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12 - Further spherically symmetric geometries

Published online by Cambridge University Press:  05 September 2012

M. P. Hobson
Affiliation:
University of Cambridge
G. P. Efstathiou
Affiliation:
University of Cambridge
A. N. Lasenby
Affiliation:
University of Cambridge
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Summary

In the preceding three chapters, we have considered in some detail the Schwarzschild geometry, which represents the gravitational field outside a static spherically symmetric object. We also considered the structure of the Schwarzschild black hole, in which the empty-space field equations are satisfied everywhere except at the central intrinsic singularity. In this chapter, we consider solving the Einstein equations for a static spherically symmetric spacetime in regions where the presence of other fields means that the energy–momentum tensor is non-zero. In particular, we will concentrate on two physically interesting situations. First, we discuss the relativistic gravitational equations for the interior of a spherically symmetric matter distribution (or star); in this case the energy–momentum tensor of the matter making up the star must be included in the Einstein field equations. Second, we consider the spacetime geometry outside a static spherically symmetric charged object; once again this is not a vacuum, since it is filled with a static electric field whose energy–momentum must be included in the field equations.

The form of the metric for a stellar interior

Most stars in the sky are nowhere near dense enough for general-relativistic effects to be important in determining their structure. This is true for main sequence stars (of which our Sun is an example), red giants and even such high-density objects as white dwarfs.

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Chapter
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General Relativity
An Introduction for Physicists
, pp. 288 - 309
Publisher: Cambridge University Press
Print publication year: 2006

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