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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


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