I How do you know if a ds^2 metric is valid?

1. Aug 2, 2016

NotASmurf

Hey all, in every theory that involves GR you see they give their space-time metric, but very few show any other math related to it, how does one know if a metric is valid?

2. Aug 2, 2016

robphy

The metric alone is not sufficient.
Conceptually, candidate solutions of the field equations have the general form (M,g,T)...
you have to spell out the manifold (set of events) M,
the metric g [as you have provided],
and a stress-energy tensor field T (describing the matter distribution.... special case: in vacuum, T=0).

Then you substitute into field equations:
from the metric tensor g, compute the Einstein Curvature Tensor G... and ask if it is equal (up to conventional constants) to the stress-energy T.

You could just work out the Einstein tensor G and declare your T to be equal to it....
However,
physically, we have to ask questions about the matter distribution T...
is this T physically reasonable? (Does it satisfy certain energy conditions, etc..? Is this realizable by real matter? or some weird exotic matter?)

Here is the source of this viewpoint: Geroch, General Relativity from A to B, p 172