Hi, I derived Raychaudhuri Equations in both (- + + +) and (+ - - -) sign conventions from metric. In Robert Wald and Sean Carroll books, (- + + +) sign convention and I derived correctly in that Sign Convention as given in the books. In the other convention, I am having one sign difference in Shear equation, because I changed the Sign in Projection Tensor (P(adsbygoogle = window.adsbygoogle || []).push({}); _{μν}) to save some properties on the whole derivation is based. These are: -

B_{μν}U^{μ}= 0 = B_{μν}U^{ν}

where U^{μ}is tangent vector field

Similar properties exist for shear (σ_{μν}) and rotation (ω_{μν})

To prove them, in (- + + +) convention

But in (+ - - -) conventionP_{μν}= g_{μν}+ U_{μ}U_{ν}

P_{μν}= g_{μν}- U_{μ}U_{ν}

Because of this, the Riemann part of Shear equation in (- + + +) convention is

C_{ταβμ}U^{μ}U^{ν}+ ½R_{αβ}

while in (+ - - -) comes up as

C_{ταβμ}U^{μ}U^{ν}- ½R_{αβ}

where R_{αβ}is spatially projected trace-less part of Ricci. I am pretty sure about the reason why the sign change is occurring (that is the change of sign in Projection Tensor in which makes it look correct thing to happen) but another possibility is that maybe Weyl tensor (C_{ταβμ}) has different signs under different conventions and I don't know about that. Please Help!

Also what are "spatial" tensors?

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# A Raychaudhuri Shear Equation under different Sign Convention

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