A General Relativity Workbook

CONTENTS
PREFACE
1. INTRODUCTION
	Concept Summary
	Homework Problems
	General Relativity in a Nutshell
2. REVIEW OF SPECIAL RELATIVITY
	Concept Summary
	BOX 2.1 Overlapping IRFs Move with Constant Relative Velocities
	BOX 2.2 Unit Conversions Between SI and GR Units
	BOX 2.3 One Derivation of the Lorentz Transformation
	BOX 2.4 Lorentz Transformations and Rotations
	BOX 2.5 Frame-Independence of the Spacetime Interval
	BOX 2.6 Frame-Dependence of the Time Order of Events
	BOX 2.7 Proper Time Along a Path
	BOX 2.8 Length Contraction
	BOX 2.9 The Einstein Velocity Transformation
	Homework Problems
eview of Special Relati
3. FOUR-VECTORS
	Concept Summary
	BOX 3.1 The Frame-Independence of the Scalar Product
	BOX 3.2 The Invariant Magnitude of the Four-Velocity
	BOX 3.3 The Low-Velocity Limit of u
	BOX 3.4 Conservation of Momentum or Four-momentum?
	BOX 3.5 Example: The GZK Cosmic-Ray Energy Cutoff
	Homework Problems
4. INDEX NOTATION
	Concept Summary
	BOX 4.1 Behavior of the Kronecker Delta
	BOX 4.2 EM Field Units in the GR Unit System
	BOX 4.3 Electromagnetic Equations in Index Notation
	BOX 4.4 Identifying Free and Bound Indices
	BOX 4.5 Rule Violations
	BOX 4.6 Example Derivations
	Homework Problems
5. ARBITRARY COORDINATES
	Concept Summary
	BOX 5.1 The Polar Coordinate Basis
	BOX 5.2 Proof of the Metric Transformation Law
	BOX 5.3 A 2D Example: Parabolic Coordinates
	BOX 5.4 The LTEs as an Example General Transformation
	BOX 5.5 The Metric Transformation Law in Flat Space
	BOX 5.6 A Metric for a Sphere
	Homework Problems
6. TENSOR EQUATIONS
	Concept Summary
	BOX 6.1 Example Gradient Covectors
	BOX 6.2 Lowering Indices
	BOX 6.3 The Inverse Metric
	BOX 6.4 The Kronecker Delta Is a Tensor
	BOX 6.5 Tensor Operations
	Homework Problems
7. MAXWELL’S EQUATIONS
	Concept Summary
	BOX 7.1 Gauss's Low in integral and Differential Form
	BOX 7.2 The Derivative of m^2
	BOX 7.3 Raising and Lowering Indices in Cartesian Coordinates
	BOX 7.4 The Tensor Equation for Conservation of Change
	BOX 7.5 The Antisymmetry of F Implies Charge Conservation
	BOX 7.6 The Magnetic Potential
	BOX 7.7 Proof of the Source-Free Maxwell Equations
	Homework Problems
8. GEODESICS
	Concept Summary
	BOX 8.1 The Worldline of Longest Proper Time in Flat Spacetime
	BOX 8.2 Derivation of the Euler-Lagrange Equation
	BOX 8.3 Deriving the Second Form of the Geodesic Equation
	BOX 8.4 Geodesics for Flat Space in Parabolic Coordinates
	BOX 8.5 Geodesics for the Surface of a Sphere
	BOX 8.6 The Geodesic Equation Does Not Determine the Scale of τ
	BOX 8.7 Light Geodesics in Flat Spacetime
	Homework Problems
9. THE SCHWARZSCHILD METRIC
	Concept Summary
	BOX 9.1 Radial Distance
	BOX 9.2 Falling from Rest in Schwarzschild Spacetime
	BOX 9.3 GM for the Earth and the Sun
	BOX 9.4 The Gravitational Redshift for Weak Fields
	Homework Problems
10. PARTICLE ORBITS
	Concept Summary
	BOX 10.1 Schwarzschild Orbits Must Be Planar
	BOX 10.2 The Schwarzschild "Conservation of Energy" Equation
	BOX 10.3 Deriving Conservation of Newtonian Energy for Orbits
	BOX 10.4 The Radii of Circular Orbits
	BOX 10.5 Kepler's Third Law
	BOX 10.6 The Innermost Stable Circular Orbit (ISCO)
	BOX 10.7 The Energy Radiated by an Inspiraling Particle
	Homework Problems
11. PRECESSION OF THE PERIHELION
	Concept Summary
	BOX 11.1 Verifying the Orbital Equation for u(Φ)
	Box 11.2 Verifying the Newtonian Orbital Equation
	Box 11.3 Verifying the Equation for the Orbital "Wobble"
	Box 11.4 Application to Mercury
	Box 11.5 Constructing the Schwarzschild Embedding Diagram
	Box 11.6 Calculating the Wedge Angle δ
	Box 11.7 A Computer Model for Schwarzschild Orbits
	Homework Problems
12. PHOTON ORBITS
	Concept Summary
	Box 12.1 The Meaning of the Impact Parameter b
	Box 12.2 Derivation of the Equation of Motion for a Photon
	Box 12.3 Features of the Effective Potential Energy Function for Light
	Box 12.4 Photon Motion in Flat Space
	Box 12.5 Evaluating 4-Vector Components in an Observer's Frame
	Box 12.6 An Orthonormal Basis in Schwarzschild Coordinates
	Box 12.7 Derivation of the Critical Angle for Photon Emission
	Homework Problems
13. DEFLECTION OF LIGHT
	Concept Summary
	Box 13.1 Checking Equation 13.2
	Box 13.2 The Differential Equation for the Shape of a Photon Orbit
	Box 13.3 The Differential Equation for the Photon "Wobble"
	Box 13.4 The Solution for u(Φ) in the Large-r Limit
	Box 13.5 The Maximum Angle of Light Deflection by the Sun
	Box 13.6 The Lens Equation
	Box 13.7 The Ratio of Image Brightness to the Source Brightness
	Homework Problems
14. EVENT HORIZON
	Concept Summary
	Box 14.1 Finite Distance to r = 2GM
	Box 14.2 Proper Time for Free Fall from r = R t o r = 0
	Box 14.3 The Future Is Finite Inside the Event Horizon
	Homework Problems
15. ALTERNATIVE COORDINATES
	Concept Summary
	Box 15.1 Calculating ∂t/∂r
	Box 15.2 The Global Rain Metric
	Box 15.3 The Limits on dr/dt Inside the Event Horizon
	Box 15.4 Transforming to Kruskal-Szekeres Coordinates
	Homework Problems
16. BLACK HOLE THERMODYNAMICS
	Concept Summary
	Box 16.1 Free-Fall Time to the Event Horizon from r = 2GM + ε
	Box 16.2 Calculating E_∞
	Box 16.3 Evaluating k_B, h, and T for a Solar-Mass Black Hole
	Box 16.4 Lifetime of a Black Hole
	Homework Problems
17. THE ABSOLUTE GRADIENT
	Concept Summary
	Box 17.1 Absolute Gradient of a Vector
	Box 17.2 Absolute Gradient of a Covector
	Box 17.3 Symmetry of the Christoffel Symbols
	Box 17.4 The Christoffel Symbols in Terms of the Metric
	Box 17.5 Checking the Geodesic Equation
	Box 17.6 A Trick for Calculating Christoffel Symbols
	Box 17.7 The Local Flatness Theorem
	Homework Problems
18. GEODESIC DEVIATION
	Concept Summary
	Box 18.1 Newtonian Tidal Deviation Near a Spherical Object
	Box 18.2 Proving Equation 18.9
	Box 18.3 The Absolute Derivative of n
	Box 18.4 Proving Equation 18.14
	Box 18.5 An Example of Calculating the Riemann Tensor
	Homework Problems
19. THE RIEMANN TENSOR
	Concept Summary
	Box 19.1 The Riemann Tensor in a Locally Inertial Frame
	Box 19.2 Symmetries of the Riemann Tensor
	Box 19.3 Counting the Riemann Tensor's Independent Components
	Box 19.4 The Bianchi Identity
	Box 19.5 The Ricci Tensor Is Symmetric
	Box 19.6 The Riemann and Ricci Tensors and R for a Sphere
	Homework Problems
20. THE STRESS-ENERGY TENSOR
	Concept Summary
	Box 20.1 Why the Source of Gravity Must Be Energy, Not Mass
	Box 20.2 Interpretation of T^{ij} in a Locally Inertial Frame
	Box 20.3 The Stress-Energy Tensor for a Perfect Fluid in Its Rest LIF
	Box 20.4 Equation 20.16 Reduces to Equation 20.15
	Box 20.5 Fluid Dynamics from Conservation of Four-Momentum
	Homework Problems
21. THE EINSTEIN EQUATION
	Concept Summary
	Box 21.1 The Divergence of the Ricci Tensor
	Box 21.2 Finding the Value of b
	Box 21.3 Showing that -R + 4Λ = κT
	Homework Problems
22. INTERPRETING THE EQUATION
	Concept Summary
	Box 22.1 Conservation of Four-Momentum Implies 0 = ∇_ν (ρ_0 u^v)
	Box 22.2 The Inverse Metric in the Weak-Field Limit
	Box 22.3 The Riemann Tensor in the Weak-Field Limit
	Box 22.4 The Ricci Tensor in the Weak-Field Limit
	Box 22.5 The Stress-Energy Sources of the Metric Perturbation
	Box 22.6 The Geodesic Equation for a Slow Particle in a Weak Field
	Homework Problems
23. THE SCHWARZSCHILD SOLUTION
	Concept Summary
	Box 23.1 Diagonalizing the Spherically Symmetric Metric
	Box 23.2 The Components of the Ricci Tensor
	Box 23.3 Solving for B
	Box 23.4 Solving for a(r)
	Box 23.5 The Christoffel Symbols with t-t as Subscripts
	Homework Problems
24. THE UNIVERSE OBSERVED
	Concept Summary
	Box 24.1 Measuring Astronomical Distances in the Solar System
	Box 24.2 Determining the Distance to Stellar Clusters
	Box 24.3 How the Doppler Shift Is Connected to Radial Speed
	Box 24.4 Values of the Hubble Constant
	Box 24.5 Every Point Is the Expansion's "Center"
	Box 24.6 The Evidence for Dark Matter
	Homework Problems
25. A METRIC FOR THE COSMOS
	Concept Summary
	Box 25.1 The Universal Ricci Tensor
	Box 25.2 Raising One Index of the Universal Ricci Tensor
	Box 25.3 The Stress-Energy Tensor with One Index Lowered
	Box 25.4 The Einstein Equation with One Index Lowered
	Box 25.5 Verifying the Solutions for q
	Homework Problems
26. EVOLUTION OF THE UNIVERSE
	Concept Summary
	Box 26.1 The Other Components of the Einstein Equation
	Box 26.2 Consequences of Local Energy/Momentum Conservation
	Box 26.3 Deriving the Density/Scale Relationship for Radiation
	Box 26.4 Deriving the Friedman Equation
	Box 26.5 The Friedman Equation for the Present Time
	Box 26.6 Deriving the Friedman Equation in Terms of the Omegas
	Box 26.7 The Behavior of a Matter-Dominated Universe
	Homework Problems
27. COSMIC IMPLICATIONS
	Concept Summary
	Box 27.1 Connecting the Redshift z to the Hubble Constant
	Box 27.2 Deriving the Hubble Relation in Terms of Redshift z
	Box 27.3 The Luminosity Distance
	Box 27.4 The Differential Equation for a(η)
	Box 27.5 How to Generate a Numerical Solution for Equation 27.18
	Homework Problems
28. THE EARLY UNIVERSE
	Concept Summary
	Box 28.1 Single-Component Universes
	Box 28.2 The Transition to Matter Dominance
	Box 28.3 The Time-Temperature Relation
	Box 28.4 Neutrino Decoupling
	Box 28.5 The Number Density of Photons
	Homework Problems
29. CMB FLUCTUATIONS AND INFLATION
	Concept Summary
	Box 29.1 The Angular Width of the Largest CMB Fluctuations
	Box 29.2 The Equation for Ω_k(t)
	Box 29.3 Cosmic Flatness at the End of Nucleosynthesis
	Box 29.4 The Exponential Inflation Formula
	Box 29.5 Inflation Calculations
	Homework Problems
30. GAUGE FREEDOM
	Concept Summary
	Box 30.1 The Weak-Field Einstein Equation in Terms of h_{μv}
	Box 30.2 The Trace-Reverse of h_{μv}
	Box 30.3 The Weak-Field Einstein Equation in Terms of H^{μv}
	Box 30.4 Gauge Transformations of the Metric Perturbations
	Box 30.5 A Gauge Transformation Does Not Change R_{αβμν}
	Box 30.6 Lorentz Gauge
	Box 30.7 Additional Gauge Freedom
	Homework Problems
31. DETECTING GRAVITATIONAL WAVES
	Concept Summary
	Box 31.1 Constraints on Our Trial Solution
	Box 31.2 The Transformation to Transverse-Traceless Gauge
	Box 31.3 A Particle at Rest Remains at Rest in TT Coordinates
	Box 31.4 The Effect of a Gravitational Wave on a Ring of Particles
	Homework Problems
32. GRAVITATIONAL WAVE ENERGY
	Concept Summary
	Box 32.1 The Ricci Tensor
	Box 32.2 The Averaged Curvature Scalar
	Box 32.3 The General Energy Density of a Gravitational Wave
	Homework Problems
33. GENERATING GRAVITATIONAL WAVES
	Concept Summary
	Box 33.1 H^{tμ} for a Compact Source Whose CM is at Rest
	Box 33.2 A Useful Identity
	Box 33.3 The Transverse-Traceless Components of A^{μv}
	Box 33.4 How to Find I^{jk}_{TT} for Waves Moving in the \vec{n} Direction
	Box 33.5 Flux in Terms of I^{jk}
	Box 33.6 Evaluating the Integrals in the Power Calculation
	Homework Problems
34. GRAVITATIONAL WAVE ASTRONOMY
	Concept Summary
	Box 34.1 The Dumbbell I^{jk}
	Box 34.2 The Power Radiated by a Rotating Dumbbell
	Box 34.3 The Total Energy of an Orbiting Binary Pair
	Box 34.4 The Time-Rate-of-Change of the Orbital Period
	Box 34.5 Characteristics of ι Boötis
	Homework Problems
35. GRAVITOMAGNETISM
	Concept Summary
	Box 35.1 The Lorentz Condition for the Potentials
	Box 35.2 The Maxwell Equations for the Gravitational Field
	Box 35.3 The Gravitational Lorentz Equation
	Box 35.4 The "Gravitomagnetic Moment" of a Spinning Object
	Box 35.5 Angular Speed of Gyroscope Precession
	Homework Problems
36. THE KERR METRIC
	Concept Summary
	Box 36.1 Expanding | \vec{R} - \vec{r} |^{-1} to First Order in r/R
	Box 36.2 The Integral for h^{tx}
	Box 36.3 Why the Other Terms in the Expansion Integrate to Zero
	Box 36.4 Transforming the Weak-Field Solution to Polar Coordinates
	Box 36.5 The Weak-Field Limit of the Kerr Metric
	Homework Problems
37. PARTICLE ORBITS IN KERR SPACETIME
	Concept Summary
	Box 37.1 Calculating Expressions for dt/dτ and dΦ/dτ
	Box 37.2 Verify the Value of [g_{tΦ}]² - g_{tt}g_{ΦΦ}
	Box 37.3 The "Energy-Conservation-Like" Equation of Motion
	Box 37.4 Kepler's Third Law
	Box 37.5 The Radii of ISCOs When a = GM
	Homework Problems
38. ERGOREGION AND HORIZON
	Concept Summary
	Box 38.1 The Radii Where g_{tt} = 0
	Box 38.2 The Angular Speed Range When dr and/or dθ≠0
	Box 38.3 Angular-Speed Limits in the Equatorial Plane
	Box 38.4 The Metric of the Event Horizon's Surface
	Box 38.5 The Area of the Outer Kerr Event Horizon
	Box 38.6 Transformations Preserve the Metric Determinant's Sign
	Homework Problems
39. NEGATIVE-ENERGY ORBITS
	Concept Summary
	Box 39.1 Quadratic Form for Conservation of Energy
	Box 39.2 The Square Root Is Zero at the Event Horizon
	Box 39.3 Negative e Is Possible Only in the Ergoregion
	Box 39.4 The Fundamental Limit on δM in Terms of δS
	Box 39.5 δM_{ir} ≥ 0
	Box 39.6 The Spin Energy Contribution to a Black Hole's Mass
	Homework Problems
Appendix: A Diagonal Metric Worksheet
Index
	A, B, C
	D, E
	F, G
	H, I
	J, K, L, M
	N, O, P
	Q, R, S
	T
	U, V, W
	X, Z