Oindrila Ganguly Department of Physics University of Calcutta September 2016

CORRECTEDCOPY

LORENTZINVARIANCEVIOLATION:ASTUDYINTHE CONTEXTOFEFFECTIVETHEORIES

Thesis submitted for the degree of Doctor of Philosophy (Science)

Physics (Theoretical) by oindrila ganguly

Department of Physics University of Calcutta September 2016

LORENTZINVARIANCEVIOLATION:ASTUDYINTHE CONTEXTOFEFFECTIVETHEORIES

oindrila ganguly

An Exploration of Spacetime Symmetries September 2016 Oindrila Ganguly: Lorentz invariance violation: A study in the context of effective theories, An Exploration of Spacetime Symmetries, © Septem- ber 2016 “You give much and know not that you give at all." — Khalil Gibran, ‘The Prophet’

Dedicated to Tattu, who taught me to look beyond.

ABSTRACT

Exact invariance of spacetime under Lorentz boosts, whether it be local or global, corresponds to a scale free nature of spacetime, locally or globally, because arbitrarily high boosts expose extremely short distances. It is possible that spacetime has a short distance cutoff of the order of the Planck length lPl and consequently a preferred frame! A novel microscopic structure of spacetime should then emerge at such resolutions. Yet, till date, there is no complete theory that can explain physics near the Planck scale, reproduce our observable low energy world consistently and make experimentally verifiable predictions. However, there is a strong support for the hypothesis that Planck scale physics can induce observable violation of Lorentz invariance. An effective field theory based phenomenological model by My- ers and Pospelov breaks boost invariance in flat spacetime through a modified Lagrangian density with new Planck suppressed third order derivative terms characterised by a constant background vector. This leads to modified non linear dispersion relations with cubic contributions to the momentum. In such theories of Lorentz violation, we 1 identify a fairly general class of field configurations (of spins 0, 2 and 1) which preserve invariance under infinitesimal Lorentz transformations and also satisfy equations of motion yielding cubic dispersion relations similar to those found earlier. These restricted fields are defined as functions of the fixed background vector in such a way that background dependence of the dynamics of the physical system is not manifest. We demonstrate that these fields can provide a field basis for the realisation of Lorentz algebra and allow the construction of a Poincaré invariant symplectic two form on the covariant phase space of the theory. Lorentz non invariant theories may alternatively be modelled by analogue systems. Sound waves in a classical, irrotational, barotropic, inviscid fluid propagate like scalar fields in a curved Lorentzian geometry. But if the fluid is viscous, Lorentz invariance of the analogue spacetime is destroyed. A draining bathtub flow gives rise to an analogue spacetime qualitatively similar to the spacetime outside a Kerr black hole. Theoretically, acoustic superradiance or superresonance has previously been shown to occur here. We explore how acoustic Lorentz violation due to dispersive and dissipative effects of viscosity affects superresonance from this fluidic analogue of a rotating axisymmetric black hole. We also discuss avenues of improvement of our result and possible diversification to other systems.

vii Lorentz invariance violation: A study in the context of eﬀective theories

Oindrila Ganguly

Abstract

Exact invariance of spacetime under Lorentz boosts, whether it be local or global, corresponds to a scale free nature of spacetime , locally or globally, because arbitrarily high boosts expose extremely short distances. It is possible that spacetime has a short distance cutoff of the order of the Planck length lP l and consequently a preferred frame! A novel microscopic structure of spacetime should then emerge at such resolutions. Yet, till date, there is no complete theory that can explain physics near the Planck scale, reproduce our observable low energy world consistently and make experimentally verifiable predictions. However, there is a strong support for the hypothesis that Planck scale physics can induce observable violation of Lorentz invariance. An effective field theory based phenomenological model by Myers and Pospelov breaks boost invariance in flat spacetime through a modified La- grangian density with new Planck suppressed third order derivative terms characterised by a constant background vector. This leads to modified non linear dispersion relations with cubic contributions to the momentum. In such theories of Lorentz violation, we identify a fairly general class of field 1 configurations (of spins 0, 2 and 1) which preserve invariance under infinitesimal Lorentz transformations and also satisfy equations of motion yielding cubic dispersion relations similar to those found earlier. These restricted fields are defined as functions of the fixed background vector in such a way that background dependence of the dynamics of the physical system is not manifest. We demonstrate that these fields can provide a field basis for the realisation of Lorentz algebra and allow the construction of a Poincaré invariant symplectic two form on the covariant phase space of the theory.

PUBLICATIONS

Some of the research leading to this thesis have appeared previously in the following publications:

Journal articles

[1] Oindrila Ganguly. ‘A realisation of Lorentz algebra in Lorentz violating theory’. In: Eur. Phys. J. C72 (2012), p. 2209. doi: 10. 1140/epjc/s10052-012-2209-5. arXiv: 1206.1695 [hep-th]. [2] Oindrila Ganguly, Debashis Gangopadhyay, and Parthasarathi Majumdar. ‘Lorentz-preserving ﬁelds in Lorentz-violating theories’. In: Europhys.Lett. 96 (2011), p. 61001. doi: 10.1209/0295- 5075/96/61001. arXiv: 1011.1206 [hep-th].

Conference proceedings

[3] Oindrila Ganguly, Debashis Gangopadhyay, and Parthasarathi Majumdar. ‘A discussion on Lorentz preserving scalar ﬁelds in Lorentz violating theory’. In: Proceedings, International Confer- ence on Modern Perspectives of Cosmology and Gravitation (COS- GRAV 12). Vol. 405. 2012, p. 012015. doi: 10.1088/1742-6596/ 405/1/012015.

Your hearts know in silence the secrets of the days and the nights. But your ears thirst for the sound of your heart’s knowledge. You would know in words that which you have always known in thought. You would touch with your ﬁngers the naked body of your dreams. — Khalil Gibran, ‘The Prophet’

ACKNOWLEDGMENTS

Here ends the journey begun many years ago with a lot of hope but no path in sight. A narrow way emerged somehow, and brought me to the present day when I am penning down my acknowledgements for my PhD thesis. I am still only a novice and have the same wonder at everything I see, as when I embarked on this journey, but may be am a little bit more mature. I thank everyone who have participated in and shared the travails of this expedition. Firstly, I owe my parents heartfelt gratitude for letting me live my dream. I know that I have tested their patience by managing to be re- markably unsuccessful compared to everyone around me! So, it must have taken a lot of effort from them to keep faith and stand by me. I thank my supervisor, Prof. Debashis Gangopadhyay, for being supportive, patient and lenient. Every discussion with him used to be an invigorating experience, as I came back laden with more diverse information than I had originally gone looking for. He is very strict about the duties of students but is at the same time concerned about their fair treatment. His straight forward attitude is admirable I thank Prof. Parthasarathi Majumdar, my joint supervisor, for being a wonderful advisor, teacher and friend to me. To refer to him just as my joint supervisor is misleading. Our association dates back to 2003, when I ﬁrst heard him lecture at the ‘Summer School’ or- ganised by Indian Association for the Cultivation of Science, Kolkata. He is one of the best teachers I have ever come across. No one can probably inspire me and make me realise my abilities like him. But I must have been his most exasperating student, not as obedient as he would have liked me to be. Yet, I respect him sincerely and I shall always remain thankful for having him as one of my advisors. Here, I should also mention my heartfelt gratitude to my thesis commit- tee members, specially Prof. Soumitra Sengupta for his critical inputs and innovative suggestions. I thank Tamoghna for being my friend. He has worked tirelessly from the very beginning for our dream mission, Aloha, an initiative to protect the native Indian dogs. He has always believed in me and also spoilt me by attending to my odd wishes. He is a person I can rely on in difﬁcult times. Though Nirupam has requested me not to, I thank Nirupam for coming back to India and easing responsibilities

xi off me just to help me concentrate on my doctoral studies. He has been what a friend is meant to be. I loved to discuss physics with him and enjoyed it the most when starting from some simple little physics problem, we eventually got stuck somewhere and then spent hours sorting things out. He also participated enthusiastically with me in my work on analogue gravity, digging up various ingenious ideas to resolve the complications. I shall not thank him any more because, I do not know how to. There is my sister, who must be reading this to check whether I have said nice and proper things about her. Yes, I do thank Mitthi earnestly for always pestering me about approaching deadlines. But, most importantly, in spite of her young age, she took upon herself a number of responsibilities to make my life smooth. Without her help I would have found it difﬁcult to carry on. In this regard, I am also grateful to Gaurab, Surasree, Aryadeep and Anirban. I am indebted to my institute Satyendra Nath Bose National Cen- tre for Basic Sciences (SNBNCBS), Kolkata for providing a beautiful natural environment for students to pursue their studies in. I joined there as an Integrated PhD student and have spent eight years in the campus. I miss the campus and whenever I walk past the gates, I feel a tinge of pain at not having the permission to just walk in as before. I refrain from writing down separately the names of students and staff who have made the days at SNBNCBS memorable from fear of making the acknowledgement too long. But, Sayani is an exception. I thank her and wish her the very best in life. The college I attended, before coming to SNBNCBS, I would rather forget. But, school before that comprised the best years of my life. I spent ﬁfteen years, from nursery to class twelve, at the same school, Bidya Bharati Girls’ High School and cherish every moment there. The acknowledgement for my thesis will remain incomplete if I do not write about our physics teacher, Sarmistha Mitra. It was a thrilling experience to be in her class. Her teaching was amazing. It was then that I made up my mind to become a physicist. She gave me and my friend Poulomee ‘Six Easy Pieces’ and ‘Six Not So Easy Pieces’ by Richard Feynman, to read in class eight. By class ten, both of us were proud owners of Feynman’s ‘Lectures on Physics’. Naturally, we could not understand many things but we read and reread with help from Sarmistha Aunty. In fact, I must separately thank the three volumes of Feynman’s ‘Lectures on Physics’ because whenever I am confused, I go back to them and am rewarded with a fresh insight into any topic. There are two other books, Roger Penrose’s ‘The Road to Reality: A Complete Guide to the Laws of the Universe’ [88] and J. B. Hartle’s ‘Gravity: An Introduction to Einstein’s General Relativity’ [52] that have proved in- valuable while writing the thesis. I have borrowed from them heavily when giving an overview of black hole radiation in chapter 4.

xii Finally, a disclaimer: I have not thanked here in writing the animals of our campus and around, who have made my life complete, given me company when I was alone and taught me about life itself. World would have been a better place if we learnt to live from the animals. I hold close to my heart every moment shared with them and those shall forever remain my happiest times.

xiii

CONTENTS i phenomenological models of lorentz violation 1 1 introduction3 1.1 Theory of relativity 3 1.2 Special theory of relativity 4 1.3 General theory of relativity 7 1.4 Lorentz violation - a window into quantum gravity 8 1.5 Parameterisation of Lorentz violation 10 1.6 Phenomenological model of Lorentz violation 10 1.7 Experimental bounds on Lorentz violation 12 1.8 Lorentz preserving fields 13 1.9 Effect of Lorentz violation on ‘superresonance’ 14 2 lorentz preserving fields in lorentz violating theory 19 2.1 Restoring Lorentz Invariance of action 21 2.1.1 Spin 0 fields 21 1 2.1.2 Spin 2 field 24 2.1.3 Maxwell (spin 1) field 25 2.2 Evaluation of Dispersion Relation 27 2.2.1 Scalar field 27 2.2.2 Spinor field 28 2.2.3 Vector field 29 2.3 Discussion 29 2.4 Possible Application in Cosmology 30 3 realisation of lorentz algebra in lorentz violating theory 33 3.1 Canonical formalism in the presence of higher derivatives 34 3.1.1 Non-relativistic systems with finite degrees of freedom 34 3.1.2 Relativistic continuous systems 35 3.2 Myers Pospelov theory 36 3.2.1 Constant timelike background vector 36 3.2.2 Constant spacelike background vector 38 3.3 Overview of covariant phase space formulation 39 3.4 Symplectic structure with Lorentz preserving fields 40 ii testing lorentz violation in analogue systems 41 4 effect of analogue lorentz violation on superresonance 43 4.1 Black hole physics in acoustic analogue models 43

xv xvi contents

4.2 Lorentz violation in acoustic analogue model of gravity 45 4.3 Radiation from gravitational black holes 46 4.4 Derivation of the acoustic metric 49 4.4.1 Case of the inviscid fluid 49 4.4.2 Case of a viscous fluid 52 4.5 Derivation of acoustic dispersion relation 53 4.6 Acoustic superradiance in a viscous fluid 54 4.7 Discussion 63 5 conclusion 67 5.1 A summary on Lorentz preserving fields 67 5.1.1 Criticism and outlook 69 5.2 Lorentz violating effects on superresonance 70 5.2.1 Observational prospects of radiation from fluidic black hole 71 5.2.2 Gravity waves and Bose Einstein condensates 71 5.2.3 Inertial frame dragging in an analogue model 72

iii appendix 75 a appendix 77 a.1 Derivation of Bernoulli equation 77 a.2 Derivation of Equation 113 in chapter 4 78

bibliography 83 LISTOFFIGURES

LISTOFTABLES

Table 1 Lorentz violating Lagrangian 12 Table 2 Phase space variables 1 37 Table 3 Phase space variables 2 38

LISTINGS

ACRONYMS

xvii

Part I

PHENOMENOLOGICALMODELSOFLORENTZ VIOLATION

INTRODUCTION 1

The workings of nature do not appear to be random. Instead, they strongly indicate that underlying patterns govern natural processes from the tiniest scales to the cosmological ones. Physics is a search for the fundamental patterns of nature, a lookout for a logically consistent structure that can explain and predict their evolution. Inspite of the efforts of thousands of people over thousands of years, this endeavour to ﬁnd order amidst the chaos is far from complete. Many remarkable and ingenious ideas have together charted parts of nature’s complex jigsaw puzzle but there remain corners where the pieces stubbornly refuse to ﬁt in.

1.1 theory of relativity

One of the most beautiful realisations of physics is the principle of relativity. It says that physical laws should appear the same to all inertial observers by which we mean an observer in whose frame all non- uniform motions can be attributed to some external force. Galilei was probably the ﬁrst person to conceive the idea centuries ago though Newton is credited for giving it a rigorous form. He also deduced the formula for transition from one set of inertial coordinates to another and these transformations were designed to agree with the observed fact that all inertial observers measure the same length between two points in space which appeared Euclidean. The Euclidean distance between two nearby points expressed in a Cartesian coordinate system is given by,

dl2 = dx2 + dy2 + dz2 = d~x2 .(1)

It remains invariant under the Galilean transformation equations re- Newton’s laws are lating two mutually parallel Cartesian reference frames, say K and invariant under K’: Galilei’s transformation rules 0 as long as the force x = x − vt depends only on y0 = y spatial distance but 0 not on velocity or its z = z derivatives. t0 = t .

Here (x, y, z, t) and their primed partners denote the three spatial coordinates and one temporal coordinate of a point in the K and K’ frames respectively when K’ moves with a velocity ~v = v xˆ along the x axis. The origins of the two frames are taken to coincide at t = 0 = t0.

3 4 introduction

As required, Galilei’s transformation rules left the laws of mechanics i. e. Newton’s laws of motion unchanged.

1.2 special theory of relativity

Almost two centuries after Newton, in 1865, the basic equations of electromagnetism were put forward by the great physicist Maxwell, thereby unifying into a beautifully simple mathematical scheme the laws of electric field, magnetic field and light. Solutions to Maxwell’s equations gave light waves or electromagnetic waves which had the surprising property that they propagated in vacuum with a fixed speed. The Dutch astronomer De Sitter was also able to show, from observation of double stars, that the velocity of light could not depend on the velocity of the emitter [39, 88]. Thus, it appeared that light propagated in vacuum with the same constant speed in all inertial frames! This led to a grave conflict. A constant speed of light in vacuum in all inertial frames contradicted with the Galilean rule for addition of velocities. Moreover, if Maxwell’s equations were indeed general laws of nature then according to the principle of relativity, they should have the same form in all inertial reference frames. But Maxwell’s equations were definitely not covariant under Galilean transformations. Understandably, the fault must lie with one of the propositions. Now, it was logically plausible that Maxwell’s equations were true only in a special reference frame and did not really constitute a general law of electromagnetism. Light waves would then not be required to propagate with the same speed in all inertial frames. But no experiment could provide any evidence in support of this assertion. On the contrary, theoretical and observational experience in this domain justified the belief in Maxwell’s equations, of which the constancy of the speed of light was a necessary consequence, as the correct laws of electrodynamics. Prominent theoretical physicists were, therefore, more inclined to question the universality of Galilean transformation rules. A careful consideration of the existing perceptions on measurement of time and space intervals helped Einstein realise that Galilean transformations were based on two unjustifiable hypotheses borrowed from classical mechanics: notion of absolute time: Time interval between two events is independent of the state of motion of the body of reference

notion of absolute spatial distance: The spatial interval or distance between two events is independent of the condition of motion of the reference frame. Once these two hypotheses were dropped, the classical theorem of addition of velocities became invalid and the long standing conﬂict 1.2 special theory of relativity 5 disappeared. Einstein, in 1905, put forth the special theory of relativity whose postulates are:

1. Physical laws stay the same in all inertial frames.

2. Light propagates in vacuum with a universal speed relative to any inertial observer.

The constant speed of light in vacuum is denoted by c. A new set of relations between the space and time coordinates of an event relative to the two inertial systems K and K’ follows.

x00 = γ(x0 − βx1) , x01 = γ(x1 − βx0) , (2) x02 = x2 , x03 = x3 .

Here (x0 = ct, x1, x2, x3) and (x00 = ct0, x01, x02, x03) are the time and space coordinates of an event measured by the two mutually parallel Cartesian frames K and K’ respectively. As before, observer K’ moves 1 along the x axis of K with a velocity ~v = vxxˆ. The new quantities β and γ are deﬁned as, v β ≡ c 2 − 1 γ ≡ (1 − β ) 2 .

K and K’ are said to be boosted with respect to each other and the Einstein was relations given in Equation 2 between the original and boosted co- completely unaware ordinates are known as the Lorentz transformations or Lorentz boosts of the coordinate transformation rules because independently, before the advent of special relativity, Voigt derived by Lorentz in 1887 and more completely Larmor in 1898 and Lorentz in 1899 had and others and separately written down these rules in their attempt to ﬁnd how co- derived them ordinates must transform to make sure that light travels in vacuum independently from the postulates of with the same constant speed in all reference frames. Under these special relativity. transformations, the form of Maxwell’s equations was naturally pre- served. However, an actual picture of the physical world this led into, remained obscure to Lorentz though Poincaré came very close to it. To sum things up, special relativity says that general laws of nature must be so constituted that they are covariant under Lorentz transformations. This is a precise mathematical demand of special theory of relativity on every natural law. Later, in 1908, Poincaré showed that the Lorentz transformations formed a group which became known as the Lorentz group. When translations and spatial rotations are also included, the group that results has been named after Poincaré. 6 introduction

Geometric perspective on special relativity

Despite the wonderful physical insight of Einstein and the profound contributions of Lorentz and Poincaré, the theory of special relativity was not yet complete until Minkowski provided his fundamental and revolutionary viewpoint spacetime [88]. In the four dimensional spacetime M of Minkowski, the invariant interval between two nearby ds2 is invariant events is given by under Lorentz 2 2 2 2 boosts, spatial ds = c dt − d~x . rotations and spacetime In natural units the speed of light in vacuum is set to unity i. e. c = 1, translations. so that

ds2 = dt2 − d~x2 .(3)

The Minkowski metric ηµν = diag(1, −1, −1, −1) has a Lorentzian signature (+ − −−). 1 From Equation 3, it is clear that unlike Euclidean space, the geometry of this spacetime is hyperbolic and the locus of all points equidistant from some other point is a hyperboloid. In this spacetime, two events are separated by an interval whose square can be either positive, negative or zero depending on which the interval is referred to as timelike, spacelike or null (or lightlike) respectively. In the tangent space Tp at point p, the locus of points null separated from it describes a cone referred to as the light cone or Often the term null null cone at the point p. A null cone or light cone is a fundamental cone is used for the structure of spacetime and is responsible for determining its causality tangent space structure. Any light cone has two parts to it - the past and the future structure while the term light cone is cones. The past cone represents the history of light rays converging reserved for the at the point p while the future cone is traced by light rays emanating actual locus of light from p. The Minkowski spacetime (M) is ﬂat and so its null cones are rays in the all parallel to each other. spacetime. In the spacetime picture, the motion of any particle, whether in inertial motion or not, is represented by a curve called the world line of the particle. A massive particle at p has a world line whose tangent vector at every point lies within the light cone, while light rays have world lines along the cone.

Symmetries of Minkowski spacetime

The symmetries of spacetime constrain and simplify the form of a physical law. Any transition involving magnitudes of space and time

2 2 2 1 We may equivalently write ds = −dt + d~x . The metric then is ηµν = diag(−1, 1, 1, 1) and it bears the Lorentzian signature (−, +, +, +). This choice of metric, referred to sometimes as the majority positive metric has the advantage that it makes the metric on the spatial hypersurface positive deﬁnite. On the other hand, the majority minus metric is preferred by many because it makes timelike intervals positive. Between the two it is just a matter of choice. In this thesis, we shall work with the metric of Equation 3. 1.3 general theory of relativity 7 of an event, that leaves physical laws unchanged is deﬁned to be a symmetry of spacetime. The Minkowski spacetime of special relativity has 10 continuous symmetry transformations

• Translations along 3 space and 1 time direction

• Rotations about the 3 spatial axes

• Lorentz boosts about 3 spatial directions

These ten symmetry transformations together constitute the inhomogeneous Lorentz group or Poincaré group and a subgroup of the Poincaré group that does not include translations is called the ho- mogenous Lorentz group or simply the Lorentz group. Lorentz symmetry plays a fundamental role in development of relativistic mechanics and quantum ﬁeld theory.

1.3 general theory of relativity

Soon, Einstein realised that gravitation could not be consistently formulated within the static spacetime structure of special relativity. The equivalence of inertial and gravitational mass struck him as intriguing and led him to frame the principle of equivalence. It can be stated as [100]: principle of equivalence: "In an arbitrary gravitational ﬁeld no local non gravitational experiment can distinguish a free falling non rotating system (local inertial system) from a uniformly moving system in the absence of a gravitational ﬁeld."

In other words, it says that locally a freely falling non rotating frame in a gravitational field is identical to an inertial system. Einstein came to the conclusion that gravitation cannot be regarded as a force because to a non-rotating observer who is falling freely, there is no ‘gravitational force’. Instead, gravitation manifests itself in the form of spacetime curvature. So, global inertial frames cannot be realised in the presence of a gravitational field though local ones are possible and relative to these frames special relativity remains valid. However, the metric is now a field varying continuously over finite regions of spacetime. This extended theory of special relativity and gravitation is known as the general theory of relativity. Mathematically, the spacetime of general relativity is a pseudo- Riemannian manifold with a metric field bearing the signature (+ − −−). By definition, such a manifold is said to be Lorentzian. The metric varies continuously over finite regions of the manifold but locally, at a point p, it is always possible to introduce normal coordinates so that the metric takes the normal form gµν(p) = ηµν(p) = diag(1, −1, −1, −1) and its components have vanishing first derivatives ∂λgµν(p) = 0. That is, in general relativity, local Lorentz frames 8 introduction

In curved spacetime, exist and there is only local Lorentz invariance. Just as in flat Minkowski the distinction spacetime, here too a null cone can be constructed in the tangent between null cone plane T at every point p such that the null cones are everywhere tan- and light cone is p more apparent than gent to the light cones described by the light rays. Massive or massless in flat spacetime. particles respectively follow timelike and null geodesics of the curved manifold. This is the geometric or kinematic picture of general relativity where the motion of matter is determined by the curvature of spacetime. But there is another part to it, the dynamics which describes how the metric field depends on the energy momentum distribution of all matter in the Universe. This dependence is governed by Einstein’s field equations - the heart of general relativity, that relate the curvature of spacetime to the energy momentum tensor of matter. Thus, Einstein’s general theory of relativity gives an elegant dynamic description of the Universe where both matter and geometry influence and are influenced by each other.

Beyond classical general relativity

Black holes constitute a class of exact solutions of vacuum Einstein equations. However, laws of black hole thermodynamics formulated by Bekenstein [19] and Bardeen, Carter, and Hawking [15] compel us to look beyond classical general relativity at a microscopic theory of spacetime. Moreover, general relativity predicts that black hole horizons contain within them spacetime singularities i. e. points where the curvature of spacetime diverges (e. g. at r = 0 in a Schwarzschild black hole described in Schwarzschild coordinates). This appearance of a singularity also indicates the inadequacy of Einstein’s classical theory of gravitation. Thus, it becomes imperative that a a quantum theory of gravitation be developed. But this endeavour has not only proved to be extremely challenging theoretically, it has also been severely hindered by the lack of observational and experimental guidance. The typical energy scale at which quantum gravitational effects are believed to become signiﬁcant is the scale at which gravitational action becomes of the order of the quantum of action h . This occurs 19 at the Planck scale MPl ≈ 1.22 × 10 GeV which is well beyond our reach even in the foreseeable future.

1.4 lorentz violation - a window into quantum gravity

Luckily, since the end of the last century, physicists have realised that the actual situation is a little less grim. A few models of gravitation beyond general relativity and quantum gravity tentatively hint that there can be low energy relic signatures of these models leading to deviations from standard predictions of general relativity and Stan- dard Model of particle physics, in special situations. One such ‘win- 1.4 lorentz violation - a window into quantum gravity 9 dow into quantum gravity’ is the violation of Lorentz invariance. This expectation is motivated by the idea that nature may have a short distance cutoff, hypothesised to be of the order of the Planck length lPl. Arbitrarily high boosts reveal arbitrarily short lengths and ultra high energies. If there is indeed a minimum allowed length scale then it becomes imperative that there will also be a preferred frame which can cause Lorentz violation. It is crucial to emphasise that none of the existing candidate theories of quantum gravity necessarily require or convincingly argue in favour of violation of Lorentz symmetry. But it is a proposition in favour of which these theories together provide enough evidence to motivate a search for observable constraints on the possibility of the same. However, interest in Lorentz violation is quite old. The possibility has been dug up time and again to complement certain ideas of cosmology, particle physics and gravitation. A number of developments during the closing years of the last century have recalled a flood of interest to research on Lorentz violation [60]. A systematic extension of the standard model of particle physics incorporating all possible renormalisable (i. e. of mass dimension 6 4) Lorentz violating operators by Colladay and Kostelecky [34] provided a way of calculating, within effective field theory, the observable consequences for many experiments and setting observational bounds on the Lorentz violating parameters appearing in the Lagrangian. On the observational side, the AGASA experiment [102] detected ultra high energy cosmic ray (UHECR) events beyond the Greisen, Zatsepin and Kuz’min (GZK) proton cutoff [110]. Coleman and Glashow developed a per- turbative framework to accommodate departures from exact Lorentz invariance based on which they explained the missing GZK cutoff [31]. Their analysis also suggested new observable consequences of renormalisable, isotropic Lorentz violation which can be interpreted in terms of differences in maximum attainable velocities of different particles [32]. This was followed by the work of Amelino-Camelia, Ellis, Mavromatos, Nanopoulos, and Sarkar where it was proposed that the sharp high energy signals of gamma ray bursts (GRBs) could reveal Lorentz violating photon dispersion suppressed by one power −3 of energy over the mass M ∼ 10 MPl, extremely close to the Planck mass [9]. The effective field theory was further extended by Myers and Pospelov to include non-renormalisable dimension five operators in the Lagrangian that produce 1 suppressed Lorentz violating cu- MPl bic modifications to the dispersion relation. This imposed certain relations between the Lorentz violating parameters for different helici- ties using which stringent bounds on Planck scale interactions from terrestrial experiments could be derived [82]. Thus, together with improvements in observational reach and precision, these developments led to a burst of activity in research on the subject. Above all, apart from any theoretical motivation, it is always interesting in its own 10 introduction

right to study the limits of validity of any symmetry, even more so in the case of Lorentz transformations because the Lorentz group is non-compact and an inﬁnite volume of the group always remains untested. For a more complete account of the history of research on Lorentz violation, the reviews [60, 72, 76] are excellent resources.

1.5 parameterisation of lorentz violation

The pertinent issue now is how are we to incorporate departure from Lorentz symmetry in existing theory? The standard special relativistic dispersion relation

p2 = m2

of a free particle of mass m and four momentum p is a statement of Lorentz invariance. Hence, it should be altered if there is deviation from Lorentz symmetry . The deformed dispersion relation of a certain massive particle can be written as [72]

E2 = |~p|2 + m2 + f(E, |~p|; µ, M) (4)

where E, ~p and m are the energy, three momentum and mass of the particle respectively. The four momentum p = (E, ~p). µ is a particle physics mass scale and M stands for the relevant quantum gravity 19 scale. Usually, it is assumed that M ∼ MPl ≈ 1.22 × 10 GeV. The function f(E, |~p|; µ, M) can be expanded in powers of |~p| (at high energies m << |~p| and E ≈ |~p|, though E, |~p| < MPl). Higher order terms in the dispersion relation are quite common when dealing with wave propagation through a material medium – low momentum, long wavelength modes travel in phase with the same velocity while high momentum modes with wavelength comparable to the atomic scale of the medium become sensitive to the microscopic structure and propagate at velocities depending on the value of the momentum.

1.6 phenomenological model of lorentz violation

Corrections to standard dispersion relations are suppressed by the The preferred frame Planck mass but tiny contributions can be magnified into a significant is usually taken to one when dealing with high energies E (E << MPl), long distances of coincide with the signal propagation and certain peculiar reactions. rest frame of the cosmic microwave For any meaningful prediction, a dynamical model of Lorentz vi- background because olation is necessary. The main support for Lorentz violation comes it is the only natural from the hope that a short distance cutoff may appear in quantum candidate for a gravity. This implies the presence of a preferred frame and threatens cosmic preferred frame. boost invariance. It is important to realise that any preferred frame breaks spatial anisotropy. Often, to simplify matters, the assumption 1.6 phenomenological model of lorentz violation 11 of rotational invariance is introduced because any constraint on pure boost violation is in all probability also a constraint on violation of boost plus rotational invariance. But it is not possible to construct a theory that breaks only rotational invariance while preserving invariance under Lorentz boosts. Various theoretical approaches have been adopted in this regard - some consider violation of only boost invariance, while others allow in addition rotational symmetry breaking. Most often Lorentz violation is described in the framework of an effective field theory but there are also propositions that do not fit into an effective field theory like doubly special relativity or Lorentz violation on D−branes. An obvious requirement for any such model is that they should agree with present experimental results and extant theory at low enough energies. It is best to refer to the review articles [60, 72, 76] and references therein for detailed discussion on different phenomenological and theoretical models of Lorentz violation and current observational constraints on effects predicted by them. During the course of the research presented in this thesis, we restricted our attention to Lorentz violation introduced into standard effective field theory through non-renormalisable operators, assuming separate Lorentz violating parameters for different particles. Ef- fective field theory is a natural choice for describing Lorentz violating physics without requiring us to know the details of the underlying fundamental theory. We did not taken into account renormalisable Lorentz non-invariant operators because, in practice, contributions of O(p) or O(p2) to the dispersion relation ought to be severely suppressed. Moreover, any new effect due to renormalisable operators are not, a priori, Planck suppressed. So, experimental tests restrict the coefficients of dimension three and four operators to be very small. On the other hand, non-renormalisable operators, of O(p3) at the low- est order, contribute only at high energies and are naturally Planck suppressed. Myers and Pospelov [82] used the following minimal set of criteria to determine O(p3) Lorentz violating operators:

a. they must be quadratic in the same ﬁeld,

b. contain one more derivative than the usual kinetic term,

c. are gauge invariant,

d. contain a constant background vector n that inadvertently breaks Lorentz invariance,

e. are not reducible to lower dimensional operators or total derivatives using equations of motion.

This means that in the scheme of Myers and Pospelov, Lorentz boost invariance in ﬂat spacetime is destroyed by bringing in a “constant vector" n in all spacetime which does not change under Lorentz transformations. Thus, by deﬁnition, n has the same components in all 12 introduction

Lorentz frames and hence is not a Lorentz four vector. Table 1 lists the Lorentz violating (LV) Lagrangian densities and modiﬁed dispersion relations (DR) of scalar, vector and spinor ﬁelds for the case when n = (1,~0). For convenience, we have adopted the shorthand nota-

field lvlagrangiandensity modifieddr

complex iκ φ∗∂3 φ ω2 ≈ |~p|2 + m2 + κ |~p|3 MPl n MPl scalar ξ nν 2 2 2ξ 3 Maxwell Fnν∂nF˜ ω ≈ |~p| ± |~p| MPl R,L MPl vector γ·n 2 2 2 2 2|~p|3 Dirac (η1 + η2γ5)∂ ψ ω = |~p| + m + ηR,L MPl n MPL spinor

Table 1: Dimension 5 Lorentz violating operators and corresponding modi- ﬁed dispersion relations

µ ρν nν tion (n · ∂) ≡ ∂n , n Fµν ≡ Fnν , nρF˜ ≡ F˜ . κ, ξ, η1, η2 or ηR, ηL are real dimensionless parameters constraining Lorentz violation. In writing the spinor dispersion relation, the spinors have been taken to be eigenstates of the chirality operator since we are working at high 2 2 energies (E >> m ): ηR,L ≡ η1 ± η2.

1.7 experimental bounds on lorentz violation

Numerous experiments aimed at constraining or observing Lorentz violation have been conducted in the past few years. Observational bounds have been imposed on the parameters ξ, η1,2 or ηR,L. But isn’t it absurd to consider detecting an effect suppressed by the Planck mass? No! Two different factors make things work in the two classes of measurements of Lorentz violation, namely astrophysical and terrestrial ones. In the former case, the enormous distances of signal propagation allow tiny Lorentz violating effects to accumulate in measurements of vacuum birefringence of photons or vacuum Cerenkovˇ radiation. Moreover, Lorentz violation scales with energy making astrophysical probes very useful in the search for modified dispersion relations of stable particles like electrons, light quarks and photons. Terrestrial experiments like those measuring sidereal variation, on the other hand, involve a large number of atoms making accurate measurements of the resonant frequency possible. The simplest astrophysical observations that provide interesting constraints on lack of Lorentz symmetry at Planck scale measure the differences in arrival times of photons emitted simultaneously from distant sources of radiation like γ-ray bursts, active galactic nuclei and pulsars [9, 23, 40–42, 44, 73, 96]. The authors of [2–4] found stringent limits on ξ by recording the timing of photons emitted during strong flares of the active galactic nucleus Markarian 501. The low- 1.8 lorentz preserving fields 13 est order corrections in the photon dispersion relation also imply the birefringence of vacuum (different group velocities for different he- licities of photons). In 2008, Maccione et al. [74] used polarimetric observations of hard x-ray from the Crab nebula to impose a bound from vacuum birefringence on Lorentz violation in quantum electrodynamics of |ξ| < 9 × 10−10 at 95% confidence level. Complementary constraints were also obtained from the threshold reactions of photon decay, fermion pair emission, synchrotron radiation, vacuum Cerenkovˇ radiation and helicity changing decays. In [57], the authors deduced a lower bound of η > −7 × 10−8 from syn- chroton radiation of the Crab nebula and an upper bound of η < 0.01 from vacuum Cerenkovˇ radiation. Clock comparison experiments can also impose stringent bounds on violation of Lorentz symmetry. They rely on the idea that the preferred cosmic frame of cosmic microwave background in which n = (1,~0) does not coincide with the laboratory frame on Earth. In −5 [82], an indirect limit of η 6 10 has been computed by a direct application of spin precession bound.

1.8 lorentz preserving fields

In a different context pertaining to the study of magnetic monopoles, Zwanziger in 1971 tried to construct a quantum ﬁeld theory of electric and magnetic charges based on a local Lagrangian density depending on a ﬁxed spacelike four vector. So, manifest isotropy of spacetime was lost and was shown to be regained only when certain conditions obtained by imposing an integrability criterion on the Poincaré Lie algebra were met [114]. This motivates us to ask whether a similar non- trivial condition could be found in Lorentz violating Myers Pospelov theory. We address this possibility in two parts, as described below.

Existence of Lorentz preserving ﬁelds

Starting with the modified effective action presented in [82], we show in chapter 2 that there exists a class of quite general field configurations that can make the Lorentz violating action invariant under infinitesimal Lorentz transformations [47]. This rather surprising outcome can be attributed to the fact that these special field configurations are themselves functions of the fixed background vector n. One way to think of this is to suppose that the Lorentz violation incorporated into the Myers Pospelov Lagrangian density manifests itself through the background dependence of the field configurations, thus making the dynamics Lorentz invariant. The field configurations belonging to this restricted class are referred to as Lorentz preserving fields. They also give a conserved Nöther current corresponding to Lorentz boosts. 14 introduction

Interestingly, certain intrinsic aspects of these ﬁelds appear suited to studying trans-Planckian modes and the growth of inhomogeneities in the early Universe. We end chapter 2 with an incipient analysis of this possibility, demonstrating how inhomogeneities may appear in the energy density in Minkowski spacetime due to Lorentz violation. This chapter is based on our article [47].

Algebra of Lorentz preserving ﬁelds

To complete the previous study it is important to explore whether a Lorentz Lie algebra can be realised on a basis of Lorentz preserving fields [46]. This is addressed in chapter 3 where we use the Lorentz preserving scalar field as a prototype. It exhibits all the crucial features, at the same time making calculations simple. A key aspect of the additional term in the Myers Pospelov modified Lagrangian density is its inclusion of third order derivatives of the field. But studies of higher time derivative theories date back to 1961, when Ostrograd- skii developed a canonical formalism for dealing with them [85]. In chapter 3, we employ his method to establish that in the two preferred frames defined by n = (1,~0) and n = (0, 0, 0, 1), the Lorentz preserving fields do indeed provide a representation of Lorentz algebra. But classical phase spaces are generally constructed by decompos- ing spacetime into space and time. An explicit choice of time coordinate from the beginning hides manifest Poincaré invariance and makes the canonical approach non-covariant. However, a covariant framework for the canonical formalism of a relativistic theory can be developed such that it preserves all relevant symmetries. The idea is quite old. Lagrange was probably the first person to recognise that phase spaces of physical systems can be constructed from the space of solutions of the dynamical equations of the theory, i. e. from dynam- ically allowed histories without ever referring to a preferred instant of time [12]. This covariant phase space is naturally endowed with a symplectic structure. With the help of the technique described in [36, 37], we conclude chapter 3 with an account of how the Lorentz preserving fields can be used to build a symplectic structure on the covariant phase space of the Lorentz violating effective field theory. This also guarantees that the Lorentz preserving fields provide a field basis for the realisation of a Lorentz algebra. The findings of this chapter have been reported in [46].

1.9 effect of lorentz violation on ‘superresonance’

Another domain of physics plagued by an absence of experimental feedback is the phenomenon of black hole radiation, which consists of spontaneous radiation (Hawking radiation) and stimulated emission (superradiance). Astrophysical black holes mostly accrete in- 1.9 effect of lorentz violation on ‘superresonance’ 15 stead of radiating spontaneously because their Hawking temperature is smaller than the cosmic microwave background temperature. This also makes superradiance hard to distinguish from within a background of X-rays emitted during accretion. However, this lacuna has driven a search for alternate physical models, where one may carry out tests to indirectly detect those phenomena associated with black holes that are insensitive to whether or not the metric satisfies Ein- stein’s equations. Among the early models, it was Unruh who proposed in 1981 an analogue gravity system based on fluid mechanics [103]. His amazing observation was that for a non-relativistic, barotropic, inviscid, irrotational fluid flowing in flat space, the equation of motion satisfied by the acoustic perturbation of the velocity potential is identical to the d’Alembertian equation of motion of a minimally coupled massless scalar field in an effective Lorentzian geometry. It is given by, 1 √ ψ = √ ∂ −g gµν ∂ ψ = 0 a −g µ ν a where gµν is the ‘acoustic’ metric and ψa denotes acoustic perturbations over the velocity potential. The acoustic geometry can be ‘felt’ only by the field ψa. Algebraically, this effective acoustic geometry depends entirely on the background fluid motion which in turn is governed by Newtonian physics and not by Einstein’s equations. So, the acoustic analogue can correctly reflect only half of general relativity - the kinematical sector, governed by the fact that general relativity holds in a Lorentzian spacetime. The dynamics, described by Einstein’s equations cannot in general be consistently mapped to any analogue model. Now, if the flow is such that it admits a surface In analogue systems, where the inward normal velocity of the fluid exceeds the local speed the acoustic horizon of sound at every point then that surface behaves as an outer trapped is similar to a black hole horizon only in surface or an acoustic analogue of a black hole horizon. It traps all its action as a one acoustic disturbances within it, creating an acoustic black hole or dumb way membrane. It hole [16, 105]. does not enclose a Unruh showed that if one has a static dumb hole then quantised physical spacetime singularity. acoustic perturbations (phonons) can be emitted from its horizon, just like the emission of photons from a black hole horizon, via Hawking radiation [103]. The only caveat is that this necessitates the quantisa- tion of linearised acoustic disturbances, which seems to be a rather artificially imposed condition on a classical fluid [16]. On the other hand, rotational superradiance from a black hole can be fully explained by classical general relativity. Discovered by Zel’dovich, superradiance appeared to be a universal phenomenon, by which any rotating body made of absorbing material is able to amplify scalar or electromagnetic waves reflecting off its surface when the following condition is satisfied [111, 112],

ω < mΩ 16 introduction

Here, ω, m are the frequency and multipole moment of the incident wave whereas Ω is the angular velocity of the rotating body. The extra energy carried away by the reﬂected wave comes from the rotational energy of the body. Surprisingly, he predicted that in a similar fash- ion, energy can also be extracted from a rotating black hole described by the Kerr geometry owing to the existence of an ergosphere. Scalar or electromagnetic waves with frequency ω and orbital momentum m can be scattered from the ergosphere with an ampliﬁcation in amplitude provided

ω < mΩH ,(5)

ΩH being the angular velocity of the black hole horizon. With the advent of fluid based analogue models, it was found that a particular kind of fluid flow known as the draining bathtub can very well mimic a Kerr black hole. Calculations by Basak and Majumdar in [16, 17] showed that acoustic superradiance or superresonance indeed occurs from such a dumb hole under the same condition as Equation 5.

Superresonance in Lorentz violating acoustic geometry

It is evident that we can, in principle, design in the laboratory simple and robust models of different classes of Lorentzian geometry in kinematic situations (where the acoustic metric is non-dynamical). But, can we incorporate departures from Lorentz invariance into such analogue models? Mais ouis! It seems so. The acoustic geometry emerging in a viscous ﬂuid is not Lorentzian as is evident from the dispersion relation satisﬁed by acoustic perturbations [105]. It is given by, v u !2 u 2ν|~k|2 2ν|~k|2 ω = ~v · ~k ± tc2|~k|2 − − i .(6) b s 3 3

Here, ν is the coefficient of viscosity and other symbols have their usual connotation. When ν = 0, the dispersion relation becomes linear. The real ν dependent term under the square root and the last imaginary term give rise to viscous dispersion and dissipation respectively. Unlike the modified dispersion relations studied in [60, 82], Equation 6 has been deduced directly from the equation of motion of acoustic disturbances in a classical fluid. We naively expect superresonance from a rotating dumb hole in a draining bathtub to be influenced by such viscous dissipation and dispersion. In chapter 4, dedicated to an investigation of this possibility, we offer an approximate solution to the problem under some simplifying assumptions. Our results say that in the viscous case, only scalar waves that have a frequency below the critical value mΩH can be scattered from the ergoregion. If this be so, then all scattered waves satisfy the condition for superresonance and travel away from the ergosurface with more energy than they originally carried in. 1.9 effect of lorentz violation on ‘superresonance’ 17

With this ends the three problems relating to Lorentz violation due to Planck scale physics that we have tried to explore during the course of this thesis. The results we have arrived at have led us on to many new questions. A discussion on limitations and possible improvements of our work together with an outline of future prospects is given in chapter 5, thereby formally concluding the thesis. The thesis is divided in two parts: chapter 1, chapter 2 and chapter 3 constitute the ﬁrst part of the thesis while chapter 4 and chapter 5 make up the second part.

LORENTZPRESERVINGFIELDSINLORENTZ 2 VIOLATINGTHEORY

Invariance under Lorentz transformation appears as a global symmetry of spacetime when gravitation is ignored. But in the quest for new physics near the Planck scale, questions have been raised about the validity of this symmetry at sufficiently high energies E, though E << MPl as we discussed in chapter 1. There has been much excite- ment over the expectation that small deviations from exact Lorentz invariance in flat spacetime may emerge as low energy (compared to MPl) signatures of quantum gravity. If there is any Lorentz violation, the standard special relativistic dispersion relation p2 = m2 E2 − ~p2 = m2 of a free particle of mass m having energy E and three momentum ~p will no longer hold exactly. Departures suppressed by the Planck mass from the above dispersion relation have been accepted as poten- tially observable signatures of Lorentz violation. Any such deviation will be negligible at most observable energy scales except very high ones. But, any ad hoc correction introduced into the standard dispersion relation due to Lorentz non-invariance must have its origin in new terms in the action of the system. A number of diverse and interesting takes on this problem can be found in the literature [1–4, 6–11, 23, 27, 32–34, 40–42, 44, 45, 50, 51, 56, 57, 59, 63–67, 69, 73, 74, 95, 96]. A ‘constant vector’ Myers and Pospelov [82] have studied this issue within the frame- or ‘fixed vector’ is work of effective field theory involving fields of spins 0, 1/2 and 1, not a Lorentz four vector at all as it has by incorporating into the action dimension five operators containing the same a constant timelike four vector n which ostensibly breaks Lorentz in- components in all variance. Choosing a Lorentz frame where nµ = (1,~0), corrections of Lorentz frames. O(p3) to the dispersion relation of each of the three fields have been obtained in [82] in the limit of relatively high energies E (MPl >> E >> m). For a complex scalar field this is given by κ ω2 ≈ |~p|2 + m2 + |~p|3.(7) MPl In the case of a Dirac spinor one gets, 3 2 2 2 2|~p| ω − |~p| − m − (η1 + η2γ5) ψ ≈ 0, MPL 3 2 2 2 2|~p| ω − |~p| − m − ηR,L ≈ 0.(8) MPL

19 20 lorentz preserving fields in lorentz violating theory

In Equation 8, the spinors have been chosen to be eigenstates of the chirality operator which is a valid assumption at high energies. ηR,L ≡ η1 ± η2. For the Maxwell ﬁeld, the dispersion relation is of the form (for circularly polarized photons) [59, 82]

2 2 2ξ 3 ωR,L ≈ |~p| ± |~p| (9) MPl It is clear from our discussion of this subject in chapter 1 that the possibility of having a deformed dispersion relation has been an object of extensive observational scrutiny [60, 72, 76]. However, does it unequivocally imply Lorentz violation? Inspired by Zwanziger’s work on the theory of magnetic monopoles [114], we explore the possibility that special field configurations exist for which the apparently Lorentz symmetry violating action of [82] may still be Lorentz invariant. In [114], a local, manifestly anisotropic Lagrangian density was shown to preserve Lorentz invariance when the fields obeyed certain constraints. In our article [47], we study the Nöther current corresponding to Lorentz transformations in the higher derivative theory proposed by Myers and Pospelov. Requiring that this Nöther current is conserved leads us to the condition that when the fields are de- composed in a particular way, the effective action remains Lorentz invariant. The initial absence of Lorentz symmetry in the action is responsible for a Lorentz non-invariant splitting of the fields. Identical configurations also appear when we demand that the action changes at most by a constant (i. e. the Lagrangian density changes by a total derivative) when the fields transform under an infinitesimal Lorentz transformation while n stays fixed. We further investigate the dispersion relation that these special field configurations satisfy. The particular decomposition referred to above may have direct application in the physics of the early Universe when energies were sufficiently high for Lorentz violation to have been possible. Usually, inhomogeneities in matter distribution are related to density perturbations that act as seeds for structure formation. In this commonly practised approach, there is no specific mention of quantum gravity effects that can lead to inhomogeneities. However, the metric fluctuations responsible for the growth of large scale anisotropies can originate from quantum fluctuations of the inflaton field. We discuss how Lorentz invariance violation near the Big Bang can be related to inhomogeneous Lorentz preserving fields which might give rise to structure formation. This scenario is developed in the context of the model constructed by Myers and Pospelov [82]. 2.1 restoring lorentz invariance of action 21

2.1 restoring lorentz invariance of action

2.1.1 Spin 0 ﬁelds

The action functional SMPφ for a complex scalar ﬁeld φ put forth in [82] is,

4 SMPφ = d x LMPφ Z iκ = d4x |∂φ|2 − m2|φ|2 + d4x φ∗(n · ∂)3φ (10) M Z Z Pl 4 4 = d x Lφ + d x LVφ Z Z = Sφ + SVφ , with κ being a real, dimensionless parameter The Lagrangian densities Lφ and LVφ and the corresponding actions Sφ and SVφ denote respectively the Lorentz invariant and Lorentz violating contributions. Under an inﬁnitesimal Lorentz transformation Λ, the coordinates {xµ} of a point in Minkowski spacetime transform as

µ µ µ x → x˜ = Λν µ µ ν 2 ν ≈ (δν + ν)x + O( )x as µν → 0. Owing to the orthogonality of the Lorentz group, µν = −νµ. Consequently, any scalar ﬁeld φ(x) undergoes the transformation,

Λ φ(x) −→ φ˜ (x˜) = φ(x) = φ(Λ−1x˜) . x˜ being a dummy variable, we may as well write

φ˜ (x) = φ(Λ−1x) .

In the limit of → 0,

µ µν 2 φ˜ (x) ≈ φ(x − xν + O( )) µν 2 ≈ φ(x) − xν∂µφ(x) + O( ) 1 ≈ φ(x) + µνx ∂ φ(x) + O(2) . 2 [µ ν]

We deﬁne the variation δαβφ(x) in a scalar ﬁeld φ(x) as,

∂φ˜ (x) δαβφ(x) ≡ αβ ∂ =0 1 = (δµδν − δµδν)x ∂ φ(x) 2 α β β α [µ ν] = x[α∂β]φ(x) µ = ∂µ(x[αδβ]φ(x)) 22 lorentz preserving fields in lorentz violating theory

which is a total derivative. The Lagrangian density Lφ being a Lorentz scalar changes by a total derivative under the action of Λ:

µ δαβLφ = ∂µ(x[αδβ]Lφ) .(11)

Therefore,

δαβSφ = 0 .

But due to the presence of the ‘ﬁxed vector’ n, the Lagrangian density

LVφ fails to transform as a Lorentz scalar. Its variation can be shown We have used the to be, notation n · ∂ ≡ ∂n. µ iκ ∗ 2 δαβLVφ = ∂µ(x[αδβ]LVφ ) + φ (x)n[α∂β]∂nφ(x) .(12) MPl The second term on the right hand side cannot be expressed as a total

derivative and so we have a non-vanishing variation of the action SVφ : iκ δ S = d4x φ∗(x)n ∂ ∂2 φ(x) . αβ Vφ M [α β] n Pl Z On the other hand, using the Euler Lagrange equation, the variation µ in the total Lagrangian density LMPφ (φ, ∂µφ, n ) can be expressed as Note that λ δαβn = 0 as n is a ‘ﬁxed vector’. ∂LMPφ δαβLMPφ = ∂µ δαβφ(x) .(13) ∂(∂µφ)

When the results of Equation 11 and Equation 12 are substituted into the left hand side of Equation 13, we have µ iκ ∗ 2 ∂LMPφ ∂µ(x[αδβ]LMPφ ) + φ (x)n[α∂β]∂nφ(x) = ∂µ δαβφ(x) . MPl ∂(∂µφ) (14)

If we deﬁne the Nöther current corresponding to Lorentz transformation as, µ ∂LMPφ µ Jαβ = δαβφ(x) − x[αδβ]LMPφ ,(15) ∂(∂µφ)

µ then from Equation 14 we ﬁnd that Jαβ has a non-vanishing four divergence:

µ iκ ∗ 2 ∂µJαβ = φ (x)n[α∂β](n · ∂) φ(x) .(16) MPl If Lorentz transformations are symmetries of the system then both the conditions

δαβSVφ = 0 2.1 restoring lorentz invariance of action 23 and

µ ∂µJαβ = 0 must be satisﬁed. Requiring either of these yields

2 n[α∂β](n · ∂) φ(x) = 0 (17) as φ∗(x) is not identically zero everywhere. A possible non-trivial solution is,

(n · ∂)2φ(x) = f(x · n) = f(σ) (18) where σ ≡ x · n. It is convenient in ﬂat spacetime to resolve the coordinate 4-vector x along and orthogonal to n:

x = xk + x⊥ (19)

x·n σ µ µ µ µ so that n · x⊥ = 0 and xk = n2 n = n2 n. Thus x = x (xk , x⊥) and

µ µ µ ∂x ν ∂x ν dx = ν dxk + ν dx⊥ , ∂xk ∂x⊥ µ ∂xν µ ν µ ∂x k ∂x ∂x⊥ ∂x α = α ν + α ν . ∂x ∂x ∂xk ∂x ∂x⊥

As this holds for all xµ, we get the operator relation

∂xν ν ∂ k ∂ ∂x⊥ ∂ α = α ν + α ν . ∂x ∂x ∂xk ∂x ∂x⊥

Now, from Equation 19 we have

α ∂x α ν = δν ∂xk and α ∂x α ν = δν . ∂x⊥ So,

∂ ν ∂ ν ∂ α = δα ν + δα ν , ∂x ∂xk ∂x⊥

∂α = ∂kα + ∂⊥α 1 = n ∂ + ∂ . n2 α σ ⊥α

∂ α ≡ ∂ ∂ α ≡ ∂ (To simplify the notation, we have deﬁned xk kα and x⊥ ⊥α.) Equation 18 can thus be restated as,

2 2 (n · ∂) φ(x) = (∂σ) φ(xk, x⊥) = f(x · n) = f(σ) (20) 24 lorentz preserving fields in lorentz violating theory

which on integration leads to a constraint on the complex scalar ﬁeld

φ(x) = φk(xk) + φ⊥(x⊥) .(21)

Here, φk and φ⊥ are arbitrary functions of their arguments. However, under a Lorentz transformation, xk and x⊥ do not transform as four vectors or remain respectively parallel or perpendicular to n because they, by definition, are functions of the ‘fixed vector’ n. Thus, the imprint of Lorentz violation introduced into the action is borne by the decomposition of the scalar field to ensure that the action retains Lorentz symmetry. If n is timelike, we can choose coordinates such that x0 lies along n. Then Equation 21 implies that when the full scalar field is a linear combination of a time-dependent, spatially homogeneous piece and a static spatially inhomogeneous piece, the theory possesses Lorentz symmetry.

1 2.1.2 Spin 2 ﬁeld

In [82], the action describing a Dirac spinor has been modiﬁed to,

4 SMPψ = d xLMPψ Z γ · n = d4x ψ¯ [(iγ · ∂ − m)ψ + (η + η γ )(n · ∂)2ψ] M 1 2 5 Z Pl ≡ Sψ + SVψ ,(22)

where Sψ is the standard Dirac action of a spinor ﬁeld ψ(x) and SVψ is the additional Lorentz violating action. The dimensionless parameters η1, η2 give measures of Lorentz violation. Here, the only source of Lorentz violation is, by assumption, the

appearance of the constant four vector n in SVψ . Thus, there are no constant vectors in the theory independent of n. It is straightforward to show that, under an inﬁnitesimal Lorentz transformation, the ac-

tion SVψ changes by, 1 δ S = d4x ψ¯ {n γ (η + η γ )∂2 ψ αβ Vψ M [α β] 1 2 5 n Pl Z + γ · n(η1 + η2γ5)n[α∂β]∂nψ} .(23)

If we set ∂nψ = χ(σ) , where σ = x · n, then the second term in Equation 23 vanishes. After a partial integration (dropping the surface term), the ﬁrst term reduces to, 1 δ S = − d4x η n χγ¯ χ + η n χγ¯ γ χ αβ Vψ M 1 [α β] 2 [α β] 5 Pl Z 1 h i = − d4x η n J (σ) + η n J5 (σ) ,(24) M 1 [α β] 2 [α β] Pl Z 2.1 restoring lorentz invariance of action 25

5 where Jα(σ) ≡ χγ¯ αχ and Jα(σ) ≡ χγ¯ βγ5χ. Now, one can decompose the currents J and J5 as n · J J = n + J n2 ⊥ and n · J5 J5 = n + J5 n2 ⊥

5 so that n · J⊥ = 0 and n · J⊥ = 0. Inserting this decomposition into Equation 24, it is clear that

4 h 5 i δαβSVψ = − d x η1n[αJ⊥β](σ) + η2n[αJ⊥β](σ) ,(25) Z that is, Lorentz violation now depends on the current four vectors J⊥ 5 and J⊥. It should be noted, however, that these current four vectors are orthogonal to n and are constants in the direction they point! If, for 5 example, n is timelike, the currents J⊥ and J⊥ must be spacelike and yet must be spatially homogeneous, being functions of σ. This makes them constant four vectors independent of n. Since, by assumption there are no constant four vectors in the problem apart from n, these currents must vanish. Thus, as illustrated for the scalar ﬁeld, requirement of Lorentz invariance of the action implies that,

ψ(x) = ψk(xk) + ψ⊥(x⊥) .(26)

~ In the preferred frame where n = (1, 0), ψk(xk) is a spatially homogeneous time dependent spinor whereas the spinor ψ⊥(x⊥) is time independent.

2.1.3 Maxwell (spin 1) ﬁeld

In this case, the usual kinetic term of the free Maxwell field and a dimension five, n dependent operator constitute the modified La- grangian density proposed in [82],

4 SMPγ = d xLMPγ Z 1 ξ = d4x − F Fµν + nµF nα∂ n F˜ρν .(27) 4 µν M µν α ρ Z Pl Here, ξ is a dimensionless parameter constraining Lorentz violation and 1 F˜µν = µνλσF . 2 λσ 26 lorentz preserving fields in lorentz violating theory

For convenience, we also deﬁne

µ n Fµν ≡ Fnν

and

ρν nν nρF˜ = F˜ .

Directly studying the variation of the action or the divergence of the Nöther current corresponding to Lorentz boosts, in parallel with the argument given in case of the scalar ﬁeld, the condition for the action to be Lorentz invariant is deduced:

nν nν nν n[αFβ]ν∂nF˜ + Fnνn[α∂β]F˜ + F ∂nn[αF˜β]ν = 0 .(28)

The last term is always zero because

λσ n[αF˜β]ν = n[αβ]νλσF

contains a fifth rank completely antisymmetric tensor in four spacetime dimensions! We transform to the Lorentz frame defined by n = (1,~0) to get a better physical picture of the problem in terms of electric and magnetic field three vectors identified as 1 E = F = F˜ , E = F˜ ; i 0i 2 ilm lm lmi i lm 1 B = F˜ = − F , − B = F . i 0i 2 ijk jk jki i jk The fourth and third rank Levi Civita tensors are related by

0ijk = ijk = −0ijk.

So, Equation 28 becomes,

ijm ∂0BjBm − Ej∂iBj = 0 , B~˙ × B~ − (~E · ∇~ )B~ = 0 .(29)

It is easy to see that if the ﬁelds are harmonic functions of spacetime as

~E = Re(~E0exp(−iωt + i~k · ~x)) ,(30)

B~ = Re(B~ 0exp(−iωt + i~k · ~x)) ,(31)

they satisfy Equation 29 when the relation ~E · B~ = 0 (deduced from Bianchi identity) is incorporated. This demonstrates that in the modi- ﬁed electrodynamics designed by Myers and Pospelov [82], it is possible to have Lorentz covariant electromagnetic ﬁelds that make the

action SMPγ Lorentz invariant. 2.2 evaluation of dispersion relation 27

2.2 evaluation of dispersion relation

Now that we have found quite general and non-trivial field configurations that make the modified scalar, spinor and vector actions of [82] Lorentz invariant, the next step entails calculating the dispersion relations obeyed by these special fields.

2.2.1 Scalar ﬁeld

The scalar ﬁeld φ(x) assumed to be given by Equation 21 allows us to write the equation of motion

2 iκ 3 ( + m )φ(x) = ∂nφ MPl as

2 2 2 2 iκ 3 (∂⊥ + m )φ⊥ = −(∂k + m )φk + ∂nφk .(32) MPl Here, we have used the decomposition

2 2 2 φ(x) = ∂ φ(x) = ∂kφk + ∂⊥φ⊥ when n is a unit vector. The left and right hand sides of Equation 32 are functions of x⊥ and xk respectively. It is obvious that to make sense of this equation, we must set both sides equal to a constant which we choose to vanish for convenience. Let us make an ansätz

φk(xk) ∼ exp(−i kk · xk) and

φ⊥(x⊥) ∼ exp(−k⊥ · x⊥) , ~ ~ where kk = (Ek, kk) and k⊥ = (E⊥, k⊥) are four momenta parallel and orthogonal to n. The dispersion relations of the ﬁelds φk and φ⊥ then become 2 ~ 2 2 κ 3 Ek = |kk| + m + (n · kk) ,(33) MPl 2 ~ 2 2 E⊥ = |k⊥| − m .(34)

In the special inertial frame deﬁned by n = (1,~0), ~ Ek ≡ E , kk = 0

E⊥ = 0 , ~k⊥ ≡ ~k and the dispersion relations (Equation 33, Equation 34) take the sim- pliﬁed form: κ E2 = m2 + E3 , MPl |~k|2 = m2 . 28 lorentz preserving fields in lorentz violating theory

~ The total four momentum k ≡ (E, k) is the superposition of kk and 2 k⊥. One can now eliminate m from these equations to get the dispersion relation in the high energy regime E ' |~k| >> m: κ E2 ' |~k|2 + |~k|3 .(35) MPl This is same as the dispersion relation of Equation 7 computed in [82].

2.2.2 Spinor ﬁeld

The equation of motion of the spinor ﬁeld ψ(x), as deduced from the

variation of the action SMPψ , has the form

2 2i 3 ( + m )ψ(x) = (η1 + η2γ5) ∂nψ(x) MPl

where terms of O(m/MPl) have been ignored [82]. If we take the spacetime dependence of the ﬁelds in Equation 26 to be

ψk(xk) ∼ exp(−i kk · xk) ,

ψ⊥(x⊥) ∼ exp(−k⊥ · x⊥) and proceed as for the scalar case, then the dispersion relations of the ﬁelds ψk and ψ⊥ turn out respectively to be, " # 2(n · k )3 2 ~ 2 2 k Ek − |kk| − m − (η1 + η2γ5) ψk = 0 ,(36) MPl h 2 ~ 2 2i E⊥ − |k⊥| + m ψ⊥ = 0 .(37) Symbols here have their obvious connotations. We are interested in high energy phenomena and at sufﬁciently high energies the massive spinors can be treated as chirality operator eigenstates. In the special Lorentz frame where n = (1,~0), ~ ~ kk = (Ek, 0) ≡ (E, 0) ,

k⊥ = (0,~k⊥) ≡ (0,~k) and the dispersion relations simplify to 3 2 2 2 E E − m − ηR,L = 0 ,(38) MPl |~k|2 − m2 = 0 .(39) If we deﬁne a four momentum k as

k = kk + k⊥ , then k = (E,~k) and the two dispersion relations (Equation 38, Equa- tion 39) can be combined to get ~ 3 2 ~ 2 2|k| E = |k| + ηR,L (40) MPl in the limit of high energy E ' |~k| >> m. 2.3 discussion 29

2.2.3 Vector ﬁeld

The equation of motion obtained by the variation of the action SMPγ is (derived in [78]),

µν ξ ρσµν 2 nν ∂µF + nρ ∂µ∂nFnσ − ∂nF˜ = 0 .(41) MPl

The above equation and the Bianchi identity ∂[µFνρ] = 0 in the chosen reference frame are equivalent to the following equations:

∇~ · ~E = 0 = ∇~ · B~ ,(42)

∇~ × ~E = −∂tB~ ,(43) ξ −~E˙ + ∇~ × B~ + (B~¨ − ∇~ × ~E˙ ) = 0 .(44) MPl These are the modiﬁed free Maxwell equations. If we take the curl of both sides of Equation 44, simplify using the Bianchi identity of Equation 42, substitute the Lorentz invariant solution for the magnetic ﬁeld and assume k = (ω, 0, 0, k3), the dispersion relations at high energy ω ' |~k| are:

2 ~ 2 2ξ ~ 3 ωR,L − |k| ' ± |k⊥| .(45) MPl The plus and minus signs appear for right and left circularly po- larised electromagnetic waves respectively.

2.3 discussion

In section 2.1, we have demonstrated how requiring invariance under Lorentz transformation of the Lorentz violating scalar or spinor action leads to nontrivial restrictions on the functional form of fields in that the fields decouple into two parts one of which is a function only of the projection of the coordinate vector along the fixed vector n, the other part being a function of projections of the coordinate vector orthogonal to n. These constraints have not been imposed by hand: they are the most general solutions of the equations that result on requiring invariance of the action under infinitesimal Lorentz transformations. We have worked out the invariance for transformations only close to the identity in the parameter space of the Lorentz group. How- ever, in dealing with Lorentz transformations in field theory, one in- variably dealing with the simply-connected universal cover SL(2, C) of the Lorentz group and so it is quite adequate to consider only the Lorentz Lie Algebra and its action on the fields. Indeed, we expect that Lorentz Lie Algebra will be realised on our special field configurations because the Nöther current appropriate to the transformations is conserved and also the full action is invariant under 30 lorentz preserving fields in lorentz violating theory

infinitesimal transformations. (We address this issue in the next chapter.) This should guarantee invariance under the connected part of the Lorentz group which is of concern here provided we adhere to the special field configurations. In fact, this is sufficient to obtain the non-standard dispersion relations as well, as has been illustrated. On the contrary, in the Lorentz violating electrodynamics, the Lorentz preserving fields are independent of n and thus are naturally Lorentz covariant. An important reminder on terminology: in this chapter, a confusion between Lorentz four vectors and “fixed vectors” may arise. The fixed vector n is not by any means a Lorentz four vector as it does not transform under an action of the Lorentz group like a four vector. It is a four component object which in any particular frame may be treated as a vector. But it has the same components in all boosted coordinates. So, any vector dotted with n or any function of n cannot be a Lorentz scalar. For want of a better name, it is conventionally referred to as a “fixed vector”.

2.4 possible application in cosmology

The field configurations that we have obtained have aspects of intrinsic interest when one considers prospective application to cosmology, as in inflationary scenarios. A peculiarity of almost all models of inflation [70, 83] is that the phase of accelerated expansion is believed to have lasted long enough for present scales of cosmological interest to be redshifted from trans-Planckian length scales at the onset of inflation. Hence, Lorentz violation may have been an intrinsic feature of the nascent Universe. One of the phenomenological approaches towards studying this era consists of modifying the standard dispersion relation of the scalar inflaton field together with the introducing a standard timelike unit vector in the effective Lagrangian to define the preferred frame. The requirement remains that the altered dispersion relation must obviously reduce to the standard linear behaviour at energies much smaller than MPl. The Myers-Pospelov theory [82] shares these criteria which makes it suitable for studying the physics of our Universe at the beginning of inflation. The fact that there is a natural decomposition in Lorentz- preserving (scalar) fields between spatially homogeneous and inhomogeneous parts implies that while the former can play the role of the inflaton field in a Friedmann-Lemaître-Robertson-Walker (FLRW) background spacetime, the latter, acting as a perturbation on the former, may provide natural seeds for the growth of inhomogeneities in the Universe. Moreover, in an FLRW spacetime, a chosen frame exists by construction. Hence, the “constant vector” n can be taken to be orthogonal to the homogeneous isotropic spatial sections in FLRW spacetime such that n = (1,~0) in the comoving frame. 2.4 possible application in cosmology 31

Just as an illustration, we demonstrate how inhomogeneities appear in the energy momentum tensor of the ﬁeld φ(x) in a Minkowski spacetime. The general formula for the energy momentum tensor is,

∂LMPφ Tµν = ∂νφ − ηµνLMPφ . ∂(∂µφ)

For ﬁelds of the form given in Equation 21 described by the La- grangian density of Equation 10, the energy density ρ and pressure p in the massless limit, as measured in the chosen frame, are:

ρ = T00 iκ ... iκ ... ˙ 2 ∗ ∗ ~ 2 = φk − φkφk − φ⊥φk + (∇φ⊥) ,(46) MPl MPl 1 p = − T 3 ii iκ ... iκ ... 1 ˙ 2 ∗ ∗ ~ 2 = φk + φkφk + φ⊥φk − (∇φ⊥) .(47) MPl MPl 3 The presence of the additional κ dependent terms in the energy density ρ indicates that Lorentz violation in an originally flat spacetime will automatically introduce non-trivial curvature. The inhomogeneous field φ⊥(x⊥) appearing in the last two terms of the expression for ρ can be envisaged as a source of perturbations over the otherwise homogeneous energy density of the field φk(xk). Such Lorentz preserving perturbations due to the field φ⊥(x⊥) have the potential to lead to growth of Lorentz invariant inhomogeneities in spacetime. This may be understood in the following way. Owing to its dependence on the static, spatially inhomogeneous field φ⊥, the energy density ρ varies for different spatial points. After the inflationary period, when fluctuations were amplified, this could make the density of matter vary slightly from place to place in the Universe. Gravity could make these tiny variations grow in amplitude and eventually lead to large scale structures we see in the Universe today.

REALISATIONOFLORENTZALGEBRAIN 3 LORENTZVIOLATINGTHEORY

We take the Lorentz preserving scalar field, derived in chapter 2, corresponding to the modified action of [82] as an illustrative example in this chapter. Given below is the by now well known extended action of a complex scalar field φ that was proposed by Myers and Pospelov,

4 SMPφ = d xLMPφ , Z iκ = d4x |∂φ|2 − m2|φ|2 + d4x φ∗∂3 φ .(48) M n Z Z Pl According to [47] and chapter 2, this action is symmetric under in- ﬁnitesimal Lorentz boosts or rotations if the complex scalar ﬁeld can be split as,

φ(x) = φk(xk) + φ⊥(x⊥) (49) where, φk and φ⊥ are arbitrary functions of their respective arguments. A key aspect of the additional term in Equation 48 is that it contains third order derivatives of the field unlike standard first order La- grangians. But higher derivative Lagrangians are not new to physics [91]. Back in 1961, Ostrogradskii had developed a canonical formalism for dealing with them [85]. Reviews and modifications of his technique study mostly systems having finite number of degrees of freedom and higher time derivatives of the generalised coordinates [14, 79, 81, 98]. An extension to special relativistic continuous systems is presented in [104], though it relies on the existence of Lorentz symmetry. Usually, the canonical formalism is understood to be non-covariant because it involves the choice of a spacelike hypersurface and its orthogonal time direction. In section 3.1, we briefly review Ostrograd- skii’s original construction and its generalisation to classical field systems, tailored to suit the Lagrangian of Equation 48 with fields that decouple as in Equation 49. Most of the peculiarities of a higher derivative theory are due only to higher order time derivatives. Spa- tial derivatives are quite benign, staying within the scope of standard first order canonical approach. So, in section 3.2, the study of the modified scalar field theory is split up into two cases distinguished by n2 being timelike and spacelike. We avoid commenting on the case when n2 = 0, a topic set aside for now. We also take the liberty of working in certain Lorentz frames that simplify calculations.

33 34 realisation of lorentz algebra in lorentz violating theory

However, a covariant framework for the canonical formulation of a relativistic theory may also be developed through the construction of a covariant phase space and a symplectic two-form on it [12, 36, 37, 109, 113]. A natural extension of this technique to higher derivative field theories appears in [5]. Below, section 3.3 contains a summary of the main features of the covariant phase space for first order as well as higher order derivative field systems. We conclude the chapter by showing how the Lorentz preserving fields facilitate the construction of a covariant phase space structure for the Lorentz violating effective field theory under consideration.

3.1 canonical formalism in the presence of higher derivatives

3.1.1 Non-relativistic systems with ﬁnite degrees of freedom

Let us consider a system described by the Lagrangian

l L = L(qa, dtqa, ..., dtqa) (50)

which is a function not only of the generalised coordinate qa(t) and 2 l the velocity dtqa(t) but of all derivatives dtqa(t), dt qa(t), ..., dtqa(t) upto order l. Here a labels the different degrees of freedom and we adopt the notation

j qa(j) ≡ dtqa, j = 0, ..., l

so that

qa(j+1) = q˙ a (j) .

a(j) Each of these qa(j) upto j = l − 1 has a conjugate momentum p . Remember though that the superscript j of pa(j) only denotes that it is conjugate to qa(j). The system is now speciﬁed by a point in the phase a(j) space spanned by qa(j), p with j = 0, ..., l − 1. We have here assumed that the highest derivative qa(l) can be written as a function of the a(0) a(l−1) other variables qa(l)(qa(0), p ; ...; qa(l−1), p ). The condition for extremisation of the action

S[q(t)] = dt L Z is,

l ∂L δ S = 0 = dt (−d )j δq q t ∂q a j=0 a(j) Z X l l−1 ∂L + dtd (−d )i−(j+1) δq . t t ∂q a(j) i=j+1 j=0 a(i) Z X X 3.1 canonical formalism in the presence of higher derivatives 35

If the boundaries are such that the variations of qa and its derivatives upto order (l − 1) vanish, the second integral goes to zero and the equation of motion becomes

j ∂L (−dt) = 0 .(51) ∂qa(j)

On the other hand, if this system undergoes a symmetry transformation then δS = 0 and substitution of the equation of motion given in Equation 51 yields,

l l−1 ∂L d (−d )i−(j+1) δq = 0 . t t ∂q a(j) = +1 =0 a(i) i Xj Xj The quantity within the square brackets is the conserved Nöther current J. From its structure, we may read off the conjugate momenta

l ( ) ∂L p j ≡ (−d )i−(j+1) , j = 0, ..., l − 1 . a t ∂q = +1 a(i) i Xj The Nöther current may then be cast into the standard form

l−1 (j) J = pa δqa(j) . =0 Xj

3.1.2 Relativistic continuous systems

Here, in place of the generalised coordinates, we work with special relativistic fields having infinite number of degrees of freedom. Let us consider a system of scalar fields φa(x). If

L = L(φa, φa,ρ1 , φa,ρ1ρ2 , ..., φa,ρ1...ρl ) be the Lagrangian density where

φa,ρ1...ρj ≡ ∂ρ1 ...∂ρj φa then by extremising the action

S[φ(x)] = d4x L Z with respect to the ﬁelds φa(x), we get the equation of motion

l j ∂L (−1) ∂ρ1 ...∂ρj = 0 .(52) ∂φ ρ ρ =0 a, 1... j Xj The necessary boundary conditions are that the ﬁelds and their derivatives upto φρ1...ρl−1 fall off at inﬁnity. It would be convenient if we 36 realisation of lorentz algebra in lorentz violating theory

could write down a general form of the Nöther current for such higher derivative systems. This is achieved by applying a symmetry transformation to the ﬁelds and consequently employing the equation of motion appearing in Equation 52 to get the Nöther current,

l l−1 ∂L ρ1 i−(j+1) J = (−1) ∂ρj+2 ...∂ρi δφρ2...ρj+1 ,(53) ∂φ ρ ρ = +1 =0 a, 1... i i Xj Xj which is locally conserved, that is,

ρ1 ∂ρ1 J = 0 . It may so happen that instead of δL being zero, the Lagrangian density varies by a total derivative. This would then contribute to the Nöther current. However, as we have already mentioned, a canonical formalism of relativistic field theories requires us to foliate spacetime into spacelike hypersurfaces. This, in turn, involves the separation of temporal and spatial derivatives of the fields 1. It is most often possible to arrange terms in the Lagrangian such that mixed derivatives of the fields as 2 in ∂t ∂kφ, do not survive. This is true not only when the Lagrangian is Lorentz invariant [104] but also when the Lorentz violating action in Equation 48 is written in terms of the Lorentz preserving fields, as illustrated in the next section. In such situations, the canonical momenta are given by

l ∂L πa(j) ≡ (−d )i−(j+1) , j = 0, ..., l − 1 .(54) t ∂φ = +1 a(i) i Xj The canonical variables satisfy the Poisson bracket

b (j) 0 b j (3) 0 {φa (i)(t,~x), π (t,~x )} = δa δi δ (~x − ~x ) .(55)

3.2 myers pospelov theory

3.2.1 Constant timelike background vector

Without loss of generality, we work in a Lorentz frame defined by ~ n = (1, 0). Then the Lorentz preserving fields φk(t), φ⊥(~x) become spatially homogeneous and static, spatially inhomogeneous respectively. This greatly simplifies the Lagrangian density: iκ ... L = φ˙ ∗φ˙ − ∇~ φ∗ · ∇~ φ + (φ∗ + φ∗ )φ ,(56) MPφ k k ⊥ ⊥ M k ⊥ k ... Pl ˙ ¨ ∗ ˙ ∗ ~ ∗ ~ ∗ = LMPφ (φk, φk, φk, φk, φk, φk, φ⊥, ∇φ⊥, φ⊥, ∇φ⊥) .(57)

1 At this stage, it is imperative that we sort the indices. Latin letters from the middle of the alphabet set, viz. i,j,k,l are used as summation indices while those from the end like r,s,...,z denote spatial components. Different ﬁelds are labelled by the alphabets a,b,c,d. Greek letters are reserved for spacetime indices. 3.2 myerspospelovtheory 37

Here we have neglected the masses of the ﬁelds as we are interested in behaviour of the system at energies much higher than the ﬁeld masses. It is now evident why our chosen Lorentz frame is particu- larly useful. Equation 56 has only higher order time derivatives of ~ φk(t) while ∇φk = 0 = ∂tφ⊥. Thus all mixed derivatives in the sense described above vanish. This permits us to safely use Equation 54 to determine the canonical momenta. We list them in Table 2. Note that

generalised coordinate generalised momentum φ = φ π = φ˙ ∗ + iκ φ¨ ∗ k(0) k k(0) k MPl k φ = φ˙ π = − iκ φ˙ ∗ k(1) k k(1) MPl k φ = φ¨ π = iκ (φ∗ + φ∗ ) k(2) k k(2) MPl k ⊥ ∗ ∗ ˙ φk(0) = φk πk∗(0) = φk

φ⊥(0) = φ⊥ π⊥(0) = 0 ∗ ∗ φ⊥(0) = φ⊥ π⊥∗(0) = 0

Table 2: Canonically conjugate phase space variables when n = (1,~0)

π⊥(0) = 0 and π⊥∗(0) = 0 are constraint relations that are imposed weakly. The equal time Poisson Bracket of Equation 55 holds for the canonical variables. Under an inﬁnitesimal Lorentz transformation, δαβφ = x[α∂β]φ. The Lagrangian density being a scalar function, also changes as

ρ1 δαβL = x[α∂β]L = ∂ρ1 (x[αδβ]L) .

µ The Lorentz preserving ﬁelds conserve the Nöther current Jαβ [47]. The Nöther charge is given by

3 ρ1 Qαβ = d σρ1 Jαβ . ZΣ Here, Σ is a three dimensional hypersurface. If we orient it orthogonal to the time axis then,

3 (0) (1) Qαβ = d ~x πk δαβφk(0) + πk δαβφk(1) ZΣ (2) ∗ 0 + πk δαβφk(2) + πk∗ δαβφk − x[αδβ]L ,

3 (j) 0 = d ~x πa δαβφa(j) − x[αδβ]L .(58) ZΣ ∗ Here φa = φk, φk. The other variables do not contribute. From the structure of Qαβ we can deduce that

∗ {Qαβ(t), φtmb(k)(t,~x)} = −δαβφb (k)(t,~x), for φb = φk, φk ; ∗ = 0, for φb = φ⊥, φ⊥ .(59) 38 realisation of lorentz algebra in lorentz violating theory

The ﬁnal step is to evaluate the algebra of the charges. A simple calculation using Equation 58, the basic Poisson bracket of Equation 55 and the relation δαβφ = x[α∂β]φ gives,

{Qαβ(t), Qρσ(t)} = ηασQβρ(t) − ηαρQβσ(t)

+ ηβρQασ(t) − ηβσQαρ(t) .(60)

Equation 59 and Equation 60 confirm that Nöther charges defined in terms of the Lorentz preserving fields generate Lorentz transformations through Poisson brackets and also form a representation of the associated Lie algebra. Thus, the special field configurations provide a valid basis for the realisation of Lorentz Lie algebra which determines the local structure of Lorentz group near the identity 2.

3.2.2 Constant spacelike background vector

Next, we go over to the Lorentz frame where n is a unit spacelike vector having components n = (0,~1). Then the Lorentz preserving ﬁelds are φk(~x), φ⊥(t) and the Lagrangian takes the form

iκ ˙ ∗ ˙ ~ ∗ ~ ∗ ∗ ~ 3 LMPφ = φ⊥φ⊥ − ∇φk · ∇φk + (φk + φ⊥)|∇| φk .(61) MPl This Lagrangian density contains only ﬁrst order time derivatives of the ﬁelds. Thus, no extra phase space variables are required. Table 3 lists the pairs of canonical coordinates and momenta. The fundamen-

generalised coordinate generalised momentum

φk πk = 0 ∗ φk πk∗ = 0 ˙ ∗ φ⊥ π⊥ = φ⊥ ∗ ˙ φ⊥ π⊥∗ = φ⊥

Table 3: Canonically conjugate phase space variables when n = (0,~1)

tal Poisson Bracket is,

0 (3) 0 {φa(t,~x), πb(t,~x )} = δab δ (~x − ~x ) .(62)

2 A generic element Λ of the Lorentz group can be obtained by exponentiating the generators and the antisymmetric parameters of a transformation as

− i Qµν Λ = e 2 µν .

Expanding the exponential in the neighbourhood of the identity, it becomes clear that the Lie algebra determines the local structure of the Lorentz group near the identity. 3.3 overview of covariant phase space formulation 39

Integration of the zeroth component of the Nöther current over an appropriately chosen three dimensional spatial slice normal to the time axis gives the Nöther charge for Lorentz transformation,

3 ∗ 0 Qαβ = d ~x π⊥δαβφ⊥ + π⊥∗ δαβφ⊥ − x[αδβ]L .(63) Z This is of the same form as the standard Nöther current obtained for a first order system of scalar fields. Hence, the Nöther charges would generate infinitesimal Lorentz transformations and satisfy the Lorentz algebra.

3.3 overview of covariant phase space formulation

The covariant construction of phase space keeps intact Poincaré invariance of a physical system [12, 36, 37, 109, 113]. As opposed to the standard decomposition of phase space in the 3 + 1 framework, the classical covariant phase space Z of a physical theory is deﬁned as the space of classical solutions of the dynamical equations of the theory. Functions φ(x), tangent vectors δφ, one forms δφ(x) and exterior derivatives δ can be deﬁned on Z as in [36, 37]. The phase space Z is naturally endowed with a closed, non-degenerate two-form ωe ( ωe is non-degenerate provided the one-form ωe(V) = 0 if and only if V = 0) called a symplectic structure. It can be writtene as an integral ofe some closed, conserved symplectic two-form current µ µ ωe (note that ωe is a two-form in phase space Z and a vector current µ µ in spacetime; ∂µω˜ = 0, δω˜ = 0) over a hypersurface Σ,

µ ωe = dσµωe (64) ZΣ Hence, the task now is to ﬁnd a suitable symplectic current, given any Lagrangian. Owing to our present interest in the covariant description of canonical formalism for higher derivative theories [5], we straight away take the general example of a Lagrangian density

L(φa, φa,ρ1 , φa,ρ1ρ2 , ..., φa,ρ1...ρl ). For φa belonging to Z and an arbitrary linear transformation δφa(x) that takes φa(x) → φa(x) + δφa(x) on the phase space, we have l l−1 i−(j+1) ∂L δL = ∂µ (−1) ∂ρj+2 ...∂ρi δφρ2...ρj+1 ∂φ µρ ρ = +1 =0 a, 2... i i Xj Xj µ = ∂µj ,(65) when the Euler Lagrange equation of motion of Equation 52 is substituted. jµ is interpreted as a pre-symplectic current because it is used µ µ to deﬁne the symplectic current ωe = δj . From Equation 65, one can see that it is obviously closed in phase space and conserved through its dependence on spacetime. Hence, the symplectic structure

µ µ ωe = dσµωe = δ dσµj (66) ZΣ ZΣ 40 realisation of lorentz algebra in lorentz violating theory

is not only closed but also exact. Moreover, the local conservation of the symplectic current in spacetime guarantees that ωe does not change with the choice of the surface of integration Σ in Equation 66 and in particular, is Poincaré invariant [37].

3.4 symplectic structure with lorentz preserving fields

The covariant version of the canonical formalism is meaningful only when the physical theory has Poincaré invariance. The original My- ers Pospelov Lorentz violating model does not meet this requirement. But when the allowed field configurations are restricted to the Lorentz preserving fields, the dynamics of the theory becomes invariant under infinitesimal Lorentz transformations enabling us to construct its covariant phase space. Only in this context, Equation 65

holds for LMPφ . One must have observed that the presymplectic current given in Equation 65 and the Nöther current given in Equation 53 have identical forms. In fact, with the interpretation of δφa(x) as a one-form on phase space, the Nöther current becomes the pre-symplectic current one form. It must also be stressed that the entire construction of the symplectic structure in Equation 66 follows from Equation 65, the validity of which is ensured by the Lorentz preserving ﬁelds. This, in turn, guarantees the existence of a Lorentz invariant (and Poincaré invariant) symplectic structure on the covariant phase space of Lorentz preserving solutions of the equation of motion of Myers Pospelov action. Part II

TESTINGLORENTZVIOLATIONIN ANALOGUESYSTEMS

EFFECTOFANALOGUELORENTZVIOLATIONON 4 SUPERRESONANCE

The simple observation that the same equations have the same solutions precisely embodies the logic behind working with analogue models. We seek simple physical systems analogous, naturally not in all, but some aspects, to a complicated system of interest to help us answer our queries pertaining to the latter. In turn, this study may also lend a different perspective on our well known simple physical system.

4.1 black hole physics in acoustic analogue models

Many predictions of general relativity associated with strong gravitational fields are well known for being inaccessible to direct detection. Notable among them is black hole radiation which may be either stimulated or spontaneous. The former is known as superradiance while the latter goes by the name of Hawking radiation, after its discoverer Stephen Hawking. Their existence has been theoretically predicted in the 1970s, but it has proved to be rather impossible to isolate traces of such radiation from a sea of cosmic microwave background radiation (CMB) and X-rays emitted during accretion by black holes. This impasse was lifted with a discovery by Unruh [103] in 1981 that a classical fluid can be used to mimic a black hole, albeit under a number of conditions. His idea was inspired by the observation that in a non-relativistic, inviscid, barotropic (pressure p is a function only of density ρ) fluid flowing with an irrotational velocity (curl of velocity ~v vanishes) in flat 3 space, the equation of motion of acoustic fluctuations in the velocity potential (ψa) is identical to the Klein Gordon equation satisfied by a massless scalar field in general curved (3 + 1) dimensional spacetime: 1 √ ψ ≡ √ ∂ −g gµν ∂ ψ = 0 .(67) a −g µ ν a Here, g acts as an ‘acoustic metric’ of an effective Lorentzian geometry to which only the acoustic fluctuations ψa in the velocity potential are sensitive. At first glance, it may seem queer that though the physics of fluid dynamics is non-relativistic (fluid velocity is much smaller than the speed of light), Newtonian and distinguishes clearly between the notions of space and time, the acoustic fluctuation field ψa ‘hears’ a curved Lorentzian spacetime. This revealation extends the field of interest of Lorentzian geometry beyond the usual scope of Einstein’s theory of gravitation. In fact,

43 44 effect of analogue lorentz violation on superresonance

the study of black hole physics in acoustic models helps us identify phenomena which are artefacts of Lorentzian geometry from those intrinsic to Einstein’s gravity. Calculation, to be presented below, show that the nature of the acoustic geometry is completely fixed by the velocity profile and physical properties of the fluid. All backreaction on the metric are ignored. Thus, by properly choosing the bulk velocity profile it becomes possible to create in the laboratory different kinds of acoustic black holes or ‘dumb holes’, as they have been aptly named. If the flow admits a surface on which at every point the inward normal component of fluid velocity exceeds the local speed of sound, then all acoustic disturbances originating within it are swept inwards by the flowing fluid. This surface consequently acts as an outer trapped surface or horizon of the dumb hole. Unruh chose a spherically symmetric, stationary, convergent background flow to construct an analogue of the spacetime outside a Schwarzschild black hole. Likewise, the fluid velocity can also be suitably tuned to reproduce the geometry outside a Kerr black hole. A particular type of velocity profile, known as the draining bathtub flow is appropriate for this purpose [105]. Unruh showed that quantised acoustic perturbations (phonons) can be emitted from the horizon of a stationary, spherically symmetric black hole with a thermal spectrum, just like the emission of photons from a black hole horizon via Hawking radiation [103] and the Hawking temperature is proportional to the surface gravity at the horizon. A drawback of this analogy is that it requires the quantisa- tion of linearised acoustic perturbations or the sound field, ψa, in a classical fluid, a step not naturally justifiable. Apart from this, studies by Zel’dovich, Misner and Starobinskii [77, 99, 111, 112] had found that scalar or electromagnetic waves scattered from the ergosphere of a rotating gravitational black hole can extract rotational energy and angular momentum from the hole through the process of superradiance, provided the following criterion is satisfied by the frequency of the incident wave:

0 < ω < mΩH .(68)

Here, m is the orbital momentum of the incoming wave and ΩH is the angular velocity of the black hole horizon. A similar phenomenon, referred to as superresonance, involving the ampliﬁcation of sound waves on reﬂection from a rotating dumb hole has been shown to occur theoretically by Basak and Majumdar in [16, 17]. The condition for superresonance deduced by them is same as Equation 68. 4.2 lorentz violation in acoustic analogue model of gravity 45

4.2 lorentz violation in acoustic analogue model of gravity

But is there an analogue model of gravitation that does not have local Lorentz invariance? In this chapter, from the outset, we have imposed three physical conditions on the fluid that it must be inviscid and barotropic with a locally irrotational flow. The last two make sure that acoustic disturbances can be described by a minimally coupled scalar field, the velocity potential ψa. The first requirement of zero viscosity is responsible for the local Lorentz covariance of Equation 67 [105]. If The local Lorentz the fluid has a non-zero viscosity, the emergent Lorentzian geometry invariance of will be lost as is apparent from the form of the viscous contributions acoustic spacetime is as real to the to Equation 67: acoustic disturbances as is 4 ν ~ 2 ψa = − ∂t +~vb · ∇ ∇ ψa (69) that of physical four 3 ρbcs dimensional spacetime to us. The Here, ρ ,~v and ν are the bulk density, bulk velocity and coefficient b b crucial difference lies of kinematic viscosity of the fluid respectively, the suffix b referring in the fact that the to bulk variables. cs stands for the local speed of sound in the fluid. field ψa feels a In the above equation, the left hand side involves the general curved globally acoustic Lorentzian acoustic metric g while terms on the right hand side cou- general relativity governed by the ple to flat metric of three space – evidently violating Lorentz covari- hydrodynamic 3 ance of the equation. Moreover, new terms of O(|~k| ) and higher equations as opposed appear in the dispersion relation of the acoustic perturbations, also to the dynamical implying a breakdown of local acoustic Lorentz invariance. Under- equations of gravitation by standably, here the acoustic analogue of Lorentz violation becomes Einstein and meaningful only at high values of |~k| because only at such large mo- Hilbert. menta the continuum picture of a classical fluid fails. It is interesting to investigate how superresonance from a rotating dumb hole is affected by the Lorentz violation due to a viscous flow. Physically, viscosity is known to have dissipative and dispersive effects on the propagation of sound waves through a fluid. So, it is expected also to modify the process of superresonance from a rotating dumb hole. This question constitutes the final problem to be addressed in this thesis. In what follows, we present an asymptotic solution to the problem in a limit where certain simplifying assumptions about the governing equations can be used. The detailed derivation presented in section 4.6 shows that a condition same as Equa- tion 68 is obtained but with some more restrictions. Prior to that, in section 4.3, a brief overview of the notable features of superradiance and Hawking radiation from gravitational black holes is presented (based mainly on the wonderful books by Hartle [52] and Penrose [88]). Then, section 4.4 gives a pedagogical account of the standard procedure for obtaining the acoustic metric and the equations of motion of acoustic perturbations in both an inviscid fluid and a viscous one (following [105]). The dispersion relations of sound waves described by a planar solution in both classes of fluids is given in sec- 46 effect of analogue lorentz violation on superresonance

tion 4.5. Finally, following an actual deduction of the new conditions for superresonance in a viscous ﬂuid in section 4.6, we end this chapter with a discussion of our result and prospective improvements in section 4.7.

4.3 radiation from gravitational black holes

Superradiance - a geometric perspective

The seemingly paradoxical proposition that black holes can emit radiation was put forward almost four decades ago by Zel’dovich [111]. As already mentioned, the radiation from black holes comprises of two distinct parts, with somewhat different origins – superradiance and Hawking radiation. Superradiance is a ubiquitous phenomenon in physics, occuring not only in black holes but also in optics and quantum mechanics. Its history dates back to 1954 when the term was introduced by Dicke to refer to ampliﬁcation of radiation due to coherence in an emitting medium [38]. Later, Zel’dovich [111, 112] observed that modes of scalar or electromagnetic radiation falling on a cylinder made of absorbing material and rotating about its axis of symmetry with a frequency Ω are ampliﬁed when

ω < m Ω .(70)

Here, ω is the frequency of the incident wave and m is the projec- While speaking of tion of its angular momentum along the axis of rotation. Zel’dovich, superradiance from Misner and Starobinskii [77, 99, 111, 112] realised that this kind of ro- black holes, we shall tational superradiance is a general feature of any macroscopic rotating always mean rotational dissipative system with internal degrees of freedom causing transfer superradiance. of energy from one medium to another, usually triggered or stimulated by wave scattering. Classically, the horizon of a black hole is a perfect absorber and they expected low frequency waves scattered off the horizon of a rotating (Kerr) black hole to be similarly amplified at the cost of black hole mass and angular momentum. Let us see how this follows from the spacetime geometry outside a rotating black hole given by the Kerr metric. This spacetime is stationary and axisymmetric and by definition contains a timelike Killing vector field κ and a spacelike axial Killing vector field ξ. In the Boyer Lindquist system of coordinates (t, r, θ, φ) for a black hole of mass M and total angular momentum J, the line element is given by [52]

2Mr 4Marsin2θ ρ2 ds2 = 1 − dt2 + dφdt − dr2 − ρ2dθ2 ρ2 ρ2 ∆2 2M r a2 sin2θ − r2 + a2 + sin2θdφ2 ,(71) ρ2 2 2 2 2 = gttdt + 2gtφdtdφ + grrdr + gθθdθ + gφφdφ .(72) 4.3 radiation from gravitational black holes 47

Here,

a = J/M , ρ2 ≡ r2 + a2cos2θ , ∆ ≡ r2 − 2Mr + a2 .(73)

In these coordinates, κ = (1, 0, 0, 0) and ξ = (0, 0, 0, 1). Consequently, there are two conserved quantities associated with test particle motion, the energy E and angular momentum L:

µ t ν φ E = pµκ = gttp , L = pνµ = gφφp ,(74) p being the four momentum of a particle. On a surface r = re(θ) in this geometry, gtt = 0 and for r < re, gtt < 0. So, beyond this surface, α no stationary observer (stationary with respect to infinity i. e. uobs = t (uobs, 0, 0, 0)) with a timelike four velocity is allowed. This surface at r = re(θ) is called the ergosurface or stationary limit surface and the region between the ergosurface and black hole horizon is known as the ergosphere. Any stationary observer coming in from infinity has to co-rotate with the black hole once he crosses the ergosurface, though he may remain at constant (r, θ) as seen from infinity. Obviously, the timelike Killing vector κ also becomes spacelike for r < re:

κ · κ = gtt < 0 .

Thus, inside the ergosphere, fields are permitted to have negative energy, as measured from infinity while still being able to communicate with distant parts of the Universe. We remind that the This remarkable feature of the Kerr spacetime is responsible for su- negative energy perradiance. It becomes possible for a scalar or electromagnetic wave wave falling into the black hole is, when to enter the ergosphere from outside and get scattered there into two viewed locally, parts such that one of the resulting waves has negative energy while nothing but an the other (reflected wave) escapes out carrying more energy than the ordinary wave with original incoming wave. The net result is to carry energy away from an ordinary four momentum. It is just the black hole, slightly reducing the energy of its rotational motion. that the quantity µ The condition for this rotational superradiance is once again the same p · κ = pµκ which as Equation 70: gives the conserved energy as measured from infinity, ω < MΩH .(75) happens to become negative which is Here, ΩH is the angular velocity of the black hole horizon. A similar perfectly permissible procedure for classically extracting energy from a rotating black hole as long as the wave was suggested by Penrose in 1969 [86, 87] involving particles instead lies within the of fields, though it is commonly believed that the negative energy ergosphere. swallowed by the hole is primarily in the form of fields instead of particles.

Hawking radiation - a geometric perspective

In his article on superradiance, Zel’dovich had also afﬁrmed that if the ﬁelds were quantised, rotating objects including black holes 48 effect of analogue lorentz violation on superresonance

would even radiate spontaneously [112]. The mathematical calculation of this effect was carried out by Hawking and he also showed that this mechanism would induce black hole evaporation very gener- ically, even in the absence of rotation [53]. We will just glean over an intuitive argument in favour of spontaneous radiation suggested by Hawking, taking the case of a Schwarzschild black hole, where the spacetime outside the horizon is static [88]. A static spacetime is characterised by a timelike Killing vector field κ. Near the horizon, virtual particle-antiparticle pairs are always being created out of the vacuum only to annihilate each other in a very short period of time. However, from time to time, one of the particles of the pair falls into the black hole, the other one escaping. This can only happen when the escaping particle becomes a real particle (i. e. ‘on shell’ as opposed to the virtual ‘off shell’ particle it started out as) and therefore, the escaping particle must have positive energy. So, from energy conservation, the particle falling into the black hole must also turn into a real particle, but with negative energy. All energies here are measured from infinity. We know that negative energies can occur for real particles inside the black hole because the Killing vector field κ becomes spacelike in the interior of the horizon and a future pointing timelike four momentum p can have a negative scalar product p · κ, this being the (conserved) energy of the particle. The Hawking process comes about because a real (as opposed to virtual) particle can have negative energy if it is inside the event horizon of a black hole. The real outgoing partner of such a particle must have positive energy, thus carrying positive energy away from the hole. The energy in both superradiance and Hawking radiation comes from negative energy fields or particles being swallowed by the black hole, which leads to positive energy escaping from the hole to infinity. However, an important difference is that in a rotating black hole, the part of spacetime where the Killing vector κ becomes spacelike extends outside the black hole horizon. In this spacetime, referred to as the ergosphere, fields or particles can have negative energy while still lying within the domain of outer communication.

A thermodynamic viewpoint on black hole radiation

Alternatively, in 1973, Bekenstein used the formula for the entropy SBH of a black hole whose horizon has an area A: A S = (76) BH 4π discovered by himself and Hawking and Hawking’s black hole area theorem [15] which states that

dA > 0 (77) 4.4 derivation of the acoustic metric 49 in any black hole process to arrive at the criterion in Equation 75 for superradiance from a rotating Kerr black hole. Moreover, in 1973 Hawking showed that a black hole must also have a temperature κ TBH = 2π (in Planck units) as measured by a static observer at infinity, κ being its surface gravity. So, the black hole is expected to emit photons as if it were a physical object in thermal equilibrium, radiating energy with the characteristic black body spectrum. It must be noted that although the Bekenstein Hawking entropy is enormous for astrophysical black holes of a plausible size, the Hawk- For a Schwarzschild ing temperature is absurdly tiny, much less than the temperature of black hole of mass M κ = 1/4M the Cosmic Microwave Background [88]. Such a black hole mostly , . So TBH = 1/8πM. accretes and Hawking radiation is ridiculously small, threatening to Primordial black relegate the latter to only a matter of theoretical interest. In contrast, holes, that may have stimulated emission of radiation from classical rotating black holes formed in the early may have significant astrophysical implications. Indeed, a probable Universe, can have much smaller masses explanation for the huge energy output of a quasar is that this en- and correspondingly ergy comes from the rotation of a gigantic black hole [18, 24, 87, 88, higher Hawking 106–108] . Still, the condition in Equation 75 for superradiance is not temperatures than frequently satisfied and X-rays emitted during the accretion process astrophysical ones. can hide its signals even when present. Observational prospects of phenomena like Hawking radiation and superradiance from physical black hole spacetimes are thus rather bleak.

4.4 derivation of the acoustic metric

Let us consider a classical, non-relativistic ﬂuid characterised by its pressure p, density ρ and velocity ~v. The fundamental equations of ﬂuid dynamics are the equation of local conservation of mass (continuity equation) ~ ∂tρ + ∇ · (ρ ~v) = 0,(78) and the equation for conservation of momentum (Newton’s law) d~v h i ρ ≡ ρ ∂ ~v + (~v · ∇~ )~v = ~F.(79) dt t We shall neglect all external body forces in this chapter.

4.4.1 Case of the inviscid ﬂuid

In the absence of viscosity and external body forces, ~F = −∇~ p . For In an inviscid fluid, our benefit, we shall impose two additional constraints on the fluid Newton’s law is whereby it is taken to be: called the Euler equation. irrotational The fluid velocity obeys ∇~ ×~v = 0 i. e. it is locally vorticity free. This allows us to define a velocity potential ψ as

∇~ ψ = ~v (80) 50 effect of analogue lorentz violation on superresonance

barotropic This effectively means that p(ρ) and ρ(p). A barotropic equation of state ensures that the pressure does not generate vorticity. So, an initially irrotational flow remains so. It is then possible to define the specific enthalpy h(p) of the fluid as p dp0(ρ0) p 1 dp0 h(p) = = dρ0 .(81) ρ0(p0) ρ0 dρ0 Z0 Zo So, dh 1 dh 1 dp = , = (82) dp ρ dρ ρ dρ and 1 ∇~ h(ρ(~x, t)) = ∇~ p(ρ(~x, t)) .(83) ρ An equivalent alternate form of the Euler equation given in Equa- tion 79 is 1 1 ∂ ~v −~v × ∇~ ×~v = − ∇~ p − ∇~ v2 .(84) t ρ 2 which for a barotropic, inviscid fluid reduces to 1 ∂ ψ + h + (∇ψ)2 = 0 ,(85) t 2 after absorbing the constant of integration in the definition of ψ.A derivation of Equation 84 and Equation 85 is given in the appendix in Acoustic section A.1. The latter, in fact, is a version of the Bernoulli’s equation. disturbances or sound waves are fluctuations It is both customary and convenient to separate defined to be low amplitude and high the fluid motion described by the exact variables (ρ, p,~v or ψ) into an frequency linearised average bulk motion given in terms of the bulk variables (ρb, pb,~vb fluctuations in the or ψb) plus low amplitude fluctuations in these variables. Since, we dynamical restrict these fluctuations to small amplitudes, we shall retain terms quantities. The suffix b and a only upto linear order in a small dimensionless parameter . denote bulk variables 2 and acoustic ρ = ρb + ρa + O( ) , fluctuations over 2 p = pb + pa + O( ) , them, respectively. 2 ~v = ~vb + ~va + O( ) , 2 ψ = ψb + ψa + O( ) . as → 0. Using these definitions, we can obtain the linearised versions of continuity equation (Equation 78) and Bernoulli equation (Equa- tion 85). If we combine these two, we get ∂ρ − ∂ ρ (∂ ψ +~v · ∇ψ ) t ∂p b t a b a

∂ρ + ∇ · − ρ ~v (∂ ψ +~v · ∇ψ ) + ρ ∇ψ = 0.(86) ∂p b b t a b a b a

4.4 derivation of the acoustic metric 51

The position and time dependent background fields ρb, pb and ~vb = ~ ∇ψb are allowed to have arbitrary temporal and spatial dependences but with the constraint that they must satisfy the equation of motion of a barotropic, inviscid fluid exhibiting an irrotational flow. Now, the local speed of sound cs is defined by 1 ∂ρ 2 ≡ . cs ∂p This brings us to the following equation describing the propagation of the linearised fluctuation field ψa:

ρb h 2 i i i j 2 ij i − 2 ∂t + vb∂t∂i + vb∂i∂t + vbvb∂i∂j − cs δ ∂i∂j ψa = 0 .(87) cs A little judicious rearrangement of the terms in the above equation casts it in the compact form

µν ∂µ (f ∂ν) ψa = 0 .

Here, we have introduced the (3 + 1) dimensional spacetime coor- Greek indices run dinates xµ = (t, xi) and fµν is a symmetric 4 × 4 matrix having the from 0–3 and denote explicit form spacetime coordinates, while   lower case Latin . j 1 . v indices except a, b ρ b µν b   run from 1–3 and f (t,~x) ≡ − 2 ···················  .(88) cs   represent the purely . j i . 2 ij i spatial degrees of vb . −(cs δ − vbvb) freedom. The similarity of the above equation with the d’Alembertian operator in a general Lorentzian manifold is obvious. So, we may write

µν √ µν ∂µ (f ∂ν) ψa ≡ ∂µ −g g ∂ν ψa = 0 , 1 √ √ ∂ −g gµν ∂ ψ = 0 .(89) −g µ ν a Therefore, √ fµν(t,~x) = −g gµν(t,~x) . g and gµν are the determinant and pointwise matrix inverse of the acoustic metric gµν. Now, if α be a constant multiplying an n dimensional matrix M then we know that

det(αM) = αndet(M) .

Moreover,

det(adj M) = (det M)n−1 .

Armed with these two useful formulae, we can calculate the determinant of fµν: 2 µν √ µν 2 adj gµν g 3 1 4 det(f ) = det( −g g ) = g det = 4 g = g = − 2 ρb . g g cs 52 effect of analogue lorentz violation on superresonance

In the last step, we have incorporated the explicit form of the matrix µν √ 2 f from Equation 88. Therefore 1/ −g = cs/ρb and   . j 1 . vb µν 1 µν 1   g = √ f = − ···················  .(90) −g ρbcs   i . 2 ij i j vb . −(cs δ − vbvb) It is now straightforward to write the acoustic metric itself:   2 2 . j (cs − vb ) . vb ρb   gµν ≡ − ··················· .(91) cs   i . vb . −δij

The effective So, the invariant acoustic interval is, acoustic geometry 2 µ ν couples only to the ds ≡ gµν dx dx scalar field ψa of ρ = − b c2 − v2 dt2 − δ dxi dxj + 2(v ) dt dxi .(92) fluctuations in the c s b ij b i velocity potential s arising due to the Equation 87 or Equation 89 are written in terms of the d’Alembertian propagation of sound operator , defined with respect to g, as waves.

ψa = 0 .(93)

Equation 93 concisely expresses the physical structure of the wave equation Equation 87, and establishes a connection between ﬂuid mechanics and (3 + 1) dimensional Lorentzian geometry. From now on, we have at our disposal the tools of Lorentzian geometry to study ﬂuid mechanics.

4.4.2 Case of a viscous ﬂuid

The governing equations of ﬂuid dynamics, the equations of conservation of mass and conservation of momentum obviously stay unaltered. The change comes through the force ~F in Equation 79 which now has an additional contribution from viscosity: 1 ~F = −∇~ p + ρ ν ∇~ 2~v + ∇~ (∇~ ·~v) .(94) 3

The equation of Here, for simplicity, the coefficient of kinematic viscosity, ν, is taken motion in presence to be position independent, though, in general it need not always of viscous stresses is be so. We have also neglected bulk viscosity of the fluid in order called the Navier Stokes’ equation. to have as few free parameters as possible. Taking the fluid to be barotropic and irrotational once again and repeating the steps that led to Equation 85, we now reduce the Navier Stokes’ equation to 1 4 ∂ ψ + h + (∇~ ψ)2 + ν ∇~ 2ψ = 0.(95) t 2 3 4.5 derivation of acoustic dispersion relation 53

If we carry out the linearisation procedure as before, it again gives us a wave equation in the scalar fluctuation ψa: 1 4 2 − ∂t 2 ρb ∂tψa +~vb · ∇ψa − ν ∇ ψa cs 3 1 4 2 + ∇ · − 2 ρb~vb ∂tψa +~vb · ∇ψa − ν ∇ ψa + ρb ∇ψa = 0. cs 3 (96) but now it includes third order derivatives of the field. After impos- ~ ing the constraint that ρb, pb and ~vb = ∇ψb satisfy the equation of motion of a viscous, barotropic, irrotational fluid, we can simplify µν Equation 96 using the matrix f and the acoustic metric gµν defined earlier. √ µν µν 4 νρb ~ 2 ∂µ (f ∂ν) ψa = ∂µ −g g ∂ν ψa = − 2 ∂t +~vb · ∇ ∇ ψa . 3 cs √ 2 Multiplying by 1/ −g = cs/ρb , we are finally led to

4 ν ~ 2 ψa = − ∂t +~vb · ∇ ∇ ψa .(97) 3 ρbcs Deﬁning an acoustic 4-velocity of the bulk ﬂuid [105] 1 Vµ = √ (1,~v) , ρbcs the above equation is concisely expressed as

4ν 1 µ ~ 2 ψa = − √ V ∂µ ∇ ψa .(98) 3 ρbcs The review article [105] contains an elaborate derivation of Equa- tion 98.

4.5 derivation of acoustic dispersion relation

A dispersion relation expresses the momentum of a wave as a function of its energy. Let us assume ψa to be a plane wave

i(ωt−~k·~x) ψa(x) = A(~x)e (99) obeying the eikonal approximation according to which the amplitude A(~x) is taken to be a slowly varying function compared to the exponential. To keep things simple, we ignore derivatives of the metric. Substituting this into Equation 87, i. e., the equation of motion of ψa in an inviscid ﬂuid, we get the linear dispersion relation

2 ~ 2 ~ 2 ω −~vb · k = cs |k| (100) ~ ~ ω = ~vb · k ± cs|k| .(101) 54 effect of analogue lorentz violation on superresonance

This equation says that there is a bulk motion of the waves due to the fluid flow together with the propagation of the sound waves in the medium with a group velocity cs. Likewise, in a viscous fluid we ~ obtain the following quadratic equation in ω −~vb · k [105]: 2 4ν ω −~v · ~k + i ω −~v · ~k |~k|2 − c2|~k|2 = 0 b 3 b s whose roots give the dispersion relation v u !2 u 2ν|~k|2 2ν|~k|2 ω = ~v · ~k ± tc2|~k|2 − − i .(102) b s 3 3

In this non-linear dispersion relation, the ν dependent term under the big square root gives rise to dispersion while dissipative effects of viscosity are brought in by the imaginary term. Just like in the previous chapters, we can interpret this modified non-linear dispersion relation as an indicator of acoustic Lorentz violation. Note that the dispersive and dissipative acoustic Lorentz violating terms in Equa- tion 102 contribute only at high momenta ~k. This agrees intuitively with the fact that at high momenta the continuum fluid model breaks down thereby invalidating the assumption of a continuous ‘acoustic spacetime’. However, unlike in the previous chapters where ad hoc modifications of the dispersion relations have been carried out, here a similar though not identical breakdown of acoustic Lorentz invariance has been deduced directly from the hydrodynamic equations by taking into account the effects of viscosity [105]. Both the viscous wave equation and the viscous acoustic dispersion relation make this breakdown of local Lorentz covariance manifest.

4.6 acoustic superradiance in a viscous fluid

Gravitational superradiance can occur only in the axisymmetric, stationary spacetime outside a rotating black hole because the ergosphere, i. e. the region where the time translation Killing vector becomes spacelike, extends outside the event horizon to the domain of outer communication. Fluid models of black hole kinematics enjoy the advantage that the effective geometry outside the sonic horizon is completely determined by the bulk velocity profile, provided the fluid is irrotational and barotropic. A flow that may suitably mimic a rotating black hole must have a radially inward plus a tangential component of the velocity such that it is locally vorticity free. These properties are inherent to the draining bathtub flow with a sink at the origin. Without loss of generality, we can work with a fluid flowing in two dimensions with a bulk velocity A B ~v = − rˆ + φˆ .(103) b r r 4.6 acoustic superradiance in a viscous fluid 55

A, B are real, positive constants and (r, φ) are plane polar coordinates. ~ ~vb is not only locally irrotational (∇ ×~vb = 0) but is also locally diver- ~ gence free (∇ ·~vb = 0). Substituting the draining bathtub background velocity (Equation 103) into Equation 92 we get the corresponding (2 + 1) dimensional rotating dumb hole metric

2 2 2 ρb 2 A + B 2 2 2 2 dsDB = − cs − 2 dt − dr − r dφ cs r 2 A − dr dt + 2B dφ dt .(104) r

Clearly, the draining bathtub metric does not exactly correspond to a Kerr geometry, not even to a section of it. What is important is that, the two are qualitatively similar. This stationary and axisymmetric metric possesses symmetries corresponding to time translations ∂ generated by a timelike Killing vector field κ = ∂t and planar rota- ∂ tions generated by a spacelike Killing vector ξ = ∂φ . On the 2-surface A at rH = in this flow, the fluid velocity is everywhere inward point- cs ing and just beyond it, the radial component of the fluid velocity exceeds the local speed of sound. In the acoustic geometry, this surface behaves as an outer trapped surface and can be identified with the future event horizon of the black hole. As in the Kerr black hole, the radius of the ergosphere is determined√ by the vanishing of the metric 2 2 component g00 which occurs at rE = A + B /cs. To keep calculations simple, we restrict the coefficient of kinematic viscosity of the fluid to be so small that terms of O(ν2) and higher may be neglected with respect to ν as ν → 0 and also assume that the background density ρb remains constant. This in turn implies the constancy of the bulk pressure (owing to barotropicity) and the local speed of sound. With this, a further simplification becomes possible. We rescale the dimensions and set the local speed of sound to unity i. e. cs = 1, for the rest of this chapter. The metric of Equation 104 then becomes The overall constant factor of ρb A2 + B2 ds2 = −ρ 1 − dt2 − dr2 − r2 dφ2 appearing in the DB b r2 metric can be safely 2 A neglected. But we − dr dt + 2B dφ dt .(105) keep it to simplify r calculations. The viscous wave equation (Equation 97)

4 ν ~ 2 ψa = − ∂t +~vb · ∇ ∇ ψa 3 ρbcs 56 effect of analogue lorentz violation on superresonance

when written out explicitly in the (t, r, φ) coordinates has the form A2 1 B2 2A 2B − ∂2 + 1 − ∂2 + 1 − ∂2 + ∂ ∂ − ∂ ∂ t r2 r r2 r2 φ r t r r2 t φ 2AB 1 A2 2A2 2AB + ∂ ∂ 1 − + ∂ − ∂ ψ r3 φ r r r2 r3 r r4 φ a

4ν A B A B = − − ∂3 + ∂3 − ∂ ∂2 + ∂2∂ 3 r r r4 φ r3 r φ r2 r φ 1 A 2A B + ∂2 ∂ + ∂2∂ − ∂2 + ∂2 + ∂ ∂ r2 φ t r t r2 r r4 φ r3 r φ 1 A + ∂ ∂ + ∂ ψ .(106) r r t r3 r a The massless Klein Gordon equation is known to allow complete separation of the variables in the Kerr metric [26, 28]. The same holds for the Klein Gordon equation with a mass term. As a result, the homogeneous part of Equation 106, which is nothing but the inviscid wave equation (Equation 93) in a draining bathtub metric, is also separable and the (φ, t) dependence is given by the usual eigenfunctions appropriate to an axially symmetric and stationary background geometry. namely, exp(−iωt + imφ) [16, 17]. Here, ω and m are the frequency and orbital momentum of the wave parallel to the symmetry axis. 3 3 In Equation 106, the highest order derivatives ∂r , ∂φ are suppressed by the parameter ν multiplied by a factor which is always less than one because we are interested in a solution outside the acoustic horizon so that r(A, ). So, let us start by assuming a separable solution to the viscous wave equation (Equation 106): ∞ ψa(t, r, φ) = T(t)R(r)Φ(φ) . If we put ν = 0 in We tune the flow parameters A, B such that B << A. This helps us go the equations 3 to the limit where the term B∂φψa in Equation 106 may be ignored in obtained in this 2A∂2 ψ section, we are taken comparison to φ a. As a result, the highest order of axial deriva- back to the standard tive occuring in Equation 106 is reduced to two. We employ the ansätz derivation of ψ (t, r, φ) = R(r)e−iωt+imφ .(107) superresonance in a an inviscid fluid. Following the standard steps, we can find the separated radial equation 4ν A A2 4ν A Bm d2 − ∂3 + 1 − − + i ω − R(r) 3 r r r2 3 r2 r2 dr2

1 A2 2A2 i2A Bm + 1 − + − ω − r r2 r3 r r2 4ν A Am2 i Bm d + + − ω − R(r) 3 r3 r3 r r2 dr

Bm2 m2 i2ABm + ω − − − r2 r2 r4 4ν2Am2 im2ω − − R(r) = 0 .(108) 3 r4 r2 4.6 acoustic superradiance in a viscous fluid 57

This is a linear third order ordinary homogeneous differential equation in R(r). Ideally, it should be possible to analytically find a general solution to this linear equation, but the problem becomes rather in- tractable. So, to get a first approximation to the complete answer, we instead study the asymptotic form of Equation 108 in the limit where 1 d3R(r) << d2R(r) . r r r This is equivalent to requiring 1 d3ψ << d2ψ . r a r r a Thus, restricting the parameter space by the criterion B << A, employing ansätz of Equation 107 and finally constraining ψa to vary slowly with respect to r, we are able to reduce Equation 108 to the following second order, linear differential equation

A2 4ν A Bm d2 1 − − + i ω − R(r) r2 3 r2 r2 dr2

1 A2 2A2 i2A Bm + 1 − + − ω − r r2 r3 r r2 4ν A Am2 i Bm d + + − ω − R(r) 3 r3 r3 r r2 dr

Bm2 m2 i2ABm + ω − − − r2 r2 r4 4ν2Am2 im2ω − − R(r) = 0 .(109) 3 r4 r2

But we can make things simpler. Let us deﬁne a new radial coordinate r∗ called the tortoise coordinate by

A2 4ν A Bm d d 1 − − + i ω − ≡ (110) r2 3 r2 r2 dr dr ∗ and a new function as

R(r∗) = R(r) .

This deﬁnition makes the tortoise coordinate a complex variable and When ν = 0, r∗ the differential equation is generalised from the real r axis to the com- remains real. plex r∗ plane. The bar over R in R is used to denote its dependence on r∗ and in the following formulae we shall place a bar over all other 58 effect of analogue lorentz violation on superresonance

functions of r∗. Under the transformation r → r∗ and R(r) → R(r∗), Equation 109 becomes,

d2 1 A2 i2A Bm 2 R(r∗) + 1 − 2 − ω − 2 dr∗ r r r r 4ν Am2 A i Bm i2Bm d + − − ω − + R(r ) 3 r3 r3 r r2 r3 dr ∗ ∗ A2 4ν A Bm Bm2 m2 + 1 − − + i ω − ω − − r2 3 r2 r2 r2 r2

i2ABm 4ν2Am2 im2ω − − − R(r ) = 0 .(111) r4 3 r4 r2 ∗

Note that the coefficients here remain functions of r. They have not been changed because we do not know explicitly r(r∗). In fact, we do not need to know this inverse transformation, as will become clear gradually. If we denote the coefficients of dR(r∗)/dr∗ and R(r∗) by a(r) = a(r∗) and b(r) = b(r∗) then Equation 111 is simply of the form d2 d 2 R(r∗) + a(r) R(r∗) + b(r)R(r∗) = 0 .(112) dr∗ dr∗ We wish to write this as a one dimensional equation with an effective potential by defining another radial function ζ(r∗). In other words, we plan to make a transformation from R(r∗) → ζ(r∗) in a way that will make the coefficient of dζ/dr∗ zero. If

1 r∗ R(r ) ≡ exp − ds a(s ) ζ(r ) ∗ 2 ∗ ∗ ∗ Z then Equation 112 is indeed reduced to

2 d 2 2 ζ + k (r∗)ζ = 0 (113) dr∗ where retaining terms upto O(ν),

2 2 k (r∗) = k (r) Bm2 A2 A2 1 m2 A2 ≈ ω − + 1 − 1 − − − r2 r2 r2 4r2 r2 r4

2ν A2 3A 2Am2 i Bm − 1 − + + ω − 3 r2 r4 r4 r2 r2

i2m2ω i6Bm Bm3 2A Bm2 − − + 2i ω − + ω − r2 r4 r2 r2 r2

Bm 4ABm i2m2 i2A2 + ω − − − (1 + m2) r2 r4 r2 r4

2Am2 2A3 − − (114) r4 r6 4.6 acoustic superradiance in a viscous fluid 59 as ν → 0. Any differential equation of the form of Equation 113 (with vanishing coefﬁcient of the ﬁrst derivative) is known to have a constant Wronskian. This allows us to solve the asymptotic forms of Equation 113 in the limiting cases of r → A (the acoustic horizon) and r → and equate the corresponding Wronskians in order to deduce the conditions for superresonance (following Basak and Ma- jumdar [16∞, 17]).

Near horizon

The surface at r = A acts as an acoustic horizon rotating at the angular velocity B Ω = . H A2 Owing to the restrictions on the parameter space (B << A), the angular velocity of the horizon is quite small. Now, as the radial coordinate r → A, let the functions ζ(r∗) → ζhor(r∗) and 2 2 k (r) → khor(r) 4ν 1 2mΩ 1 + m2 ≈ (ω − mΩ )2 − (ω − mΩ )2 + (ω − mΩ ) H − H 3 H A H A A3 1 + 2m2 + i(ω − mΩ )3 − i(ω − mΩ ) (115) H H A2 as ν → 0, neglecting terms of O(ν2) and above. Therefore, 2ν 1 + m2 k ≈ (ω − mΩ ) − (ω + mΩ ) − hor H 3A H A2(ω − mΩ ) H 2ν 1 + 2m2 + i − (ω − mΩ )2 .(116) 3 A2 H

A solution to the approximate near horizon differential equation

2 d 2 2 ζhor + khorζhor = 0 (117) dr∗ can easily be written down as

−ikhorr∗ ζhor1(r∗) = Tωme .(118)

Here, Tωm is the transmission coefﬁcient and we have incorporated the boundary condition that the group velocity of the wave for r → A is directed towards the trapping surface. Another linearly independent solution to Equation 117 can be

∗ ikhorr∗ ζhor2(r∗) = Tωme .(119)

The corresponding Wronskian Whor is given by,

d d 2 Whor = ζhor1 ζhor2 − ζhor2 ζhor1 = 2ikhor|Tωm| .(120) dr∗ dr∗ 60 effect of analogue lorentz violation on superresonance

Near inﬁnity

In the asymptotic region where r → , the function ζ(r∗) → ζinf(r∗) and 4ν ∞ 2ν 2 k2(r) → k2 (r) ≈ ω2 − i ω3 ≈ ω − i ω2 (121) inf 3 3 as before keeping only upto linear terms in ν as ν → 0. So, 2ν k ≈ ω − i ω2 .(122) inf 3 A solution to the approximate differential equation

2 d 2 2 ζinf + khorζinf = 0 (123) dr∗ is

ikinfr∗ −ikinfr∗ ζinf1(r∗) = Rωme + e .(124)

Here, Rωm is the reﬂection coefﬁcient in the sense of potential scattering and the amplitude of the incident wave is normalised to unity. Another linearly independent solution to Equation 123 is

∗ −ikinfr∗ ikinfr∗ ζinf2(r∗) = Rωme + e .(125) So, the Wronskian

d d 2 Winf = ζinf1 ζinf2 − ζinf2 ζinf1 = 2ikinf(1 − |Rωm| ) .(126) dr∗ dr∗

Comparison of Wronskians

The Wronskian of Equation 114 being constant, we have

Whor = Winf , 2 2 2ikhor|Tωm| = 2ikinf(1 − |Rωm| ) , 2 1 − |Rωm| khor 2 = .(127) |Tωm| kinf It is ﬁrst necessary to calculate the asymptotic formula for the ratio khor/kinf as ν → 0 using Equation 116 and Equation 122. k 1 2ν 1 + m2 hor ≈ (ω − mΩ ) − ω + mΩ − k ω − i 2ν ω2 H 3A H A2(ω − mΩ ) inf 3 H 2ν 1 + 2m2 + i − (ω − mΩ )2 . 3 A2 H

Now, as ν → 0, 1 1 1 2ν = ≈ + i . 2ν 2 2ν ω 3 ω − i 3 ω ω(1 − i 3 ω) 4.6 acoustic superradiance in a viscous fluid 61

So, in this limit, when O(ν2) or higher contributions are neglected,

k 1 2ν 1 + m2 hor ≈ ω − mΩ − ω + mΩ − k ω H 3A H A2(ω − mΩ ) inf H 2ν 1 + 2m2 + i − (ω − mΩ )2 + (ω − mΩ )ω .(128) 3ω A2 H H

However, from Equation 127 it is clear that khor/kinf must be real. So, In an inviscid fluid, no such condition 1 + 2m2 appears and mΩ (ω − mΩ ) = − .(129) H H A2 scattering can occur for the entire range As the expression to the right of the equality is always negative, this of frequencies. But equation puts the following constraint on ω: there will be superresonance only when ω < mΩH . 0 < ω < mΩH. This must hold to allow scattering of acoustic perturbations off the ergosphere of a dumb hole in a fluid with a small kinematic viscosity ν if the conditions specified above are true. Now, when superresonance occurs, intensity of the reflected wave 2 is greater than that of the incident wave i. e. |Rωm| > 1. Equation 127 implies that a necessary condition for superresonance is, k hor < 0 .(130) kinf So, since ω 6= 0,

2ν 1 + m2 ω − mΩ − ω + mΩ − < 0 . H 3A H A2(ω − mΩ ) H Substituting for the last term using Equation 129 we have,

2ν m2 ω − mΩ − ω + 2mΩ + < 0 . H 3A H A2(ω − mΩ ) H

Now, we multiply by (ω − mΩH) but as ω < mΩH, this reverses the sign of the inequality.

2ν m2 (ω − mΩ )2 − (ω + 2mΩ )(ω − mΩ ) + > 0 , H 3A H H A2

2ν m2 ω2 + m2Ω2 − 2ωmΩ − ω2 − 2m2Ω2 + ωmΩ + > 0 , H H 3A H H A2 2ν ν 4ν 2νm2 1 − ω2 − 2 1 + mΩ ω + 1 + m2Ω2 − > 0 . 3A 3A H 3A H 3A3 (131)

The roots of the corresponding equation

2ν ν 4ν 2νm2 1 − ω2 − 2 1 + mΩ ω + 1 + m2Ω2 − = 0 3A 3A H 3A H 3A3 62 effect of analogue lorentz violation on superresonance

are given by Sridhar Acharya rule:

2ν −1 ν ω = 1 − 1 + mΩ 3A 3A H s ν 2 2ν 4ν 2νm2 ± 1 + m2Ω2 − 1 − 1 + m2Ω2 − , 3A H 3A 3A H 3A3

r 2ν ν 2νm2 ω ≈ 1 + 1 + mΩ ± , 3A 3A H 3A3 r ν m 2ν ω ≈ 1 + mΩ ± . A H A 3A

In the last two steps we have ignored terms of O(ν2) or above as ν → 0. Inserting this into the inequality of Equation 131, we have r r ν m 2ν ν m 2ν ω − 1 + mΩ − ω − 1 + mΩ + > 0 . A H A 3A A H A 3A

Since, ω < mΩH, the expression within the ﬁrst set of brackets is negative for all values of ω. As a result, we must have r ν m 2ν ω − 1 + mΩ + < 0 , A H A 3A r ν m 2ν ω < 1 + mΩ − .(132) A H A 3A

We must convince ourselves that this result is consistent with Equa- tion 129 ( ω < mΩH). It will be so if r m 2ν ν > mΩ , A 3A A H 2ν ν2B2 > . 3A A4 The last step is obtained after dividing by the common non-zero factor m/A and taking the square of both sides. We have also substituted the formula for ΩH. This leads to the relation 2 νB2 > 3 A3 which obviously is true because we have made the choice that ν << 1 and B/A << 1. Hence, we can conclude that in the limit where our assumptions hold good, sound waves will be scattered off the ergosphere of a rotating dumb hole when ω < mΩH and there will be superresonance for all such waves. As we have come almost to the end of this section, it is best to summarise the assumptions we have made and the conclusions they lead us to. Below is a list of the assumptions we have employed to simplify the problem: 4.7 discussion 63

1. Kinematic viscosity ν << 1 i. e. ν → 0 so that terms higher than O(ν) can be neglected.

2. Solutions to Equation 106 are separable.

3. Bulk fluid flow is such that B << A (tangential fluid velocity B 3 is much smaller than radial component) implying r4 ∂φψa << 2A 2 r4 ∂φψa , r ∈ (A, ). 4. ψ = R(r)exp[−iωt + imφ], ω, m > 0 and m takes integral val- a ∞ ues

5. We consider regions of slow variation of R(r). So, Equation 106 3 reduces approximately to Equation 109. Here, dr R has been ne- 2 glected in comparison to (1/r)dr R. Under such conditions, a physical solution to the approximate radial equation near the horizon is given by

−ikhorr∗ ζhor1(r∗) = Tωme 2ν m2 −i (ω−mΩ )− (ω+2mΩ )+ r∗ H 3A H A2(ω−mΩ ) = Tωme H 2ν mΩ −ωωr × e 3 H ∗ (133) where we have made use of Equation 116, Equation 118 and Equa- tion 129. Though, there appears an exponentially growing term in the solution, it does not pose any threat because the solution itself is valid only very close to the dumb hole horizon at r = A. Similarly, a solution to the approximate radial equation near r → is given by

ik r∗ −ik r∗ ζ 1(r∗) = Rω e inf + e inf inf m ∞ 2ν 2 2ν 2 iωr∗ ω r∗ −iωr∗ − ω r∗ = Rωme e 3 + e e 3 .(134) Equating the Wronskians of the two approximate equations, we have ﬁrst deduced that ω must always be less than mΩH for the scattering process to occur and then showed that all such solutions to Equa- tion 106 under the above conditions will be ampliﬁed by extracting rotational energy from the dumb hole. The assumptions made by us have somehow conspired to always suppress the frequency ω below mΩH. We believe this constraint will be abolished once we can afford a more general analysis of Equation 106.

4.7 discussion

Our motivation for addressing this problem has been to know how emergent Lorentz violation of the acoustic wave equation given in Equation 69 affected superresonance of acoustic disturbances from a rotating acoustic black hole. Upto a ﬁrst approximation, when conditions 1 − 5 given above are satisﬁed, acoustic Lorentz violation re- stricts the range of allowed frequencies for a scattering process to be 64 effect of analogue lorentz violation on superresonance

always lower than mΩH. Under such circumstances, the criterion for superresonance remains unaltered. In fact, all sound waves that get scattered off the ergosphere of a rotating dumb hole are now amplified at the cost of rotational energy of the dumb hole. Naturally, we wish to improve our result and not only see how the criterion for superresonance is modified but also check whether the upper bound on values of ω is relaxed. However, it will be foolhardy to immediately start off on such a course by keeping terms upto the next higher order i. e. O(ν2). Remember that we have not yet solved Equation 108 for all values of r. Undoubtedly, an exact or approximate global solution will be most informative. The question remains, how should we proceed! When ν = 0 in Equation 97 or Equation 106, the problem is solv- able as r → A and r → [16, 17]. Moreover, if we are content to restrict ourselves to very small values of ω, then the full radial solution is known in terms of∞ the hypergeometric function [16]. Thus, we may be tempted to straightaway apply regular perturbation theory to get a solution to our third order linear differential equation (Equa- tion 106). But there are some issues demanding attention. The (small) parameter ν here multiplies the highest derivative in Equation 106, apart from introducing additional ν dependent contributions to the lower derivatives. The former is a signature of a singular perturbation problem as discussed in [21]. The nature of the solution is completely different as ν → 0 and at ν = 0. A solution to the differential equation vanishes when ν = 0 because the order of the differential equation is reduced from three to two. Moreover, Equation 106 is a partial differential equation and it is not possible to separate the equations for each of the three variables (T(t), R(r), Φ(φ)) because of the presence of mixed derivative terms. Even so, we expect exponential corrections to appear in the solution to the azimuthal part. There is much interesting feature in this problem to warrant future studies beyond just a first approximation. A WKB analysis will hopefully be useful in arriv- ing at a global approximate solution because the problem has all the appropriate characteristics [21]: a. highest derivative is multiplied by a small parameter ν

b. solution oscillates rapidly, globally

c. the problem describes a dissipative as well as a dispersive phenomenon Fortunately, WKB theory is known to provide a simple yet powerful global approximation technique for linear differential equations with these characteristics. An insightful exposition on the WKB theory is given by Bender and Orszag in [21]. Apart from being a study into Lorentz violation, it is by itself interesting to know the effects of viscosity on superresonance. But the interest is not only theoretical. Owing to the requirement of zero viscosity, it is not possible to use real fluids as black hole models. One 4.7 discussion 65 is forced to look into ideal fluids like superfluid helium. But, if we know how viscosity modifies the outcome in a real fluid, it may be straightforward to deduct viscous effects and extrapolate a result to the real, inviscid case.

CONCLUSION 5

5.1 a summary on lorentz preserving fields

Inspired by the idea that Planck scale physics may lead to departures from exact Lorentz invariance at low energies E (E << MPl), physicists have invested huge effort in the theoretical, phenomenological and experimental exploration of this possibility. With no es- tablished theory of quantum gravity on hand, a lack of theoretical principles to guide this enthusiasm has been badly felt. Still, the wis- dom gained from our knowledge of quantum field theory, general relativity and Lorentzian geometry has helped immensely in identi- fying underlying discrepancies of some radically new propositions. Phenomenological models of Lorentz violation, on the other hand, have been numerous and diverse (a review can be found in [60]). But most of them share a common feature – they assume that spacetime is granular and there is a minimum length scale given by the Planck length lPl which brings in higher order (i. e. of order greater than two in momentum) corrections to the dispersion relations of free elementary particles at energies E << MPl. So, the Lagrangian density must be accordingly modified to account for the extra terms in the dispersion relation. Some of the proposed Lorentz violating La- grangian densities incorporate only renormalisable operators while others prefer only non-renormalisable ones. Obviously, any effect of Lorentz violation at observable energies must be very tiny. In this regard, non-renormalisable operators have the advantage that they can be suppressed by inverse powers of the Planck mass, MPl. Usually, in all such phenomenological models, Lorentz invariance is explicitly destroyed by assuming that spacetime has a preferred direction. The Lorentz non-invariant phenomenological model propounded by Myers and Pospelov and discussed in this thesis has mass dimension five operators in which matter fields of spins 0, 1/2 and 1 couple to a constant vector n in flat spacetime. Here, the fields themselves transform under appropriate representations of the Lorentz group. However, the Lagrangian density does not transform as a Lorentz scalar, unlike in standard field theory. In chapter 2, we have described how a class of scalar and spinor fields can be chosen that make their corresponding Lorentz violating Lagrangian densities transform as four scalars under infinitesimal Lorentz boosts [47]. But these ‘Lorentz preserving fields’ have the odd property that their coordinate dependence is separable into dependence on parallel and orthogonal co-

67 68 conclusion

ordinates, xk and x⊥, defined with respect to n. That is, the Lorentz preserving scalar field φ(x) and spinor field ψ(x) must have the form,

φ(x) = φk(xk) + φ⊥(x⊥) ,

ψ(x) = ψk(xk) + ψ⊥(x⊥) .

Clearly, (φk, φ⊥) and ψk, ψ⊥ do not confirm to the definitions of a Lorentz scalar or spinor. If the coordinate axes are oriented such that n = (1,~0), the Lorentz preserving fields become the sum of a spatially homogeneous part and a time dependent spatially inhomogeneous part. The scenario is rather different for the deformed Maxwell electrodynamics [82] – the Lorentz preserving electric and magnetic field three vectors here have to be harmonic functions of spacetime coordinates and satisfy the Bianchi identity (chapter 2). Irrespective of their particular functional dependence, the Lorentz preserving fields give us the benefit of using standard tools of field theory, relying on Lorentz symmetry of background spacetime, to study Lorentz violating dynamics. A reformulation of ordinary electrodynamics in terms of gauge free potentials has been carried out in [22, 75] by pruning away the redun- dant degrees of freedom of the electromagnetic potential from the outset, all the while retaining manifest Lorentz invariance and local- ity in four dimensional Minkowski spacetime. It can be an insightful exercise to design a Lorentz violating term to be added to the gauge free electrodynamics action following the principle of Myers and Pospelov and see how the Lorentz preserving fields turn out. Moving on, in chapter 3, we have adopted Ostrogradskii’s canonical formalism for a higher derivative theory to demonstrate that Lorentz algebra can indeed be realised on a basis of Lorentz preserving scalar fields [47]. For this purpose, we studied two situations distinguished by a unit timelike and a unit spacelike fixed background vector and derived the canonically conjugate pairs of variables in each case. Our result has been further verified by constructing a covariant phase space with a Poincaré invariant symplectic structure for the Lorentz violating scalar dynamical system by restricting the allowed field configurations to the Lorentz preserving fields. Due to the presence of higher time derivatives, negative norm states appear which allow the probability to be negative and break unitarity. In fact, Os- trogradskii had showed that the presence of time derivatives greater than two generally leads to the problem of ghosts. It appears that one can set up a well posed initial value formalism of the Myers Pospelov higher derivative theory with respect to our field solutions thus elim- inating ghost states. 5.1 a summary on lorentz preserving fields 69

5.1.1 Criticism and outlook

All experiments till date put very stringent constraints on departures from Lorentz symmetry in the matter sector of the standard model. The parameters ξ, η1, η2 quantifying Lorentz violation of the photons and fermions respectively are not only many orders of magnitude less than unity, they are also restricted to extremely narrow regions of the parameter space. It is very uncommon for dimensionless parameters emerging from a theory with only one scale to differ much from O(1). But this is what is happening here with the parameters even after the energy scale of quantum gravity, MPl, has been factored out [57, 72]! There is another problem with dimension five Lorentz non-invariant operators. Radiative corrections due to particle interactions do not preserve a dispersion relation of the form given in Equation 7, Equa- tion 8 and Equation 9. Instead they naturally induce lower dimensional Lorentz violating terms of O(p) and O(p2) which are not suppressed by the Planck mass and so dominate even over the standard p2 term in the dispersion relation [35, 101]. This problem can be avoided only by an extreme fine tuning of the parameters which is rather un- natural in the absence of a custodial symmetry. Thus, Lorentz violation of standard model fields by dimension five operators of the form predicted by Myers and Pospelov is effectively ruled out. Attempts to use CPT even, dimension six operators instead have also failed because they too suffer from the same disease [60, 72, 76]. Lorentz violation in other areas of physics like gravitation [92], dark matter [25] and inflation [62] are somewhat less con- strained. But their study requires us to include local Lorentz violation in curved spacetime. The two well known propositions in this regime are Einstein aether theory [58] and Hoˇrava Lifshitz gravity [54, 80]. Re- sults following from Einstein aether theory show that contribution of Lorentz violation to inflationary dynamics can only affect the cosmic microwave background (CMB) by an unobservably small amount. Still, it is a possibility that the technique of constructing Lorentz preserving fields φ(x) through a superposition of a spatially homogeneous and an inhomogeneous piece may prove useful in describing the growth of inhomogeneities in the early Universe. An inhomogeneous contribution to the energy density of the inflaton field, similar to the one described in Equation 46 at the end of chapter 2, may also lead to Jeans instability or some other aspect of structure formation in a Friedmann-Lemaître-Robertson-Walker background. Lastly, there is another very important issue that needs to be taken care of. Three general algebraic conditions on every classical dispersion relation have been derived by Raetzel, Rivera, and Schuller [93] starting just from the fundamental requirements that

1. the matter ﬁeld dynamics must be predictive 70 conclusion

2. there must be a well deﬁned observer independent notion of positive energy.

Restrictions on dispersion relations understandably translate also to the underlying spacetime geometries. In particular, the dispersion relation associated with the deformed Maxwell theory by Myers and Pospelov does not meet these criteria [93]. This supports our previous observation that dispersion relations with cubic modiﬁcations are not physically viable models of Planck scale induced Lorentz violation.

5.2 lorentz violating effects on superresonance

It is intriguing that sound waves in a flowing medium evolve under the massless Klein Gordon equation like minimally coupled scalar fields in a curved, non-dynamic, Lorentzian geometry. If the flow becomes hypersonic in a region bounded by a closed surface which marks the transition from subsonic to supersonic flow, then no sound wave can travel outside this surface from within it against the flow. This surface acts as an acoustic horizon just like the event horizon of a black hole. So, we have an acoustic black hole or a dumb hole. The theoretical construction of a fluidic black hole, one of the first analogue gravity systems, is based on three assumptions about the nature of the fluid, namely, that it must be inviscid, irrotational and barotropic. A zero viscosity ensures that an originally vorticity free flow does not develop any local vorticity due to viscous effects. Vis- cosity further complicates affairs because it leads to a dispersive and dissipative dispersion relation for the acoustic perturbations, as explained in chapter 4. The new terms are of O|~k|3 and higher and their appearance implies that local Lorentz invariance of the effective acoustic spacetime is ruined by viscosity. This complicated state of affairs is just what we wanted because it gives us an opportunity to study violation of Lorentz invariance in a relatively simple model in the laboratory. The primary target of the analogue gravity programme has been to look for black hole radiation, both superradiance and Hawking radiation. This motivated us to investigate how superresonance is modified in a spacetime geometry without local Lorentz invariance. To this end, we have explicitly derived the criterion for superresonance of acoustic perturbations in the velocity potential ψa from the ergosphere of a (2 + 1) dimensional rotating dumb hole, in the limit where tangential velocity of the fluid is much less than its radial velocity. The coefficient of kinematic viscosity has also been restricted to be so small that terms of O(ν2) and higher can be neglected in comparison with other parameters describing the system. In addition, our results hold only in the regime where the radial part R(r) of the field ψa(t, r, φ) is slowly varying. With all these simplifying conditions in place, we have deduced that in this case an incoming scalar wave will be scattered from the ergosphere, 5.2 lorentz violating effects on superresonance 71 giving rise to an infalling transmitted wave and an outgoing reflected wave only when

0 < ω < mΩH .(135)

But this is also the well known criterion for superresonance. Thus, it so happens that in this system, all reflected waves will be superresonant i. e. they will extract energy from the black hole and will carry away more energy than they originally brought in. The introduction of the new upper bound on the frequency ω is somewhat counterintuitive. We think that the assumption of a slowly varying R(r) ceases to hold as we approach the ergosphere. It is thus important to get an asymptotic global solution of the problem taking into account the highest derivative radial contribution. As discussed at the end of chapter 4, WKB analysis seems to provide a suitable tool for this purpose. However, a study of superresonance in a viscous fluid not only is relevant from the perspective of understanding Lorentz violation, it also helps us predict how realistic classical fluids (as opposed to inviscid, ideal ones) may be used in experiments.

5.2.1 Observational prospects of radiation from ﬂuidic black hole

Successful experimentation on analogue systems have been possible only in the last few years. The classical fluid based model usually relying on the assumption of an inviscid flow is difficult to realise in practice because no classical fluid has a zero viscosity. Moreover, Hawk- ing radiation is a consequence of the physics of quantum fields on a curved spacetime. Thus to be able to reproduce Hawking radiation in the laboratory, we must have a quantum analogue model such that its description can be separated into a classical effective background spacetime plus some some standard relativistic quantum fields living in it [13]. A classical fluid does not fit the bill. Alternatively, superresonance can be described fully by classical general relativity though it is technically very difficult to achieve the level of precision needed for the measurement of tiny signals of superresonance from a fluid exhibiting draining vortex flow.

5.2.2 Gravity waves and Bose Einstein condensates

But all hope is not lost. During the last few years, the analogy has been extended to many other condensed matter systems ranging from gravity waves, superﬂuid helium, Bose Einstein condensates (BEC) to nonlinear electrodynamic systems [13, 43, 84]. A characteristic showed by all these models is that they consist of a ﬂowing medium through which some kind of perturbation or excitation is propagated. Under certain conditions determined by the precise nature of the system, it appears that these perturbations propagate over effective Lorentzian 72 conclusion

geometries, at least at low momenta. An analogue of a black hole horizon can be formed at the surface just inside which the ingoing normal velocity of the flow becomes greater than that of the perturbations or excitations because no signal from inside the horizon will be able to travel against the flow and reach the outer world. In a shallow basin filled with liquid, gravity acts as a restoring force on surface waves formed in the liquid. Here, we are talking about a classical liquid in a Newtonian gravitational field. When the wavelength of gravity waves is long compared to the depth of the basin and the liquid has zero viscosity and no vorticity, the gravity waves “see" an effective Lorentzian metric [13, 94]. By changing the depth of the basin and the background flow velocity of the liquid from point to point, it is relatively easy to map various Lorentzian geometries. In fact, the analogue of a black hole horizon has been formed very recently in the laboratory using gravity waves on water [97]. A nu- merical simulation also shows that considering realistic experimental parameters, gravity waves with frequency in the range given by Equa- tion 135 can be amplified due to superresonance [94]. In the past few years, Bose Einstein condensates have emerged as one of the preferred systems to serve as analogue models of semi- classical gravity phenomena. Their high degree of quantum coherence, very low temperatures minimising background noise and low speed of sound are favourable characteristics. Starting with the time dependent Gross-Pitaevski equation [89] which follows from the microscopic Bogoliubov equations for the order parameter of a Bose Einstein condensate exhibiting off diagonal long range order in the mean field approximation [90], the identification of the gradient of the phase of the order parameter with the velocity of the fluid leads to the Euler hydrodynamic equations [13, 48, 49]. In this hydrodynamic limit, under appropriate conditions, if the Bose Einstein condensate admits an acoustic horizon, an emission of thermal distribution of phonons via the process of Hawking radiation can occur. The recent identification of an acoustic horizon in such a system in the laboratory[68] lends more hope to this possibility. Additionally, if a draining vortex flow and an ergosphere can be formed, there must also be a superresonant current of phonons from the ergosphere when the frequency of phonons coming in from infinity satisfies necessary conditions. But how can we model a Lorentz violating geometry by a Bose Einstein condensate? This is an open question which has not yet received much careful attention.

5.2.3 Inertial frame dragging in an analogue model

But black hole radiation is not the only strong gravity phenomenon whose chances of direct detection have improved with the development of analogue models. Soon after the discovery of general rela- 5.2 lorentz violating effects on superresonance 73 tivity by Einstein, Lense and Thirring had predicted that in a stationary spacetime with angular momentum, gravity will cause the dragging of inertial frames due to which there will be a precession of the angular momentum or spin vector of any test gyroscope placed in the spacetime [71, 100]. Despite being discovered almost a century ago, the Lense-Thirring precession has been observed and the frequency of precession measured only in spacetimes with curvatures small enough to justify the weak gravity approximation [55, 61]. The processes of inertial frame dragging and the resulting Lense- Thirring precession are of kinematic origin and so are perfectly suitable for reproduction in an analogue model. Very recently, an exact formula for the precession frequency of a gyroscope due to inertial frame dragging has been deduced close to the ergosphere of a (2 + 1) dimensional draining bathtub acoustic geometry (stationary and axisymmetric) in an inviscid, barotropic ﬂuid [30, 100]. For the line element of Equation 104, given in chapter 4, the Lense Thirring precession frequency is given by

Br2 r2 −1 ΩDB = − E 1 − E .(136) 2+1 r4 r2

2 2 1/2 Here, rE = (A + B ) is the radius of the ergosphere. From the above equation, it is clear that the precession becomes dominant near the ergosphere, i. e., r → rE, just as in a physical, stationary, axisymmetric spacetime. The real and analogue gyroscopic precession frequencies also exhibit similar behaviour in a weak gravitational ﬁeld far away from the ergosphere [29, 30]. Thus, it is evident that the analogue models of gravity are quite robust systems able to map various geometric, non-dynamic effects of general relativity. Different models are possible, each with their own virtues and drawbacks. This is an active and interesting ﬁeld of study which promises practical realisation of many otherwise inaccessible domains of nature. Hopefully, there will be many new exciting research in this domain in the years to come.

Part III

APPENDIX

APPENDIX A

In the chapters of this thesis, we have sometimes omitted details of calculation that took us from one step to the next to preserve the continuity of the text/ to avoid frequent breaks in the ﬂow of the story we have tried to present. This place, the appendix, is where we give those left out calculations, but only those we think may pose a little difﬁculty. a.1 derivation of bernoulli equation

It is just a small step from the Euler equation (Equation 79) to Bernoulli equation (Equation 85). In Equation 79, we have given the Euler equation in absence of external body forces and viscosity d~v ∇~ p ≡ ∂ ~v + (~v · ∇~ )~v = − .(137) dt t ρ Now,

~v × ∇~ ×~v|i = ijkvjklm∂lvm

= (δilδjm − δimδjl)vj∂lvm

= vj∂ivj − vj∂jvi . So, 1 ~v × ∇~ ×~v = ∇~ (~v2) − (~v · ∇~ )~v . 2 Substituting for (~v · ∇~ )~v in Equation 137, 1 ∇~ p ∂ ~v + ∇~ (~v2) −~v × ∇~ ×~v = − , t 2 ρ ∇~ p 1 ∂ ~v −~v × ∇~ ×~v = − − ∇~ ( ~v2) . t ρ 2 Since the ﬂuid is barotropic, ∇~ p(ρ(~x, t) ∇~ h(ρ(~x, t)) = . ρ and as it is locally vorticity free, we also have ~v = ∇~ ψ. Thus, 1 ∇~ {∂ ψ + h + (∇~ ψ)2} = 0 , t 2 1 ∂ ψ + h + (∇~ ψ)2 = 0 t 2 where the constant of integration has been absorbed into the deﬁni- tion of ψ. This is the Bernoulli equation given in Equation 85 in a form useful for our purpose.

77 78 appendix

a.2 derivation of Equation 113in chapter 4

Our starting point is the radial differential Equation 109: A2 4ν A Bm d2 1 − − + i ω − R(r) r2 3 r2 r2 dr2

1 A2 2A2 i2A Bm + 1 − + − ω − r r2 r3 r r2 4ν A Am2 i Bm d + + − ω − R(r) 3 r3 r3 r r2 dr

Bm2 m2 i2ABm + ω − − − r2 r2 r4 4ν2Am2 im2ω − − R(r) = 0 .(138) 3 r4 r2 Equation 109 or Equation 138 is of the general form d2R(r) dR(r) P (r) + P (r) + P (r)R(r) = 0 .(139) 0 dr2 1 dr 2 We deﬁne a new radial coordinate called the tortoise coordinate as, d d P0 ≡ ,(140) dr dr∗ dr P0(r) = .(141) dr∗ Therefore, 2 2 d 2 d dP0 d 2 = P0 2 + P0(r) . dr∗ dr dr dr

Let us also deﬁne a new radial function R(r∗) = R(r). Remember from section 4.6 that a bar is placed over functions of r∗. Now, to express Equation 139 in terms of the new variables R and r∗, we multiply it by P0(r): d2R dR P2 + P P + P P R(r) = 0 0 dr2 0 1 dr 0 2 d2 dP d dP d P2 + P (r) 0 R(r) + P − 0 P R(r) + P P R(r) = 0 0 dr2 0 dr dr 1 dr 0 dr 0 2 2 d dP0 d d 2 R(r∗) + P1 − R(r∗) + P0(r)P2(r)R(r∗) = 0 . dr∗ dr dr dr∗ (142)

dR(r ) If we denote the coefﬁcient functions of ∗ and R(r ) by a(r) = dr∗ ∗ a(r∗) and b(r) = b(r∗) respectively, then Equation 142 takes the compact form: d2 d 2 R(r∗) + a(r) R(r∗) + b(r)R(r∗) = 0 ,(143) dr∗ dr∗ d2 d 2 R(r∗) + a(r∗) R(r∗) + b(r∗)R(r∗) = 0 .(144) dr∗ dr∗ A.2 derivation of Equation 113in chapter 4 79

It will become evident that it may not always be possible to convert the coefﬁcient functions of r into functions of r∗ because the relation in Equation 140 deﬁning r∗ is not always invertible. The explicit expressions for the functions a(r), b(r) are:

dP a(r) = P − 0 1 dr 1 A2 i2A Bm = 1 − − ω − r r2 r r2 4ν Am2 A i Bm i2Bm + − − ω − + (145) 3 r3 r3 r r2 r3 and

b(r) = P0(r)P2(r) A2 4ν A Bm = 1 − − + i ω − r2 3 r2 r2

Bm2 m2 i2ABm 4ν2Am2 im2ω × ω − − − − − . r2 r2 r4 3 r4 r2 (146)

Substituting these formulae for a(r) and b(r) into Equation 143 we get Equation 111. Now, we intend to make a change of the dependent variable from R(r∗) → ζ(r∗) in such a way that the coefﬁcient of dζ/dr∗ vanishes. For this purpose, we deﬁne a function " # 1 r∗ U(r ) = exp − ds a(s ) . ∗ 2 ∗ ∗ Z Therefore,

dU(r∗) a(r∗) = − U(r∗) dr∗ 2 and

2 2 ! d U(r∗) a 1 da 2 = − U(r∗) . dr∗ 4 2 dr∗

Now, we use U(r∗) to deﬁne the function ζ(r∗) through the formula

R(r∗) ≡ U(r∗)ζ(r∗) .

So,

dR dU dζ = ζ + U dr∗ dr∗ dr∗ ! dζ a = U − ζ dr∗ 2 80 appendix

and d2R d2U dU dζ d2ζ 2 = 2 ζ + 2 + U 2 dr∗ dr∗ dr∗ dr∗ dr∗ " ! # d2ζ dζ a2 1 da = U 2 − a + − ζ . dr∗ dr∗ 4 2 dr∗ Inserting these into Equation 144 and dividing the entire equation by U(r∗) we arrive at a linear, second order ordinary differential equation in ζ(r∗) where the coefﬁcient of the linear derivative dζ/dr∗ is indeed zero. ! d2ζ a2 1 da 2 + b(r∗) − (r∗) − ζ = 0 , dr∗ 4 2 dr∗ 2 d ζ 2 2 + k (r∗)ζ(r∗) = 0 . dr∗ Here, we have introduced the function 2 2 a 1 da k (r∗) = b(r∗) − (r∗) − 4 2 dr∗ a2 1 dr da = b(r) − (r) − 4 2 dr∗ dr = k2(r) . In order to deduce the detailed functional form of k2(r) we need to 2 know the explicit expressions for a (r), da/dr and dr/dr∗. 1 A2 i2A Bm 2 a2(r) = 1 − − ω − r r2 r r2

8ν 1 A2 i2A Bm + 1 − − ω − 3 r r2 r r2

Am2 A i Bm i2Bm × − − ω − + r3 r3 r r2 r3

16ν2 Am2 A i Bm i2Bm 2 + − − ω − + . 9 r3 r3 r r2 r3

As ν → 0, we keep terms only upto O(ν), so that 1 A2 2 4A2 Bm2 i4A A2 Bm a2(r) ≈ 1 − − ω − − 1 − ω − r2 r2 r2 r2 r2 r2 r2 8ν 1 A2 i2A Bm + 1 − − ω − 3 r r2 r r2

Am2 A i Bm i2Bm × − − ω − + .(147) r3 r3 r r2 r3

Next, da 2A2 1 A2 i2A Bm i4ABm = − 1 − + ω − − dr r4 r2 r2 r2 r2 r4 4ν 3A 3Am2 i8Bm i Bm + − − + ω − .(148) 3 r4 r4 r4 r2 r2

A.2 derivation of Equation 113in chapter 4 81

Finally,

dr A2 4ν A Bm = P (r) = 1 − − + i ω − .(149) dr 0 r2 3 r2 r2 ∗ We substitute Equation 146, Equation 147, Equation 148 and Equa- tion 149 in the formula for k2(r) and simplify retaining terms only upto O(ν) in the limit of ν → 0. At the end we get back Equation 114:

2 2 k (r∗) = k (r) Bm2 A2 A2 1 m2 A2 ≈ ω − + 1 − 1 − − − r2 r2 r2 4r2 r2 r4

2ν A2 3A 2Am2 i Bm i2m2ω i6Bm − 1 − + + ω − − − 3 r2 r4 r4 r2 r2 r2 r4

Bm3 2A Bm2 + 2i ω − + ω − r2 r2 r2 Bm 4ABm i2m2 i2A2 + ω − − − (1 + m2) r2 r4 r2 r4

2Am2 2A3 − − .(150) r4 r6

This completes the derivation of Equation 113.

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The following minor corrections have been made to this thesis:

1. page 47, paragraph 2, line 3 after Equation 75: ‘Sir Roger Pen- rose’ has been replaced by ‘Penrose’;

2. page 49, paragraph 1, line 3: the part ‘... have a temperature, which turns out to be proportional to its surface gravity’ has κ been replaced by ‘... have a temperature TBH = 2π (in Planck units) as measured by a static observer at inﬁnity, κ being its surface gravity’;

3. page 49, paragraph 2, line 2: the part ‘... for black holes of a plausible size’ has been changed to ‘... for astrophysical black holes of a plausible size’;

4. page 49, paragraph 2: the dependence of Hawking temperature on the mass of a black hole has been illustrated in a comment in the margin;

5. page 49, paragraph 2, line 10: references [18, 24, 87, 106–108] have been added, thereby 5 new entries ([18, 24, 106–108]) appear in the bibliography;

6. typo in reference [88] has been corrected. colophon

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