Deriving the Metric from the Energy-Momentum Tensor

atyy

Doesn't the energy-momentum tensor contain the metric?

Nugatory

No. They are related by the Einstein Field Equation, which is a set of differential equations for the metric in terms of the stress-energy tensor, but the EFE alone does not contain enough information to completely specify the metric. Like any reasonable differential equation, the EFE gives you a family of solutions; the boundary conditions are required to know the specific member of that family that describes the physical situation at hand.

atyy

How about Eq 8.15 of http://ned.ipac.caltech.edu/level5/M.../Carroll8.html? Isn't that the metric in the stress-energy tensor?

K^2

This has to do with choice of perfect fluid as the source. What this really tells you is that when you'll be solving for metric, you'll be solving it as a system. You'll be trying to find T and g that satisfy both the equation for fluid and the Einstein Field Equation.

atyy

But isn't that the OP's question? I guess I don't understand what he means by "expression for the energy momentum tensor" since the expression seems to already contain the metric.

PAllen

The original question was can you guess the metric from T, for some simple cases, without solving differential equations (and I consider guessing a method of solution; perhaps the most common method - guess and verify). I sidetracked that by suggesting it was an implausible expectation, because it couldn't be done for the trivial case of T=0.

As for g within T, the physical parts of it (e.g. pressure and density) do not contain the metric. This is what you might specify; then the EFE contain g on both sides as an undetermined variable to solve for. Alternatively, if you are given the full T as function in some arbitrary coordinates, you have no idea a priori how the the metric figures in T.

I guess one special case is you know the form of T in terms of g and physical quantities, you are given the physical quantities and the complete expression of T. Then, you could get g. Of course, if you pick these things arbitrarily, the chance of satisfying the EFE is zero.

atyy

So is the last what the OP had in mind, since he did specify that it's the stress tensor of a perfect fluid?

PAllen

Well, we'll have to hope the OP answers that.

I assumed you had just T, not T and and density and pressure functions that you knew all constituted a valid solution. With just T, you've got no way to read off the metric. And the only way you have pressure, density, and T that satisfy the EFE is to first solve the EFE with metric undetermined. I believe my interpretation of the question is the most likely - but again, we'll never know without input from OP.