For a congruence of observers (or particles or photons) with four velocity
, the kinematic invariants fully describe
their relative motion. Consider those that, in a particular moment
, occupy the surface of an infinitesimally small sphere.
Now, consider the deformation of that surface at a later moment
. The volume change
is called
expansion. The shear tensor
measures the ellipsoidal distortion of this sphere, and the vorticity tensor
describes
its rotation (i.e., three independent rotations around three perpendicular axes). Expansion, shear, and vorticity are determined
by the tensor
in the following way:
,
, and
. Here
is the projection tensor. The acceleration
is also considered
a kinematic invariant.