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Spherically Symmetric Black Holes

Valeri P. Frolov and Andrei Zelnikov

in Introduction to Black Hole Physics

Published in print:
2011
Published Online:
January 2012
ISBN:
9780199692293
eISBN:
9780191731860
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199692293.003.0006
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

To derive spherically symmetric solutions of the Einstein equations the dimensional reduction is used. The Birkhoff theorem is proved, which shows that any vacuum spherically symmetric solution ... More


An introduction to black holes

Piotr T. Chruściel

in Geometry of Black Holes

Published in print:
2020
Published Online:
December 2020
ISBN:
9780198855415
eISBN:
9780191889233
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198855415.003.0004
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

In this chapter the basics of the geometry of stationary black-hole spacetimes are presented. We start in Section 4.1 with a brief review of astrophysical black holes. We continue in Section 4.2 with ... 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


Dynamical black holes

Piotr T. Chruściel

in Geometry of Black Holes

Published in print:
2020
Published Online:
December 2020
ISBN:
9780198855415
eISBN:
9780191889233
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198855415.003.0008
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

In this chapter we review what is known about dynamical black hole-solutions of Einstein equations. We discuss the Robinson–Trautman black holes, with or without a cosmological constant. We review ... More


The Schwarzschild black hole

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0047
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses the Schwarzschild black hole. It demonstrates how, by a judicious change of coordinates, it is possible to eliminate the singularity of the Schwarzschild metric and reveal a ... More


Particles and Light Motion in Schwarzschild Spacetime

Valeri P. Frolov and Andrei Zelnikov

in Introduction to Black Hole Physics

Published in print:
2011
Published Online:
January 2012
ISBN:
9780199692293
eISBN:
9780191731860
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199692293.003.0007
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This Chapter studies particle and light motion near the Schwarzschild black hole. We focus mainly on those aspects of the motion that are specific for the black holes and that are of the most ... 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


The Schwarzschild solution

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0046
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter deals with the Schwarzschild metric. To find the gravitational potential U produced by a spherically symmetric object in the Newtonian theory, it is necessary to solve the Poisson ... More


General relativistic time

George Jaroszkiewicz

in Images of Time: Mind, Science, Reality

Published in print:
2016
Published Online:
January 2016
ISBN:
9780198718062
eISBN:
9780191787553
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198718062.003.0019
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses the transition from special relativistic (Minkowski) spacetime to general relativistic spacetimes. It reviews the concept of timelike, spacelike, and null (or lightlike) vector ... More


Black holes

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/9780198822158.003.0010
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Chapter 3 used the Schwarzschild metric to obtain predictions for the Solar System. In this chapter, that metric is derived as the unique static, spherically symmetric solution of the vacuum Einstein ... More


Black Holes

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/9780199658633.003.0020
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Black holes seem like something out of science fiction, but they are real. Now that we understand the relevant properties of gravity we can separate fact from fiction. We start by defining black ... More


Schwarzschild’s Spacetime

Prasenjit Saha and Paul A. Taylor

in The Astronomers' Magic Envelope: An Introduction to Astrophysics Emphasizing General Principles and Orders of Magnitude

Published in print:
2018
Published Online:
July 2018
ISBN:
9780198816461
eISBN:
9780191858246
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198816461.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

The concept of a metric is motivated and introduced, along with the introduction of relativistic quantities of spacetime, proper time, and Einstein’s field equations. Geodesics are cast in basic form ... More


The Cartan structure equations

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0065
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the ... More


Geodesics in the Solar System

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/9780198822158.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Given the spacetime geometry of the Solar System, the geodesic equation can be used to derive the motion of massive objects and light. In this chapter, starting with the Schwarzschild metric, the ... More


Geometry of Black Holes

Piotr T. Chruściel

Published in print:
2020
Published Online:
December 2020
ISBN:
9780198855415
eISBN:
9780191889233
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198855415.001.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

There exists a large scientific literature on black holes, including many excellent textbooks of various levels of difficulty. However, most of these prefer physical intuition to mathematical rigour. ... More


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