A relativistic cat paradox.

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).

8 thoughts on “A relativistic cat paradox.

  1. 22 years is the total time elapsed for the observer on Earth. It is not the so-called proper time (co-moving time in the earlier, and more evocative Elie Cartan semantic). Thus the proper could be as little as a month or so (ten days for each acceleration at one g within 99.9% of the speed of light).

    The reason time slows down in the frame which is moving back and forth at nearly the speed of light is that a huge amount of acceleration has been piled up into it. As a result, the moving mass has become enormous. By the principle of equivalence, it’s as if the moving mass was sitting deep in a gravitational well, and time slows down in a gravitational well.

    I always thought the “Equivalence of All Uniformly Moving Frames” was an hypocrisy, experimentally speaking. The Cosmic Background Radiation defines indeed a state of immobility at any point in space. Moreover, so do galaxies at a distance of one billion light year, and they coincide.

    However, the problem is deep. It’s related to the problem of how much energy (hence mass) is piled up in a given volume. Basically there is just enough, and no more, so that space, as defined by Einstein’s Gravitation Theory, is flat.

    That density, whatever QED and QFT say, or want to say, or could say is low. If low it defines a low acceleration. Meaning all and any mass moved fast relative to it would bend the universe into a ball.

    The latter argument, to my knowledge is new. (It’s related to the Unruh-Hawking effect: T ~ (h/kc) (acceleration).

    • Hello Patrice. The problem is if Einstein was correct in saying there is no preferred reference frame, and that is what I am objecting to, then the time dilation or the mass enhancement depends on the value of the velocity you put into the Lorentz contraction (γ). If there is no preferred reference frame, or no fixed background, then that velocity is arbitrary and dependent on the observer. So, in my opinion Einstein was wrong on that. Of course it makes no difference to tests of his equations because in practice any observer is in a frame of reference that is moving at trivial velocities with respect to any fixed background compared with the velocity of light. In my mind, there is no doubt about the mass enhancement, but of course the accelerations in any experiment are provided in Earth’s frame of reference, and there is no doubt about the time dilation effect, as shown by muon enhancement. In the example above, Horatio would have been a little older.

      I deliberately left acceleration out of the thought experiment to simplify it and to focus on the “velocity only” dilation effect. That is totally impractical, but then thought experiments permit that.

      Thanks for the comment.

      • Thanks for the answer Ian!
        1) The “Theory of Relativity” under that name was due to Poincare’, 1904, who used these exact words, as he was touring American universities at the time.
        2) For the universe to be big as observed, it has to be flat. If it is flat, its average density of matter-energy is very low: otherwise it would curve. So now consider a blob, and accelerate it at great speed: it will acquire great mass, hence curve the universe. Therefore, since that is found, experimentally, not to be the case, the universe defines a state of absolute rest. This argument is new, and of my own invention.
        3) Acceleration is fundamental. Arguments similar to those basic to Relativity shows it slows down time, shrinks space. Acceleration is why there is no twin paradox. Acceleration is the equivalent (principle of equivalence!) to a heavy gravity field. So it slows down clocks. Just as a heavy gravitational field does. That argument of mine too is new, and I am surprised neither Einstein nor Feynman, nor any of the countless physicists who wrote on the subject, thought of it.

        And now for the big question: are you persuaded?

  2. Thanks for the further comment, Patrice. Am I persuaded?
    (1) re Poincaré, I believe you, if you mean relativity with the Lorentz contraction. Strictly speaking, the concept of frames of reference was shown by Galileo, and maybe earlier.
    (2) Only to a point. The blob has to be big enough to start with. For example, we accelerate protons to almost light speed in the LHC but they are too small. I know in principle you could keep at it and built their mass to infinity but I don’t believe so in practice. If you did you would curve the Universe. Interestingly, in Smoot’s paper (ref above) he measured the rotational angular velocity of the Universe, at least at the time of the formation of the CMB, so it is rotating! Or so Smoot claims.
    (3) I agree that acceleration will cause a time dilation, as if it were a gravity field. nevertheless, in my paradox, there are four separate states of the trip: two of acceleration, and two in an inertial frame of reference. The actual time lost is the sum of the four dilations. (Agree?) Both observers will agree what happens during the acceleration, so I discounted that for the purposes of the paradox. The two stages where the two ships comply with the requirement of inertial frames of reference still have ADDITIONAL time dilation, and in principle these stages can be as long as you like, and it is the additional stages that are paradoxical, and cannot be accounted for by acceleration as that is undisputed. I made the change of state instantaneous in the paradox, simply to keep the argument focused on the inertial frame of reference issue.

    As an aside, since I believe in the Universal frame of reference, strictly speaking for me there is no paradox, but Einstein was certain there is no p-referred frame. As an aside, a fixed background would bring relativity much closer to quantum mechanics.



  3. Question; our velocity as relative to the CBR is about 300-600 km/sec, or in other words CBR exceeds from us in this speed. This is far behind speed of light. Since CBR represents an event 350,000 years after the big bang and presumably after the inflation, at the “time” the velocity of the universe expansion was close to speed of light. If so, at this early stage of Universe i would expect much higher velocity than that measured. What’s the explanation to this discrepancy?
    Other question is, there are galaxies exceeding from us at speed close to the speed of light, close to the end of visible horizon. It means they should be farther away than the CBR. How is it possible if CBR is the ultimate frame of reference, or is it?

    • Much confusion reigns relative to the speed of light among the peons. The speed of light cannot be used to measure distance one way. It has no meaning, because the integrated speed of light is trajectory dependent.

      The speed of light is thus a holonomic notion.

      So what does it mean that the speed of light is constant (if, considered globally, it’s not constant… as Einstein himself convened)? Well the speed of light is measured not just locally, but at a point. At each and every single point.

      One can view Gravitation Theory as a theory of embedding of a three dimensional space foliation in a seven dimensional space. Then absolute distances can be determined within each leaf, and how they change. That defines a SPEED OF SPACE (so to speak).

      The speed of space knows no bound. That’s the whole idea of Cosmological Inflation.

      • Speed, or velocity, being a differential, can only be measured at a point. The velocity of light is constant, according to maxwell, but only as long as the permittivity and permeability are constant, and as usual, we assume that everywhere in space, but since we have never been there, we don’t know.

        The speed of space! Great phrase!

  4. As I understand it, our velocity relative to what I call the CMB is about 400 km/sec, and yes, this is far below the speed of light. This si good because it means Einstein’s equations work well. The concept, as I understand it, is that about 350,000 yrs after the big bang, energy/mass decoupled and temperature had a meaning. That temperature was uniform throughout the Universe because prior to that the situation could be best explained as space expanding – i.e everything expanded uniformly, although as you note, the further away you were from a point, the higher the relative velocity. The CMB is that temperature having cooled, and it is supposed to be uniform throughout the Universe, with one discrepancy, namely due to the Uncertainty Principle there were fluctuations and there were also sound waves, each of which led to slight matter variations in concentration. Because it was everywhere uniformly, and still is, it is a preferred background, in my opinion anyway.

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