Since last week's announcement of the direct detection of gravitational waves, several little papers have shown up on the arXiv about what it means.
He's a nice one, constraining the difference in speeds of gravitational waves relative to light speed.
Einstein didn't assume the gravitational waves predicted by his equations moved at the speed of light -- his equations predicted it.
How? What did his equations of gravity know about light? Nothing, per se. But he did know his equations had to match Newton's equation for the force of gravity, the familar F=Gm1m2/r2 for weak gravitational fields.
Sure enough, for low speeds and in small gravitational fields, Einstein's equations reduced to Newton's equation to a very close good approximation, provided a free parameter of Einstein's, a constant, was a certain simple function of G and c, Newton's gravitational constant and the speed of light.
That let Einstein write his equations as
There's no point here in getting into what these 16 equations mean (only 10 after symmetries) -- basically they say the geometry of spacetime is proportional to the energy density in that spacetime.
Anyway, the 8*pi*G/c4 was the constant Einstein deduced from comparing the low energy form of his equations to Newton's.
Given that, he shows his equations predicted gravitational waves with a speed of c -- though these waves aren't the simple sinusoids we are familar with from electromagnetism.
The LIGO observation of gravitational waves allowed this to be checked. They did this by comparing the timing of LIGO's wave with independent observations of gamma ray photons from the same black hole-black hole merger, seen in the Fermi Gamma-Ray Burst Monitor and published only four days ago, seen 0.4 seconds after the gravitational wave event was detected.
The black hole merger was 1.3 billion light years away, and the grav waves showed up 0.4 seconds before the time that EM wave took to get here -- about 4e16 seconds. That's about one part in 1017. So the Ellis et al paper concluded the speed of gravitational waves and electromagnetic waves differ by, at most, only 1 part in 1e17:
It really is amazing how much Einstein was able to deduce -- well, make very smart guesses about -- from thinking clearly and deeply of very general principles about how light and gravity should behave.