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.
- \(T^{00}\) is energy density.
- \(T^{0i}\) is energy flux.
- \(T^{i0}\) is momentum density.
- \(T^{ij}\) is momentum flux. In this sense \(T{ii}\) has the meaning of pressure.
For perfect fluid, the definition that satisfies the requirements is