Some pedantic caveats
TheMuses said:
Since nothing can go faster then the speed of light, how about gravity? While the Earth is rotating around the sun, we are being influenced by its gravity and light. What if the Sun suddenly disappears?
I'm guessing ... gravity is as fast as the speed of light.
1. According to gtr (and most related theories of gravitation), gravitational radiation and electromagnetic radiation move at the same speed (at least, in "empty space", i.e. region not filled with matter). In particular, in gtr one can study both gravitational and electromagnetic plane wave solutions, or plane waves combining both types of radiation, and in the latter case, the wavefronts of the gravitational and the EM radiation are the same.
2. Even in flat spacetime, there are many distinct but operationally significant notions of distance, and thus of velocity, over noninfinitesimal regions. For this reason, you should speak of the "infinitesimal speed" or some such phrase signalling this caveat.
3. In gtr, the principle of conservation of mass-energy prevents one from studying a scenario in which the Sun simply "vanishes". One can however study a scenario which which an isolated object suddenly and asymmetrically redistributes its matter (e.g. an exploding bomb in deep space). In such a case, we would expect the explosion to produce both gravitational and electromagnetic radiation, and the wavefronts would expand at the same speed.
Arian said:
If the Sun dissappeared, it would take approx. 8 minutes for the light to stop and the sun's gravity to take no more effect.
Which should be understood as shorthand for: if our Sun exploded, it would take 8 minutes for the bad news to reach us. (As a practical concern, the EM radiation would be far more dangerous than any gravitational radiation. But while the Sun is expected to eventually misbehave badly, this is not expected to happen for some billions of years, and the expected scenario would initially pose problems for life on Earth rather different from what one might guess from this "eight minute scenario".)