If gravity "travels" at $c$, and the sun is travelling "forward", does it mean the planets are actually orbiting various points "behind" the center of the sun? Does it also mean the sun is interacting with various points behind the planets? A line from the center of a planet, to a point behind the sun, would not be the same line as from the center of the sun to a point behind the planet. Does it mean there actually is no "common" point they both "orbit"? How is this stable?
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@Prahar asked, "What do you mean by saying that the sun is travelling forward?"
That's easy. "Forward" is an arbitrary name for the direction in which the Sun is moving in the arbitrary coordinate system that you have chosen. We can imagine drawing a big letter "F" on the Sun ("F" for "front"), and "Forward" is motion in the direction of the "F".
The real problem is, suppose I choose a different coordinate system. In my coordinate system, the Sun is moving "Backward," (i.e., toward the letter "B" that I drew on the Sun directly opposite to your "F".)
If the point that the Earth orbits is somewhere on the the "B" side of the center of the Sun in your coordinate system, does that mean that the point the Earth orbits in my coordinate system is a different point? A point somewhere on the "F" side of the center?
How can the arbitrary choice of a coordinate system change the reality of what's happening?
It can't.
The answer is, Gravity does not move at the speed of light. Gravity does not move at all. What moves at the speed of light are disturbances in the gravitational field. If you grab hold of the Sun and give it a little shake, it will take about eight minutes at the speed of light before we here on Earth will feel the shake.
Otherwise, the Sun, and the shape that the Sun gives to the gravitational field, and the Earth, and the barycenter around which the Earth and the Sun mutually orbit all are co-moving, in whatever arbitrary coordinate system you choose.
Solomon Slow
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