The so-called 'speed of gravity' has been the source of some controversy over the years. The anti-relativity fraternity maintains that the speed of gravity must be infinite, i.e., that gravity acts instantaneously over any distance. Relativists, on the other hand, maintain that no influence propagates faster than the speed of light and that gravity is no exception.
The Moon's Orbit. We know that gravity keeps the Moon in a stable orbit around Earth, despite the fact that Earth is moving around the Sun at the enormous speed of 30 km per second. What is more, the Moon and Earth are actually both orbiting around their common center of mass, also called their barycenter.
The Gravitational Field. So what happens to the gravitational field in this scenario? It is reasonable to expect that the field must be varying continuously at every point in a coordinate system centered on the barycenter. According to Newton's orbital theory, these variations in the gravitational field must occur simultaneously over the whole field.
If this is a not so, orbits cannot be stable. For example, light takes about 1.25 seconds to travel from the Earth to the Moon. In that time, the Earth/Moon barycenter system has moved some 37 km in its orbit around the Sun. If the gravitational effect propagated at the speed of light, the gravitational force would not be pointing at the barycenter, causing an unstable orbit.
Since we observe stable orbits, we must conclude that nature does not work in this way - the speed of gravity is apparently near infinity. The problem is that Einstein's relativity theory forbids any effect to propagate faster than the speed of light - and that includes gravity.
So how can Einstein's theory of gravity produce stable orbits? Further, since Einstein's gravity reduces to Newton's gravity in the limiting case of low fields and low velocity, how can the two theories be reconciled?
The answer lurks in the depths of solutions to Einstein's field equations, which are unfortunately outside of the scope of this article. Loosely stated, it tells us that for any mass that moves uniformly relative to an inertial frame, (i.e., a non-accelerated mass), its gravitational field appears 'static' relative to the mass itself - i.e., it moves as if attached to the mass.
A test particle that is kept stationary in the inertial frame and then released will immediately start to curve towards the proper position of the moving mass and not towards the position where the mass was, taking into account the finite speed of light.
Now this is totally unremarkable, because we could just as well have viewed the mass as stationary and the test particle as moving relative to the mass. The gravitational acceleration of such a particle would surely be towards the proper position of the stationary mass.
What is remarkable is this: general relativity tells us that if we could somehow abruptly stop the movement of the mass relative to the inertial frame, the gravitational field will also stop moving, but the 'stop effect' will propagate at the speed of light from the mass outwards.
This means that the field will be deformed for a period of time, with some (outer) parts of the field still moving at the original speed and some (inner) parts having already stopped moving. See figure above.
The semi-circle is to where the 'stopped' gravitational field has had time to propagate. The bent lines illustrate the instantaneous directions that particles will follow in different regions of the gravitational field.
A test particle at some distance from the mass will continue to be curved towards the extrapolated position of the moving mass, until such time as the change in the gravitational field reaches the particle. At that time the particle will start to curve towards the proper position of the now stationary mass.
Maxwell. If the above sounds weird, take comfort from the fact that Maxwell's equations for electromagnetic radiation predicts exactly this behavior for the field around a moving charge.
The foregoing is readily comprehensible for a uniformly moving mass, but what about two bodies in orbit around their common center of gravity? Surely, in the barycentric inertial reference frame, both masses are constantly being accelerated towards the barycenter.
Acceleration? But do the objects experience acceleration in the sense that a force is acting upon them? Not quite. The two objects are in free-fall and are for all practical purposes inertially moving masses. More technically correct, they are moving along space-time geodesics, which can be thought of as moving in straight lines through curved space-time.
Their respective gravitational fields are moving with them, just as in the inertially moving mass in the absence of gravity. Relative to their space-time geodesics they experience no acceleration, so their gravitational fields are not deformed, except for small higher order effects that will be discussed below.
If we consider one mass as the 'test particle', that mass will always curve directly towards the proper position of the other mass and not towards it's 'visible' position. If the bodies have sufficient transverse momentum, so that they do not crash into each other, they will stay in stable orbits around the barycenter.
It gets Tougher! The above is an oversimplification of the real situation, because general relativity also tells us that there is a higher order interaction between the gravitational fields of the two very massive, closely orbiting objects. This interaction causes some deformation of the gravitational fields of both objects and the deformation continuously robs them of a little orbital energy.
For normal orbiting objects the effect is negligible because of the relatively large distance between them. For binary neutron stars that are very massive and very close to each other, the effect becomes appreciable.
Gravitational Waves. This loss of orbital energy is radiated as gravitational waves. The effect is observable when the binary neutron stars also happen to be pulsars. The orbital periods of the binary pulsars can be measured very accurately from their highly stable 'light house' effect. This can also be used to measure the 'speed of gravity'.
The rate of orbital decay agrees with Einstein's theory to within 1%, which is the experimental uncertainty. So it seems that Einstein was right again! The eBook Relativity 4 Engineers contains a lot on gravity, orbits, gravitational waves, the 'speed of gravity' and more...