The speed of gravity

The internet is full of cats and porn. No breaking news, here. There are a lot of other places with interesting information, though. One of them is the askscience of reddit. If you follow the link, you'll find this one too.

The question stated is: if gravity is transferred at $c$ (speed of light), and Sun is 8 minutes away at $c$, regarding the fact the Sun moves, does it mean Earth is attracted to the position the Sun had 8 minutes ago?

Video is from a guy named DjSadhu. Music is cool, if you ask me.

Didn't a guy named Kepler fix this already?

The Sun moves, of course. It orbits the center of our galaxy, like a lot of other stars, so it moves. Depending on the frame of reference, you would be able to see the Sun (and the solar system), moving along a path. In the case of planets, because they are orbiting the sun, they would follow a helical trajectory. This is tricky. Technically, you can choose any frame of reference that can make follow any body any trajectory, but it seems reasonable to say Sun orbits the galaxy center so, follow me on this.

Kepler said Earth sees the Sun static. That was the reason my first thought regarding the question was, well, since the Earth and the Sun are moving at the same velocity respect galactic center, it doesn't really matter, right? Earth was originated already orbiting the Sun, sharing any other velocity it may have, so we can say the spacetime curvature bubble "travels" with the gang. So the answer to the question would be: "Earth is atracted to the actual position of the Sun".

Turns out that's the answer; but the explanation for it can be explained better in terms of spacetime.

What does the paper say, then?

The paper starts by pointing at something called velocity-dependant terms. It examplifies first with electrodynamics: it turns out, a charged particle moving exerts a force over a test particle in rest, exactly to its instantaneous position. There is no magic here and no instantaneous transmission of information. It turns out the delayed term of the electromagnetic field (the thing that tells the test particle to feel attracted to the old position), cancels with the velocity of the field itseld. Since the delay in the field due to distance is called aberration, this effect is known as aberration cancelling.

Stelar aberration. Aberration takes the meaning of deviation. Really tempted to post some Lovecraft monster, but not Creative Commons found. Image from here.

The image above is an example of non-cancelling aberration. Although its effect is small, the fact that the Earth moves respect to the Sun, makes Sun's light look like coming from another place. It's a small effect because $\frac{v}{c}$ is small. Change Sun with any other fixed star.

Wow. I just said a charged particle has an aberration cancelling and after that I post a diagram about light being aberrated. WTF? Answer seems easier than a paper, but since I didn't find any reference, you'll have to take my word (maybe wrong) for it. We just said a moving charged particle cancels aberration in its field because of the velocity, but we didn't say a thing about the test particle in rest. It turns out, the moving particle feels the aberration produced by the rest particle. Which doesn't convince too much... :|

Coming back to the aberration cancelling, in GR it turns out it's almost the same. It's a little bit trickier and the demonstration requires to obtain the affine connection (Christoffel symbols, nothing new) to a gendanken called photon rocket. In a similar manner to electromagnetism, a velocity-dependant term cancels the aberration.

Paper ends finding a reason for this appropiate cancellation occur. I mean, it's too good to be truth that the velocity-dependant terms appear to solve things up: they must come from some important nature relevant happening. The answer is Noether's theorem. Conservation laws, you know. Trippy.

Truth be said, I preffer keeping with the spacetime-bubble-travelling-with-the-mass explanation. The fact that a charged particle moving sees aberration but a test particle doesn't, makes me a little bit nervous. Something is wrong there, you know. Maybe light is not subject to the aberration cancelling for whatever reason, or maybe I've just missed some concept very important in this...

Anyway, and BTW, not really my business but if you're in America, I really do hope you vote for the right candidate.