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In modern parlance we talk about two different speeds of light in general relativity. We distinguish between the local speed of light, which is always $c$, and the coordinate speed of light, which can have any value from zero upwards without limit.

In Einstein's early papers he simply refers to the speed of light and by this he means what we would today call the coordinate speed of light. I'm curious to know if at any point in his published work he started making the same distinction that we do today, and if so whether he used the modern terminology.

I've attempted to investigate this using the Einstein Digital Archive, but with 5,000 documents it's proving a bit impenetrable. I wonder if someone more familiar with Einstein's key papers would be able to answer this more easily.

If it is of interest there is a related question in the Physics Stack Exchange: GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light

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    $\begingroup$ IMO the first paragraph presents a false assumption, that a certain distinction is important and widely recognized. The way a relativist would normally look at this is that coordinates and coordinate velocities are of little interest and have no direct physical interpretation. Since Einstein designed GR to have general covariance, I think he'd see it the same way. A modern relativist would seldom have any need to write the symbol c, because they would work in units where c=1, and they would be conscious of the fact that calling it the "speed of light" is basically a historical mistake. $\endgroup$
    – user466
    Dec 13, 2016 at 23:56

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I think the concept of "local speed of light" is (deliberately perhaps) confused in today's literature. "Local" should mean "as measured by a freely falling observer", and some authors do define it in this way, but if the definition is not explicit, "local" becomes misleading.

The concept of "coordinate speed of light" is also confused. "Coordinate" should mean "as measured by an observer at rest relative to the source of gravity" but authors introduce some measuring of the speed of light from a distance, which is absurd, to say the least.

There is a third confusion. Einstein's 1911 formula for the coordinate speed of light, c'=c(1+φ/c^2), is similar to the formula predicted by Newton's emission theory of light, and one sometimes teaches that they are identical. Actually Newton's formula is different: c'=c(1-φ/c^2). According to Einstein, the speed of light DECREASES as photons approach the source of gravity while according to Newton it INCREASES, like the speed of ordinary falling bodies.

In the final version of general relativity Einstein changed his formula - it became c'=c(1+2φ/c^2) - but the change was not essential. The sign remained "plus" and the speed of light continued to DECREASE as the source of gravity was approached.

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    $\begingroup$ Hi Pentcho. I don't see how this answers my question. $\endgroup$
    – John Rennie
    Dec 13, 2016 at 7:22
  • $\begingroup$ It does answer the question indirectly. "Speed of light as measured by a freely falling observer" was not something Einstein would have paid special attention to - this speed was obviously constant, c, to him. The problem is only existent nowadays because "speed of light as measured by a freely falling observer" is replaced by the misleading "local speed of light", and the definition of "coordinate speed of light", too, is often confused. $\endgroup$
    – Pentcho Valev
    Dec 13, 2016 at 8:34
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    $\begingroup$ I asked if Einstein ever explicitly made the distinction in his published work. You haven't answered this question. Unless you're claiming the answer is "no". $\endgroup$
    – John Rennie
    Dec 13, 2016 at 8:42

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