# Hidden assumption in SRT?

Einstein used an unstated assumption in SRT theory that seems like it cannot be correct. In his clock synchronization of sending a signal from A to a mirror at B with the timed return of the signal, etc., he assumes that the reflection at the mirror is instantaneous. I am sure it is PDQ, but can it be *instant*, with absolutely no lag between the arrival and departure of a photon? If that photon carries momentum, then an instant turnaround would deliver an infinite impulse to the electrons in the mirror. It probably doesn’t make any difference in the theory, but I am just curious.

Is his synchronization scheme actually used in practice? If someone could do a Mossbauer type experiment with the photons / gammas reflected between emission and reception, would there be a Doppler type redshift of the spectral line due to recoil of the electrons in the reflector?

A.T.
If that photon carries momentum, then an instant turnaround would deliver an infinite impulse to the electrons in the mirror.
The impulse is finite just like the momentum. Impulse is just the amount of transferred momentum.
http://en.wikipedia.org/wiki/Impulse_(physics)

Nugatory
Mentor
Einstein used an unstated assumption in SRT theory that seems like it cannot be correct. In his clock synchronization of sending a signal from A to a mirror at B with the timed return of the signal, etc., he assumes that the reflection at the mirror is instantaneous. I am sure it is PDQ, but can it be *instant*, with absolutely no lag between the arrival and departure of a photon? If that photon carries momentum, then an instant turnaround would deliver an infinite impulse to the electrons in the mirror. It probably doesn’t make any difference in the theory, but I am just curious.
You are right that the turnaround is not instantaneous for any real-world mirror, but that doesn't stop us from reasoning as if we had an ideal mirror. When it comes to a real setup, we can make the effect as small as we want by increasing the distance between the mirrors while the turnaround time remains constant. Make it small enough and we can treat it as if it is zero, and its actual non-zero value is just another contributor to the innumerable other effects that contribute to the error bars that you see in published descriptions of real-world experiments.

• PAllen
Nugatory
Mentor
I am sure it is PDQ, but can it be *instant*, with absolutely no lag between the arrival and departure of a photon?
Intuitively, you would expect the time to be proportional to ##1/\omega##, which does qualify as PDQ.

When working a relativity problem, it's almost always better (more descriptive, less likely to lead to misconceptions that must be unlearned before you move on to quantum mechanics) to think of light as waves instead particles. When you hear the word "photon" in a relativity problem, chances are that the speaker really meant "light signal" or "flash" or "pulse of light".

Dale
Mentor
2020 Award
Force isn't a concept that is used much at all in quantum mechanics. You can, of course, make a force operator and calculate the expectation value. In all likelihood it would not be in an eigenstate, so there would be some uncertainty.

pervect
Staff Emeritus
My understanding is that reflection can be modeled like refraction at the microscopic level using the extinction theorem.
http://en.wikipedia.org/wiki/Ewald–Oseen_extinction_theorem

Also see "penetration depth" and "skin depth".

Basically I believe that if you shine a laser at a front surface mirror, the pulse will get smeared out a bit ,due to the laser penetrating the mirror (the amount of energy that penetrates drops exponentially with the penetration depth). I'd expect that the light that penetrates a bit before being reflected will have additional delays, so that a narrow pulse will get smeared out. The front edge of the pulse shouldn't experience additional delays though.

On a quantum level, I'd expect there would be some average time for a photon to be re-emitted after it was scattered. The first photons should get scattered by the front layer of atoms. I've never seen this calculated, but I'd guess that it would be really really small. I think a theoretical estimate of this would be interesting - my QM / QED isn't good enough to do this.

I would regard these effects as experimental details, subject to experimental tests, and not as fundamental challenges to SR. I'd expect the effects to be very minor and additionally able to be tested and/or removed by using alternate synchronization setups such as the one mentioned by ghwellsjr - though the effects are so small one might only be able to put an upper bound on them rather than measure their value. Since I regard this as an experimental detail, rather than a fundamental objection, I would think it would be wise to look at the actual techniques used in the actual experiments rather than to get too tied up into analyzing Einstein's theoretical method of synchronization for flaws rather than regarding it as an attempt to communicate the basic idea clearly rather than an experimental prescription.

More fundamentally, I view the whole issue of one-way light speed is a red herring. The only place we really need to address synchronization issues at the level needed to demonstrate SR is when we measure velocity. If we measure velocities by measuring proper velocity with a clock onboard the moving object over a course of known proper length, we can develop the framework of relativity without worrying about synchronization issue at all. The reasons for the way we define velocity as being measured with two synchronized clocks and not one proper clock are , in my opinion, again experimental reasons , due basically to the fact that vibration issues affectd\ the precision of moving clocks so it was preferred to eliminate this by having the clocks stationary and well isolated from vibration.

PAllen
Well, proper velocity is a funny concept in that it mixes quantities from two frames (if you treat an inertial 'mover' as a frame). Alternatively, it combines an invariant (proper time along a world line) with a frame variant quantity (distance), with the distance measured NOT according to any natural coordinates associated with the clock.

[Edit: On the other hand, on more thought, it seems equivalent to time dilation (derivative of proper time by coordinate time). This is frame dependent but useful, so I guess why not proper velocity (spatial coordinates derivative by proper time)? Just not something I'm used to. Yet, I still wonder that it is too close to so much confusion and false paradoxes that come from mixing quantities from different frames.]

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pervect
Staff Emeritus
Lets me present an example to illustrate the idea. Suppose we observe that, according to the wall clock, it takes around 8 hours to fly from Los Angeles to New York, but only about 2 hours to fly from New York to Los Angeles. And we want to find out, by experiment, how much of this is due to physical effects like headwinds, and how much is due to effects like the wall clocks not being synchronized. We will pretend we don't already know that New York is three hours ahead of LA. and think about experiments we might do.

Well, we could do a lot of testing to send signals that we thought should travel at a constant rate between the two cities, though we'd wind up worrying about the same basic issues - how do we know the winds don't affect the light signals, too? What bout "ether winds?" How do we really know that the signal takes the same time to travel both ways.

Or we could just put a clock on the plane.. When the plane clock shows that it's really about 5 hours each way, we know that most of the issue is the wall clocks not "fairly" synchronized, without debating the issue extensively and trying to find loopholes in what we mean by "fair synchronization". There's only one clock, so the synchronization issues become non-issues.

With more precise readings, we could come up with a good estimate of the headwinds that affected the groundspeed.

As far as the physics go, we might also be interested in the momentum of the planes, and/or their energy, while they are flying. If we didn't account for the unfair clock synchronization, we'd get silly results like the momentum being 4x the value in one direction than the other. If we did some studies with crashing planes into each other and studying the debris field (for momentum) or crashing the planes into calorimeters (for energy) we could eventually disprove that the momentum and/or energy was so greatly different depending on the direction of flight.

Or we could just use the proper velocity to calculate the theoretical value of momentum via m*v_proper, and get the right value for it without worrying about the clock synchronization at all. Happily, this formula would be relativistically correct. There would be a similar formula for the energy in terms of proper velocity (I haven't worked out the details, not sure it matters), and we still wouldn't have to worry about whether our clocks were synched fairly or not, because there would only be one clock.

What we would have to worry about instead is the effect of acceleration on the travelling clock causing it to not keep good time. For instance, the atoms in an atomic clock wouldn't be affected by jerks and jolts, but the phase locked loops used to get the time readout would be.

We might still wind up with some notion of clock synchronization to be able to chart the course of the plane more easily, but it wouldn't affect things like the values for momentum and energy if we did it unfairly, so we wouldn't have to spend a lot of time developing the concepts of fairness and isotropy.

• TumblingDice and Nugatory
Lets me present an example to illustrate the idea. Suppose we observe that, according to the wall clock, it takes around 8 hours to fly from Los Angeles to New York, but only about 2 hours to fly from New York to Los Angeles. And we want to find out, by experiment, how much of this is due to physical effects like headwinds, and how much is due to effects like the wall clocks not being synchronized. We will pretend we don't already know that New York is three hours ahead of LA. and think about experiments we might do.[..]

When the plane clock shows that it's really about 5 hours each way, we know that most of the issue is the wall clocks not "fairly" synchronized, without debating the issue extensively and trying to find loopholes in what we mean by "fair synchronization". There's only one clock, so the synchronization issues become non-issues. [..]

If we did some studies with crashing planes into each other and studying the debris field (for momentum) or crashing the planes into calorimeters (for energy) [..]

[..] we could just use the proper velocity to calculate the theoretical value of momentum via m*v_proper, and get the right value for it without worrying about the clock synchronization at all. Happily, this formula would be relativistically correct.
[..]
This is becoming rather off-topic, but I'd appreciate a precision here. For while it avoids the synchronization errors, your illustration looks rather classical (non-relativistic) to me, and of course airplane travel involves hardly measurable relativistic effects. In particular, what do you mean with "relativistically correct" m*v_proper? To make it more tangible, let's say an "airplane" of 10'000 kg with a ground speed of 0.8c. With your method and equation, how do you correctly predict the released energy upon a crash?

PAllen
This is becoming rather off-topic, but I'd appreciate a precision here. For while it avoids the synchronization errors, your illustration looks rather classical (non-relativistic) to me, and of course airplane travel involves hardly measurable relativistic effects. In particular, what do you mean with "relativistically correct" m*v_proper? To make it more tangible, let's say an "airplane" of 10'000 kg with a ground speed of 0.8c. With your method and equation, how do you correctly predict the released energy upon a crash?
I guess you just use:

E2 = p2c2 + m2c4

I don't think Pervect meant to suggest that proper velocity allows you to use Newtonian formulas in general.

However, I would pose: How do you compute differential aging with proper velocity? Well you integrate coordinate velocity / proper velocity by coordinate time. Hmm, what have we gained here?

pervect
Staff Emeritus
This is becoming rather off-topic, but I'd appreciate a precision here. For while it avoids the synchronization errors, your illustration looks rather classical (non-relativistic) to me, and of course airplane travel involves hardly measurable relativistic effects. In particular, what do you mean with "relativistically correct" m*v_proper? To make it more tangible, let's say an "airplane" of 10'000 kg with a ground speed of 0.8c. With your method and equation, how do you correctly predict the released energy upon a crash?
The illustration is rather classical. If you use a literal airplane, going at about one millionth of the speed of light or a little less ##10^{-6}c##, then the difference between the relativistic momentum and the ordinary momentum will be the factor of gamma, only be about 5 parts in ##10^{13}##. You probably wouldn't be able to measure this because momentum measurements (especially from debris fields) aren't so precise.

However, modern prototype atomic clocks have been reported (see http://www.nist.gov/pml/div688/mercury_atomic_clock.cfm ) with accuracies of 1 part in ##10^{16}##, so if you did a round-trip airplane flight you could measure the difference between the round trip proper time and the round trip ground time with such a clock. You'd have to account for additional GR effects in such an experiment though.

The airplane can also be interpreted as a metaphor, it doesn't really have to literally be an airplane, I chose an airplane because it was so familiar. The wind doesn't have to be a headwind, either, the wind can be a metaphor as well. The metaphor is intended to be precise enough that it addresses the fundamental arguments even when one replaces the airplane by an ion, for instance.

The actual predictions of relativity, which I just "tossed in" without explaining them, are that the Newtonian momentum, mv, would be replaced by the relativistic momentum m v_proper = gamma m v, and the Newtonian energy, (1/2) m v^2, would be replaced by
E_kinetic = ##\sqrt{(pc)^2 + (mc^2)^2} - mc^2##, where the momentum p = m v_proper.

A series expansion gives the kinetic energy ##E_k## in terms of the proper velocity ##v_p## as
##E_k = \frac{1}{2} m v_p^2 \left(1 - \frac{1}{8}(\frac{v_p}{c})^2 + \frac{1}{16} (\frac{v_p}{c})^4 + ... \right)##

Also
##v_p = \gamma \, v \approx 1 + \frac{1}{2} (\frac{v}{c})^2 + \frac{3}{8} (\frac{v}{c})^4 + ...##

The illustration is rather classical.

[..]

The actual predictions of relativity, which I just "tossed in" without explaining them, are that the Newtonian momentum, mv, would be replaced by the relativistic momentum m v_proper = gamma m v, and the Newtonian energy, (1/2) m v^2, would be replaced by
E_kinetic = ##\sqrt{(pc)^2 + (mc^2)^2} - mc^2##, where the momentum p = m v_proper.
[..]
OK, that's immediately the standard equation for momentum, as determined with a standard inertial frame... thus your v_proper = gamma v.
: ah now I see, with v_proper you mean celerity! And of course you use the adapted equations. Thus you could for example measure your clock times when passing certain points and use a pre-established map or ask someone on the ground to measure the distance between those points and radio that value to you. ..

In practice synchronization methods turned out to be very useful, for example for "universal time". Bringing the discussion back to the topic, typical modern synchronization methods don't use mirrors but identical cables as ghwellsjr mentioned (and how could "proper time" be used for Bertozzi's experiment?), or one-way radio signals. For equal path lengths one then assumes the times to be equal, else a correction is made using the known speed of light (nowadays the speed of light in vacuum is even "known" by definition, and the meter has been redefined accordingly for increased precision). That avoids any possible imprecision from delay at a mirror surface.

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