advanced mechanics and general relativity j franklin

This page intentionally left blank ADVANCED MECHANICS AND GENERAL RELATIVITY Aimed at advanced undergraduates with background knowledge of classical mechanics and electricity and magnetism, this textbook presents both the particle dynamics relevant to general relativity, and the field dynamics necessary to understand the theory Focusing on action extremization, the book develops the structure and predictions of general relativity by analogy with familiar physical systems Topics ranging from classical field theory to minimal surfaces and relativistic strings are covered in a consistent manner Nearly 150 exercises and numerous examples throughout the textbook enable students to test their understanding of the material covered A tensor manipulation package to help students overcome the computational challenge associated with general relativity is available on a site hosted by the author A link to this and to a solutions manual can be found at www.cambridge.org/9780521762458 joel franklin is an Assistant Professor in the physics department of Reed College His work spans a variety of fields, including stochastic Hamiltonian systems (both numerical and mathematical), modifications of general relativity, and their observational implications ADVANCED MECHANICS AND GENERAL RELATIVITY JOEL FRANKLIN Reed College CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521762458 © J Franklin 2010 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2010 ISBN-13 978-0-511-77654-0 eBook (NetLibrary) ISBN-13 978-0-521-76245-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate For Lancaster, Lewis, and Oliver Contents Preface Acknowledgments Newtonian gravity 1.1 Classical mechanics 1.2 The classical Lagrangian 1.2.1 Lagrangian and equations of motion 1.2.2 Examples 1.3 Lagrangian for U (r) 1.3.1 The metric 1.3.2 Lagrangian redux 1.4 Classical orbital motion 1.5 Basic tensor definitions 1.6 Hamiltonian definition 1.6.1 Legendre transformation 1.6.2 Hamiltonian equations of motion 1.6.3 Canonical transformations 1.6.4 Generating functions 1.7 Hamiltonian and transformation 1.7.1 Canonical infinitesimal transformations 1.7.2 Rewriting H 1.7.3 A special type of transformation 1.8 Hamiltonian solution for Newtonian orbits 1.8.1 Interpreting the Killing vector 1.8.2 Temporal evolution Relativistic mechanics 2.1 Minkowski metric vii page xiii xviii 1 2 12 16 21 25 26 30 33 34 37 37 39 43 46 46 52 56 57 viii Contents 2.2 Lagrangian 2.2.1 Euclidean length extremization 2.2.2 Relativistic length extremization 2.3 Lorentz transformations 2.3.1 Infinitesimal transformations 2.4 Relativistic Hamiltonian 2.5 Relativistic solutions 2.5.1 Free particle motion 2.5.2 Motion under a constant force 2.5.3 Motion under the spring potential 2.5.4 The twin paradox with springs 2.5.5 Electromagnetic infall 2.5.6 Electromagnetic circular orbits 2.5.7 General speed limit 2.5.8 From whence, the force? 2.6 Newtonian gravity and special relativity 2.6.1 Newtonian gravity 2.6.2 Lines of mass 2.6.3 Electromagnetic salvation 2.6.4 Conclusion 2.7 What’s next Tensors 3.1 Introduction in two dimensions 3.1.1 Rotation 3.1.2 Scalar 3.1.3 Vector (contravariant) and bases 3.1.4 Non-orthogonal axes 3.1.5 Covariant tensor transformation 3.2 Derivatives 3.2.1 What is a tensorial derivative? 3.3 Derivative along a curve 3.3.1 Parallel transport 3.3.2 Geodesics Curved space 4.1 Extremal lengths 4.2 Cross derivative (in)equality 4.2.1 Scalar fields 4.2.2 Vector fields 4.3 Interpretation 4.3.1 Flat spaces 61 61 63 67 67 71 74 75 76 78 80 82 83 85 86 87 87 89 92 93 94 98 99 99 101 102 104 106 111 112 117 117 122 127 128 129 130 130 132 134 9.4 A relativistic string solution 353 Notice that we have not imposed arc-length parametrization at this point, the magnitude of X is as yet unspecified Our equations of motion simplify through their momenta – with our current choices, we have: (X · X ) X˙ µ T0 c τ µ = σ µ T0 =− c (c2 − v ) X · X (c2 − v ) Xµ (c2 − v ) X · X (9.82) , ˙ · X ˙ using v = X As usual, this is a set of four equations For the µ = case, we know that X0 = 0, so the equation of motion here is: ∂ ∂τ T0 (X · X ) (c2 − v ) X · X = 0, (9.83) telling us that the expression in parentheses is τ -constant, i.e equal to some arbitrary f (σ ) We can use this to define the σ -parametrization One choice, as we just saw, is arc-length, then X · X = But in general, we have the full class: √ √ f (σ ) c2 − v X ·X = (9.84) T0 We can input this into the spatial part of (9.79): ∂ ∂τ ˙ − ∂ f (σ ) X c ∂σ T02 X c f (σ ) = (9.85) Remember, f (σ ) is up to us – we can take any function of σ only Here’s a good choice: f (σ ) = f0 , a constant Then we have a familiar reduction, the wave equation: − f02 ¨ X + X = T02 (9.86) To complete the traditional picture, let’s set the constant f0 = Tc0 , then we recover the wave equation with propagation speed c We have a few side-constraints, now, so let’s tabulate our results: − ¨ X+X =0 c2 ˙ ·X =0 X X ·X =1− ˙ ˙ X · X c2 (9.87) 354 Additional topics Here we can see that a choice for σ has been made (fixed by f (σ )) – whatever you want to call that choice, we are clearly not using arc-length parametrization at this stage The wave equation for the relativistic string does not appear in the above form in arc-length parametrization, so to make this wave association, we had to use σ such that X · X = − vc2 Take free endpoint boundary conditions (the string is not fixed) We are not in arc-length parametrization, and the spatial constraint that σµ = defining this ˙ · X = 0), boundary condition (see (9.69)) automatically has σ0 = (because X leaving: σ =− T0 (c2 − v ) X =0 c2 v⊥ − c2 (9.88) so that at the endpoints, we must have X (σ = σ0 ) = X (σ = σf ) = 0.6 There are now sufficient boundary and initial conditions to actually solve the string equations of motion Since we have fixed all of the available gauge freedom, our solution should have direct physical interpretation Rather than discuss the general case, let’s work our specific rotating string example, and see how the above equations constrain the solutions 9.4.5 A rotating string Consider a string that rotates with some constant angular velocity – we can make left and right-traveling ans¨atze in the usual way: X(t, σ ) = δ (A cos(κ (σ − c t)) + B cos(κ (σ + c t))) xˆ + δ (F sin(κ (σ − c t)) + G sin(κ (σ + c t))) yˆ (9.89) for constants {A, B, F, G} We have automatically solved the wave equation with our choice of left and right-traveling waves Now we have the boundary condition X (t, σ = 0, σf ) = Taking the σ = case, we learn that B = A and G = −F Then imposing the boundary condition at σf puts a constraint on κ: π κ σf = n π −→ κ = , (9.90) σf where we have chosen to set n = (so that as σ = −→ σf , we go across the string once, our choice) Our current solution, then, is: X(t, σ ) = δ cos(π σ/σf ) A cos(π c t/σf ) xˆ − F sin(π c t/σf ) yˆ (9.91) Remember that we saw this same sort of equation (9.78), in arc-length parametrization There, we know that X = 0, and we concluded that v = c at the ends of the string In the current setting, where we know that X is ˙ we cannot conclude that X = 0, nor we know that v = c at the ends perpendicular to X, 9.4 A relativistic string solution 355 ˙ = gives the relation A = ±F , and taking the Now setting the condition X · X negative sign to adjust the phase, we have F = −A above Finally, we use the last of the relations in (9.87) to fix the magnitude of A: X ·X + σf ˙ ˙ , X · X = −→ A = ± c 2δπ (9.92) and our final solution reads: X(t, σ ) = σf πσ cos π σf cos π ct σf π ct σf xˆ + sin yˆ (9.93) From this solution, we can obtain the perpendicular component of velocity – since the motion is transverse to the string already, the velocity is perpendicular to the string everywhere, and has value: v⊥ = ˙ ·X ˙ = c cos π σ X σf (9.94) As expected, then, the ends of the string travel at the speed of light Rather than continue in this parametrization, we’ll now develop the same solution in arc-length parametrization, this provides a way to interpret the string equations of motion classically, in addition to highlighting the interpretation of the same solution in different gauges 9.4.6 Arc-length parametrization for the rotating string We’ll start with X(t, σ ) from (9.93) and impose arc-length parametrization using its definition and a change of variables We could also go back to (9.84) and develop/solve the equations of motion in this parametrization (we’ll that to make contact with a classical string) Arc-length parametrization is defined by s(σ ) such that ∂X has unit magnitude ∂s Given that we know the magnitude in σ -parametrization from (9.87), we can ): perform the change of variables (let X = ∂X ∂σ X ·X = ∂X ∂X · ∂s ∂s ∂s ∂σ =1− ∂s ˙ ˙ πσ X · X −→ = sin c ∂σ σf , (9.95) =1 ˙ ˙ ∂s where we have simplified the requirement ∂σ = − X·c2X by using our solution (9.93) The ODE above is easily solved – we require s(0) = − L2 and s(σf ) = L2 356 Additional topics to get s ∈ [− L2 , L2 ], and then the solution is: s(σ ) = − πσ L cos σf (9.96) with σf = π2L In this s-parametrization, the string solution is, from (9.93): X(t, s) = −s cos 2ct L xˆ + sin 2ct L yˆ (9.97) From this form, it is clear that the magnitudes of the velocity and spatial derivatives are: 2 ˙ ·X ˙ = 4c s X L2 X ·X =1 (9.98) (primes now refer to s-derivatives) Once again, the endpoints travel at the speed of light – in this case, we have a constant (unit) magnitude s derivative along the curve, so the speed of the endpoints must be c Finally, we can return to the canonical momenta τµ and σµ – it is easy to verify that (9.79) is satisfied, and our immediate concern is the energy density of this configuration By analogy with the point particle, the energy of points along the string should be related to the t-canonical momentum’s zero component We can calculate the full tensor:   − T0 s  c 1−  τ (9.99) ˙  T0 X˙ L  µ= c2 1− s2 L For particles, the analogous tensor would be pµ , and we know that p0 = − Ec for E the energy density – then in this case, we should associate: E= T0 1− (9.100) s2 L2 with the energy per unit length of the string (shown in Figure 9.6) Notice that it is a function of s – the endpoints carry infinite energy, apparently, but the total energy, given by: E= L/2 −L/2 E ds = π L T0 , (9.101) 9.4 A relativistic string solution 357 E in units of T0 2.5 2.0 1.5 0.4 0.2 0.2 0.4 s in units of L Figure 9.6 The energy density of the rotating string (9.100) depends only on the “tension” and length, and is finite (its units are correct, as well, with T0 having units of force) 9.4.7 Classical correspondence To understand what this relativistic string corresponds to classically, we will redevelop the string equations of motion in the static (temporal) gauge with σ taken to be the arc-length parameter In a lab frame, we measure time as coordinate time (for the lab), so static gauge is appropriate In addition, we typically define, say, transverse oscillation in terms of an x-coordinate – this amounts to arc-length parametrization in our current language So we want to write (9.79) in terms of t and s, where: X ·X =1 (9.102) ˙ = here For a background Minkowski space We still have X · X for X ≡ ∂X ∂s time written in Cartesian coordinates, the relevant canonical momenta are (returning to (9.80)): T0 c −X˙ µ τ µ =− σ µ T0 (c2 − v ) Xµ =− c (c2 − v ) (c2 − v ) (9.103) 358 Additional topics Now we have four equations, only three of which contain the information of interest For µ = 0, we learn that: ∂ ∂t T0 = 0, (c2 − v ) (9.104) which tells us that the term in parentheses is a t-constant of the motion For the µ = 1, 2, terms: ∂ ∂t ˙ T0 X √ c c2 − v − T0 c ∂ ∂s c2 − v X = (9.105) Note that these can be obtained directly from the general requirement in (9.84) with the arc-length definition for f (σ ): f (σ ) = √cT2 −v Using the information from the µ = equation, we can write (9.105) suggestively: T0 c2 ¨ − ∂ X ∂s − vc2 T0 1− v2 X c2 = (9.106) Referring to our discussion of field Lagrangians in Section 5.1.1 (and (5.14) in ¨ is what we would call the “effective particular), the term sitting in front of X mass density” of the classical string The additional term in the ∂s∂ derivative is the effective tension: T0 µeff = c2 1− v2 c2 Teff = T0 1− v2 c2 (9.107) Thinking of our rotating string solution, we see that v = v⊥ , and the effective string tension vanishes at the endpoints, while the effective mass density becomes infinite there As a subject that ties together our work on extremization of an action, gauge choices, and the metric, classical strings are interesting theoretical constructs We have only touched on the solution of the equations of motion, and this introduction is meant to provide only the highlights of these familiar elements The idea of an extended, relativistic body is interesting, and the action is natural (as the relativistic analogue of a soap film) The equations of motion emphasize the importance of fixing gauges prior to physical interpretation, and we can familiarize ourselves with the content of the classical strings by comparing them to waves in a strange medium To go beyond these observations, and in order to appreciate the role of string theory in modern physics, one must make the move to quantum mechanics, which is not our current subject 9.4 A relativistic string solution 359 Problem 9.9 Starting from (9.80), we’ll use static gauge and arc-length parametrization directly to arrive at (9.97) (a) Simplify σµ and τµ from (9.80) using the following gauge choice: X˙ µ = ˙ c ˙ X Xµ= ˙ X (9.108) ˙ ·X =0 X X · X = ∂ σ (b) Show, from the µ = equation of ∂τ∂ τµ + ∂σ µ = 0, that the combination T ˙ · X), ˙ and hence v is independent of t √ is independent of t (here, v = X c2 −v (c) With v = v(σ ) a function of σ only, we can write: ˙ = v(σ ) [cos(f (σ, t)) xˆ + sin(f (σ, t)) yˆ ] X (9.109) We know that X should be perpendicular to this vector and have unit magnitude Take: X = [− sin(f (σ, t)) xˆ + cos(f (σ, t)) yˆ ] (9.110) to enforce these final two gauge requirements Using the integrability condition, ∂ ˙ X = ∂t∂ X , find f (σ, t) and v(σ ) Assume we require v( /2) = v(− /2) = c, ∂σ appropriate for arc-length parametrization (see (9.78)) In the end, you should recover (9.97) (up to direction of rotation and starting point) Problem 9.10 What we have, in (9.87), is a set of spatial functions of σ and τ that satisfy: ✷2 Xi = i = 1, 2, 3, (9.111) ∂ where ✷2 = − ∂τ∂ + ∂σ in units where c = Note that, from our choice of τ = t, it is also the case that ✷ X = 0, so each element of the four-vector Xµ satisfies the wave equation We know how to write an action that leads to the wave equation for a scalar (Klein–Gordon), and in this problem, we’ll start with four copies of that action, and recover the wave equation in τ –σ coordinates by introducing an auxiliary metric field (a) Using: ˙ hab = −1 0 (9.112) and its inverse hab (to raise and lower indices referring to the two-dimensional τ –σ space, we’ll use roman indices for this space), show that the action: S= Xµ,a ηµν Xν,b hab dσ dτ (9.113) 360 Additional topics gives back ✷2 Xµ = when varied with respect to Xν – commas here refer to the µ µ µ µ and X ,1 = ∂X ) derivatives of Xµ with respect to τ and σ (so that X ,0 = ∂X ∂τ ∂σ The metric ηµν is the usual four-dimensional Minkowski metric in Cartesian coordinates (b) We can promote the metric hab to dynamical variable, and use it to recover the auxiliary conditions Remember that the wave equation is not enough for a relativistic string, we have to enforce all of the gauge conditions in the solution By making hab itself a field, we can return those gauge conditions as field equations (a setting reminiscent of Lagrange multipliers) The “Polyakov” action is: √ −h hab Xµ,a ηµν Xν,b dσ dτ, (9.114) SP = varying with respect to both Xµ and hab (don’t forget our result ∂h∂hab = −h hab for the derivative of a matrix determinant with respect to the matrix elements from Problem 3.13) This time, then, we are assuming that hab (σ, τ ) is itself an undetermined function of σ and τ (c) We know, from Problem 4.7, that any two-dimensional metric can be written in conformally flat form – here, the statement is hab = f (σ, τ ) η¯ ab where η¯ ab is just the two-dimensional Minkowski metric of (9.112) Use this conformal flatness in your field equations to show that you recover the wave equation for each of the Xµ , and a set of two constraints representing gauge conditions: X˙ µ ηµν X ν = and X˙ µ X˙ ν + X µ X ν ηµν = Show, finally, that the gauge conditions in (9.87) satisfy this set Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] Arnold, V I Mathematical Methods of Classical Mechanics Springer-Verlag, 1989 Bergmann, P G Introduction to the Theory of Relativity Prentice-Hall, 1947 Bondi, H “Negative mass in general relativity”, Rev Mod Phys 29(3): 423–8, 1957 Carroll, S M Spacetime and Geometry: An Introduction to General Relativity Addison-Wesley, 2004 Carroll, S M Lecture Notes on General Relativity gr-qc/9712019 Chandrasekhar, S The Mathematical Theory of Black Holes Oxford University Press, 1992 Deser, S “Self-interaction and gauge invariance”, Gen Rel Grav 1: gr-qc/0411023, 1970 Deser, S Lecture Notes on General Relativity (unpublished) Orsay, 1972 D’Inverno, R Introducing Einstein’s Relativity Oxford University Press, 1992 Dirac, P A M General Theory of Relativity John Wiley & Sons, 1975 Einstein, A., Lorentz, H A., Weyl, H and Minkowski, H The Principle of Relativity Dover Publications, Inc., 1952 Eisenhart, L P Riemannian Geometry Princeton University Press, 1926 Flanders, H Differential Forms with Applications to the Physical Sciences Dover Publications, Inc., 1989 Fock, V The Theory of Space, Time and Gravitation Macmillan Company, 1964 Franklin, J and Baker, P T “Linearized Kerr and spinning massive bodies: an electrodynamics analogy”, Am J Phys 75(4): 336–42, 2007 Goldstein, H., Poole, C and Safko, J Classical Mechanics Addison-Wesley, 2002 Griffiths, D J Introduction to Electrodynamics Prentice-Hall, 1999 Hartle, J B Gravity: An Introduction to Einstein’s General Relativity Addison-Wesley, 2003 Hatfield, B Quantum Field Theory of Point Particles and Strings Westview Press, 1992 Hobson, M P., Efstathiou, G P and Lasenby, A N General Relativity: An Introduction for Physicists Cambridge University Press, 2006 http://einstein.stanford.edu/ http://www.ligo.caltech.edu/ http://lisa.nasa.gov/ Jackson, J D Classical Electrodynamics John Wiley & Sons, 1999 Kreyszig, E Differential Geometry Dover Publications, Inc., 1991 361 362 Bibliography [26] Lanczos, C The Variational Principles of Mechanics (fourth edition) Dover Publications, Inc., 1986 [27] Landau, L D and Lifshitz, E M The Classical Theory of Fields Butterworth-Heinenann, 1975 [28] Levi-Civita, T The Absolute Differential Calculus (Calculus of Tensors) Dover, 1977 [29] Lightman, A P., Press, W H., Price, R H and Teukolsky, S A Problem Book in Relativity and Gravitation Princeton University Press, 1975 [30] Milne-Thomson, L M Theoretical Hydrodynamics Dover, 1996 [31] Misner, C W., Thorne, K S and Wheeler, J A Gravitation W H Freeman and Company, 1973 [32] Panofsky, W K H and Phillips, M N Classical Electricity & Magnetism Dover, 2005 [33] Polchinksi, J String Theory, Vol Cambridge University Press, 2005 [34] Rubakov, V (transl by S Wilson) Classical Theory of Gauge Fields Princeton University Press, 2002 [35] Synge, J L and Schild, A Tensor Calculus Dover Publications, Inc., 1949 [36] Schutz, B A First Course in General Relativity Cambridge University Press, 2009 [37] Schwinger, J Particles, Sources, and Fields Addison-Wesley, 1970 [38] Thirring, W E Classical Mathematical Physics: Dynamics Systems and Fields Springer, 1997 [39] Wald, R M General Relativity The University of Chicago Press, 1984 [40] Weinberg, S Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity John Wiley & Sons, 1972 [41] Wheeler, N Analytical Dynamics of Fields Reed College, 1999 [42] Zwiebach, B A First Course in String Theory Cambridge University Press, 2004 Index action area, 342, 343 classical, 3, coordinates and conservation, 189 E&M, 212 Einstein–Hilbert, 237, 243, 253, 293 field, 183 first-order, 200–2 first-order form, 234, 243 geodesic, 331 length, 62 mass term, 201, 228 Nambu–Goto, 347f Palatini, 242f Polyakov, 360 relativistic, 63ff reparametrization invariance, 62, 71, 331, 342 scalar field, 174f, 183, 201, 226 second-order form, 213, 253 source term, 214, 240 tensor field, 241 variation, 3f, 175, 177, 178, 180 vector field, 206–8, 226 advanced solution, 309f analytic extension, 282 angle between vectors, 121 angular momentum, 48, 69, 71 conservation, 47 scalar field, 195f Schwarzschild geodesics, 266 aphelion, 21 apparent singularity, 294 arc-length parametrization, 129, 138 area minimization, 340–2 relativistic, 345 axial symmetry, 258 balls and springs, 171 basis vectors, 99, 102, 109, 289f, 294 coordinate, 115, 117 spatial, 287 transformation, 102–4, 109 bending of light, 272, 276f Bianchi identity, 149, 162f for E&M, 166 linearized, 240 Birkhoff’s theorem E&M, 259 black holes, 285 Born–Infeld E&M, 259 bound orbit, 51 boundary condition Dirichlet, 349 free endpoint, 349 Boyer–Lindquist coordinates, 331, 334 canonical infinitesimal transformations, 37 canonical momentum, 30f, 73, 193 field, 193, 200 relativistic, 71, 73 scalar field, 234 strings, 348, 352 tensor field, 233, 241 vector field, 206 canonical transformations, 33 generating functions, 34, 35 identity, 38 infinitesimal, 38 infinitesimal generator, 39 Carter’s constant, 335 central potential, 7, 16, 45 centripetal acceleration, 84 charge conservation, 162, 191, 213ff Chern–Simon term, 247 Christoffel connection, 115f, 122, 135, 241 circular light-like orbits for Kerr, 336 circular orbit, 316 comma notation, 22, 32 complex scalar fields, 220 conformal flatness, 151 conjugate momentum, 18 connection, see Christoffel connection, 14, 44, 114–16, 123 transformation, 114 363 364 conserved current, 220 continuity, 199 continuity equation, 191, 223 contravariant transformation, 98 coordinate basis vectors, see basis vectors, 115, 117, 292 coordinate differential, 22 coordinate invariance, 217 coordinate singularity, 282, 294 coordinate time, 76, 78 coordinate transformation, 21, 23 cosmological constant, 293f cosmological model, 326 Coulomb gauge, 297 Coulomb potential, 253, 255 covariant derivative, 44, 113f, 122, 125, 130f, 145, 245 density, 185 E&M, 223 covariant transformation, 98, 106 cross polarization, 305 cross-derivative equality, 130f current, 191, 213 from fields, 220 curvature curves, 139 Ricci scalar, 140f curved space, 134 cylindrical coordinates, 5, 123 D’Alembertian, 306, 308 deflection angle, 277 density, 184 derivative along a curve, 119 determinant, 126, 230 determinant of the metric, 182 dipole moment, 313 dipole radiation, 314f dipole source, 305 Dirac delta function, 198, 306, 308, 310 Dirichlet boundary conditions, 349 divergence theorem, 180, 186, 192 dummy index, 13 eccentricity, 269 Eddington coordinates, 283–285 Einstein summation notation, 11 dummy index, 13 Einstein tensor, 163 linearized, 260f Einstein’s equation, 153, 157, 161–3, 237 Einstein–Hilbert action, 237, 243, 253 ellipse, 16, 19–21, 50 eccentricity, 269 Kepler’s laws, 52 period, 52 semi-major axis, 52 elliptical orbit, 19 energy Schwarzschild geodesics, 266 energy and momentum conservation, 162 Index energy density, 95, 193, 200 field, 194 particle, 199 energy–momentum tensor, see stress energy tensor, see stress tensor, 191, 197, 200 density, 198 E&M, 211f particles, 197 equations of motion, 2–5, 14, 31 continuum limit, 170 relativistic, 75 equivalence principle, 152 escape speed, 286 Euler’s method, 337 Euler–Lagrange equations, 3–5, 178, 181 field equations, 180 event horizon, 280, 285 extended body infall, 287 extremal length, 128 extrinsic curvature, 141 field equations first-order, 201 linearized, 260 tensor, 232 vector, 208 field-strength tensor, 206, 210 finite difference, 337 first-order form, 200 flat space, 134 Fourier transform, 298, 308, 310, 312f charge conservation, 312 Dirac delta function, 310 Lorentz gauge, 312 free endpoint boundary conditions, 349 free scalar field, 178, 193 gauge choice, 300 E&M, 297f linearized GR, 299–301 string, 349 gauge field, 229 gauge freedom, 217, 354 charge conservation, 215 E&M, 209 gauge invariance, 213 general covariance, 181 generating function, 34f geodesic, 14, 122–4, 129, 132, 145, 161 action, 331 effective force, 262 Euclidean, 123 Kerr, 334 light, 277, 280f, 284 light-like, 272–4, 276f on cylinder, 124 on sphere, 123 perturbation, 268, 274 radial, 277, 281, 284 Schwarzschild, 265 geodesic deviation, 158–161, 288, 292, 303 Index geodesic equation, 98, 246, 266 gradient, 22, 108f gravitational field, 153 general relativity, 94 Newtonian, 88, 96 gravitational radiation, 318 energy, 318 energy loss, 321 gravitational redshift, 286f gravitational wave detector, 317 gravitational waves, 302, 305, 316 sources, 305 gravitomagnetic force, 93, 264, 328 gravity as geometry, 154 Newtonian, 19, 87, 152 Gravity Probe B, 93, 333 Green’s function, 306–10, 322 Green’s theorem, 307 Hamiltonian, 28, 30 canonical infinitesimal transformations, 38 canonical transformations, 37 equations of motion, 30–2 field, 200 relativistic, 71, 73 total energy, 73 transformation, 39 Helmholtz equation, 309 Hulse, Taylor, 321 hyperbolic motion, 76 identity transformation, 38 ignorable coordinate, 18, 33 indefinite length, 58 induced metric, 341, 346 inertial mass, 153 infall extended body, 287 light, 280 light and coordinates, 281, 284 massive, 277 infinitesimal Lorentz boost, 70 infinitesimal rotation, 48f, 53, 67f generator, 69 integration by parts, 3, 178, 180, 313 intrinsic curvature, 141 isometry, 40 Jacobian, 182–4 Kepler’s second law, 52 Kepler’s third law, 52 Kerr metric, 328, 331, 333 equatorial geodesic, 339 geodesics, 334–6, 339 linearized, 332 singularity, 337 Killing tensor, 125f, 335 Killing vector, 44, 46, 125 Killing’s equation, 44, 46, 53, 125f, 335 Klein–Gordon equation, 196, 201–203 Kronecker delta, 13, 24, 126 Lagrange density, 175, 179, 181, 189, 200 particles, 198 Lagrange multiplier, 74 Lagrangian, 2, 5, 61 central potential, classical, 12 classical mechanics, for fields, 167, 173, 181 relativistic, 65, 78 scalar field, 174, 226 tensor field, 236 vector field, 226 Legendre transform, 25–8, 30, 35, 73, 200 length-extremizing curve, 124 Lense–Thirring precession, 93, 333 Levi–Civita symbol, 69 Levi–Civita tensor density, 229f light-like, 58, 272, 299 LIGO, 317, 321 line element, 11f, 58, 63, 128 linear ODE, 274 linearized Einstein tensor, 261 linearized gravity, 260 LISA, 321 local gauge invariance, 218, 222, 224f local rest frame, 64 longitudinal waves, 172 Lorentz boost, 59 Lorentz gauge, 192, 209, 229, 297f, 312 GR, 306 Lorentz gauge condition, 192 Lorentz transformation, 58f boost, 59, 70 infinitesimal, 70 infinitesimal generator, 71 rapidity, 60, 70 lowering, 22, 107 mass current, 262, 264 mass distribution, 262 mass in meters, 165 massive scalar field, 196, 201 massive vector field, 228 matrix, 16 antisymmetric, 15 determinant, 126 inverse, 22 orthogonal, 101 rotation, 100f symmetric, 16 matrix–vector notation, Maxwell stress tensor, 162, 200, 212 Mercury, 270 metric, 9, 10, 12, 23, 107 as tensor field, 237, 242 Christoffel connection, 115f, 122 connection, 14, 123 derivative, 32 365 366 metric (cont.) derivative identity, 43 determinant, 182, 184 Euclidean, indefinite, 58, 63 induced, 341 Kerr, 328, 331–3 line element, 11 Minkowski, 57f perturbation, 251, 260, 301 Reissner–Nordstrom, 295 Schwarzschild, 257, 264 signature, 134, 183 spherically symmetric ansatz, 252, 255 transformation, 105 Weyl, 258 metric connection, 115 metric density, 242 metric field, 157 metric perturbation, 251, 301 minimal substitution, 119, 224, 237, 245 minimal surface, 342 Minkowski diagram, 60 Minkowski distance, 82 Minkowski metric, 58 Minkowski space-time, 57 momentum conservation, 33 momentum density, 195, 199f scalar field, 194 momentum for relativistic particles, 199 monopole source, 305 Nambu–Goto action, 347, 348 static gauge, 350 Newton’s second law, Newtonian gravity, 16, 87, 152f potential, 16, 19 Newtonian deviation, 158, 161 Newtonian gravity, 152f Newtonian orbits Hamiltonian, 46 Noether’s theorem, 40, 173 fields, 190 particles, 40 non-tensorial derivative, 111 normal coordinates, 146–9, 151 null vector, 272, 281, 299 observer at infinity, 287 open index, 14 orbital motion, 16, 46, 264 orbital period, 52 orthonormal basis, 292 Palatini form, 242 parallel transport, 117–20, 129, 131, 133f, 145, 289 passive mass, 154 path dependence for parallel transport, 131 perihelion, 21 perihelion precession, 265, 268, 270 period, 269 Index perturbation bending of light, 274 perihelion precession, 268 plane-wave solution, 302 exact, 323 plus polarization, 305 Poisson bracket, 39 polar coordinates, 6, 24, 183 polarization tensor, 299, 302f polarization vector, 298 Polyakov action, 360 post-Newtonian approximation, 271 Poynting vector, 162, 200, 300 gravitational, 318 precession, 265, 268 Proca equation, 229 projection, 106, 289 transverse, traceless gauge, 319 prolate spheroidal coordinates, 165 proper time, 64, 72, 138 light, 272 pseudo-Riemannian manifold, 128 quadrupole moment, 314–6 quadrupole radiation, 315 radial geodesic (Schwarzschild), 280f, 284 radial geodesics (Schwarzschild), 277 radial infall, 20 radial infall, massive, 278 radius of curvature, 139 raising, 22, 106 rapidity, 60, 70 redshift of light, 286 Reissner–Nordstrom metric, 295 relativistic circular orbits, 83, 85 constant force, 76f infall, 82 spring force, 78ff relativistic momentum, 72, 76 relativistic string, 345f reparametrization invariance of action, 62, 71, 331, 342 reparametrization-invariant, 349 rest frame, 64, 290 retarded solution, 309f Ricci curvature, 137 Ricci scalar, 137, 140f, 143, 256 Ricci tensor, 141, 143, 162, 241 Bianchi identity, 163 linearized, 238, 240, 249 Riemann tensor, 129, 132, 134, 136f, 142–6, 149, 161 Bianchi identity, 149 contractions, 140 counting components, 149 for flat space, 134 geodesic deviation, 161 linearized, 249, 303 normal coordinates, 148 symmetries, 146, 148 Riemann tensor (defined), 131 Index Riemannian manifold, 127, 150 Robertson–Walker cosmological model, 326 Rosen gravitational waves, 324 rotation, 67f, 99–101 Runge–Kutta, 337–9 Runge–Lenz vector, 126 scalar, 101 scalar density, 184 scalar field, 171, 173f massive, 196, 201 Schwarzschild metric, 251, 257, 264 Eddington coordinates, 283, 285 light, 272–4, 276f orbital motion, 268 Schwarzschild radius, 280, 286 second-rank tensor field, 231 self-coupling, 241 semi-major axis, 269 semicolon notation, 44 separation of variables, 41 additive, 41 multiplicative, 41 separation vector, 304 signature (metric), 134 sources in E&M, 213 sources in general relativity, 244 space-like, 58, 272 space-time, 57, 82 special relativity, 58 speed limit, 85 spherical coordinates, spherically symmetric potential, 19 static gauge, 347, 352 stress energy tensor, see energy–momentum tensor, see stress tensor, 161–3 stress tensor, see energy–momentum tensor, see stress energy tensor, 95, 192f, 195, 240, 244, 246, 261 conservation, 162, 192 continuity, 191 density, 246 E&M, 211 field definition, 191 fields, 192, 199 for linearized gravity, 239 for metric perturbation, 318 particle, 197 perfect fluid, 293 scalar field, 193 string arc-length parametrization, 350, 357 boundary conditions, 351 effective mass density, 358 effective tension, 358 energy density, 356 equations of motion, 352 equations of motion (arc-length), 355 gauge choice, 349 momentum, 356 Nambu–Goto action, 347 Polyakov action, 360 static gauge, 350, 357 temporal parametrization, 350 wave equation, 353 strong equivalence principle, 156f, 237, 245 superposition, 226 tensor, 22f antisymmetric, 15 contravariant, 22f, 98, 102, 103 covariant, 22f, 98, 106 covariant derivative, 113f, 125 covariant derivative (covariant form), 125 definition, 21 density, 184 derivatives, 111f lowering, 22 matrix form, 15 mixed, 23 raising, 22 rank, 23 second-rank, 107, 108 symmetric, 15 tensor density, 184 derivative, 185 transformation, 184 weight, 184 tensor field equations, 232 tidal force, 292 time-like, 58, 272 torsion, 116, 128, 130 trace, 166, 232 stress tensor, 166 trace-reversed form, 232 traceless, 301f transverse, 299, 301 transverse velocity, 351 transverse, traceless gauge, 301, 304, 316, 319, 325 twin paradox, 80f units Gaussian, 213 SI, 212 variation, 3f variational derivative, vector field, 205 vector plane wave, 299 vector potentials, 209 wave equation, 172, 178, 180, 298 weak equivalence principle, 153f Weyl method, 251, 253–5, 295 E&M, 254f general relativity, 255 Weyl tensor, 151 world-line, 60 world-sheet, 346 Young’s modulus, 172 zoom-whirl, 339 367 [...]... 0 and t = T : j sin j Tπ t by appropriate choice of the coefficients { j }∞ j =1 So “any” trajectory for the particle could be represented by: ∞ x(t) = xf t j t j sin + T T j =1 (1.25) T Find the value of the action S = 0 L dt for this arbitrary trajectory, show (assuming j ∈ IR) that the value of the action for this arbitrary trajectory is greater than the value you get for the dynamical trajectory... book, and his comments and scathing criticism have been addressed in part – his help along the way has been indispensable Finally, Professor Deser introduced me to general relativity, and I thank him for sharing his ideas, and commentary on the subject in general, and for this book in particular Much of the presentation has been informed by my contact with him – he has been a wonderful mentor and teacher,... type of second semester of classical mechanics geared toward senior physics majors In the back of most classical mechanics texts, there is a section on field theory, generally focused on fluid dynamics as its end goal I again chose general relativity as a target – if geodesics and geometry can provide an introduction to the motion side of GR in the context of advanced mechanics, why not use the techniques... solution manual with me, and provided a careful, critical reading of the text as it was prepared for publication The Reed College physics department has been a wonderful place to carry out this work – my colleagues have been helpful and enthusiastic as I attempted to first teach, and then write about, general relativity I would like to thank Johnny Powell and John Essick for their support and advice Also within... Doniach, and Scott Hughes, for their thoughtful advice, gentle criticism, not-so-gentle criticism, and general effectiveness in teaching me something (not always what they intended) I have benefitted greatly from student input,1 and have relied almost entirely on students to read and comment on the text as it was written In this context, I would like to thank Tom Chartrand, Zach Schultz, and Andrew Rhines... telling, and says more about the structure of the Lagrangian and its quadratic dependence on velocities than anything else Setting aside the details of spherical coordinates and central potentials, we can gain insight into the classical Lagrangian by looking at it from a slightly different point of view – one that will allow us to generalize it appropriately to both special relativity and general relativity. .. equation again) end of the same subject? This is done by many authors, notably Thirring and Landau and Lifschitz I decided to focus on the idea that, as a point of physical modelbuilding, if you start off with a second-rank, symmetric tensor field on a Minkowski background, and require that the resulting theory be self-consistent, you end up, almost uniquely, with general relativity I learned this wonderful... “straight lines” in general Certainly in the current setting, if we take U = 0 and work in Cartesian coordinates, where ανγ = 0 (since the metric does not, in this case, depend on position), the solutions are manifestly straight lines Later on, in special and general relativity, we will lose the familiar notion of length, but if we accept a generalized length interpretation, we can still understand solutions... Preface So my first goal was to exploit students’ familiarity with classical mechanics to provide an introduction to the geometric properties of motion that we find in general relativity We begin, in the first chapter, by reviewing Newtonian gravity, and simultaneously, the role of the Lagrangian and Hamiltonian points of view, and the variational principles that connect the two Any topic that benefits... tensors in the context of these flat spaces, introducing definitions and examples meant to motivate the covariant derivative and associated Christoffel connection These exist in flat space(-time), so there is an opportunity to form a connection between tensor ideas and more familiar versions found in vector calculus To understand general relativity, we need to be able to characterize space-times that are