DOCSLIB.ORG

Linearized Einstein Field Equations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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). Finally, there is a \tensor" mode hij , which is the transverse-trace-free part of hij, ~ and is obtained by double-projecting hij transverse to k, and subtracting the trace. This piece has 2 independent components. Explicitly, we have TT TT hij = Pijmnhmn; (1) where the TT-projection operator is given by 1 PTT ≡ P P − P P ;P ≡ δ − k^ k^ : (2) ijmn im jn 2 ij mn ij ij i j Let us write explicitly the gauge transformation equations in Fourier space: replace @j by ikj: h00 ! h00 + 2@0ξ0; h0j ! h0j + @0ξj + ikjξ0; hjl ! hjl + 2ik(jξl): (3) GAUGE-INVARIANT METRIC PERTURBATIONS While a gauge transformation in electromagnetism amounts to providing one scalar function, a gauge transforma- tion in linearized GR amounts to providing 4 functions ξ0; ξ~. These 4 functions can be decomposed in 2 scalars 0 ^ i ^j k ξ ; kiξ and 1 transverse vector ijkk ξ . Therefore, we expect that out of the 4 scalars components, only 2 linear combinations are gauge-invariant. Similarly, out of the 2 vector modes, only 1 linear combination is gauge-invariant. Finally, since there is no way to make a TT mode out of scalars and vectors, we expect, and will show explicitly, that the \tensor" mode is gauge-invariant. The two scalar gauge-invariant variables are not unique (any linear combination is also gauge-invariant). We'll see that these two make expressions particularly simple (for reference these are related to the gauge-invariant Bardeen potentials in cosmological perturbation theory): " # 1 1 2ik^ 3 1 Ψ ≡ k^ k^ h − h ; Φ ≡ − h + j @ h − @2 k^ k^ h − h : (4) 4 j l jl jj 2 00 k 0 0j 2k2 0 j l jl 3 jj I encourage you to explicitly check gauge-invariance { and please report likely typos! 2 There is only one gauge-invariant transverse vector mode, defined up to a normalization constant: i vi ≡ ilmk^ h + @ h k^ (5) l 0m k 0 mj j Indeed, under a gauge transformation, the change in the parenthesis is a pure gradient, so its curl is zero. Finally, since a gauge transformation cannot add a TT part to the metric: the TT part of the metric pertur- bation is gauge-invariant, much like the transverse-vector part of the vector potential is in electromagnetism. I encourage you to show this explicitly using the TT projection operator applied to a gauge transformation. So, to summarize, gauge freedom implies that there are only 6 physical degrees of freedom in the metric perturbation (that we could tell right away just from counting the number of coordinates). For linearized GR, we can moreover explicilty identify these degrees of freedom and classify them as 2 scalar modes, 1 transverse vector mode, and 1 transverse-traceless \tensor" mode. Note: it is always possible to set the metric to be Minkowski and with vanishing first derivatives at any given point by using a LICS. So it is no surprise that the gauge-invariant variables are defined with at least two derivatives of the metric. LINEARIZED EINSTEIN TENSOR The Riemann tensor takes the form Riemann ∼ @Γ + ΓΓ, where Γ are the Christoffel symbols. To linear order in hµν , we only need to keep the first term. Furthermore, we only need to compute the Christoffel symbol at linear order in hµν . This implies 1 R = (@ @ h + @ @ h − @ @ h − @ @ h ) ; (6) αµβν 2 µ β αν ν α βµ µ ν αβ α β µν which is the same expression as that in a LICS, derived in lecture 11. From this we obtain the Ricci tensor, by contracting the first and third indices: 1 R = Rα = (@ @αh + @ @αh − @ @ h − h ) ; (7) µν µαν 2 µ αν ν αµ µ ν µν µ µν µ µν where h ≡ h µ ≡ η hµν is the trace of hµν (obtained using the Minkowski metric) and ≡ @ @µ = η @µ@ν is the D'Alembertian operator. The Ricci scalar is obtained by taking the trace: µ α β R = R µ = @ @ hαβ − h: (8) From this we obtain the Einstein tensor at linear order in metric perturbations: 1 G = @ @αh + @ @αh − @ @ h − h − η (@α@βh − h) : (9) µν 2 µ αν ν αµ µ ν µν µν αβ EINSTEIN FIELD EQUATIONS (EFES) We first note that we can also decompose Gµν in scalars, vectors and tensors, and that each kind can only include the components of hµν of the same kind: at linear order one cannot make a transverse vector mode out of a scalar, and vice-versa. Let us now explicitly write the EFEs. The 00 equation can be rewritten in terms of our gauge-invariant variable Ψ, 2 r Ψ = 4πT00: (10) This is just Poisson's equation. As in the case of electromagnetism, the (linearized) Bianchi identity G0i;i = G00;0, ,ν consistent with the conservation of the stress-energy tensor Tµν = 0 (at linear order), implies that the divergence of the 0i equation is nothing but the time-derivative of the 00 equation, and therefore does not carry any additional information. The curl of the 0i equation can be written in terms of the gauge-invariant vector field as 2 i ilm^ r v = 16π klT0m (11) 3 Out of the 6 purely spatial equations, 3 are redundant with the G0i equations, again, from the contracted Bianchi 1 identity. We are left with three independent equations. The first one can be taken to be Gij − 3 δijGkk, which, upon taking the double gradient, gives us [please report typos!] 1 r4(Φ − Ψ) = −12π@ @ T − δ T : (12) i j ij 3 ij kk The right-hand-side is proportional to the anisotropic stress, i.e. the stress tensor Tij minus its isotropic (i.e. diag- onal) part, proportional to the pressure. This equation shows that Φ = Ψ if there are no anisotropic stresses. Finally, the transverse-trace-free part of Gij gives us the following equation for the TT part of the metric pertur- bation: TT TT TT TT hij = −16πTij ;Tij ≡ PijmnTmn: (13) As expected, the EFEs only provide information about the 6 gauge-invariant, physical degrees of freedom. To fix all 10 components of hµν one needs to impose 4 additional and freely specifiable gauge (or coordinate) conditions. Constraints and dynamics Now, let us consider the character of these equations. The first three are constraint equations: they do not involve any time derivatives. It is easier to see this explicitly in the transverse gauge, which is the generalization of the Coulomb gauge in electromagnetism: this gauge is defined by the 4 conditions 1 @ h = 0 = @ h − δ h : (14) i 0i i ij 3 ij kk First, one needs to show explicitly that such a gauge choice is indeed possible, and it is: starting from a coordinate 2 i 1 system in which the gauge condition is not satisfied, make a gauge transformations such that r ξ + 3 @jξl;l = 1 2 0 i −2@j(hij − 3 δijhkk) and r ξ = @ih0i + @0ξ;i. The metric perturbation in these new coordinates will satisfy the transverse gauge conditions. 1 1 i ilm^ In the transverse gauge, we have Ψ = − 6 hkk and Φ = − 2 h00, and v = klh0m, i.e. no time derivatives appear, and the equations for Φ; Ψ and vi are clearly purely spatial, constraint equations. TT Finally, the TT part is a truly dynamical equation for the gauge-invariant tensor mode hij . Therefore, just like electromagnetism, GR has two dynamical degrees of freedom, the TT part of the metric. These are the famous gravitational waves, which propagate at the speed of light (they satisfy the wave equation in vaccum), and can exist even in vacuum, while all other components can be set to zero in vacuum by appropriate gauge choices. While we have shown all this in linearized gravity, let's now mention how this carries over to non-linear GR. First, it remains true at the non-linear level that there are 6 physical degrees of freedom, due to the 4 coordinate degrees of freedom.
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]
  • MITOCW | 14. Linearized Gravity I: Principles and Static Limit
    MITOCW | 14. Linearized gravity I: Principles and static limit. [SQUEAKING] [RUSTLING] [CLICKING] SCOTT All right, so in today's recorded lecture, I would like to pick up where we started-- HUGHES: excuse me. I'd like to pick up where we stopped last time. So I discussed the Einstein field equations in the previous two lectures. I derived them first from the method that was used by Einstein in his original work on the subject. And then I laid out the way of coming to the Einstein field equations using an action principle, using what we call the Einstein Hilbert action. Both of them lead us to this remarkably simple equation, if you think about it in terms simply of the curvature tensor. This is saying that a particular version of the curvature. You start with the Riemann tensor. You trace over two indices. You reverse the trace such that this whole thing has zero divergence. And you simply equate that to the stress energy tensor with a coupling factor with a complex constant proportionality that ensures that this recovers the Newtonian limit. The Einstein Hilbert exercise demonstrated that this is in a very quantifiable way, the simplest possible way of developing a theory of gravity in this framework. The remainder of this course is going to be dedicated to solving this equation, and exploring the properties of the solutions that arise from this. And so let me continue the discussion I began at the end of the previous lecture. We are going to find it very useful to regard this as a set of differential equations for the spacetime metric given a source.
    [Show full text]
  • Rotating Sources, Gravito-Magnetism, the Kerr Metric
    Supplemental Lecture 6 Rotating Sources, Gravito-Magnetism and The Kerr Black Hole Abstract This lecture consists of several topics in general relativity dealing with rotating sources, gravito- magnetism and a rotating black hole described by the Kerr metric. We begin by studying slowly rotating sources, such as planets and stars where the gravitational fields are weak and linearized gravity applies. We find the metric for these sources which depends explicitly on their angular momentum J. We consider the motion of gyroscopes and point particles in these spaces and discover “frame dragging” and Lense-Thirring precession. Gravito-magnetism is also discovered in the context of gravity’s version of the Lorentz force law. These weak field results can also be obtained directly from special relativity and are consequences of the transformation laws of forces under boosts. However, general relativity allows us to go beyond linear, weak field physics, to strong gravity in which space time is highly curved. We turn to rotating black holes and we review the phenomenology of the Kerr metric. The physics of the “ergosphere”, the space time region between a surface of infinite redshift and an event horizon, is discussed. Two appendices consider rocket motion in the vicinity of a black hole and the exact redshift in strong but time independent fields. Appendix B illustrates the close connection between symmetries and conservation laws in general relativity. Keywords: Rotating Sources, Gravito-Magnetism, frame-dragging, Kerr Black Hole, linearized gravity, Einstein-Maxwell equations, Lense-Thirring precession, Gravity Probe B (GP-B). Introduction. Weak Field General Relativity. In addition to strong gravity, the textbook studied space times which are only slightly curved.
    [Show full text]
  • 9 Linearized Gravity and Gravitational Waves
    9 Linearized gravity and gravitational waves 9.1 Linearized gravity 9.1.1 Metric perturbation as tensor field 1 We are looking for small perturbations hab around the Minkowski metric ηab, gab = ηab + hab , hab 1 . (9.1) ≪ These perturbations may be caused either by the propagation of gravitational waves through a detector or by the gravitational potential of a star. In the first case, current experiments show that we should not hope for h larger than (h) 10−22. Keeping only terms linear in h O ∼ is therefore an excellent approximation. Choosing in the second case as application the final phase of the spiral-in of a neutron star binary system, deviations from Newtonian limit can become large. Hence one needs a systematic “post-Newtonian” expansion or even a numerical analysis to describe properly such cases. We choose a Cartesian coordinate system xa and ask ourselves which transformations are compatible with the splitting (9.1) of the metric. If we consider global (i.e. space-time inde- b ′a a b pendent) Lorentz transformations Λa, then x = Λb x . The metric tensor transform as ′ c d c d c d ′ c d gab = ΛaΛb gcd = ΛaΛb (ηcd + hcd)= ηab + ΛaΛb hcd = ηab + ΛaΛb hcd . (9.2) Thus Lorentz transformations respect the splitting (9.1) and the perturbation hab transforms as a rank-2 tensor on Minkowski space. We can view therefore hab as a symmetric rank-2 tensor field defined on Minkowski space that satisfies the linearized Einstein equations, similar as the photon field is a rank-1 tensor field fulfilling Maxwell’s equations.
    [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]