski spacetime. In fairness, using the term ( + ) − “gravity” at this point may be slightly misleading: right now the effective metric that we defined has no curvature and so the effective spacetime has no real analogy to any gravitational phenomena. Such effects can be introduced, however: we will discuss how this can be done at the end of this section.

Our system admits (at least mathematically) a relativistic description. We should therefore be able to utilize the guide for building a quantum field theory from a classical field theory that was laid out in Subsection 2.3.1 to obtain a quantum model from our classical one. In fact, if we make the definition

<222 $2 s := 0 (2.5.40) 2 2 ~ 2s

then we can rewrite the Lagrangian density from Equation 2.5.38 as follows:

2 2 ! 1 < 2s 2 = %)% ) ) . (2.5.41) ℒ 2T − ~2

This form of the Lagrangian density is identical in form to Equation 2.3.10, the Lagrangian density that we used to demonstrate the quantization procedure in Subsection 2.3.1. Therefore, the quantization procedure for this Lagrangian density is identical to that as laid out in Subsection 2.3.1, the only diﬀerences

being cosmetic in nature ( and 2 2s), and the fact that we only have ↔ T ↔ one spatial component rather than three. After quantization, our Klein–Gordon 74 Theory |

equation acts on an operator-valued ﬁeld rather than a classical ﬁelds: ! $2 % % 0 ) = 0. (2.5.42) 2 ˆ − 2s

Initially our classical ﬁeld ) described sound waves within our system, and so the operator valued ﬁeld )ˆ describes the quanta of sound: phonons.

2.5.4 Including curvature (and thus some analogy to gravity)

Consider the case in which our system is described by a position-dependent linear mass-density, tension, and natural frequency by making the following changes to the continuum limit Lagrangian Equation 2.5.21: G G 1 ( ) , G, ( ) 1 (2.5.43) → + ∀ G G 1 T( ) , G, T( ) 1 (2.5.44) T → T + ∀ T T A G A G $2 $2 1 ( ) , G, ( ) 1. (2.5.45) 0 → 0 − A ∀ A where the change to the square of the angular frequency can be thought of to arise as a result of letting the arm lengths of pendula vary as a function of position in the following way: A G A A 1 ( ) . (2.5.46) → + A In other words, the parameters describing our physical system are everywhere a small perturbation away from their original values. Making these changes, performing some algebraic manipulations (which includes making use of the binomial approximation), and discarding any terms that are second order or higher in our small ratios, we can write the Lagrangian density corresponding to our modiﬁed form of Equation 2.5.21 in the following way:

2 2 ! G %) G %) G A G = T 1 ( ) 1 T( ) 1 T( ) ( ) $2)2 . ℒ 2 + %C − + %G − + − A 0 T T T (2.5.47) Provided that we take G A G T( ) = ( ) , (2.5.48) A T An interesting toy model 75 | this further simpliﬁes to:

2 2 ! 1 G %) G %) = 1 ( ) 1 T( ) $2)2 . (2.5.49) ℒ 2T + %C − + %G − 0 T T Perturbing our system in the way that we have corresponds to linearizing around some background solution analogous to the discussion in Subsection 2.4.1.2, except our choices here turn out to correspond to linearizing our metric tensor about the 1 1 -D Minkowski metric, whereas in Subsection 2.4.1.2 the background ( + ) metric was in principle allowed to be arbitrary with only the acoustic ﬂuctuations in the medium being constrained to be linearized about some background.30

Using the deﬁnition of the speed of sound 2s from our unperturbed system (Equation 2.5.31, repeated here for convenience), s 2 B T , (2.5.50) s we can deﬁne the following:

1 % % % B %0, %G , %0 B , %G B , (2.5.51) ( ) 2s %C %G G ©1 ( ) 0 ª + ® , B ®. (2.5.52) G ® 0 1 T( ) ® − + « T ¬ Our Lagrangian density Equation 2.5.49 can then be expressed as follows: ! 1 $2 = ,% )% ) 0 )2 . (2.5.53) 2 ℒ 2T − 2s

In our particular case, the metric tensor g can be decomposed into two parts in a way that is consistent with the approach taken in studying linearized gravity:

, = ℎ. (2.5.54) − taking the metric signature of the Minkowski metric to be , , the tensor h can (+ −)

30We have already restricted our acoustic ﬂuctuations to be small by requiring the angular displacement of our pendula in our microscopic model to be very small. 76 Theory |

be given as follows:

G © ( ) 0 ª ℎ = − ® . (2.5.55) G ® 0 T( )® « T ¬ Within the regime of linearized gravity the Ricci scalar (a measure of the scalar curvature of a manifold) takes on a simple expression that we can readily evaluate. With the empty box symbol denoting the d’Alembert operator ( = % %) we have: ' = %% ℎ ℎ , (2.5.56) − In our particular case, this leads to the following:

1 %2 G ' = ( ). (2.5.57) − %G2

The presence of a non-zero Ricci scalar is associated with spacetime curvature, and so allowing the linear mass-density to be a position dependent quantity takes our system from being merely an analogue to special relativity to being an actual analogue gravity model (albeit, a highly restricted analogue-gravity model that can only describe an analogue to weak gravitational eﬀects in 1 1 -dimensions). ( + ) In principle, we could also allow the perturbations to our system to vary in time too, and this would also contribute to the Ricci scalar (and thus to some eﬀective notion of curvature). Part II

Analogue-gravity universes and in-universe observers: Published/Submitted for publication

SOUNDCLOCKSANDSONICRELATIVITY

• This chapter is a minorly modiﬁed version of a paper that has been published in the journal Foundations of Physics [2]. This paper was co-authored with Nicolas C. Menicucci.

Sound propagation within certain non-relativistic condensed matter models obeys a relativistic wave equation despite such systems admitting entirely non- 3 relativistic descriptions. A natural question that arises upon consideration of this is, “do devices exist that will experience the relativity in these systems?” We describe a thought experiment in which ‘acoustic observers’ possess devices called sound clocks that can be connected to form chains. Careful investigation shows that appropriately constructed chains of stationary and moving sound clocks are perceived by observers on the other chain as undergoing the relativistic phenomena of length contraction and time dilation by the Lorentz factor, , with 2 the speed of sound. Sound clocks within moving chains actually tick less frequently than stationary ones and must be separated by a shorter distance than when stationary to satisfy simultaneity conditions. Stationary sound clocks appear to be length contracted and time dilated to moving observers due to their misunderstanding of their own state of motion with respect to the laboratory. Observers restricted to using sound clocks describe a universe kinematically consistent with the theory of special relativity, despite the preferred frame of their universe in the laboratory. Such devices show promise in further probing analogue relativity models, for example in investigating phenomena that require careful consideration of the proper time elapsed for observers.

3.1 Introduction

The repeated null results from experiments to detect the luminiferous aether towards the end of the 19th century—most notably the null result of the Michelson– Morley experiment [112]—culminated with many physicists, most notably George FitzGerald, Hendrik Lorentz, and Henri Poincaré, proposing mechanisms by which the aether was undetectable. FitzGerald [121] and Lorentz [122, 123] independently (1889 and 1892 respectively) suggested that objects contract parallel to their direction of motion. Woldemar Voigt [123, 124] suggested modiﬁcation of the time coordinate to ensure that the wave equation for light worked in all reference frames, and Lorentz [123, 125] also later introduced this same notion of ‘local time,’ though unlike length contraction he did not assign any physical importance to it. Poincaré [126], however, realised a physical signiﬁcance of this

79 80 Sound clocks and sonic relativity |

notion of local time as suggested by Lorentz in that it would be the time recorded on clocks synchronised using light signals. Eventually, aether theory fell victim to Ockham’s razor1: Einstein’s theory of special relativity sufficed to explain all of the same phenomena with the added simplification of not requiring an undetectable aether. It is important to realise that while the theory of special relativity won out over any of the aether theories, aether-based models still produce the exact same kinematic results as the theory of special relativity when treated correctly due to both theories exhibiting the exact same mathematical formalism [127]. In an effort to describe black holes, Unruh [103] came up with an analogy in terms of sound waves propagating up a waterfall: if at some location along the waterfall the flow speed of the water exceeds the speed of sound in the water, it becomes impossible to sonically signal upstream anymore. This model is analogous to a black hole, except here sound takes the place of light. Such objects are called acoustic black holes or dumb holes (where ‘dumb’ is a synonym for mute). It is worth noting that such a model possesses a preferred reference frame: the reference frame in which the water is stationary. Under several assumptions (no gravitational back-reaction, an unquantised gravitational field, and that at the Planck scale the wave equation for quantum fields is still applicable), Stephen Hawking famously demonstrated that black holes may be expected to evaporate by radiating at a characteristic temperature [128]. Whether the assumptions that Hawking made are valid is still unknown, and when Hawking’s assumptions were scrutinised by Unruh [77], the result of black hole evaporation was initially put into doubt. Acoustic black holes possess comparative problems: at sufficiently small length scales the continuum description breaks down (the expected equivalent of the Planck scale for spacetime), the field fluctuations (phonons) interact with the background that they propagate on (in analogy to gravitational back-reaction), and the analogy to the gravitational field itself (the fluid flow) is unquantised. Unruh, following the same process as Hawking, theorised that acoustic black holes should emit an acoustic analogue of Hawking radiation [1]. Fortunately, in stark contrast to physics at the Planck scale, molecular physics, atomic physics, and fluid mechanics are well understood. While the validity of the underlying assumptions in Hawking’s derivation of black hole radiation could not be directly tested, the comparative assumptions in the acoustic black hole model could be. Unruh [105] and others (Brout et al. (analytically) [108], Corley (analytically) [109, 110], and Corley and Jacobson (numerically) [106, 107]) were able to show that

1Named in honour of William of Ockham, though often spelt “Occam’s razor”. Introcution 81 | acoustic black holes should radiate at a temperature as calculated using Hawking’s approach for black holes even when the comparative assumptions in the acoustic black hole model were broken. This result demonstrates that the blackbody temperature of black holes as calculated by Hawking will not necessarily break down when the contributions of Planck scale physics are taken into account, providing us with some evidence that our current understanding of black hole thermodynamics could well be correct [103]. Models such as acoustic black holes are inherently nonrelativistic, yet nevertheless they appear to share some properties with relativistic systems [93]. This class of models are referred to as analogue gravity models. There are many examples of analogue gravity models. For example, quasiparticle production has been theorised [129] to occur in expanding Bose–Einstein condensates in analogy to particle production due to cosmic inflation. Bose–Einstein condensates have also been used to study analogue Hawking radiation, both theoretically [130, 131] and experimentally [94, 132]. An extensive list of the analogue gravity models known to exist up until 2011 can be found in the Living Reviews in Relativity article by Barceló, Liberati, and Visser [93]. It is natural to wonder how far these analogies can be pushed before they break down. How many of the features of general relativity appear in analogous form within these models? If all of the features of general relativity emerge in some analogue form, then under what assumptions does this occur? Is there a microscopic mechanism behind the analogue form of gravity, and can we use this knowledge to infer anything about the origins of gravitation in our universe? If it is not possible to make all of the features of general relativity emerge in some analogous form in these systems, then why do we find the emergence of some aspects of general relativity and not others? In order to answer these questions it is useful to understand the experience of observers within analogue relativity systems since observers were crucial in the understanding of general relativity, specifically in explaining the physical interpretation of general covariance [43]. The principle of general covariance states that the laws of physics should be independent of choice of coordinate system, as coordinates do not exist in nature a priori. It can be shown, however, that smoothly dragging the gravitational field around on a background spacetime manifold (a mathematical process called an active diffeomorphism) can result in a physical description of reality that is mathematically indistinguishable from a mere change in coordinates (a passive diffeomorphism). Therefore, if it is impossible to distinguish between an active diffeomorphism and a passive diffeomorphism, and if the principle of general covariance is taken to be true (that 82 Sound clocks and sonic relativity |

coordinates are indeed unphysical), then it follows that active diffeomorphisms must also be unphysical. Consequently, the spacetime manifold itself must be seen as being unphysical. It is only with respect to events or physical objects (in other words, observable quantities) that locations in spacetime have any meaning at all. The interpretation here is that spacetime has no real, physical meaning independent of coincidences. What is meant by this is that points in spacetime are only defined by two or more events or objects coinciding, such as two particles passing through the same point in space simultaneously. With this in mind, the existence of more than one solution to a covariant set of field equations is not seen to be problematic. If one metric tensor solves the field equations and yields spacetime paths for multiple observers that coincide at certain proper times for each observer, then all metric tensors consistent with the same initial conditions will yield spacetime paths for these observers that coincide at the same proper times. Einstein remarks on this, “... the requirement of general covariance takes away from space and time the last remnant of physical objectivity” [43]. The principle of general covariance and its interpretation on the meaning (or rather lack of meaning) of spacetime naïvely appears to be a problem that will be impossible to overcome in analogue gravity models because such models do possess an objective, physical analogue of spacetime; there is a preferred reference frame. In order to investigate if or how general covariance manifests in analogue gravity systems, we will need to consider the proper time elapsed by observers in analogue gravity models. Barceló and Jannes [133] described the physics of ‘natural’ interferometers in analogue relativity models. Such devices would be constructed out of quasiparticles, themselves made up from the particles of the medium under consideration. Observers within these analogue universes, using these interferometers to perform Michelson–Morley type experiments, would find the exact same results that we found in our universe: that the aether is undetectable. The reason for this is that the medium itself obeys a relativistic wave equation, and thus the resulting kinematics of the medium are subject to the symmetries of the Lorentz group. For example, it has been shown how electromagnetic-like theories based on models that admit a privileged reference frame can, in the low-energy limit, appear to obey Lorentz invariance to internal observers [134]. Consequently, any object constructed from the medium of the analogue universe will inherit the symmetries of the medium, i.e. the Lorentz group. Interferometers built this way will possess arms that will shrink in their direction of motion, and thus without any way to measure velocity relative to their aether, observers who Approach 83 | only have access to these devices will come to believe the postulates of relativity via Ockham’s razor. Additional discussions on the emergence of Lorentz symmetry and relativity in physical models that admit a rest frame can be found in the literature. For example, see Liberati, Sonego, and Visser [135], Volovik [136], and Nandi [137]. While quasiparticle interferometers are sufficient in demonstrating the emergent relativity of these systems, we would like to ask, are such constructs necessary to demonstrate this relativity? An observer within such a system who is free to perform any experiment they would like using an interferometer made from a material other than the one used to construct the medium of their universe will be able to infer motion relative to their aether and will not come to believe the postulates of relativity. However, if certain restrictions are placed on which experiments such observers are allowed to perform with non-quasiparticle interferometers, can such observers come to believe the postulates of relativity through their observations? Operationally speaking, constructing (or even describing how to construct) quasiparticles that would then in turn be used to build devices such as interferometers seems difficult, though such devices would most likely prove to be invaluable additions to the tool-kit of experimental physicists seeking to test analogue relativity systems. To this end, we turn our attention towards devices that are inserted into analogue relativity systems from the laboratory in order to determine if, with certain constraints, such devices can appropriately act as relativistic observers in such analogue systems. With certain restrictions placed on what type of experiments can be conducted, we show here that sound clocks – the equivalent of light clocks in systems obeying sonic relativity – can be used as appropriate relativistic observers for a medium in which sound obeys a relativistic wave equation.

3.2 Approach

As a preface to what we will discuss here, it should be noted that the work carried out here is essentially equivalent to the work performed by Poincaré at the beginning of the 20th century in his eﬀort to incorporate his philosophy of relativity into the aether based model prevalent at the time. Here, our aether is that of a condensed matter system (such as a large slab of solid matter or a perfect liquid, either of which must be homogeneous and isotropic), and light is replaced by disturbances within our medium: sound. While Poincaré sought a description for the way in which mechanical bodies naturally behaved in order 84 Sound clocks and sonic relativity |

to obscure detection of the aether, we are, in some sense, attempting to do the reverse: how can we manufacture obfuscation of an aether from observers within it in such a way that the observers come to believe the postulates of relativity? What are the minimum constraints that we are required to put on experimental equipment in an aether-based system such that observers who only have access to that equipment have the existence of that aether hidden from them? We ask this questions in the context of analogue gravity models from which we aim to, if possible, make analogies to phenomena in our own universe. However, let it be made abundantly clear that we are not attempting to revive aether-based models as a basis for how our own universe works. We go through a detailed analysis of the experience of observers who are both stationary and moving with a constant velocity with respect to the medium that they are confined to (which is at rest within the laboratory). The results obtained for stationary observers are in no way surprising, and calculations done from the laboratory frame without considering the technicalities of stationary observers’ experience will yield the same results. However, going through the more painstaking analysis is instructive in how to approach the problem of analysing the experience of moving observers. The results found for the experience of moving observers are indeed surprising at first glance and are not what is expected from the naïve, simple calculation done from the laboratory (which just happens to work for the stationary observers because they share the reference frame of the laboratory, even if they are unaware of it). We will discuss the specifics of the analysis and the restrictions that are necessary as they become important.

3.3 Simple sound clocks

Here we describe devices called sound clocks for systems in which sound obeys a relativistic wave equation, i.e. systems that exhibit sonic relativity. A sound clock is a device that is analogous to the light clock used in thought experiments to show time dilation in special relativity. Sound clocks consist of a clock mechanism out from which an arm, called the timing arm, is extended. At both ends of the timing arm, sound can be detected and emitted. To record time, a sound pulse is emitted from the end of the timing arm in contact with the clock mechanism, and at some later point in time, part of the wavefront will intersect the far end of the timing arm and be detected. Immediately following detection, a sound pulse is then emitted back in the same fashion. Upon reception of this second sound pulse – what we will herein refer to as the ‘echo’ – the clock mechanism advances its reading forward by one tick. Simple sound clocks 85 |

Figure 3.1: The operation of a sound clock is, in principle, straightforward: a sound pulse is emitted by the clock when an observer wishes to begin recording time, and after travelling to the end of the sound clock arm it is detected, whereupon a new sound pulse (the ‘echo’) is then emitted back towards the central clock mechanism. Upon reception of the echo, the clock reading is advanced forward by one tick. Note that this ﬁgure depicts one sound clock at three distinct moments in time as opposed to three sound clocks at one moment in time.

A diagrammatic outline of this process can be seen in Figure 3.1. We could also imagine additional shorter timing arms being present to increment the sound clock by fractions of a tick, but for the sake of clarity in figures, we shall not include these. Observers thus have some sense of local time by counting ticks of their clock. However, if an observer wishes to know when an event occurred at some other location, then they must have access to the reading on the clock positioned at that location (for example, by sending a message to the observer at that location and requesting the reading for when a specified event happened). Let us first consider a single sound clock as observed from the laboratory that is limited in the possible trajectories it can take: it is only allowed to travel in the direction perpendicular to its timing arm. If in the laboratory the length of the sound clock’s timing arm is known, and the velocity of the sound clock with respect to the medium is known, then the distance that = sound pulses have travelled can be determined via simple geometric arguments. Furthermore, if one knows the speed of sound in the medium, then the time it took for these = sound pulses to trace out their paths can be determined. From Figure 3.2, it can be determined that, in the laboratory frame, the total distance travelled by = sound pulses that are emitted and received by a sound clock 86 Sound clocks and sonic relativity |

EΔC=

Figure 3.2: A sound clock travelling with velocity E for some time ΔC= in which time = sound pulses are emitted and returned to the clock.

travelling at velocity E for time ΔC= (ΔC= for the duration of time that is required for = sound pulses to be emitted and received) in the direction perpendicular to its timing arm is given by q 2 2 B = 2!= EΔC= . (3.3.1) ( ) + ( ) Dividing the distance that the = sound pulses have travelled by their speed, the

speed of sound, 2s, yields the time, ΔCB (ΔCB for the time it took to trace out the path of length B), that it has taken for these = sound pulses to trace out their paths through space, s 2 2 B 2! E ΔCB = = = ΔC= . (3.3.2) 2s 2s + 2s

Note that ΔCB and ΔC= are equal, which is easiest to see for integer values of =: when = is integer, a sound pulse has just been detected; in order to detect a sound pulse, the sound clock and the sound pulse that is being detected must be at the

same place at the same time. Deﬁning some new variable ΔC B ΔC= = ΔCB (how

long the clock has been recording ticks for), and also deﬁning B E 2s, we can / determine exactly how long it takes for a sound clock travelling at any velocity

less than 2s to record = ticks of the clock,

2! = ΔC = p . (3.3.3) 2s 1 2 − Sound clock chains 87 |

Downchain Upchain

2 1 0 1 2 ! − − E ì

Figure 3.3: A chain of sound clocks whose clocks have been separated and synchronised under the assumption that they are stationary. The length of the vertical timing arms (!) of all sound clocks in the chain is equal, as is the length of the horizontal spacing arms (!), though the lengths ! and ! are not necessarily equal (only at rest is this the case).

The tick frequency, or the period, of any sound clock is therefore given by,

ΔC 2! 1 ) = = p . (3.3.4) = 2s 1 2 − p 1 The Lorentz factor, = 1 2− , has appeared for the tick frequency of a − moving sound clock.

Note that for a sound clock at rest with respect to the medium in the laboratory frame we have E = 0, and thus = 0, from which we obtain

2! ΔC = =, (3.3.5) 2s which corresponds to a period of

2! ) = . (3.3.6) 2s

This is exactly the time we expect it to take for = sound pulses to bounce along and back an arm of length ! at rest. 88 Sound clocks and sonic relativity |

3.4 Sound clock chains

Consider now multiple sound clocks at different locations in space. A chain of regularly spaced sound clocks is the easiest such example of multiple sound clocks to consider. The sound clocks that form a chain are connected by arms of tunable length to their neighbours and are synchronised with the use of a sound pulse from some agreed-upon clock (call it the origin clock). Consider the sound clocks to be labelled with integer values corresponding to how many steps away from the origin clock they are, with the origin clock itself being labelled clock 0. In general, there can be sound clocks to either side of the origin clock, with clocks on one side possessing positive integer labels, and clocks on the other side possessing negative integer labels. We shall adopt the following convention: when labelled from the laboratory, the sound clock with the largest positive-integer label is at the front of the chain if the chain is in motion; if the chain is stationary, we can freely label either side positive or negative. Relative to a given sound clock, we call clocks closer to the front of a moving chain ‘upchain’ and clocks closer to the back of a moving chain ‘downchain’: from the definition of the labelling scheme outlined above this means that the direction ‘upchain’ is parallel to the sound clock chain’s velocity vector, and the direction ‘downchain’ is anti-parallel to the sound clock chain’s velocity vector. The labelling convention is shown in Figure 3.3. We wish for all of the sound clocks within a given chain to share a common coordinate system, so sound clocks within a chain must tick at the same frequency. The timing arms of adjacent sound clocks are assumed to be exactly parallel, and so to fulfil the requirement that they tick synchronously, the timing arms must be of equal length. This requirement and others will be discussed further in what follows.

3.4.1 Calibrating clock separation

A chain of sound clocks can be seen in Figure 3.3, where ! refers to the length of

the ‘vertical’ arms (i.e. the timing arms), and ! is the length of the horizontal arm separating sound clocks, which we also call the spacing arms. Note that for what follows the terms ‘vertical arms’ and ‘horizontal arms’ are interchangeable with ‘timing arms’ and ‘spacing arms’, respectively. The vertical arm of each sound clock is used to measure time directly in the same method as described for the single sound clock in Section 3.3, while the horizontal arm is used to space the sound clocks in an appropriate manner. For the purposes of our discussions here, the timing arms and spacing arms are considered to always be perpendicular to Sound clock chains 89 | one another. We focus on the case where chains of sound clocks are only allowed to travel in the direction of the axis in which they are connected.

Observers who possess sound clocks and who are limited in their ability to make measurements as previously detailed have no ability to detect motion with respect to their medium. Any inertial motion within an analogue relativity system should therefore be indistinguishable from rest since observers who only possess sound clocks have no way to tell the diﬀerence between zero velocity with respect to the medium and constant non-zero velocity with respect to the medium.

Observers travelling with constant velocity separate sound clocks within a chain by simultaneously sending sound pulses along their timing arms and spacing arms. Believing themselves to be at rest, when the spacing arms are tuned to such a length that the two sound pulses return simultaneously, observers believe that the separation of their sound clocks (i.e. the spacing-arm length) is exactly the same as the length, !, of their timing arms. This belief happens to be true when a chain of sound clocks is actually at rest, but this is not the case when a chain of sound clocks is moving with a constant velocity.

By what distance should sound clocks travelling at a constant velocity be separated in order for this simultaneity condition to hold? Consider a simple chain of two sound clocks as seen in Figure 3.4. The observer located at one of the sound clocks is able to adjust the separation distance of the two sound clocks by extending the arm that connects them or by reeling it back in.

Within a chain of sound clocks, the path taken by the sound pulse within the timing arms on its outbound journey and its inbound journey is symmetric for any motion perpendicular to the timing arms. This, however, is not the case for a sound pulse in the spacing arms. For sound pulses propagating between clocks in the spacing arms, there are two distinct paths taken when the clocks are travelling with non-zero velocity. As can be seen in Figure 3.4, there is a downchain journey for which the sound pulse is travelling in the opposite direction to the sound clocks, and there is an upchain journey for which the sound pulse is travelling in the same direction as the sound clocks. The downchain journey, as seen in Figure 3.4, takes less time to complete as E grows, while the upchain journey takes longer. The time it takes for a sound pulse to travel between two adjacent clocks downchain (ΔC3, which is greater than zero) and the time it takes to travel between two adjacent clocks upchain (ΔCD, which is greater than zero), as shown in Figure 3.4, can be seen to obey the following relationships for a separation 90 Sound clocks and sonic relativity | E ì 0 1 0 1

Figure 3.4: The spacing arms between adjacent sound clocks in moving sound clock chains must be shorter than the timing arms by a factor of , to ensure that sound pulses that are emitted simultaneously along both arms also return simultaneously. Note that the readings of both clocks are advanced by exactly one tick when the sound pulses return (this is obscured partially for clock 0).

length of !,

! EΔC = 2sΔC , (3.4.1) − 3 3 ! ∴ ΔC3 = , (3.4.2) 2s E + and,

! EΔCD = 2sΔCD , (3.4.3) + ! ∴ ΔCD = . (3.4.4) 2s E − For simultaneously emitted timing and synchronisation sound pulses to be detected simultaneously, we don’t need to know exactly how long it takes the sound pulse to make either the downchain or upchain journey alone. We know that, by construction, the sum of the upchain and downchain times must be the same as the time for which one tick of the clock occurs as dictated by the timing arm, and one tick of the clock is given by Equation 3.3.3 with = = 1. With the length of the vertical arm labelled ! we have the relationship

2! 1 ΔC = ΔC3 ΔCD = p . (3.4.5) + 2s 1 2 − Sound clock chains 91 |

Substituting in the relationships for ΔC3 and ΔCD allows us to solve for the separation distance, !, which we ﬁnd to be

! ! = . (3.4.6)

3.4.2 Synchronisation of clocks

When the origin clock (clock 0) in a sound clock chain begins to record time, it simultaneously sends sound pulses along every arm connected to it. The sound pulse sent down its own timing arm is used to advance its own clock, whereas the pulses sent along the sound clock chain via the spacing arms are used to synchronise directly adjacent clocks (these are the synchronisation pulses). Upon receiving the synchronisation pulse, a given clock will simultaneously begin to record its own time and propagate the synchronisation pulse further along the chain. By this manner, all of the clocks in the chain can be synchronised with respect to the origin clock. We can construct expressions for the time taken for the synchronisation pulse emitted from the origin clock to travel to another clock: ΔC+ = : ΔCD is the time B | | it takes for the synchronisation pulse to reach clock : (: steps in the upchain direction), while ΔC− = : ΔC3 is the time it takes for the synchronisation pulse B | | to to reach clock : (: steps in the downchain direction). When at rest, ΔC and − B+ ΔCB− are equal. The relationships for the downchain and upchain synchronisation times can be combined into a single expression, and substituting in the expression for ! from Equation 3.4.6 we ﬁnd s ! 1 ΔCB± = ± : . (3.4.7) 2s 1 | | ∓ Despite the fact that the sound clocks are travelling in a medium with a preferred reference frame, the relativistic Doppler factor, s 1 = + , (3.4.8) 1 − has appeared instead of the non-relativistic one, where is the fractional speed of the sound clock chain with respect to sound. This is a result of observers within the sound clock chain setting the separation between adjacent clocks in such a way that the simultaneity of returned sound pulses occurs. In setting their separation in such a way, they have not only made their chain appear to exhibit the relativistic phenomenon of length contraction to observers in the laboratory, 92 Sound clocks and sonic relativity |

but they have also made their system appear to display the relativistic Doppler shift in regards to how long it takes sound to propagate between adjacent clocks to observers within the laboratory.

We now know how long it takes for any given clock, :, to tick = times (=:) once it begins its clock: this is given by Equation 3.3.3. We also know how long it takes for clock : to start its clock with respect to some initial clock (i.e. the time until it receives the synchronisation pulse): this is given by the appropriate choice th of synchronisation time from Equation 3.4.7. For = ticks of the : clock (=:), the total amount of time that has transpired since the lead clock ﬁrst began recording time is simply given by the sum of these two times.

For a clock 0 in the upchain direction we have the expression s ! 1 2! =0 C = + 0 p , (3.4.9) 2s 1 | | + 2s 1 2 − − and for a clock 1 in the downchain direction we have the expression s ! 1 2! =1 C = − 1 p . (3.4.10) 2s 1 | | + 2s 1 2 + − Let us also rearrange these equations for the number of ticks recorded by any clock post synchronisation. Upchain, the number of ticks recorded by clock 0 after time C is p 2 2sC 1 0 =0 = − | | 1 , (3.4.11) 2! − 2 ( + ) while downchain the number of ticks recorded by clock 1 after time C is

p 2 2sC 1 1 = = − | | 1 . (3.4.12) 1 2! − 2 ( − )

Adding to Equation 3.4.11 and Equation 3.4.12 the oﬀsets that the observers located at clocks 0 and 1 will add to their clocks to account for the believed synchronisation time ( 0 2 and 1 2, respectively) yields the proper clock | | / | | / reading, denoted , for clocks that are upchain and downchain, respectively. Note that, by the labelling scheme, we can express the proper clock reading of any clock in the chain, : (where : is any integer), at an instant in time with a single formula,

p 2 : 2sC 1 : B = | | = − . (3.4.13) : : + 2 2! − 2 Sound clock chains 93 |

From this equation we can rearrange for C again, yielding ! 2! 2! : C = : : =: = : . (3.4.14) 2s | | + + 2s 2s + 2

Equation 3.4.9 and Equation 3.4.10 are limiting cases of Equation 3.4.14 and are now superﬂuous.

From Equation 3.4.13 or Equation 3.4.14, one can obtain the useful relationships for the diﬀerence in time as a function of the diﬀerence of proper clock reading of any clock or clocks,

2! ! ΔC Δ = C ; C : = Δ ; : , (3.4.15) ( ) ( ) − ( ) 2s + 2s ( − ) and for the diﬀerence in proper clock reading for any clock or clocks as a function of a diﬀerence in time,

2s 1 Δ ΔC = ; C1 : C0 = ΔC : ; . (3.4.16) ( ) ( ) − ( ) 2! + 2 ( − )

These two relationships, Equation 3.4.16 and Equation 3.4.15, are of crucial importance in Section 3.5 and Section 3.6. It is important to note that the quantity Δ present in Equation 3.4.15 and Equation 3.4.16 is entirely general and can correspond to the difference in proper clock reading as recorded by a single clock (: = ;), which must of course occur at two different instances in time (ΔC ≠ 0), or to the difference in proper clock reading as recorded by different clocks (: ≠ ;), which can be calculated for any difference in time.

When observers – moving or not – request information from another observer on their chain for a believed simultaneous point in time, they will request information recorded at the same proper clock reading (i.e. Δ = 0). This is because all observers in inertial motion believe themselves to be at rest and so believe that all synchronised clocks possess the same proper clock readings at a given point in time. In reality, a moment in time is given by ΔC = 0 for which, in general, diﬀerent clocks will not read a diﬀerence in proper clock reading of Δ = 0 (which only happens to be true for chains of sound clocks at rest).

Figure 3.5 shows two chains of sound clocks, one at rest and one travelling with a velocity of E parallel to the stationary chain, at an instant in time. It can be seen that, as per Equation 3.4.16, the stationary chain ( = 0) possesses clocks that all have the same reading at an instant in time, whereas the moving chain possesses clocks whose readings diﬀer as a function of their separation from the origin clock. 94 Sound clocks and sonic relativity |

E ì

0 1 2 3 4 5 6

0 1 2 3

Figure 3.5: Two chains of sound clocks instantaneously adjacent to one another after having carried out their calibration and synchronisation procedures. The sound clock chain at the bottom is stationary, while the sound clock chain at the top is travelling with a fractional velocity of = √3 2, corresponding to a Lorentz / factor of = 2. Simultaneity requirements lead to the moving chain possessing asynchronous clocks that are separated by a distance of ! , where ! is the length / of the vertical timing arms.

3.5 Relativistic eﬀects observed by stationary sound clocks

Consider the following scenario: a moving chain of sound clocks passes by a stationary chain of sound clocks with a velocity of E, which corresponds to a fractional speed of , with respect to sound. The moving chain travels in the direction parallel to its own spacing arms, and passes by the stationary chain parallel to it and close enough to it that the travel time for sound between clock faces on adjacent chains is small with respect to the time it takes clocks on either chain to tick. Observers located at each clock can only record the information Relativistic effects observed by stationary sound clocks 95 | accessible from their immediate surroundings: they can read their own clock face, they can count how many clock faces they have passed by on the adjacent chain, and they can read the value recorded on clocks in the adjacent chain when they are sufficiently close (i.e. next to them). Observers within a chain then have to talk to one another and exchange their own measurements in order to come to some understanding of whatever experiment they conducted. As we shall demonstrate, observers located on the stationary chain of sound clocks determine that the sound clocks within the moving chain appear to be both separated by a shorter distance and tick less frequently than their own sound clocks. This is in keeping with what is seen in the laboratory. Later, when we consider measurements made by observers on a moving sound clock chain of a stationary chain we find that, contrary to what occurs in the laboratory, moving observers also determine that the clocks in the stationary chain are separated by a shorter distance than their own and ticks less frequently than their own. This only happens when both chains are treated equally in that observers are only allowed to use their own clocks as time references, observers can only signal with sound pulses, and observers have no means by which to detect motion with respect to the medium that they are embedded within.

3.5.1 Time dilation as seen by stationary observers

Imagine that observers within the stationary chain of sound clocks decide to focus on the lead sound clock of the moving chain as it passes by, as seen in Figure 3.6. The ﬁrst experiment they wish to conduct is: “How many times do moving clocks appear to tick for every tick of stationary ones?” To determine how many times sound clocks in a measuring chain believe themselves to tick for every one tick of a sound clock in a diﬀerent chain, we use the following procedure:

1. Determine the separation in time, ΔC, for some clock, I, in the chain that is being measured to increment its clock reading once. This is obtained using

Equation 3.4.15 with I = ; = : and ΔI = 1. 2. Determine which clocks, : and ;, in the chain performing measurements clock I is next to at two points in time separated by ΔC. This is done by determining how far clock I has moved in ΔC as calculated in Step1 . 3. Determine the proper clock reading on clock : when the clock that is being measured, I, is next to it, and determine the proper clock reading on clock ; when the clock that is being measured, I, is next to it. The diﬀerence in these two proper clock times is how many ticks observers within a measuring chain believe to have occurred for them for one tick of the clock they were 96 Sound clocks and sonic relativity |

0 1 0 1 E ì

0 1 2 3

Figure 3.6: The observers in the stationary chain focus only on the lead sound clock of the moving chain. The moving chain has a fractional velocity of = 3 √13, / with respect to sound, corresponding to a Lorentz factor of = √13 2 1.8. / ≈ With this velocity, the moving chain of sound clocks travels a distance of 3! in the laboratory for every tick of its clocks. In this example, the clocks labelled : and ; in the stationary chain are clocks 0 and 3, respectively.

measuring. In other words, evaluate Equation 3.4.16 for clocks : and ; as determined in Step2 using time diﬀerence obtained in Step1 . The diﬀerence in these proper clock readings gives the perceived number of ticks that have transpired in the chain performing measurements for one tick of the clock being measured.

Consider that some clock, :, in the stationary chain is next to a clock in the M moving chain, I, when the moving clock has a proper clock reading I (where the superscript ‘M’ denotes that the quantity pertains to the moving chain). At some later point in time, clock I in the moving chain has moved next to some other clock in the stationary chain, ;, at the moment that clock I advances its proper M clock reading forward one tick to I 1. From Equation 3.4.15, the time it takes + M for a given clock in the moving chain, I, to tick once (Δ = 1) is 2! 2s where I / is the sonically relativistic Lorentz factor of the moving clock. The distance covered by a moving clock ticking once is then given by

2! GM = EΔC = E = 2!. (3.5.1) 2s Relativistic eﬀects observed by stationary sound clocks 97 |

Stationary clocks are separated by a length of !, so the number of stationary-clock spacing-arm lengths that the moving clock has travelled in this time is given by 1!, where 1 is just some number. Equating these two distances and solving for 1, we ﬁnd, 1 = 2. (3.5.2)

We have then that ; = : 1. What is the difference in the proper clock reading, + ΔS = S S (the superscript ‘S’ denotes that the quantity pertains to the ; − : stationary chain), corresponding to the difference in time, ΔC, that it takes for clock I in the moving chain to travel between clocks : and ; in the stationary chain? From Equation 3.4.16 (with = 0 as we are considering the proper clock reading difference of the stationary chain), we find the difference in proper clock readings to be,

S S S Δ ΔC = C1 C0 = . (3.5.3) ( ) ; ( ) − :( )

Observers in the stationary chain determine that their clocks have all ticked times for one tick of the moving clock. This is, in fact, true. In the laboratory, all clocks within the stationary chain possess the same proper clock reading at the same instant in time, and all stationary clocks have indeed ticked times for one tick of the moving clock.

3.5.2 Length contraction as seen by stationary observers

Now imagine that observers located in the stationary chain decide to focus on more than just the ﬁrst sound clock of the moving chain. The next experiment they wish to perform is: “How are clocks in the moving chain spaced, relative to clocks in the stationary chain?”

When we want to determine the length of an object in a laboratory (at least for any experiment that takes place over reasonable distances and times) we will typically find the positions of both ends of the object in question at an instant in time and then determine the separation of these points in space. This is done with respect to some fixed coordinate system parallel to the object (e.g. a ruler). The definition of ‘an instant in time’, a notion that is crucial in operationally determining lengths, is a velocity-dependant quantity for observers who only have access to sound clocks. The only measure of time that observers with sound clocks have access to is their proper clock reading as given by Equation 3.4.13. Within a given chain, observers at different clocks will possess a different proper clock reading at a given point in time if their chain is travelling with respect to 98 Sound clocks and sonic relativity |

0 1 2 E ì

0 1 2 3

Figure 3.7: The observers in the stationary chain of sound clocks determine the separation of sound clocks in the moving chain by determining which of their own clocks are simultaneously next to a given pair of clocks in the moving chain. From this information, they are able to determine how many spacing arms in the stationary chain are simultaneously parallel to a given number of spacing arms in their own chain, and with the knowledge that their own spacing arms are of length L, they can determine how long the spacing arms in the moving chain are. The moving chain is travelling with a fractional velocity of = √3 2, / corresponding to a Lorentz factor of = 2. In this example, clocks : and ; in the stationary chain are clocks 0 and 1, respectively.

the medium (i.e. ≠ 0 in Equation 3.4.16). To determine how observers with sound clocks measure distance, we must ask what measurements they make when their clocks have the same readings. The procedure we follow to determine what distance one chain of sound clocks measures another chain’s spacing arms to be is:

1. Determine which two clocks, : and ; (where : ≠ ;), in a given chain are going to be used to perform measurements on lengths in another chain. 2. Determine the diﬀerence in time that is required for these two recording clocks to have the same proper clock reading, i.e. determine which two times

correspond to : = ; using Equation 3.4.13. The diﬀerence in time between these two clock readings is given by Equation 3.4.15 for Δ = 0 and : ≠ ;. Relativistic eﬀects observed by stationary sound clocks 99 |

3. Determine which clock, :0, in the chain that is being measured is next to clock : in the chain performing measurements at the time value corresponding to

:, and which clock, ;0, in the chain that is being measured is next to clock ; in the chain that is performing measurements at the time value corresponding

to ;. 4. Determine how many spacing arms, 1, separate the clocks in the chain performing measurements ( 1 = ; : ). Determine how many spacing | | | − | arms, 10, separate the clocks in the chain being measured ( 10 = ;0 :0 ). | | | − | Using this information, determine how many spacing arms in the chain being measured simultaneously appear to be parallel to one spacing arm in the chain performing the measurements.

Consider that, at some time, one of the clocks in a moving chain, :0, is next to one of the clocks (call it :) in the stationary chain: we shall label this time C S . ( :) Which clock in the moving chain, ;0, is next to some other clock in the stationary chain, call it ; = : 1, when clock ; has the same proper clock reading as clock : + at time C S ? From Equation 3.4.15, with = 0, the diﬀerence in time between ( :) the moments when clocks : and ; have the same reading is 0. That is to say,

C S = C S . (3.5.4) ( :) ( ; )

As expected, the clocks in the stationary chain possess the same proper clock reading at the same instant in time. This means that the moving chain has not moved relative to the stationary chain when clocks : and ; perform their measurements at equal clock readings, as can be seen in Figure 3.7, for 1 = 1. Clock ; is 1 spacing arms away from clock :, and the spacing arms in the stationary chain have length !; therefore clocks : and ; are separated by a distance of 1!. Clock :0 in the moving chain is next to clock : in the stationary chain, while simultaneously clock ;0 in the moving chain is next to clock ; in the stationary chain; the number of spacing arms separating clocks :0 and ;0 in the moving chain is labelled 10. Clocks in the moving chain are separated by spacing arms of length ! , so we have the equality / ! 1! = 10 . (3.5.5) This leads to the relationship

10 = 1. (3.5.6)

Directly adjacent clocks in the stationary chain (when 1 = 1 in Equation 3.5.6) will (correctly) conclude that, parallel to the single spacing arm that separates them, there are spacing arms in the moving chain. 100 Sound clocks and sonic relativity |

Furthermore, if one were to extend the formalism and imagine that there were clocks at every point in space along a chain of sound clocks, the hypothetical clock with label : 1 would be next to some clock in the moving chain, whose + / neighbour would be next to clock : in the stationary chain.

3.6 Relativistic eﬀects observed by moving sound clocks

Consider that observers located on a moving chain of sound clocks wish to perform the same experiments as the stationary chain did in Section 3.5. To reiterate: a moving chain of sound clocks passes by a stationary chain of sound clocks with a velocity of E, which corresponds to a fractional speed of , with respect to sound. The moving chain travels in the direction parallel to its own spacing arms and passes by the stationary chain parallel to it and close enough to it that the time it takes sound to propagate between the two chains is small with respect to the time it takes clocks on either chain to tick. Observers located at each clock only have at hand the information accessible from their immediate surroundings; they can count how many clock faces they pass by on the adjacent chain, and they can read the value off of adjacent clock faces when they are sufficiently close. Observers within the chain then have to talk to one another and exchange their own measurements in order to come to some understanding of whatever experiment they conducted. As foreshadowed in Section 3.5, we find that observers located on moving chains of sound clocks find that sound clocks within a stationary chain appear to be both separated by a shorter distance and tick less frequently than their own sound clocks. It is not obvious that this result should appear, and in fact it only does so when the observers are constrained to use their own clocks as time references, can only signal with sound pulses, and cannot detect motion with respect to the medium that they are embedded within. These constraints lead observers to ask for measurements made at the wrong laboratory time when they wish to aggregate and compare data recorded ‘simultaneously’ because the clocks in the moving chain do not actually possess the same reading at an instant in time.

3.6.1 Time dilation as seen by moving observers

Imagine that observers in the moving chain decide to focus on the lead sound clock of the stationary chain as they pass it, as seen in Figure 3.8. The ﬁrst experiment they wish to conduct is: “How many times do stationary clocks appear to tick for every tick of moving ones?” Relativistic eﬀects observed by moving sound clocks 101 |

E ì

0 1 2 3 0 1 2 3

0 1

Figure 3.8: The observers in the moving chain focus only on the lead sound clock of the stationary chain in order to determine how many times their own clocks tick for every one tick of a clock in the moving chain. Due to the asynchronicity of clocks in the moving chain, the observers within the moving chain come to the incorrect conclusion that their clocks tick times for every 1 tick of a clock in the stationary chain. The moving chain is travelling with a fractional velocity of = 3 √13, with respect to sound, corresponding to a Lorentz factor of / = √13 2 1.8. In this example, the clocks labelled : and ; in the moving chain / ≈ are clocks 3 and 0, respectively.

The method used here is exactly the same as was outlined in Subsection 3.5.1. Consider that some clock in the moving chain, :, is next to a clock in the stationary S chain, I, when the stationary clock has a proper clock reading I . At some later point in time, the moving chain has moved such that some other clock in the moving chain, ;, is next to clock I at the moment that clock I advances its proper S clock reading forwards one tick to I 1. The diﬀerence in time, ΔC, that it takes S + for clock I to tick once, ΔI = 1 can be obtained using Equation 3.4.15 (with = 0 because we are considering the stationary clock). The time taken for a stationary sound clock to advance its proper clock reading by one tick is found to be 2! 2s (as expected from Equation 3.3.6). The distance that the moving chain / 102 Sound clocks and sonic relativity |

has travelled in this time is given by

GM = EΔC = 2!. (3.6.1)

Sound clocks within the moving chain are separated by a distance of ! . The / distance that the moving sound clock chain has travelled, GM, is equal to some multiple, 1, of its own sound clocks’ separation length,

! 1 = 2!. (3.6.2)

From this, we can easily solve for 1:

1 = 2 (3.6.3)

Note that the value of 1 determined here is exactly the same as 1 in Equation 3.5.2. The moving sound clock chain has travelled a distance of 1 multiples of its own spacing arm length in the time that it has taken the stationary sound clock, I, to tick once. Therefore, clock ; = : 1 is next to the stationary clock, I, when it − advances its time forward by one tick. The diﬀerence between the proper clock

readings of clock : at some time C0 and clock ; at some later time C1 = C0 2! 2s + / can be obtained from Equation 3.4.16:

M M M Δ ΔC = C1 C0 = . (3.6.4) ( ) ; ( ) − : ( ) Where, again, the superscript M indicates that these are quantities pertaining to the moving chain. Even though the chain of moving sound clocks actually ticks less frequently than the chain of stationary sound clocks, observers travelling along with the moving chain believe that the stationary chain ticks less frequently than their own due to their incorrect belief that they are at rest, which results in their clocks being asynchronous.

3.6.2 Length contraction as seen by moving observers

The observers in the moving chain now decide to focus on more than just the ﬁrst sound clock of the stationary chain in order to perform their next experiment: “How are clocks in the stationary chain spaced, relative to clocks in the moving chain?” We take the same approach as with the stationary chain’s experiment, except with the roles of the stationary and moving chain reversed: which two clocks in the moving chain are two adjacent clocks in the stationary chain simultaneously Relativistic eﬀects observed by moving sound clocks 103 |

0 1 E 0 1 ì

0 1 2 3

Figure 3.9: The observers in the moving chain determine what they believe to be the separation of sound clocks in the stationary chain by determining how many sound clocks are between two of their own sound clocks at an instant in time (as they understand it). The moving chain has a fractional velocity of = √3 2, / with respect to sound, corresponding to a Lorentz factor of = 2. At this velocity, adjacent clocks in the moving chain only have the same clock reading when separated by a period of time that corresponds to having travelled a distance of ! = 2! in the laboratory. In this example, clocks : and ; in the moving chain are clocks 0 and 1, respectively.

next to (where simultaneity is deﬁned as equal proper clock readings)? The method used here is exactly the same as was outlined in Subsection 3.5.2.

Consider the scenario in which at some time, C0, some clock in the moving chain, M :, has proper clock reading : while next to a clock in the stationary chain. After what period of time does another clock ;, that is 1 spacing arms away (; = : 1), have the same proper clock reading as clock : (M = M)? With use of + ; : Equation 3.4.15 we can ﬁnd the diﬀerence in time, ΔC, for ΔM = 0 for clocks : and ; = : 1: + M M M ! ΔC Δ = C ; C : = 1 . (3.6.5) ( ) − ( ) 2s In this time the moving chain has moved a total distance of,

GM = EΔC = 1!2. (3.6.6) 104 Sound clocks and sonic relativity |

Clocks in the moving chain are separated by a distance of ! , so clock ; = : 1 / + is 1 multiples of ! away from clock :. Clock ; has also travelled a total distance / of GM (from Equation 3.6.6) after the moment in time when clock : made its

measurement, so the location of clock ; at time C1 as compared the location of

clock : at time C0 is given by

M ! 2 G C1 = 1 1! . (3.6.7) ; ( ) +

This distance corresponds to some multiple, 10, of the stationary sound clocks’ M spacing arm length. Equating 1 ! with G C1 yields 0 ; ( )

10 = 1. (3.6.8)

Clocks : and ; = : 1 in the moving chain register the same proper clock reading + at two diﬀerent instants in time. At these instants in time, clock : is situated over some clock in the stationary chain, and clock ; = : 1 is situated over some other + clock that is 1 spacing arms away from the clock that : was situated over, as can be seen in Figure 3.9, for 1 = 1. Directly adjacent neighbours in the moving chain (i.e. 1 = 1 in Equation 3.6.8) will (incorrectly) conclude that there are exactly clock arms simultaneously parallel to their single clock arm, a result that arises due to the asynchronicity of their clocks.

3.7 Sonic relativity

We have now seen what look like relativistic effects from the perspective of internal observers: time dilatation appears to be given by Equation 3.5.3 and Equation 3.6.4, while apparent length contraction appears to be described by equations Equation 3.5.6 and Equation 3.6.8. However, none of these relationships explicitly deal with lengths or times. Ticks and numbers of clocks are both unitless, and furthermore, while as it appears in these equations is a quantity that we can calculate in the laboratory, internal observers as we have currently described them can only measure it (by counting how many clocks they pass in a given period of time or by comparing clock readings on their own chain to another chain). We shall now cast all previous formulae in terms of quantities that internal observers themselves can measure and demonstrate the appearance of sonic relativity. Let us start with some defined quantities based on beliefs held by internal observers. Assume that observers operationally define a unit of length called an ‘arm’ and a unit of time called a ‘tic’. A timing arm of length ! (as measured in the laboratory) is operationally defined by internal observers to be 1 arm long, and Sonic relativity 105 |

1 tic is operationally deﬁned to be the time it takes for a sound clock with a timing arm of length 1 arm to advance its clock reading forward once. All observers in inertial motion believe themselves to be at rest, and so they believe 1 tic of time to be the time it takes for a sound pulse to travel 2 arm (to the end of the timing arm and back again). The speed of sound to any internal observer is then deﬁned to be

arm 2 B 2 . (3.7.1) ˜s tic

With the belief that they are at rest, all observers co-moving with a chain of sound clocks that are calibrated and synchronised as per the procedures outlined in Subsection 3.4.1 and Subsection 3.4.2 believe that their spacing arms are exactly 1 arm in length each, as it takes exactly 1 tic of time for the echo of a sound pulse propagated between adjacent clocks in a chain to return. This requirement itself formed the basis of the calibration procedure as outlined in Subsection 3.4.1. With these deﬁnitions, we can determine at what velocity – in units of arm/tic – observers in a given chain believe another chain to be travelling.

In 1 tic of time for a stationary clock, 2 clock arms in the moving chain pass by (as per Equation 3.6.3). Recalling from Equation 3.5.6 that stationary observers believe that spacing arms in the moving chain are simultaneously next to one of their own, observers on the stationary chain come to conclude 1 that clocks in the moving chain are separated by − arm per clock. Stationary observers therefore (correctly) believe that 2 clocks in the moving chain have a length of 2 arm. These 2 sound clocks of length 2 arm take 1 tic to pass by. Thus, the perceived velocity of the moving chain, E, in units of arm/tic (up to a ˜ sign) is given by 2 arm E arm 2s E = = 2 = ˜ E = 2s. (3.7.2) | ˜ | 1 tic 2s tic 2s ˜

The perceived fractional velocity, B E 2s, of the moving chain with respect to ˜ ˜/ ˜ sound using only measurements of quantities available to internal observers is related to the actual fractional velocity with respect to sound by

˜ = . (3.7.3)

Moving observes determine the same relationships. A moving clock passes by 2 stationary clocks in 1 tic of time (as per Equation 3.5.2), and from Equation 3.6.8 moving observers deduce that spacing arms in the stationary chain lay simultaneously parallel to one of their own (i.e. moving clocks appear to 1 be separated by − arm per clock). Because the moving observers think that they are the ones who are stationary and that the stationary observers are moving, 106 Sound clocks and sonic relativity |

they believe that 2 clocks have passed by them, with a total length of 2 arm in a time of 1 tic. This leads to the (believed) velocity of the moving chain, E, in units ˜ of arm/tic (up to a sign) of

2 arm E arm 2s E = = 2 = ˜ E = 2s. (3.7.4) | ˜ | 1 tic 2s tic 2s ˜

The believed fractional velocity, B E 2s, of the stationary chain with respect to ˜ ˜/ ˜ sound using only measurements of quantities available to internal observers is given by

˜ = . (3.7.5) Both stationary and moving observers believe that the other chain of sound clocks is moving with a fractional velocity with respect to sound that is equal in magnitude to the value of the moving chain’s fractional velocity with respect to sound in the laboratory frame. Utilising an agreed-upon coordinate system,

observers in the moving chain will report the value of ˜ that they measure to be diﬀerent than that as measured by stationary observers by a sign. Both stationary and moving observers deﬁne

1 B . (3.7.6) ˜ q 1 2 − ˜

A given observer measures length by counting the number of spacing arms between simultaneous measurements of the endpoints of an object: 1 arm is exactly the length of one spacing arm. We deﬁne ℓ˜0 to be the length, as measured by a stationary observer, of an object whose length is measured to be ℓ˜ by an observer at rest with respect to the object. Both stationary and moving observers believe that 1 clocks in the other chain lay simultaneously next to 1 of their own, as per Equation 3.5.6 and Equation 3.6.8, respectively. Thus, observers in each frame will state that the total length spanned by some number of their own spacing arms is equal to the length spanned by times as many spacing arms of the other chain, leading to the relationship

ℓ ℓ 0 = ˜ , (3.7.7) ˜ ˜ where has been replaced by the internal-observer-deﬁned . ˜ A given observer measures elapsed time by counting ticks of their own clock: 1 tic is exactly the time it takes for one’s own clock to tick once. We deﬁne to be ˜0 the duration, as measured by a stationary observer, of a localised process whose Discussion 107 | duration is measured to be by an observer at rest with respect to the process. ˜ As per Equation 3.5.3 and Equation 3.6.4, both stationary and moving observers believe their own clock readings to have advanced times in the time it takes a ˜ given clock in the other chain to advance its clock reading once. Therefore, both stationary and moving observers believe that a clock in the other chain takes as ˜ much time to tick once as their own clock does, leading to the relationship

0 = , (3.7.8) ˜ ˜ ˜ where, as before, the internal-observer-deﬁned Lorentz factor is used due to its ˜ equality with .

3.8 Discussion

Through an operational approach, it has been shown that it is possible for a certain class of inertial observers to deduce the existence of two key phenomena from special relativity – length contraction and time dilation – in condensed-matter systems for which the speed of sound plays an analogous role to the speed of light within our universe. The observers we discuss are significantly restricted in their ability to assign temporal and spatial values to events. They are only able to claim when events occur relative to their own clock, can only make claims about events that are sufficiently local such that the time it takes for the signal to reach them is negligible, and must confer with one another after taking local measurements to come to an understanding of the events that transpired. The observers travelling at a constant velocity have no way to tell if they are stationary or moving. With no way to tell who is in motion, this question would become a philosophical one for internal observers, and some internal observers may even come to the same conclusions that we in our universe have: that all motion is relative. We have seen that when internal observers assume the former state of motion – that they are indeed stationary – then those who are in motion incorrectly set the separation of their clocks, a process that was achieved by fulfilling the requirement that sound simultaneously emitted along two objects of equal length will result in the simultaneous reception of echoes. This simultaneity condition results not only in the incorrect separation of clocks within a moving chain; it also results in the asynchronicity of those clocks. In constructing their chains of sound clocks in such a way that local simultaneity conditions hold, observers who are stationary see the moving chain to be length contracted exactly as one would expect from a naïve application of relativistic formulae, with 2s being the speed of sound instead of the speed of light. 108 Sound clocks and sonic relativity |

The clocks within the moving chain also appear to be time dilated as would be expected from special relativity. This is due to the use of sound pulses to advance clock readings, and moving clocks increase the path length, thus increasing the time it takes for a sound pulse to return to the clock mechanism by exactly the Lorentz factor again. More curiously, the observers in moving chains also witness stationary sound clock chains as being length contracted: this is not actually the case and is again a result of making use of simultaneity arguments. Moving observers think that the clocks within their own chain are synchronous, and thus when they wish to know what happened at some distant clock simultaneous with their own clock, they ask the observer located at that distant clock to provide them with information recorded when the distant clock’s reading was the same as their own. These clocks are not actually synchronous, however, so the observers in the moving chain are actually comparing information from two separate instances in time. This happens to work out in exactly the right way to make the observations of moving observers and stationary observers symmetric: moving observers also perceive stationary sound clock chains to be length contracted and time dilated exactly as one would expect from a naïve application of relativistic formulae, with

2s being the speed of sound instead of the speed of light. We see that the ‘in-universe’ experience – the internal observers’ description of their universe – can be described by the mathematical formalism of special relativity provided that such observers believe the postulates of relativity, a conclusion that they would reasonably come to when given no ability to detect their aether. It is merely a misunderstanding of the Newtonian mechanics at play that results in the appearance of relativistic effects to these internal observers. Given the ability to detect their own state of motion relative to their aether, moving observers would quickly come to understand that they have incorrectly separated and calibrated their clocks, and fixing this problem would result in the disappearance of the apparent relativistic effects that are witnessed by moving observers. We have intentionally remained within the realm of discussing devices that are operationally controlled by observers, and the relativity within the system described appears as a result of the belief that observers have about their state of motion. Nevertheless, it is worth noting that if we had access to devices that were built from quasiparticles made up from the medium itself, then sonic relativity would emerge naturally. As described by Barceló and Jannes [133], a device constructed from these quasiparticles (such as a sound clock chain) would

shrink naturally as E approaches 2s, just as physical objects held together by Conclusion 109 | the electromagnetic force do when travelling close to the speed of light [138]. This would entirely remove the role of the observers in tuning the separation of neighbouring clocks, and therefore the belief held by the observers on their state of motion would become inconsequential. Furthermore, while we have restricted our analysis to chains of sound clocks that must keep their timing and spacing arms perpendicular, and that are only able to move in the direction perpendicular to their timing arms, devices built from quasiparticles would not be limited in these ways. If the sound clock chains that we describe here were permitted to travel with a velocity possessing a non- zero component in the direction parallel with the timing arms, then any change in the direction of motion mid-journey to include such a component of velocity would lead to the asynchronous return of echoes in the timing and spacing arms. Additionally, if these devices could alter the angle between their arms, this would again lead to the asynchronous return of sound pulses in the timing and spacing arms, and if the angle between the arms was closed to 0◦, observers could quite easily determine that the timing and spacing arms are different lengths. These effects would be naturally taken care of in quasiparticle devices, however: the entire device’s dimensions would change accordingly with its velocity, conspiring to make the effects of length contraction impossible to detect to the observers located on the chain.

3.9 Conclusion

It is perhaps not too surprising that stationary observers in these systems infer that moving observers undergo relativistic time dilation and length contraction, but given the presence of a preferred reference frame it may not be immediately obvious that moving observers should see stationary observers subject to these same eﬀects (although if one has a suﬃcient understanding of the history of special relativity, especially in the context of aether theory, then this may not actually be too surprising). Given that relativity is seen in both directions just as in our universe, it can be seen that the existence of a preferred reference frame is not immediately prohibitive in the emergence of a self-consistent description of relativity by internal observers in analogue gravity systems. While some aspects of relativity can be made to appear, it is not clear to what extent relativistic physics can be made to manifest in such systems. If the role of observers in analogue gravity systems is taken more seriously, investigating the observations made by such observers might give us some insight into how many of the phenomena described by general relativity can be seen 110 Sound clocks and sonic relativity |

to arise in analogue gravity models in a self-consistent manner. Are there any analogue gravity models that appear to possess all of the phenomena of general relativity (in analogous forms) to internal observers? If this is not the case, then why not? Why should some of the phenomena of relativity emerge in a self- consistent manner in such models, but not others? Considering the experience of observers may help to answer these questions. PARTICLESCATTERINGIN ANALOGUE-GRAVITYMODELS

• This chapter is a minorly modiﬁed version of a paper that is currently under review for publication with Physical Review D. A preprint version of this paper is accessible on arXiv [3]. This paper was co-authored with Giacomo Pantaleoni, Valentina Baccetti, and Nicolas C. Menicucci, all of whom provided 4 major contributions to Section 4.5.

We investigate a simple toy model of particle scattering in an analogue-gravity model. The analogue-gravity medium is treated as a scalar field of phonons that obeys the Klein–Gordon equation and thus admits a Lorentz symmetry with respect to 2s, the speed of sound in the medium. The particle from which the phonons are scattered is external to the system and does not obey the sonic Lorentz symmetry that the phonon field obeys. In-universe observers who use the exchange of sound to operationally measure distance and duration find that the external particle appears to be a sonically Lorentz-violating particle. By performing a sonic analogue to Compton scattering, in-universe observers can determine if they are in motion with respect to their medium. If in-universe observers were then to correctly postulate the dispersion relation of the external particle, their velocity with respect to the medium could be found.

4.1 Introduction

Analogue-gravity models provide an indirect way to probe the physics of gravitational systems for which the actual (i.e., non-analogue) experiments are currently inaccessible. Perhaps most well known are the acoustic analogues to black holes (also called dumb holes1), originally proposed by Unruh [1] to provide a theoretical and experimental testbed for the study of Hawking radiation [79, 128] in a system where the microscopic physics is understood. Subsequently, analogue models for a variety of general relativistic and semi-classical gravitational phenomena have been discovered, and many such models are now being experimentally realized— notably acoustic analogues for Hawking radiation [94–96, 139], optical-media analogues for Hawking radiation [97], and analogues for cosmological expansion and particle production [98]. A comprehensive list of analogue-gravity models and the research into them as of 2011 can be found in the extensive review article by Barceló, Liberati, and Visser and the references therein [93]; a few noteworthy theoretical results since 2011 include [140–145]. 1‘Dumb,’ in this case, being a synonym for ‘mute.’

111 112 Particle scattering in analogue-gravity models |

Studying relativistic phenomena with analogue models is, of course, only a valid approach provided that the analogue system can be faithfully mapped back to the actual physical system of interest. One obvious—and seemingly detrimental—way in which the mapping between analogue models and real physical systems of interest seems to fail is that analogue models are not truly relativistic. In the most obvious case, there exists a preferred reference frame: the rest-frame of the analogue-gravity medium itself. Previous work by Barceló and Jannes [133] has highlighted that the natural way to consider analogue-gravity systems as genuine relativistic analogues is to use devices and observers that are internal to the analogue medium (e.g., they are constructed from quasiparticle excitations of the medium itself) to make internal measurements of phenomena within the analogue universe. Phenomena like the Lorentz-FitzGerald contraction can be operationally shown to appear in a relativistically reciprocal manner by utilizing this notion of internal devices and observers: internal observers who are at rest with respect to the medium will measure moving observers’ devices to be length contracted (they are), and internal observers who are moving with respect to the medium will measure stationary observers’ devices to be length contracted (even though they are not) [2]. We call such observers in-universe observers. One way to understand the operational emergence of relativity from an analogue-gravity model is to consider the flat-spacetime limit: for an acoustic system, the flat-spacetime limit is exactly a sonic analogue to Lorentz ether theory,2 and it is known that Lorentz ether theory and special relativity are observationally equivalent [138, 146]. Therefore, from the operational viewpoint that considers only the measurements made by in-universe observers [2], the sonic analogue to Lorentz ether theory can be seen to faithfully map to a sonic analogue of special relativity, which we call sonic relativity. In attempting to utilize analogue-gravity models as a means to study phenomena within the overlap of relativistic physics and quantum physics, it would prove useful to understand how operational measurements of quantum mechanical phenomena can be performed in such systems. In the real world, quantum field theory (QFT) is the most well-developed theory incorporating the effects of special relativity and quantum mechanics, and within the context of QFT, scattering experiments are one of our most valuable experimental tools for probing the dynamical interactions of physical phenomena.3 If we are to attempt to utilize analogue-gravity models to their full potential

2Provided that one only considers the motion of sound waves belonging to the linear part of the medium’s dispersion relation. 3See almost any textbook on QFT for a discussion on scattering and its relevance to experiments, e.g., [119, 147–149]. For a more in-depth discussion on the nature of detectors themselves, see [150, 151]. Introduction 113 |

(i.e., as a method by which to carry out actual experiments), then we should—as a first step—seek to describe and understand those detector models that can conceivably be experimentally realized and that can be used in conjunction with analogue-gravity systems. This leads to several questions. For example, what constitutes an appropriate model of a particle detector within an analogue-gravity model? How much of our understanding of quantum detectors from actually relativistic theories can we apply to analogue-gravity systems? If we consider deviations away from the idealized case in which we only consider that which is internal to an analogue-gravity model, how is our understanding of detector models impacted?4 In particular, this last question provides the basis of the work presented in this paper. We will herein consider scattering experiments within analogue-gravity models. We do so in keeping with the philosophy of a previous paper that was published by two of the present authors [2]: we consider the measurements made by in-universe observers within an analogue-gravity system, and we restrict these observers to measuring duration and distance solely through the exchange of sound pulses. This ensures their measuring devices obey sonic Lorentz symmetry. We assume that in-universe observers have access to the apparatus required to do scattering experiments with phonons. That is, they can produce phonons, and they can detect recoiling phonons through a “click” of a particular detector within an array of such detectors. We neglect the detailed questions of how such items are constructed, understanding that our assumptions are enough to meaningfully discuss scattering experiments in this setting. We seek to determine the results that in-universe observers measure of phonon scattering experiments, specifically in the case that phonons are scattered from an external particle—that is, one whose dispersion relation is non-relativistic. The contents of this paper are as follows: In Section 4.2 we review the notion of in-universe observers [2]. In Section 4.3 we elucidate further on our aims, provide a sketch of the approach that is used throughout this paper, state the conventions that we shall use, and give a schematic of the scattering events experiments that we consider. In Section 4.4 we determine the kinematic expressions of a toy model of phonon scattering from two types of particles: one type of particle is sonically Lorentz obeying (internal particle), and the other is sonically Lorentz violating (external particle). In both cases, we first obtain the kinematic description of scattering in the laboratory frame and then in the frame of an in-universe observer who is co-moving with the particle from which the phonons scatter. (This is the approach laid in out in Section 4.3.) In Section 4.5 we consider phonon

4Where the notion of internal is as per the discussions of Barceló and Jannes [133]. 114 Particle scattering in analogue-gravity models |

scattering from a ﬁrst-quantized particle with a Newtonian dispersion relation (an external particle) and present the results that demonstrate that these scattering experiments reveal information about the existence of a preferred rest frame to in-universe observers. In Section 4.6 we explicitly explain in what way the results presented in the previous section reveal information about the existence of a preferred rest frame. In Section 4.7 we summarize our results and demonstrate that the Standard-Model Extension (SME) might provide a way to model phonon scattering from external particles as described by in-universe observers.

4.2 In-universe observers

Assume that there are in-universe observers in an analogue-gravity universe who believe in the principle of special relativity, with the exception that sound takes the place of light [2]. Taking the coordinates G, H, I, and C to be measured operationally by some in-universe observer via the exchange of sound, and deﬁning x = G,H,I ( ) as a notational shorthand, we can construct the following four-vector ! 2sC - B . (4.2.1) x

This four-vector is a sonically Lorentz covariant object, and in keeping with special relativity we denote this the sonic four-position. Of course, there is nothing special about Cartesian coordinates: the sonically Lorentz covariant nature of the sonic four-position is true for any set of orthogonal coordinates. We have merely made the choice to pick Cartesian coordinates for the purposes of demonstration. Consider now two sets of in-universe observers who are initially travelling in the same direction, which we shall call the I-direction, by convention, with respect to their sound-carrying medium. The ﬁrst set of observers are travelling with

velocity v1 = 0, 0,E1 with respect to the medium, and the second set of observers ( ) are travelling with a velocity of v2 = 0, 0,E2 with respect to the medium. Note ( ) that the operational measurements of distance and duration allow for in-universe observers to operationally calculate velocities [2]. In the particular case that we are considering, the operationally determined sonic fractional velocity of the second set of observers as measured by the ﬁrst set of observers will be given (in terms of the laboratory-frame values of quantities) by the following expression:

2 1 = − , (4.2.2) 1 21 − where 1 = E1 2s and 2 = E2 2s. The quantity is the boost parameter of the / / In-universe observers 115 | second frame with respect to the first frame. Each set of observers can describe the sonic four-position from their own operational measurements of distance and duration, as discussed above. Denote - to be the components of the sonic four-position as operationally determined by the first set of observers, and -0 to be the components of the sonic four-position as operationally determined by the second set of observers. The first set of observers can relate the components -0 to - with the following familiar equation:

-0 = Λ - . (4.2.3)

5 For the particular case that we have been discussing Λ is given by

0 0 © − ª ® 0 1 0 0 ® Λ = ® , (4.2.4) 0 0 1 0 ® ® 0 0 «− ¬ where is the operationally determined sonic fractional velocity of the second set of observers with respect to the first set of observers, given by Equation 4.2.2, p and B 1 1 2. / − Another way to see that measurements of duration and distance made using only the exchange of sound signals can be grouped together into a Lorentz covariant object, -, is that the object being used to define spatial and temporal measurements (the phonon) obeys, in our simplified toy model, the Klein–Gordon equation [2]. The Klein–Gordon equation admits a Lorentz symmetry, and as a consequence of this, any measurements that are made using phonons inherit the Lorentz symmetry that the phonons themselves obey. A consequence of the existence of the sonic four-position vector and its associated Lorentz transformation is that all physical quantities whose values can be determined geometrically—that is to say, from the components of the sonic four- position—transform exactly as expected from special relativity. For our purposes, we note that measurements of spatial and/or temporal coordinates (i.e., the components - or -0 ) can be used by in-universe observers to operationally determine the wavelength and frequency of sound waves, and also as a way to operationally determine angles: as a result, between any pair of in-universe-

5Note that this definition is not simply some vacuous one that we merely assert to be true axiomatically. While we do not offer a derivation within this paper, it is indeed possible to take the content of [2] and formulate a proper four-vector description of coordinates from the operational measurements made by in-universe observers. In doing so, the Lorentz transformation matrix naturally arises as the transformation relating the four-position between different in-universe- observer frames. 116 Particle scattering in analogue-gravity models |

observer reference frames the wavelengths and frequencies of sound waves obey the relativistic Doppler shift formula, and the angle of propagation of an acoustic ray will change via the relativistic aberration formula (where, in both cases, the speed of sound replaces the speed of light). The relativistic Doppler shift formula and the relativistic aberration formula are given, respectively, by:

$ = 1 cos 0 $0 (4.2.5) + cos 0 cos = + , (4.2.6) 1 cos + 0 where the last relation leads to the following useful expression,

1 1 cos = − 1 cos 0 , (4.2.7) ( − ) 1 cos ( − ) + 0 and where in all cases is the boost parameter of the second frame with respect to the first (given by Equation 4.2.2), an unprimed symbol denotes the in-universe observer measured value of some physical quantity in the first frame, and a primed symbol denotes the value of that same physical quantity as measured by in-universe observers in the second frame. Note that in the specific case that we

are considering, the angle is measured relative to the I-axis and the angle 0 is

measured relative to the I0-axis. For notational brevity in what follows, deﬁne the Doppler factor to be: s 1 B + . (4.2.8) 1 −

In the particular case that 0 = 0, the Doppler factor can be used to simplify

Equation 4.2.5 to $ = $0.

4.3 Aim and Approach

4.3.1 Aim

Our purpose is to investigate a particular type of scattering experiment within the context of analogue-gravity models. In particular, we choose to investigate a scattering process that is in some sense analogous to Compton scattering. In true Compton scattering [147, 152], a photon is scattered from a charged particle. By analogy we choose to analyze the scattering of phonons within an analogue-gravity system. We consider phonon scattering from two diﬀerent types of particle, which we label as either sonically Lorentz-obeying or sonically Aim and Approach 117 |

Lorentz-violating based on their dispersion relation. Our aim is the following: Characterize the in-universe-observer perspective of these scattering experiments, with the speciﬁc goal of determining what in- universe observers can learn from the results of scattering experiments involving sonically Lorentz-violating particles, including what they can learn about their state of motion with respect to the medium.

4.3.2 Sketch of our approach

In a typical derivation of scattering within quantum ﬁeld theory, one often makes use of relativistic arguments to simplify the derivation by, for example, moving into the center-of-mass frame of the system [147]. While such an approach indeed simpliﬁes the derivation for phonon scattering from sonically Lorentz-obeying particles [133], this is certainly not the case for phonon scattering from sonically Lorentz-violating particles. In this paper we take the following approach to our derivations:

1. We anchor ourselves to the laboratory frame, which is the frame in which the analogue-gravity medium is assumed to be at rest. In this particular frame, we know the actual dynamics, and we know that energy and momentum must be conserved. 2. We identify, in the laboratory frame, the dispersion relations that apply to the phonon and to the particle from which the phonon will scatter. 3. Utilizing the relevant dispersion relations, we obtain the laboratory-frame kinematic expression for scattering by insisting on energy and momentum conservation. 4. With the laboratory-frame kinematics determined, we utilize the appropriate transformation rules to obtain the kinematic description of scattering from the perspective of the in-universe-observer frame that is co-moving with respect to the particle prior to phonon scattering.

This procedure grounds our calculation in the frame for which we know the dynamics: the laboratory frame. We do all of our dynamical calculations from this frame. The final step then determines how this behavior would appear to an in-universe observer who perceives the particle to be initially at rest in their own frame (even though both the observer and the particle may in fact be moving with respect to the medium). This allows us to tease out the differences between experiments that look the same to in-universe observers—particle initially at rest in the observer’s frame—but that actually differ in their initial velocity with respect to the analogue-gravity medium—i.e., with respect to the sonic ether. 118 Particle scattering in analogue-gravity models |

As a sanity check, we ﬁrst utilize this approach to demonstrate that the kinematic description of phonon scattering from a sonically Lorentz-obeying particle shows no dependence on the initial motion with respect to the medium, as would be the case for ordinary scattering in a fully Lorentz-obeying model [133] and in ordinary QFT [147]. We then proceed to use the same approach to arrive at a kinematic description for phonon scattering from a sonically Lorentz-violating particle. A mathematical sketch of this approach is the following. The energy and momentum conservation in the laboratory frame is simply:6

i ~$i = f ~$f, (4.3.1) + + pi ~ki = pf ~kf, (4.3.2) + + where the notation, which is used throughout, is deﬁned in Table 4.1. The dispersion (energy-momentum) relations allow us to express energies as functions of momenta:

= p , ~$ = ~$ k , (4.3.3) ( ) ( )

from which we can rewrite the conservation of energy and momentum like

i pi ~$i ki = f pf ~$f kf , (4.3.4) ( ) + ( ) ( ) + ( ) pf = pi ~ kf ki . (4.3.5) − ( − ) From here, we explicitly substitute our dispersion relations into Equation 4.3.4, using the conservation of momentum [Equation 4.3.5] in the explicit functional form of p . (The specific functional forms that we will use will be given in f( f) subsequent sections.) Finally, we perform any valid algebraic steps that are required to obtain the desired kinematic expression for the Compton-like scattering of phonons. The resulting expression, by construction, obeys energy and momentum conservation, and is written entirely in terms of the the initial parameters of our system—which, in principle, we are free to control—and the final state of the phonon—which we can envisage experimentally detecting using some appropriate apparatus, e.g., a detector array comprised of some sonic analogue to photomultiplier tubes. Our final step is to utilize the laboratory-frame kinematic description of scattering—along with our understanding of how the values of physical quantities

6Note that we have restricted our considerations to the linear part of the medium’s dispersion relation through our particular choices for the phonon’s energy (~$) and momentum (~k). Aim and Approach 119 |

Phonon Particle Meaning

~ki pi Initial 3-momentum (lab frame) ~kf pf Final 3-momentum (lab frame)

: = k ? = p 3-vector magnitude (lab frame) | | Table 4.1: Notation for momenta of the phonon and particle. We restrict our initial setup to the case where ki and pi are in the positive-I direction. as measured by in-universe observers change between diﬀerent reference frames— to determine the in-universe-observer description of the kinematics. Speciﬁcally, we choose to obtain the kinematic description of scattering in the frame of an in-universe observer who is initially co-moving with the particle from which the phonon will scatter—i.e., we are interested in the kinematic description as seen by the observer who initially believes the particle to be at rest. Before we can explicitly list the transformation rules for the values of physical quantities, it is important to recall a few key facts and to have a schematic of the scattering experiment in mind.

4.3.3 Schematic of our scattering experiment

Recall that the specific form of the transformation equations that we established in Section 4.2 were predicated on the assumption that pairs of in-universe-observer reference frames had coordinate axes that were aligned, and that the relative motion between pairs of in-universe-observer reference frames was constrained to be in the I-direction. For the purposes of this paper, we choose only to consider experiments for which these assumption hold true. Also note that—as explicitly demonstrated in [2]—operational measurements of distance and duration that are made by in-universe observers who are stationary with respect to the analogue- gravity medium coincide with the equivalent measurements as made in the laboratory frame (up to unit conversions). That is to say, the laboratory frame is an in-universe-observer reference frame; it simply happens to be the in-universe- observer reference frame corresponding to observers who are stationary with respect to the medium. With all of this in mind, we present in Figure 4.1 a schematic of the scattering experiments that we will consider throughout the remainder of this paper. We note that our scattering experiments are initialized such that the trajectories of our particles are parallel with respect to the incoming phonons; we define these trajectories to point in the positive I-direction. Denoting velocities v = EG ,EH ,EI , ( ) we can write the initial velocity of our particle vi = 0, 0,Ei , where Ei 0 by fiat ( ) ≥ 120 Particle scattering in analogue-gravity models |

(a) (b) λ λf f0 ~ θ Λµ λ λi βi ν i0 θ0 φ φ0 − − ~ ~ β0 βf f

Figure 4.1: A scattering event as seen in (a) the laboratory frame and (b) the co-moving in-universe-observer frame. All quantities labelled in the figure are measured operationally with sound-based clocks and rulers [2] and thus their values as measured by in-universe observers transform with respect to a sonic Lorentz transformation. We envisage the particle from which scattering occurs to be centered with respect to a hollow shell of detectors that can detect phonons incident upon them. To in-universe observers in the co-moving frame (b), the detector array appears to form a spherical shell defined by constant spatial coordinates; in the laboratory frame (a), the array of detectors appears to be Lorentz-contracted in the direction of motion and the spatial coordinates defining this shell change between the scattering event and the detection event. The relativistic Doppler formula and the relativistic aberration formula relate the operationally measured values of wavelengths and angles between reference frames, respectively. The operationally measured value of the final velocity of the particle in different reference frames is related by the general form of the relativistic velocity composition formula [153] (see [154] for an English translation).

of our assumptions; consequently, we have Ei = Ei. Denoting sonic fractional ì velocities to be B v 2s, we can write i B i = Ei 2s. Restricting ourselves / ì / to cases in which the particle’s initial and ﬁnal speeds are slower than 2s, we

always have that < 1; taken in combination with Ei 0, we have then that ≥ 0 i < 1. The ﬁnal velocity of the particle f is allowed to have components in ≤ any direction, provided that f < 1. As it turns out, though, we will ultimately

be able to remove any reference to the ﬁnal state of the particle (and thus to f) from the kinematic expressions that we obtain. We previously provided equations for the relativistic Doppler shift of sound waves [Equation 4.2.5], and for the relativistic aberration formula [Equation 4.2.6]. As we have emphasized, in our particular scattering experiments we wish to consider the operational measurements made in the co-moving in-universe-

observer frame. In the laboratory frame, we therefore have that = i—that is, the boost parameter of the co-moving in-universe-observer frame is numerically equal to the initial velocity of the particle, and thus the values of quantities in Phonon scattering kinematics 121 | the laboratory frame (unprimed symbols) are related to the values of quantities in the co-moving in-universe-observer frame (primed symbols) in the following speciﬁc ways:

$i = i 1 i $0 = i$0, (4.3.6) + i i $ = i 1 i cos 0 $0, (4.3.7) f + f cos 0 i cos = + , (4.3.8) 1 i cos + 0 1 i 1 cos = − 1 cos 0 , (4.3.9) ( − ) 1 i cos ( − ) + 0 where Equations 4.3.7–4.3.9 follow directly from Equations 4.2.5–4.2.7, respectively, and Equation 4.3.6 follows from Equation 4.2.5 with = 0 = 0 because the phonon is always initially travelling in the positive I-direction.

4.4 Phonon scattering kinematics

We now proceed to derive the kinematic expressions of Compton-like phonon scattering within an analogue-gravity model. As we stated in Section 4.3, we ﬁrst obtain the kinematic expression for phonon scattering for the case of sonically Lorentz-obeying particles (internal particles). Once we have obtained both the laboratory frame and co-moving in-universe-observer frame kinematics for phonon scattering from internal particles, we will then proceed to do the same for sonically Lorentz-violating particles (external particles).

4.4.1 Phonon scattering from internal particles

4.4.1.1 The laboratory frame kinematics of phonon scattering from internal particles

To provide some context, we imagine that, following Ref. [133], internal particles are collective-excitation quasiparticles whose dynamical description is covariant with respect to the same sonic Lorentz symmetry that in-universe-observer reference frames transform under. In other words, these quasiparticles are relativistic with respect to the speed of sound 2s of the analogue-gravity medium. Most obviously, one might imagine these particles to arise from the analogue-gravity medium itself (hence the use of the word ‘internal’). However, there is no particular reason that these particles could not also arise in a separate medium with the same speed of sound, and to which the analogue-gravity medium is somehow coupled. 122 Particle scattering in analogue-gravity models |

For the purposes of this paper, we choose the following simple and familiar energy-momentum relations for our internal particles:

2 = <2s , p =

In order to be able to cleanly substitute all of our quantities into Equation 4.3.4, we desire to be able to write = p . Noting that both p and are functions of v, ( ) we can perform some simple algebra to obtain the following familiar expression for = p : ( ) s ?2 p = 1 ; (4.4.2) 2 2 ( ) + < 2s where ?2 = p p. Substituting this expression for into our expression for energy · in Equation 4.4.1, we obtain = p . It proves more useful for our purposes to ( ) consider 2 p , however (the form of which should also be quite familiar): ( )

2 p = ?222 <224. (4.4.3) ( ) s + s

Now that we have = p , we can proceed precisely as we described in ( ) Section 4.3. Rather than performing all of our substitutions and then algebraically simplifying afterwards, we choose to perform the reverse process. Square the energy and momentum conservation equations [Equations 4.3.4 and 4.3.5] to obtain the following:

2 2 n o pf = i pi ~ $f kf $i ki , f ( ) ( ) − ( ) − ( ) 2 2 2 2 = 2~i $f $i ~ $ 2$i$f $ ; (4.4.4) i − ( − ) + f − + i 2 2 ? = pi ~ kf ki , f − ( − ) 2 = ? 2~?i :f cos 2~?i :i i − + 2 2 2 2 ~ : : 2~ G:f :i cos . (4.4.5) + f + i −

Note that the cos terms in Equation 4.4.5 arise as a result of the experimental scenario that we are considering as per Figure 4.1. In Equation 4.4.4 we have suppressed the explicit functional dependency on momentum for notational brevity: Phonon scattering kinematics 123 | this dependency still applies, of course. From here, we can directly substitute Equation 4.4.3 into Equation 4.4.4 wherever the square of the particle’s energy appears. Doing so, and utilizing the square of the conservation of momentum [Equation 4.4.5], we obtain the following expression:

2s?i~$i ~i$i − 2 = ~$f 2s?i ~$i cos i~$f ~ $i$f, (4.4.6) + − − where we have made use of the fact that $ = 2s k . From here, it is a simple | | matter to isolate :f. Doing so yields the following expression: i 2s?i :i :f = − . (4.4.7) i 2s?i 1 cos ~$i 2s?i − + ( − ) + Some simple algebraic manipulations also allow us to phrase this in the following way:

1 " #− $ ~$i 2s?i f = 1 + 1 cos . (4.4.8) $i + i 2s?i ( − ) − Using the explicit forms of and p as given in Equation 4.4.1, and recalling that

our experiment is initialized such that ìi = i (with i > 0), we can further rewrite Equation 4.4.8 entirely in terms of dimensionless quantities:

1  ! − $f  ~$i 1 cos  = 1 . (4.4.9)  2 i −  $i  + 2 + 1 i   i s −    This is just the kinematic relationship for ordinary Compton scattering [152] as viewed from a boosted frame, except the boost is with respect to the speed of sound 2s rather than the speed of light. In other words, this is the kinematic description of a sonic analogue to Compton scattering.

4.4.1.2 The co-moving in-universe-observer frame kinematics of phonon scattering from internal particles

We now possess an unambiguous kinematic description of phonon scattering from an internal particle in the laboratory frame. Here we re-express this from the perspective of an in-universe observer who is in the frame that is co-moving 124 Particle scattering in analogue-gravity models | with the particle prior to scattering. We use Equations 4.3.6 and 4.3.7 to write $0 1 i $ f = + f . (4.4.10) $0 1 i cos $i i + 0 Plugging in Equation 4.4.9 and using Equations 4.3.6–4.3.9 to rewrite all quantities from the co-moving frame, this becomes $0 1 i f = + $0 1 i cos i + 0 1  ! −  ~$i0 1 cos 0  1 1 . (4.4.11)  2 i i −  ×  + 2 ( + ) + 1 i cos   s + 0    Straightforward algebra simpliﬁes this to

1 ! − $   f0  ~$i0  = 1 1 cos 0  . (4.4.12) $  + 2 −  i0  2s   

Notice that all dependence on i has disappeared in the ﬁnal form, as it must since both types of particle respect the sonic Lorentz symmetry. While Equation 4.4.12 may not be in its most familiar form, this is precisely the kinematic description of Compton scattering in the rest-frame of a particle prior to scattering (with, of

course, the understanding that all references to 2 are replaced by 2s). Recalling that our dispersion relation for phonons is taken to be linear, this expression can be easily cast into perhaps its most well known form:

ℎ 0f 0i = 1 cos 0 . (4.4.13) − <2s −

Note that the co-moving in-universe-observer frame’s kinematic description of phonon scattering from internal particles makes no explicit reference to any velocity. This should not be too surprising, given the relativistic form of the energy-momentum relation that we chose for internal particles. It is important

to note that we did not manually set i = 0 to obtain this result, though: all

instances of i and i were removed from the kinematic expression purely via the use of valid algebraic manipulations. The resulting expression, however, is

entirely equivalent to taking Equation 4.4.9, setting i = 0 (and hence, i = 1), and then appending primes to the remaining operationally determined quantities—a method that should be familiar to any student of special relativity.

At this point, the reader may find themselves wondering why we seem Phonon scattering kinematics 125 | to be re-deriving results that are well known in QFT [147]. While the naïve method of setting i = 0 in Equation 4.4.9 does indeed provide a shortcut to the co-moving in-universe description of scattering [i.e., without explicitly needing to consider the transformation equations Equations 4.3.6–4.3.9], this only works for scattering from particles that are sonically Lorentz obeying in nature. Specifically, the equivalence of these two methods is a result of the fact that the energy-momentum relation for internal particles transforms covariantly under the same sonic Lorentz transformation that applies to in-universe-observer reference frames.7 External particles, however, are not sonically Lorentz obeying in nature—their energy-momentum relations do not transform covariantly under the sonic Lorentz transformation—and so the shortcut that happened to work with internal particles does not apply when we are considering external particles. Re-obtaining the familiar expression for Compton scattering by utilizing the simple approach that we first detailed in Section 4.3 provides a sanity check for our approach, and it provides a familiar example through which to elucidate some important details regarding the derivation.

4.4.2 Phonon scattering from external particles

4.4.2.1 The laboratory frame kinematics of phonon scattering from external particles

As we have stated before, external particles are particles that are not sonically Lorentz obeying. That is to say, the dynamical description of external particles is not covariant with respect to the sonic Lorentz symmetry of the analogue- gravity medium. This, of course, leaves a wide range of possibilities in terms of selecting energy-momentum relations for external particles. For our purposes here, we choose speciﬁcally to consider our external particle to be an ordinary quantum-mechanical particle with the usual Newtonian energy-momentum relation:

?2 = p =

Following the approach described in Section 4.3, we directly substitute our expression for energy into Equation 4.3.4 and utilize the conservation of momentum

[Equation 4.3.5] to eliminate any reference to kf. This leads to an equation that is

7As noted in passing earlier, the internal particle’s energy and momentum can be collected into 2 the four-momentum % . With % B 2s, p , one can use Equation 4.4.1 ( = 2 and p =

quadratic in :f:

2 2 " # ~ :f ~:i ?i cos ~:f + 2s 2< − < − ~:i 2?i ~:i + 2s = 0. (4.4.15) + 2< −

In the fully sonically Lorentz obeying case (that is, phonon scattering from internal particles) the conservation of energy and momentum lead to a linear equation

in :f, and thus for a given set of initial experimental parameters and for a given

scattering angle there existed only one possibly value of :f. In this case, however, we have a quadratic equation, and so for a given set of initial experimental parameters and for a ﬁxed scattering angle of , there are two possible values

that :f can take. We can solve Equation 4.4.15 for ~:f and multiply the solutions by 2s to obtain solutions of the following form for the ﬁnal energy of the phonon: p 2 2s~:f, 1,2 = ~$f, 1,2 = 2s 2s . (4.4.16) ( ) ( ) ¯ ± ¯ − ¯ where 2s~:f, 1 (or equivalently, ~$f, 1 ) is taken to be the solution with the positive ( ) ( ) sign, 2s~:f, 2 (or equivalently, ~$f, 2 ) is taken to be the solution with the negative ( ) ( ) sign, and

= ~:i ?i cos <2s, (4.4.17a) ¯ ( + ) − = ~:i ~:i 2?i 2<2s . (4.4.17b) ¯ + −

Note that the bar ( ) serves only as a notational label; it has no particular physical or ¯ mathematical meaning. The quantity ¯ has units of a three-momentum, whereas the quantity ¯ has units of the square of a three momentum.

The term 2s¯ in Equation 4.4.16 expands to produce terms of the form 2s~:i, 2 2s and 2s?i. The ﬁrst of these three terms is precisely the initial energy of the phonon, ~$i. The remaining two terms have the structure of relativistic terms: by analogy to actual relativity, the second term appears to have the structure of a sonic analogue to rest mass-energy, and the third term has the structure of the initial (relativistic) kinetic energy of the particle. With that said, these ﬁnal two terms are not actually descriptions of the external particle’s energy because the external particle is not sonically relativistic. Nonetheless, it is interesting to note that terms of a relativistic form appear naturally in the description of our non-relativistic external particle.

As for the case of phonon scattering from internal particles, we also express Phonon scattering kinematics 127 | the kinematic description of phonon scattering from external particles in terms of the ratio $ $i: f/ $f, 1,2 ( ) = √2 , (4.4.18) $i ± − where, as per above, $f, 1 is taken to be the positive solution and $f, 2 is taken ( ) ( ) to be the negative solution. The quantities and are related to their barred versions and are given as follows:

2 2 2s?i <2 = s ¯ = 1 cos s ; (4.4.19a) ~$i + ~$i − ~$i 2 2 2 22s?i 2<2 = s ¯ = 1 s . (4.4.19b) 2 ~$i + ~$i − ~$i ( ) The expressions and will prove useful when we move into the co-moving in- universe-observer reference frame. Before proceeding to determine the description of kinematics from that frame, we shall take a moment to address the nature of the two solutions for the kinematic description of phonon scattering from external particles.

4.4.2.2 Physical meaning of the two solutions

We will now focus on understanding the physical meaning of the two solutions given by Equation 4.4.16. Scattering from either type of particle should always yield a real and positive value of the ratio of the ﬁnal-to-initial phonon energies. In the case of scattering from the external particle, the requirement that the ratio of phonon energies be real puts restrictions on the allowed values of for given initial ?i and ~$i = 2s~:i since the discriminant of the square-root in Equation 4.4.16 must be non-negative. From Equation 4.4.18 we see that this leads to two conditions:

Condition 1. Solutions 1 and 2 are both guaranteed to be real and positive when the following conditions are both satisﬁed: p 0 and . (4.4.20) ¯ ≥ ¯ ≥ ¯ 128 Particle scattering in analogue-gravity models |

This condition directly translates into an inequality in cos , restricting the scattering angle range as follows:

q <2B ~:i ~:i 2?i 2<2B 1 cos + + − , (4.4.21) ≥ ≥ ~:i ?i ( + ) with

~:i 2?i ~:i ~:i 2?i 2<2s 0 = <2s + . (4.4.22) + − ≥ ⇒ ≤ 2

Condition 2. If 0 then solution 1 is always real and positive (note that 2 is ¯ ≤ ¯ always positive). This means that there aren’t any restrictions on the scattering angle, and so cos can take the usual range of values:

1 cos 1. (4.4.23) ≥ ≥ −

Restrictions are however placed on the absolute values of the three-momenta ~:i and ?i such that:

~:i 2?i ~:i ~:i 2?i 2<2s 0 = <2s + . (4.4.24) + − ≤ ⇒ ≥ 2

These conditions will play a very important role in the calculation of the scattering cross section in Section 4.5.

4.4.2.3 The co-moving in-universe-observer frame kinematics of phonon scattering from external particles

As we have discussed previously, the kinematic description of phonon scattering from external particles in the co-moving in-universe-observer frame is not simply

given by taking the laboratory frame description and setting i = 0. The observers’ geometric measurements constitute a Lorentz-covariant object (-) but the energy and momentum of the external particle cannot be collected into a sonically Lorentz-covariant object. As a result of this mismatch in symmetry groups between the dynamically relevant physical quantities (energy and momentum) and the reference frames of observers, the kinematic description of phonon scattering from external particles is not Lorentz covariant. We take then the approach that we discussed in Section 4.3 (and which we previously applied to the kinematics for phonon scattering from internal particles in Section 4.4.1.2) and use the relations given by Equations 4.3.6–4.3.9 to replace the value of all Phonon scattering kinematics 129 | laboratory frame quantities with their co-moving in-universe-observer frame values. Substituting these formulas into Equation 4.4.16, some basic algebraic manipulations lead to the following description of phonon scattering from external particles in the co-moving in-universe-observer frame:

2s~:f0, 1,2 = ~$f0, 1,2 ( ) ( ) p 2s 2 = ¯0 ¯0 ¯ 0 , (4.4.25) i 1 i cos ± − + 0 where cos 0 i 0 = ~:i0 <2si + <2s, (4.4.26a) ¯ + 1 i cos − + 0 0 = ~:0 ~:0 2<2si 2<2s . (4.4.26b) ¯ i i + −

¯0 and ¯ 0 are just ¯ and ¯, respectively, written in the co-moving (0) coordinates using Equations 4.3.6–4.3.9. Note that we have used the Doppler factor as deﬁned in Equation 4.2.8, with = i, for notational brevity in these expressions. In the in-universe-observer frame, the dimensionless ratio of the ﬁnal to initial frequencies of the phonon can be given by the following relation:

$0 1 f, 1,2 i √ 2 ( ) = ( + ) 0 0 0 , (4.4.27) $0 1 i cos ± − i ( + 0) where here 0 and 0 are <22 1 2 cos 0 i s − i 0 = + , (4.4.28a) 1 i cos 0 − ~$0 1 i cos + i + 0 2 2<2s 0 = 1 1 i . (4.4.28b) − ~$i0 ( − )

In fact, the reader can verify, using Equations 4.3.6–4.3.9, that 0 and 0 are just and expressed in co-moving (0) coordinates. The kinematic description of phonon scattering from an internal particle and the co-moving in-universe- observer frame [Equation 4.4.12] made no reference to the velocity of the particle (and hence the co-moving observer), whereas the kinematic description of phonon scattering from an external particle [Equation 4.4.27] does retain reference to the velocity of the particle (and hence the co-moving observer). The presence of velocity dependency in the co-moving in-universe observer’s description of phonon scattering from external particles means that, in principle, an in-universe 130 Particle scattering in analogue-gravity models |

observer could use such scattering experiments to identify their state of motion with respect to their analogue universe. We shall come back to this point in Section 4.5.5.5. The forbidden scattering angles and three-momenta that arise in the laboratory frame description of phonon scattering from external particles take on the following forms in the co-moving in-universe-observer frame:

Condition 1. Solutions 1 and 2 are both guaranteed to be real and positive when the following conditions are both satisﬁed: p 0 0 and 0 . (4.4.29) ¯ ≥ ¯ ≥ ¯ 0

This condition translates into the following restrictions on the scattering angle

cos 0:

2 p <2s 1 i~:0 − i − i + ¯ 0 1 cos 0 , (4.4.30) ≥ ≥ p ~:0 i i − ¯ 0 with

2 ~$i0 <2B . (4.4.31) ≤ 2 1 i ( − )

Condition 2. If 0 then solution 1 is always real and positive [note that 2 ¯ 0 ≤ ( ¯0) is always positive]. In this case, there are no restrictions on the scattering angles, but the absolute values of the three-momenta are constrained:

1 cos 0 1, (4.4.32) ≥ ≥ − with

2 ~$i0 <2B . (4.4.33) ≥ 2 1 i ( − )

4.5 Scattering cross sections

We now have an understanding of the kinematics of scattering for both the case of a phonon scattering from an internal particle (obeying sonic Lorentz symmetry), and the case of a phonon scattering from an external particle (not obeying sonic Lorentz symmetry). Our goal now is to understand whether, by analyzing the cross sections for phonons scattering on external particles, in-universe observers Scattering cross sections 131 | can infer anything about their velocity with respect to the laboratory reference frame. In pursuit of this, we need to devise some appropriate quantum toy model for sonic particle scattering that captures the important features of actual particle scattering, such as Compton scattering [147, 152]. The physics in the laboratory frame is unambiguous when we have such a model, and the method by which we determine the co-moving in-universe-observer description of events is by transforming observable quantities appropriately. We also continue to define all measurements operationally: we imagine that in-universe observers take an operational approach to detecting scattered phonons by using arrays of detectors that are, in their own frame, seen to be spherical and seen to be centred on the scattering event. Figure 4.2 highlights what this looks like in the laboratory frame. By performing several scattering events to create a statistical distribution and counting how many times any single detector clicks, we can obtain the reaction rate —i.e., the number of scattering events per unit of volume per unit of time ℛ in a particular direction—to obtain the differential cross section. In the following, we will start by carefully deriving the scattering cross section equations for external particles. Our system is a “hybrid” one, composed of three parts: the reference frames that we use to operationally perform measurements, the phonon, and the external particles. Only two of the three—the phonon and the reference frames—are always fully sonically relativistic, hence we cannot simply apply the QFT definition of the cross section since Lorentz covariance is deeply embedded in the derivation [147]. On the other hand, we have to upgrade the ordinary (non-relativistic) quantum-mechanical scattering theory as this is usually derived only in the laboratory reference frame [116], while we want to be able to transform between Lorentz-obeying reference frames.8

4.5.1 General cross section deﬁnition

In quantum physics, the cross section measures the transition probability for a speciﬁc process to happen in the scattering of two or more particles [155]. The total cross section is deﬁned as

= ℛ , (4.5.1) Φ

8The operational method of using arrays of detectors for the derivation of scattering cross section is a useful tool in our hybrid system, as we can easily transform the position of the detectors (with the transformation rules derived in Section 4.4) and use them to easily redeﬁne a reaction rate in the new reference frame. 132 Particle scattering in analogue-gravity models |

~ βi θ

Figure 4.2: Phonon scattering from an external particle as viewed in the laboratory frame. The large circle centred on the scattering event (clipped at the top and bottom) represents a 2D slice of a spherical array of detectors that is stationary with respect to the laboratory frame. The two ellipses represent 2D slices of a single array of detectors (co-moving with the particle prior to scattering) at two diﬀerent moments in time. Ticks on the circumference of both detector arrays are separated by 15◦ in the frame of that detector array. The scattering event occurs at the moment in time for which the geometric centres of both detector arrays are coincident. At some later point in time, the scattered phonon is again coincident with both detector arrays: the laboratory frame detector array detects the phonon at the detector located at an angle above the I-axis (here 17.8 ), which in the ∼ ◦ co-moving frame corresponds to the detector located at the angle 0 above the I0-axis (here 45◦, indicated by the shaded wedge in the moving detector array). In this particular example i = 0.75, corresponding to i = 4 √7 1.5 (which / ≈ is the factor by which the moving detector array is contracted in its direction of motion). Figure 4.1(a) shows the same scattering event indicated here, whereas Figure 4.1(b) shows this scattering event in the frame of the moving detector array.

where is the reaction rate (transition probability per unit time per unit volume), ℛ and Φ is the incident flux of particles. The cross section has units of area and qualitatively represents the effective cross-sectional area (hence the name) that the particle presents to the incoming beam of particles to be scattered, and it will depend on the momentum of the incoming particles. The cross section as defined in Equation 4.5.1 can be expressed more explicitly as

1 Õ Fi f 1 Õ Õ Fi f i = → = → , (4.5.2) [S| ] Φ !3 Φ !3 f kf pf ∈S Scattering cross sections 133 | where i and f represent the initial and final states, respectively (often shown | i | i with ket symbols omitted), and where is the subset of all final states that S are different from i (in order to exclude the case of no scattering). We use box normalization (volume !3) for our particles, so that the reaction rate , which is a 3 ℛ transition rate per unit volume, becomes Fi f ! , with → /

2 2 Fi f = f int i fi , (4.5.3) → ~ |h | ˆ | i| ( ) being the transition rate from an initial state i at C = to a final state f at | i −∞ | i time C = ), given some interaction Hamiltonian ˆ int. The symbol fi represents the change in total energy of the system (as measured in the lab), and the states and the interaction Hamiltonian are all defined in Schrödinger picture. For more details on how to derive Equation 4.5.3, please see Ref. [116]. The notation i represents the cross section for scattering to any final state [S| ] f (different from the initial one) given the particular initial state i . The last | i ∈ S | i form of Equation 4.5.2 is the one we will use for our calculation.

4.5.2 Cross section in the in-universe observers’ reference frame

Before going into the details of the cross section derivation, we want to clarify the behaviour of the cross section when viewed from the frame of an in-universe observer in motion with respect to the laboratory frame. Since our model lacks full (sonic) Lorentz invariance, we must reexamine some of the basic assumptions about cross sections—most especially, how it behaves when viewed from a moving frame. In ordinary QFT, which has full Lorentz invariance, the cross section behaves exactly like a cross-sectional area: it is invariant under boosts in the direction of motion of the incoming particle to be scattered (assuming a stationary target particle) but not generally in other directions [147]. We cannot rely on this being true in our model. To examine how the cross section behaves in this hybrid model in which some pieces obey sonic Lorentz symmetry, we return to its basic form, Equation 4.5.1, which is a ratio of the reaction rate to the incoming ﬂux Φ. We will examine how each of ℛ these would be perceived by an in-universe observer co-moving with the external particle.

4.5.2.1 Reaction rate

The reaction rate is a transition rate per unit volume. This can be reexpressed ℛ as the total transition probability per unit spacetime volume. 134 Particle scattering in analogue-gravity models |

A spacetime volume element is invariant under a Lorentz transformation, and thus it remains invariant when measured by in-universe observers. Thus, the denominator of is invariant. ℛ Now let us examine the numerator. The probability of some particular event happening also remains invariant, even though the different observers may disagree on the particular descriptors used to specify the event (e.g., direction, duration, etc.). If a given detector clicks, then all observers will agree that the detector clicked. At the end of a given set of experiments, all observers—regardless of state of motion—will therefore agree on the list of detectors that clicked, and how many times each detector clicked. Thus, the probability of a physical detector clicking must also be independent of an observer’s state of motion. This means that the transition probability for any given given event i f → must be invariant. A moving observer would interpret this as probability for the transition i f , where i and f are the initial and final states as interpreted by 0 → 0 0 0 the moving observer, but the numerical value of this probability would be the same for both observers. The total transition probability is just the sum of all relevant individual transition probabilities of this form. Since this sum includes all possible final states (except the initial one), all observers must agree on this value. Thus the total transition probability (numerator of ) is invariant. ℛ Combining the above two results, we see that the reaction rate is itself ℛ invariant, being the ratio of two invariant quantities: total transition probability and amount of spacetime volume.

4.5.2.2 Flux

The behavior of the flux Φ under a Lorentz transformation is the subject of much discussion, even in a fully relativistic theory such as QFT. A recent article [155] explains clearly the subtleties of this topic and provides the definitive resolution. The key results of that work also apply to our setup, and we repeat them here, along with a discussion of how our particular setup modifies them (or not). The approach will be to write the flux in a manifestly Lorentz-invariant form based on the sonic analogue to the four-current, which we denote ! =2s B , (4.5.4) =v with = being the number density of particles (in some frame) and v the three- velocity of the particle (as measured from the same frame). has units of speed times number density (or speed per unit volume), which has the intuitive interpretation of number of particles passing through a unit of area per unit of Scattering cross sections 135 | time. is a sonic Lorentz-covariant four-vector and will transform as such when changing from one in-universe observer’s frame to that of another. Crucially, notice that this is a kinematical quantity, not a dynamical one. It is based on a description of the motion of the particle and makes no reference to its particular dispersion relation.

The crucial observation from Ref. [155] is that the ﬂux for a two-particle scattering experiment can be written in an invariant form using the four-currents 1 9 and 2 for the two particles:

1p 2 2 2 Φ = 2− 1 2 1 2 , (4.5.5) s ( · ) − ( ) ( ) where 1 and 2 are the two scattering particles, the dot ( ) is the four-vector dot 2 · product (in our case, using 2s instead of 2), and = . Importantly, every ( ) · in-universe observer will agree on the numerical value of this quantity, so we only need to calculate it in one frame: the laboratory frame.

In the laboratory frame, the four-currents for the phonon and the external particle, respectively, are

2 2 © sª © sª ® ® 0 ® 0 ® s = =s ® , p = =p ® , (4.5.6) 0 ® 0 ® ® ® 2s Ei « ¬ « ¬ where =s and =p are the initial number density of the phonon and of the particle. Then,

2 s p = =s=p2 1 i , (4.5.7) · s ( − ) 2 s = 0, (4.5.8) ( ) 2 2 2 2 p = = 2 1 . (4.5.9) ( ) p s ( − i )

Plugging these into Equation 4.5.5 gives the formula for the ﬂux in the laboratory frame:

Φ = =s=p 2s Ei , (4.5.10) | − |

9 1 Reference [155] uses natural units. We have included the prefactor 2s− to get the units right in our case. 136 Particle scattering in analogue-gravity models |

10 3 For our choice of normalization, the number densities are =p = 1 ! and 3 / =s = 1 ! , so the ﬂux in the laboratory frame is /

2s Ei 1 i Φ = | − | = 2s − . (4.5.11) !6 !6 This is the value of Φ in all frames. Nevertheless, a quick conﬁrmation shows that this is indeed what in-universe observers co-moving with the particle would calculate. Simply apply the ap- 1 ) propriate Lorentz transformation to the currents, giving s 0 = − =s 2s, 0, 0, 2s 1 ) ( ) and 0 = − =p 2s, 0, 0, 0 , and plug into Equation 4.5.5. The result for Φ is p i ( ) unchanged.

4.5.2.3 Total cross section

The total cross section is the ratio of two sonic Lorentz scalars: and Φ. ℛ The numerical value of each of these quantities is therefore agreed upon by all in-universe observers, and so, therefore, is their ratio = Φ. The total cross ℛ/ section is a sonic Lorentz scalar with respect to all in-universe-observer frames moving in the I direction. Importantly, however, note that we have taken care to only assert that these quantities are invariant with respect to diﬀerent states of motion of the observers. The fact that the external particle violates the sonic Lorentz symmetry of the system means that the quantities in question are not necessarily invariant with respect to simultaneous boosts of all objects with respect to the medium. In fact, as we will discover in the coming pages, experiments that appear to in-universe observers to have the same initial conditions—assuming the observers in each case are co-moving with the particle—will nevertheless lead to diﬀerent outcomes depending on their initial velocity with respect to the medium. In the language of Lorentz-violating extensions to the Standard Model [156], the cross section is invariant with respect to observer boosts but not necessarily so with respect to particle boosts.

4.5.2.4 Diﬀerential cross section

While we have shown above that the numerical value of the total cross section is agreed upon by all observers, we are actually interested in the diﬀerential cross

10 There is much discussion about choosing a Lorentz-invariant normalization in QFT textbooks (e.g., Ref. [147]), but this is merely a convention that has some calculational and conceptual utility in a fully relativistic setting. Importantly, it is not required for obtaining physically valid results—even in a fully relativistic setting. Any normalization will do as long as it is treated consistently throughout the calculation. And that is what we do. Scattering cross sections 137 | section 3 3 cos with respect to scattering angle in the laboratory frame and / 3 3 cos with respect to scattering angle in the co-moving frame. / 0 0 We can illustrate this situation as follows. We imagine a large spherical array of detectors that is stationary with respect to the laboratory frame and centred on a phonon scattering event, as illustrated in Figure 4.2. Each detector subtends a small solid angle 3Ω , ) at a given orientation , ) as measured in the ( ) ( ) laboratory frame. Whether a particular detector has clicked or not is manifestly invariant, as is the probability of any physical detector clicking, as discussed above. So too would this reasoning apply to a spherical array of detectors co-moving with the particle, as illustrated in Figure 4.2. Each of these detectors would subtend a solid angle 3Ω , ) at a given orientation , ) as measured in the co-moving 0( 0 0) ( 0 0) frame.

The total cross section can be split up into inﬁnitesimal pieces 3, corresponding to scattering (in the laboratory frame) at angle into a narrow ring of detectors for all azimuthal angles ). We can reassemble these pieces into the total cross section:

¹ ¹ 3 i = 3 = 3 cos [ | ] , (4.5.12) 3 cos where we make explicit the final scattering angle and initial state i. Since the probability that some detector in that ring clicks is agreed upon by all observers, it is merely a question of kinematics as to how that ring appears to in-universe observers in a different state of motion. To find this, we first find 0 and i0 using Equation 4.3.8 and Equation 4.3.6, and then we note that

3 i = 3 0 i0 (4.5.13) [ | ] [ | ] by the scalar nature of and the argument above about the invariance of probabilities.

The co-moving frame description of the scattering event is therefore obtained solely by appropriately transforming the laboratory-frame kinematic quantities into the co-moving frame of the particle. From Equation 4.5.13 we get the diﬀerential cross section in the co-moving reference frame from the laboratory one [157]:

3 0 i0 3 i [ | ] = [ | ] , 0 , (4.5.14) 3 cos 0 3 cos [ ] where , is the Jacobian of the coordinate transformations between reference [ 0] 138 Particle scattering in analogue-gravity models |

frames,

3 cos 1 2 , = = − i , (4.5.15) 0 2 [ ] 3 cos 1 i cos 0 ( + 0) obtained using the transformation Equation 4.3.8. With these tools in hand, our approach is to calculate the diﬀerential cross section in the laboratory frame (since that is the frame in which we know the dynamics) and then use Equation 4.5.14 to express it from the co-moving observers’ perspective.

4.5.3 Phonons, quantized external particles, and the interaction Hamiltonian in the laboratory frame

In the laboratory frame, we assume that phonons are excitations of the analogue- gravity medium; for simplicity, we treat the phonon field as a scalar field.11 We use standard quantum mechanics to describe external particles, as this suffices to capture all of their relevant quantum mechanical degrees of freedom. Note that the only degree of freedom that we endowed our external particle with (see Section 4.4.2) is its centre-of-mass energy/momentum—we have not endowed the external particle with any additional degrees of freedom such as angular momentum. So in summary, our recipe for a quantum-mechanical interaction Hamiltonian includes:

1. First-quantized matter that describes the external particle. 2. A single excitation of a second-quantized phonon ﬁeld that describes the scattering phonon. 3. An interaction Hamiltonian depending on the phonon’s ﬁeld amplitude at the (quantized) position of the external particle.

We deﬁne the phonon ﬁeld as r 1 Õ ~2s 8k x 8k x ) x = 0k4 · 0† 4− · . (4.5.16) ˆ( ) 3 2 2: ˆ + ˆk ! / k p Its units are energy length . ( )/( )

11Treating phonons as a scalar ﬁeld is appropriate in systems that are isotropic. For systems that exhibit anisotropies, e.g., any system with a crystal structure, a vector ﬁeld description must be used to fully capture all appropriate degrees of freedom [120]. Scattering cross sections 139 |

4.5.3.1 Interaction Hamiltonian

We want to build a toy-model for phononic interaction with external particles. The Hamiltonian we will consider is

, 2 int = ) x , (4.5.17) ˆ 2 ˆ ( ) where , is the “charge” of the interaction with units of length that we can, with foresight, use to deﬁne a dimensionless coupling constant , s B , (4.5.18) 4p in terms of a reduced sonic Compton wavelength of the external particle12

~ 4p B . (4.5.19) <2s

The cross section is evaluated by standard perturbation theory with the quantized interaction term

, 1 Õ ~2s 8k x 8k x int = 0k4 ·ˆ 0k† 4− ·ˆ ˆ 2 !3 2√:: ˆ + ˆ k,k0 0 8k0 x 8k0 x 0k 4 ·ˆ 0† 4− ·ˆ . (4.5.20) × ˆ 0 + ˆk0

The Hamiltonian is discrete as we are considering quantization in a ﬁnite “sonic universe” of volume !3. Important to notice is that the interaction Hamiltonian in Equation 4.5.20 is not Lorentz invariant—neither light, nor sonic—as the operator x cannot be ˆ applied in a relativistic scenario. Hence, it is valid only for external particles. We will elaborate more in the next subsection, Section 4.5.4.

4.5.4 Internal particle cross section

We are now equipped with the relevant mathematical machinery and physical understanding to compute scattering cross sections for phonon scattering experiments involving external particles. Before we proceed to do so, we will discuss the nature of the scattering cross section for phonon scattering experiments involving internal particles, as we will make qualitative comparisons between these two types of scattering experiments in what follows.

12 Note that we are free to deﬁne the numerical quantity 4p despite the fact that the external particle is not a ﬁeld excitation. 140 Particle scattering in analogue-gravity models |

In Section 4.4.1 we demonstrated that the kinematic description of phonon scattering from internal particles takes on the same mathematical form in both the laboratory frame and the co-moving in-universe observer frame. In other words, we can refer to the kinematic description of phonon scattering from internal particles as being sonically Lorentz covariant. The types of particles that we would expect to behave in this manner would be collective excitation quasiparticles [134, 136] belonging either to the analogue gravity medium itself (in addition to the phonons, which are also collective excitation quasiparticles) or to some other analogue gravity medium with the same characteristic speed of sound to which the phonon bearing medium is coupled. Condensed-matter quantum field theory [119] is the appropriate way to study collective excitation quasiparticles, and so it is a reasonable hypothesis that the appropriate quantum description of sonically relativistic particles would therefore be given by a sonically relativistic condensed matter field theory, analogous to quantum field theory but with the speed of sound taking the place of the speed of light. There is in fact evidence to support this hypothesis: for example, both Volovik [136] and Barceló et al. [134]13 have demonstrated the emergence of quantum electrodynamics as an effective dynamical description of certain collective excitation quasiparticles within condensed matter systems. We will therefore assume that this hypothesis is correct, and so—in analogy to quantum field theory—the dynamical equations of motion governing the scattering of phonons from internal particles will be sonically Lorentz covariant. In-universe observers’ measurements of kinematics are also sonically Lorentz covariant, and so the differential cross section for phonon scattering from internal particles will trivially be sonically Lorentz invariant from the perspective of in-universe observers, akin to how the differential cross section for actual Compton scattering (described by the Klein–Nishina formula [147, 158]) is actually Lorentz invariant. When considering the differential cross section of phonon scattering from external particles we are not afforded the same convenience. We imagine our external particle to be some regular particle free of any inherent association to the analogue gravity medium: its classical description is merely that of a Newtonian particle, and its corresponding quantum description is given by regular quantum mechanics. We must therefore take a careful and considered approach in calculating the differential cross section of phonon scattering from external particles, and to this end we apply the same general procedure that we applied in calculating the co-moving in-universe observer frame description of the kinematics.

13The latter work being based, in part, on the former. Scattering cross sections 141 |

Particle Type One-Particle State Commutation Relations Inner Product One-Particle Identity Í Phonon ~k = 0† 0 s 0k, 0† = k,k ~k ~k0 = k,k I = k ~k ~k | i ˆk| i ˆ ˆk0 0 h | i 0 | i h |

¹ 8p x 1 3 · ~ I Í External p = 3 2 3 G4 x N/A p p0 = p,p0 = p p p ! / | i ! | i h | i | i h | Table 4.2: Representation of single-particle states of the phonon and external particle. The phonon is treated as single excitations (second quantized) of a sonically relativistic field—i.e., one with a dispersion relation $ = 2s :. The external particle is treated as an ordinary quantum-mechanical particle (first quantized). Our choice of normalization allows all types of particle to have the same form of the inner product and resolution of the (one-particle) identity operator. Our choice of normalization for the phonon differs from that of the usual one in quantum field theory [147]. This has no effect on observable quantities, including cross sections—see Footnote 10.

4.5.5 External particle cross section

To compute the diﬀerential cross section of phonon scattering from external particles, we apply Equations 4.5.2 and 4.5.3 to the speciﬁc experimental scenario that we have in mind, which is discussed in Section 4.3 and shown in Figure 4.1.

4.5.5.1 Initial and ﬁnal states

The quantum toy model we are considering is composed of two independent systems: the phonon and the external particle. Hence the initial and ﬁnal states of the system can be written as follows:

i = pi ~ki , f = pf ~kf . (4.5.21) | i | i ⊗ | i | i | i ⊗ | i pi f are the initial/final states of the external particle, and ~ki f are the | / i | / i initial/final states of the phonon. In Table 4.2 we illustrate the details of the states, where we have use the subscript 9 = i, f for the final and initial states In line with the interaction Hamiltonian Equation 4.5.20, states are discrete as they are defined in a finite “universe” of volume !3, and the delta functions are Kronecker-deltas. Note that we are using a different normalisation than what is usually used in literature, for example in Peskin [147]. However, this does not have any effects on measured quantities (see Footnote 10.)

4.5.5.2 Cross section derivation

We now have all the general tools we need to calculate the diﬀerential and the total cross section in the laboratory frame and in the in-universe observers’ reference frame. We will start with calculating the transition rate for Hamiltonian 142 Particle scattering in analogue-gravity models |

Equation 4.5.20, and the initial and ﬁnal states in Table 4.2:

f int i . (4.5.22) h | ˆ | i

If we plug in the expressions for the interaction Hamiltonian Equation 4.5.20, and initial and ﬁnal states we obtain , f int i = p 0 0k h | ˆ | i 2 h f| ⊗ sh | ˆ f 1 Õ ~2s 8k x 8k x 0k4 ·ˆ 0k† 4− ·ˆ × !3 2√:: ˆ + ˆ k,k0 0 8k0 x 8k0 x 0k 4 ·ˆ 0† 4− ·ˆ pi 0† 0 s. (4.5.23) × ˆ 0 + ˆk0 | i ⊗ ˆki | i

By considering the commutation relations in Table 4.2, the orthogonality of the particle states, and well known quantum mechanics relations14, the transition amplitude can be rewritten as

, ~2s 1 f i = pf,pi, ~ki ~kf (4.5.24) ˆ 3 + − h | | i 2 ! √:i :f

Given the expression Equation 4.5.3 we can rewrite the transition rate as

2 , ~2s 1 Fi f = pf,pi ~ki ~pf fi 3 + − → ~ 2 ! √:i :f ( ) 2 2 2 2 , ~2s 1 = ( 6) pf,pi ~ki kf fi ~ 4 ! :i :f + − ( ) 2 2 2 , ~2s 1 = , (4.5.25) ( 6) pf,pi ~ki kf fi ~ 4 ! :i :f + − ( ) where

ﬁ = f ~$f i ~$i + − − = f :f,:i,?i ~2s :f i ~2s :i (4.5.26) ( ) + − −

14For the full derivation see AppendixA . Scattering cross sections 143 |

If we want to re-express the cross section Equation 4.5.2 in terms of continuous variables for the ﬁnal state of the phonon, we can use the relation

Õ 1 Õ = Δ : 3 3 ~ f Δ~:f ( ) kf ( ) kf 1 ¹ !3 ¹ 33 : = 33 : , (4.5.27) 3 f 3 f ≈ Δ: R3 2 R3 ( f) ( ) so that the total cross section is

3 ¹ 1 ! 3 Õ Fi f i = 3 : → . (4.5.28) 3 f 3 [S| ] Φ 2 R3 ! ( ) pf

Equation 4.5.28 is very useful to deduce the diﬀerential cross section with respect to the scattering angle cos

3 ¹ ¹ 2 3 1 ! +∞ 2 Õ Fi f = 3: : 3) → . (4.5.29) 3 f f 3 3 cos Φ 2 0 0 ! ( ) pf

Since we have assumed that the scattering processing we are considering is co-planar and co-linear, we know that the quantity Fi f does not depend on the → angle ). Hence we can solve the integral in ) straightaway

3 ¹ 3 2 ! +∞ 2 Õ Fi f = 3: : → . (4.5.30) 3 f f 3 3 cos Φ 2 0 ! ( ) pf

Therefore the diﬀerential cross section Equation 4.5.30 becomes

3 ¹ 3 i 2 ! +∞ = 3: :2 [S| ] 3 f f 3 cos Φ 2 0 2( ) 2 Õ 1 2 , ~2s 1 3 ( 6) pf,pi ~ki ~kf fi × ! ~ 4 ! :i :f + − ( ) pf 6 3 ¹ ! ! +∞ = 2 3: :2 3 f f 2s Ei 2 | − | 0 (2 ) 2 Õ 1 2 , ~2s 1 3 ( 6) pf,pi ~ki ~kf fi × ! ~ 4 ! :i :f + − ( ) pf 2 2 1 ~2s s 4p = 1 i 2 4 ¹ − +∞ 2 1 3:f :f fi , (4.5.31) × :i :f ( ) 0 pf=pi ~ki ~kf + − 144 Particle scattering in analogue-gravity models |

Equation 4.5.31 can be further simpliﬁed by rewriting the Dirac delta as

Õ :f :0 fi = ( − ) , (4.5.32) ( ) 3fi 3:f : =:0 :0 ker f ∈ ( fi) ( / ) where ker B :0 : :0 = 0 , and as before, is the analytic expression ( fi) { fi( ) } fi for the change in the total energy of the system (i.e., the difference between the final and the initial energies of the system). The differential cross section Equation 4.5.30 is

242 ¹ 3 1 ~2s s p ∞ 2 = 3:f :f 3 cos 1 i 2 4 0 − ( ) 1 Õ :f :0 ( − ) (4.5.33) : : × i f 3ﬁ 3:f : =:0 p =p k k :0 ker f f i ~ i ~ f ∈ ( ﬁ) ( / ) + −

4.5.5.3 Scattering cross sections in the laboratory frame

We can now proceed to derive the diﬀerential cross section for phonon scattering from external particles in the laboratory frame. From the kinematic derivation

in Section 4.4.2.1 we know that we have two solutions ~:f,1 and ~:f,2, see Equa- tion 4.4.16, that satisfy the condition = tot tot = 0, see Equation 4.4.15. The ﬁ f − i diﬀerential cross section Equation 4.5.33 becomes

2 2 3 i 1 ~2s s 4p 1 [S| ] = 3 cos 1 2 4 :i i ( ) ¹ − +∞ Õ : :0 3: : ( f − |) . (4.5.34) × f f 0 3ﬁ 3:f :f=:0 :0 :f,1,:f,2 ( / ) ∈{ } The denominator is easily calculated from Equation 4.4.15

3ﬁ ~ = ~:f ~:i ?i cos <2s . (4.5.35) 3:f < − + + Scattering cross sections 145 |

By expanding the sum and considering the two solutions of ker the diﬀerential ( ﬁ) cross section becomes 2 2 ¹ 3 i 1 ~2s s 4p 1 +∞ [S| ] = 3:f :f 3 cos 1 i 2 4 :i 0 − ( )    :f :f,1  ( − ) ×  1 : : ? cos <2  < ~ ~ f,1 ~ i i s  − + +   :f :f,2  ( − )  (4.5.36) + 1 : ? ? cos <2  < ~ ~ f,2 ~ i i s  − + + 

When we perform the integral in :f we ﬁnd

2 2 3 i 1 ~2s s 4p 1 < [S| ] = 3 cos 1 i 2 4 :i ~ − ( ) :f,1 :f,1 :f,2 :f,2 ( [ p ] + [ ]) , (4.5.37) × 2 ¯ − ¯ where and are deﬁned in Equation 4.4.17. The functions : and : ¯ ¯ [ f,1] [ f,2] are the Heaviside step functions that ensure that the two solutions :f,1 and :f,2 are real and positive, as we have already seen in the kinematic Section 4.4.2.1. By inserting the two solutions from Equation 4.4.16 into Equation 4.5.37, the explicit expression for the diﬀerential cross section is

2 3 i 1 ~2s , 1 < = [S| ] 2 3 cos 1 i 2 4 :i ~ − ( ) p 2 1 , (4.5.38) × [ ¯ − ¯] [ ¯] + ( + ) (− ¯) where we have deﬁned

¯ B p . (4.5.39) 2 ¯ − ¯ p The three conditions speciﬁed by the three Heaviside step functions , ( ¯ − ¯) , and are the same as those in Equation 4.4.20. The diﬀerential cross ( ¯) (− ¯) section has units of area, as it must, since the quantity ~2s < is dimensionless. :i ~2 146 Particle scattering in analogue-gravity models |

4.5.5.4 Scattering cross sections in the co-moving in-universe observer frame

As we have seen multiple times throughout this paper, the co-moving in-universe observer frame description of the scattering event is obtained by appropriately transforming the laboratory frame kinematic quantities into the co-moving frame of the particle. We have already performed all of the heavy lifting required to obtain the differential cross section in the frame of the co-moving in-universe observer. Our final task is to transform the kinematic quantities in Equation 4.5.37 [or, alternatively, Equation 4.5.38] according to the transformations specified

in Equations 4.3.6–4.3.8. Substituting :i = :i0, where is the Doppler factor [Equation 4.2.8], and applying Equation 4.5.14, the diﬀerential cross section becomes

2 3 i 1 i , ~2s < 1 0 0 = + [S | ] 2 2 3 cos 0 1 i cos 0 4 2 ~ :i0 ( + ) ( ) p 20 0 0 0 1 0 , (4.5.40) × ( ¯ − ¯ 0) ( ¯ ) + + (− ¯ ) where ¯0 and ¯ 0 are just ¯ and ¯ re-expressed in the co-moving frame, as shown in Equation 4.4.26, and 0 is

¯0 0 = p . (4.5.41) 2 ¯0 − ¯ 0

4.5.5.5 Using Lorentz-violating sonic Compton scattering to determine absolute motion

The diﬀerential cross section written in terms of the values of quantities as measured by co-moving in-universe observers is given by Equation 4.5.40. In principle, in-universe observers could use this expression to make qualitative statements regarding their state of motion by performing several scattering

experiments on external particles with diﬀerent initial velocities Ei = i2s, using

phonons with diﬀerent initial energies ~$i0 (or equivalently ~2s :i0). To facilitate the comparison between diﬀerent energy and velocity regimes, we introduce the dimensionless quantity ~$0 ~:0 = i = i = 4 : , (4.5.42) 0 2 p i0 2s <2s that represents the ratio of the initial energy of the phonon to the “sonic rest-mass energy” that in-universe observers would think to associate to the particle. Scattering cross sections 147 |

90° 105° 75° The ratio 0 is directly analogous to a ratio 2.5 120° 60°

2. that appears in the description of actual Compton 135° 45°

1.5 scattering, which we shall here explicitly denote15 150° 30° 1. , the value of which has particular physical im- 165° 15° ζ'= 0.001 0.5 β= .0 plications. The limit 1 corresponds to Thom- 180° 0. 0 β=.35 β= .6 son scattering [159], which describes the classical 195° 345° β= .7 and non-relativistic scattering of electromagnetic 210° 330° waves from charged particles, whereas the limit 225° 315° 240° 300° 255° 285° & 1 corresponds to scenarios in which both rel- 270° ativistic and quantum theoretic eﬀects are promi- (a) In-universe co-moving frame: 0 = 0.001. nent: as a result, a full quantum ﬁeld theoretic

90° 105° 75° description is necessary in the limit 1 [160]. 1.5 & 120° 60° 1.25 In the following we demonstrate what in- 135° 45° 1. 150° 30° universe observers will measure if they were to 0.75 0.5 165° 15° ζ'=1 perform several scattering experiments for the 0.25 β= .0 180° 0. 0 same value of 0 and several values of i and vice β=.25 β=.35 versa. We provide the diﬀerential cross sections as 195° 345° viewed from the co-moving frame for two types 210° 330° 225° 315° of experiments: in one type of experiment, the 240° 300° 255° 285° 270° value of 0 is held constant while i is varied; in the other type of experiment, is held constant i (b) In-universe co-moving frame: 0 = 1. while 0 is varied. 90° 105° 75° 4. 120° 60°

135° 3. 45° Fixed 0, varying i. For this case we present the 150° 30° 2. scattering cross sections for = 0.001, 1, 1.5 . ζ'= 1.5 0 165 15 { } ° 1. ° β= .0 In Figure 4.3(a) we can already see that, even β=.1 180° 0. 0 β=.25 for a very small value of 0, the cross section β=.35 195° 345° shows a marked dependence on diﬀerent values β= .6 210° 330° of i. We can qualitatively compare this result 225° 315°

240° 300° with the Klein-Nishima differential cross section 255° 285° 270° formula [158, 161] for unpolarized photons, where it is easy—and expected—to see that there is no (c) In-universe co-moving frame: 0 = 1.5. dependency of the cross section on the state of Figure 4.3: Differential scattering cross sections for motion of the rest frame of the particle (for in phonon scattering from external particles for fixed 0 actual relativity, there is no meaningful notion in and varying i of the external particle. which different inertial states of motion differ).

15 The expression for is identical in form to that of 0 except all references to sound are replaced by their corresponding references to light. That is to say, 2 takes the place of 2s, and the frequencies $i0 and $f0 correspond to frequencies of light, not sound. 148 Particle scattering in analogue-gravity models |

90° 105° 75° For higher values of 0, another noteworthy fea- 2. 120° 60°

135° 1.5 45° ture becomes apparent: in Figure 4.3(b) and Fig-

150° 30° 1. ure 4.3(c) we can see that, as the initial velocity of β= 0.0 165 15 ° 0.5 ° ζ'= 0.001 the particle increases, the range of the scattering ζ'= 0.5 180° 0. 0 ζ'= 1.7 angle becomes smaller, eventually leading to a ζ'= 3. 195° 345° ζ'= 7. scenario in which the particle is prohibited from 210° 330° scattering outside of some given angular window. 225° 315° 240° 300° This eﬀect is due to the conditions Equation 4.4.29, 255° 285° 270° which place restrictions on the allowed angles of

(a) In-universe co-moving frame: i = 0. scattering as per Equation 4.4.30.

90° 105° 75° 3. 120° 60° 2.5 135° 45° Fixed i, varying 0. The dependency of the scat- 2.

150° 30° 1.5 tering angle on the initial energy of the phonon

1. β= 0.5 165° 15° ζ'= 0.001 becomes clearer when we consider the in- 0.5 0 ζ'= 0.5 180° 0. 0 ζ'= 1.7 universe diﬀerential cross section for two values ζ'= 3. 195° 345° ζ'= 7. of i = 0, 0.5 and several values of . In Fig- { } 0 210° 330° ure 4.4(a) and Figure 4.4(b) we can see that as

225° 315° 0 increases, the scattering angle becomes more 240° 300° 255° 285° 270° forward, tending towards the direction of motion

(b) In-universe co-moving frame: = 0.5. of the particle in the laboratory frame. This effect becomes more pronounced with increasing i, Figure 4.4: Differential scattering cross sections for as one can see by comparing Figure 4.4(a) and phonon scattering from external particles for fixed i Figure 4.4(b). of the external particle and varying 0.

4.6 Discussion

In considering phonon scattering from external particles we have restricted our considerations to that of non-relativistic quantum mechanics for all values of

0 because, by construction, our external particle is a non-relativistic quantum mechanical object. By analogy to true Compton scattering, phonon scattering from internal particles requires a full quantum field theoretic description16 because relativistic effects (with respect to sound) are important, and relativistic quantum mechanics is inapplicable because it is a fundamentally inconsistent theory [160]. We can however make qualitative comparisons between phonon scattering from internal particles and external particles in the limit 1, as in this limit 0 relativistic effects become unimportant for internal particles. Strictly speaking, any comparisons of this type should be made in the specific case for which both

16As previously discussed, a condensed matter quantum ﬁeld theory. Conclusion 149 |

types of particle are initially travelling very slowly in the laboratory frame (i 1) as it is in this limit that the energy-momentum relations for internal and external particles coincide and thus the limit in which a quantum field theoretic description should coincide with an ordinary quantum mechanical description. With this in mind, Figure 4.3(a) shows the differential cross sections for phonon scattering from external particles characterized by = 10 3: the limit 1 is respected 0 − 0 here, and while the total amount of scattering is a function of i, the overall form of the differential scattering cross section (i.e., its angular dependency) is not. That the angular dependency of the differential scattering cross section is insensitive to

i in this case is what we expect from the expected equivalence between phonon scattering from internal and external particles for low 0. Any qualitative similarities between phonon scattering from internal and external particles vanishes for higher values of 0. In Figure 4.3(b) and Figure 4.3(c) the diﬀerential scattering cross sections for various values of i are plotted for

0 = 1 and 0 = 1.5, respectively. In both cases not only does the amount of scattering vary with initial i, but the overall form of the diﬀerential scattering cross sections is sensitive to changes in i too: this is in stark contrast to what would occur for scattering from internal particles which, again, must be insensitive to the value of i due to sonic Lorentz covariance that is inbuilt into internal particles.

The effect of varying 0 for constant values of i should be expected to alter the angular distribution of scattering, even for internal particles. In real Compton scattering, forward scattering becomes preferentially favoured as the initial photon energy is increased (see, for example, [162]). The fact that there exists angular dependency in the differential scattering cross section of phonons from external particles for fixed i in Figure 4.4(a), and Figure 4.4(b) is therefore not in and of itself surprising or unexpected. With that said, for phonon scattering from external particles, increasing 0 for fixed i eventually reveals the presence of forbidden scattering angles: this has no qualitative similarity to Compton scattering, and thus no qualitative similarity to phonon scattering from internal particles.

4.7 Conclusion

Provided that in-universe observers in an analogue-gravity universe are allowed to interact with Newtonian particles external to their own medium, then the sonic analogue to Compton scattering—in which phonons scatter from these external particles—can be used by in-universe observers to infer that there must exist 150 Particle scattering in analogue-gravity models | some preferred rest frame. In all but the most restrictive cases (i.e., unless 0 ≈ and 1), external particles result in qualitatively different scattering profiles 0 than occur in fully relativistic scattering, such as Compton scattering. In the most dramatic cases, scattering from external particles results in differential scattering cross sections with forbidden angles, and as the energy of the phonon increases relative to the “sonic rest-mass energy” of the particle (0 increases) the window of allowed scattering angles becomes more tightly concentrated in the direction of the trajectory of the particle prior to scattering. Even in the cases for which scattering occurs at all angles, scattering for fixed values of 0 shows a preference towards forward scattering for increasing i.

In principle, in-universe observers could conceivably utilize phonon scattering experiments from external particles to identify not only that a preferred rest frame must exist, but speciﬁcally which frame is the rest frame of their analogue universe. The ability to resolve which frame is actually the laboratory frame is fundamentally constrained by the mass of external particles and the energies of phonons that in-universe observers have access to. If in-universe observers are only able to probe the parts of parameter space corresponding to 1 then 0 the angular distribution of scattering will not be considerably aﬀected by their state of velocity (see Figure 4.3(a)) and thus they will only be able to detect their state of motion provided that they correctly deduce the relationship between

i and the magnitude of the diﬀerential scattering cross section (that is, lower magnitudes correspond to lower i, as per Figure 4.3(a)). If, on the contrary, in-universe observers are able to probe regions of parameter space corresponding to 0 & 1, then the presence of forbidden scattering angles could be utilized to locate the medium’s rest frame. In order for in-universe observers to utilize forbidden scattering angles to their advantage, they would either have to reverse engineer the energy-momentum relation for external particles, or postulate the correct energy-momentum relation and conﬁrm it experimentally. Given that the correct energy-momentum relation for external particles corresponds to the

i 1 limit of internal particles, it is not unreasonable to think that they would eventually postulate the correct relation.

Scattering experiments performed from internal particles must ultimately be equivalent to scattering experiments performed within a truly relativistic theory due to the fact that a Lorentz symmetry (with respect to 2s) is inbuilt into both internal particles themselves and the reference frames of in-universe observers. The interpretation here is that when internal particles are used, every part of the system is sonically Lorentz invariant, and when this is true the whole analogue-gravity system can be treated as being a sonic analogue to something Conclusion 151 | like Lorentz ether theory, which is operationally indistinguishable from special relativity. Thus, when in-universe observers are only allowed to interact with internal particles, they cannot determine the presence of a preferred rest frame. It is only when the symmetry groups obeyed by the particle and the phonons are different that the analogy between our model and a Lorentz ether theory (and hence a relativistic theory) breaks down and the presence of the medium can be detected. As a final note, to in-universe observers the sonic analogue to Compton scattering from external particles constitutes a breaking of the sonic Lorentz symmetry that they would otherwise believe in without access to external objects. From this point of view, the Standard Model Extension [156] might provide a natural way to further investigate such scenarios from the perspective of in- universe observers who want to believe that the sonic Lorentz symmetry of their universe is fundamental. To highlight this point, consider that an excitation of a truly relativistic field (that is, relativistic with respect to the speed of light) would also constitute an example of an external particle from the perspective of in-universe observers. The Lagrangian density describing a real scalar field ℒ; that is actually relativistic (the subscript ; denotes that the Lagrangian is invariant under Lorentz transformations with respect to 2 the speed of light) can be written in terms of some sonically relativistic Lagrangian B (the subscript B denotes that ℒ the Lagrangian is invariant under Lorentz transformations with respect to 2s the speed of sound) with some additional term to account for the difference: K

= B . (4.7.1) ℒ; ℒ + K

Viewed this way, in-universe observers might be able to describe certain external particles as though they were external particles that were coupled to some background vector ﬁeld deﬁning a preferred frame (the rest frame of the medium). This is precisely the type of scenario that the standard model extension deals with, though in the standard model extension the Lorentz symmetry is taken to be fundamental at high energies and spontaneously broken at low energies, whereas the sonic Lorentz symmetry is emergent rather than fundamental.

Part III

Work-in-progress and additional unpublished work

FROMSOUNDCLOCKSTOTHELORENTZ TRANSFORMATION

5.1 Deﬁning four-vectors using sound clocks

In Chapter3 we explicitly demonstrated the emergence of sonically relativistic measurements of space and time within an analogue gravity system, where 5 space and time are deﬁned operationally by constructing a lattice of clocks whose operation, separation, and synchronization is tied to the sending and receiving of sound. Such a scenario is in direct analogy to that of Lorentz aether theory, and it is known that Lorentz aether theory and the theory of special relativity are operationally indistinguishable from one another. Therefore, operational kinematic measurements made by in-universe observers within a sonically relativistic setting must admit a four-vector description identical to that of special relativity, but with the appropriate replacements of the speed-of-light,

2, with the speed-of-sound, 2s. In this chapter, we provide a derivation of this. Before we continue, let us recall a few key assumptions from Chapter3 . Within a given sound clock chain we imagine there to be an observer located at every clock who can make only local measurements. By local measurements, we mean that an observer at a given sound clock in one chain can measure only the following things:

1. The reading (i.e., time) of their own clock; 2. The reading (i.e., time) of any clock within an adjacent chain that is spatiotemporally coincident with their own clock; 3. The clock number of any clock within an adjacent chain that is spatiotemporally coincident with their own clock (or alternatively: the number of clocks in another chain that have passed directly by their own clock).

For completeness we will also note that a given observer knows their own clock number within their own chain. Also note that throughout this chapter we deﬁne the fractional velocity with respect to sound of any object, , and the corresponding sonic Lorentz-factor, , to be:

E B , (5.1.1) 2s 1 B p , (5.1.2) 1 2 −

155 156 From sound clocks to the Lorentz transformation |

where E is the laboratory velocity of the object in question, and 2s is the speed of sound within the analogue gravity medium as measured from the laboratory frame (also note that the analogue gravity medium is taken to be at rest within the laboratory frame, and so the laboratory frame and the medium’s rest frame coincide).

5.1.1 Clock-readings

In Chapter3 we derived the following equation (Equation 3.4.16):

2s 1 Δ ΔC = ; C1 : C0 = ΔC : ; . (5.1.3) ( ) ( ) − ( ) 2! + 2 ( − )

This equation relates the diﬀerence in clock reading (Δ) of two clocks within the same chain (indexed by ; and :) at two diﬀerent points in time in the laboratory

(C1 and C0, separated by the duration ΔC). We restate this equation here with the following notational changes: First, we append a subscript to all quantities that can vary between sound clock chains so that when we are discussing more than one chain of sound clocks there is no confusion about which quantity th pertains to which chain. Second, we deﬁne Δ#8 := ;8 :8 (where ;8 is the ; sound th − clock in chain 8, and :8 is the : sound clock in chain 8). With these changes, Equation 3.4.16 can be written as:

2sΔC 1 Δ8 = 8Δ#8 . (5.1.4) 2!8 − 2

5.1.2 In-universe observer measurement of velocity

The operational definition of time is simply given by the reading of a clock, and so in a sonically relativistic setting in-universe observers measure time using sound clocks. The operational definition of distance is given by multiples (or fractions) of some standard length, and so in a sonically relativistic setting in- universe observers measure distance in multiples of the span between clocks. In-universe observers in a given sound clock chain can then use their operational definitions of distance and time to operationally measure the velocity of some target object. The operational velocity that in-universe observers would measure of a non-accelerating target object that is moving collinearly with respect to their sound clock chain is given by the ratio

Δ# E = 8 , (5.1.5) ˜ Δ8 Defining four-vectors using sound clocks 157 | where the symbol E is chosen to agree with the symbol used for the in-universe ˜ speed of sound (2 ) in Chapter3 , Δ# is the difference in clock number between ˜s 8 the clock that records the final position of the target object and the clock that records the initial position of the target object, and Δ8 is the difference in clock reading between the clocks that record the target object’s final and initial positions (where each clock takes its reading when the target object is spatiotemporally coincident with the clock).

Substituting Equation 5.1.4 into Equation 5.1.5, we obtain the following expression for the in-universe observer measured velocity of some target object:

2!8Δ#8 E = . (5.1.6) ˜ 2sΔC 8 !8Δ#8 −

Denote the actual laboratory velocity of the target object E9. Now consider that the relative laboratory velocity of the target object with respect to the sound clock chain is given by ΔE = E9 E8, and so in a time ΔC the sound clock chain and the − target object will have become relatively displaced in the laboratory by an amount equal to ΔEΔC. This relative displacement between the sound clock chain and the target object means that, in general, the target object will be next to diﬀerent clocks at the beginning and the end of the elapsed time ΔC; therefore, express this relative displacement as some multiple of the separation length of sound clocks within the measuring chain (that separation being ! 8): / ! ΔEΔC = Δ#8 . (5.1.7) 8

This expression can then be rearranged into a slightly more helpful form:

2 !8Δ#8 = 8 ΔEΔC. (5.1.8)

Substitute this expression into Equation 5.1.6, and deﬁne Δ B ΔE 2s. We can / 158 From sound clocks to the Lorentz transformation | then write:

22ΔEΔC E = 8 , (5.1.9) 2 ˜ 2sΔC 8 ΔEΔC − 8 2ΔE = , (5.1.10) 2 2s− 2s8Δ 8 − 2Δ = , (5.1.11) 2 1 8 9 8 − 8 − − 2 9 8 = − . (5.1.12) 1 89 − Finally, divide both sides of this expression by the in-universe observer speed of sound, recalling from Equation 3.7.1 in Chapter3 that this speed is deﬁned to have a value of 2 arm/tic. Doing so, we obtain an expression for the fractional velocity with respect to sound of the object as measured by in-universe observers:

E 9 8 := ˜ = − . (5.1.13) ˜ 2s 1 98 ˜ − This is related to the composition law for velocities from special relativity. This expression is completely general in regards to the nature of the target object that the in-universe observers are measuring the velocity of; the target object (whose fractional velocity with respect to sound is 9) could be another sound clock chain, a wooden metre stick, the crests of a passing wave, or virtually any other object. As a ﬁnal note, if in-universe observers deﬁne the in-universe Lorentz factor of the target object to be 1 B (5.1.14) ˜ q 1 2 − ˜ then using Equation 5.1.13 one can show that the following identity is true:

= 12 1 12 . (5.1.15) ˜ −

5.1.3 Parametrising coordinates in one frame in terms of coordinates in another, or how I learned to stop worrying and love the Lorentz transformation

In keeping with Chapter3 , we are going to consider a scenario in which two sound clock chains pass relative to one another on collinear trajectories; two moments in time of one such scenario can be seen in Figure 5.1. Our goal is Defining four-vectors using sound clocks 159 | to obtain a mathematical description of the coordinate system defined by one sound clock chain in terms of the coordinate system defined by another sound clock chain. Denote E1 and E2 to be the laboratory velocities of the two sound

! / 2

3 2 1 0 1 2 3 3 2 1 0 1 2 3 ! − − − − − − E ì2 E ì1

! 2 1 0 1 2 2 1 0 1 2 − − − −

! / 1 Figure 5.1: Two sound clock chains pass by one another in the laboratory, with the clocks at the origin of each chain being instantaneously synchronous. Some period of time later, the two chains have moved relative to one another. If observers in each sound clock chain parametrize their measurements of the other chain (adjacent clock numbers and clock readings) in terms of their measurements of their own chain (their own clock numbers and clock readings) then they come to determine that the coordinate systems deﬁned by clock numbers and clock readings are related via a Lorentz transformation. In the particular example shown here 1 = √5 3 0.745 (1 = 3 2) and 2 = √3 2 (2 = 2 0.866). Note / ≈ / / ≈ that the in-universe observer notion of simultaneity does not agree with the actual notion of simultaneity in the laboratory; the ﬁgure shows two actually simultaneous moments in the laboratory, but one can note, for example, that the observers in chain 1 at clock 0 and clock 2 would believe that these two distinct moments in time are actually simultaneous (as at these distinct points in time, their clock readings are identical). clock chains that are labelled chain 1 and chain 2, respectively. In keeping with this labelling scheme, 1 and 2 are the corresponding fractional velocities with respect to sound, and 1 and 2 are the corresponding Lorentz factors. To be explicit, we are going to determine the following things:

1. A parameterization of the time coordinate in chain 2—defined by the readings of clocks within chain 2—entirely in terms of operationally defined measurements made by observers in chain 1. 2. A parameterization of the spatially coordinate in chain 2—defined by the clock numbers within chain 2—entirely in terms of operationally defined measurements made by observers in chain 1. 160 From sound clocks to the Lorentz transformation |

The ﬁnal results are independent of the choice of which chain measures which; the choice to parameterize the coordinate system deﬁned by chain 2 in terms of operational measurements made by observers in chain 1 is made only to prevent any confusion in the derivation. That is to say, the entire derivation is symmetric under the swapping of the indices 1 and 2 throughout this derivation.

To begin, recall that Equation 5.1.4 links the diﬀerence in the in-universe

observer measured value of time, Δ8, at sound clocks separated by Δ#8 spacing arms when in-universe observer’s measurements are actually separated by in the laboratory by a time ΔC. We can invert this expression to obtain

2! ! ΔC = 8Δ8 8 8Δ#8 , (5.1.16) 2s + 2s which is merely a relabelled version of Equation 3.4.15 from Chapter3 . For a ﬁxed value of ΔC, we can equate the right-hand-side of Equation 5.1.16 for our chain indexed by 8 = 1 to the right-hand-side of Equation 5.1.16 for our chain indexed by 8 = 2, and in doing so we remove any explicit reference of laboratory coordinates (though, of course, laboratory coordinates are still implicitly present

in the deﬁnitions of E, 2s, and ; any reference to ! is unimportant as this is a constant prefactor on both sides of the resulting expression):

2! ! 2! ! 1Δ1 11Δ#1 = 2Δ2 22Δ#2. (5.1.17) 2s + 2s 2s + 2s

Note that this expression relating the measurements made by observers in one sound clock chain to the measurements made by observers in another sound clock chain is true for any fixed value of ΔC, even if the initial time and the final time are different. However, we wish to parametrize the measurements made by in-universe observers in one sound clock chain in terms of measurements made by in-universe observers in another sound clock chain, and we have constrained in-universe observers to only be able to make local measurements. Therefore, for what follows, not only is ΔC equal for both chains, but the laboratory times corresponding to either side of ΔC are also equal.

For the purpose of what we wish to demonstrate—that being the emergence of a four-vector description of in-universe observer coordinates, and the existence of a sonic Lorentz-transformation— it is instructive to invert this expression

for either Δ1 or Δ2. As we have already mentioned, it turns out that the choice is unimportant, however, we have previously decided to parametrize the measurements in chain 2 in terms of chain 1, and so here we invert the expression Deﬁning four-vectors using sound clocks 161 |

for Δ2. Doing so yields the following expression: " # Δ1 1 Δ#1 1 Δ#2 Δ2 = 12 1 2 . (5.1.18) 2 + 2 2 − 2 2 2 1 2

Note that this is not the best possible parametrization that one can obtain. The quantity Δ#2 can itself be parametrized in terms of Δ#1 and Δ1, and thus a parametrization of Δ2 that is entirely in terms of clock readings and clock numbers corresponding to chain 1 can be obtained. To obtain such an expression, consider the following scenario:

1. At some initial point of time, the clocks :1 and :2—belonging to chain 1 and 2, respectively—are directly adjacent to one another. At this initial point in

time, the observer at clock :1 makes note of their own time, the time on clock

:2, and the clock number corresponding to clock :2;

2. At some later point in time, the clocks ;1 and ;2—belonging to chain 1 and 2, respectively—are directly adjacent to one another. At this later point in time,

the observer at clock ;1 makes note of their own time, the time on clock ;2,

and the clock number corresponding to clock ;2;

3. After the experiment ends, the observers at clocks :1 and ;1 share their measurements (and their own clock numbers). In the laboratory frame, deﬁne spatial coordinates such that the initial measurement (i.e., the measurement made by observer at clock :1) occurs at the origin. At the instant in time corresponding to the initial measurement, clock ;1 was Δ#1 spacing arms distance away from clock :1 (Δ#1 = ;1 :1), and clock ;2 was Δ#2 − spacing arms distance away from clock :2 (Δ#2 = ;2 :2). At the instant in time − corresponding to the ﬁnal measurement, chain 1—and therefore clock ;1—has travelled a distance E1ΔC in the laboratory; similarly, chain 2—and therefore clock

;2—has travelled a distance E2ΔC in the laboratory. In the laboratory frame, the spatial coordinates of clocks ;1 and ;2 are identical when the observer at clock ;1 takes their measurements. Recalling—in the laboratory frame—that the spacing arms within a given sound clock chain are of length ! 8, we then have that the / following must be true:

! ! Δ#2 E2ΔC = Δ#1 E1ΔC. (5.1.19) 2 + 1 +

If one utilizes Equation 5.1.16 to again express ΔC in terms of measurements made by in-universe observers in chain 1 (i.e., set 8 = 1 in Equation 5.1.16), then Equation 5.1.19 can be rearranged to provide the following expression which makes no explicit reference to laboratory coordinates (again, implicit reference 162 From sound clocks to the Lorentz transformation |

exists via the deﬁnitions of E, 2s, and ):

Δ1 1 Δ#1 1 Δ#2 1 1 2 = Δ1 1 12 Δ#1 2 1 (5.1.20) 2 + 2 2 − 2 − − 2 − 2 2 1 2

The left-hand-side of this expression is just the bracketed term in Equation 5.1.18.

Substituting this into Equation 5.1.18 we remove any reference to Δ#2 and arrive

at the following parameterization of Δ2: 1 2 1 Δ2 = 12 1 12 Δ1 Δ#1 − . (5.1.21) − − 2 1 12 − Multiply both sides of this expression by the in-universe speed-of-sound, recalling that 2 B 2 arm/tic, and also recall both Equation 5.1.13—relating the in-universe ˜s observer measured velocity to laboratory velocities—and Equation 5.1.15—the definition of the in-universe observer Lorentz factor. We can then write: 2sΔ2 = 2sΔ1 Δ#1 , (5.1.22) ˜ ˜ ˜ − ˜ where we have evaluated 2s = 2 arm/tic in the right hand term in the parentheses. This expression that we have arrived at parametrizes the difference in clock readings of clocks in chain 2 entirely in terms of quantities that observers in chain 1 can operationally measure or define. No explicit reference to any outside quantity exists at all in the right-hand-side of Equation 5.1.22! Let us now make some notational changes so that we can rewrite Equation 5.1.22 in a more familiar form. With the understanding that all quantities in our expression are made by in-universe observers in chain 1, drop the tilde everywhere that it appears. Additionally, take the operational definition that time is what a clock reads, and that distance is what you measure with a ruler and apply the following relabelling:

Δ1 C, Δ2 C , and Δ#1 G. With this, we ﬁnally obtain: → → 0 → 2sC0 = 2sC G . (5.1.23) −

This equation should be quite familiar; it has the exact same structure as the Lorentz transformation for the (fully unitful1) time-component of the four-position

in special relativity (but with 2s taking the role of 2)! To obtain the full Lorentz transformation, we also need to parametrize the spatial coordinates of chain 2 entirely in terms of quantities that can be operationally measured (or deﬁned) by in-universe observers in chain 1. To this end, recall 5.1.19; this equation can be directly manipulated into the following

1i.e., where 2 is explicitly present and not set equal to 1. Deﬁning four-vectors using sound clocks 163 |

2 expression for Δ#2:

" # 2 2 1 Δ#2 = 12 1 12 Δ#1 Δ1 − . (5.1.24) − − 1 12 − Recalling Equation 5.1.12 and Equation 5.1.15—expressions for the in-universe observer measured velocity and the in-universe observer deﬁned Lorentz-factor, respectively—-this can be written as:

Δ#2 = Δ#1 EΔ1 . (5.1.25) ˜ [ − ˜ ]

If one performs the relabelling Δ#2 G , and also makes the same notational → 0 changes that led from Equation 5.1.22 to Equation 5.1.23—that is, drop the tilde everywhere and perform the relabellings Δ1 C, Δ2 C , Δ#1 G—then → → 0 → one can rewrite Equation 5.1.25 as

G0 = G EC , (5.1.26) ( − ) or equivalently, G0 = G 2sC . (5.1.27) − Now that we have a parametrization of one set of operationally deﬁned in- universe coordinates entirely in terms of another (a parametrization that makes no reference—explicitly or implicitly—to laboratory coordinates), we can explicitly demonstrate that in-universe observers’ coordinate systems are indeed related by a Lorentz transformation. The two-vector formed by 2sC and G transforms like: ! ! ! 2sC0 2sC = − . (5.1.28) G G 0 −

This is—with the exception that all instances of 2 are replaced by 2s—identically the action of the Lorentz transformation in one spatial dimension from special relativity! For a full four-vector description, observers using sound clocks would need to build entire three-dimensional lattice of clocks and rulers. Provided that in-universe observers orient their sound clock lattice appropriately, the synchronisation and calibration schemes outlined in Chapter3 would result in a shortening of the spacing arms separating clocks in the direction of motion relative to the spacing arms separating clocks in the other two dimensions (which are both perpendicular to the direction of motion). Provided that in-universe observers

2We could have presented it this way in the ﬁrst place, however, we chose to originally present this expression in the form that was given in 5.1.19 such that it made the algebra easier to follow. 164 From sound clocks to the Lorentz transformation | align their lattices properly (i.e., so that the G, H, and I axes are all instantaneously aligned at some point in time), and provided that the relative motion between two lattices is entirely within the G direction, in-universe observer coordinate − systems constructed from appropriate operational deﬁnitions of spatial and temporal measurements would obey the following transformation law:

This is—again, with the exception that all instances of 2 are replaced by 2s— identically the action of the Lorentz transformation on the (fully unitful) four- position from special relativity! LORENTZVIOLATINGSCATTERING

6.1 Notation and terminology

In this chapter, we shall see equations for both actually-relativistic (i.e.photonically- relativistic, or relativistic with respect to the speed-of-light) systems and for sonically-relativistic systems. In order to prevent confusion, we choose to adopt 6 the following notational convention within this chapter: Greek indices—such as and —shall be used to denote the components of objects that are actually relativistic (i.e., photonically relativistic), whereas Latin indices from the start of the English alphabet—such as 0 and 1—shall be used to denote the components of objects that are sonically relativistic. For example, in the expression %)% ), the 1 % four-derivatives % and % are photonically-relativistic, and thus have %0 = 2 %C ; 01 by contrast, in the expression %0 )%1 ), the four-derivatives %0 and %1 are sonically-relativistic, and thus have % = 1 % . 0 2s %C Furthermore, we shall append the subscripts B and ; to any symbols whose lack of a subscript may cause confusion. For example, throughout this chapter fractional velocities with respect to both sound and light will appear; we choose to denote these symbols B and ;, respectively. The speed of light shall always be denoted 2 with no subscript, while the speed of sound shall be denoted 2s, as has been the case throughout this thesis. As a quick recap of terminology: in Chapter4 we dubbed collective-excitation quasiparticles of the analogue-gravity medium (such as phonons) to be internal particles, while all other particles were referred to as external particles. Furthermore, we used—in reference to these two types of particles—the adjectival phrases sonically Lorentz-obeying and sonically Lorentz-violating as synonyms for internal and external, respectively. We shall adopt the same terminology throughout this chapter.

6.2 The Lorentz symmetry

Let us ﬁrst recap the Lorentz symmetry brieﬂy. The Lorentz symmetry admits a group theoretic derivation that can be obtained from four axioms (which here have been collected into two supersets): 1. The laws of physics are temporally and spatially invariant. a) Homogeneity of space; b) Isotropy of space; c) Homogeneity of time.

165 166 Lorentz violating scattering |

2. The laws of physics take the same form in all inertial frames of reference (the special relativity principle).

Note that the constancy of the speed-of-light is not axiomatic in this derivation. It is a consequence of this derivation that there must exist some characteristic speed that relates the coordinates of any pair of inertial reference frames: when this characteristic speed is taken to be positive and ﬁnite1 one obtains the Lorentz transformation2 [164–167]. The Lorentz symmetry as derived from these four axioms (though not explicitly stated as such) was ﬁrst published by Vladimir Ignatowski [165], and consequently this derivation of the Lorentz symmetry is often referred to as the Von Ignatowski theorem . These four axioms have been grouped into two supersets to highlight a particular point: respectively, the two supersets above can be viewed as encapsulating the following ideas:

1. The laws of physics are invariant under geometric transformations (a one-oﬀ, static, or non-continuous transformation). 2. The laws of physics are invariant under the physical boosting of systems (a perpetual, non-static, or continuous transformation).

Note that this particular derivation of the Lorentz transformations makes no statements about the speed of light. Rather, during the course of the derivation itself, one is forced to introduce a constant whose dimensions are that of speed; if one sets the value of this speed to be real and positive, the Lorentz transformations that we are familiar with follow naturally.

6.3 Scattering from ﬁelds with diﬀerent relativities

6.3.1 In-universe observers and external particles

In Chapter4 we determined what in-universe observers would measure of a scattering event between a sonically Lorentz-obeying particle (a phonon) and a sonically Lorentz-violating particle. In our treatment of this scattering problem, all calculations were performed from the laboratory frame due to the fact that—in the laboratory frame—the physical arguments are unambiguous (e.g., momentum and energy are conserved as we expect them to be); only after we determine the outcome of experiments in the laboratory frame do we move into the in-universe observer frame to determine what co-moving in-universe observers witness of

1There is no reason a prioi that the characteristic speed cannot be infinitely large; in fact, taking this limit returns exactly the transformations of Galilean relativity. 2There does not however appear to be any theoretical reason that this speed should correspond to the speed of light. See [163] for a discussion on the technicalities of this matter. Scattering from fields with different relativities 167 | sonically Lorentz-violating scattering experiments. We now seek to formalize such scattering experiments completely from the perspective of in-universe observers. Recall that external particles—such as subatomic particles, atoms, or even molecules—are, at least in principle, described by quantum field theory3. Internal particles also admit a quantum-field-theoretic description, albeit at an effective level4 and with the speed-of-sound in the analogue-gravity medium playing the role that the speed-of-light plays in the quantum-field-theory for a comparable external particle. With these considerations in mind, assume that we have at our disposal an external particle that admits an actual quantum-field-theoretic description. For the sake of simplicity, assume that this external particle can be modelled as being an excitation of a real scalar field, ). The Lagrangian for such a particle—in the absence of any interactions—is a familiar one that we have encountered several times already in the course of this thesis:

2 2 ! 1 < 2 2 ext = %)% ) ) , (6.3.1) ℒ 2 + ~2

2 2 2 ! 1 1 %) 2 < 2 = ) )2 (6.3.2) 2 − 22 %C + ∇ + ~2

This Lagrangian is relativistic with respect to light rather than with respect to sound, however, we could express the Lagrangian as a sum of a sonically- relativistic Lagrangian—one that in-universe observers would think to propose for a scalar ﬁeld—and some additional terms to make up the diﬀerence:

2 2 2 2 2! 1 1 %) 2 < 2 1 %) 1 %) = ) )2 , (6.3.3) ext 2 2 2 2 ℒ 2 − 2s %C + ∇ + ~ + 2s %C − 2 %C 2 2! 1 <222 1 %) 1 %) = 01 % )% ) )2 , (6.3.4) 0 1 2 2 2 2 + ~ + 2s %C − 2 %C

! 2 1 <222 22 1 %) = ©01 % )% ) )2 1 s ª . (6.3.5) 0 1 2 2 2 ® 2 + ~ + − 2 2s %C « ¬ Where, if we recall our choice of notation, the four-gradient with Latin indices

3Though (and in contrast to this footnote’s sister-footnote), in the case of atoms or molecules, a description at an effective level is far more useful and practical. 4Though (and in contrast to this footnote’s sister-footnote), internal particles are collective excitations of their medium, and their medium is made up from actually fundamental particles. Resultantly, one could—in-principle—attempt to describe an internal-particle fully with an actual (rather than an effective) quantum field theory; doing so, however, would be a fool’s (or perhaps a masochist’s) errand. 168 Lorentz violating scattering |

from the start of the English alphabet is the sonic four-gradient, i.e., % = 1 % . 0 2s %C With the exception of the final term in the Lagrangian ext, everything is written ℒ in an explicitly sonically Lorentz-covariant way. Now define a tensor b having the following representation in the frame denoted by 0: 1 © ª ® 0 0® 1 B ® . (6.3.6) 0® ® 0 « ¬ With this definition, the Lagrangian for our external particle, ext, can be further ℒ rewritten in the following way:

2 2 2 ! 1 01 < 2 2 2s 0 2 ext = © %0 )%1 ) ) 1 1 %0 ) ª . (6.3.7) ℒ 2 + ~2 + − 22 ® « ¬ As an aside, the coefficient of the final term of the Lagrangian can be viewed as being related to the Lorentz-factor for a photonically relativistic theory, which is given by: 1 ; E B r , (6.3.8) ( ) E2 1 − 22 where the subscript ; is used to remind us that this is the Lorentz factor as defined with respect to the speed of light. Using this definition, we can express the Lagrangian in a slightly more compact form.

2 2 ! 1 < 2 1 2 = 01 % )% ) )2 10 % ) . (6.3.9) ext 0 1 2 2 0 ℒ 2 + ~ + 2s l ( ) We have not done anything controversial in obtaining the particular expressions for our Lagrangian as given in Equations 6.3.7 and 6.3.9; we have taken the Lagrangian of an actually relativistic field (Equation 6.3.2), added zero to it (Equation 6.3.5), and then we have made the notational choice to define the tensor b (Equation 6.3.6) such that the entire Lagrangian can be expressed in terms of tensors—the transformation rules of which we understand within a relativistic theory. The net result, however, is that we have taken a Lagrangian (Equation 6.3.2) whose constituent components are actually Lorentz-covariant objects (that is, with respect to light) and—through some very simple mathematics—we appear to have rewritten it in a way such that it is expressed entirely in terms of sonically Lorentz-covariant objects. The Lagrangian that we started with applied to an Scattering from fields with different relativities 169 | external particle, however. We know that an external particle is not sonically relativistic, so what has gone on here? In our short derivation above, we implicitly assumed the equivalence between the in-universe observer time coordinate and the time coordinate used in the description of the Lagrangian for the external particle.5 Based on our previous considerations of in-universe observer reference frames throughout this thesis, the in-universe observer time coordinate is only equal to the laboratory (or external) time coordinate at all locations for the in-universe observer frame that happens to coincide with the laboratory frame (i.e., for in-universe observers who are at rest with respect to the analogue gravity medium), and so the specific form of the Lagrangian as given in Equations 6.3.3, 6.3.4, and 6.3.5 is only valid for in-universe observers who are at rest within the laboratory frame. With all of this in mind, we can observe that the particular definition of the tensor b that was chosen (10) in Equation 6.3.6 only applies to the laboratory frame; in all other frames the tensor b has non-zero spatial components, and in this way, the tensor b singles out the laboratory frame as a preferred reference frame. The sonically Lorentz-violating nature of the Lagrangian as given in Equations 6.3.7 and 6.3.9 is manifest in the fact that the tensor b points in a specific direction through sonic spacetime. While it is true that the external particle may not be sonically relativistic, in-universe observer reference frames are! The thing that can be demonstrated by writing the Lagrangian in terms of the tensor b (Equations 6.3.7 and 6.3.9) is that every term of the Lagrangian—and hence, the Lagrangian as a whole—transforms appropriately between different in-universe observer frames, where by appropriately it is meant under the action of a sonic Lorentz transformation. To clarify, for a fixed state of the field ), and for any pair of in-universe observer inertial frames, the components of the tensorial quantities that make up the Lagrangian will be related by the sonic Lorentz transformation that connects the pair of reference frames in question. Said another way, the Lagrangian is invariant under a passive coordinate transformation (which is what in-universe observer reference frames are related by), as one should expect. Given that the tensor b singles out the laboratory frame (or, equivalently, the rest frame of the medium), it must represent something about the analogue gravity medium itself. What, then, does the tensor b represent? As it turns out, the tensor b can be viewed as being related to the sonic four-velocity of the medium itself from the perspective of in-universe observers! To see this, first define

5This occurred when we added and subtracted % ) 2 to our Lagrangian in Equation 6.3.3 and ( 0 ) chose to use the time coordinate C rather than some other time coordinate such as C0. 170 Lorentz violating scattering |

the in-universe sonic four-velocity6 of some object travelling in the G-direction to be:

1 © ª ® 2 ˜B® *˜ = B 2s ® , (6.3.10) ˜ 0 ® ® 0 « ¬ where—as in Chapters3 and 5—a symbol with a tilde over it denotes that the quantity is operationally measured/deﬁned by in-universe observers themselves.

In this case, ˜B is the in-universe observer operationally measured sonic fractional- velocity of the object under consideration, is the in-universe observer deﬁned ˜B Lorentz factor that applies to the object under consideration, and 2s is the speed- of-sound in the medium (for which, as per previous discussions, the tilde can be omitted). Let us now write all of the in-universe observer values of quantities in terms of the laboratory frame values of those same quantities. For velocities, we can utilize Equation 5.1.13 to express in-universe observer measured values of velocities

in terms of actual laboratory velocities. Recall that, in Equation 5.1.13, 8 is the laboratory frame sonic fractional velocity of the observer who is performing some

measurement, 9 is the laboratory frame sonic fractional velocity of the object whose velocity is being measured, and ˜—which we are, in this chapter, denoting ˜B—is the operationally measured in-universe value of the sonic fractional velocity for the object being measured. Therefore, an in-universe observer whose sonic

fractional velocity in the laboratory is B (8 = B) will operationally measure the sonic fractional velocity of the medium—which is actually at rest in the lab

(9 = 0)—to be B = B. We can also show from the deﬁnition of B—denoted ˜ − ˜ ˜ in Equation 5.1.15—that = . With these expressions, the in-universe sonic ˜B B four-velocity of the medium itself is, in-terms of laboratory frame values, given by the following: 1 © ª ® 2 B® *˜ = B 2s − ® . (6.3.11) 0 ® ® 0 « ¬ This, of course, agrees with our intuition from the reciprocity of both special relativity and sonic relativity. If an observer is travelling at some velocity with respect to some object, then from the perspective of the observer, the object appears to be travelling in the opposite direction with the same magnitude of

6That is, the sonic-four velocity as defined by in-universe observers. Scattering from fields with different relativities 171 | velocity. With this in mind, let us now return our focus to the tensor b. Under a sonic Lorentz-boost from the 0 frame to the 2 frame, the components of the tensor b transform like: 2 2 0 1 = Λ 01 . (6.3.12)

In the laboratory frame the components of the tensor b are 10, as given in Equation 6.3.6. In the boosted frame corresponding to an in-universe observer who is travelling in the G-direction with sonic fractional-velocity B (with respect to the laboratory frame), the tensor b will have components 12 as given by:

B B B 0 0 1 © − ª © ª ® ® 2 B B B 0 0® 0® 1 = − ® ® (6.3.13) 0 0 1 0® 0® ® ® 0 0 0 1 0 « ¬ « ¬ 1 © ª ® B® = B − ® . (6.3.14) 0 ® ® 0 « ¬

2 From the form of *˜ in terms of laboratory frame values (Equation 6.3.11), we then have that the following is true:

2 2 *˜ = 2s1 . (6.3.15)

That is to say, from the perspective of in-universe observers, the tensor b is a unit vector that points in the direction of the four-velocity of the medium. If in-universe observers correctly posited that external particles were sonically Lorentz-symmetry breaking then they could identify the rest frame of their analogue gravity medium by identifying the frame in which the tensor b is entirely time-like (where by “time-like” it is meant in terms of their own operational deﬁnition of time7). How might in-universe observers measure the components of b in order to achieve this? One method of doing so would be by performing scattering experiments from external particles and then, by then determining how much their experiments deviate from fully sonically relativistic scattering events (i.e., how they deviate from scattering events involving internal particles), they could infer the components of the tensor b within their frame.

7Though of course, the frame in which this happens to be true is the laboratory frame, and in this particular frame the in-universe operational deﬁnition of time happens to coincide with the laboratory deﬁnition of time anyway. 172 Lorentz violating scattering |

Using the relation between the in-universe four-velocity of the medium and the tensor b, we can write the Lagrangian for an external particle in any in-universe observer frame in the following way: ! 1 <222 1 2 = 01 % )% ) )2 * 0 % ) . (6.3.16) ext 0 1 2 2 2 ˜ 0 ℒ 2 + ~ + 2 2s s l ( )

6.3.2 External observers and internal particles

In the previous subsection we outlined a simple procedure to take an actually relativistic Lagrangian (i.e., a photonically-relativistic Lagrangian) and rewrite it in the form of a sonically-relativistic part—as would be naturally proposed by in-universe observers—plus some sonically Lorentz-violating part. For the sake of completion, let us now demonstrate that the complimentary procedure can be applied to a sonically-relativistic Lagrangian; an observer in the laboratory can express the effective condensed-matter field-theory Lagrangian that applies to an internal particle in terms of an actually relativistic part (i.e., with respect to light) and some actually Lorentz violating part (also with respect to light). In this sense, a reciprocity exists between the two relativities as viewed by observers of the other relativity. As per above, we assume for simplicity that the particle under consideration can be modelled as being an excitation of a real scalar field, !. In terms of the laboratory-frame coordinate system—and in the absence of any interactions—the effective condensed-matter field-theory Lagrangian describing the field ! is: ! 1 $2 = 01 % !% ! 0 !2 , (6.3.17) int 0 1 2 ℒ 2 + 2s 2 2 ! 1 1 %! 2 $ = ! 0 !2 . (6.3.18) 2 2 2 − 2s %C + ∇ + 2s

Via the complimentary process to Subsection 6.3.1, we can rewrite this as a photonically-relativistic Lagrangian, plus the terms that make up the diﬀerence:

! 2 1 $2 22 1 %! = ©% !% ! 0 !2 1 ª . (6.3.19) int 2 2 2 ® ℒ 2 + 2s + − 2s 2 %C « ¬ Where, if we recall our choice of notation, the four-gradient with Greek indices is 1 % the photonic four-gradient, i.e., %0 = 2 %C . As per Subsection 6.3.1, define a tensor that allows us to express the last term Scattering from fields with different relativities 173 | of our Lagrangian in a way that is Lorentz-covariant with respect to different observer reference frames. In this case, we define the tensor f which in the frame has the following representation:

1 © ª ® 0® 5 B ® . (6.3.20) 0® ® 0 « ¬ With this, the Lagrangian of the internal particle can be rewritten the following way:

! 1 $2 22 2 = ©% !% ! 0 !2 1 5 % ! ª . (6.3.21) int 2 2 ® ℒ 2 + 2s + − 2s « ¬ We could also, in a complimentary way to Subsection 6.3.1, write the coeﬃcient of the ﬁnal term of the Lagrangian in terms of the sonic Lorentz-factor:

1 B E B , (6.3.22) ( ) s E2 1 2 − 2s where the subscript B is used to remind us that this is the Lorentz factor as deﬁned with respect to the speed of sound. Using this deﬁnition, we can express the Lagrangian in the following, slightly more compact form. ! 2 2 1 $0 2 1 int = %!% ! ! 5 %! . (6.3.23) ℒ 2 + 22 + 2 2 s s ( )

2 Note that for all physical materials we have that B 2 is negative, while the 2 ( ) corresponding quantity from Subsection 6.3.1— 2s —is positive for all physical ; ( ) materials. The transformation properties of the tensor f with respect to the actually Lorentz-covariant reference frames of laboratory observers are complimentary to the transformation properties of the tensor b with respect to in-universe observer reference frames. That is to say, for external observers, the tensor f transforms in the same way that the actually relativistic four-velocity of the analogue gravity medium transforms: * = 2 5 . (6.3.24)

Where * is the four-velocity of the analogue gravity medium as deﬁned by an 174 Lorentz violating scattering |

external observer in the frame, deﬁned in terms of their measured fractional

velocity of the medium with respect to light, ;:

1 © ª ® ;® * = ; 2 − ® . (6.3.25) 0 ® ® 0 « ¬ We can then express the Lagrangian of an internal particle from the perspective of any external observer frame in the following way: ! 2 2 1 $0 2 1 int = %!% ! ! * %! . (6.3.26) ℒ 2 + 22 + 222 2 s s ( ) This Lagrangian—for an internal particle, written entirely in terms of actually Lorentz-covariant tensors—has precisely the same form as the Lagrangian for an external particle as expressed entirely in terms of sonically Lorentz-covariant tensors. While the structure of the Lagrangian for an external particle as written by in-universe observers—Equation 6.3.16—appears to be identical to the structure of the Lagrangian for an internal particle as written by external observers— Equation 6.3.26—there does exist a subtle diﬀerence between the two. As has 2 2 already been mentioned, the quantities 2s and 2 possess diﬀerent signs, ; ( ) ( ) which is to say that the corresponding Lagrangians themselves possess Lorentz violating terms of opposite signs.

6.3.3 Sonic relativity, photonic relativity, and the standard-model extension

The standard model of particle physics is our current best theory incorporating both the principles of quantum mechanics with the principles of special relativity. Consequently, the standard model of particle physics is built in such a way that it is inherently Lorentz obeying from the beginning. It is, however, possible to construct a theory that limits to the standard-model of particle physics in the appropriate regimes, while also allowing one to describe Lorentz violating interactions in other regimes; this is the case in the standard-model extension (SME) [156]. The standard-model extension is the result of assuming that there exists some fundamentally Lorentz-invariant theory that allows for spontaneous Lorentz-symmetry breaking under the demand that the standard-model is the approximate low-energy effective theory that results in the limit that the Lorentz- Scattering from fields with different relativities 175 | symmetry breaking effects are small. In essence, the standard-model is contained within the standard-model extension, which is itself contained within some Lorentz obeying theory-of-everything.8 As we have seen in Subsection 6.3.1 and Subsection 6.3.2, the Lagrangians for both internal particles and external particles can be expressed—in terms of the other relativity (i.e., the relativity not obeyed by the particle itself)—as the sum of a Lorentz-covariant part and a Lorentz violating part. As will be discussed below, Lagrangians of these form appear in the standard-model extension. Given that the standard-model extension is equipped—in fact, constructed—to deal with scenarios involving Lorentz violation, and given that Lagrangian terms contained within the standard-model extension are consistent with the terms from our toy-model Lagrangian, then it would appear that the framework of the standard-model extension is a promising way to describe interactions between the internal particles of analogue gravity systems, and ordinary (or external) particles such as electrons and the likes. In this chapter we have chosen to consider particles that can be modelled as excitations of real scalar fields. As a result, the terms from the standard-model extension that are of interest to us for our current purposes are those terms that describe scalar fields and their interactions. The Higgs field is the scalar field—albeit, a complex scalar field rather than a real scalar field—that features in the standard-model, and so the Lagrangian terms from the standard-model extension that are most appropriate for our current considerations are those describing the Higgs field, ). There are two terms within the standard-model extension that involve the Higgs field—one CPT-even and one CPT-odd—and they are:

CPT odd − = 8 : )† ) H.C., (6.3.27) ℒHiggs ( )) + CPT even 1 † − = : ) ) H.C. ℒHiggs 2 )) + : )†) : , )†, ) . (6.3.28) − ) − )

The operator is the usual gauge-covariant derivative operator, and the tensorial quantities and , are the field-strength tensors corresponding to the U(1) and SU(2) gauge fields and ,, respectively. In natural units (2 = ~ = 1), the coefficient :)) is dimensionless, can have real-symmetric and imaginary- antisymmetric components , and is taken to be traceless on account of the

8The standard-model extension also allows for the possibility of charge-parity-time (CPT) violation as well as Lorentz violation, however, for our current purposes it is only important to know that the model allows for the description of Lorentz violating interactions. 176 Lorentz violating scattering |

fact that the pure trace component maintains Lorentz invariance9 . In natural

units (2 = ~ = 1), the coeﬃcients :) and :), have dimensions of mass , are traceless , and must have real-antisymmetric parts. The CPT-odd (or CPT-violating)

coefficient :) has, in natural units (2 = ~ = 1), dimensions of mass and can be an arbitrary complex number. H.C. denotes the Hermitian conjugate of the preceding term. CPT even For the purposes of comparison, treat the field ) in − as being a purely ℒHiggs real scalar field. In this case, the gauge-covariant derivative in the Lagrangian could be expanded to reveal a term of the following form in flat spacetime:

:)) %)% ). (6.3.29)

With this in mind, note that the sonically Lorentz-violating Lagrangian for an external particle as viewed by an in-universe observer contains the following term: 2 0 *˜ %0 ) . (6.3.30) This term can be rewritten in the following form:

2 0 1 2 *˜ %0 ) = *˜ %1 ) *˜ %2 ) , (6.3.31) 1 2 = *˜ *˜ %1 )%2 ). (6.3.32)

Being somewhat loose with notation, one can deﬁne a second-rank tensor B)) in the following way: 12 1 2 )) ≔ *˜ *˜ ; (6.3.33) with this deﬁnition, the sonically Lorentz-violating term in the in-universe Lagrangian for an external particle can be rewritten in the form of Equation 6.3.29:

12 )) %1 )%2 ). (6.3.34)

By inspection, our sonically Lorentz-violating Lagrangian contains a term— Equation 6.3.34—that can be expressed in the same form as the Lorentz violating term that would apply to a real scalar field as per the standard model extension— Equation 6.3.29. That is, we have the contraction between a rank-2 tensor and the product of derivatives of a scalar field. This is, of course, a rather generic structure. In order for us to have confidence in utilizing the framework of the standard model extension for our purposes, we need to demonstrate that our

9The trace component of a Lorentz-covariant tensor is, under observer Lorentz boosts, the invariant of that tensor. Scattering from ﬁelds with diﬀerent relativities 177 |

sonically Lorentz-violating tensor B)) has the same form as the corresponding

Lorentz violating tensor in the standard model extension, k)).

The components of the tensor B)) in the frame denoted by the indices 1 and 2—and in terms of the laboratory frame values of quantities—are given as follows:

1 © ª 12 ® 2 2 B® )) = B 2s − ® 1 B 0 0 , (6.3.35) 0 ® − ® 0 « ¬ 1 B 0 0 © − ª 2 ® 2 2 B B 0 0® = B 2s − ® . (6.3.36) 0 0 0 0® ® 0 0 0 0 « ¬

Inspecting the components of our tensor B)) in its most general form, we can easily note that it obeys two of the three requirements put on the tensor k)) from the standard model extension: ﬁrst, in the natural units of in-universe observers (2s = 1), the tensor B)) is dimensionless; second, the tensor B)) has real-symmetric parts and trivially obeys the requirement of possessing antisymmetric imaginary components by having an imaginary part equal to zero.

The tensor B)), however, is not traceless, while in the standard model extension the tensor k)) is. Nevertheless, this is not a problem; we can extract a traceless tensor from B)) by decomposing it into its irreducible parts, and once we have done this, we can manipulate the Lagrangian into a form that is mathematically consistent with the standard model extension via a particular choice of ﬁeld and parameter redeﬁnitions.

Consider that any rank-2 tensor )89 can be decomposed into symmetric and anti-symmetric parts in the following way:

89 89 89 ) = )( ) )[ ], (6.3.37) +

89 89 where )( ) and )[ ] are, respectively, the symmetric and anti-symmetric parts of the tensor )89. These are deﬁned by:

89 1 89 98 )( ) B ) ) , (6.3.38) 2 + 89 1 89 98 )[ ] B ) ) . (6.3.39) 2 −

The symmetric part of the tensor can be further decomposed into a trace part, 178 Lorentz violating scattering |

and a trace-free symmetric part. In ﬂat spacetime, this yields the decomposition: 89 1 89 :; 89 1 89 :; 89 ) = ) )( ) ) )[ ], (6.3.40) = :; + − = :; + where the ﬁrst term is the trace of the tensor, the second term is the trace-free symmetric part of the tensor, the third term is the anti-symmetric part of the tensor, and = is equal to the dimensionality of the vector space (= = 4 in our case).

In our particular case, the tensor B)) is completely symmetric, so we have 01 = 0. The trace of the tensor B is simply 22, so the trace part of the ( )))[ ] )) − s tensor is equal to 1 4 2201. Finally, the trace-free symmetric part of the tensor −( / ) s is:

2 1 2 ©B B B 0 0ª − 4 − ® 2 2 2 1 ® 01 23 B 0 0® ( ) 1 01 ( ) 2 − B B B + 4 ® )) 23 )) = 2s 1 ® , (6.3.41) − 4 0 0 0® 4 ® 1® 0 0 0 ® 4 « ¬ This trace-free symmetric tensor is—by construction—traceless, has real-symmetric components (and, again, trivially has antisymmetric imaginary components), and

in the natural units of in-universe observers (2s = 1) is dimensionless. This fulfils the requirements made of the tensor :)) in the standard model extension, and so, to notationally reflect the standard model extension, we define a new

tensor K)) in the following way:

01 1 01 1 23 ( ) 01 ( ) , (6.3.42) )) B 2 )) 23 )) 2s − 4

With this deﬁnition, we can express the tensor B)) in the following way:

01 1 01 = 2201 22 . (6.3.44) )) −4 s + s ))

Inserting this back into the Lagrangian for an external particle as written by an Scattering from ﬁelds with diﬀerent relativities 179 | in-universe observer, we have: " # 1 <222 1 01 1 = 01 % )% ) )2 % )% ) 01 % )% , ext 0 1 2 2 )) 0 1 0 1 ℒ 2 + ~ + 2s − 4 l ( ) (6.3.45) !  2 2 01  1  1 01 < 2 2 1  =  1 %0 )%1 ) ) )) %0 )%1 ) . 2  − 42 2 + ~2 + 2 2   ; s l s   ( ) ( )  (6.3.46)

Deﬁning the following: ! 1 1 , (6.3.47) 0 B 2 − 4 2s ; ( ) tv 2 4 2s < ; ( ) <, (6.3.48) 0 B 2 4 2s 1 ; ( ) − 01 4 01 ; (6.3.49) 0)) B 2 )) 4 2s 1 ; ( ) − the Lagrangian for an external particle as written by an in-universe observer can be further rewritten in the following form: " # 2 2 01 1 01 <0 2 2 ext = 0 %0 )%1 ) ) 0 %0 )%1 ) . (6.3.50) ℒ 2 + ~2 + ))

The choice to define the quantity 0 is mathematically equivalent to either defining a new field or to defining a new metric-tensor; regardless of which of these three terms term the prefactor in Equation 6.3.47 is absorbed into, the final Lagrangian has the exact same mathematical structure.

This Lagrangian is consistent with the Higgs sector of the standard model extension when the Higgs is taken to be a real scalar field, rather than a complex scalar field. For in-universe observers who (erroneously) come to believe that their universe fundamentally obeys the sonic Lorentz-symmetry, such a description of an external particle (described by a scalar field) would be a natural one. While the sonic Lorentz-symmetry of analogue gravity models is not actually fundamental, we have demonstrated here that the Lagrangian for an external particle can be manipulated into a form that is mathematically consistent with a formalism—the standard model extension—that is, in part, founded on the assumption that such a symmetry is indeed fundamental. It seems, therefore, that the standard model 180 Lorentz violating scattering |

extension should provide a natural way to investigate the in-universe perspective of analogue gravity systems when interactions with external particles are present. Given what we have demonstrated within this chapter so far, a natural question to ask next would be, “can other types of ﬁelds (vectorial, spinorial, etc.) external to analogue gravity systems also be put into correspondence with the standard model extension?”. While this is no doubt an important question to ask and answer, we instead choose to close this chapter with a diﬀerent consideration.

6.3.3.1 Analogue gravity and the standard model extension in the laboratory

Within the context of analogue gravity systems, is it possible to leverage the formalism of the standard model extension in such a way as to be of actual utility? At the present time, we do not have—and therefore, we can not offer—a definitive answer to this question. However, the musings that follow below may help motivate the idea that there could indeed be some utility to this approach, and so future work to this end may indeed prove worthwhile. As we have demonstrated in Section 6.3.3, the in-universe observer Lagrangian for a free photonically relativistic scalar field can be written as a linear sum of a sonically Lorentz-obeying part and a sonically Lorentz-violating part (Equation 6.3.50). In other words, in-universe observers can express the Lagrangian for an external particle in terms of the Lagrangian for some corresponding internal particle in the following way:

ext = int L.V., (6.3.51) ℒ ℒ + ℒ where int is the sonically relativistic Lagrangian of some internal particle, and ℒ L.V. is the Lorentz violating term. ℒ The separability of the Lagrangian in this way might then lead one to ask the following question: “can the results of experiments that involve external particles also be separated into a simple linear sum of Lorentz obeying and Lorentz violating parts?”. If the answer to this question is in the aﬃrmative then it might prove possible to interpolate between sonically Lorentz-violating and sonically Lorentz-obeying systems. That is to say, the following two experimental scenarios may prove to be equivalent10:

1. An experiment is performed using a sonically relativistic scalar ﬁeld; 2. An equivalent experiment is performed using a photonically relativistic scalar ﬁeld, but the resultant experimental data is corrected in the following two ways:

10In the sense that the experimental data from the two scenarios can be brought into one-to-one correspondence with one another. Scattering from ﬁelds with diﬀerent relativities 181 |

a) Any contributions to the experimental data that result from the Lorentz violating part are set to zero; b) The appropriate scaling factors are applied to any parts of the data that require it (for example, the scaling factor for masses as given in Equation 6.3.48 is applied to any part of the data involving the mass of an external particle). If this hypothetical equivalency turns out to indeed be true, then the experimental physicist who finds themselves interested in analogue gravity systems will, in principle, have at their disposal two methods by which to investigate relativistic phenomena. Internal particles could be utilized by the experimentalist to directly perform tests of sonically Lorentz-obeying systems, whereas external particles could be utilized to indirectly test sonically Lorentz-obeying systems. A first step towards answering the question of “can the results of experiments that involve external particles also be separated into a simple linear sum of Lorentz obeying and Lorentz violating parts?” would be to answer the following question “can the dynamical equations that govern systems containing external particles be separated into a simple linear sum of Lorentz obeying and Lorentz violating parts?” For a free photonically relativistic scalar field, a naïve (though not necessarily wrong) first guess might be that the answer to this question is in the affirmative; after all, the equations of motion of any system that is described by a Lagrangian can be obtained through the Euler–Lagrange equation, and the Euler–Lagrange equation acts linearly on the Lagrangian. If the Lagrangian is itself a linear sum of sonically Lorentz-obeying and sonically Lorentz-violating parts, then the Euler–Lagrange equation separates into sonically Lorentz-obeying and sonically Lorentz-violating parts: ! % ext % ext 0 = ℒ %0 ℒ , (6.3.52) %) − % %0 ) ! % int L.V. % int L.V. = (ℒ + ℒ ) %0 (ℒ + ℒ ) , (6.3.53) %) − % %0 )  !  ! % int % int  % L.V. % L.V.  =  ℒ %0 ℒ   ℒ %0 ℒ  . (6.3.54)  %) − % % )  +  %) − % % )   0   0      For a free particle, it appears that this should lead to equations of motion that are indeed separable into Lorentz obeying and Lorentz violating parts (though complications may arise in the definition and/or interpretation of the canonically conjugate momentum, and thus in definition and/or interpretation of the Hamiltonian). However, it is not entirely clear that a straightforward 182 Lorentz violating scattering |

separability of Lorentz obeying and Lorentz violating terms will always apply when specific experimental scenarios are considered. For example, in attempting to describe interacting quantum fields—such as the scattering of phonons from external particles, as per our considerations in Chapter4 —these modifications may manifest in the dynamics in non-trivial ways that ultimately spoil the naïve hope of being able separating experimental results into Lorentz obeying and Lorentz violating parts in a simple and useful way. A thorough and detailed investigation will therefore be required to determine if there is any merit to this idea. TOWARDSSONICATOMS

7.1 Sonically-relativistic quantum detectors

From our results in Chapter3 , we have shown that there exists a simple operational procedure with which to build a classical relativistic-detector in an analogue gravity universe: a collection of sound-clocks—analogous to Einstein’s light- 7 clocks—are spatially organized and have their clock-readings synchronized via an operational procedure involving the exchange of sound. The logical next-step from here is to try to develop some model for a quantum relativistic- detector—that is, a device that obeys sonic relativity and that can interact with excitations of some background quantum field, e.g., something akin to an Unruh– deWitt (UdW) detector [77, 168–170]. Equipped with some model of a quantum relativistic-detector we would be more suitably equipped to ask questions about the in-universe perspective of analogue-gravity models that are expected to—if the analogy is a good one—incorporate the effects of semi-classical or quantum gravity. At perhaps the most basic level, the type of physical system that we want is one that possesses states with a discrete energy spectrum that can be changed through interactions with sonically relativistic particles such as phonons—that is to say, we want a system for which absorbing or emitting sonically relativistic particles allows us to move up or down the energy-ladder. The energy spectrum of free particles is continuous, while the energy spectrum of spatially bound particles is always discrete; at a slightly less basic level then, the type of system that we want is a bound system that can interact with sonically relativistic particles (such as phonons) to alter the energy level of the system. If we choose phonons to be the particles that can alter the energy level of our bound system, then in some sense what we desire is a system that is analogous to an atom but for which the electromagnetic interaction is replaced by some sonic—or phonon-mediated—one. That is to say, we desire some model of a sonic atom. Do such systems exist? There certainly exist quasiparticles in condensed matter systems that can be treated, at an effective level, as being bound by phonons: Cooper pairs (within a conductor) are the obvious example. Cooper pairs are composite particles (a type of quasiparticle) that consist of a pair of fermionic particles bound into a bosonic state. The most common example of Cooper pairing is found in condensed matter systems in which pairs of electrons in a conductor are—at an effective level—bound by phonons to form a bosonic state; this is the basis for the mechanism by which superconductivity is described in Bardeen–

183 184 Towards sonic atoms |

Cooper -Schrieffer (BCS) theory [171, 172] . The phenomenon of Cooper paring is, however, quite general and applies to other fermionic systems and other binding mechanisms as well: for example, in ultracold helium-3, helium-3 atoms—which are fermionic—pair up through spin fluctuations to form Cooper pairs, and it is by this mechanism that superfluidity in ultracold helium-3 emerges [173, 174]. Cooper pairs—at least those within a conductor—seem to meet the major conceptual requirements of a sonic atom: they are a bound system and their binding interaction can be treated as being mediated by phonons. Being a bound system, Cooper pairs should possess a discrete energy spectrum, and by analogy to actual atoms and the electromagnetic interaction, interactions between Cooper pairs and phonons should provide a mechanism by which Cooper pairs can change energy states. It would appear then that—in-principle—it is possible to map the behaviour of sonic atoms onto Cooper pairs. While Cooper pairs provide an example of a physical system that appears to possess the basic features that we desire of a sonic atom, they are not necessarily the only such physical system with these properties. In order to gain some understanding on what kind of physical systems may appropriately represent a sonic atom, we propose in Section 7.2 a toy model for a sonic atom that is—to first order—a one to one analogue of a real atom. Via some back of the envelope calculations, we are able to place some experimental constraints on any physical system that might meaningfully be used as a sonic atom.

7.2 A toy model for a sonic atom

Let us proceed then with our optimistic hope that there does indeed exist some physical system that could be used as a sonically-relativistic quantum detector and, to this end, let us consider a semi-realistic toy model. The use of the descriptor “semi-realistic” is for a particular reason: this model does not, a priori, represent any particular real physical system—the particles being bound by phonons may, in principle, be anything—and so we choose to ignore certain potential problems that might arise in the attempt to experimentally realize such a system. For example, if we imagined that our model was meant to represent a Cooper pair, then the strong correlations between Cooper pairs within a Cooper pair condensate is ignored. In effect, we are treating the model of our sonic atom in isolation. Of course, we want the toy model to represent something that could actually exist in an analogue-gravity system in principle, and so we must take into account certain physical features of analogue gravity models—in particular, the existence of a minimum distance scale (the intermolecular or interatomic spacing within the A toy model for a sonic atom 185 | analogue gravity medium itself). For the sake of illustration, we are going to choose to consider—as we briefly touched on earlier—some analogue to an atom that has its constituent parts bound, in some effective sense, by phonons. We will refer to this as a sonic atom. Our sonic atom toy model possesses the following features:

• Two objects/particles are bound—at an effective level—by phonons. • The sonic atom possesses a discrete energy spectrum. • Interactions with phonons can be used to change the internal state of the sonic atom, in analogy with interactions between atoms and photons. The objects/particles that we choose to bind using phonons can be sorted into two categories1. The first category is sonically-Lorentz obeying objects, while the second category is sonically-Lorentz violating objects. To understand what we mean when we talk about an object as being sonically-Lorentz obeying or sonically-Lorentz violating in this context, let us recall the Lorentz transformation from special relativity. The Lorentz transformation can be derived from four axioms, that can be collected into two supersets: 1. The laws of physics are temporally and spatially invariant. a) Homogeneity of space; b) Isotropy of space; c) Homogeneity of time. 2. The laws of physics take the same form in all inertial frames of reference (the special relativity principle). The Lorentz transformation being derived from these four axioms is often referred to as the Von Ignatowski theorem. These four axioms have been grouped into two supersets to highlight a particular point: respectively, the two supersets above can be viewed as encapsulating the following ideas: 1. The laws of physics are invariant under geometric transformations (a one-off, static, or non-continuous transformation). 2. The laws of physics are invariant under the physical boosting of systems (a perpetual, non-static, or continuous transformation). Written this way, the first point tells us something intrinsic about the universe itself—in a sense, this tells us about some external, or global symmetry of the universe. The second point tells us something about objects within the universe, and the relation of those objects to the universe—this second point, in a sense, tells us something about the internal symmetries of a particle. 1Where when we say “category” we mean this in the commonly understood, non-technical sense of the word. That is to say, we are not discussing the categories of category theory. 186 Towards sonic atoms |

So far, all experimental indications in our own universe point towards both of these postulates being fundamentally true; resultantly, we treat the Lorentz symmetry as being fundamental. However, when we consider the sonic-relativity of analogue-gravity universes, it is clear that the first point is only true in appropriate limits—when the objects under consideration are sufficiently larger than the discretization-length of the medium such that the medium appears to be a continuum—and the second point can only be guaranteed for certain quasiparticle excitations of the medium. Since very early in this thesis, we have always—unless otherwise specificied—worked in the limit that the first of these two points is true (for example, when discussing sound propagation in an analogue gravity model we have always worked in the limit that the sound waves are characterized by a wavelength much larger than the discretization-length of the medium): that shall be no different here. For our present purposes then, when we refer to the constituent objects/particles that are used to construct our sonic atom as being sonically-Lorentz obeying/violating, what we are referring to in particular is the obedience/violation of this second point: i.e., we are discussing the nature of the internal symmetries of the particle, in particular, their energy-momentum/dispersion relation. If we imagine binding a pair of sonically-Lorentz obeying objects together with phonons to form a sonic atom, then every constituent part of the system will be sonically-Lorentz obeying, and hence, the entire system collectively will behave in a sonically-Lorentz obeying manner. If, on the other hand, one or both of the objects that are bound by phonons are sonically-Lorentz violating, then the system will not collectively behave in a sonically-Lorentz obeying manner. The difference between a sonically-Lorentz obeying and a sonically-Lorentz violating atom is highlighted diagrammatically2 in Figure 7.1 and Figure 7.2: in both cases we have, on the left, a Bohr-model like representation of a sonic atom at rest in the laboratory frame, while on the right, we have the same species of sonic atom travelling at some sonic fractional-velocity with respect to the laboratory frame.

2It is important to note that Figure 7.1 and Figure 7.2 are meant for the purposes of conceptual illustration only. The sonic analogues to the nucleus and the electron should not be thought of as hard-spheres that actually length contract when moved through their medium with a non-zero velocity. The choice to draw the sonic atoms as such is simply a result of the fact that it is diﬃcult to geometrically represent transformations on internal-symmetries like energy and momentum. As a result, a transformation that is easier to graphically depict—that is, length contraction—is substituted in place of these more abstract symmetries. A toy model for a sonic atom 187 |

%4 %4 = Λ %?

%? %? = Λ %? ì

Figure 7.1: A diagrammatic representation of a Lorentz-obeying sonic atom at rest in the laboratory frame (left) and boosted to fractional velocity (right). The sonically-Lorentz obeying nature of the phonon ﬁeld is indicated by the Lorentz contraction of the orbital of the sonic atom in the direction of motion when boosted. Similarly, the sonically-Lorentz obeying nature of the internal symmetries of the sonic-nucleus and the sonic-electron—that is, their energy-momentum relation— are indicated analogously.

%4 %4 ≠ Λ %?

%? %? ≠ Λ %? ì

Figure 7.2: A diagrammatic representation of a Lorentz-violating sonic atom at rest in the laboratory frame (left) and boosted to fractional velocity (right). The sonically-Lorentz obeying nature of the phonon ﬁeld is indicated by the Lorentz contraction of the orbital of the sonic atom in the direction of motion when boosted. On the contrary, the sonically-Lorentz violating nature of the internal symmetries of the sonic-nucleus and the sonic-electron—that is, their energy- momentum relation—are indicated by the fact that they have not been contracted when boosted. 188 Towards sonic atoms |

We have discussed Cooper pairs before, and so it is natural to wonder which category Cooper pairs would fall into. In the case of Cooper pairs as we have previously discussed them, the particles being bound—be it electrons or helium-3 atoms—are actual particles that obey real (or photonic) relativity, and as a result their internal symmetries are not sonically-relativistic. We should therefore expect Cooper pairs to be best described by a model of a sonically Lorentz-violating atom3. In fact, it is not clear that any real physical system would ever exhibit complete sonic-Lorentz symmetry. In order for this to be the case, the particles being bound by phonons would themselves be required to be collective-excitation quasiparticles [133] of the analogue-gravity medium that obey a sonically-relativistic dispersion/energy-momentum relation; this would require us to have, for example, some second kind of collective-excitation quasiparticle distinct from phonons but also possessing a Lorentz-symmetry with respect to the same speed-of-sound characteristic to phonons in our analogue-gravity medium.

7.2.1 General requirements for sonic atoms

The ﬁrst tangible bound that we can place on our sonic atom is one that we have discussed many times before: the characteristic length scale of our sonic atom should be much larger than the discretization-length of the analogue gravity medium that it is within. This requirement is made in order to suppress any sonically Lorentz-violating eﬀects that would arise from the discrete nature of our analogue gravity medium and can be characterized simply by the mathematical relation:

;sonic atom ;min, (7.2.1) where ;sonic atom is some characteristic length-scale associated with the sonic atom,

and ;min is some characteristic discretization length-scale of the analogue gravity medium within which our sonic atom resides.

What is the physical nature of the length scales ;sonic atom and ;min? In gen-

eral, the exact physical nature of the length scales ;sonic atom and ;min will be model-speciﬁc. For the speciﬁc case that we have been discussing—that of a

bound structure analogous to a hydrogen atom—one might expect ;sonic atom to characterize something like the radial distance from the centre of the sonic atom at which the sonic-electron’s probability density is highest. In an actual hydrogen atom, the Bohr radius approximately characterizes this very length

3In reality, due to the identical mass of the particles in both Cooper pair examples, Cooper pairs would actually appear to be better analogues for something like positronium than a proper atom. Do not get lost in this detail, however: the important point that we are considering is whether the model exhibits complete or only partial sonic-Lorentz symmetry, and in the case of the Cooper pairs that have been discussed, it is only partial sonic-Lorentz symmetry. A toy model for a sonic atom 189 |

scale (when the electron is in the ground-state), so we might think to call ;sonic atom the sonic Bohr-radius in this particular case. As to the length scale ;min, if our analogue gravity medium is a simple condensed matter medium with no large scale structures then the length scale ;min would be proportional to the average interatomic or intermolecular spacing of the medium.

Not only should the entire sonic atom be subject to such a constraint on length- scales, but similar constraints should also be placed on the constituent parts of the sonic atom—that is, the objects/particles that constitute our sonic analogues to the proton and the electron should themselves be characterized by length scales that are much larger than the characteristic discretization length-scale of the analogue gravity medium. What characteristic length scale is associated to the sonic analogues of the proton and the electron in our sonic atom? Assuming that our sonic-electron and our sonic-proton can be modelled as massive particles, the relevant physical length scale that should characterize them is their de Broglie wavelength. Therefore, in order for the sonic-electron and the sonic-proton to behave as though their medium is a continuum, we should expect their de Broglie wavelenghs to be much larger than the discretization length-scale of the medium. This is just the relations:

sonic-proton ;min, (7.2.2) ;min, (7.2.3) sonic-electron where sonic-proton is the de Broglie wavelength of the analogue to the proton in our sonic atom, and sonic-electron is the de Broglie wavelength of the analogue to the electron in our sonic atom.

These statements on length scales are the most general statements about length scales that we can make about sonic atoms as a whole without having to go into too many system-specific or model-specific details. These statements must be obeyed by both sonically Lorentz-obeying atoms and sonically Lorentz-violating atoms in order to suppress the sonically Lorentz-violating effects that arise for the discretized nature of any real condensed matter medium.

For the speciﬁc case of a sonically Lorentz-violating sonic atom—i.e., if external particles are used construct our sonic atom—we would also like to insist that the magnitude of any sonically Lorentz-violating eﬀects are small for values of 1, where is the fractional sonic-velocity of any external particles. 190 Towards sonic atoms |

7.2.2 An approximately one to one analogue of an actual atom

Let us insist that we have a sonic atom that is approximately (i.e., to first-order) a one to one analogue of an actual hydrogen atom. More specifically, insist that there exists an exact one to one sonic analogue to electromagnetism—or more specifically, quantum electrodynamics (QED)—from which we can construct bound structures such as a sonic analogue to a hydrogen atom.4 Is this reasonable? That such an analogue could exist is perhaps most easy to accept for the case of a fully sonically Lorentz-obeying sonic atom; in this case, all of the constituent parts of the sonic atom are quantum mechanical objects that are sonically Lorentz-obeying, and therefore there ought to be an effective condensed-matter field-theory Lagrangian underlying our toy model. We should, therefore, be able to utilize all of the relevant mathematical machinery of quantum field theory to describe our sonic atom. In such a scenario, it may well be possible to find (or engineer) physical systems whose behaviour is described by the mathematical machinery of quantum electrodynamics (QED), but with all instances of the speed-of-light replaced by the speed-of-sound. In fact, Barceló et al.have demonstrated that certain collective-excitation quasiparticles within a superfluid helium-3 condensate can indeed be described within the framework of quantum electrodynamics5, and so there does appear to be some hope that a one to one sonic analogue to an atom could, in-principle, be realized. Whether or not it is reasonable to expect that such an analogue to electromagnetism could exist for a sonically Lorentz-violating atom is something that we shall discuss later on in this chapter; for now, we assume this to be true (or at least true in certain limits). Insisting that our sonic atom is described by a one to one analogue of electromagnetism allows us to determine somewhat explicit bounds on such structures and the experimental scenarios in which they could be realized. The fact that electromagnetism permits the existence of bound structures like atoms at all—and the fact that it is possible to interact with these structures in a way that

4We insist on the sonic analogue to electromagnetism being “exact” in the sense that we demand that its mathematical structure is identical to actual electromagnetism; that is to say, the sonic analogue to QED can be represented by a * 1 abelian gauge theory. Our insistence that our ( ) sonic atom only be approximately a one to one analogue of an actual hydrogen atom should be understood to mean that the specific characteristic parameters of our sonic theory that apply to our sonic atom are not constrained to be exactly identical to the equivalent parameters in QED; for example, the sonic equivalent to the electromagnetic fine-structure constant may be different than the electromagnetic fine-structure constant (this is discussed in more detail throughout this chapter). 5It should be noted that this particular work only applies to quasiparticles that can be modelled as massless Dirac fermions; a full description involving massive fermions (such as electrons) is beyond the scope of the work. A toy model for a sonic atom 191 | doesn’t immediately destroy them—is fundamentally tied to the strength of the electromagnetic interaction: this strength is characterized by the fine-structure constant6 whose value is approximately 1 137. The electromagnetic fine-structure / constant can be expressed entirely in terms of the physical constants 4, 0, ~, and 2: respectively, these are the charge of an electron, the permittivity of free space, (the reduced) Planck’s constant, and the speed of light. In terms of these constants, is 42 = . (7.2.4) 40~2 The fine-structure constant can also be expressed more simply in terms of the

Bohr radius, 00, and the Compton wavelength of the electron, 4 . The Bohr radius is deﬁned by the following expression:

2 40~ 0 ≔ , (7.2.5) 0 2 <4 4 where <4 is the mass of the electron, and all other quantities on the right-hand- side are as previously deﬁned. The reduced Compton wavelength of the electron

44 (where 244 = 4 ) is deﬁned by the following expression:

~ 44 ≔ , (7.2.6) <4 2 where all quantities on the right-hand-side are as previously deﬁned. In terms of the Bohr radius and the reduced Compton wavelength of the electron, the ﬁne-structure constant is then simply

4 = 4 . (7.2.7) 00

By analogy with the electromagnetic interaction, there must exist some sonic

fine-structure constant B that describes the strength of the interaction between sonic-charges (e.g., the sonic-electron or the sonic-proton) and the phonon field. Furthermore, if the analogy between electromagnetism and its sonic analogue is one to one then there must exist sonic-equivalents to the above relations for the sonic fine-structure constant, where all quantities are replaced by their in-universe

6In perturbative quantum ﬁeld theory, as the relevant dimensionless coupling constant for an interacting theory tends to 0, the theory becomes a free theory (no interactions, and thus no bound structures); in the limit that the relevant dimensionless coupling constant tends to 1, the theory becomes less and less amenable to perturbation theory. If the value of the relevant, dimensional coupling constant is greater than or equal to 1, then perturbation theory cannot be used to obtain sensible results. In quantum electrodynamics (QED), is the relevant dimensionless coupling constant and the smallness of this value ( 1 137) allows for perturbative methods to obtain models ' / that accurately reﬂect reality to lowest order in for many electromagnetic processes. 192 Towards sonic atoms |

sonic equivalent. Denoting the in-universe observer measured/defined/derived value of a quantity by the usual symbol with a tilde over it, the sonic fine-structure constant would be defined by in-universe observer as follows:

2 4B B = ˜B . (7.2.8) 4 ~2s ˜0 ˜ Where 4 is the in-universe sonic-electron charge, B is the in-universe permittivity ˜B ˜0 of the sonic vacuum, and 2s is the in-universe speed-of-sound. Note that B does not require a tilde because it is a dimensionless constant and so is the same to all observers—in-universe or otherwise—in all reference frames. We have also—as we have throughout this entire thesis—worked under the assumption that ~˜ = ~. Deﬁne the in-universe observer sonic Bohr-radius 0B and (reduced) sonic Compton ˜0 wavelength for the sonic-electron 4˜B as follows:

4B~2 0B ≔ ˜0 , (7.2.9) ˜0 < 42 ˜ 4 ˜B ~ 4B ≔ , (7.2.10) ˜ <4 2s where < is the in-universe sonic-electron mass and all other quantities are as ˜ 4 previously deﬁned. With these deﬁnitions, in-universe observers have:

4 = ˜ B . (7.2.11) B 0B ˜0 For the particular case of in-universe observers who happen to be at rest within the laboratory frame (i.e., the in-universe observer is at rest with respect to the analogue gravity medium), the in-universe observer measured value of any geometric quantity is exactly equal to the actual laboratory value of that same geometric quantity. As a result, for a sonic atom that is at rest within the laboratory frame we can drop the tildes on the geometric quantities 0B and 4 allowing us to ˜0 ˜ B write Equation 7.2.11 in the following way:

4B B = B . (7.2.12) 00

Note also that, in the laboratory frame, we have that 2 = 2 . When this observation ˜s s is combined with the fact that 4˜ B = 4B for a sonic-electron that is at rest within the laboratory frame, we can use the deﬁnition of the (reduced) sonic Compton wavelength (Equation 7.2.10) to deduce that < = < for a sonic-electron that is ˜ B B at rest within the laboratory frame. A toy model for a sonic atom 193 |

For a sonic atom at rest within the laboratory frame, the ratio of the sonic ﬁne-structure constant to the electromagnetic ﬁne-structure constant can be given, rather trivially, by the following expression:

B 4B 00 = B , (7.2.13) 44 00 <4 2 00 = B (7.2.14)

All of the non-sonic quantities ( , 44 , 00, 2, <4 ) in these expressions have known B values. The sonic quantities ( B, B, 00, 2s, <) can be given reasonable values via educated guesses/reasoning, or can be inferred through their relation to those quantities whose values are known and those quantities whose values can be reasonably selected. While most of the sonic quantities are constrained in some B obvious general sense (2s < 2, B < 1, 0 “physical size of experimental system”), 0 ≤ additional and specific arguments are required to obtain experimentally useful restrictions in regards to which regions of parameter space may possibly give rise to sonic atoms as we have discussed. A sketch of some of these considerations and their implications is offered at the end of this chapter. At this point, however, B let us simply consider a specific set of reasonable values for B, 00, and 2s. For our sonic analogue to electromagnetism to yield a sonic atom that is approximately a one to one analogue of a hydrogen atom, we might reasonably expect that B . ; for the purpose’s of this back-of-the-envelope calculation, we B 2 shall specifically use B . Additionally, we might also expect that 0 & 10 00, ∼ 0 a statement that can be justified by recalling the following two things: first, the actual Bohr-radius will define an absolute minimum length scale of any reasonable analogue gravity medium; second, in order for our sonic atom to be agnostic about the existence of its medium, then the sonic Bohr-radius should be much larger than the minimum length scale of its analogue gravity medium. It seems entirely unreasonable for the length-scale requirements to be obeyed for a sonic Bohr-radius that is only a single order of magnitude larger than the actual Bohr radius, so we choose two orders of magnitude as a slightly less unreasonable lower bound.

The value 2s is obviously system specific, though it is worth noting that superfluid helium-3 provides a particularly good candidate system for our current purposes. To this end, it has previously been demonstrated [134, 136] that certain collective-excitation quasiparticles within superfluid condensate of helium-3 can be treated—at an effective level—within the framework of quantum field theory. While the work by Barceló et al. [134] does not reproduce the electrodynamics of massive particles (such as our sonic-electron and our sonic-proton would be) it 194 Towards sonic atoms | does recover a subset of QED, and this provides some suggestion that it might be possible to leverage such systems to our advantage. If we take 2s to be the speed of 2 sound within a helium-3 condensate, then we have that 2s 10 m/s [175, 176]. B ∼ Taking the values of B and 00 as stated above, we have B 1 and B 2 B 2 / ∼ 00 0 . 10− (or, alternatively, 0 00 & 10 ). With these values, Equation 7.2.13 / 0 0/ can be used to obtain the following relation:

2 4B & 10 44 . (7.2.15)

2 6 Taking 2s 10 m/s (or 2 2s = 10 ), this expression (or equivalently, Equa- ∼ / tion 7.2.14) can then be used to obtain the following:

Provided that the sonically Lorentz-violating eﬀects related to external particles are negligible for objects with 1 within the laboratory frame, then all of these relations should (to ﬁrst order in for the sonic-electron) apply to both internal and external particles. It is worth noting then that for the particular values that have been chosen, external particles that could be utilized as a sonic-electron include helium atoms themselves: the mass of the helium-3 atom is close to 5498 electron masses, while the mass of the helium-4 atom is just slightly greater than 7296 electron masses. It seems then that it may be possible to form a sonic atom in a condensate of helium-3 using one of the helium-3 atoms itself as the sonic-electron. Part IV

Retrospective and outlook

CONCLUSION

The inception of this thesis was a single question: “What devices or observers experience the relativity of analogue gravity models?” To tackle this problem, we decided that our aim should be to develop in- principle experimentally-realizable operational models for devices/detectors that constitute observers in analogue gravity models; the ultimate end goal of 8 this endeavour being the development of models with which one could probe analogue gravity models from an “in-universe” perspective—that being, in a way that maps naturally to our probing of our own universe. This task of developing models for detectors in an analogue gravity universe was split up into two major subtasks. First, develop an operational model (or class of operational models) for a classical detector that would naturally respect the relativity inherent to analogue gravity models. Second—once we had an operational model for a classical detector—develop an appropriate operational model for a quantum detector: in analogy to our classical detector, this too should also naturally respect the relativity inherent to analogue gravity models, however, it should also be able to interact with the natural quantum objects inherent to such systems too, such as phonons. The development of an operational model for classical detectors in analogue gravity models was approached in Chapter3 in which we described sound- clocks—analogous to Einstein’s light-clocks—and showed that, under certain restrictions, one-dimensional chains of sound-clocks can be constructed in such a way that they constitute simple sonically-relativistic reference frames. This idea was expanded on in Chapter5 in which we showed that—again, under certain restrictions—a four-vector description of a sonic analogue to spacetime can be obtained from a three-dimensional lattice of sound clocks. These results show that, at least in principle, devices that constitute classical observers and that respect the relativity inherent to analogue gravity models can indeed be constructed. The development of an operational model for quantum detectors in analogue gravity models has proven to be somewhat more complicated. As we previously stated, the original requirements that we wanted to impose on such an object is that it must respect both the relativity and quantum excitations natural to an analogue gravity system; in particular, we wanted to consider objects that would be able to act analogously to an Unruh–DeWitt detector, a model of a particle detector that has been used to demonstrate interesting theoretical results in the areas of curved spacetime quantum ﬁeld theory and relativistic quantum information theory. Unruh–DeWitt detectors, as typically considered, have two

197 198 Conclusion |

distinct energy levels, with transitions between these energy levels driven through interactions with a massless scalar field; in certain analogue gravity models (or, at least, in certain limiting cases), phonons can be treated as a massless scalar field, and so a sonic Unruh-DeWitt detector should be an appropriate model for a phonon detector. A two-level system that can interact with the phonon field to change energy states has the qualitative feeling of some analogue to an atom (or an atom-like system) where the field binding the two constituent parts of the structure is the phonon field, rather than the electromagnetic field. The development of an operational model for quantum detectors in analogue gravity models then, effectively, becomes the question: “what is an appropriate operational model of a sonic analogue to an atom (or a sonic-atom)?”

The goal of having some bound system that obeys both quantum mechanics and the sonic relativity inherent to analogue gravity models should simply lead us to some kind of sonic analogue for quantum ﬁeld theory; something like a condensed matter ﬁeld theory in which the characteristic speed is that of sound

rather than light. While this is mathematically a trivial swap to make (2 2s) this → does not in-and-of-itself tell us anything about which physical systems behave in the desired ways, and which particular objects—which particular quasiparticles, for example—exhibit the properties that we might desire. The parameter space of hypothetical sonically-relativistic quantum field theories is simply far too large to explore in a bottom-up approach in any kind of sensible and directed manner, and so, to make any progress on finding a system (or systems) to act as candidates for a sonic-atom, we decided that we needed to take a more top- down approach; we decided that we needed to ask if specific systems can be mapped to a sonically-relativistic quantum field theory. In a further effort to make our research goals less nebulous and more concrete, our previous question of “what is an appropriate operational model of a sonic analogue to an atom (or a sonic-atom)?” was further refined to be, “what objects can appropriately be bound with phonons?”. In the most ideal case, all of the constituent components of a sonic-atom would be ‘natural’ to the analogue gravity universe itself—i.e., quasiparticles that arise from collective excitations of the medium—such that they all inherit the desired symmetries (namely, covariance under sonic Lorentz-transformations). But again, without having a specific system in mind that admits several distinct types of quasiparticle excitations, all we would be doing by investigating such a hypothetical scenario is making quantum field theory calculations with all instances of the speed-of-light being swapped for the speed-of-sound; this would be unlikely to get us any closer to our goal of being able to propose an actual in-principle experimentally-realizable quantum detector. In order to have a concrete model to work with, we decided to focus on modelling a system in which two ordinary quantum mechanical particles/objects (i.e.regular non–sonically-relativistic particles) are bound into an atom-like structure by phonons. Such particles/objects would not have degrees-of-freedom that obey sonic Lorentz-transformations—for example, the energy and momentum of these particles would transform under Galilean boosts in the velocity regimes that we would be considering—and so these objects would themselves be sonically Lorentz-violating. While the phonon field binding these objects would transform appropriately under sonic Lorentz-transformations, we expected the overall collective sonic-atom to be sonically Lorentz-violating, due to the sonically Lorentz-violating nature of the analogues to the nucleus and the electron. At face value, this seems quite bad; such a model of a sonic-atom violates half of the requirements that we had for a sonically-relativistic quantum detector (!), and therefore, this system cannot be directly mapped to a sonically-relativistic quantum field theory. Nevertheless, this may turn out to be an acceptable approach in light of the existence of the standard-model extension, a framework that—in principle—should allow us to map between Lorentz-violating and Lorentz-obeying theories. We expect that should be able to model a Lorentz- violating sonic-atom using the Lorentz-violating quantum electrodynamics (QED) sector of the standard-model extension (where we make the swap 2 2s), and → so, on the advice of a colleague1, we set out to investigate the sonic analogue to a Lorentz-violating version of the most simple light-matter interaction in QED: Compton scattering. These considerations led to the work in Chapter4 . Chapter6 sketches an outline for how one might express the toy-model of phonon scattering from Newtonian particles in an analogue gravity model (as first seen in Chapter4 ) in terms of the standard-model extension. We show that the Laboratory frame Lagrangian for such a scattering process can be manipulated into a form that has terms consistent with the standard-model extension. We hope to pursue this idea further in the near-future. Some supplementary calculations on sonic-atoms are included in Chap- ter7 . Here we show—via some fairly straightforward back-of-the-envelope calculations—that there are some fairly stringent bounds placed on sonic-atoms if such structures are to arise from a direct sonic analogue to QED.

1The wonderfully insightful Sundance Osland Bilson-Thompson

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220 Part V

Appendix

GENERALTRANSITIONPROBABILITY DERIVATION A

From the deﬁnition of cross-section in Equation 4.5.2, or equivalently from Equation 4.5.28, we see that we need to evaluate the amplitude

f int i , (A.0.1) h | ˆ | i where the initial and ﬁnal states are, respectively,

i = pi ~ki , f = pf ~kf , (A.0.2) | i | i ⊗ | i | i | i ⊗ | i and where the states for external particles are deﬁned in Table 4.2. We consider the states represented in momentum space, so the phonon initial and ﬁnal states can be written as

, 1 Õ ~2s f int i = h | ˆ | i 2 !3 2√:: × k,k0 0 8k x 8k x 8k0 x 8k0 x pf s 0 0kf 0k4 ·ˆ 0† 4− ·ˆ 0k 4 ·ˆ 0† 4− ·ˆ pi 0† 0 s, h | ⊗ h | ˆ ˆ + ˆk ˆ 0 + ˆk0 | i ⊗ ˆki | i , 1 Õ ~2s = 2 !3 2√:: × k,k0 0 8 k k0 x 8 k k0 x pf s 0 0kf 0k0† 4 ( − )·ˆ 0† 0k 4− ( − )·ˆ pi 0† 0 s, h | ⊗ h | ˆ ˆ ˆk0 + ˆk ˆ 0 | i ⊗ ˆki | i , ~2s Õ 1 = 2 !3 2√:: × k,k0 0 8 k k0 x 8 k0 k x pf s 0 0kf 0k0† 4 ( − )·ˆ 0† 0k4− ( − )·ˆ pi 0† 0 s. (A.0.4) h | ⊗ h | ˆ ˆ ˆk0 + ˆk0 ˆ | i ⊗ ˆki | i | {z } (∗)

223 In the term we have swapped k and k as these are dummies variables. Using (∗) 0 the commutation relations mentioned in Table 4.2 we obtain

, Õ ~2s 1 8 k k0 x f int i = pf s 0 0kf 20k† 0k k,k0 4 ( − )·ˆ pi 0k† 0 s, h | ˆ | i 2 !3 2√:: h | ⊗ h | ˆ ˆ 0 ˆ + | i ⊗ ˆ i | i k,k0 0

, Õ ~2s 1 8 k k0 x = pf s 0 0kf 20k† 0k4 ( − )·ˆ pi 0k† 0 s 2 !3 2√:: h | ⊗ h | ˆ ˆ 0 ˆ | i ⊗ ˆ i | i k,k0 0 | {z } (∗∗) ! 8 k k0 x p f 0 0 k k , k 4 ( − )· ˆ p i 0 † 0 . (A.0.5) + h | ⊗ sh | ˆ f 0 | i ⊗ ˆki | is | {z } (∗∗∗) We evaluate ﬁrst: (∗∗)

8 k k0 x = 2 pf 4 ( − )·ˆ pi s 0 0kf 0† 0k0† 0 s. (A.0.6) (∗∗) h | | i h | ˆ ˆk0 ˆ ˆki | i

8k x Considering that 0p0† = 0† 0p p,p , and that 4 ·ˆ p = p ~k , becomes ˆ ˆp0 ˆp0 ˆ + 0 | i | + i (∗∗) = 2 pf pi ~k ~k0 s 0 0† 0kf kf,k 0† 0k k,ki 0 s (∗∗) | + − h | ˆk0 ˆ + 0 ˆki ˆ + | i

= 2pf,pi ~k ~k kf,k k,ki , (A.0.7) + − 0 0 and becomes (∗ ∗ ∗)

= pf pi 0 0k 0† 0 k,k (∗ ∗ ∗) | sh | ˆ f ˆki | is 0

= pf,pi kf,ki k,k0 . (A.0.8)

The amplitude Equation A.0.1 is therefore

      , ~2s Õ 1   f int i = 2pf,pi ~k ~k kf,k k,ki pf,pi kf,ki k,k . (A.0.9) h | ˆ | i 2 !3  + − 0 0 + 0  2√::0   k,k0 | {z } | {z }  1 2   ( ) ( )  Evaluating 2 ﬁrst we obtain ( )

, ~2s Õ 1 2 = pf,pi kf,ki k,k0 ( ) 2 !3 2√:: k,k0 0 , ~2s Õ 1 = p ,p k ,k . (A.0.10) 2 !3 f i f i 2: k

224 This divergent term represents the case of no scattering. Since we are only interested in the set of ﬁnal states that do not contain the initial states (i.e., we S are excluding the case of no scattering), we will not consider this term. Term 1 ( ) instead is

, ~2s Õ 1 1 = 2pf,pi ~k ~k0 kf,k0 k,ki ( ) 2 !3 2√:: + − k,k0 0 , ~2s 1 = pf,pi ~ki ~kf . (A.0.11) 3 + − 2 ! √:i :f

The amplitude is therefore

, ~2s 1 f int i = pf,pi ~ki ~kf . (A.0.12) ˆ 3 + − h | | i 2 ! √:i :f

The transition rate [Sakurai:1167961] for an initial state i at time and a ﬁnal | i −∞ state f at time ) is | i

2 2 Fi f = f int i ﬁ , (A.0.13) → ~ |h | ˆ | i| ( ) where the states and the interaction Hamiltonian are deﬁned in Schrödinger picture. Substituting Equation A.0.12 into Equation A.0.13 gives the explicit form of the transition rate:

2 , ~2s 1 Fi f = pf,pi ~ki~kf fi 3 + → ~ 2 ! √:i :f ( ) 2 2 2 2 , ~2s 1 = ( 6) pf,pi ~ki ~kf fi ~ 4 ! :i :f + − ( ) 2 2 2 , ~2s 1 = . (A.0.14) ( 6) pf,pi ~ki ~kf fi ~ 4 ! :i :f + − ( )

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