Anisotropic stress (${\pi }^{ij}=T^{\hat{i}\hat{j}}-\frac{1}{3}{\delta }^{ij}{\delta }_{kl}{T}^{\hat{k}\hat{l}}$)

In a measuring observer's orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor. Using the positive Euclidean spatial metric ${\delta }_{ij}$, defineThe trace here is the spatial Euclidean trace. Writing a Lorentzian mixed trace $T^{\hat{k}}{}_{\hat{k}}$ without its signature convention can reverse its sign in signature $(+---)$.

An isotropic perfect fluid in general relativity has $T^{\hat{i}\hat{j}}=P{\delta }^{ij}$ in its rest frame and therefore zero anisotropic stress. Different directional pressures or an angular quadrupole of particle momenta can give nonzero ${\pi }^{ij}$. In particular, the photon quadrupole determines the linear photon contribution.

Some linear cosmological perturbation theory conventions use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{\hat{i}\hat{j}}=P{\delta }^{ij}-{\Pi }^{ij}$. This sign must accompany a photon Euler equation; trace removal and the choice of sign are separate conventions.