Energy Momentum Tensor

Energy momentum tensor is an important concept when dealing with continuum media.

In general, what we would like to define is a tensor that contains the energy density.

First of all, energy density obviously is not a conserved quantity. As an example, we consider a number of particles with number density $n$ and each with mass $m$. In its comoving frame, we would define the energy density as $\rho=n m$ since every single particle is stationary. When we transform to another frame, say $\bar O$ frame, $\bar\rho = \gamma^2 n m$, which indicates that this quantity is not a scalar.

So to achieve this goal of an invariant quantity, we need a tensor. Suppose its components are denoted as $T^{\alpha\beta}$, we need to find a definition that carries the following meanings.

1. $T^{00}$ is energy density.
2. $T^{0i}$ is energy flux.
3. $T^{i0}$ is momentum density.
4. $T^{ij}$ is momentum flux. In this sense $T{ii}$ has the meaning of pressure.

For perfect fluid, the definition that satisfies the requirements is

$$T^{ab} = (\rho+p) U^a U^b + p g^{ab}.$$