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A Raychaudhuri Shear Equation under different Sign Convention

  1. Apr 4, 2016 #1
    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 (Pμν) to save some properties on the whole derivation is based. These are: -

    BμνUμ = 0 = BμνUν

    where Uμ is tangent vector field
    Similar properties exist for shear (σμν) and rotation (ωμν)

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

    Pμν = gμν + UμUν
    But in (+ - - -) convention

    Pμν = gμν - UμUν

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

    CταβμUμUν + ½Rαβ

    while in (+ - - -) comes up as

    CταβμUμUν - ½Rαβ

    where Rαβ 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 (Cταβμ) has different signs under different conventions and I don't know about that. Please Help!

    Also what are "spatial" tensors?
     
  2. jcsd
  3. Apr 9, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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