Which form of Einstein's stress-energy tensor $T$ is physical, the covariant or the contravariant?

Question

Using Einstein's Field Equations for a specific metric I am interested in, I have calculated the first diagonal term of the stress energy tensor $T$, which is the energy density. In my calculations, the covariant density ($T$ with lower indices 00) does not show a singularity, but the contravariant density ($T$ with upper indices 00) does show a singularity. I need to know whether this singularity is physical. Which form of the stress energy tensor represents actual physical density, the covariant or the contravariant?

Answer 1

Generally speaking the contravariant form $T^{\alpha\beta}$ is the one most naturally related to what we think of as physical observables. My favourite way of thinking about this is to start with the stress-energy tensor of a point particle, and we can understand the more complicated forms as made up from point particles.

But we should note Bob Bee's comments. When you write down $\mathbf T$ you have to choose a coordinate system to do so, and the individual components of the tensor will depend on the coordinates you choose. If your choice of coordinates results in coordinate singularity then you may well find your stress-energy tensor behaves strangely at that point.

It's hard to comment without knowing the details of the geometry you are studying, but if $T_{\alpha\beta}$ isn't singular this suggests very strongly that you have a coordinate singularity rather than a real one. Try calculating the trace of $\mathbf T$ to see if it diverges.