# Einstein Equation Quick sign question

Some sources seem to have: ##G_{uv}=8\pi G T_{uv} ##

Whereas others have: ##G_{uv}=-8\pi G T_{uv} ##

I thought that it may have been covered by how ##G_{uv}## is defined on the sources, but in both cases it is given as ## G_{uv}=R_{uv} - \frac{1}{2}g_{uv}R ##

I'm confused.
Thanks.

Matterwave
Gold Member
Have you checked how ##R^a_{~~bcd}##, ##R_{ab}## etc., are defined in the different sources?

There are a lot of sign conventions in GR, for all the different tensors, and even the metric signature, etc., so it's generally not so surprising that an equation would differ by a negative sign between 2 sources. Finding out exactly where that negative sign came from might take a bit of effort.

This Wikipedia article gives a good brief overview of the sign conventions that affect the form of the Einstein Field Equation:

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

Thanks, so it says due to differently defining the sign of the Ricci Tensor - but isn't this defined as a contraction of the Riemannian tensor, and surely the definition of this contraction would not change, so is it due to a different sign in the Riemannian tensor? Thanks.

PeterDonis
Mentor
2020 Award
When they talk about a convention for the Ricci tensor, they are talking about whether the Ricci tensor is defined as the contraction of the Riemann tensor, or as minus the contraction of the Riemann tensor. The contraction of the Riemann tensor is fixed, as you say.

When they talk about a convention for the Ricci tensor, they are talking about whether the Ricci tensor is defined as the contraction of the Riemann tensor, or as minus the contraction of the Riemann tensor. The contraction of the Riemann tensor is fixed, as you say.

Thanks, both sources I'm looking at have defined it to be positive contraction, but one uses a (+,-,-,-) signature and one (-,+,+,+) so this explains the sign difference.

PeterDonis
Mentor
2020 Award
both sources I'm looking at have defined it to be positive contraction, but one uses a (+,-,-,-) signature and one (-,+,+,+) so this explains the sign difference.

I'm not sure it does, unfortunately. The metric signature sign convention tells you whether timelike squared intervals are considered positive or negative. But changing that convention does not change the form of the Einstein Field Equation.

The metric signature sign convention is [S1] in the notation of the Wikipedia article I linked to; the EFE sign convention is [S3], and the Ricci tensor sign convention is the product of [S2] and [S3]. So if both sources define the Ricci tensor to be the positive contraction of the Riemann tensor, then they both have [S2] x [S3] = 1. That means one of your sources, which has [S3] = 1 (no minus sign in the EFE), must also have [S2] = 1 (Riemann tensor defined in terms of the Christoffel symbols as given in MTW), while the other source, which has [S3] = -1 (minus sign in the EFE), must also have [S2] = -1 (Riemann tensor defined in terms of Christoffel symbols with opposite sign from MTW). Changing [S1] does not change any of this.

I'm not sure it does, unfortunately. The metric signature sign convention tells you whether timelike squared intervals are considered positive or negative. But changing that convention does not change the form of the Einstein Field Equation.

The metric signature sign convention is [S1] in the notation of the Wikipedia article I linked to; the EFE sign convention is [S3], and the Ricci tensor sign convention is the product of [S2] and [S3]. So if both sources define the Ricci tensor to be the positive contraction of the Riemann tensor, then they both have [S2] x [S3] = 1. That means one of your sources, which has [S3] = 1 (no minus sign in the EFE), must also have [S2] = 1 (Riemann tensor defined in terms of the Christoffel symbols as given in MTW), while the other source, which has [S3] = -1 (minus sign in the EFE), must also have [S2] = -1 (Riemann tensor defined in terms of Christoffel symbols with opposite sign from MTW). Changing [S1] does not change any of this.

Okay thanks. But I dont understand how ##R_{uv} ## can depend on the sign of the EFE equation, I was looking to order these arguments differently, if possible, to the sign convention of EFE depending on other things, not the Ricci tensor - I think this stems from what I've been introduced to in GR is deriving a metric (from certain symmetries we desire it to hav) and then solving for ##R_{uv}## with the metric at hand, and finally plugging into Einstein equation, as a pose to solving Einstein's equation for the metric, in which case it , as I understand, would then make sense to have the Ricci tensor sign convention being the 'dependent' one.

PeterDonis
Mentor
2020 Award
I dont understand how ##R_{uv}## can depend on the sign of the EFE equation

The sign convention relationships don't imply any "dependence" in the sense of which one gets determined first. They're just relationships that have to exist between the sign conventions we adopt. Changing sign conventions doesn't change anything about the physics; it just changes how the same physics is represented in the math. So picking sign convention [S2] or [S3] doesn't mean deciding how the Riemann tensor or the Ricci tensor works physically, or what kinds of solutions exist to the field equation; it just means deciding how the same physics is going to appear when we write the equations down.