About an Einstein equation

Question

This is a question about an historical theory of gravitation, studied by Einstein quite a bit before he settled on General Relativity. At that time, Einstein did not know that gravity was a consequence of curved space-time. He identified the variations of gravity with the variations of light speed in a gravitational field.

In March 1912, Einstein postulated a first equation for static gravitational field, derived from the Poisson equation $$\Delta c = kc\rho \tag{1}~,$$ where $c$ is light speed, $\rho$ is mass density and $\Delta$ is Laplacian.

Two weeks later, he modified this equation by adding a nonlinear term to satisfy energy-momentum conservation : $$\Delta c = k\big(c\rho+\frac{1}{2kc} (\nabla c)^2\big)~. \tag{2}$$ Einstein's argument is the following:

The force per unit volume in terms of the mass density $\rho$ is $f_a$ $= \rho \nabla c$. Substituting for $\rho$ with $\frac{\Delta c}{kc}$ [equation (1)], we find $$f_a = \frac{\Delta c}{kc} \nabla c~.$$

This equation must be expressible as a total divergence (momentum conservation) otherwise the net force will not be zero (assuming $c$ is constant at infinity). Einstein says:

"In a straightforward calculation, the equation (1) must be replaced by equation (2)."

I never found the straightforward calculation. That's something that's actually hard for me!

addendum
The solution given and explained by @Gluoncito (see below) answers perfectly my question. However, it is likely that it is not the demonstration of Einstein for at least one reason : It is not a straightforward calculation.
Historically, Abraham, a german physicist, was the first to generalize the Poisson equation by adding a term for the energy density of the gravitational field (coming from $E=mc^2$). He published a paper in january 1912 containing a static field equation with the term : $\frac{c^2}{\gamma}(\nabla c)^2 $ different but not far away from the Einstein term. After the publication of Einstein, Abraham claimed That Einstein copied his equation. I believe Einstein was at least inspired by Abraham. To what extent, I don't know.

Answer 1

There is a derivation of the equations above given by Giulini (may be more pedagogical?), You can look at it at :

http://ae100prg.mff.cuni.cz/presentations/Giulini_Domenico.pdf

As you will see he arrives at the same equation (2) assuming the "variable speed of light" is actually proportional to the gravitational potential, as I first assumed, (no need of variable speed of light). Please note that in general relativity the speed of light is constant in local charts, and that`s enough for the theory.

ok, as asked by the operators I copy the main parts of the demonstration: the field equation for the gravitational field in Newtonian mechanics is: $$\Delta \phi = 4πG \rho$$,

the Newtonian force per unit volume (mass density x acceleration) is: $$f = −\rho \nabla \phi$$. Now, the work done against gravity to assemble a piece of matter $\delta \rho$ (along an incremental change $\delta \xi$ along the flow) is: $$\delta A=-\int \delta \vec{\xi}.\vec{f} =\int \phi \delta \rho$$,

a small change in the density of matter produce a change in the gravitational potential: $$\Delta \delta \phi = 4πG \delta \rho$$ with this the work can be written: $$\delta A=\int \phi \delta \rho=\delta ( \frac{-1}{8 \pi G} \int (\nabla \phi)^2)$$ where the equality is given integrating by parts the lhs. Thus it is possible to find the energy density of the gravitational field as: $$\epsilon=\frac{-1}{8 \pi G} (\nabla \phi)^2$$ Now, the important point is that any source of gravitational field must be compatible with the principle $E=m c^2$. Thus, (I jump to eq. 9) the mass equivalence of the gravitational field is: $$\delta M_g=\frac{1}{4 \pi G} \int \Delta \delta \phi$$ * Newtonian gravity fail this principle since the rhs is zero in absence of matter. So Einstein added the energy of the gravitational field (the $\epsilon$ calculated before) as a source. $$\Delta \phi=4 \pi G (\rho-\frac{1}{8 \pi G c^2} (\nabla \phi)^2)$$ Then computes the mass term (a complicated integral I`m a bit lost here, eq. 11), and redefines for consistency with the work $\delta A$ the field: $$\phi \rightarrow \Phi=c^2 exp(\phi/c^2)$$ with this definition the equation becames: $$\Delta \Phi=\frac{4 \pi G}{c^2} (\Phi \rho +\frac{c^2}{8 \pi G \Phi} (\nabla \Phi)^2)$$ The redistribution of the c-factors is due to the redefinition of $\phi$. The $\Phi$ multiplying $\rho$ cancels when you replace everything by $\Phi$ and gets the original eqs. * above.

Well, this was the derivation of Giulini, not very pedagogical because of eq. 11. If I understand it I tell you. It could be better to read the original Einstein paper but I have it only in German. I'm sure Einstein was clearer at the end...