= Static perfect-fluid rest-frame criterion
{title2=$T^{0i}=0,\quad\rho c^2+p\ne0\ \Longrightarrow\ u^i=0$}
In a diagonal static metric, an ordinary <perfect fluid in general relativity> with a timelike <four-velocity> has $T^{0i}=(\rho+p/c^2)u^0u^i$. Since $u^0\ne0$, vanishing mixed time-space components imply a static fluid if the enthalpy is nonzero. For vacuum-like matter with $p=-\rho c^2$, the tensor is $p g^{ab}$ independently of the velocity; its rest frame cannot be recovered from that tensor.
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