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

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