Einstein Equation and the Stress-Energy-Momentum Tensor

Let G denote Newton’s constant and c denote the speed of light. The Einstein equation is

    \[  G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]

where G_{\mu\nu} is the Einstein tensor and T_{\mu\nu} is the stress-energy-momentum tensor. These equations extremize the action

    \[ S[g_{\mu\nu},\Phi] = \frac{c^3}{16\pi G}\int d^4x \sqrt{-g} R + S_{matter}[g_{\mu\nu},\Phi] \]

where g is the determinant of the spacetime metric g_{\mu\nu} and R is the curvature scalar. The matter fields are denoted collectively by \Phi. The stress-energy-momentum tensor is obtained from the matter action by

    \[ T^{\mu\nu} \equiv \frac{2c}{\sqrt{-g}} \frac{\delta S^{\rm matter}}{\delta g_{\mu\nu}}  \]

Let U^\mu denote the velocity of an observer and let e^\mu_i denote a set of spatial, orthonormal basis vectors. We have the normalization conditions U\cdot U = -c^2, U\cdot e_i = 0, and e_i\cdot e_j = \delta_{ij}. The 4-momentum density and the 4-momentum flux as seen by the observer are:

(1)   \begin{eqnarray*} {\mathcal P}^\mu & \equiv & -\frac{1}{c^2} T^\mu_\nu U^\nu \\ {\mathcal F}^\mu_i & \equiv & T^\mu_\nu e^\nu_i \end{eqnarray*}

From the 4-momentum density we define the energy density and 3-momentum density:

(2)   \begin{eqnarray*} (energy\ density) & \equiv & -{\mathcal P}\cdot U = \frac{1}{c^2}  T_{\mu\nu} U^\mu U^\nu \\ (3momentum\ density)_i & \equiv & {\mathcal P}\cdot e_i = -\frac{1}{c^2} T_{\mu\nu} U^\mu e^\nu_i  \end{eqnarray*}

From the 4-momentum flux we define the energy flux and the 3-momentum flux:

(3)   \begin{eqnarray*} (energy\ flux)_i & \equiv & -{\mathcal F}_i\cdot U = - T_{\mu\nu} U^\mu e^\nu_i  \\ (3momentum\ flux)_{ij} & \equiv & {\mathcal F}_i\cdot e_j = T_{\mu\nu} e^\mu_i e^\nu_j \end{eqnarray*}

Note that (energy\ flux)_i = c^2 (3momentum\ density)_i. The 3-momentum flux is also identified as the spatial stress.