For a congruence of observers (or particles or photons) with four velocity Uμ, the kinematic invariants fully describe their relative motion. Consider those that, in a particular moment s0, occupy the surface of an infinitesimally small sphere. Now, consider the deformation of that surface at a later moment s0 + ds. The volume change dV∕ds = Θ is called expansion. The shear tensor σμν measures the ellipsoidal distortion of this sphere, and the vorticity tensor ωμν describes its rotation (i.e., three independent rotations around three perpendicular axes). Expansion, shear, and vorticity are determined by the tensor X μν = ∇ μUν in the following way: Θ = (1∕3)Xμμ, σ μν = hαμhβν(1∕2)(X αβ +X βα)− Θh μν, and ωμν = hαμhβν((1∕2)(X αβ − X βα). Here hαμ = δαμ + U αUμ is the projection tensor. The acceleration aμ is also considered a kinematic invariant.