The equation is important as a fundamental lemma for the Penrose–Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass–energy in general relativity, as it is in Newton's theory of gravitation.
(which can be interpreted as a family or congruence of nonintersecting world lines via the integral curve, not necessarily geodesics), Raychaudhuri's equation can be written where are (non-negative) quadratic invariants of the shear tensor and the vorticity tensor respectively.
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.
The expansion scalar measures the fractional rate at which the volume of a small ball of matter changes with respect to time as measured by a central comoving observer (and so it may take negative values).
If the derivative (with respect to proper time) of this quantity turns out to be negative along some world line (after a certain event), then any expansion of a small ball of matter (whose center of mass follows the world line in question) must be followed by recollapse.
The shear tensor measures any tendency of an initially spherical ball of matter to become distorted into an ellipsoidal shape.
The vorticity tensor measures any tendency of nearby world lines to twist about one another (if this happens, our small blob of matter is rotating, as happens to fluid elements in an ordinary fluid flow which exhibits nonzero vorticity).
This balance may be: Suppose the strong energy condition holds in some region of our spacetime, and let
be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal.
For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of the Einstein field equation (provided that these world lines are not twisting about one another, in which case the congruence would have nonzero vorticity).
Then Raychaudhuri's equation becomes Now the right hand side is always negative or zero, so the expansion scalar never increases in time.
When the vorticity is zero, then assuming the null energy condition, caustics will form before the affine parameter reaches
The event horizon is defined as the boundary of the causal past of null infinity.