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Raychaudhuri equation for shear

by julian
Tags: equation, raychaudhuri, shear
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julian
#1
Mar6-12, 04:55 AM
PF Gold
P: 316
Following Wald I have nearly got the right answer out for time derivative for shear...what I am left with is showing that [itex]R_{cbad} V^c V^d + h_{ab} R_{cd} V^c V^d / 3[/itex] (which is obviously symmetric and trace-free) can be written as [itex]C_{cbad} V^c V^d + \tilde{R}_{ab} / 2[/itex] where [itex]\tilde{R}_{ab}[/itex] is the spatial, trace-free part of [itex]R_{ab}[/itex], i.e. [itex]h_{ac} h_{bd} R^{cd} - h_{ab} h_{cd} R^{cd} / 3[/itex].

Is there an easy way of proving this?
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bloby
#2
Mar6-12, 05:07 PM
P: 112
Is the Riemann tensor symmetric in ba?
julian
#3
Mar7-12, 04:26 AM
PF Gold
P: 316
It is when contrcted by [itex]V^c V^d[/itex] cus that means you can take it to be symmetric over c and d, this plus the usual symmetries of [itex]R_{cbad}[/itex] makes [itex]R_{cbda} V^c V^d[/itex] symmetric over a and b.

bloby
#4
Mar8-12, 06:36 AM
P: 112
Raychaudhuri equation for shear

Right
atrahasis
#5
Mar8-12, 10:01 PM
P: 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.


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