DOCSLIB.ORG

Post-Newtonian Gravity in Cosmology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Post-Newtonian Gravity in Cosmology Astronomy Unit School of Physics and Astronomy Queen Mary University of London Post-Newtonian Gravity in Cosmology Viraj A A Sanghai Submitted in partial fulfillment of the requirements of the Degree of Doctor of Philosophy 1 Declaration I, Viraj Alias Adhir Prabhas Sanghai, confirm that the research included within this thesis is my own work or that where it has been carried out in collaboration with, or supported by others, that this is duly acknowledged below and my contribution indicated. Previously published material is also acknowledged below. I attest that I have exercised reasonable care to ensure that the work is origi- nal, and does not to the best of my knowledge break any UK law, infringe any third party's copyright or other Intellectual Property Right, or contain any confidential material. I accept that the College has the right to use plagiarism detection software to check the electronic version of the thesis. I confirm that this thesis has not been previously submitted for the award of a degree by this or any other university. The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author. Details of collaboration and publications: Part of this work is done in collabo- ration with Timothy Clifton and Pierre Fleury. I have made a major contribution to the original research presented in this thesis. It is based on the following papers that have been published or submitted for publication: Ray tracing and Hubble diagrams in post-Newtonian cosmology • V. A. A. Sanghai, P. Fleury and T. Clifton, JCAP 07 (2017) 028, Arxiv: 1705.02328 [astro-ph.CO] Parameterized post-Newtonian cosmology • V. A. A. Sanghai and T. Clifton, Class. Quantum. Grav. 34 (2017) 065003, Arxiv: 1610.08039 [gr-qc] 2 3 Cosmological backreaction in the presence of radiation and a cosmological constant • V. A. A. Sanghai and T. Clifton, Phys. Rev. D 94, 023505 (2016), Arxiv: 1604.06345 [gr-qc] Post-Newtonian cosmological modelling • V. A. A. Sanghai and T. Clifton, Phys. Rev. D 91, 103532 (2015), Erratum: Phys. Rev. D 93, 089903 (2016), Arxiv: 1503.08747 [gr-qc] Post-Newtonian cosmological models • V. A. A. Sanghai, Contribution to 14th Marcel Grossmann proceedings (2015), to be published in World Scientific proceedings, Arxiv: 1512.04824 [gr-qc] Signature: Viraj A A Sanghai Date: 12/08/2017 Abstract The post-Newtonian (PN) perturbative framework has been successful in understand- ing the slow-motion, weak field limit of Einstein's theory of gravity on solar system scales, and for isolated astrophysical systems. The parameterized post-Newtonian (PPN) formalism extended the PN framework to put very tight constraints on deviations from Einstein's theory on the aforementioned scales and systems. In this work, we extended and applied the post-Newtonian formalism to cosmological scales. We first used it to construct a cosmological model to understand the effect of regularly arranged point sources on the background expansion. Here we found that at higher orders we obtained a small radiation-like correction to the standard Friedmann-Lema^ıtre-Robertson-Walker (FLRW) equations, for a matter-dominated universe. This radiation-like correction was purely due to the inhomogeneity of our model, and the non-linearity of Einstein's field equations. We also extended the post-Newtonian formalism to include other forms of matter that are cosmologically relevant, such as radiation and a cosmological constant, and studied the non-linear effects they might have on the background expansion. Then we constructed an extension of the parameterized post-Newtonian formalism (PPN) to cosmological scales. We used it to parameterize the background expansion of the universe as well as first-order perturbations in cosmology, using four functions of time. In the future, this could allow us to put constraints on deviations from Einstein's theory of gravity on cosmological scales. We gave examples of how our parameterization would work for dark energy models and scalar-tensor and vector-tensor theories of gravity. In the final part of this work, we studied how light propagation behaves in an inhomogeneous post-Newtonian cosmology with matter and a cosmological constant. We used it to understand the effect that inhomogeneities would have on observables such as angular diameter distances as compared to those that are expected from a homogeneous and isotropic FLRW universe. 4 Acknowledgements I would like to begin by thanking my supervisor, Timothy Clifton, for his constant encouragement and support throughout my PhD. I would also like to thank my grandfather, Santosh Maskara, who sadly passed away during my PhD, but not before inspiring me to always pursue what I am passionate about. I also thank my parents, Prabhas and Aneetha Sanghai, my brother, Devesh, and my family in the UK and in India, without whom I wouldn't be where I am today. I thank Karim Malik, Pierre Fleury, David Mulryne, Richard Nelson, Juan Valiente Kroon, Phil Bull and Chris Clarkson for their guidance and support at various times during my PhD. I would especially like to thank Pierre Fleury for a fruitful collaboration. Finally I also thank my friends, especially Pedro Carrilho, John Ronayne, Shailee Imrith, Sophia Goldberg, Charalambos Pittordis, Zac Kenton, Mike Cole, Andries Waelkens, Sara Ricc`o, Gerben Oling, Dominic Dold, Patrick Mangat, Matt Mutter, Sanson Poon, Domenico Trotta, Clark Baker, Jorches Fuentes, Kit Gallagher, Louis Coates and Jessie Durk for many enjoyable work and non-work related conversations. This work was supported by the Science and Technology Facilities Council (STFC), grant number ST/K50225X/1. 5 6 To the loving memory of my grandfather, Santosh Maskara Contents Abstract4 Acknowledgements5 List of Figures 10 List of Tables 13 1. Introduction 14 1.0.1. Notations and Conventions . 18 2. Background 19 2.1. General Relativity and its Alternatives . 19 2.1.1. Test of the Weak Equivalence Principle . 20 2.1.2. Test of Local Position Invariance . 21 2.1.3. Test of Local Lorentz Invariance . 22 2.1.4. General Relativity . 22 2.1.5. Lovelock's Theorem . 23 2.1.6. Scalar-Tensor Theories of Gravity . 24 2.1.7. Vector-Tensor Theories of Gravity . 26 2.1.8. Birkhoff's Theorem . 28 2.1.9. Post-Newtonian Formalism . 29 2.1.10. Parameterized Post-Newtonian Formalism . 36 2.2. Standard Approaches in Cosmology . 41 2.2.1. Friedmann-Lema^ıtre-Robertson-Walker Background . 41 2.2.2. Cosmological Perturbations . 44 2.2.3. Newtonian N-body Simulations . 49 2.3. Inhomogeneous Cosmology . 50 2.3.1. The Cosmological Principle . 50 2.3.2. Backreaction Problem in Cosmology . 50 2.3.3. Inhomogeneous Models of Cosmology . 53 2.3.4. Optics in Inhomogeneous Cosmologies . 56 3. Post-Newtonian Cosmological Modeling 62 3.1. Introduction . 62 3.2. Applying the Post-Newtonian Formalism to Cosmology . 63 3.3. Building a Cosmology Using Junction Conditions . 65 3.4. Comparison with other Approaches . 70 7 Contents 8 3.5. Governing Equations . 71 3.5.1. Einstein's Field Equations . 72 3.5.2. Extrinsic Curvature Equations . 74 3.6. Cosmological Expansion . 76 3.6.1. Newtonian Order . 77 3.6.2. Post-Newtonian Order . 79 3.7. Post-Newtonian Cosmological Solutions . 84 3.7.1. The General Solution: An Application of the Green's Function Formalism............................. 85 3.7.2. A Special Case: Point Sources . 88 3.8. Relationship with FLRW models . 94 3.8.1. The Proper Length of Cell Edges . 94 3.8.2. Transformation to a Time-dependent Background . 96 3.9. Discussion . 100 4. Post-Newtonian Cosmological Model with Radiation and Λ 102 4.1. Introduction . 102 4.2. The Geometry of a Lattice Cell . 103 4.2.1. Post-Newtonian Expansion . 104 4.2.2. Matter Content . 105 4.2.3. Einstein's Field Equations . 106 4.3. Cosmological Expansion . 109 4.3.1. Newtonian Accuracy . 110 4.3.2. Post-Newtonian Accuracy . 111 4.4. Point Sources with Radiation, Spatial Curvature and Λ . 114 4.4.1. Solutions.............................. 115 4.4.2. Results............................... 117 4.5. Discussion . 125 5. Parameterized Post-Newtonian Cosmology 127 5.1. Introduction . 127 5.2. From Weak Fields to Cosmology . 129 5.2.1. Expanding and Non-Expanding Coordinate Systems . 129 5.2.2. Junction Conditions . 130 5.2.3. Emergent Cosmological Expansion . 132 5.3. An Extended PPN formalism . 134 5.3.1. Review of the Standard PPN formalism . 134 5.3.2. Additional Matter Content . 136 5.3.3. Additional Potentials . 137 5.3.4. Additional Time Dependence . 138 5.4. A Parameterized Approach to Cosmology . 139 5.4.1. Review of the Conservation Equations . 139 5.4.2. Background Expansion . 140 5.4.3. First-order Perturbations . 142 5.5. Worked Examples . 143 5.5.1. Dark Energy Models . 144 Contents 9 5.5.2. Scalar-Tensor Theories of Gravity . 146 5.5.3. Vector-Tensor Theories of Gravity . 150 5.6. Discussion . 157 6. Ray Tracing and Hubble Diagrams in Post-Newtonian Cosmology159 6.1. Introduction . 159 6.2. Method .................................. 160 6.2.1. A Universe in a Cell . 161 6.2.2. Initial Conditions . 164 6.2.3. Reflection of Light at the Cell's Boundary . 165 6.2.4. The Ray-Tracing Code . 167 6.3. Results . 169 6.3.1. Redshift . 170 6.3.2. Angular Distance . 172 6.3.3. HubbleDiagram ......................... 174 6.4. Comparisons . 176 6.4.1. Weinberg-Kibble-Lieu Theorem . 177 6.4.2. Comparison with Stochastic Lensing . 178 6.4.3. Comparison with Numerical Relativity . 181 6.5. Discussion . 184 7. Conclusions and Future Work 185 7.1. FutureWork................................ 187 A.
Load more

Recommended publications
  • A Mathematical Derivation of the General Relativistic Schwarzschild
    A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components .................
    [Show full text]
  • Einstein's Mistakes
    Einstein’s Mistakes Einstein was the greatest genius of the Twentieth Century, but his discoveries were blighted with mistakes. The Human Failing of Genius. 1 PART 1 An evaluation of the man Here, Einstein grows up, his thinking evolves, and many quotations from him are listed. Albert Einstein (1879-1955) Einstein at 14 Einstein at 26 Einstein at 42 3 Albert Einstein (1879-1955) Einstein at age 61 (1940) 4 Albert Einstein (1879-1955) Born in Ulm, Swabian region of Southern Germany. From a Jewish merchant family. Had a sister Maja. Family rejected Jewish customs. Did not inherit any mathematical talent. Inherited stubbornness, Inherited a roguish sense of humor, An inclination to mysticism, And a habit of grüblen or protracted, agonizing “brooding” over whatever was on its mind. Leading to the thought experiment. 5 Portrait in 1947 – age 68, and his habit of agonizing brooding over whatever was on its mind. He was in Princeton, NJ, USA. 6 Einstein the mystic •“Everyone who is seriously involved in pursuit of science becomes convinced that a spirit is manifest in the laws of the universe, one that is vastly superior to that of man..” •“When I assess a theory, I ask myself, if I was God, would I have arranged the universe that way?” •His roguish sense of humor was always there. •When asked what will be his reactions to observational evidence against the bending of light predicted by his general theory of relativity, he said: •”Then I would feel sorry for the Good Lord. The theory is correct anyway.” 7 Einstein: Mathematics •More quotations from Einstein: •“How it is possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” •Questions asked by many people and Einstein: •“Is God a mathematician?” •His conclusion: •“ The Lord is cunning, but not malicious.” 8 Einstein the Stubborn Mystic “What interests me is whether God had any choice in the creation of the world” Some broadcasters expunged the comment from the soundtrack because they thought it was blasphemous.
    [Show full text]
  • Linearized Einstein Field Equations
    General Relativity Fall 2019 Lecture 15: Linearized Einstein field equations Yacine Ali-Ha¨ımoud October 17th 2019 SUMMARY FROM PREVIOUS LECTURE We are considering nearly flat spacetimes with nearly globally Minkowski coordinates: gµν = ηµν + hµν , with jhµν j 1. Such coordinates are not unique. First, we can make Lorentz transformations and keep a µ ν globally-Minkowski coordinate system, with hµ0ν0 = Λ µ0 Λ ν0 hµν , so that hµν can be seen as a Lorentz tensor µ µ µ ν field on flat spacetime. Second, if we make small changes of coordinates, x ! x − ξ , with j@µξ j 1, the metric perturbation remains small and changes as hµν ! hµν + 2ξ(µ,ν). By analogy with electromagnetism, we can see these small coordinate changes as gauge transformations, leaving the Riemann tensor unchanged at linear order. Since we will linearize the relevant equations, we may work in Fourier space: each Fourier mode satisfies an independent equation. We denote by ~k the wavenumber and by k^ its direction and k its norm. We have decomposed the 10 independent components of the metric perturbation according to their transformation properties under spatial rotations: there are 4 independent \scalar" components, which can be taken, for instance, ^i ^i^j to be h00; k h0i; hii, and k k hij { or any 4 linearly independent combinations thereof. There are 2 independent ilm^ ilm^ ^j transverse \vector" components, each with 2 independent components: klh0m and klhmjk { these are proportional to the curl of h0i and to the curl of the divergence of hij, and are divergenceless (transverse to the ~ TT Fourier wavenumber k).
    [Show full text]
  • Equivalence Principle (WEP) of General Relativity Using a New Quantum Gravity Theory Proposed by the Authors Called Electro-Magnetic Quantum Gravity Or EMQG (Ref
    WHAT ARE THE HIDDEN QUANTUM PROCESSES IN EINSTEIN’S WEAK PRINCIPLE OF EQUIVALENCE? Tom Ostoma and Mike Trushyk 48 O’HARA PLACE, Brampton, Ontario, L6Y 3R8 [email protected] Monday April 12, 2000 ACKNOWLEDGMENTS We wish to thank R. Mongrain (P.Eng) for our lengthy conversations on the nature of space, time, light, matter, and CA theory. ABSTRACT We provide a quantum derivation of Einstein’s Weak Equivalence Principle (WEP) of general relativity using a new quantum gravity theory proposed by the authors called Electro-Magnetic Quantum Gravity or EMQG (ref. 1). EMQG is manifestly compatible with Cellular Automata (CA) theory (ref. 2 and 4), and is also based on a new theory of inertia (ref. 5) proposed by R. Haisch, A. Rueda, and H. Puthoff (which we modified and called Quantum Inertia, QI). QI states that classical Newtonian Inertia is a property of matter due to the strictly local electrical force interactions contributed by each of the (electrically charged) elementary particles of the mass with the surrounding (electrically charged) virtual particles (virtual masseons) of the quantum vacuum. The sum of all the tiny electrical forces (photon exchanges with the vacuum particles) originating in each charged elementary particle of the accelerated mass is the source of the total inertial force of a mass which opposes accelerated motion in Newton’s law ‘F = MA’. The well known paradoxes that arise from considerations of accelerated motion (Mach’s principle) are resolved, and Newton’s laws of motion are now understood at the deeper quantum level. We found that gravity also involves the same ‘inertial’ electromagnetic force component that exists in inertial mass.
    [Show full text]
  • Derivation of Generalized Einstein's Equations of Gravitation in Some
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 February 2021 doi:10.20944/preprints202102.0157.v1 Derivation of generalized Einstein's equations of gravitation in some non-inertial reference frames based on the theory of vacuum mechanics Xiao-Song Wang Institute of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, Henan Province, 454000, China (Dated: Dec. 15, 2020) When solving the Einstein's equations for an isolated system of masses, V. Fock introduces har- monic reference frame and obtains an unambiguous solution. Further, he concludes that there exists a harmonic reference frame which is determined uniquely apart from a Lorentz transformation if suitable supplementary conditions are imposed. It is known that wave equations keep the same form under Lorentz transformations. Thus, we speculate that Fock's special harmonic reference frames may have provided us a clue to derive the Einstein's equations in some special class of non-inertial reference frames. Following this clue, generalized Einstein's equations in some special non-inertial reference frames are derived based on the theory of vacuum mechanics. If the field is weak and the reference frame is quasi-inertial, these generalized Einstein's equations reduce to Einstein's equa- tions. Thus, this theory may also explain all the experiments which support the theory of general relativity. There exist some differences between this theory and the theory of general relativity. Keywords: Einstein's equations; gravitation; general relativity; principle of equivalence; gravitational aether; vacuum mechanics. I. INTRODUCTION p. 411). Theoretical interpretation of the small value of Λ is still open [6]. The Einstein's field equations of gravitation are valid 3.
    [Show full text]
  • Formulation of Einstein Field Equation Through Curved Newtonian Space-Time
    Formulation of Einstein Field Equation Through Curved Newtonian Space-Time Austen Berlet Lord Dorchester Secondary School Dorchester, Ontario, Canada Abstract This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out that, the vital assumptions would not be realized. Note: A number inside of a double square bracket, for example [[1]], denotes an endnote found on the last page. 1. Introduction The purpose of this paper is to show a way to rediscover Einstein’s General Relativity. It is done through analyzing Newton’s equations and making the conclusion that space-time must not only be realized, but also that it must have curvature in the presence of matter and energy. 2. Principal of Least Action We want to show here the Lagrangian action of limiting motion of Newton’s second law (F=ma). We start with a function q mapping to n space of n dimensions and we equip it with a standard inner product. q : → (n ,(⋅,⋅)) (1) We take a function (q) between q0 and q1 and look at the ds of a section of the curve. We then look at some properties of this function (q). We see that the classical action of the functional (L) of q is equal to ∫ds, L denotes the systems Lagrangian.
    [Show full text]
  • Einstein's 1916 Derivation of the Field Equations
    1 Einstein's 1916 derivation of the Field Equations Galina Weinstein 24/10/2013 Abstract: In his first November 4, 1915 paper Einstein wrote the Lagrangian form of his field equations. In the fourth November 25, 1915 paper, Einstein added a trace term of the energy- momentum tensor on the right-hand side of the generally covariant field equations. The main purpose of the present work is to show that in November 4, 1915, Einstein had already explored much of the main ingredients that were required for the formulation of the final form of the field equations of November 25, 1915. The present work suggests that the idea of adding the second-term on the right-hand side of the field equation might have originated in reconsideration of the November 4, 1915 field equations. In this regard, the final form of Einstein's field equations with the trace term may be linked with his work of November 4, 1915. The interesting history of the derivation of the final form of the field equations is inspired by the exchange of letters between Einstein and Paul Ehrenfest in winter 1916 and by Einstein's 1916 derivation of the November 25, 1915 field equations. In 1915, Einstein wrote the vacuum (matter-free) field equations in the form:1 for all systems of coordinates for which It is sufficient to note that the left-hand side represents the gravitational field, with g the metric tensor field. Einstein wrote the field equations in Lagrangian form. The action,2 and the Lagrangian, 2 Using the components of the gravitational field: Einstein wrote the variation: which gives:3 We now come back to (2), and we have, Inserting (6) into (7) gives the field equations (1).
    [Show full text]
  • Post-Newtonian Approximations and Applications
    Monash University MTH3000 Research Project Coming out of the woodwork: Post-Newtonian approximations and applications Author: Supervisor: Justin Forlano Dr. Todd Oliynyk March 25, 2015 Contents 1 Introduction 2 2 The post-Newtonian Approximation 5 2.1 The Relaxed Einstein Field Equations . 5 2.2 Solution Method . 7 2.3 Zones of Integration . 13 2.4 Multi-pole Expansions . 15 2.5 The first post-Newtonian potentials . 17 2.6 Alternate Integration Methods . 24 3 Equations of Motion and the Precession of Mercury 28 3.1 Deriving equations of motion . 28 3.2 Application to precession of Mercury . 33 4 Gravitational Waves and the Hulse-Taylor Binary 38 4.1 Transverse-traceless potentials and polarisations . 38 4.2 Particular gravitational wave fields . 42 4.3 Effect of gravitational waves on space-time . 46 4.4 Quadrupole formula . 48 4.5 Application to Hulse-Taylor binary . 52 4.6 Beyond the Quadrupole formula . 56 5 Concluding Remarks 58 A Appendix 63 A.1 Solving the Wave Equation . 63 A.2 Angular STF Tensors and Spherical Averages . 64 A.3 Evaluation of a 1PN surface integral . 65 A.4 Details of Quadrupole formula derivation . 66 1 Chapter 1 Introduction Einstein's General theory of relativity [1] was a bold departure from the widely successful Newtonian theory. Unlike the Newtonian theory written in terms of fields, gravitation is a geometric phenomena, with space and time forming a space-time manifold that is deformed by the presence of matter and energy. The deformation of this differentiable manifold is characterised by a symmetric metric, and freely falling (not acted on by exter- nal forces) particles will move along geodesics of this manifold as determined by the metric.
    [Show full text]
  • Einstein's $ R^{\Hat {0}\Hat {0}} $ Equation for Non-Relativistic Sources
    Einstein’s R0ˆ0ˆ equation for nonrelativistic sources derived from Einstein’s inertial motion and the Newtonian law for relative acceleration Christoph Schmid∗ ETH Zurich, Institute for Theoretical Physics, 8093 Zurich, Switzerland (Dated: March 22, 2018) With Einstein’s inertial motion (freefalling and nonrotating relative to gyroscopes), geodesics for nonrelativistic particles can intersect repeatedly, allowing one to compute the space-time curvature ˆˆ ˆˆ R00 exactly. Einstein’s R00 for strong gravitational fields and for relativistic source-matter is iden- tical with the Newtonian expression for the relative radial acceleration of neighbouring freefalling test-particles, spherically averaged. — Einstein’s field equations follow from Newtonian experiments, local Lorentz-covariance, and energy-momentum conservation combined with the Bianchi identity. PACS numbers: 04.20.-q, 04.20.Cv Up to now, a rigorous derivation of Einstein’s field structive. Newtonian relative acceleration in general La- equations for general relativity has been lacking: Wald [1] grangian 3-coordinates (e.g. comoving with the wind) writes “a clue is provided”, “the correspondence suggests has the same number of 106 uninstructive terms. the field equation”. Weinberg [2] takes the ”weak static The expressions for Einstein’s R 0ˆ (P ) and the New- 0ˆ limit”, makes a ”guess”, and argues with ”number of tonian relative acceleration are extremely simple and ex- derivatives”. Misner, Thorne, and Wheeler [3] give ”Six plicitely identical with the following choices: (1) We work Routes to Einstein’s field equations”, among which they with Local Ortho-Normal Bases (LONBs) in Cartan’s recommend (1) “model geometrodynamics after electro- method. (2) We use a primary observer (non-inertial or dynamics”, (2) ”take the variational principle with only inertial) with worldline through P , withu ¯obs =e ¯0ˆ, and a scalar linear in second derivatives of the metric and no with his spatial LONBse ¯ˆi along his worldline.
    [Show full text]
  • Relativity Theory, We Often Use the Convention That the Greek6 Indices Run from 0 to 3, Whereas the Latin Indices Take the Values 1, 2, 3
    Relativity Theory Jouko Mickelsson with Tommy Ohlsson and H˚akan Snellman Mathematical Physics, KTH Physics Royal Institute of Technology Stockholm 2005 Typeset in LATEX Written by Jouko Mickelsson, 1996. Revised by Tommy Ohlsson, 1998. Revised and extended by Jouko Mickelsson, Tommy Ohlsson, and H˚akan Snellman, 1999. Revised by Jouko Mickelsson, Tommy Ohlsson, and H˚akan Snellman, 2000. Revised by Tommy Ohlsson, 2001. Revised by Tommy Ohlsson, 2003. Revised by Mattias Blennow, 2005. Solutions to the problems are written by Tommy Ohlsson and H˚akan Snellman, 1999. Updated by Tommy Ohlsson, 2000. Updated by Tommy Ohlsson, 2001. Updated by Tommy Ohlsson, 2003. Updated by Mattias Blennow, 2005. c Mathematical Physics, KTH Physics, KTH, 2005 Printed in Sweden by US–AB, Stockholm, 2005. Contents Contents i 1 Special Relativity 1 1.1 Geometry of the Minkowski Space . 2 1.2 LorentzTransformations. 4 1.3 Physical Interpretations . 5 1.3.1 LorentzContraction ............................ 7 1.3.2 TimeDilation................................ 7 1.3.3 Relativistic Addition of Velocities . 7 1.3.4 The Michelson–Morley Experiment . 8 1.3.5 The Relativistic Doppler Effect . 10 1.4 The Proper Time and the Twin Paradox . 11 1.5 Transformations of Velocities and Accelerations . 12 1.6 Energy, Momentum, and Mass in Relativity Theory . 13 1.7 The Spinorial Representation of Lorentz Transformations . 16 1.8 Lorentz Invariance of Maxwell’s Equations . 17 1.8.1 Physical Consequences of Lorentz Transformations . 20 1.8.2 TheLorentzForce ............................. 22 1.8.3 The Energy-Momentum Tensor . 24 1.9 Problems ...................................... 27 2 Some Differential Geometry 39 2.1 Manifolds .....................................
    [Show full text]
  • The Confrontation Between General Relativity and Experiment
    The Confrontation between General Relativity and Experiment Clifford M. Will Department of Physics University of Florida Gainesville FL 32611, U.S.A. email: [email protected]fl.edu http://www.phys.ufl.edu/~cmw/ Abstract The status of experimental tests of general relativity and of theoretical frameworks for analyzing them are reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the E¨otv¨os experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse–Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves. arXiv:1403.7377v1 [gr-qc] 28 Mar 2014 1 Contents 1 Introduction 3 2 Tests of the Foundations of Gravitation Theory 6 2.1 The Einstein equivalence principle . .. 6 2.1.1 Tests of the weak equivalence principle . .. 7 2.1.2 Tests of local Lorentz invariance . .. 9 2.1.3 Tests of local position invariance . 12 2.2 TheoreticalframeworksforanalyzingEEP. ....... 16 2.2.1 Schiff’sconjecture ................................ 16 2.2.2 The THǫµ formalism .............................
    [Show full text]
  • Doing Physics with Quaternions
    Doing Physics with Quaternions Douglas B. Sweetser ©2005 doug <[email protected]> All righs reserved. 1 INDEX Introduction 2 What are Quaternions? 3 Unifying Two Views of Events 4 A Brief History of Quaternions Mathematics 6 Multiplying Quaternions the Easy Way 7 Scalars, Vectors, Tensors and All That 11 Inner and Outer Products of Quaternions 13 Quaternion Analysis 23 Topological Properties of Quaternions 28 Quaternion Algebra Tool Set Classical Mechanics 32 Newton’s Second Law 35 Oscillators and Waves 37 Four Tests of for a Conservative Force Special Relativity 40 Rotations and Dilations Create the Lorentz Group 43 An Alternative Algebra for Lorentz Boosts Electromagnetism 48 Classical Electrodynamics 51 Electromagnetic Field Gauges 53 The Maxwell Equations in the Light Gauge: QED? 56 The Lorentz Force 58 The Stress Tensor of the Electromagnetic Field Quantum Mechanics 62 A Complete Inner Product Space with Dirac’s Bracket Notation 67 Multiplying quaternions in Polar Coordinate Form 69 Commutators and the Uncertainty Principle 74 Unifying the Representations of Integral and Half−Integral Spin 79 Deriving A Quaternion Analog to the Schrödinger Equation 83 Introduction to Relativistic Quantum Mechanics 86 Time Reversal Transformations for Intervals Gravity 89 Unified Field Theory by Analogy 101 Einstein’s vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions 115 Einstein’s vision II: A unified force equation with constant velocity profile solutions 123 Strings and Quantum Gravity 127 Answering Prima Facie Questions in Quantum Gravity Using Quaternions 134 Length in Curved Spacetime 136 A New Idea for Metrics 138 The Gravitational Redshift 140 A Brief Summary of Important Laws in Physics Written as Quaternions 155 Conclusions 2 What Are Quaternions? Quaternions are numbers like the real numbers: they can be added, subtracted, multiplied, and divided.
    [Show full text]