A general relativity workbook

440 Pages·2010·6.327 MB·English

by  Moore T.A.

A General Relativity
Workbook
Credit: NASA
Thomas A. Moore
Version b0.92 © January 2010
GR Units: c = 1 (with time measured in meters, mass-energy in kg or eV)
Important Conversion Factors for GR Units
1m / (1/299,792,458)s= 3.33564095#10-9s . 3.34ns
1 ns = 0.2998 m (of time) ≈ 1 ft
1 µs = 299.8 m (of time)
1 ms = 299.8 km
1 s = 299,800 km
1 min = 17.99 × 106 km
1 hr = 1.079 × 109 km
1 day = 25.90 × 109 km
1 y = 365.25 d = 9.461 × 1015 m = 1 ly
1J =1.1126501#10-17kg (energy)
1 kg (energy) = 8.98755179 × 1016 J
1 kg (momentum) = 299,792,458 kg·m/s
1 eV = 1.602 × 10–19 J = 1.782 × 10–36 kg (energy)
1 eV (momentum) = 5.34 × 10–28 kg·m/s
Useful Data in GR Units
1AU = 1.495 978 706 91(30) × 1011 m
1 pc = 30.857 × 1012 m = 3.2616 ly
1 M = mass of earth = 5.9736 × 1024 kg (GM = 4.4361 mm)
e e
1 M9 = 1 solar mass = 1.9891 × 1030 kg (GM9 = 1477.1 m)
m (electron) = 9.109 382 15(45) × 10–31 kg = 0.510 998 910(13) MeV
e
m (proton) = 1.672 621 637(83) × 10–27 kg = 938.272 013(23) MeV
p
m (neutron) = 1.674 927 211(84) × 10–27 kg = 939.565 346(23) MeV
n
t = age of universe = 13.73(0.16) Gy = 1.30 × 1026 m
0
H = Hubble constant at present = 70.4(1.5) km s–1 Mpc–1 = 7.61 × 10–27 m–1 = [13.9(0.3) Gy]–1
0
t = critical density at present = 9.32(0.20) × 10–27 kg m–3
c
X = baryon density fraction = 0.0441(23)
b
X = matter density fraction = 0.268(18) (includes baryons and dark matter)
m
X = vacuum energy density fraction = 0.732(18)
v
Important Constants in SI and GR Units
Symbol Value in SI Units* Value in GR Units Comments
g 9.8 m/s2 1.09 × 10–16 m–1 = (0.97 ly)–1
G 6.674 28(67) × 10–11 m3 kg–1 s–2 7.426 × 10–28 m/kg = 1477 m / solar mass
& 1.054 571 628(53) × 10–34 J·s 3.518 × 10–43 kg·m
&c 197.326 9631(49) eV·nm same (= &) (when using eV)
k 1.380 6504(24) × 10–23 J K–1 1.536 × 10–40 kg K–1
B
k 8.617 343(15) × 10–5 eV K–1 same (when using eV)
B
v 5.670 400(40) × 10–8 W m–2 K–4 2.105 × 10–33 kg m–3K–4 Stefan-Boltzmann
SB
k 8.987 551 788 × 109 kg m3 s–2 C–2 1.000 × 10–7 kg m C–2 (exact)
*Most SI constants from http://physics.nist.gov/cuu/Constants/index.html, 2006 values
Values in parentheses are the uncertainties (68% confidence range) in the last two digits.
Sign Conventions (Landau-Lifshitz Spacelike Convention):
Metric Signature: – + + +, Riemann tensor: Rabno=+2nCbao-f, Einstein tensor: Gno=+8rGTno
Useful Formulae:
R V
c -cb 0 0
S W
Lorentz Transformation: Kno=SSS-0cb c0 01 00WWW fcor/b[1=-^+b2b]-,01/2,0h
S 0 0 0 1W
T X
Flat Space Metric: ds2=-dt2+dx2+dy2+dz2/hnodxndxo
(in spherical coordinates): ds2=-dt2+dr2+r2di2+r2sin2idz2
Spherical surface metric: ds2=R2di2+R2sin2idz2
dxn
Four-velocity definition: un/ dx (note that -1=u:u/gnounuo)
Energy seen by observer: E=-p:u where p for object /mu
obs
2xn 2xn2xb
Tensor Transformations: Ul =U, Anl= 2xaAa, Bnol= 2xa 2xoBab, etc.
2xn 2xa
Fundamental Identity: 2xa 2xo =dno, where dno=1if n=o, 0otherwise
Raising, Lowering Indices: Bno=gnaBao, Bno=gnaBao, etc.
R V
0 E E E
S x y z W
S-E 0 B -B W
EM Field Tensor: Fno=S-Ex -B 0z B yW
S y z x W
S-E B -B 0 W
z y x
T X
dpn
Maxwell’s Equations: 2oFno=4rkJn, 2aFno+2oFan+2nFoa=0 Lorentz force law: dx =qFnouo
Christoffel Symbols: Cnao= 21gav 2ngov+2ogvn-2vgno
7 A
d dxb dxn dxo d2xa dxn dxo
Geodesic Equation: 0= dxcgab dxm- 21(2agno) dx dx or 0= dx2 +Cnao dx dx
Absolute Gradient: daBno=2aBno+CanvBvo-CacoBnc (one Christoffel symbol per index)
Riemann Tensor: Rabno=2nCbao-2oCban+CnacCbco-CoavCbvn
(in a LIF, where Cnao=0): Rabno= 21 2b2ngao+2a2ogbn-2b2ogan-2a2ngbo
^ h
Riemann Symmetries: Rabon=-Rabno, Rbano=-Rabno, Rnoab=+Rabno, Rabno+Ranob+Raobn=0
Bianchi Identity: dvRabno+doRabvn+dnRabov=0
Ricci Tensor: Rbo=Rabao Curvature Scalar: R=gboRbo=Roo
Stress-Energy: perfect fluid: Tno=(t0+p0)unuo+p0gno vacuum: Tvnaoc=-Kgno/8rG
Einstein Tensor: Gno=Rno- 1gnoR
2
Einstein Equation: Gno=8rGTno or Rno=8rG(Tno- 21gnoTaa)
Linear Approximation: 2a2aHno=-16rGTno, with 2nHno=0, and
Hno/hno- 21hnohvv where gno=hno+hnowith hno <<1
(raise and lower H and h indices using the flat space metric)
2GM 2GM -1
Schwarzschild Metric: ds2=-b1- r ldt2-b1- r l dr2+r2di2+r2sin2idz2
Cosmic Metric: ds2=-dt2+[a(t)]2[drr2+q2(di2+sin2idz2)]
where q(rr)=Rsinh(rr/R), rr,orRsin(rr/R), with R=H0 Xk 1/2= constant
1 da 2 X X
Friedman Equation: cH0 dtm =Xk+ am + a2r +Xva2, where Xk=1-Xm-Xr-Xv
2GMr r2+a2cos2i
Kerr Metric: ds2=- 1- dt2+ dr2+(r2+a2cos2i)di2
c r2+a2cos2im dr2-2GMr+a2n
2GMra2sin2i 4GMrasin2i
+ r2+a2+ sin2idz2- dzdt
d r2+a2cos2i n dr2+a2cos2in
(This page is intentionally blank)
A General Relativity
Workbook
Credit: NASA
Thomas A. Moore
Pomona College
Version b0.92 © January 2010
ii
For Joyce, whose miraculous love always supports me and
allows me to take risks with life that I could not face alone.
and for Edwin Taylor, whose book with Wheeler set me on this path decades ago,
and whose gracious support and friendship has kept me going.
Page Layout: Thomas Moore (using Adobe® InDesign®, MathMagic®,
and Virginia Systems InSeq® and InDex Pro®)
Copyright © 2009 Thomas A. Moore
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Table of Contents iii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
General Relativity in a Nutshell: Outline . . . . . . . . . . . . . . . . . .2
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Review of Special Relativity . . . . . . . . . . . . . . . . . . . .11
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Box 2.1 Overlapping IRFs move with constant relative velocities . . . 16
Box 2.2 Unit conversions between SI and GR units . . . . . . . . . . 17
Box 2.3 One derivation of the Lorentz Transformation . . . . . . . . . 18
Box 2.4 Lorentz Transformations and Rotations . . . . . . . . . . . . 21
Box 2.5 Frame-independence of the Spacetime Interval . . . . . . . . 22
Box 2.6 Frame-Dependence of the Time Order of Events . . . . . . . 22
Box 2.7 Proper time along a path . . . . . . . . . . . . . . . . . . . . 23
Box 2.8 Length contraction . . . . . . . . . . . . . . . . . . . . . . . 23
Box 2.9 The Einstein Velocity Transformation . . . . . . . . . . . . . 24
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Box 3.1 The Frame-Independence of the Scalar Product . . . . . . . . 30
Box 3.2 The Invariant Magnitude of the Four-Velocity . . . . . . . . . 30
Box 3.3 The Low-Velocity Limit of u . . . . . . . . . . . . . . . . . 31
Box 3.4 Conservation of Momentum or Four-momentum? . . . . . . . 32
Box 3.5 Example: The GZK Cosmic-Ray Energy Cutoff . . . . . . . . 33
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Box 4.1 Behavior of the Kronecker Delta . . . . . . . . . . . . . . . . 42
Box 4.2 EM Field Units in the GR Unit System . . . . . . . . . . . . 42
Box 4.3 Example: The GZK Cosmic-Ray Energy Cutoff . . . . . . . . 43
Box 4.4 Identifying Free and Bound Indices . . . . . . . . . . . . . . 44
Box 4.5 Rule Violations . . . . . . . . . . . . . . . . . . . . . . . . . 44
Box 4.6 Example Derivations . . . . . . . . . . . . . . . . . . . . . . 45
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Arbitrary Coordinates . . . . . . . . . . . . . . . . . . . . . . . . .47
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Box 5.1 The Polar Coordinate Basis . . . . . . . . . . . . . . . . . . 52
Box 5.2 Proof of the Metric Transformation Law . . . . . . . . . . . . 53
Box 5.3 A 2D Example: Parabolic Coordinates . . . . . . . . . . . . . 54
Box 5.4 The LTEs as an Example General Transformation . . . . . . . 56
Box 5.5 The Metric Transformation Law in Flat Space . . . . . . . . . 56
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 56
iv Table of Contents
6 Tensor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Box 6.1 Example Gradient Covectors . . . . . . . . . . . . . . . . . . 62
Box 6.2 Lowering Indices . . . . . . . . . . . . . . . . . . . . . . . . 63
Box 6.3 The Inverse Metric . . . . . . . . . . . . . . . . . . . . . . . 64
Box 6.4 The Kronecker Delta is a Tensor . . . . . . . . . . . . . . . . 65
Box 6.5 Tensor Operations . . . . . . . . . . . . . . . . . . . . . . . 65
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 66
7 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . .67
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Box 7.1 Gauss’s Law in Integral and Differential Form . . . . . . . . 72
Box 7.2 The Derivative of m2 . . . . . . . . . . . . . . . . . . . . . . 73
Box 7.3 Raising and Lowering Indices in Cartesian Coordinates . . . . 73
Box 7.4 The Tensor Equation For Conservation of Charge . . . . . . . 74
Box 7.5 The Antisymmetry of F Implies Charge Conservation . . . . . 75
Box 7.6 The Magnetic Potential . . . . . . . . . . . . . . . . . . . . . 76
Box 7.7 Proof of the Source-Free Maxwell Equations . . . . . . . . . 77
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 78
8 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Box 8.1 The Worldline of Longest Proper Time in Flat Spacetime . . . 83
Box 8.2 Derivation of the Euler-Lagrange Equation . . . . . . . . . . 84
Box 8.3 Deriving the Second Form of the Geodesic Equation . . . . . 85
Box 8.4 Geodesics for Flat Space in Parabolic Coordinates . . . . . . 86
Box 8.5 Geodesics for the Surface of a Sphere . . . . . . . . . . . . . 88
Box 8.6 The Geodesic Equation Does Not Determine the Scale of x . 90
Box 8.7 Light Geodesics in Flat Spacetime . . . . . . . . . . . . . . . 90
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 91
9 The Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . . 93
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Box 9.1 Radial Distance . . . . . . . . . . . . . . . . . . . . . . . . . 98
Box 9.2 Falling from Rest in Schwarzschild Spacetime . . . . . . . . 99
Box 9.3 GM for the Earth and the Sun . . . . . . . . . . . . . . . . 100
Box 9.4 The Gravitational Redshift for Weak Fields . . . . . . . . . 100
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 102
10 Particle Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Box 10.1 Schwarzschild Orbits Must be Planar . . . . . . . . . . . . 108
Box 10.2 The Schwarzschild “Conservation of Energy” Equation . . . 109
Box 10.2 Deriving Conservation of Newtonian Energy for Orbits . . . 110
Box 10.4 The Radii of Circular Orbits . . . . . . . . . . . . . . . . . 110
Box 10.5 Kepler’s Third Law . . . . . . . . . . . . . . . . . . . . . . 112
Box 10.6 The Innermost Stable Circular Orbit (ISCO) . . . . . . . . . 113
Box 10.7 The Energy Radiated by an Inspiraling Particle . . . . . . . 113
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 114
Table of Contents v
11 Precession of the Perihelion . . . . . . . . . . . . . . . . . . . 115
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Box 11.1 Verifying the Orbital Equation for u(z) . . . . . . . . . . . 120
Box 11.2 Verifying the Newtonian Orbital Equation . . . . . . . . . . 120
Box 11.3 Verifying the Equation for the Orbital “Wobble” . . . . . . 121
Box 11.4 Application to Mercury . . . . . . . . . . . . . . . . . . . . 121
Box 11.5 Constructing the Schwarzschild Embedding Diagram . . . . 122
Box 11.6 Calculating the Wedge Angle d . . . . . . . . . . . . . . . 123
Box 11.7 A Computer Model for Schwarzschild Orbits . . . . . . . . 123
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 126
12 Photon Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Box 12.1 The Meaning of the Impact Parameter b . . . . . . . . . . 132
Box 12.2 Derivation of the Equation of Motion for a Photon . . . . . 132
Box 12.3 Features of the Effective Potential Energy Function for Light 133
Box 12.4 Photon Motion in Flat Space . . . . . . . . . . . . . . . . . 133
Box 12.4 Evaluating 4-Vector Components in an Observer’s Frame . 134
Box 12.6 An Orthonormal Basis in Schwarzschild Coordinates . . . . 134
Box 12.7 Derivation of the Critical Angle for Photon Emission . . . . 135
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 136
13 Deflection of Light . . . . . . . . . . . . . . . . . . . . . . . . . . .137
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Box 13.1 Checking Equation 13.2 . . . . . . . . . . . . . . . . . . . 143
Box 13.1 The Differential Equation for the Shape of a Photon Orbit . 144
Box 13.3 The Differential Equation for the Photon “Wobble” . . . . . 144
Box 13.4 The Solution for u(z) in the Large-r Limit . . . . . . . . . 145
Box 13.5 The Maximum Angle of Light Deflection by the Sun . . . . 145
Box 13.6 The Lens Equation . . . . . . . . . . . . . . . . . . . . . . 146
Box 13.7 The Ratio of Image Brightness to the Source Brightness . . 147
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 148
14 Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Box 14.1 Finite Distance to r = 2GM . . . . . . . . . . . . . . . . . 154
Box 14.2 Proper Time for Free Fall from r = R to r = 0 . . . . . . . . 156
Box 14.3 The Future is Finite Inside the Event Horizon . . . . . . . . 157
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 158
15 Alternative Coordinates . . . . . . . . . . . . . . . . . . . . . . .159
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Box 15.1 Calculating 2tc/2r . . . . . . . . . . . . . . . . . . . . . . . 164
Box 15.2 The Global Rain Metric . . . . . . . . . . . . . . . . . . . 165
Box 15.3 The Limits on dr/dtc Inside the Event Horizon . . . . . . . 165
Box 15.4 Transforming to Kruskal-Szekeres Coordinates . . . . . . . 166
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 168
vi Table of Contents
16 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . .169
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Box 16.1 Free-fall Time to the Event Horizon from r = 2GM + f . . . 174
Box 16.2 Calculating E3 . . . . . . . . . . . . . . . . . . . . . . . . 175
Box 16.3 Evaluating k , &, and T for a Solar-Mass Black Hole . . . . 176
B
Box 16.4 Lifetime of a Black Hole . . . . . . . . . . . . . . . . . . . 177
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 178
17 The Absolute Gradient . . . . . . . . . . . . . . . . . . . . . . . . 179
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Box 17.1 Absolute Gradient of a Vector . . . . . . . . . . . . . . . . 184
Box 17.2 Absolute Gradient of a Covector . . . . . . . . . . . . . . . 184
Box 17.3 Symmetry of the Christoffel Symbols . . . . . . . . . . . . 185
Box 17.4 The Christoffel Symbols in Terms of the Metric . . . . . . . 185
Box 17.5 Checking the Geodesic Equation . . . . . . . . . . . . . . . 186
Box 17.6 A Trick for Calculating Christoffel Symbols . . . . . . . . . 186
Box 17.7 The Local Flatness Theorem . . . . . . . . . . . . . . . . . 187
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 190
18 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . . . .191
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Box 18.1 Newtonian Tidal Deviation Near a Spherical Object . . . . 196
Box 18.2 Proving Equation 18.9 . . . . . . . . . . . . . . . . . . . . 197
Box 18.3 The Absolute Derivative of n . . . . . . . . . . . . . . . . 197
Box 18.4 Proving Equation 18.14 . . . . . . . . . . . . . . . . . . . 198
Box 18.5 An Example of Calculating the Riemann Tensor . . . . . . 198
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 200
19 The Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . .201
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Box 19.1 The Riemann Tensor in a Locally Inertial Frame . . . . . . 204
Box 19.2 Symmetries of the Riemann Tensor . . . . . . . . . . . . . 205
Box 19.3 Counting the Riemann Tensor’s Independent Components . 206
Box 19.4 The Bianchi Identity . . . . . . . . . . . . . . . . . . . . . 207
Box 19.5 The Ricci Tensor is Symmetric . . . . . . . . . . . . . . . . 208
Box 19.6 The Riemann and Ricci Tensors and R a Sphere . . . . . . . 208
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 210
20 The Stress-Energy Tensor . . . . . . . . . . . . . . . . . . . . . 211
Concept Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Box 20.1 Why the Source of Gravity Must be Energy, not Mass . . . 216
Box 20.2 Interpretation of Tij in a Locally Inertial Frame . . . . . . . 216
Box 20.3 The Stress-Energy Tensor for a Perfect Fluid in its Rest LIF 217
Box 20.4 Equation 20.16 Reduces to Equation 20.15 . . . . . . . . . 219
Box 20.5 Fluid Dynamics from Conservation of Four-Momentum . . 219
HOMEWORK PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 220