## 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

## Harmonic morphisms with one-dimensional fibres

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0012
- Subject:
- Mathematics, Pure Mathematics

This chapter shows that a harmonic morphism from a manifold of dimension n+1 to a manifold of dimension n is, locally or globally, a principal bundle with a certain metric. When n = 3, in a ... 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

## PHYSICAL BACKGROUND

*Eduard Feireisl*

### in Dynamics of Viscous Compressible Fluids

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528388
- eISBN:
- 9780191713590
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528388.003.0001
- Subject:
- Mathematics, Applied Mathematics

This chapter is devoted to a review on the underlying physical theory. Besides the basic notions of reference configurations, kinematics, constitutive equations, and balance laws, this part includes ... More

## St And Brs Symmetries, Stochastic Field Equations

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Section 15.3 introduced a transformation depending on anticommuting parameters, to prove the geometric stability of homogeneous spaces under renormalization. There is a set of topics, stochastic ... 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

## Generalization of Gravitation Theory

*Hanoch Gutfreund and Jürgen Renn*

### in The Formative Years of Relativity: The History and Meaning of Einstein's Princeton Lectures

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174631
- eISBN:
- 9781400888689
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174631.003.0022
- Subject:
- Physics, History of Physics

This chapter attempts to formulate a consistent extension of the theory of general relativity. The starting point of the general theory of relativity is the recognition of the unity of gravitation ... More

## Relativistic Theory of the Non-symmetric Field

*Hanoch Gutfreund and Jürgen Renn*

### in The Formative Years of Relativity: The History and Meaning of Einstein's Princeton Lectures

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174631
- eISBN:
- 9781400888689
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174631.003.0023
- Subject:
- Physics, History of Physics

This chapter attempts to find a measure for the “strength” of a system of field equations, which is determined by the amount of free data consistent with the system. It introduces the infinitesimal ... More

## Collective dynamics in plasmas—II. Some basic fluid modes

*Abraham Bers*

### in Plasma Physics and Fusion Plasma Electrodynamics

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780199295784
- eISBN:
- 9780191749063
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199295784.003.0003
- Subject:
- Physics, Nuclear and Plasma Physics, Particle Physics / Astrophysics / Cosmology

This chapter continues the discussion of collective dynamics in plasmas by exploring linear collective modes, as well as some of their nonlinear couplings. The modes are derived from simplified ... More

##
The General Theory of Relativity *(Continued)*

*Hanoch Gutfreund and Jürgen Renn*

### in The Formative Years of Relativity: The History and Meaning of Einstein's Princeton Lectures

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174631
- eISBN:
- 9781400888689
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174631.003.0017
- Subject:
- Physics, History of Physics

This chapter shows that in the limit of weak fields and low velocities, the equation of the geodesic line reduces to Newton's equation of motion. It proceeds to derive the gravitational field ... More

## Langevin Field Equations

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0014
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

In this chapter, Langevin field equations (or Ito stochastic field equations, SFEs) are derived that are equivalent to functional Fokker–Planck equations (FFPEs) for bosons and fermions. Phase space ... More

## Equations of motion

### in Advanced General Relativity: Gravity Waves, Spinning Particles, and Black Holes

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680696
- eISBN:
- 9780191760662
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680696.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

A technique for extracting the relativistic equations of motion of Schwarzschild, Reissner–Nordström, or Kerr particles moving in external fields, from the vacuum Einstein or Einstein–Maxwell field ... More

## Einstein Equation and its Spherical Solution

*Ta-Pei Cheng*

### in A College Course on Relativity and Cosmology

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780199693405
- eISBN:
- 9780191803130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199693405.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

The general relativity (GR) field equation relates spacetime curvature to mass/energy distribution. Its solution is the metric, determining spacetime geometry. This chapter introduces curvature in ... More

## The Riemann curvature tensor

*Clifford Henry Taubes*

### in Differential Geometry: Bundles, Connections, Metrics and Curvature

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0016
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter first examines some fundamental examples where the Riemann curvature tensor has an especially simple form. It then discusses the ways in which the Riemann and Ricci curvatures are used ... More

## Field Equation of Nonlocal Gravity

*Bahram Mashhoon*

### in Nonlocal Gravity

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198803805
- eISBN:
- 9780191842313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198803805.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

In extended general relativity (GR), Einstein’s field equation of GR can be expressed in terms of torsion and this leads to the teleparallel equivalent of GR, namely, GR||, which turns out to be the ... More

## Plane gravitational waves

### in Advanced General Relativity: Gravity Waves, Spinning Particles, and Black Holes

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680696
- eISBN:
- 9780191760662
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680696.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Starting with plane gravitational waves of arbitrary profile as solutions of Einstein’s vacuum field equations in the linear approximation a sequence of gauge transformations results in a metric ... More

## Application to Multi-Mode Systems

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0015
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter presents two examples of applying phase space methods in systems where large numbers of modes may be involved, such as in fermion cases with large particle numbers. Functional ... More

## Schwinger Action Principle and Variational Calculus

*Norman J. Morgenstern Horing*

### in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198791942
- eISBN:
- 9780191834165
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198791942.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory ... More

## The Einstein field equations

*Steven Carlip*

### in General Relativity: A Concise Introduction

- Published in print:
- 2019
- Published Online:
- March 2019
- ISBN:
- 9780198822158
- eISBN:
- 9780191861215
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822158.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The Einstein field equations are the fundamental equations of general relativity. After a brief qualitative discussion of geodesic deviation and Newtonian gravity, this chapter derives the field ... More

## The General Theory of Relativity and Gravitation

*David D. Nolte*

### in Introduction to Modern Dynamics: Chaos, Networks, Space, and Time

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198844624
- eISBN:
- 9780191880216
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198844624.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The intrinsic curvature of a metric space is captured by the Riemann curvature tensor, which can be contracted to the Ricci tensor and the Ricci scalar. Einstein took these curvature quantities and ... More