I have put forward my relativistic cat paradox both in a previous post here, but also elsewhere, and I received various comments. Accordingly, I thought I should resurrect it, and address some of the points made that are not recorded in the original post. At the risk of being repetitive, the essence of the paradox, which is a variant of the twin paradox, is as follows:
To test the premise that all inertial reference frames are of equal standing, Fred and my five-year-old cat Horatio participate in a thought experiment. Horatio is put into a cat friendly space ship (SS1), I put myself into SS2, and the two ships accelerate rapidly then coast as close as possible to light speed in the direction of Epsilon Eridani, which is about 10.5 light years away, leaving observer Fred behind. The ships loop around the back of Epsilon eridani, then returns to Earth, landing where we took off. Fred and I ach calculate what should happen to Horatio, bearing in mind a cat seldom lives more than 20 years, then we open the hatch to SS1, and the question then is, which calculations are correct?
First, the issue is not, can we agree what we shall see before we open the hatch? Of course we can! The question is, can we agree on what we shall see when we view a third object from two inertial frames of reference without assigning a special preference for one, because Einstein argued that all inertial frames of reference are equally valid. Actually, that comes straight from Galilean relativity, but in that there is no effect on time or space from velocity.
Reviewing the situation, both parties agree there are three phases of acceleration, and two of coasting. I left acceleration out of the original post because all observers agree on the rate of acceleration. (That is an oddity, if you like, of all relativity, including Galilean.) Acceleration will cause time dilation, but since all parties agree on the rate of acceleration, both Fred and I agree what that will be. Accordingly we focus on the inertial stages. Here, time is dilated through a factor of γ where γ = 1/√(1 – v2/c2). Accordingly, Fred sees Horatio sailing through space where v ≈ c, and hence concludes Horatio should hardly age. Fred predicts Horatio will be alive. I, however, see v = 0 in my frame of reference, and Horatio ages about 11 years on reaching Epsilon Eridani, and another 11 years on the way back. (If I look forward, from the Doppler effect on the spectra I see Epsilon Eridani hurtling towards me at near light speed, but that only means that to me, anyone in that star system is not aging.) Accordingly, I predict Horatio will be dead.
This is similar to the twin paradox, for which there are various explanations. The twin paradox is often explained by the twin on Earth sending signals to the ship, and applying something similar to the Doppler effect. Such signals are irrelevant; whatever happens should be irrespective of any signalling. Some other people considered the time dilation was due to acceleration, and we all agree on what has accelerated. Thus following acceleration, there is time dilation caused by “enhanced mass”, because mass is enhanced by γ. We could also say the traveller knows he has accelerated, as has Horatio, and therefore he knows he is the one sustaining the time dilation. Another comment was that the distance has shortened, so the amount of time evolved must be less. That is all very correct, but there is a flaw in the argument.: these explanation still ultimately depend on γ, which in turn depend on the value of v put into the term. If you are coasting, or you are observing Horatio, in your frame of reference, v = 0. What has happened is that these explanations, while they have the virtue of each party getting the same result, which also is most likely to be correct, they only get it because the traveller is still using the stationary observer’s frame of reference. Following the acceleration, they know they are moving at near light speed, therefore they get the same answer by putting in the same value of v but that means there is only one frame of reverence: Fred’s. This avoids the problem of the paradox, but only by deleting the cause of the paradox and ignoring the point the paradox was addressing.
A true theory should always give the correct answer, irrespective of who is calculating, but in this case that can only happen if both use the same value of v because v is the only variable in the equation that is used to calculate the effect. If you do not accept v as a variable, then you reject the fundamental theory. But v depends on the frame of reference, as a velocity represents the change of distance between the object and a reference point. Accordingly, you cannot get a constant value of v unless both parties use the same frame of reference.
If you accept a preferred frame of reference, you have to explain why it is preferred. In the above, the preferred frame of reference was where the journey started and ended, and that is logical because then the time dilation is proportional to the kinetic energy imparted by the applied force. An alternative is that there is indeed a preferred background frame of reference. If so, we could not discern this as yet because the Earth would be moving too slowly with respect to the background to make any discernible difference. But either way, in my opinion you cannot have all frames of reference being equally valid.