Yes well they are actually related, although in different ways. Special Relativity relates to 'object in comparison to...', whereas General Relativity relates to 'object is not at rest as it is accelerating, when in a gravity field'.
So imagine a 'photonic clock', ie, 2 mirrors locked in place with a photon bouncing between them. By moving the clock, the photon will need to 'travel' a longer distance and the effect is:
- Under Special Relativity: if you are inside the clock, you notice no difference. If you are outside the clock moving at a different speed, looking at it, it will tick at a dilated rate because the photon will still travel from your point of view at the same speed, so the 'ticking' will be slower.
- If it accelerates, Special Relativity still occurs but at the same rate as the new speed. The photon will begin it's journey as normal in the clock from one of the mirrors, but by the time it gets to the other mirror, the clock has accelerated and thus the distance is larger. So, in regards to Special Relativity, there is no difference between velocity and acceleration. In your example above, if you stop accelerating, your dilation will now remain the same from then on, relative to someone else at a different speed.
Now consider the clock also is affected by General Relativity, if it is in a gravitational field (which all objects in the universe actually is). This is an additional effect, so you could say both forms of time dilation exist at the same time on the clock. But there are some differences:
- The clock is inside a gravitation field, and time elapses differently in the space around mass. This is not a function of the clock's velocity, but rather its location close to mass. Yes, it is experiencing 'acceleration' if it is close to a mass, however to clarify my simplistic post above it is not actually due to acceleration (the acceleration is the manifestation) but more accurately its 'gravitational potential'. So if the clock is on a planet, it looks like it is at rest, however it has 'gravitational potential' so if a hole suddenly appears under it and it starts to fall, this only manifests as acceleration.
- It is this 'potential' that causes the time dilation, different to special relativity's 'velocity'.
- I have omitted the obvious next step here, which is to do with 'spacetime', worthy perhaps of another thread. This explains why dilation would occur in the middle of the planet too, which theoretically there is no 'acceleration' because mass is all around you, yet the dilation would be the strongest here because that is where the field is the strongest.
SO, there are some key differences between SR and GR dilation:
- SR dilation is dependant on someone's viewpoint traveling at a different speed to yours.
- GR dilation is dependant on if someone is closer to mass.
- The difference means that if we all had clocks, in SR, our clocks would be seen as slower by each other, as we are moving away 'from each other'.
- Whereas the GR effect (which is in addition to SR), our clocks are moving at different rates by each if we are in different areas of Gravitational Potential (distance from mass) so this means a clock in orbit, as seen by a clock on the planet, could be faster, and a clock on the planet, as seen by a clock out in orbit, would be slower. This is a different result to SR where both would be slower as seen by each other.