Raychaudhuri Equation - Focusing Theorem

Focusing Theorem

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, so even if the expansion scalar is initially positive (if our small ball of dust is initially increasing in volume), eventually it must become negative (our ball of dust must recollapse).

Indeed, in this situation we have

Integrating this inequality with respect to proper time gives

If the initial value of the expansion scalar is negative, this means that our geodesics must converge in a caustic ( goes to minus infinity) within a proper time of at most after the measurement of the initial value of the expansion scalar. This need not signal an encounter with a curvature singularity, but it does signal a breakdown in our mathematical description of the motion of the dust.