Derivations of Einstein field equations

[cr7einstein]

Hello Everyone,
I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $$c^4$$ in the denominator. the 8πG term can be obtained from Poisson's equation, but how does c^4 pop up? Most of the books just derive it with $$8\pi G$$, and say that in units where c is not equal to 1, you get $$8πG/c^4$$, even though there is no mention of an explicit assumption that c=1. They kind of just bring it up suddenly, and there is no prior need to assume c=1 anyway. I don't want to do it with Einstein-Hilbert action, but the standard $$G_{\mu\nu}=kT_{\mu\nu}$$ approach, where I need to show that $$k=\frac{8\pi G}{c^4}$$.
Thanks in advance!

 

[PeterDonis]

the 8πG term can be obtained from Poisson's equation, but how does c^4 pop up?

It's just correcting the units. Normally $T_{\mu \nu}$ is in units of energy density, and $G_{\mu \nu}$ is in units of curvature, i.e., inverse length squared. The conversion factor between these units is $G / c^4$. (The $8 \pi$ part comes from matching the prediction for static, weak-field gravity with Poisson's equation, as you say.) If you choose to use different units, that will change the conversion factor; for example, if you use units of mass density instead of energy density for $T_{\mu \nu}$, then the factor will be $G / c^2$.

 

[cr7einstein]

So are both (energy density and mass density) equally valid?

 

[PeterDonis]

So are both (energy density and mass density) equally valid?

Sure, it's just a matter of which you prefer, or which is more suitable for whatever problem you're working on.