Sign conventions in general relativity

[Nabigh R]

Hi guys... I was wondering if anyone have a sort of a summary of sign conventions in general relativity books. By convention I mean the definition of Riemann tensor, Ricci tensor and signs of stress-energy tensor and signs of einstein field equations for a given sign of metric tensor... I heard that there is a table that shows something like that in Misner, Thorne, Wheeler... can somebody please upload it as I don't have access to that book.

Thanks.

 

[UltrafastPED]

See http://en.wikipedia.org/wiki/Sign_convention

And http://en.wikipedia.org/wiki/Einstein_field_equations#Sign_convention

 

[Nabigh R]

Thanks ultrafast... I have already seen those and in fact it's because of the latter link only I posted this question... it's really hard to keep track of changes to the field equations with different conventions... I mean in one, the constant on right hand side is negative, other positive, even the form of stress-energy tensor of perfect fluid say, depends on sign convention and so forth. So I was wondering if anyone have a summary of how everything looks like with different conventions.

 

[robphy]

Someone had scanned in this famous chart:

http://itp.epfl.ch/webdav/site/itp/users/166340/public/Misner%28SignConvention%29.pdf

which is the origin of part of that wikipedia entry.

 

[Nabigh R]

Thanks a lot robphy. That really helps.

 

[Nabigh R]

As an aside to the original question: the Christoffel symbols doesn't depend on the sign of metric, right?
What about other tensors found using them?

 

[waitedavid137]

As an aside to the original question: the Christoffel symbols doesn't depend on the sign of metric, right?
What about other tensors found using them?

For example, the Riemann tensor is defined in terms of them, but is anti-symmetric in its last two indeces and as such defining the Ricci-tensor in terms of contraction over the 1st and 3rd Vs 1st and 4th indeces yields a conventional sign difference relating the Einstein tensor to the stress-energy tensor. This is why you sometimes see a minus sign in front of the stress-energy tensor side even though its usually a plus.

 

[Nabigh R]

For example, the Riemann tensor is defined in terms of them, but is anti-symmetric in its last two indeces and as such defining the Ricci-tensor in terms of contraction over the 1st and 3rd Vs 1st and 4th indeces yields a conventional sign difference relating the Einstein tensor to the stress-energy tensor. This is why you sometimes see a minus sign in front of the stress-energy tensor side even though its usually a plus.

Yeah David, I think even MAXIMA defines Ricci tensor in terms of contraction over 1st and 4th indices since it defines the Riemann tensor as in Weinberg (1972), but the components of Ricci tensor have opposite signs to the ones given by Weinberg.