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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:

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:

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:

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:

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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