Mathematical Statement
Given a timelike unit vector field (which can be interpreted as a family or congruence of nonintersecting world lines, not necessarily geodesics), Raychaudhuri's equation can be written
where
are (non-negative) quadratic invariants of the shear tensor
and the vorticity tensor
respectively. Here,
is the expansion tensor, is its trace, called the expansion scalar, and
is the projection tensor onto the hyperplanes orthogonal to . Also, dot denotes differentiation with respect to proper time counted along the world lines in the congruence. Finally, the trace of the tidal tensor can also be written
This quantity is sometimes called the Raychaudhuri scalar.
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