# 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

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