A General Relativity Workbook

Description

A General Relativity Workbook is a textbook intended to support a one-semester undergraduate course on general relativity. Through its unique workbook-based design, it enables students to develop a solid mastery of both the physics and the supporting tensor calculus by guiding them to work through the implications. The mathematics is introduced gradually and in a completely physical context. Each chapter, which is designed to correspond to one class session, involves a short overview of the concepts without obscuring derivations or details, followed by a series of boxes that guide students through the process of working things out. This active-learning approach enables students to develop a more secure mastery of the material than more traditional approaches. More than 350 homework problems support further learning. 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