A colleague helped me to better understand the Tolman paradox
Around four years ago I read a physics paper written by Moses Fayngold entitled “A possible resolution of the Tolman Paradox as a Quantum Superposition”. Both the topic as well as the depth of ideas Fayngold employed to assemble his paper fascinated me. Naturally enough I could only understand snippets to what the author was talking about. I passed the document along to my colleague and friend Peter to help me better understand it. I now share this interesting information with my readers. I am also paying tribute to Peter for the enormous effort he rendered in responding to my question. Thank you Peter!
The Fayngold essay is attached to this blog in the pdf file below.
On 7/Jan/2013 I sent an email to Peter. This was his response:
As a layperson I find this article interesting. Obviously I understand mere snippets of it but I have zeroed on to the closing sentence that “… QM could be nature’s device against violations of relativistic causality” May I ask has this guy got a point?
Thanks. This is very interesting and it is actually a new paper (1104 means 2011 04, ie April 2011). But his argument is in relationship to the claims of Special Relativity. For the benefit of both of us I will try to explain.
Prior to Special Relativity, people assumed that light waves traveled through a medium (eg ether, or Cahill’s dynamical 3-space) at a fixed speed ‘c’. That implied that if you were moving through the medium at say speed v, then the speed of light waves relative to yourself would be c + v if you were headed into the waves, or c-v if you were headed away and being overtaken by the waves.
This in turn implied that people could determine the speed of the earth through the “ether” by measuring the speed of light in different directions. If they got a maximum speed say of c+v in one direction, and a minimum of c-v in the opposite direction, then the earth would have a speed of v relative to the ether.
Such an experiment was done by Michelson and Morley in 1887, however it gave a value for v of only about 8 km/s. Now since the earth was known to have an orbital speed around the sun of about 30 km/s, its speed through the ether would have to be at least as fast as that, so something appeared to be wrong.
In response to this, a theory was developed that motion through the ether caused matter to contract along its direction of motion and caused its internal physical processes to slow down, in such a way as to cause laboratory instruments to always measure the speed of light as being equal c, even if in reality it differed from c. This became known as Lorentzian Relativity Theory and it implied that it might be completely impossible to experimentally detect motion of matter relative to ether.
However, Cahill argues that if laboratory instruments get distorted by motion through ether, then to get the true value of v, one needs to multiply the experimentally determined value of 8 km/s by a scale factor, which then gives a value of about 400 km/s, which accounts for the orbital velocity of the earth plus the velocity of the sun as it orbits our galaxy etc. See:
The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion
However, if one assumes, as most physicists came to do, that motion of matter relative ether can not be experimentally detected, then it should be possible to make the same predictions as Lorentzian Relativity, using simpler equations that don’t include terms related to velocity through ether.
That is, it should be possible to develop a mathematically simpler alternative to Lorentzian Relativity that would work just as well for practical purposes.
This was achieved by Einstein’s theory of Special Relativity, which became popular because of its greater simplicity.
However, by leaving out the ether, Special Relativity allows paradoxes to arise such as Tolman’s paradox, which I will try to illustrate in a simple way.
Suppose we have observers A and B each with physically identical clocks.
And suppose A is at rest in ether and that B brushes past A and then away from A at a speed through the ether that causes B’s clock to run at half speed.
Then according to Lorentzian Relativity, A can say that the motion of B through the ether causes B’s clock to run at half speed relative to A’s clock and B can agree that this is so.
However, for the same situation Special Relativity asserts that A can say that the motion of B relative to A causes B’s clock to run at half speed relative to A’s clock but that since there is no ether, B has an equal right to say that the motion of A relative to B causes A’s clock to run at half speed relative to B’s clock.
At first sight this appears a ridiculous contradiction, but in practice, it is not possible for A and B to compare the times shown by their clocks without sending signals to each other and it turns out that if the signals do not exceed the speed of light, inconsistencies do not arise when comparisons are made. So because we currently have no way to send signals faster than light, A and B are each entitled to claim that his own clock is running normally and that it is the clock of the other that is slow and there is no practical way to prove that either is wrong.
But, suppose it was possible for each observer to remotely stop the clock of the other using a signal of infinite speed?
Eg suppose at the instant that B brushes past A, A and B zero their clocks, and then after two seconds A stopped B’s clock and then in response B stopped A’s clock. What would be the result?
Lorentzian Relativity theory predicts that when A stops B’s clock, B’s clock will only be showing a time of one second because it runs at half the rate of A’s clock. And when B stops A’s clock in response to that, A’s clock will be showing a time of two seconds, because from B’s point of view, A’s clock runs at twice the speed of B’s clock. This is also what A would expect, because it would take zero time for a signals of infinite speed to travel from A to B and back to A again. So A would expect his clock to stop as soon as he stops B’s clock.
So Lorentzian Relativity predicts a logically consistent result.
Special relativity theory predicts that when A stops B’s clock, B’s clock will only be showing a time of one second because it runs at half the rate of A’s clock. But when B stops A’s clock in response to that, A’s clock will only be showing a time of half of a second because from B’s point of view, A’s clock runs at half the speed of B’s clock.
But if A’s clock stops when it is showing only half a second, it could never get to two seconds to allow A to send the signal to stop B’s clock !
So in this example, Special Relativity results in paradox if we consider signals that travel at infinite speed. (The paradox can also arise if the signal speed is less than infinite but greater than the speed of light, but that is more difficult to reason about).
For people who prefer Special Relativity, the usual way to avoid this paradox is to assume that it is impossible for anything to travel faster than light, because if anything could do so, it could be used to send signals between observers such as A and B. That rules out tachyons so far as such people are concerned.
However, the paper you referenced suggests another way to avoid the paradox. The argument seems to be like this.
Suppose we assume that two versions of A (and his clock) can exist in a state of superposition, eg A1 and A2. Then when the clock of A1 shows two seconds, A1 sends a an infinitely fast signal to stop B’s clock. B then sends an infinitely fast signal to stop A’s clock which owing to the claims of Special Relativity arrives when A’s clock shows half a second. This would result in a paradox if it stopped the clock of A1, but thanks to the existence of A2, this signal can stop the clock of A2 rather than the clock of A1.
However, the situation of having two versions of A in superposition cannot continue for ever. At some point, one or other of the states must end up as the one that is observed to be real. If A1 is manifested, then A sent a signal to stop B’s clock, but did not receive a signal back to stop his clock, so that signal was in effect lost in quantum noise. If A2 is manifested, then A received a signal from B to stop his clock, but did not send a signal to stop B’s clock so the signal that stopped B’s clock was in effect spontaneously generated by quantum noise.
So this method of avoiding the paradox allows signals (eg using tachyons) that are faster than light, but does not allow such signals to be used for reliable communication.
On the other hand, as illustrated above, Lorentzian Relativity does not result in paradoxes when we consider signals (eg using tachyons) that move faster then light so adopting Lorentzian Relativity instead of Special Relativity provides a simpler way to avoid the paradox.
tachyons and superposition pdf