If I got a diagonal stress energy tensor $T_{\alpha \alpha} = x_{\alpha}$ for some coefficients $x_{\alpha}$, could anyone tell me how can I extract the four components of the stress energy tensor? Also, is there a way to derive the metric for such a case?
1 Answer
Also, is there a way to derive the metric for such a case.
No, the metric isn't uniquely defined by the sources. For example, if the stress-energy tensor is zero everywhere, your metric could be Minkowski, or it could be something with gravitational waves in it. For example, here is a vacuum solution, expressed in coordinates where it's diagonal, that isn't Minkowski:
$$ds^2 = d t^2 - p(z-t)^2 d x^2 - q(z-t)^2 d y^2 - d z^2 $$
Here $p$ and $q$ are any functions that satisfy the differential equation $\ddot{q}/q+\ddot{p}/p=0$.