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General Relativity says that gravity moves at the speed of c. How precise are the measurements?

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The accuracy is extremely good thanks to LIGO and a binary neutron star merger. It's accurate to about 3 parts in 1 quadrillion!

See the paper Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A

The paper constrains "the difference between the speed of gravity and the speed of light to be between $−3×10^{-15}$ and $+7×10^{−16}$ times the speed of light."

Older response before the LIGO observation:

One of the more recent of gravity was first measured in 2003 using Jupiter and a quasar. The measurements were for the purpose of proving a finite speed and weren't very precise. See the paper for more.

More recently in 2013 a Chinese group built a model using Earth's tides that helped them narrow it down.

... [T]he speeds of gravity are from 0.93 to 1.05 times the speed of light with a relative error of about 5%. This provides first set of strong evidences to show that the speed of gravity is the same as the speed of light.

This is so far the most accurate measurement I've seen. See the paper for more.

In the near future, LIGO may be able to provide more accurate measurements by comparing the distance among detectors and the delay of observation.

Also related, see the question "How is the most accurate value of G measured?"

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How precise are the observational measurements for the speed of gravity?

Not very. See the Wikipedia article on the speed of gravity which refers to dissent surrounding the claims made by Sergei Kopeikin and Edward Fomalont. The dissenters include Clifford M Will and Steve Carlip who are fairly well respected. The 2016 paper about the LIGO detection said the measured speed was less than 1.7c. The more recent paper by Neil Cornish and others said it was between 0.55c and 1.4c. This is not at all precise.

Note that in seismology p waves and s waves travel at different speeds. So IMHO it wouldn't be unreasonable to expect different types of waves in space to travel at different speeds. As to whether one should expect light waves and gravitational waves to travel at exactly the same speed is a slightly different matter, complicated by the way a gravitational wave is thought to be a quadrupole wave.

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    $\begingroup$ I don't know if referring to p and s waves in seismology is an apt comparison. $\endgroup$ – zephyr Sep 1 '17 at 15:02
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    $\begingroup$ @zephyr It's just an analogy that highlights the idea that there are many different kinds of waves. I like it. $\endgroup$ – Florin Andrei Sep 1 '17 at 19:16
  • $\begingroup$ @zephyr : IMHO when you read Einstein's old material there's a definite sense of continuum mechanics. Mind you, this is an interesting discussion following a lecture. $\endgroup$ – John Duffield Sep 2 '17 at 17:44
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    $\begingroup$ Quite out of date. $\endgroup$ – Rob Jeffries Aug 5 at 20:49

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