General relativity and the Einstein equations / Yvonne Choquet-Bruhat.

Format:

Book

Author/Creator:

Choquet-Bruhat, Yvonne.

Series:

Oxford mathematical monographs.

Oxford mathematical monographs

Language:

English

Subjects (All):

General relativity (Physics)--Mathematics.

General relativity (Physics).

Einstein field equations.

Genre:

Electronic books.

Physical Description:

xxiv, 785 p. : ill.

Edition:

1st ed.

Place of Publication:

Oxford ; New York : Oxford University Press, 2009.

Summary:

Aimed at researchers in mathematics and physics, this monograph, in which the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

Contents:

Intro

CONTENTS

I: Lorentz geometry

1 Introduction

2 Manifolds

3 Differentiable mappings

4 Vectors and tensors

4.1 Tangent and cotangent space

4.2 Vector fields

4.3 Tensors and tensor fields

5 Pseudo-Riemannian metrics

5.1 General properties

5.2 Riemannian and Lorentzian metrics

6 Riemannian connection

7 Geodesics

8 Curvature

9 Geodesic deviation

10 Maximum of length and conjugate points

11 Linearized Ricci and Einstein tensors

12 Second derivative of the Ricci tensor

II: Special Relativity

1 Newton's mechanics

1.1 The Galileo-Newton spacetime

1.2 Newton's dynamics - the Galileo group

2 Maxwell's equations

3 Minkowski spacetime

3.1 Definition

3.2 Maxwell's equations on M[sub(4)]

4 Poincaré group

5 Lorentz group

5.1 General formulae

5.2 Transformation of electric and magnetic vector fields (case n = 3)

5.3 Lorentz contraction and dilatation

6 Special Relativity

6.1 Proper time

6.2 Proper frame and relative velocities

7 Dynamics of a pointlike mass

7.1 Newtonian law

7.2 Relativistic law

7.3 Equivalence of mass and energy

8 Continuous matter

8.1 Case of dust (incoherent matter)

8.2 Perfect fluids

III: General relativity and Einstein's equations

2 Newton's gravity law

3 General relativity

3.1 Physical motivations

4 Observations and experiments

4.1 Deviation of light rays

4.2 Proper time, gravitational time delay

5 Einstein's equations

5.1 Vacuum case

5.2 Equations with sources

6 Field sources

6.1 Electromagnetic sources

6.2 Electromagnetic potential

6.3 Yang-Mills fields

6.4 Scalar fields

6.5 Wave maps

6.6 Energy conditions

7 Lagrangians

7.1 Einstein-Hilbert Lagrangian

7.2 Lagrangians and stress energy tensors of sources

7.3 Coupled Lagrangian.

8 Fluid sources

9 Einsteinian spacetimes

9.1 Definition

9.2 Regularity hypotheses

10 Newtonian approximation

10.1 Equations for potentials

10.2 Equations of motion

11 Gravitational waves

11.1 Minkowskian approximation

11.2 General linear waves

12 High-frequency gravitational waves

12.1 Phase and polarizations

12.2 Radiative coordinates

12.3 Energy conservation

13 Coupled electromagnetic and gravitational waves

13.1 Phase and polarizations

13.2 Propagation equations

IV: Schwarzschild spacetime and black holes

2 Spherically symmetric spacetimes

3 Schwarzschild metric

4 Other coordinates

4.1 Isotropic coordinates

4.2 Wave coordinates

4.3 Painlevé-Gullstrand-like coordinates

4.4 Regge-Wheeler coordinates

5 Schwarzschild spacetime

6 The motion of the planets and perihelion precession

6.1 Equations

6.2 Results of observations

6.3 Escape velocity

7 Stability of circular orbits

8 Deflection of light rays

8.1 Theoretical prediction

8.2 Results of observation

8.3 Fermat's principle and light travel parameter time

9 Red shift and time delay

10 Spherically symmetric interior solutions

10.1 Static solutions. Upper limit on mass

10.2 Matching with an exterior solution

10.3 Non-static solutions

11 The Schwarzschild black hole

11.1 The event horizon

11.2 The Eddington-Finkelstein extension

11.3 Eddington-Finkelstein white hole

11.4 Kruskal complete spacetime

11.5 Observations

12 Spherically symmetric gravitational collapse

12.1 Tolman metric

12.2 Monotonically decreasing density

13 The Reissner-Nordström solution

14 Schwarzschild spacetime in dimension n + 1

14.1 Standard coordinates

14.2 Wave coordinates

V: Cosmology

2 Cosmological principle.

3 Isotropic and homogeneous Riemannian manifolds

3.1 Isotropy

3.2 Homogeneity

4 Robertson-Walker spacetimes

4.1 Space metrics

4.2 Robertson-Walker spacetime metrics

4.3 Robertson-Walker dynamics

4.4 Einstein static universe

4.5 Cosmological red shift and the Hubble constant

4.6 De Sitter spacetime

4.7 Anti de Sitter (AdS) spacetime

5 Friedmann-Lemaître models.

5.1 Equation of state

5.2 General properties

5.3 Friedmann models

5.4 Some other models

5.5 Confrontation with observations

6 Homogeneous non-isotropic cosmologies

7 Bianchi class I universes

7.1 Kasner solutions

7.2 Models with matter

8 Bianchi type IX

9 The Kantowski-Sachs models

10 Taub and Taub NUT spacetimes

10.1 Taub spacetime

10.2 Taub NUT spacetime

11 Locally homogeneous models

11.1 n-dimensional compact manifolds

11.2 Compact 3-manifolds

12 Recent observations and conjectures

VI: Local Cauchy problem

2 Moving frame formulae

2.1 Frame and coframe

2.2 Metric

2.3 Connection

2.4 Curvature

3 n + 1 splitting adapted to space slices

3.1 Adapted frame and coframe

3.2 Structure coefficients

3.3 Splitting of the connection.

3.4 Extrinsic curvature

3.5 Splitting of the Riemann tensor

4 Constraints and evolution

4.1 Equations. Conservation of constraints

5 Hamiltonian and symplectic formulation

5.1 Hamilton equations

5.2 Symplectic formulation

6 Cauchy problem

6.1 Definitions

6.2 The analytic case

7 Wave gauges

7.1 Wave coordinates

7.2 Generalized wave coordinates

7.3 Damped wave coordinates

7.4 Globalization in space, ê wave gauges

7.5 Local in time existence in a wave gauge

8 Local existence for the full Einstein equations

8.1 Preservation of the wave gauges

8.2 Geometric local existence.

8.3 Geometric uniqueness

8.4 Causality

9 Constraints in a wave gauge

10 Einstein equations with field sources

10.1 Maxwell constraints

10.2 Lorentz gauge

10.3 Existence and uniqueness theorems

10.4 Wave equation for F

VII: Constraints

2 Linearization and stability

2.1 Linearization of the constraints map, adjoint map

2.2 Linearization stability

3 CF (Conformally Formulated) constraints

3.1 Hamiltonian constraint

3.2 Momentum constraint

3.3 Scaling of the sources

3.4 Summary of results

3.5 Conformal transformation of the CF constraints

3.6 The momentum constraint as an elliptic system

4 Case n = 2

5 Solutions on compact manifolds

6 Solution of the momentum constraint

7 Lichnerowicz equation

7.1 The Yamabe classification

7.2 Non-existence and uniqueness

7.3 Existence theorems

8 System of constraints

8.1 Constant mean curvature &amp

#915

sources with York-scaled momentum

8.2 Solutions with &amp

&amp

#8802

constant or J[(0)] &amp

0

9 Solutions on asymptotically Euclidean Manifolds

10 Momentum constraint

11 Solution of the Lichnerowicz equation

11.1 Uniqueness theorem

11.2 Generalized Brill-Cantor theorem

11.3 Existence theorems

12 Solutions of the system of constraints

12.1 Decoupled system

12.2 Coupled system

13 Gluing solutions of the constraint equations

13.1 Connected sum gluing

13.2 Exterior (Corvino-Schoen) gluing

VIII: Other hyperbolic-elliptic well-posed systems

2 Leray-Ohya non-hyperbolicity of [sup((4))] R[(ij)] = 0

3 Wave equation for K

3.1 Hyperbolic system

3.2 Hyperbolic-elliptic system

4 Fourth-order non-strict and strict hyperbolic systems for g

5 First-order hyperbolic systems

5.1 FOSH systems

6 Bianchi-Einstein equations.

6.1 Wave equation for the Riemann tensor

6.2 Case n = 3, FOS system

6.3 Cauchy-adapted frame

6.4 FOSH system for u = (E, H, D, B, g, K, &amp

)

6.5 Elliptic-hyperbolic system

7 Bel-Robinson tensor and energy

7.1 The Bel tensor

7.2 The Bel-Robinson tensor and energy

8 Bel-Robinson energy in a strip

IX: Relativistic fluids

2 Case of dust

2.1 Equations

2.2 Motion of isolated bodies (Choquet-Bruhat and Friedrichs 2006)

3 Charged dust

3.1 Equations

3.2 Existence and uniqueness theorem in wave and Lorentz gauges

3.3 Motion of isolated bodies

4 Perfect fluid, Euler equations

5 Energy properties

6 Particle number conservation

7 Thermodynamics

7.1 Definitions. Conservation of entropy

7.2 Equations of state

8 Wave fronts, propagation speeds, shocks

8.1 General definitions

8.2 Case of perfect fluids

8.3 Shocks

9 Stationary motion

10 Dynamic velocity for barotropic fluids

10.1 Fluid index and equations

10.2 Vorticity tensor and Helmholtz equations

10.3 Irrotational flows

11 General perfect fluids

12 Hyperbolic Leray system

12.1 Hyperbolicity of the Euler equations.

12.2 Reduced Einstein-Euler entropy system

12.3 Cauchy problem for the Einstein-Euler entropy system

12.4 Motion of isolated bodies

13 First-order symmetric hyperbolic system

14 Equations in a flow adapted frame

14.1 n + 1 splitting in a time adapted frame

14.2 Bianchi equations (case n = 3)

14.3 Vacuum case

14.4 Perfect fluid

14.5 Conclusion

15 Charged fluids

15.1 Equations

15.2 Fluids with zero conductivity

16 Fluids with finite conductivity

17 Magnetohydrodynamics

17.1 Equations

17.2 Wave fronts

18 Yang-Mills fluids

19 Dissipative fluids

19.1 Viscous fluids

19.2 The heat equation.

X: Relativistic kinetic theory.

Notes:

Formerly CIP.

Includes bibliographical references (p. [771]-779) and index.

Description based on publisher supplied metadata and other sources.

ISBN:

9780191552267

0191552267

0-19-157885-1

0-19-171087-3

9786611998707

0-19-155226-7

1-281-99870-2

OCLC:

317496332