## General Relativity and Einstein's Equations

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0003
- Subject:
- Mathematics, Applied Mathematics

This chapter begins with a discussion of Newton's gravity law. It then covers general relativity, observations and experiments, Einstein's equations, field sources, Lagrangians, fluid sources, ... More

## Local Cauchy Problem

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0006
- Subject:
- Mathematics, Applied Mathematics

This chapter begins with a discussion of moving frame formulae. It then covers n + 1 splitting adapted to space slices, constraints and evolution, Hamiltonian and symplectic formulation, Cauchy ... More

## Global Existence Theorems: Asymptotically Euclidean Data

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0015
- Subject:
- Mathematics, Applied Mathematics

This chapter shows how the Penrose transform can be used to prove global existence of solutions of various semilinear field equations. It outlines the foundation points of Friedrich's conformal ... More

## Relativistic Kinetic Theory

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0010
- Subject:
- Mathematics, Applied Mathematics

This chapter presents the basic equations of the relativistic kinetic theory; it proves local existence theorems for Einstein equations coupled with kinetic matter. It then provides thermodynamic ... More

## BRIEF REVIEW OF GENERAL RELATIVITY

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter presents some of the basic concepts of general relativity. Topics discussed include notation and conventions, special relativity, manifolds and tensors, the metric tensor, Lie ... More

## GR as a geometric theory of gravity — II

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0014
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

The mathematical realization of equivalence principle (EP) is the principle of general covariance. General relativity (GR) equations must be covariant with respect to general coordinate ... More

## 5 GW generation by post-Newtonian sources

*Michele Maggiore*

### in Gravitational Waves: Volume 1: Theory and Experiments

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198570745
- eISBN:
- 9780191717666
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570745.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses how to go beyond the limit of sources moving in flat space-time. For a self-gravitating system such as a binary star, assuming that space-time is flat means that its dynamics ... More

## The Einstein‐Maxwell Equations, Extremal Kähler Metrics, and Seiberg‐Witten Theory

*Claude LeBrun*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0003
- Subject:
- Mathematics, Geometry / Topology

Nigel Hitchin has played a key role in the exploration of four-dimensional Riemannian geometry, and in particular has made foundational contributions to the theory of self-dual manifolds, ... More

## THE 3+1 FORMALISM

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

There are several different approaches to the problem of separating the Einstein field equations in a way that allows us to give certain initial data, and from there obtain the subsequent evolution ... More

## Other Hyperbolic-Elliptic Well-Posed Systems

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0008
- Subject:
- Mathematics, Applied Mathematics

This chapter presents well-posed hyperbolic or hyperbolic-elliptic systems that lead to the same local existence andgeometric uniqueness theorems as the wave gauge choice. However, these different ... More

## GR as a geometric theory of gravity – I

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

A geometric description of equivalence principle physics of gravitational time dilation is presented. In this geometric theory, the metric plays the role of relativistic gravitational potential. ... More

## Stationary Spacetimes and Black Holes

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0014
- Subject:
- Mathematics, Applied Mathematics

This chapter presents the proofs of fundamental uniqueness theorems for complete stationary solutions of the vacuum, or electrovac, solutions of the Einstein equations. It studies the properties of ... More

## Formulations of the Einstein–Euler Equations

*Luciano Rezzolla and Olindo Zanotti*

### in Relativistic Hydrodynamics

- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780198528906
- eISBN:
- 9780191746505
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528906.003.0007
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter, which opens the second part of the book devoted to the numerical solution of the relativistic-hydrodynamics equations, presents in a concise, yet complete, way the several formulations ... More

## Linearized theory and gravitational waves

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0015
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

In the weak-field limit Einstein's equation can be linearized and it takes on form of the familiar wave equation. Gravitational waves may be viewed as ripples of curvature propagating in a background ... More

## Why quantum gravity?

*Claus Kiefer*

### in Quantum Gravity: Third Edition

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199585205
- eISBN:
- 9780191739378
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199585205.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This introductory chapter presents the general motivations for constructing a quantum theory of gravity. The main argument is the conceptual incompleteness of present theoretical physics. The chapter ... More

## Covariant approaches to quantum gravity

*Claus Kiefer*

### in Quantum Gravity: Third Edition

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199585205
- eISBN:
- 9780191739378
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199585205.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces the important concept of the graviton. It discusses at some length linearized quantum gravity and its implications. A central role is played by path-integral quantization from ... More

## INITIAL DATA

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Only six out of ten Einstein field equations contain time derivatives and therefore represent the true evolution equations of the spacetime geometry. The remaining four equations are constraints that ... More

## On the Bounded L2 Curvature Conjecture

*Sergiu Klainerman*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

### in Advances in Analysis: The Legacy of Elias M. Stein

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0010
- Subject:
- Mathematics, Numerical Analysis

This chapter deals with a fundamental application of new methods to a geometric quasilinear equation to settle an important conjecture in General Relativity. According to the bounded L² curvature ... More

## General Relativity and the Schwarzschild Metric

*David M. Wittman*

### in The Elements of Relativity

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780199658633
- eISBN:
- 9780191863653
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199658633.003.0018
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Previously, we saw that variations in the time part of the spacetime metric cause free particles to accelerate, thus unifying gravity and relativity; and that orbits trace those accelerations, which ... More

## General Relativity

*Nicholas Manton and Nicholas Mee*

### in The Physical World: An Inspirational Tour of Fundamental Physics

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198795933
- eISBN:
- 9780191837111
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198795933.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter presents the physical motivation for general relativity, derives the Einstein field equation and gives concise derivations of the main results of the theory. It begins with the ... More