Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 34A / Solution

Use $x^0=ct$. The stress-energy tensor component $T^{00}$ is energy per unit volume; $c{T}^{i0}$ is energy flux in direction $i$; $T^{0j}/c$ is momentum density in direction $j$; and $T^{ij}$ is the flux of momentum component $j$ through a surface normal to direction $i$. In a symmetric tensor, energy flux equals $c^2$ times momentum density. Thus ${\partial }_{\mu }{T}^{\mu 0}=0$ is energy conservation and ${\partial }_{\mu }{T}^{\mu j}=0$ is momentum conservation.