Spherically Symmetric Black Holes
We begin the analysis of the physical properties of black holes with the simplest case
in which both the black hole and its gravitational field are spherically symmetric. The
spherically symmetric gravitational field (spacetime with spherical three-dimensional
space) is described in every textbook on general relativity [see, e.g., Landau and
Lifshitz (1975), Misner, Thorne, and Wheeler (1973)]. Therefore, here we will only
reproduce the necessary results. Mathematical details connected with definition and
properties of a spherically symmetric gravitational field can be found in Appendix B.
Let us write the expression for the metric1 in a region far from strong gravitational
fields (i.e., where special relativity is valid), using the spherical spatial coordinate
system (г, в, ф):
where с is the speed of light, and dl is the distance in three-dimensional space.
Consider now a curved spacetime but preserve the condition of spatial spherical
symmetry. Spacetime is not necessarily empty, it may contain matter and phys-
ical fields (which are, of course, also spherically symmetric if their gravitation is
considered). It can be shown (see, e.g., Misner, Thorne, and Wheeler (1973), and
Appendix B) that there exist coordinates (x0x1,в, ) in a spherically symmetric
spacetime such that its metric is of the form
where xl = x1(x°,x1) is the solution of B.1.3) forx1. It describes the radial motion
of the points of the new reference frame (with the coordinate x-1 = const) with respect
to the older one.
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in which both the black hole and its gravitational field are spherically symmetric. The
spherically symmetric gravitational field (spacetime with spherical three-dimensional
space) is described in every textbook on general relativity [see, e.g., Landau and
Lifshitz (1975), Misner, Thorne, and Wheeler (1973)]. Therefore, here we will only
reproduce the necessary results. Mathematical details connected with definition and
properties of a spherically symmetric gravitational field can be found in Appendix B.
Let us write the expression for the metric1 in a region far from strong gravitational
fields (i.e., where special relativity is valid), using the spherical spatial coordinate
system (г, в, ф):
where с is the speed of light, and dl is the distance in three-dimensional space.
Consider now a curved spacetime but preserve the condition of spatial spherical
symmetry. Spacetime is not necessarily empty, it may contain matter and phys-
ical fields (which are, of course, also spherically symmetric if their gravitation is
considered). It can be shown (see, e.g., Misner, Thorne, and Wheeler (1973), and
Appendix B) that there exist coordinates (x0x1,в, ) in a spherically symmetric
spacetime such that its metric is of the form
where xl = x1(x°,x1) is the solution of B.1.3) forx1. It describes the radial motion
of the points of the new reference frame (with the coordinate x-1 = const) with respect
to the older one.
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