Schwarzschild solution

The Schwarzschild solution is a special case within the general framework of General Relativity.

What kind of system are we modeling?
The Schwarzschild solution models empty space outside a static, spherically symmetric mass (e.g., the space around a star, planet, or black hole).

So in this case:

• The matter fields $\psi$ are absent in the region of interest (vacuum).
• The energy-momentum tensor is zero:$$T_{\mu \nu }=0$$

Let’s walk through these steps in the Schwarzschild case.

1. Matter → Geometry

• Since we’re in vacuum: $T_{\mu \nu }=0$.
• The Einstein field equations become:$$G_{\mu \nu }=0$$
• This is still a nonlinear PDE system, but now significantly simpler due to the absence of matter.
• Imposing staticity and spherical symmetry, you solve $G_{\mu \nu }=0$ and find the unique solution (up to mass parameter):$$d{s}^2=(1-\frac{2GM}{c^{2}r})(cdt{)}^2-\left(\frac{1}{1-\frac{2GM}{c^{2}r}}\right)d{r}^2-r^{2}d{\theta }^2-r^2{\sin }^2\theta d{\varphi }^2$$This is the Schwarzschild metric, and the parameter $M$ enters as an integration constant, interpreted as the mass of the source.

2. Geometry → Matter Motion

• You can now compute how test particles move in this fixed background metric by solving the geodesic equation.
• This describes motion of bodies or light around the massive object (e.g., orbits, gravitational lensing, time dilation).

3. Back to Step 1?

• In the Schwarzschild setup, you assume the matter doesn't change (e.g., the central mass is static), so $T_{\mu \nu }$ remains zero and the metric doesn’t evolve.
• So it’s not a dynamic simulation, just a static solution to the Einstein equations for a fixed configuration.