Synchronized Clocks in Frames boosted by Acceleration

Jorrie

I have often thought about how the one-way speed of light can be measured without having to use synchronized clocks, which tends to be controversial because the clocks are normally synchronized by using one-way light in the first place. Recently, Don Lincoln of Fermi-Lab suggested a method to me. To convince myself, I tested it in a thought experiment, using two clocks and two cables in a frame that is boosted by accelerating it to a new inertial frame. I thought it useful to share it and get other opinions.

First, Don Lincoln's test.
I expanded Don's test as follows:
Consider two spaceships connected by a taut cable of 600 units long. Add Don’s second identical cable, folded back and strapped at the half-distance point (for ease of presentation) so that the setup becomes equivalent to his scenario, with the cables calibrated for identical transmission times. Accelerate the whole ‘lab’ lengthwise to 0.6c in such a way that the cables do not stretch (Born-rigidity) and let it coast again.

I used the Rindler coordinates equation [itex]t = \sqrt{x^2-\sigma^2}[/itex], where [itex]\sigma[/itex] is the distance of the start of the curve from the origin (1 and 1.6 units respectively). The origin coincides with the common light cone of Rindler observers with constant proper acceleration ([itex]a[/itex]), where [itex]a =1 / \sigma[/itex]. The other relevant equations are: [itex]\tau = \sigma[/itex]asinh(a t) and v/c = tanh(a [itex]\tau[/itex]).
The spacetime diagram is a tad busy, but gives an overview of all the values calculated for the acceleration phase and the later cruise phase.


Click on the thumbnail below if you do not see the diagram.

For simplicity, use a cable with a signal speed of exactly 0.6c, so that a signal takes exactly one time unit to travel the length of the cable. After the acceleration, the speed of the pulse relative to the original reference frame is [itex](0.6\pm0.6)/(1+0.36)[/itex]c, i.e. 0.882c in the forward direction and zero in the return direction.

In the reference frame the acceleration lasts for t=0.75 units for the blue ship and t=1.2 units for the front ship. The corresponding ship proper times are [itex]\tau_{blue}[/itex]=0.69 and [itex]\tau_{red}[/itex]=1.11 units. The latter clock must be set back by 0.42 units to [itex]\tau'_{green}[/itex]=0.69 units in order to be synchronized with blue again.

The pulse in cable 1 is transmitted simultaneously with the laser pulse and arrives back at the oscilloscope at [itex]\tau_{blue}[/itex]=1.69, while the cable 2 pulse arrives at [itex]\tau_{blue}[/itex]=2.29. This gives a time delay of 0.6 units, which is the travel time of the laser pulse. Since we expect the proper length of the cable to remain the same before and after acceleration, this seems to indicate that it is a true clock-sync-independent measurement of the one-way speed of light in an inertial frame. If correct, this also means that calibrated cables can be used to synchronize distant clocks without slow transport involved.

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ghwellsjr

You should read the wikipedia article on the one-way speed of light. You will see that you have misunderstood what Don was telling you or else he has fooled himself and you. You cannot measure the one-way speed of light, period. All attempts, no matter how complicated or subtle, either are measuring the two-way speed of light or are in fact using the equivalent of synchronized clocks.

You and/or Don have made the faulty assumption that the propagation of an electrical signal down a cable is a constant no matter the orientation of the cable. Suppose that in his setup, you start with the two 1000-foot cables stretched out 500 feet away and back. You are assuming that the time it takes for the signals to get 500 feet away is identical to the time it takes for the signals to get back so that when you stretch out one of the cables the full length it will take the same time to go one-way as it did to go both ways for half the distance. Can you see that this is flawed?

Jorrie

I think you misinterpret. The equal time for equal distance (both ways) in the 'test' is measured in the frame of the moving medium, not the reference frame. I fail to see any flaw in that.

mfb

You need the signal speed in the cable to be the same both ways. This is similar to moving objects (or sending light) between the spaceships.

ghwellsjr

I was talking only about Don's original setup, not your expanded test. His original setup does not measure the one-way speed of light. I haven't paid any attention to your expanded test. It's unnecessary if you see the problem with Don's setup.

Nugatory

Don Lincoln's version and OP Jorrie's version are interestingly different, and OP's mistake may be in not appreciating this difference.

In the original form, everything is being done in the lab frame, and the calibration process IS a clock synchronization process - the cables are clocks (easily synchronized clocks, but impractical for general use because they only tick twice so can only make a single measurement). The only possibly questionable assumption is the assumption that the signal propagation time won't depend on whether the cable is laid straight or coiled; that assumption can be verified experimentally.

This synchronization process falls apart completely as soon as the cable is set in motion, as in OP's spaceship version. So, although I see nothing wrong with Don Lincoln's one-way measurement, it's still a synchronized-clock measurement; and OP's spaceship variant does not provide a counterexample to teh need for two-way measurements.

mfb

It cannot, unless you use some other method to sync clocks (like light... but then your sync relies on the one-way-speed of light again).

ghwellsjr

The synchronization process also falls apart completely as soon as you move one end of the cable, going from coiled (or just folded in half as I suggested) to stretched out, as in Don's original setup. There is no point in proposing an expanded test if you understand the flaw in the original test setup.

Jorrie

I would like to hear more about why the sync process falls apart if I move one end of the cable. Apart from the difficulty of measuring the actual speed in cables without synchronized clocks, is there a technical reason why the signal speed (relative to cable) in a perfect cable in free space would be dependent on direction of the signal?

Nugatory

Why so? I colocate the two ends so that I can use the same clock to measure the send and receive events, drop the signal into one end of the cable and see how long it takes to come out the other... Try switching the sending and receiving ends, drag the cable around my lab, coil it ten times, a hundred times, coil it clockwise and counterclockwise, or both at the same time, rerun the same tests again with my lab rotated 90 degrees... and my one and only lab clock gives me the same transit time every time. Furthermore, I go to the trouble of moving the ends to different locations, use the non-controversial two-way speed of light to establish a simultaneity convention, and still measure the same transmission time.

So I decide that a reasonable simultaneity convention is that the transmission time of a signal (the signal speed need not be, and probably isn't, the speed of light) in the cable is constant provided the cable remains at rest relative to me. Hence my claim that the cable, after appropriate testing, IS a synchronized clock; and in the future I can use my calibrated cable as a pair of synchronized clocks without repeating the calibration process.

But of course all this does is prove that the detectors at the ends of my cables can be used as synchronized clocks. There's no challenge here to the fact that a one-way measurement requires a synchronized clock; instead I claim that a synchronized clock is hidden in Lincoln's formulation of the thought experiment, and OP has inadvertently lost it.

ghwellsjr

Let's say we take two identical parallel straight 500-foot perfect cables that propagate an electrical signal at c. We send a signal down one of them at the same time that we generate a flash of light. At the other end of the 500-foot cables we have a light detector which goes into our scope on one channel and the electrical signal from the cable goes into a second channel of the scope. We see that the light signal arrives simultaneously with the electrical signal. We also put a mirror at the detector to reflect the light back to the source and we put a tee in the cable to transmit the electrical signal back down the other cable. Back at the source we have a second light detector which we put into one channel of another scope along with the signal from the second cable in a second channel of the scope. We see that the light signal and the electrical signal arrive simultaneously.

Why did we bother with the electrical cables? We already know that when we do this experiment with light and a mirror that we cannot say that the time it takes for the light to propagate in both directions is the same unless we arbitrarily define them to be the same. If we cannot say that they are the same for light in free space, then we cannot say that they are the same for the electrical signals in cables.

ghwellsjr

I agree with everything you said here.

The problem is that you earlier said in post #6 that you saw nothing wrong with Don Lincoln's one-way measurement but his claim was that the one-way speed of light can be measured without having to use synchronized clocks. It would have made more sense if you had pointed out that there was something wrong with Don's claim because he was in fact not measuring the one-way speed of light but rather defining it by using the virtual synchronized clocks of his cables.

Jorrie

I understand what you are saying for light. My understanding is that the speed of a signal in a cable is some factor (less than unity) of c, with c the two-way speed of light. This forms the basis for one argument that I have heard for why a measurement using two cables actually constitutes a two-way measurement (e.g. Zhang 1997). It does not sound very convincing to me.

Secondly, propagation speed in a real cable has different dynamics than a pulse of light in vacuum, e.g. a real observer can co-move with a pulse in a cable. Is it correct to sweep this difference under the carpet when replacing the cables with light pulses? I'm happy either way, but I'm simply not convinced by your argument up to this point.

ghwellsjr

A person can co-move with a slowly transported clock but that doesn't mean that it also isn't following the same standard arbitrary synchronization process that we always use in Special Relativity. Whenever you do any of these things, you are defining time at a remote location, you're not measuring it. Zhang is right, you should pay attention to what he says.

Jorrie

OK, I agree that if we stay strictly in one inertial frame, there is no difference between Lincoln's two cables and two synchronized clocks. However, I have attempted to show that when we boost the final setup lengthwise to a new inertial frame, there is a difference in that the clocks go out of sync, while the cables apparently do not lose their definition of simultaneity.

To further illustrate, if we bring the two clocks slowly together after the acceleration, we expect them to be out of sync. If we bring the two ends of the straightened cable together again, I expect the cables to still have the same simultaneity as before the acceleration, i.e. I do not expect to have to adjust their lengths to show local simultaneity, a-la Lincoln's test. I may be wrong on the latter, but then I want to understand why.

Jorrie

I'm afraid when I paid my dues to Zhang's chapter 8, especially his equations 8.3.18 to 8.3.25 (and some references to earlier equations), I was left with the distinct impression that he has shown that the one-way speed of the gamma rays were within very close bounds to the two-way speed of light. His conclusion on the Alvager-Farley-Kjellman-Wallin experiment (eq. 3.1.25):

Yes, it determined [itex]c'[/itex] in terms of [itex]c[/itex], not absolutely, but does this not reveal a deep truth? Isn't any arbitrary selection of simultaneity convention, other than Einstein's, in conflict with experiment?

ghwellsjr

As Nugatory said in post #6, the cables act as a clock that ticks only twice--good for a single measurement. They are not producing a continual stream of ticks which are being counted which is what happens in a real clock. If they did that, they would go out of sync just like your real clocks, meaning the values of the counters will be different after you accelerate them and bring them together.