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Tensors and Relativity: Chapter 8

General discussion of the Schwartzschild solution

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In order to understand the physical properties of the Schwartzschild solution we first have to clarify the physical significance of the integration constant tex2html_wrap_inline1281 . This is best done through a comparison with Newtonian theory. For large values of the coordinate r, the line element deviates only a little from that of flat space, and from our analysis of the Newtonian limit we see that

equation1037

so

equation1039

We thus have to interpret the Schwarzschild solution as the gravitational field outside a spherically symmetric mass distribution whose [ Newtonian ] mass is M.

Notice that the line element is singular when tex2html_wrap_inline1287 . This is the Schwartzschild radius or the gravitational radius of the source. For example for the Sun tex2html_wrap_inline1289 km and for the Earth it is tex2html_wrap_inline1291 mm.

From the line element, we can immediately deduce some of the effects we derived using the equivalence principle.