# 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:

$$ds^2 \sim -c^2 dt^2 + dr^2 + r^2 d \Omega^2$$

• When you take a limit like $r \to \infty$ the metric should not depend on $dr$ anymore. Commented Aug 9, 2020 at 11:56
• @JamalS I mean't it was asymptotic to that that. To be more precise: $\lim_{r \to \infty} \frac{-c^2 dt^2 + dr^2 +r^2 d \Omega^2}{ds^2} = 1$ Commented Aug 9, 2020 at 12:01
• One obvious constraint is that since the Einstein tensor goes to zero, so does the energy-momentum tensor. Commented Aug 9, 2020 at 15:15
• @Javier I agree however I'm unable to show this is the only constraint it implies Commented Aug 9, 2020 at 16:37