Subject: Re: speed of gravity
Date: 8 Nov 1995 19:43:30 GMT
Organization: University of California, Davis
Oz (Oz@upthorpe.demon.co.uk) wrote:
: I know noting about GR. However I was under the impression : that it explained gravitational movements by a distortion of : spaceYes. (Well, a distortion in spacetime, strictly speaking.)
. I presumed that since the distortion was in a way : 'set up in advance' (ie statically) there was no real : requirement for the gravitational force to 'travel' from : body to body, since the distortion was there (so to speak) : before the test body arrived.This is true in the approximation that the sources of the curvature---i.e., matter---are static. For example, if you're dealing with honest "test bodies" orbiting the Sun, that is, bodies who have a negligible back-reaction on the Sun, then this view is right. But in that limit, the speed of propagation of gravity is undetermined even in the Newtonian theory; a static field doesn't propagate, after all.
The complaints about speed-of-light propagation come from situations in which the system is not static. In the real Solar System, for example, the Sun is not stationary; in Newtonian language, it orbits the center of mass of the Solar System, which is not the same as the center of the Sun. So it is meaningful to ask how fast gravity propagates in such a system.
This is tricky, though, since (as you know from other threads) there isn't a preferred coordinate system in general relativity, so there's quite a bit of ambiguity in what you mean by speed. (Speed is distance/time, but there's no unique way to measure distances or synchronize clocks at different locations.)
Still, though, Newtonian gravity is a good approximation to GR, and this, in a sense, gives you a special coordinate system---the coordinates in which GR looks the most like Newtonian gravity. There's a lot of difficult work that goes into defining those coordinates, most of which I've never looked at in detail, but once you've done this, it turns out that you can rewrite general relativity in a language of forces and accelerations. It then makes sense to ask in this language how fast gravity propagates, and that's the sense that my answer assumed.
You can also ask this question much more directly: given a distribution of matter and the corresponding spatial geometry on an arbitrary spacelike surface (i.e., an instant of time in an arbitrary coordinate system), imagine perturbing the matter distribution. This will lead to a perturbation in the geometry that propagates forward in time; what is the spacetime path of this propagation? The answer in GR is that it is the same as the path of a light ray. This is an invariant (coordinate- independent) statement, whose meaning is that "gravity propagates at light speed." It *should* be enough to answer the question. But people who know about the problems that light-speed propagation causes in Newtonian gravity will still, quite reasonably, ask how GR avoids these problems; for that, you need to translate back into Newtonian language.
: >This cancellation is not quite exact, although the effect is not : >observable in the Solar System. The slow change in the periods of : >binary pulsar systems can, in fact, be described as the result of : >the failure of propagation-delay effects and noncentrality and : >velocity-dependence of the gravitational "force" to exactly cancel. : Is this difference related to gravity waves in some way?Yes. It's equivalent to saying that the nonconserved "mechanical" angular momentum is compensated for by angular momentum carried away by gravitational waves.
To quote from the sci.astro FAQ: Consider two bodies--- call them A and B---held in orbit by either electrical or gravitational attraction. As long as the force on A points directly towards B and vice versa, a stable orbit is possible. If the force on A points instead towards the retarded (propagation-time-delayed) position of B, on the other hand, the effect is to add a new component of force in the direction of A's motion, causing instability of the orbit. This instability, in turn, leads to a change in the mechanical angular momentum of the A-B system. But *total* angular momentum is conserved, so this change can only occur if some of the angular momentum of the A-B system is carried away by electromagnetic or gravitational radiation.
Now, in electrodynamics, a charge moving at a constant velocity does not radiate. (Technically, the lowest order radiation is dipole radiation, which depends on the acceleration.) So to the extent that that A's motion can be approximated as motion at a constant velocity, A cannot lose angular momentum. For the theory to be consistent, there *must* therefore be compensating terms that partially cancel the instability of the orbit caused by retardation. This is exactly what happens; a calculation shows that the force on A points not towards B's retarded position, but towards B's "linearly extrapolated" retarded position. Similarly, in general relativity, a mass moving at a constant acceleration does not radiate (the lowest order radiation is quadrupole), so for consistency, an even more complete cancellation of the effect of retardation must occur. This is exactly what one finds when one solves the equations of motion in general relativity.
Steve Carlip
carlip@dirac.ucdavis.edu