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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?

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    $\begingroup$ 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
    Commented Jul 27, 2020 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
    Commented Jul 27, 2020 at 18:13

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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.

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