# Raychaudhuri equation for shear

1. ### julian

402
Following Wald I have nearly got the right answer out for time derivative for shear...what I am left with is showing that $R_{cbad} V^c V^d + h_{ab} R_{cd} V^c V^d / 3$ (which is obviously symmetric and trace-free) can be written as $C_{cbad} V^c V^d + \tilde{R}_{ab} / 2$ where $\tilde{R}_{ab}$ is the spatial, trace-free part of $R_{ab}$, i.e. $h_{ac} h_{bd} R^{cd} - h_{ab} h_{cd} R^{cd} / 3$.

Is there an easy way of proving this?

Last edited: Mar 6, 2012
2. ### bloby

112
Is the Riemann tensor symmetric in ba?

3. ### julian

402
It is when contrcted by $V^c V^d$ cus that means you can take it to be symmetric over c and d, this plus the usual symmetries of $R_{cbad}$ makes $R_{cbda} V^c V^d$ symmetric over a and b.

Last edited: Mar 7, 2012

112
Right

5. ### atrahasis

11
You have to replace the Riemann by it's decomposition into Weyl tensor ... which is given by the eq. 3.2.28 in Wald's book.

Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?