Revisiting the Relativistic Cat Paradox

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.

14 thoughts on “Revisiting the Relativistic Cat Paradox

  1. I deduced mass dilation from time dilation [original research!), so I feel rather fierce.

    And I agree there is a rest mass problem. Poincare’, the founder of Relativity, saw it, and Einstein, Poincare”s parrot, did not. Something about the brain limitations of parrots.

    This being said, I don’t see the problem here. First, cats are nice. So why don’t you put Horatio in your spaceship? it would provide you with company, and would make become batty more difficult (cats eat bats).

    Let’s put numbers on it. Say one accelerates at g, Earth gravity. After a day or so, the rocket ship will get so close to light speed that its gamma factor, 1/square root [1-vv/cc] would be around 100. That means a round trip to Eridani will take 21 years, Earth time, but only two months and a half, ship time. So Horatio will be just be fine.

    I will write two essays on the general subject. First, I agree there is a preferred rest frame (so did Einstein, decades later, if one reads between his lines!)
    Second, the acceleration is truly a General Relativity effect. So the Twin Paradox belongs to GR, not SR.
    Third, Quantum Physics proves Einstein wrong on time. And Bergson right (people can google the names).

    • I put Horatio in a different ship to ensure there was a third object viewed externally from two different frames of reference. The point I was trying to make is there is a preferred reference frame; of course you can calculate the right end point, BUT according to most text books, and as far as I am aware, Einstein, all frames of reference are suppose to be of equivalent status.

      Acceleration belongs to GR, but the coasting still belongs to SR. Time is additive. Mass is an interesting concept, and thoughts belong elsewhere.

  2. The point is that at 100 gamma the PROPER TIME back and forth to Eridani is only two and a half month. At the speed of light, it takes zero time (OK, it’s impossible to get there, agreed).
    The question of having an absolute rest frame at any point of space is a different matter entirely.

  3. I guess my mind is too cluttered at the moment to do more than offer a comment;

    Way back when I first studied Relativity it was explained that since nothing can travel faster than light in a vacuum, then in any frame moving in that vacuum, the motion of the frame has to be subtracted from the physical actions in that frame, both light and atomic activity. Otherwise it would have to move faster than light in the larger vacuum. So the physical structure is compressed and the light is still relative to the rest vacuum. Thus in the moving frame, both measures of time and distance are compressed equally.
    While I don’t find this explanation in current descriptions, it would still seem there is the assumption of a fundamental rest frame running through all descriptions.
    I also argue that as time is simply a measure of rate of change, ie. frequency, in space, it is inconsequential that different clocks should run at different rates, as they are separate actions.
    For instance, an animal with faster metabolism and heart rate ages quicker than one with a slower rate. To wit, the tortoise is still plodding along, long after the hare has died.
    The twin in the rest frame ages quicker, but then the one in the moving frame likely encounters much more damaging radiation and stress of compressed structure.
    They all still exist in the physical state of the present, where the conserved energy exists.
    The argument against simultaneity is based on observed information, not the energy manifesting it. Yes, we see events relative to our point of view, but what is the mystery? Looking up at the night sky, we see the moon as it was a moment ago, but stars as they were years ago. The light still strikes our eye simultaneously. The events recorded no longer exist, or they wouldn’t have radiated away the energy by which we observed them. Past and future are prior and succeeding configurations of the same, conserved energy. They can’t physically exist, as that require entire universe worth of energy to manifest every moment and we couldn’t observe them, as that energy wouldn’t transfer to our moment.

    • Yes, the Lorentz contraction really needs another post. The simplest explanation I have seen comes from Feynman, who just shows that if light speed is constant, and it goes the same “rest distance” in two directions, this is only possible with the contraction.
      My cat paradox demands simultaneity because the traveller returns and stands beside the rest observer, and they observe a single event when they open the hatch. A true theory must predict what each observer sees, otherwise it is falsified. But if each observer has calculated in the frames of reference they were in, they cannot agree. The net result is you need a preferred frame for the two calculations; the problem then is, which frame and why? It is the why that is difficult. (It has to be for some other reason than to get the right answer if you want enlightenment.)

  4. Ian,

    There does seem to be a tendency to dismiss space as simply an effect of measurement and so say all frames are relative to each other, with no absolute frame. Yet what is being measured?
    With time, it is cycles of action that are measured, but with space, whether distance, area or volume, it is space that is being measured, not anything else.
    So if you accept space as an elemental dimensionality, which is only measured and otherwise delineated, but not created by this mapping, then it is the rest frame. As I said previously, if it has no physical properties to define it, then it isn’t limited, curved or unstable, so its non-physical attributes would be equilibrium and infinity.

    As a sidenote, in cosmology, when it was discovered that we appear at the center of this expanding universe, because galaxies are redshifted proportional to distance, with no lateral motion, it was argued that because of “spacetime,” space itself is expanding and so every point would appear as the center. This is utterly illogical, because the very premise of spacetime as the physical explanation for Relativity is the speed of light remaining Constant to the distance, as both are dilated equally, yet if it is redshifted because the light is taking longer to cross between galaxies moving away from each other, then it isn’t Constant to that distance! As Einstein said, “Space is what you measure with a ruler.” The ruler, the speed of light in a vacuum, is not being stretched, only more units added. So this would be increased distance, not expanding space.
    Now we are at the center of our view of the universe, so an optical explanation for redshift would be an entirely reasonable proposition.

    Which ties back to my point about how physics tries to dismiss space, but overlooks the fact it still underlays everything.

    • Yes, I am quite happy to have space merely as what dimensions lead to, nevertheless that by itself is not really a rest frame because you cannot say (or at least I don’t think you can) “I am travelling at x velocity with respect to length. I am with Einstein – space is what you measure with a ruler.

      However, with that view, something is superimposed on space that does have properties. It has a permittivity, a permeability, and the equivalent for storing gravitational energy. Exactly what that is is something of a puzzle.

  5. Ian,

    If we can determine the clock in one frame runs faster than another, because it is closer to rest, couldn’t we keep placing frames around and find the one with the fastest clock and that would be closest to the equilibrium of the vacuum? Now there might always be another, with a slightly faster clock, but the difference would keep diminishing, thus implying the existence of the rest frame, even if it can’t be exactly specified.

  6. Ian,
    Yes. Time is an effect of action, not a fundamental dimension, so while you can change the ways energy acts, such as converting atomic activity/clock rate to linear G force, through acceleration, you still don’t escape the present state.

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