From what I've read only objects that undergo acceleration experience a slowing of time compared to a reference point that they are accelerating from. The example I read talked about a spaceship with unlimited power that would leave the Earth and accelerate at 1 g for about a year, reaching 1/2 c as observed from the earth.
Even though from the perspective of the ship that the Earth is accelerating away, the only real acceleration occurring here is the spaceship and it's occupants, and only the spaceship experiences a slowing of time.
Now assuming that spaceship decelerates for another year, then reverses path back towards earth, accelerating for a year, the decelerating for another year as it returned to earth, a clock on the spaceship would show less time had passed than a clock on the earth.
Regarding a situation where two object pass by each other at high speed, and neither is accelerating, each object sees the other objects clock as being slow. This is based on the theory that the speed of light is constant regardless of the frame of reference (as long as it's not accelerating), so if each object had a pulse of light bouncing vertically between two mirrors, the other object would see the pulse of light traveling farther because instead of a vertical path, the other object would see the pulse of light traveling relative to itself as it bounced between the mirrors moving relative that object.
Observed distances also seem to be shorter. Assume two long objects passing each other at high speed. By the time the light from the front of an object reaches the other object, the object has moved significantly, so that the rear of the object appears sooner, and the object appears to be shorter.
To back up this theory, there are high speed particles with a high decay rate that reach the Earth surface that shouldn't be able to due to their decay rate. From an Earth observer, it's because the particle's rate of time is slower, from the particle's point of reference, it's because the distance is shorter.
I've also read that accelerating away from a gravitational field speeds up the relative rate of time for the accelerating object.
As far as the speed of light being a constant regardless of the frame of reference, I don't know if there's any way to actually prove this.
For example what about photons? From the perspective of a photon, it's own speed is 0, while that of other photons from other beams wouldn't be zero. So the concept of the speed of light being a constant may only be true for objects with rest mass and sub-light speeds.
Acceleration affects the apparent speed of light, and then there's the theory that within a closed object you can't tell the difference between gravitation and acceleration, except for the obvious fact that gravity changes in strength depending on distance from the source, so if the object is "tall" enough, and had pulses of light traveling back and forth between the bottom of the object and the top of the object, the difference in gravity would cause the beams to travel a different path relative to each other, as opposed to the same tall object just undergoing acceleration.
Beams of light are know to bend around large masses, but is this because gravity is accelerating the light inwards, or is it because gravity warps space?
