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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.

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  • $\begingroup$ Hi user27423, and welcome to Physics Stackexchange! As we use Latex notation for mathematical symbols here, implemented via Mathjax, I edited your question to make it format better. (For an overview of Mathjax see this guide.) You should make sure I didn't inadvertently change the meaning of your post. $\endgroup$ – user10851 Jul 23 '13 at 22:12
  • $\begingroup$ Is there a typo in the equations? $\nabla c=0$, since c is the speed of light (a constant). Could you please give the bibliographic reference/s. thankx. $\endgroup$ – Gluoncito Jul 23 '13 at 23:44
  • $\begingroup$ Dear Gluoncito - the question is about Einstein's earlier theories, before GR came into being. And even in GR, if your co-ordinates are not locally inertial, you'll measure different values of $c$. Before you say "Duh - that's just a co-ordinate system change", see physics.stackexchange.com/q/33816/26076 - the Rindler metric defines what could be construed as a "plausibly real laboratory" - i.e. real clocks and measurement instruments have nonzero extent and the load-bearing matter within them can impose accelerations of parts of your laboratory relative to the (spatially varying) ... $\endgroup$ – WetSavannaAnimal Jul 24 '13 at 0:56
  • $\begingroup$ .... freefalling frame. For a fuller description, see the description of "spaghettification" in the section "Where does force come into it?" in my answer physics.stackexchange.com/a/71831/26076 (skip the other sections - oh my, I do tend to bang on a bit when having fun, so apologies for my wordiness). Anyhow, the OP's model (which is altogether unwonted to me, but see en.wikipedia.org/wiki/Variable_speed_of_light for some history) is Einstein a long way off publishing GR, when he still thought variable lightspeed theories might be THE theory. $\endgroup$ – WetSavannaAnimal Jul 24 '13 at 0:59
  • $\begingroup$ Dear user27423 (I would like to address you with a real-sentient being name!) - welcome to physics SE. I added an opening sentence to your question to make sure people understand that this is an historical model considered by Einstein before GR. One of the answerers clearly missed this point (also, I nearly did too, because I am pretty ignorant of history). $\endgroup$ – WetSavannaAnimal Jul 24 '13 at 1:08
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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...

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  • $\begingroup$ Can you add some parts of the proof. Link-only answers are discouraged. $\endgroup$ – Ali Jul 24 '13 at 8:52
  • $\begingroup$ @Giuoncito Thank you very much for this link. It is exactly what i looked for. Cannot be better. I fully agree with you, we don't need speed of light but I stuck on Einstein view. 3 months later, he gave up this approach for the metric tensor $\endgroup$ – user27423 Jul 24 '13 at 8:59
  • $\begingroup$ @user27423: I found an arxiv:1306.5966v1 from Giulini in which he explains in more detail his derivation. In particular is now clear to me his derivation of the equation 11 of his presentation, that I didn't understand before. $\endgroup$ – Gluoncito Jul 26 '13 at 19:45
  • $\begingroup$ Giulini also talks about the theory of Abraham a rival theory , published in the same year: "Einstein’s heuristics indicated clearly that Special Relativity had to be abandoned, in contrast to the attempts by Max Abraham (1875-1922), who published a rival theory [2][1] that was superficially based on Poincare invariant equations (but violated Special Relativity in abandoning the condition that the four-velocities of particles had constant Minkowski square). In passing I remark that Einstein’s reply [11]" $\endgroup$ – Gluoncito Jul 26 '13 at 19:46
  • $\begingroup$ Saddly, I didn't find the English translations of these papers, and Hawking's compilation of Einstein's papers do not contain the paper in question. $\endgroup$ – Gluoncito Jul 26 '13 at 19:49

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