Stress-energy tensor

The stress-energy tensor is a symmetric tensor which describes the spacetime density and flux of energy and momentum.

\begin{align*} T^{\alpha\beta} \compeq \mat{ {\color{red} T^{00}} & \color{orange} T^{01} & \color{orange}T^{02} & \color{orange}T^{03} \\ \color{orange} T^{10} & \color{green}T^{11} & \color{blue}T^{12} & \color{blue}T^{13} \\ \color{orange} T^{20} & \color{blue}T^{21} & \color{green}T^{22} & \color{blue}T^{23} \\ \color{orange} T^{30} & \color{blue}T^{31} & \color{blue}T^{32} & \color{green}T^{33} }. \end{align*}

$\color{red} T^{00}$ is the energy density, or the density of $p^0$

flowing in the $x^0$ direction.

$\color{orange} T^{0a}$ is the energy flux, or the density of

$p^0$ flowing in the $x^a$ direction. Actually, it’s missing a $c$ factor, since $p^0 = E/c$ .

$\color{orange} T^{a0}$ is the momentum density, or the densit

of $p^a$ flowing in time. Note $T^{0\alpha} = T^{\alpha0}$ . Here we are missing a $1/c$ factor because $x^0 = ct$ .

$\color{green} T^{aa}$ is the flux of momentum $p^a$

in the $x^a$ direction, also known as stress or pressure.

$\color{blue} T^{ab}$ is the flux of momentum $p^a$ in the $x^b$

direction. When talking about a fluid, these components represent the non-normal flows.

We can derive the continuity equation from understanding $T^{\alpha\beta}$ as the $p^\alpha$ momentum flux in the $x^\beta$ direction.

Let $\Delta p^\alpha$ be the change in 4-momentum in the $x^\alpha$ direction in some time $\Delta t$ over a volume $V$ . We can write this using the $T^{\alpha0}$ components as

\begin{align*} \Delta p^\alpha &= \Delta t \int_V \frac{\partial T^{\alpha 0}}{\partial x^0} \d V. \end{align*}

We can also write this by computing the surface over the enclosing surface $S$ , and apply the divergence theorem

\begin{align*} \Delta p^\alpha &= - \Delta t \oint_S T^{\alpha i} \d A^i \\ &= - \Delta t \int_V \frac{\partial T^{\alpha i}}{\partial x^i} \d V. \end{align*}

We set these two integrals and simplify to find

\begin{align*} \int_V \frac{\partial T^{\alpha 0}}{\partial x^0} \d V &= - \int_V \frac{\partial T^{\alpha i}}{\partial x^i} \d V \\ \frac{\partial T^{\alpha 0}}{\partial x^0} + \frac{\partial T^{\alpha i}}{\partial x^i} &= 0 \\ \partial_\beta T^{\alpha\beta} &= 0. \end{align*}

This equation encodes conservation of momentum and energy.