Stellar pressure-energy tensor (${\Pi }_{ij}=\int \rho {\sigma }_{ij}{d}^{3}x$)

This tensor integrates the local pressure tensor, where ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$ is the velocity covariance. Half its trace is the random kinetic energy. It enters the tensor virial theorem with coefficient one.