Static perfect-fluid rest-frame criterion ($T^{0i}=0,\rho c^2+p\ne 0⟹u^i=0$)

In a diagonal static metric, an ordinary perfect fluid in general relativity with a timelike four-velocity has $T^{0i}=(\rho +p/c^2)u^0{u}^i$. Since $u^0\ne 0$, 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.