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 outside the mass:

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

1. Matter → Geometry

• Since we’re in vacuum, outside the central mass: $T_{\mu \nu }=0$.
• 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.

Justification of the assumptions

I wonder:

If the metric is present, some matter or energy must be sourcing it—so how can we justify treating the metric as “given” while relegating everything else to test particles that don’t affect it?

Let’s address this clearly. There are two layers of interpretation at work here: physical realism versus modeling abstraction.

1. Physically: No metric without sources

Einstein’s equations:

tie the geometry directly to matter and energy. The metric is determined (up to initial/boundary conditions) by the full energy–momentum content of spacetime. So physically, my intuition above is correct:

“If a metric exists, it must be due to something. There’s no geometry ‘floating in space’ without some kind of source.”

But in many contexts, we do exactly that. Why?

2. Modeling: The test-particle regime

In practice, we often distinguish between:

(a) The source matter

This is the matter content that actively contributes to the stress–energy tensor $T_{\mu \nu }$ and thus determines $g$. For example, in the solar system:

• The Sun, planets, gas, radiation, etc., all contribute to $T_{\mu \nu }$.
• The metric $g$ is a solution to Einstein’s equations sourced by this content.

(b) The probe matter

This refers to small, negligible-energy entities (like spacecraft or single particles) whose mass–energy is too small to significantly alter the geometry. They are used to test the geometry by following its geodesics.

This modeling split is entirely approximate, but often extremely accurate. The approximation consists of:

• Fixing the metric as the result of some large but unspecified or idealized $T_{\mu \nu }$.
• Neglecting the energy–momentum of the test bodies.
  So, the geometry is not “ungrounded”—it is assumed to be sourced, but the sources are idealized or not explicitly modeled.

You’re not really dropping all matter. You’re just choosing not to model the matter that determines $g$ within the same field-space as the matter you’re probing with.

3. Why the modeling still makes sense

There are deep justifications for this in practice:

● Effective theories

In effective field theory (EFT), you often integrate out high-energy degrees of freedom to work with low-energy approximations. Similarly here:

• The full theory is $g$ and $\varphi$.
• The effective theory is $g$ fixed and $X$ as a test object.

● Separation of scales

If there is a strong hierarchy between the mass/energy of the geometry-determining source (e.g., a star) and the test particle, the latter has negligible backreaction. This justifies keeping $g$ fixed.

5. Summary:

It’s a two-level model:

• The metric $g$ is assumed to have been selected by some selection criterion (e.g., solution to Einstein’s equations with some matter you’re not modeling).
• The worldline $X$ is then selected by a second, subordinate criterion: extremizing the action in that background.