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As far as I understand Wikipedia, Robert Dicke in 1957 proposed a flat spacetime with a variable speed of light such that gravity works similar to a refractive index, slowing down the speed of light. The formula given in Wikipedia is $$ n = \frac{c}{c_0} = 1 + \frac{2GM}{rc^2}$$ but one has to guess whether $c_0$ is the speed of light absent of gravity or whether it is $c$. But both versions don't seem to make sense. If $c_0$ is the constant, $c$ should not appear on the right side. With $c$ the constant, we get $c/2$ as the speed of light at $r=$Schwarzschild radius. I thought it would turn out as zero in this theory.

Should $c$ and $c_0$ exchanged? Is $r$ implicitly negative?

I cannot get hold of the original paper of Dicke (35$ to buy). What is the formula of Dicke really?

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  • $\begingroup$ If you really want I can send you the Dicke paper by email, if it is the "Gravitation Without Equivalence Principle" one, or tell you... A way to get it. But I can't private message people here, or find a email so... I can't do anything. $\endgroup$ – Vendetta Sep 26 '17 at 18:41
  • $\begingroup$ @Vendetta thanks for the offer, but in fact I got the paper send to me shortly after I posted the question. $\endgroup$ – Harald Sep 26 '17 at 19:45
  • $\begingroup$ @Harald Note to self: physics.stackexchange.com/a/77280/73067 likely contains what Dicke proposed and what should be in the Wikipedia, amounting to $c/c_0 = 1- 2GM/(c_0^2r)$. The minus sign was forgotten when someone replaced the potential $\Phi$ with $2GM/r$ and the $c^2$ comes from a Schwarzschild Radius which, in this context, should be $c_0^2$. $\endgroup$ – Harald Nov 24 '18 at 15:34
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Dicke's formula is actually

$$\epsilon=\mu=1+\frac{2GM}{r},$$

and, of course, $c=1/\sqrt{\epsilon\mu}$ is then the speed of light in that medium (eqn (4) in Dicke's article).

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