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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




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.