# Einstein Equation and the Stress-Energy-Momentum Tensor

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

where is the Einstein tensor and is the stress-energy-momentum tensor. These equations extremize the action

where is the determinant of the spacetime metric and is the curvature scalar. The matter fields are denoted collectively by . The stress-energy-momentum tensor is obtained from the matter action by

Let denote the velocity of an observer and let denote a set of spatial, orthonormal basis vectors. We have the normalization conditions , , and . The 4-momentum density and the 4-momentum flux as seen by the observer are:

(1)

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

(2)

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

(3)

Note that . The 3-momentum flux is also identified as the spatial stress.