= Anisotropic stress
{title2=$\pi^{ij}=T^{\hat i\hat j}-\tfrac13\delta^{ij}\delta_{kl}T^{\hat k\hat l}$}
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 $\delta_{ij}$, define
$$
P=\frac13\delta_{kl}T^{\hat k\hat l},\qquad
\pi^{ij}=T^{\hat i\hat j}-P\delta^{ij},\qquad
\delta_{ij}\pi^{ij}=0.
$$
The 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.
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