General relativity : a graduate course / Horatiu Nastase.

Format:

Book

Author/Creator:

Năstase, Horațiu, 1972- author.

Language:

English

Subjects (All):

General relativity (Physics).

Physical Description:

1 online resource (xiii, 385 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2025.

Summary:

This text on general relativity and its modern applications is suitable for an intensive one-semester course on general relativity, at the level of a Ph.D. student in physics. Assuming knowledge of classical mechanics and electromagnetism at an advanced undergraduate level, basic concepts are introduced quickly, with greater emphasis on their applications. Standard topics are covered, such as the Schwarzschild solution, classical tests of general relativity, gravitational waves, ADM parametrization, relativistic stars and cosmology, as well as more advanced standard topics like vielbein-spin connection formulation, trapped surfaces, the Raychaudhuri equation, energy conditions, the Petrov and Bianchi classifications and gravitational instantons. More modern topics, including black hole thermodynamics, gravitational entropy, effective field theory for gravity, the PPN expansion, the double copy and fluid-gravity correspondence, are also introduced using the language understood by physicists, without too abstract mathematics, proven theorems, or the language of pure mathematics.

Contents:

Cover

Half-title page

Title page

Copyright page

Dedication

Contents

Preface

Acknowledgments

Introduction

1 General relativity, kinematics: metric, parallel transport, and general coordinate invariance

1.1 Lightning review of SR

1.2 General relativity: curved spacetimes

1.3 Intrinsically curved spaces and non-Euclidean geometry

1.4 Einstein's theory of general relativity

1.5 Kinematics

1.6 Motion of free particles

Important concepts to remember

References and further reading

Exercises

2 General relativity, dynamics: curvature, the Einstein-Hilbert action, and the Einstein equation

2.1 Curvature

2.2 Properties of the Riemann tensor

2.3 Actions in general relativity

2.4 The Einstein-Hilbert action

2.5 Einstein's equations

2.6 Energy-momentum tensor and general form of the Einstein's equations

2.7 Interpretation of Einstein's equations

3 Perturbative gravity: Fierz-Pauli action and gauge conditions

3.1 Fierz-Pauli action for small fields

3.2 Gauge invariance and the de Donder gauge

3.3 Transverse-traceless (TT) gauge

3.4 Synchronous gauge

4 Gravitational waves: perturbation, exact solutions, generation, and multipole expansion

4.1 Radiation in TT gauge

4.2 The gravitational field of a mass distribution

4.2.1 The electromagnetic multipole expansion

4.2.2 Gravity: multipole expansion

4.3 Gravitational radiation from a source

4.4 The pseudotensor of the gravitational field

4.5 The power radiated through gravitational waves

4.6 Exact cylindrical gravitational waves (Einstein-Rosen) solution

Exercises.

5 Nonperturbative gravity: the vacuum Schwarzschild solution

5.1 Newtonian limit

5.2 Ansatz and equations of motion

5.2.1 Ansatz

5.2.2 Christoffel symbols

5.2.3 Ricci components

5.3 Solution to the equations of motion

6 Deflection of light by the Sun and comparison with special relativity

6.1 Motion of light as motion in a medium with small, position-dependent index of refraction

6.2 Formal derivation using the Hamilton-Jacobi equation

6.2.1 Quick review of the Hamilton-Jacobi theory

6.2.2 Deflection of light by the Sun

6.3 Comparison to special relativity

7 The other classical tests of general relativity: the gravitational redshift, the perihelion precession, and the time delay of radar

7.1 Gravitational redshift

7.2 Geodesic radial motion

7.3 Time delay of the radar signal

7.4 Precession of the perihelion of the ecliptic

7.5 Motion in the Schwarzschild metric by nonrelativistic analogy

7.5.1 Null geodesic motion

8 Vielbein-spin connection formulation of general relativity: gravity vs. gauge theory, in four dimensions and three dimensions

8.1 Vielbein-spin connection formulation

8.2 Fermions in general relativity

8.3 Gravity vs. gauge theory, in four dimensions and in three dimensions

9 Gravity and geometry, Lovelock and Chern-Simons, topological terms, extensions, and anomalies

9.1 Gravity and geometry

9.2 Lanczos-Lovelock Lagrangian

9.3 Anomalies

10 The ADM parametrization and applications

10.1 ADM parametrization

10.2 Extrinsic curvature

10.3 Gauss-Codazzi equations for embedding

10.4 Killing vectors

10.5 Asymptotic flatness and the BMS group

10.5.1 Asymptotic flatness and definitions of mass

10.6 Boundary terms in the gravitational action

11* Canonical formalism for gravity, the Wheeler-de Witte quation, and the canonical quantization of gravity

11.1 Canonical formalism for gravity

11.2 A quick review of the Dirac formalism

11.3 Hamiltonian constraint and momentum constraint

11.4 The Brown-York stress tensor

11.5 The Wheeler-de Witt equation

11.5.1 Interpretation and wave functions

11.6 *Ashtekar variables (difficult subject, can be skipped)

12 Gravitoelectric and gravitomagnetic fields and applications

12.1 Electromagnetic analogy

12.2 Covariant form

12.3 Electromagnetism and tidal forces and tensors

12.3.1 Maxwell's equations as equations for tidal tensors

12.4 Gravitational analog of the above

12.5 Application: the Lense-Thirring effect and "frame dragging"

12.6 Application: the clock effect

13 Penrose diagrams and black holes: the Schwarzschild example

13.1 Penrose diagrams: definition

13.2 Example 1: two-dimensional Minkowski

13.3 Example 2: d-dimensional Minkowski

13.4 Example nr. 3: Anti-de Sitter space in Poincaré coordinates (the Poincaré patch)

13.5 Black holes and the Schwarzschild example

14 Reissner-Nordstrom black hole spacetime and extremal black holes.

14.1 The Reissner-Nordstrom solution

14.2 Horizons and extremality

14.3 Penrose diagram of the Reissner-Nordstrom solution

15 Kerr and Kerr-Newman black hole spacetimes and the Penrose process

15.1 The Kerr-Newman solution

15.2 Symmetries

15.3 Causal structure

15.4 Penrose process

16 Trapped surfaces, event horizons, causality, and topology

16.1 General definitions

16.2 Congruence, convergence, and trapped, and marginally trapped surfaces

16.3 Example of marginally trapped surface different from the event horizon

16.4 Horizons: de Sitter spacetime

16.5 Rindler spacetime: accelerated spacetime with event horizon

17 The Raychaudhuri equation

17.1 Description of event horizons

17.2 Surface gravity (of the horizon)

17.3 Horizon formulae

17.4 The Raychaudhuri equation

17.5 Application: horizons with null geodesics

18 The laws of black hole thermodynamics and black hole radiation

18.1 Laws of regular thermodynamics

18.2 Laws of black hole thermodynamics

18.3 (Partial) Proofs

18.3.1 Zeroth law

18.3.2 First law

18.3.3 Second law

18.3.4 Third law

18.4 Thermodynamic potential

19 Wald entropy and Sen's entropy function formalism

19.1 Wald entropy

19.2 Sen's entropy function formalism

19.3 Effective potential and existence of attractor = horizon

20 Energy conditions, singularity theorems, and wormholes

20.1 Energy conditions

20.2 Energy conditions and singularities

20.3 Hawking and Penrose theorems on the existence of singularities

20.4 Wormholes

21 Relativistic stars and gravitational collapse to black holes

21.1 Relativistic stars

21.2 The Tolman-Oppenheimer-Volkoff equation

21.3 Stellar models

21.4 The Chandrasekhar limit

21.4.1 The Tolman-Oppenheimer-Volkoff limit

21.5 Gravitational collapse to black holes

22 Effective field theory from gravity and black holes

22.1 Quantum field theory concepts

22.2 Worldline formalism and gravity

22.2.1 Electromagnetic analogy

22.2.2 Gravity

22.3 Nonrelativistic regime

22.4 Example: scalar gravity with scalar field φ and interacting with a source current

22.5 Non-relativistic general relativity (NRGR)

23 General relativity solutions and the gauge theory double copy

23.1 Motivation: KLT relations and BCJ relations

23.2 Double copy in Kerr-Schild coordinates

23.3 Examples

23.3.1 Schwarzschild black hole

23.3.2 Kerr black hole

23.3.3 PP waves

23.3.4 The Taub-NUT solution

23.4 The Weyl double copy

23.5 General Petrov type D example

24 The fluid-gravity correspondence

24.1 Viscous relativistic fluids

24.2 Conformal fluids

24.3 Conformal fluid from asymptotically AdS space

24.4 Membrane paradigm

24.4.1 Observation

24.5 The Navier-Stokes scaling limit

25 Fully linear gravity example: parallel plane (pp) wave and gravitational shockwave solutions.

Notes:

Title from publisher's bibliographic system (viewed on 24 Apr 2025).

Description based on publisher supplied metadata and other sources.

ISBN:

1-009-57572-4

1-009-57573-2

OCLC:

1517895194