Instructor's Solution Manual To Accompany Modern General Relativity: Black Holes, Gravitational Waves, and Cosmology (Solutions)

Half Title page
Title page
Copyright page
Dedication
Brief Contents
Contents
Preface
Part I General Relativity
	1 Introduction
		1.1 Gravity and the Universe on Large Scales
		1.2 Classical Newtonian Gravity
		1.3 Transformations between Inertial Systems
		1.4 Maxwell, the Aether, and Galileo
		1.5 The Special Theory of Relativity
		1.6 Minkowski Space
		1.7 A New Theory of Gravity
		1.8 The Equivalence Principle
		1.9 General Relativity
		Background and Further Reading
		Problems
	2 Coordinate Systems and Transformations
		2.1 Coordinate Systems in Euclidean Space
			2.1.1 Parameterizing in Different Coordinate Systems
			2.1.2 Basis Vectors
			2.1.3 Expansion of Vectors and Dual Vectors
			2.1.4 Vector Scalar Product and the Metric Tensor
			2.1.5 Relationship of Vectors and Dual Vectors
			2.1.6 Properties of the Metric Tensor
			2.1.7 Line Elements
			2.1.8 Euclidean Line Element
		2.2 Integration
		2.3 Differentiation
		2.4 Non-euclidean Geometry
		2.5 Transformations
			2.5.1 Rotational Transformations
			2.5.2 Galilean Transformations
		Background and Further Reading
		Problems
	3 Tensors and Covariance
		3.1 Invariance and Covariance
		3.2 Spacetime Coordinates
		3.3 Vectors in Non-euclidean Space
		3.4 Coordinates in Spacetime
			3.4.1 Coordinate and Non-coordinate Bases
			3.4.2 Utility of Coordinate and Non-coordinate Bases
		3.5 Tensors and Coordinate Transformations
		3.6 Tensors as Linear Maps
			3.6.1 Linear Maps to Real Numbers
			3.6.2 Vectors and Dual Vectors
			3.6.3 Tensors of Higher Rank
			3.6.4 Identification of Vectors and Dual Vectors
			3.6.5 Index-free versus Component Transformations
		3.7 Tensors Specified by Transformation Laws
			3.7.1 Scalar Transformation Law
			3.7.2 Dual Vector Transformation Law
			3.7.3 Vector Transformation Law
			3.7.4 Duality of Vectors and Dual Vectors
		3.8 Scalar Product of Vectors
		3.9 Tensors of Higher Rank
		3.10 The Metric Tensor
		3.11 Symmetric and Antisymmetric Tensors
		3.12 Summary of Algebraic Tensor Operations
		3.13 Tensor Calculus on Curved Manifolds
			3.13.1 Invariant Integration
			3.13.2 Partial Derivatives
			3.13.3 Covariant Derivatives
			3.13.4 Absolute Derivatives
			3.13.5 Lie Derivatives
		3.14 Invariant Equations
		Background and Further Reading
		Problems
	4 Lorentz Covariance and Special Relativity
		4.1 Minkowski Space
			4.1.1 The Indefinite Metric of Spacetime
			4.1.2 Scalar Products and the Metric Tensor
			4.1.3 The Line Element
			4.1.4 Invariance of the Spacetime Interval
		4.2 Tensors in Minkowski space
		4.3 Lorentz Transformations
			4.3.1 Rotations in Euclidean Space
			4.3.2 Generalized 4D Minkowski Rotations
			4.3.3 Lorentz Spatial Rotations
			4.3.4 Lorentz Boost Transformations
		4.4 Lightcone Diagrams
		4.5 The Causal Structure of Spacetime
		4.6 Lorentz Transformations in Spacetime Diagrams
			4.6.1 Lorentz Boosts and the Lightcone
			4.6.2 Spacelike and Timelike Intervals
		4.7 Lorentz Covariance of the Maxwell Equations
			4.7.1 Maxwell Equations in Noncovariant Form
			4.7.2 Scalar and Vector Potentials
			4.7.3 Gauge Transformations
			4.7.4 Maxwell Equations in Manifestly Covariant Form
		Background and Further Reading
		Problems
	5 Lorentz-Invariant Dynamics
		5.1 A Natural Set of Units
		5.2 Velocity and Momentum for Massive Particles
		5.3 Geodesics and a Variational Principle
		5.4 Light and other Massless Particles
			5.4.1 Affine Parameters
			5.4.2 Energy and Momentum
		5.5 Observers
		5.6 Isometries and Killing Vectors
			5.6.1 Symmetries of the Metric
			5.6.2 Quantities Conserved along Geodesics
		Background and Further Reading
		Problems
	6 The Principle of Equivalence
		6.1 Einstein and Equivalence
		6.2 Inertial and Gravitational Mass
		6.3 The Strong Equivalence Principle
			6.3.1 Elevators, Gravity, and Acceleration
			6.3.2 Alternative Statements of the Equivalence Principle
			6.3.3 Equivalence and the Path to General Relativity
		6.4 Deflection of Light in a Gravitational Field
			6.4.1 A Thought Experiment
			6.4.2 Curvature Radius and the Strength of Gravity
		6.5 The Gravitational Redshift
			6.5.1 A Second Thought Experiment
			6.5.2 The Total Redshift in a Gravitational Field
			6.5.3 Gravitational Time Dilation
		6.6 Equivalence and Riemannian Manifolds
		6.7 Local Inertial Frames and Inertial Observers
			6.7.1 Locality and Tidal Forces
			6.7.2 Inertial Observers
			6.7.3 Definition of Local Inertial Frames
		6.8 Lightcones in Curved Spacetime
		6.9 The Road to General Relativity
		Background and Further Reading
		Problems
	7 Curved Spacetime and General Covariance
		7.1 General Covariance
		7.2 Curved Spacetime
			7.2.1 Coordinate Systems
			7.2.2 Gaussian Curvature
			7.2.3 Distance Intervals
		7.3 A Covariant Description of Matter
			7.3.1 Stress–Energy for Perfect Fluids
			7.3.2 Local Conservation of Energy
		7.4 Covariant Derivatives and Parallel Transport
			7.4.1 Parallel Transport of Vectors
			7.4.2 The Affine Connection and Covariant derivatives
			7.4.3 Absolute Derivatives and Parallel Transport
			7.4.4 Geometry and Covariant Derivatives
		7.5 Gravity and Curved Spacetime
			7.5.1 Free Particles
			7.5.2 The Geodesic Equation
		7.6 The Local Inertial Coordinate System
		7.7 The Affine Connection and the Metric Tensor
		7.8 Uniqueness of the Affine Connection
		Background and Further Reading
		Problems
	8 The General Theory of Relativity
		8.1 Weak-Field Limit
		8.2 Recipe for Motion in a Gravitational Field
		8.3 Towards a Covariant Theory of Gravitation
		8.4 The Riemann Curvature Tensor
		8.5 The Einstein Equations
		8.6 Limiting Behavior of the Einstein Tensor
		8.7 Sign Conventions
		8.8 Solving the Einstein Equations
			8.8.1 Solutions in the Limit of Weak Fields
			8.8.2 Solutions with a High Degree of Symmetry
			8.8.3 Solutions by Numerical Relativity
		Background and Further Reading
		Problems
	9 The Schwarzschild Spacetime
		9.1 The Form of the Metric
			9.1.1 The Schwarzschild Solution
			9.1.2 The Schwarzschild Radius
			9.1.3 Measuring Distance and Time
			9.1.4 Embedding Diagrams
		9.2 The Gravitational Redshift
			9.2.1 Exploiting a Symmetry of the Metric
			9.2.2 Recovering the Weak-Field Limit
		9.3 Particle Orbits in the Schwarzschild Metric
			9.3.1 Conserved Quantities
			9.3.2 Equation of Motion
			9.3.3 Classification of Orbits
			9.3.4 Stable Circular Orbits
		9.4 Precession of Orbits
			9.4.1 The Change in Perihelion Angle
			9.4.2 Testing the Prediction
		9.5 Escape Velocity
		9.6 Radial Fall of a Test Particle
		9.7 Orbits for Light Rays
		9.8 Deflection of Light in the Gravitational Field
		9.9 Shapiro Time Delay of Light
		9.10 Gyroscopes in Curved Spacetime
		9.11 Geodetic Precession
		9.12 Gyroscopes in Rotating Spacetimes
			9.12.1 Slow Rotation in the Schwarzschild Metric
			9.12.2 Dragging of Inertial frames
		Background and Further Reading
		Problems
	10 Neutron Stars and Pulsars
		10.1 A Qualitative Picture of Neutron Stars
		10.2 Solutions inside Spherical Mass Distributions
			10.2.1 Simplifying Assumptions
			10.2.2 Solving the Einstein Equations
			10.2.3 The Oppenheimer–Volkov Equations
			10.2.4 Interpretation of Oppenheimer–Volkov Equations
		10.3 Interpretation of the Mass Parameter
			10.3.1 Total Mass–Energy for a Relativistic Star
			10.3.2 Gravitational Mass and Baryonic Mass
		10.4 Pulsars and Tests of General Relativity
			10.4.1 The Binary Pulsar
			10.4.2 Precision Tests of General Relativity
			10.4.3 Origin and Fate of the Binary Pulsar
			10.4.4 The Double Pulsar
			10.4.5 The Pulsar–White Dwarf Binary PSR J0348+0432
			10.4.6 The Pulsar–WD–WD Triplet PSR J0337+1715
		Background and Further Reading
		Problems
Part II Black Holes
	11 Spherical Black Holes
		11.1 Schwarzschild Black Holes
			11.1.1 Event Horizons
			11.1.2 Approaching the Horizon: Outside View
			11.1.3 Approaching the Horizon: Spacecraft View
		11.2 Lightcone Description of a Trip to a Black Hole
			11.2.1 Worldline Exterior to the Event Horizon
			11.2.2 Worldline Interior to the Event Horizon
			11.2.3 You Can’t Get There From Here
		11.3 Solution in Eddington–Finkelstein Coordinates
			11.3.1 Eddington–Finkelstein Coordinates
			11.3.2 Behavior of Radial Light Rays
			11.3.3 The Event Horizon
		11.4 Solution in Kruskal–Szekeres Coordinates
			11.4.1 Kruskal–Szekeres Coordinates
			11.4.2 Kruskal Diagrams
			11.4.3 The Event Horizon
		11.5 Black Hole Theorems and Conjectures
		Background and Further Reading
		Problems
	12 Quantum Black Holes
		12.1 Geodesics and Uncertainty
		12.2 Hawking Radiation
			12.2.1 4-Momentum Conservation
			12.2.2 Black Hole Evaporation
			12.2.3 Relative Importance of Quantum Fluctuations
		12.3 Black Hole Temperatures
		12.4 Miniature Black Holes
		12.5 Black Hole Thermodynamics
			12.5.1 Entropy of a Black Hole
			12.5.2 The Generalized Second Law
			12.5.3 The Four Laws of Black Hole Dynamics
		12.6 The Planck Scale and Quantum Gravity
		12.7 Black Holes and Information
			12.7.1 The Holographic Principle
			12.7.2 The Holographic Universe
		Background and Further Reading
		Problems
	13 Rotating Black Holes
		13.1 The Kerr Solution
			13.1.1 The Kerr Metric
			13.1.2 Extremal Kerr Black Holes
			13.1.3 Cosmic Censorship
			13.1.4 The Kerr Horizon
		13.2 Particle and Photon Motion
			13.2.1 Orbits in the Kerr Metric
			13.2.2 Frame Dragging
			13.2.3 The Ergosphere
			13.2.4 Motion of Photons in the Ergosphere
		13.3 Extracting Rotational Energy from Black Holes
			13.3.1 Penrose Processes
			13.3.2 Practical Energy Extraction
		Background and Further Reading
		Problems
	14 Observational Evidence for Black Holes
		14.1 Gravitational Collapse and Observations
		14.2 Singularity Theorems and Black Holes
			14.2.1 Global Methods in General Relativity
			14.2.2 Singularities and Trapped Surfaces
			14.2.3 Generalized Singularity Theorems
		14.3 Observing Black Holes
		14.4 Stellar-Mass Black Holes
			14.4.1 Masses for Compact Objects in X-Ray Binaries
			14.4.2 Masses from Mass Functions
			14.4.3 An Example: A0620–00
			14.4.4 Some Black Hole Candidates
		14.5 Supermassive Black Holes
			14.5.1 The Black Hole at Sgr A*
			14.5.2 The Water Masers of NGC 4258
			14.5.3 The Virial Theorem and Gravitating Mass
		14.6 Intermediate-Mass Black Holes
		14.7 Black Holes in the Early Universe
		14.8 Show Me an Event Horizon!
		14.9 A Circumstantial but Strong Case
		Background and Further Reading
		Problems
	15 Black Holes as Central Engines
		15.1 Black Hole Energy Sources
		15.2 Accretion and Energy Release for Black Holes
			15.2.1 Maximum Energy Release for Spherical Accretion
			15.2.2 Limits on Accretion Rates
			15.2.3 Accretion Efficiencies
			15.2.4 Accretion onto Rotating Black Holes
		15.3 Jets and Magnetic Fields
		15.4 Quasars
			15.4.1 “Radio Stars” and a Spectrum in Disguise
			15.4.2 Quasar Characteristics
			15.4.3 Quasar Energy Sources
		15.5 Active Galactic Nuclei
			15.5.1 Radio Galaxies
			15.5.2 Seyfert Galaxies
			15.5.3 BL Lac Objects
		15.6 A Unified Model of AGN and Quasars
			15.6.1 The AGN Black Hole Central Engine Model
			15.6.2 Anisotropic Ionization Cones
			15.6.3 A Unified Model
			15.6.4 Example: Feeding a Nearby Monster
			15.6.5 High-Energy Photons from AGN
		15.7 Gamma-Ray Bursts
			15.7.1 The Gamma-Ray Sky
			15.7.2 Two Classes of Gamma-Ray Bursts
			15.7.3 Localization of Gamma-Ray Bursts
			15.7.4 Necessity of Ultrarelativistic Jets
			15.7.5 Association of GRBs with Galaxies
			15.7.6 Long-Period GRBs and Supernovae
			15.7.7 Characteristics of Gamma-Ray Bursts
			15.7.8 Mechanisms for the Central Engine
			15.7.9 Gamma-Ray Bursts and Gravitational Waves
		Background and Further Reading
		Problems
Part III Cosmology
	16 The Hubble Expansion
		16.1 The Standard Picture
			16.1.1 Mass Distribution on Large Scales
			16.1.2 The Universe is Expanding
			16.1.3 The Expansion Is Governed by General Relativity
			16.1.4 There is a Big Bang in Our Past
			16.1.5 Particle Content Influences the Evolution
			16.1.6 There is a Cosmic Microwave Background
		16.2 The Hubble Law
			16.2.1 The Hubble Parameter
			16.2.2 Redshifts
			16.2.3 Expansion Interpretation of Redshifts
			16.2.4 The Hubble Time
			16.2.5 A 2-Dimensional Hubble Expansion Model
			16.2.6 Measuring the Hubble Constant
		16.3 Limitations of the Standard Picture
		Background and Further Reading
		Problems
	17 Energy and Matter in the Universe
		17.1 Expansion and Newtonian Gravity
		17.2 The Critical Density
		17.3 The Cosmic Scale Factor
		17.4 Possible Expansion Histories
		17.5 Lookback Times
		17.6 The Inadequacy of Dust Models
		17.7 Evidence for Dark Matter
			17.7.1 Rotation Curves for Spiral Galaxies
			17.7.2 The Mass of Galaxy Clusters
			17.7.3 Hot Gas in Clusters of Galaxies
			17.7.4 Gravitational Lensing
			17.7.5 Dark Matter in Ultra-diffuse Galaxies
		17.8 The Amount of Baryonic Matter
		17.9 Baryonic Candidates for Dark Matter
		17.10 Candidates for Nonbaryonic Dark Matter
			17.10.1 Cold Dark Matter
			17.10.2 Hot Dark Matter
		17.11 Dark Energy
		17.12 Radiation
		17.13 The Scale Factor and Density Parameters
		17.14 The Deceleration Parameter
			17.14.1 Deceleration and Density Parameters
			17.14.2 Deceleration and Cosmology
		17.15 Problems with Newtonian Cosmology
		Background and Further Reading
		Problems
	18 Friedmann Cosmologies
		18.1 The Cosmological Principle
		18.2 Homogeneous and Isotropic 2D Spaces
		18.3 Homogeneous and Isotropic 3D Spaces
			18.3.1 Constant Positive Curvature
			18.3.2 Constant Negative Curvature
			18.3.3 Zero Curvature
		18.4 The Robertson–Walker Metric
		18.5 Comoving Coordinates
		18.6 Proper Distances
		18.7 The Hubble Law and the RW Metric
		18.8 Particle and Event Horizons
			18.8.1 Particle Horizons in the RW Metric
			18.8.2 Event Horizons in the RW Metric
		18.9 Einstein Equations for the RW Metric
			18.9.1 The Metric and Stress–Energy Tensor
			18.9.2 The Connection Coefficients
			18.9.3 The Ricci Tensor and Ricci Scalar
			18.9.4 The Friedmann Equations
			18.9.5 Static Solutions and the Cosmological Constant
		18.10 Resolution of Newtonian Difficulties
		Background and Further Reading
		Problems
	19 Evolution of the Universe
		19.1 Friedmann Cosmologies
			19.1.1 Reformulation of the Friedmann Equations
			19.1.2 Equations of State
		19.2 Friedmann Equations in Concise Form
			19.2.1 Evolution and Scaling of Density Components
			19.2.2 A Standard Model
		19.3 Flat, Single-Component Universes
			19.3.1 Special Solution: Vacuum Energy Domination
			19.3.2 General Solutions
			19.3.3 Flat Universes with Radiation or Matter
		19.4 Full Solution of the Friedmann Equations
			19.4.1 Evolution Equations in Dimensionless Form
			19.4.2 Algorithm for Numerical Solution
			19.4.3 Examples: Single Component with Curvature
			19.4.4 Examples: Multiple Components
			19.4.5 Parameters for a Realistic Model
			19.4.6 Concordance of Cosmological Parameters
			19.4.7 Calculations with Benchmark Parameters
		Background and Further Reading
		Problems
	20 The Big Bang
		20.1 Radiation- and Matter-Dominated Universes
			20.1.1 Evolution of the Scale Factor
			20.1.2 Matter and Radiation Density
		20.2 Evolution of the Early Universe
			20.2.1 Thermodynamics of the Big Bang
			20.2.2 Equilibrium in an Expanding Universe
			20.2.3 A Timeline for the Big Bang
		20.3 Nucleosynthesis and Cosmology
			20.3.1 The Neutron to Proton Ratio
			20.3.2 Elements Synthesized in the Big Bang
			20.3.3 Constraints on Baryon Density
		20.4 The Cosmic Microwave Background
			20.4.1 The Microwave Background Spectrum
			20.4.2 Anisotropies in the Microwave Background
			20.4.3 The Origin of CMB Fluctuations
			20.4.4 Acoustic Signature in the CMB
			20.4.5 Acoustic Signature in Galaxy Distributions
			20.4.6 Precision Cosmology
			20.4.7 Seeds for Structure Formation
		20.5 Accelerated Structure Formation
		20.6 Dark Matter, Dark Energy, and Structure
		Background and Further Reading
		Problems
	21 Extending Classical Big Bang Theory
		21.1 Successes of the Big Bang Theory
		21.2 Problems with the Big Bang
			21.2.1 The Horizon Problem
			21.2.2 The Flatness Problem
			21.2.3 The Magnetic Monopole Problem
			21.2.4 The Structure and Smoothness Dichotomy
			21.2.5 The Vacuum Energy Problem
			21.2.6 The Matter–Antimatter Problem
			21.2.7 Modifying the Classical Big Bang
		21.3 Cosmic Inflation
			21.3.1 The Basic Idea and Generic Consequences
			21.3.2 Taking the Inflationary Cure
			21.3.3 Inflation Doesn’t Replace the Big Bang
		21.4 The Origin of the Baryons
			21.4.1 Conditions for a Baryon Asymmetry
			21.4.2 Grand Unified Theories
			21.4.3 Leptogenesis
		Background and Further Reading
		Problems
Part IV Gravitational Wave Astronomy
	22 Gravitational Waves
		22.1 Significance of Gravitational Waves
			22.1.1 Unprecedented Tests of General Relativity
			22.1.2 A Probe of Dark Events
			22.1.3 The Deepest Probe
			22.1.4 Technology and the Quest for Gravitational Waves
		22.2 Linearized Gravity
			22.2.1 Linearized Curvature Tensor
			22.2.2 Wave Equation
			22.2.3 Coordinates and Gauge Transformations
			22.2.4 Choice of Gauge
		22.3 Weak Gravitational Waves
			22.3.1 Polarization Tensor in TT Gauge
			22.3.2 Helicity Components
			22.3.3 General Solution in TT Gauge
		22.4 Gravitational versus Electromagnetic Waves
			22.4.1 Interaction with Matter
			22.4.2 Wavelength Relative to Source Size
			22.4.3 Phase Coherence
			22.4.4 Field of View
		22.5 The Response of Test Particles
			22.5.1 Response of Two Test Masses
			22.5.2 The Effect of Polarization
		22.6 Gravitational Wave Detectors
			22.6.1 Operating and Proposed Detectors
			22.6.2 Strain and Frequency Windows
			22.6.3 Detecting Very Long Wavelengths
			22.6.4 Reach of Advanced LIGO and Advanced VIRGO
		Background and Further Reading
		Problems
	23 Weak Sources of Gravitational Waves
		23.1 Production of Weak Gravitational Waves
			23.1.1 Energy Densities
			23.1.2 Multipolarities
			23.1.3 Linearized Einstein Equation with Sources
			23.1.4 Gravitational Wave Amplitudes
			23.1.5 Amplitudes and Event Rates
			23.1.6 Power in Gravitational Waves
		23.2 Gravitational Radiation from Binary Systems
			23.2.1 Gravitational Wave Luminosity
			23.2.2 Gravitational Radiation and Binary Orbits
			23.2.3 Gravitational Waves from the Binary Pulsar
		Background and Further Reading
		Problems
	24 Strong Sources of Gravitational Waves
		24.1 A Survey of Candidate Sources
			24.1.1 Merger of a Neutron Star Binary
			24.1.2 Stellar Black Hole Mergers
			24.1.3 Merger of a Black Hole and a Neutron Star
			24.1.4 Core Collapse in Massive Stars
			24.1.5 Merging Supermassive Black Holes
			24.1.6 Sample Gravitational Waveforms
		24.2 The Gravitational Wave Event GW150914
			24.2.1 Observed Waveforms
			24.2.2 Source Localization
			24.2.3 Comparisons with Candidate Events
			24.2.4 Binary Black Hole Mergers
		24.3 Additional Gravitational Wave Events
			24.3.1 GW151226 and LVT151012
			24.3.2 Matched Filtering
			24.3.3 Binary Masses and Inspiral Cycles
			24.3.4 Increasing Sensitivity
			24.3.5 LIGO–Virgo Triple Coincidences
		24.4 Testing General Relativity in Strong Gravity
		24.5 A New Window on the Universe
		24.6 Multimessenger Astronomy
		24.7 Gravitational Waves from Neutron Star Mergers
			24.7.1 New Discoveries Associated with GW170817
			24.7.2 The Kilonova
		24.8 Gravitational Waves and Stellar Evolution
			24.8.1 A Possible Evolutionary Scenario for GW150914
			24.8.2 Measured Stellar Black Hole Masses
			24.8.3 Are Stellar and Supermassive Black Holes Related?
		Background and Further Reading
		Problems
Part V General Relativity and Beyond
	25 Tests of General Relativity
		25.1 The Classical Tests
		25.2 The Modern Tests
			25.2.1 The PPN Formalism
			25.2.2 Results of Modern Tests
		25.3 Strong-Field Tests
		25.4 Cosmological Tests
		Background and Further Reading
		Problems
	26 Beyond Standard Models
		26.1 Supersymmetry
			26.1.1 Fermions and Bosons
			26.1.2 Normal Symmetries
			26.1.3 Symmetries Relating Fermions and Bosons
		26.2 Vacuum Energy from Quantum Fluctuations
			26.2.1 Vacuum Energy for Bosonic Fields
			26.2.2 Vacuum Energy for Fermionic Fields
			26.2.3 Supersymmetry and Dark Energy
		26.3 Quantum Gravity
			26.3.1 Superstrings and Branes
			26.3.2 How Many Dimensions?
			26.3.3 Spacetime Foam, Wormholes, and Such
			26.3.4 The Ultimate Free Lunch
			26.3.5 Does the Planck Scale Matter?
		Background and Further Reading
		Problems
Appendix A Constants
Appendix B Natural Units
Appendix C Einstein Tensor for a General Spherical Metric
Appendix D Using arXiv and ADS
References
Index