# Different forms of the Einstein field equation

I am working my way through the wonderfully written introduction "General relativity for mathematicians" by Sachs & Wu. I am indeed a mathematics student and find this book to be well suited to the biases that entails. However there is one detail I have observed in which it differs from all expositions of general relativity I have seen and this is what I would like to get clarified.

The Einstein field equations are stated there in the form $$\mathbf G =\mathbf T + \mathbf E$$ where $\mathbf G = \mathbf{Ric} - \frac 12 \mathbf{g} S$ is the Einstein tensor, $\mathbf T$ is the stress-energy tensor corresponding to matter (and hence depending on the particular matter model chosen) and $\mathbf E$ is the stress-energy tensor induced by an electromagnetic field $\mathbf F$ as $$\widetilde{\mathbf E} = \frac{1}{4\pi}\left(\frac 14\operatorname{tr}(\widetilde{\mathbf F}^2)I-\widetilde{\mathbf F}^2\right)$$ where I am keeping to the notational conventions of the book and $\widetilde {\mathbf F}$ is the $(1,1)$-tensor physically equivalent to the $(0,2)$-tensor field $\mathbf F,$ viewed point-wise as a linear map $TM\to TM.$

However in nearly all other sources I checked the equation had the form $$\mathbf{G}=8\pi \mathbf T$$ for some stress-energy field $\mathbf T,$ not necessarily corresponding only to matter. Consering this I have the following questions:

Q1. If I understand correctly, this $\mathbf T$ is just the sum $\mathbf T +\mathbf E$ above. So Sachs and Wu are just writing out this tensor in two parts to account for the contribution of two distinct soures of effects on spacetime; that of matter and that of electromagnetism. Why is that not the standard practice among physicists? Are there perhaps in general more than just these two sources of effects on spacetime?

Q2. There is no factor $8\pi$ in the Wu-Sachs version of the field equation. Why is it missing? I do not suppose this has to do with the systematic use of geometric units? Or maybe Wu and Sachs are just considering all the basic objects such as $\mathbf E$ and $\mathbf T$ scaled by $8\pi$ in comparison to the standard conventions? But that does not seem to be the case either, because I have seen the very same definition of the electromagnetic stress-energy tensor in sources which include $8\pi$ in the field equation ...

Any help in clearing up this matter will be deeply appreciated!

P.S: I hope this is not an inappropriate question for this site and if it is, I apologize in advance.

## 1 Answer

Q1: I do not know why Sachs and Wu are treating the electromagnetic contribution separately from the other matters sources, but this is indeed non-standard. I think the physics convention is more natural, since the Einstein field equations are $G=T$, and the words associated with this equation is that the presence of matter ($T$) warps spacetime ($G$).

Q2: It's fairly common in mathematical physics for factors of $2$ and $\pi$ to be dropped by Mathematicians. The factor of $8\pi G$ perhaps make the field equations a bit messy, but it's natural from a physics point of view. See, for example: http://www.preposterousuniverse.com/blog/2014/03/13/einstein-and-pi/ for a nice take on this.

• Thank you for the quick and to-the-point answer! The Einstein and pi article was also a pleasent little read. So if I understand clearly, a physicist would consider electromagnetism as somehow coming from matter and not having an independent existence? – A Rock and a Hard Place Feb 19 '15 at 3:34
• Ah, welcome! Yes, a physicist would say that the system is Einstein gravity plus whatever matter you'd like to consider. Electromagnetism is simply one of many options (and I suspect Sachs and Wu treated it differently because it's one of the more well studied fields regarding it's coupling to gravity). – Surgical Commander Feb 19 '15 at 3:37