Einstein's Field Equations differ by a factor of $frac{1}{c^4}$. Why is that?

Question

I am trying to familiarize myself with General Theory of Relativity. I am by no means an expert in the field, and I am doing this as my own hobby. At any rate, I have come across Einstein's Field Equations. I understand the derivations of them are not simple and it consists of ten independent components. I have found formulas are different depending on the source. For instance, here and here, the field equation is given as
$$R_{mu v} - frac{1}{2}Rg_{mu v} = 8 pi G T_{mu v}.$$
In other sources, like the one on Wikipedia, it is given as
$$R_{mu v} - frac{1}{2}Rg_{mu v} = frac{8 pi G}{c^4} T_{mu v}.$$
So it seems they differ by a factor of $frac{1}{c^4}$. I was wondering what the explanation is for this difference.
Also when people refer to Einstein's Law of Gravity, is this the equation they have in mind? For example, when people talk about Newton's Law of Gravitation, the equation $F=G frac{m_1m_2}{r^2}$ comes to mind. In case of general relativity, is there a single equation that is associated with it or are there multiple equations and there is no single formula?

FGSUZ · Accepted Answer

Well, many physicists like working with a different convention.
They spent so many years teaching students to "always use the international system". Otherwise, formulae will not match, and that's too bad!
But then formulas start becoming quite complicated, and they say: "well, let's assume we're so good that we never commit a mistake. So we have to presume all formulae are correct. If so, I can get rid of all $c$ appearing in my formula, because everybody else will know what I mean".
So yes, they just don't write c's. They just get rid of them. It is affordable, because "anyone who looks at this equation sees that units do not match, so they are supposed to know what to do: people only have to add as many c's as they need in order to fix units". So it is presumed that you have to put as many $c$ factors as you need, wherever you need inside the formula, so that units finally fix.
This is equivalent as saying "I set $c=1$, and that's the same as saying "I'm not using the international system anymoer, I'm using different lenght and time units, such that $c=1$.
So yes, the only reason for this change of units is getting rid of tedious $c$ factors. It is common to do the same with $G$ and $hbar$.
This habit is really extended, but I certainly think it is quite bad for people who are not experts in that field. Even more, I think it is also bad for the writers themselves; but if you learn physics to a certain level, you must know that the vast majority of articles are written this way.

J. Murray · Answer

It's very common in GR to work in natural units where $c=1$. Some people take it a step further and set $G=1$ as well. With a bit of practice, it's easy to use dimensional analysis to re-insert factors of $c$ and $G$ anytime you'd like to use e.g. SI units.

Answered by J. Murray on May 27, 2021

joseph h · Answer

I was wondering what the explanation is for this difference.

Your first equation is the one where the speed of light $c$ is taken to be $1$. This is just a handy convention. See natural units for more.

Also when people refer to Einstein's Law of Gravity, is this the equation they have in mind?

When people talk about this, they probably refer to the fact that in Einstenian gravity, the force of gravity is a result of spacetime curvature. That is, gravitation is a result of spacetime being curved by matter and energy. In the Newtonian view, there is the view that gravity is caused by mass attracting mass. Of course this idea is superseded by Einstein's gravitation, which is the most accurate description we have for gravity.

In case of general relativity, is there a single equation that is associated with it or are there multiple equations and there is no single formula?

The Einstein field equations above are tensor equations, which are actually a set of 10 (distinct) differential equations (see the wiki article you cited and the links inside). The Einstein field equations equate local spacetime curvature with the local mass/energy/momentum within that spacetime.

Einstein's Field Equations differ by a factor of $frac{1}{c^4}$. Why is that?

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