Week 01: Deriving the Schwarzschild Radius

This week, I revisited the exact solution to the Einstein Field Equations for a non-rotating spherical mass.

The Metric

The line element for the Schwarzschild metric is given by:

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

Where $r_s$ is the Schwarzschild radius.

Calculation

To find the event horizon, we look for the singularity in the $g_{rr}$ component. This occurs when:

$$ 1 - \frac{r_s}{r} = 0 \implies r = r_s $$

Substituting the constants, we arrive at the classic formula:

$$ R_s = \frac{2GM}{c^2} $$

Next Week

I plan to extend this to the Kerr Metric to account for angular momentum $J$.