# The Schwarzschild metric

## Main Question or Discussion Point

The solution for the Schwarzschild metric is stated from reference 1 as:

$$ds^2=- \left(1-\frac{r_s}{r}\right) c^2 dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2(d \theta^2 +\sin^2 \theta d \phi^2)$$

The solution for the Schwarzschild metric is stated from references 2 as:

$$ds^2 = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1 - \frac{r_s}{r} \right)^{-1} dr^2 - r^2 \left(d \theta^2 + \sin^2 \theta d \phi^2 \right)$$

The solution for the Schwarzschild metric is stated from references 3 as:
$$ds^2 = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1 - \frac{r_s}{r} \right)^{-1} dr^2 - r^2 \left(d \theta^2 - \sin^2 \theta d \phi^2 \right)$$

$$r_s$$ - Schwarzschild radius

There is a difference in the sign of the elements between the stated solutions.

Which is the correct solution?

Reference:
http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution
http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_equation#_ref-ov_3
http://en.wikipedia.org/wiki/Schwarzschild_metric

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Did you check any references outside of wikipedia?

By the looks, the only difference is the sign convention: is it spacelike or timelike intervals that are represented by real lengths? For what purpose do you want one to be "correct"?

SpaceTiger
Staff Emeritus
Gold Member
The solution for the Schwarzschild metric is stated from references 3 as:
$$ds^2 = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1 - \frac{r_s}{r} \right)^{-1} dr^2 - r^2 \left(d \theta^2 - \sin^2 \theta d \phi^2 \right)$$
Might want to double-check your signs on this one. The third reference looks like it gives the same as the second reference (but distributing the last parenthetical). The differences between the first two are probably just convention, as cesiumfrog said.

CompuChip