# Special Relativity: Interpretation of the partial derivate of Stress-Energy Tensor

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?

• 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. – Brick Jul 27 '20 at 18:11
• 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. – Brick Jul 27 '20 at 18:13

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.