When travelling near the speed of light, time goes more slowly, by a factor of γ, where
γ = 1/√(1 – v2/c2)
Here, c is the velocity of light, which is constant for all observers, while v is the velocity. The problem with the velocity is that it depends on the frame of reference used, which in turn depends on the motion of the observer, and according to Einstein, there is no preferred frame of reference. That means any frame of reference is as good as another, and apparently Einstein illustrated the principle of relativity once by remarking at the end of a train journey, “Ha, the Zurich railway station is approaching, and will soon stop outside the train.”
Consider this problem. I have a friend Fred and a five-year old cat Horatio, and these two have agreed to participate in a thought experiment to test Einstein’s argument that there is no preferred frame of reference. Horatio is put into a cat friendly space ship (SS1), I put myself into SS2, and the two ships travel as close as possible to light speed in the direction of Epsilon eridani, leaving observer Fred behind. The ships loop around the back of Epsilon eridani, then head back to Earth, landing where we took off. Fred and I open the hatch to SS1, and the question then is, what do we see? Before opening the hatch, we can use the time dilation equation to make our prediction, but we get different answers.
From Fred’s point of view, the two space ships have been in flight for twenty-two years, say, but they sustain the relativistic time dilation effect because v ≈ c, and time should almost stop. Accordingly, following Einstein’s equations, Horatio will leap out, a little older than when he entered SS1. However, from my point of view, once underway, I look out the window of SS2, and see SS1 stationary beside me, and Epsilon eridani hurtling towards me at just under the speed of light. However it does not reach me for a bit under 11 years, and the same thing happens on the way back, except Earth is now hurtling towards me at almost light speed instead of receding. Accordingly, Horatio should have experienced a bit under 22 years of travel, but since cats do not live longer than about 18 years, and given his first five years were over before he started, I expect to see a long dead residue of Horatio.
Two adjacent observers must see the same thing. What do they see when they open the hatch of SS1, and why?
The paradox goes away if there is a preferred frame of reference, and the velocity both use in the equation is the velocity with respect to that frame of reference. Note that one can argue that there is a preferred frame of reference in the cosmic microwave background, and motion of our solar system relative to that has been found to be approximately 390km/sec. (Smoot et al., 1977 Phys. Rev. Lett. 39: 898).