Question concerning the speed of gravity: does gravity propagate at the speed of light?
Empty space has certain physical properties, including springiness and inertia, and it momentarily compresses or stretches when an oscillating gravitational wave passes through it. (These properties of space are related to Newton’s universal gravitational constant, and to a gravitational constant that is the gravitational analogue of the magnetic force constant in electromagnetism, and they can be written in terms of these constants). The speed of gravitational waves through empty space is: s = square root of the ratio of the springiness of space (in its spatial dimensions) to its inertia (See, for example, chapter 3, pg 32 of the teacher’s edition of the fascinating little book entitled: Similarities in Wave Behavior, by Dr. John N Shive, director of education and training Bell Telephone Laboratories, Waverly Press, Inc., third printing 1964. So the speed of gravitational waves has to be finite. Although we have not yet detected or measured the speed of gravitational waves (we have one experiment, but it is in dispute), some believe its speed to be the speed of light on theoretical grounds, as noted elsewhere in these replies. (If it is the speed of light, that has some important physical ramifications for the relationship between fields). But the important point here is that its speed has to be finite.
But, as has also been noted in these replies, it has been documented in the literature that the force of gravity, say between the Earth and the Sun, is along the direction of a line connecting the current position of the Sun with the current position of the Earth, and that if the Earth were attracted to the retarded position of the Sun (to the point occupied by the Sun 8 minutes earlier or whatever) instead of the current position, the solar orbits of the Earth and planets would be unstable. But we know from experience that they are not unstable, all of which seems to imply that the gravitational influence of the Sun reaches the Earth instantaneously. So we have a paradox (or at least an apparent paradox).
No reason to despair, though, for the laws of physics are incomplete, and, as such, there exist physical anomalies that cannot be explained with the current laws of physics. Unfortunately, such things are often swept under the rug because they cannot be explained, and are even suppressed, but paradoxes and anomalies lead to important advances in the laws of physics--they are critically important clues.
UltraPi1 is correct, it is possible to have gravitational influences traveling at finite speed while at the same time having planets respond to the instantaneous position of the Sun. It works like this: the Earth is attracted to both the retarded position of the Sun and to its advanced position (that is, to where it was eight minutes ago, and to where it will be eight minutes from now)--the gravitational influence of the Sun travels forwards in time to the Earth from the Sun’s retarded position at the speed of light, it also travels backwards in time to the Earth from the Sun’s advanced position at the speed of light. And the vector sum of the two pulls is towards the instantaneous position of the Sun, with a magnitude equal to one half of the sum of the two pulls (at least when the sun is moving at constant speed). This gives the uncanny appearance of action at a distance when in fact it is not. It was Wheeler and Feynman who first proposed this theory, except I think they had electromagnetic interactions in mind rather than gravitational ones, and it was referred to as “The Wheeler-Feynman absorber theory”--a theory of interaction between particles that radiate a traveling wave and particles that absorb that wave no matter how far apart they may be. A brief general discussion of it can be found at
http://en.wikipedia.org/wiki/Wheeler-Feynman_absorber_theory (This paradox, by the way, is similar to the Einstein-Podolsky-Rosen paradox in quantum mechanics, which also gives the appearance of action at a distance, but which can be resolved in the same way--quantum mechanics involves backwards in time causality, which accounts for some of its strangeness).
The laws of physics are symmetric with respect to things traveling forward in time and those traveling backwards in time. The Maxwell’s equations and the wave equation for electromagnetic waves and Schrödinger’s equation for matter waves, for example, all have two possible solutions: a retarded solution and an advanced one--the total solution being the sum or superposition of the two. But as Feynman emphatically points out in his Lectures on Physics, electromagnetic forces between electric charges in our world involve the retarded position only, and never the advanced position. Therefore the advanced solution is normally discarded since it is assumed that it does not physically exist in our world. And the Wheeler-Feynman absorber theory, when applied to electromagnetic forces, has to be adjusted to make the advanced solution cancel out. This time asymmetry for electromagnetic phenomena begs for an explanation--but that is still a work in progress. In any case, gravitational forces between gravitational charges (masses) appear to involve both advanced and retarded solutions in the real world. Which is interesting.
In conclusion, gravitational influences, including gravitational waves, propagate at a finite speed (probably at the speed of light), but gravitational influences, unlike electromagnetic influences, are accounted for by both retarded and advanced waves, giving the uncanny appearance of action at a distance and instantaneous propagation for gravitational disturbances.
