Our physics predicted nothing could travel faster than the speed of sound. One argument against it was that if you were the target of a supersonic bullet, you would hear the bullet hit, then you would hear the bullet travelling downrange, then you would hear the bang of the bullet leaving the gun; you would witness events going backwards in time, which seemed to present a temporal paradox. This is exactly what happens. The target witnesses the arrival of the bullet, and can even have the time to radio back to the gunner and announce that the firing is still in the target's future (measurable and recordable as such with a tape recorder). But of course the firing is in the gunner's past, so it is meaningless to ask the gunner not to fire.
The best argument against gravity having speed that I've read was by Sir Aurthur Eddington: "If the Sun attracts Jupiter towards its present position S, and Jupiter attracts the Sun towards its present position J, the two forces are in the same line and balance. But if the Sun attracts Jupiter toward its previous position S', and Jupiter attracts the Sun towards its previous position J', when the force of attraction started out to cross the gulf, then the two forces give a couple. This couple will tend to increase the angular momentum of the system, and, acting cumulatively, will soon cause an appreciable change of period, disagreeing with observations if the speed is at all comparable with that of light."
