1
$\begingroup$

This question is based on Carroll's book Spacetime and Geometry, specifically from page 33 to page 36.

In the upper mentioned section we define the Stress-Energy Tensor as:

The flux of the four momentum $p^\mu$ across a surface of constant $x^\nu$

Here lies the first problem: canonically the flux of a vector across a surface is a scalar, not a matrix or a tensor. So I don't get this definition at all.

But nevertheless I understand that the Stress-Energy Tensor represent a generalization of the concept of mass and energy, so we can move forward for now. We then get the form of the Stress-Energy Tensor for a perfect fluid: $$T^{\mu\nu}=(\rho+p)U^\mu U^\nu+p\eta^{\mu\nu}$$ (where $\rho$ is the energy density, $p$ si the pressure and $U$ is the four-velocity; keep in mind that we are working in flat spacetime). This is fine for me, however then we come to the following expression: $$\partial _\mu T^{\mu\nu}=0$$ It is stated that the $\nu=0$ component corresponds to the conservation of energy and the other three components correspond to the conservation of momentum; but no direct proof, or justification, of this statement is given. Using the definition of $T^{\mu\nu}$ previously cited: how can we show that the upper mentioned statement is true, or at least plausible?

$\endgroup$
2
  • 1
    $\begingroup$ I don't understand your comment "canonically the flux of a vector across a surface is scalar...". See en.wikipedia.org/wiki/… and compare to en.wikipedia.org/wiki/Flux#Flux_as_a_surface_integral. Are you mixing the two definitions? For the former, the "scalar" case is a very special case. $\endgroup$ – Brick Jul 27 '20 at 18:11
  • $\begingroup$ Secondly, this seems like two distinct questions to me. Really you should ask one question at a time, and the second one, at least, probably has an answer somewhere on this site already. $\endgroup$ – Brick Jul 27 '20 at 18:13
0
$\begingroup$

The fundamental tensor equation of relativistic mechanics of continous matter is $$ K^\mu = \partial_\mu T^{\mu\nu} $$ where $K^\mu$ is the 4-force-density acting on material medium and $T^{\mu\nu}$ is the energy-momentum-stress tensor of the system. Consistently for $\nu=0$ we obtain the equation of continuity and for $\nu=1,2,3$ the 3-vector equation of motion. Of course, in absence of external forces ($K^\mu=0$), the scalar relation corresponds to conservation of energy and the vector relation becomes the law of conservation of linear momentum.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.