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The spin of a small torque-free test gyroscope is represented by a spacelike vector orthogonal to its four-velocity, $s^{\mu }{u}_{\mu }=0$. Its temporal component vanishes in its own rest frame. Along a geodesic, $D{s}^{\mu }/d\tau =0$: parallel transport preserves the orthogonality and the invariant spin magnitude. For an accelerated observer the corresponding nonrotating transport is Fermi-Walker transport.

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.

For relativistic dust with nowhere-zero density, stress-energy conservation implies zero four-acceleration. If its normalized four-velocity $u$ has hypersurface orthogonality, then $u^{♭}\wedge d{u}^{♭}=0$ and ${\iota }_{u}d{u}^{♭}=0$. The interior product of the first identity gives $d{u}^{♭}=0$. The Poincaré lemma supplies $u^{♭}=-dt$ locally; spatial coordinate functions constant along the smooth flow of $u$ give $u={\partial }_t$, $g_{tt}=-1$ and $g_{ti}=0$. The resulting synchronous coordinates describe the dust until the local coordinate construction fails.

For the Schwarzschild metric with $\mu =GM/c^2$, let $k=(1-2\mu /r)\dot{t}$ and $h=r^2\dot{\varphi }$ on an equatorial timelike geodesic. The normalization of the four-velocity gives ${\dot{r}}^2+V(r)=c^2(k^2-1)$, where $V(r)=(1-2\mu /r)h^2/r^2-2\mu c^2/r$. A circular geodesic obeys $V^{\prime }(R)=0$, giving the displayed constants and requiring $R>3\mu$. Its coordinate angular velocity is ${\Omega }^2=GM/R^3$, and its proper clock rate is $d\tau /dt=\sqrt{1-3\mu /R}$. Differentiating $V$ twice at the circular solution gives stability for $R>6\mu$.

For a perfect fluid in general relativity in metric signature $(-,+,+,+)$, the weak energy condition is equivalent to $\rho \ge 0$ and $\rho +p\ge 0$. A normalized observer's four-velocity $W$ has Lorentz factor $\gamma =-u\cdot W\ge 1$ relative to the fluid, and $T_{ab}{W}^a{W}^b=\rho +(\rho +p)({\gamma }^2-1)$. The case $\gamma =1$ forces the first inequality and arbitrarily large $\gamma$ forces the second; conversely the same expression proves sufficiency.