Raychoudhuri's equation in abstract notation

1. Nov 5, 2014

center o bass

In GR an important, purely geometric equation is called Raychoudhuri's equation governing the behaviour of geodesic congruences which states that
$$\frac{d\theta}{d\tau} = - \frac{1}3 \theta^2 - \sigma^{ab}\sigma_{ab} + \omega^{ab}\omega_{ab} -R_{ab} u^a u^a$$
where $R_{ab}$ is the ricci tensor, $\theta = \nabla_a u^a$ $\omega_{ab}$ and $\sigma_{ab}$ respectively are the trace, the antisymmetric and the symmetric bart of $\nabla_a u_b$ and u is the tangent vector field to the congruence. In other words this equation governs the spread in the geodesic congruence as a result of curvature.

Since this is a purely geometric result I wondered what this equation is called in the differential geometry literature, and I wondered if there were a formulation of this result that did not use index notation, but rather abstract notation.

2. Nov 10, 2014