Suppose we have a general timelike congruence of curves with tangent vector field ##V##, then the standard decomposition of the covariant derivative in index form (see e.g. Hawking and Ellis' "Large scale structure of space and time" equation 4.17) is given by(adsbygoogle = window.adsbygoogle || []).push({});

$$V_{a;b} = \omega_{ab} + \sigma_{ab} + \frac{1}{3} \theta h_{ab} - \dot U_a U_b$$

where ##\omega_{ab}, \ \sigma_{ab}, \ \theta, \ \dot U_a## respectively represent the rotation, shear, expansion and acceleration of the congruence.

QUESTION: Is there a decomposition analogous to this but in an index-free language? If so I would very much appreciate a reference.

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# Index free decomposition of derivative timelike congruence

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