Interpretation of Stress-Energy Tensor and Dust

Question

MTW (chapter 5) and others state that $-T^a_b v^b $ should be interpreted as the four-momentum density in the reference frame of an observer with four-velocity $v^a$, where $T^a_b$ is the stress-energy tensor. This makes sense as a useful machine we'd really like to have.

Okay, very first/simplest test case: Take the stress energy of dust in coordinates where it is at rest, so that $T^{ab} = \rho u^a u^b $ with $u^a = (1,0,0,0)$. Now here comes an observer with four-velocity $v^a = (\gamma, 0,0,v \gamma) $. Since $u_a v^a = - \gamma$, the four-momentum density in the observer's frame is supposed to be $(\gamma \rho,0,0,0)$? What happened to the momentum of the dust in the observer's frame? He should see dust particles zipping by, right?

I must be missing something extremely basic for this to be bugging me so much. But what?

Answer 1

You are calculating the momentum in the dust rest frame. In the observer rest frame we have $v^a = (1,0,0,0)$ and $u^a = (\gamma,0,0,-v\gamma)$ so that $-T^a{}_bv^a = \rho\gamma^2(1,0,0,-v)$. Note that this, of course is the same expression we would get if we transformed your result to the observer restframe.

In other words, the correct statement is that $-T^a{}_bv^b$ is the 4-momentum observed by the observer, expressed in whatever frame we use for our calculations.