Raychaudhuri Equation - Mathematical Statement

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.