How do you calculate the speed of gravity using general relativity?
It is assumed that gravity travels at the speed of light. But I don't know of any mathamatics or experiments that varify it.
Basically, you look at the equations for gravitational waves (in the linearized theory) and show that they travel at the speed of light. At least that's the usual and simplest approach. This is similar to the approach of finding the speed of electromagnetic radiation, and saying that this is the speed of electromagnetism in general.
which, while it is a good treatment, isn't likely to make that much sense to someone who isn't already reasonably familiar with general relativity.
It might be just as well to skip to equation 16,17, and 18 and believe the text when it says
I believe that under some extreme conditions one can get gravitational waves in the right sort of dense medium to travel slower than 'c' - however, if one assumes that one has weak waves in empty space, one gets a speed of propagation of 'c' for gravity waves as the above paper illustrates.
BTW, it is possible to take a more advanced approach, and to pose Einstein's equations as a very complicated non-linear differential equation. It is possible to show that solutions to these must obey causality (no propagation of the solution faster than 'c') - this is known as the "intial value formulation" of GR. Wald does this in his book, "General Realtivity", but it's fairly involved.
Note that in both of these cases, one provides a disturbance, and times how long it takes that disturbance to actually produce a measuarable effect. Methods such as looking at the instantaneous direction of gravity (or the instantaneous direction of the Columb force) without a disturbance of the charge being examined are not sophisticated enough to determine the speed of gravity (or of electromagnetic radiation).
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