# Einstein Equation Quick sign question

• binbagsss
In summary: G_{uv}## would be purely determined by the symmetry of the metric.The sign convention of the Einstein Field Equation does not depend on the Ricci tensor. The sign convention of the Einstein Field Equation 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

#### binbagsss

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.

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.

PeterDonis said:
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.

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.

PeterDonis said:
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.

binbagsss said:
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.

PeterDonis said:
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 don't 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.

binbagsss said:
I don't 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.

## 1. What is the Einstein equation?

The Einstein equation, also known as the Einstein field equations, is a set of ten equations that describe the relationship between the curvature of space-time and the distribution of matter and energy.

## 2. What is the significance of the Einstein equation?

The Einstein equation is significant because it forms the basis of Einstein's theory of general relativity, which revolutionized our understanding of gravity and the structure of the universe.

## 3. How is the Einstein equation derived?

The Einstein equation is derived from the principle of equivalence, which states that the effects of gravity are equivalent to the effects of acceleration. This leads to the concept of space-time curvature and the equation that describes it.

## 4. How is the Einstein equation used in modern science?

The Einstein equation is used in modern science to understand and predict the behavior of large-scale objects, such as planets, stars, and galaxies, as well as the overall structure and evolution of the universe. It is also used in the development of technologies, such as GPS, that rely on precise measurements of space-time.

## 5. Can the Einstein equation be simplified or solved?

The Einstein equation is a complex set of equations that cannot be easily simplified or solved. However, it can be used to make predictions and calculations, and it has been extensively tested and verified through experiments and observations.