The first person to say "you may as well say the speed of light changes" was Einstein himself, in 1911. And it gave him half the bending
Episode 7 (the finale) rewrote gravity as flat space with a position-dependent speed of light. This bonus piece props that episode up from behind ── because the first person to make that rewriting was Einstein himself. Four years before general relativity was finished, in a paper of 1911, he wrote explicitly that the speed of light depends on the gravitational potential. And the light bending he got out of it was exactly half the right answer. The reason it was half is hiding inside that single letter \(n(x)\) from Episode 7. "Just vary the speed of light" is fine as far as it goes ── but deciding how much to vary it needs something deeper than the speed of light.
The paper is "On the influence of gravitation on the propagation of light." Fresh from the equivalence principle and the accelerating-box argument, Einstein concluded that clocks run slow where gravity is strong ── and then took the obvious next step: if the clock is slow there, then light there is slow too, as seen from far away. In symbols:
(For attraction \(\Phi=-Gm/\rho<0\), so \(n>1\) and light is slow. I write \(m\equiv Gm/c^2\).) This is exactly the refractive-index language of Episode 7 ── except look at the coefficient. It is 1.
Feeding that \(n\) into the bending of starlight grazing the Sun, the number he published was 0.83 arcsec. The value the 1919 eclipse confirmed was, as everyone knows, 1.75 arcsec.
This is not a near miss. It is precisely a half. With modern solar values:
| description | index | bending | for the Sun |
|---|---|---|---|
| 1911 version (clock only) | \(1+m/\rho\) | \(2m/b\) | 0.8758 arcsec |
| general relativity (right) | \(1+2m/\rho\) | \(4m/b\) | 1.7516 arcsec |
| ratio | — | — | 0.500000 |
The whole difference is whether the coefficient is 1 or 2. And you can see where the 2 comes from by taking Episode 7's \(n(x)\) apart: the speed of light has both a clock and a ruler in it.
Take the picture from Episode 7 ── a flat lattice with light moving at different speeds in different places ── and split it one level further. A coordinate speed of light is a coordinate distance divided by a coordinate time. So gravity gets to act on the numerator (the ruler) and the denominator (the clock), both.
\(m/\rho\) arrives twice. Once from the clock (time runs slow there) and once from the ruler (space is stretched there). One plus one is two. Einstein in 1911 only had the clock, so his coefficient stayed at 1 ── and that is the missing half.
Which makes the point sharper than Episode 7 could put it: saying "the speed of light varies" smuggles the curvature of space in. The \(B^2\) in the denominator of \(c=A/B^2\) is the stretching of space. So Episode 7's claim ── curvature and variable \(c\) describe the same thing ── is true precisely because the variable-\(c\) side is quietly carrying the curvature. Throw the curvature away and keep only the clock, and the answer halves. Nothing escaped; it was rewritten.
Words are cheap, so I integrated the rays. Put a refractive index \(n(\rho)\) on flat space and get the path from Fermat's principle ── exactly the picture of Episode 7. Here \(b\) is the impact parameter in units of \(m\).
| b | clock only | ruler only | clock + ruler | right answer 4m/b |
|---|---|---|---|---|
| 200 | 0.010079 | 0.010059 | 0.020300 | 0.020000 |
| 1000 | 0.002003 | 0.002002 | 0.004012 | 0.004000 |
| 5000 | 0.00040013 | 0.00040010 | 0.00080047 | 0.00080000 |
\(b=200\): clock only 0.503967 / clock+ruler 1.014998
\(b=1000\): clock only 0.500787 / clock+ruler 1.002956
\(b=5000\): clock only 0.500157 / clock+ruler 1.000589
The further out the ray passes (the weaker the field), the more cleanly these settle onto 0.5 and 1.0. Passing closer, the second column creeps above 1 ── because \(4m/b\) is only the leading weak-field term, and the integral is the more accurate of the two. Which raises the question of whether this index still works when the field is strong. It does.
An approximation that works in the weak field proves little. But \(n=B^2/A\) is not an approximation ── it is exact (space is stretched isotropically, i.e. conformally flat, so for light it really is equivalent to a single refractive index). So it has to pass strong-field tests too. Here is one: get the size of a black hole's shadow out of the index alone.
The minimum sits at \(\rho=1.866025389\,m\), which is areal radius \(3m\) ── the photon sphere. Twelve digits of \(\sqrt{27}\). Turn the Sun into a black hole and this says the shadow is \(2\sqrt{27}\,m=15.35\) km across. A "weak-field figure of speech" landing the edge of a shadow is what it means for this to be an equivalence rather than a turn of phrase.
Below, a mass with a refractive index around it, and light coming from the left, integrated for real. Dashed = the 1911 version (clock only), solid = the right answer (clock and ruler). Same mass, same entry height, twice the bend. Slide the mass down and watch the ratio settle onto 2.000; push it up and it leaves the weak field behind, where the factor of two was never promised.
The opposite. People have said it continuously. Honestly laid out:
・Einstein 1911: first. Clock only, hence half.
・Isotropic coordinates: the \(A,B\) form used above is standard textbook material. Initial data in numerical relativity (Brill–Lindquist, Bowen–York punctures) is built in this chart.
・The lapse \(\alpha\) of the 3+1 split: it is by definition a position-dependent speed of light. In numerical relativity, choosing a gauge is choosing a speed of light, and that has been the daily job since 1962.
・Gordon 1923: a moving dielectric is exactly an effective metric ── "variable speed of light ↔ curved metric" is a theorem, not a suggestion.
・Gravitational lensing: lens equations are routinely written with an index \(n\). Analogue gravity and transformation optics run the same equivalence backwards.
So the rare thing is not the idea. The rare thing is the restraint ── "this claims nothing; it just makes the arithmetic cheaper sometimes." Textbooks introduce isotropic coordinates and then decline to push the variable-\(c\) reading, because it gets confused with variable-speed-of-light cosmology.
"It claims nothing but the arithmetic sometimes gets easier" is a modest statement ── and a measurable one. Take the same black hole, the same observable (the round-trip time of a radar pulse), compute it numerically in two charts, and compare the bill. Park a mirror just outside the horizon (areal radius 2.05\(m\)) and bounce a pulse off it:
| chart | out-going light speed (dish → mirror) | squeeze | levels needed | flat grid points |
|---|---|---|---|---|
| flowing grid (PG) | 0.800 → 0.0123 | 65.2x | 8 | 24006 |
| still grid, variable c | 0.960 → 0.0522 | 18.4x | 6 | 5993 |
The observable itself ── round trip 123.367733137848 ── agrees to twelve digits in both charts. Only the bill differs, and in this setup the variable-\(c\) chart is four times cheaper. The "sometimes" is measurable too: move the mirror out to \(6m\) and the difference nearly disappears. The price depends on the problem; the answer does not. That is the exact worth of this rewriting.
Established: (1) on a static, conformally flat slice (isotropic coordinates) the propagation of light is described exactly by the single index \(n=B^2/A\); (2) that index is \(1+2m/\rho\) in the weak field and gives the correct \(4m/b\), the Shapiro delay and the gravitational redshift; (3) Einstein's 1911 \(n=1+m/\rho\) halves the bending ── historical fact, and reproducible (table above); (4) the locally measured speed of light is always \(c\). What varies is the coordinate speed seen from far away; no dimensionless quantity moves.
Only a rewriting: (5) "gravity = variable light speed" predicts nothing by itself. To change a prediction you have to move a dimensionless quantity such as \(\alpha\), and atomic clocks and BBN constrain that tightly. Variable-speed-of-light cosmology (VSL) is a different thing ── an actual physical claim. Do not conflate the two.
Out of range: (6) one index carries light. Matter motion and rotation (frame dragging) need the whole metric tensor. For Kerr there is no axisymmetric conformally flat slicing with a smooth Schwarzschild limit (Garat–Price), so this picture does not transfer to a rotating black hole as it stands. (7) The "four times cheaper" of section 08 is a measurement for this problem and this mirror. It is not a law.
Episode 7's "gravity can be written as a variable speed of light" is right, but the index has to be \(n=1+2m/\rho\). The \(m/\rho\) arrives twice ── once from the clock and once from the ruler. Einstein in 1911 slowed only the clock, and so predicted 0.83 arcsec of bending: half of the 1.75 that the sky shows.
And far from nobody having thought of "just vary the speed of light," Einstein thought of it first, it sits in the textbooks as isotropic coordinates, and it is the daily bread of numerical relativity in the form of the lapse. The rare thing is not the idea but the restraint ── "this claims nothing; the arithmetic just gets cheaper sometimes." And that restraint is measurable: the round trip agrees to twelve digits while the bill differs by a factor of four.