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Raychoudhuri's equation in abstract notation

  1. Nov 5, 2014 #1
    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. jcsd
  3. Nov 10, 2014 #2
    Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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