Raychaudhuri equation for shear

[julian]

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?

 

[bloby]

Is the Riemann tensor symmetric in ba?

 

[julian]

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.

 

[bloby]

Right

 

[atrahasis]

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.