- #1

StateOfTheEqn

- 73

- 0

[tex]d\tau^2=(1-r_s/r)dt^2-(1-r_s/r)^{-1}dr^2-r^2d\Omega^2[/tex]

where c is set to 1, r is the scalar distance, [tex]r_s[/tex] is the 'event horizon' radius, and [tex]d\Omega^2=d\theta^2+sin^2\theta d\phi^2[/tex].

In Schwarzschild's original paper from 1916 he does not use r the same way. His equation is equivalent to:

[tex]ds^2=(1-r_s/R)dt^2-(1-r_s/R)^{-1}dR^2-R^2d\Omega^2[/tex] where [tex]R=(r^3+r_s^3)^{1/3}[/tex] with (+,-,-,-) as the Minkowski signature. See equation (14) in reference linked below. BTW, In his paper he uses the notation [tex]\alpha=r_s[/tex]

Note: [tex]ds^2=d\tau^2[/tex] for a free falling test particle.

One implication of his original formulation is that the coordinate singularity at [tex]r_s[/tex] gets removed which has implications for the theory of black holes. So, the two metrics obviously cannot be equivalent.

Another important question is: how can two contradictory metrics for a central gravitational field both be consistent with the GR field equations?

Reference in German but the mathematics is quite clear: http://de.wikisource.org/wiki/Über_..._Massenpunktes_nach_der_Einsteinschen_Theorie