Raychaudhuri Shear Equation in Different Sign Conventions

[sharmax]

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_μν) to save some properties on the whole derivation is based. These are: -

B_μνU^μ = 0 = B_μνU^ν​

where U^μ is tangent vector field
Similar properties exist for shear (σ_μν) and rotation (ω_μν)

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

P_μν = g_μν + U_μU_ν​

But in (+ - - -) convention

P_μν = g_μν - U_μU_ν​

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

C_ταβμU^μU^ν + ½R_αβ​

while in (+ - - -) comes up as

C_ταβμU^μU^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?

 

[Ambitwistor]

Hi there,

Thank you for sharing your findings on the Raychaudhuri Equations in different sign conventions. It is interesting to see that the change in sign of the Projection Tensor can have an impact on the Shear equation.

To address your question about the Weyl tensor, it does in fact have different signs under different conventions. In the (- + + +) convention, the Weyl tensor has the form Cταβμ = Rταβμ - ½(Rαβgμτ - Rαμgβτ + Rβμgατ), while in the (+ - - -) convention, it has the form Cταβμ = Rταβμ + ½(Rαβgμτ - Rαμgβτ + Rβμgατ). This difference in sign is due to the different sign conventions used for the Riemann tensor.

As for "spatial" tensors, these are tensors that are defined with respect to a specific spatial coordinate system. In general relativity, the spacetime is described by a 4-dimensional manifold, with 3 dimensions representing space and 1 dimension representing time. A spatial tensor is one that is defined only in the 3 spatial dimensions, and does not involve the time dimension. This is in contrast to a spacetime tensor, which is defined in all 4 dimensions.

I hope this helps to clarify your questions. Keep up the good work in your research!