Q:

Geometry constraining the Stress-energy tensor?

Question
Let's say I have a metric in radial coordinates such that at $r \to \infty$ we find flat spacetime:
$$ds^2 \sim -c^2 dt^2 + dr^2 + r^2 d \Omega^2$$
where $ds^2$ is the line element and $t$ is time cordinate and $r$ is the radial coordinate? What constraint does this impose on the stress-energy tensor?
For example
The Schwarzschild metric is given by:
$$ ds^2 = - (1- \frac{r_s}{r})c^2 dt^2 + (1-\frac{r_s}{r})^{-1}dr^2 + r^2 d\Omega^2 $$
Now, for large $r$ we see:
$$ ds^2 \sim -c^2 dt^2 + dr^2 + r^2 d \Omega^2$$

A:

The only constraint is that the stress-energy tensor vanishes at infinity. To see this, try approaching the problem from the other side: Given any arbitrary stress-energy tensor which vanishes at infinity, you can show that there exists a corresponding metric which satisfies your condition.

