# How can I use Einstein's field equations to find the metric tensor? [duplicate]

I have watched and read a lot on the topic of General Relativity and the geometry behind it. I am confident that I can derive an approximation of the the stress-energy-momentum tensor with just the metric tensor, but if I was given the stress-energy-momentum tensor I'm not sure how I would go back to find the metric. How can I use Einstein's field equations to find the metric tensor? Would I have to expand all of the notation? Or is there some equation I can plug into my computer to get it? Or is there some other way?

## marked as duplicate by Kyle Kanos, Qmechanic♦Mar 4 '15 at 1:16

Let's look at a simple but troublesome example. Your manifold is $\mathbb{R}^4$, your stress-energy tensor is $T^{\mu \nu}=0$. There are many possible metrics. You could have a metric corresponding to a gravitational wave travelling left, one travelling right, one up, one down, one forwards, one backwards. Or just an empty flat space, the spacetime of special relativity. This is entirely like the situation with no charge and no current, you could have no fields, or an electromagnetic wave going in any direction. It's just not enough information.