B Proving Reciprocity Of Time Dilation

[DanAil]

Hopefully the answer on this question is positive.

This forum has a very good reference about the Experimental Basis of Special Relativity. The tests of the Twin Paradox conclude that faster moving clocks tick/run slower, however this seems to be an 'absolute' fact. They do not show symmetry or reciprocity. A simple example is the Hafele–Keating experiment where, with reference to the center of the earth, the eastbound airplane moves faster than the westbound, and as a result its clock runs slower.

The question: Is there a way to create an experiment that could prove the following: Two objects A and B moving relative to each other. A observes that clocks on B run slower, while B observes that clocks on A run slower.

Showcasing the above should eliminate many confusions about the Special Relativity.

 

[Vanadium 50]

It will not convince anyone who isn't already convinced. Nothing will.

 

[Nugatory]

Every GPS position fix is a successful experimental test of this reciprocity (and many other predictions of relativity).

However, if you want to prove it, the easiest way is mathematical: It follows directly from the Lorentz transformations, which in turn follow from the postulates of SR - accept these postulates and the reciprocity follows as surely as geometrical theorems follow from Euclid's axioms.

 

[FactChecker]

Suppose that we leave aside, for a second, that the mathematics and logic used to accurately calculate SR time dilation is completely symmetric for two observers traveling relative to each other. Is there something that makes you think that the Earth's reference frame is unique in being the one that does not have time dilation when observed from other inertial reference frames and that this unique property holds even as the Earth orbits the sun, traveling in different directions?

 

[Ibix]

I'd think you could do it with something like Pound-Rebka turned on its side. Measure the Doppler shift between relatively moving sources amd confirm that it matches the relativistic formula, which is the naive distance-change Doppler formula multiplied by the kinetic time dilation factor.

Actually, with modern clocks you could probably do this without mucking around with the Mossbauer effect.

 

[Ibix]

I'd think you could do it with something like Pound-Rebka turned on its side. Measure the Doppler shift between relatively moving sources amd confirm that it matches the relativistic formula, which is the naive distance-change Doppler formula multiplied by the kinetic time dilation factor.

Actually, with modern clocks you could probably do this without mucking around with the Mossbauer effect.

Note, when I say "you" here, I mean skilled experimentalists with expensive kit. I don't think this is in Joe Bloggs' reach yet.

 

[DanAil]

However, if you want to prove it, the easiest way is mathematical: It follows directly from the Lorentz transformations, which in turn follow from the postulates of SR

Yes, there is no question that the theory states that, however it is definitely more convincing when we observe it - experimentally or in nature.

 

[PeterDonis]

The tests of the Twin Paradox conclude that faster moving clocks tick/run slower, however this seems to be an 'absolute' fact.

No, that's not the "absolute fact". The "absolute fact" is that the elapsed time on a clock depends on the path it takes through spacetime. There are some heuristic rules that apply to certain special cases (such as, in special relativity, i.e., in flat spacetime, the clock that remains inertial, in free fall, all the time will have more elapsed time than any other clocks between a given pair of events), but there is no rule that generalizes to all cases other than the one I stated at the start of this paragraph.

Is there a way to create an experiment that could prove the following: Two objects A and B moving relative to each other. A observes that clocks on B run slower, while B observes that clocks on A run slower.

That depends on what you mean by "run slower". If two clocks start out together, separate for a while, and then come back together, it is impossible for each to have "run slower" (i.e., show less elapsed time) than the other: either both will show the same elapsed time (in certain edge cases where the two clocks take just the right paths through spacetime), or one will show more elapsed time and the other will show less (i.e., an asymmetrical result). But if you have clocks that only meet once and then separate, you can easily construct scenarios in which, by choosing different simultaneity conventions, each clock can "observe" (which really means "calculate based on observed data") the other "running slower" relative to it.

 

[Orodruin]

The tests of the Twin Paradox conclude that faster moving clocks tick/run slower, however this seems to be an 'absolute' fact.

As Peter mentioned, this is a misrepresentation of what the twin paradox states. The twin paradox deals with differential aging, not time dilation per se. Time dilation is not even that central to relativity except in introductory textbooks and popular science. The important thing is proper time along a worldline, which is a concept of spacetime geometry.

 

[Dale]

Is there a way to create an experiment that could prove the following: Two objects A and B moving relative to each other. A observes that clocks on B run slower, while B observes that clocks on A run slower.

I think a relativistic Doppler experiment would do that. An experiment showing the moving emitter is relativistic Doppler shifted would show that they are time dilated to the lab. An experiment showing a moving absorber is relativistic Doppler shifted would show the the lab is time dilated to them.

 

[PeterDonis]

I think a relativistic Doppler experiment would do that. An experiment showing the moving emitter is relativistic Doppler shifted would show that they are time dilated to the lab. An experiment showing a moving absorber is relativistic Doppler shifted would show the the lab is time dilated to them.

This is an example of the kind of scenario I described in the last part of post #8: the "moving emitter" and the "lab" are using different simultaneity conventions to make their conclusions about who is time dilated relative to who.

 

[Orodruin]

This is an example of the kind of scenario I described in the last part of post #8: the "moving emitter" and the "lab" are using different simultaneity conventions to make their conclusions about who is time dilated relative to who.

Except that it has nothing to do with actually measuring time dilation as what you measure are physical invariants and then is based on calculations with underlying conventions.

 

[Sagittarius A-Star]

The question: Is there a way to create an experiment that could prove the following: Two objects A and B moving relative to each other. A observes that clocks on B run slower, while B observes that clocks on A run slower.

I think that should be possible with the transverse Doppler effect.

Direct measurement of transverse Doppler effect
The advent of particle accelerator technology has made possible the production of particle beams of considerably higher energy than was available to Ives and Stilwell. This has enabled the design of tests of the transverse Doppler effect directly along the lines of how Einstein originally envisioned them, i.e. by directly viewing a particle beam at a 90° angle. For example, Hasselkamp et al. (1979) observed the Hα line emitted by hydrogen atoms moving at speeds ranging from 2.53×108 cm/s to 9.28×108 cm/s, finding the coefficient of the second order term in the relativistic approximation to be 0.52±0.03, in excellent agreement with the theoretical value of 1/2.[p 10]

Source:
https://en.wikipedia.org/wiki/Relat...rect_measurement_of_transverse_Doppler_effect

There is no contradiction with reciprocal measurement of the transverse Doppler effect, because of aberration. The angle is frame-dependent: If the angle between the light and the velocity is 90° in the lab system, it is not 90° in the moving system.

If you observe the light under 90° in the lab system, then the observed frequency of the moving clock will be lower than that of an optical clock at rest, according to the factor $1/\gamma$.

If you observe the light under an angle in the lab system, that makes the angle in the moving system to be 90°, then the observed frequency of the moving clock will be higher than that of an optical clock at rest, according to the factor $\gamma$. That means that the clock at rest in the lab system is regarded as time dilated in the frame of the moving clock.

With modern aluminium-ion clocks, time dilation was measured already at 5 m/s. That should it make possible to have measurement equipment also in the moving frame.

To measure the effect of motion, rather than move the whole clock, the team applied an electric field, making the aluminium ions vibrate. Again, the apparatus was sensitive enough to measure a time dilation, even when the atoms were moving at just $5 ms^{-1}$ — slower than a running athlete.

Source:
https://www.nature.com/articles/nphys1848

 

[Orodruin]

I think that should be possible with the transverse Doppler effect.Source:
https://en.wikipedia.org/wiki/Relat...rect_measurement_of_transverse_Doppler_effect

There is no contradiction with reciprocal measurement of the transverse Doppler effect, because of aberration. The angle is frame-dependent: If the angle between the light and the velocity is 90° in the lab system, it is not 90° in the moving system.

If you observe the light under 90° in the lab system, then the observed frequency of the moving clock will be lower than that of an optical clock at rest, according to the factor $1/\gamma$.

If you observe the light under an angle in the lab system, that makes the angle in the moving system to be 90°, then the observed frequency of the moving clock will be higher than that of an optical clock at rest, according to the factor $\gamma$. That means that the clock at rest in the lab system is regarded as time dilated in the frame of the moving clock.

With modern aluminium-ion clocks, time dilation was measured already at 5 m/s. That should it make possible to have measurement equipment also in the moving frame.Source:
https://www.nature.com/articles/nphys1848

Again, this is based on calculations that presume a simultaneity convention - just as time dilation does. The measurement itself is something invariant. A priori, you cannot measure time dilation without also setting up and synchronising the clocks of the frame you are planning to use.* That is fine though, time dilation is not that central to SR - differential aging and other invariant measurements are.

* By this I mean setting up an invariant measurement that would correspond to measuring the the elapsed time in a particular frame based on a particular simultaneity convention. The measurement itself however being frame independent. This could be done by, for example, Einstein synchronising a set of watches.

 

[vanhees71]

The socalled "twin paradox" can be completely avoided by simply specifying accurately what's objectively described by it: You take two accurate clocks starting at some time at a common place and synchronize them. Then these clocks are moving for a while and then their readings are compared at some time at the same place again.

The standard hypothesis of (special as well as general) relativity is that each clocks show the proper time,
$$\tau_{j}=\int_0^{\lambda} \mathrm{d} \lambda \sqrt{g_{\mu \nu}[x(\lambda)] \dot{x}_j^{\mu}(\lambda) \dot{x}_j^{\nu}(\lambda)}.$$
Here $x_{j}^{\mu}(\lambda)$ is the world line of clock $j$, ($j \in \{1,2\}$).

As you see, the proper times meausured by these clocks is completely independent on any arbitrary choice: It's independent of the choice of the reference frame and the coordinates you use to define the world lines of these clocks as well as the parametrization (i.e., which arbitrary world-line parameter, $\lambda$, you choose). These kinds of frame- and parametrization invariant quantities are the physically observable quantities within relativistic physics, and nothing else. Keeping this in mind avoids any possible "paradoxa".

This effect of the dependence of proper times on the worldline the corresponding "clock" takes, has been confimed with very high accuracy for many kinds of "clocks", among them also using the lifetime of unstable particles or nuclei in accelerators or directly "macroscopic clocks" in the Hafele-Keating experiment.

 

[Sagittarius A-Star]

A priori, you cannot measure time dilation without also setting up and synchronising the clocks of the frame you are planning to use.*

* By this I mean setting up an invariant measurement that would correspond to measuring the the elapsed time in a particular frame based on a particular simultaneity convention. The measurement itself however being frame independent. This could be done by, for example, Einstein synchronising a set of watches.

That's correct. On the other hand, it is possible to measure $d\tau / dt$ without synchronized clocks, by using the transverse Doppler effect. The easiest example showing this would be a moving optical clock circling around an inertial observer at constant angular velocity with a constant radius. The observer in the center of the circle compares the received frequency with that of a clock at rest near to him. The observer needs no second clock at rest in his frame, to carry-out this measurement.

 

[Nugatory]

On the other hand, it is possible to measure dτ/dt ….

I would much prefer calling that a measurement of the ratio of the proper time between two emission events and the proper time between two reception events. Sure, we can introduce a coordinate time $t$ that is, for events on the worldline of one of the clocks, equal to the proper time along that worldline…. But does that add anything?

 

[Orodruin]

On the other hand, it is possible to measure dτ/dt without synchronized clocks, by using the transverse Doppler effect.

That is still something computed based on a simultaneity convention. The measurement itself measures the ratio of proper time between signal receptions and that between signal emissions which is invariant. The return of the clock to the initial position after a full turn if anything is a measure of differential aging.

 

[Sagittarius A-Star]

That is still something computed based on a simultaneity convention. The measurement itself measures the ratio of proper time between signal receptions and that between signal emissions which is invariant.

Computed, yes. My example shows, that not always two physical clocks at rest are needed as part of the relevant measurement.

Something computed based on a simultaneity convention is needed to argue, that the $dt$ between the two reception events is the same as the $dt$ between the two related emission events: The delay caused by the light is the same for both reception events, because no longitudinal Doppler effect is involved.

The elapsed proper time of the moving clock between any coordinate times $t_1$ and $t_2$ is (measurement result marked in red):

$\require{color} \Delta \tau = \int_{\tau_1}^{\tau_2} \color{black}\mathrm{d}\tau = \int_{t_1}^{t_2} \color{red}\frac{\mathrm{d}\tau}{\mathrm{d}t}\color{black}\mathrm{d}t = \int_{t_1}^{t_2} \color{red}\frac{1}{\gamma}\color{black}\mathrm{d}t$.

 

[Orodruin]

My example shows, that not always two physical clocks at rest are needed as part of the relevant measurement.

That is true also for the longitudinal Doppler effect. You just get a different computation based on the simultaneity convention as you need to also take the increasing/decreasing distance into account.

 

[vanhees71]

Computed, yes. My example shows, that not always two physical clocks at rest are needed as part of the relevant measurement.

Something computed based on a simultaneity convention is needed to argue, that the $dt$ between the two reception events is the same as the $dt$ between the two related emission events: The delay caused by the light is the same for both reception events, because no longitudinal Doppler effect is involved.

The elapsed proper time of the moving clock between any coordinate times $t_1$ and $t_2$ is (measurement result marked in red):

$\require{color} \Delta \tau = \int_{\tau_1}^{\tau_2} \color{black}\mathrm{d}\tau = \int_{t_1}^{t_2} \color{red}\frac{\mathrm{d}\tau}{\mathrm{d}t}\color{black}\mathrm{d}t = \int_{t_1}^{t_2} \color{red}\frac{1}{\gamma}\color{black}\mathrm{d}t$.

It's much more elucidating to emphasize the invariance of proper time than to apply it only to some coordinate time.

 

[Sagittarius A-Star]

It's much more elucidating to emphasize the invariance of proper time than to apply it only to some coordinate time.

Referencing coordinate times is needed to answer the OP's time-dilation question:

The question: Is there a way to create an experiment that could prove the following: Two objects A and B moving relative to each other. A observes that clocks on B run slower, while B observes that clocks on A run slower.

 

[Orodruin]

Referencing coordinate times is needed to answer the OP's time-dilation question:

I disagree here. You really do not need to do anything to accommodate OP’s time dilation question because time dilation is reciprocal by definition of the standard simultaneity convention. It is also not reciprocal and takes a different form with some other conventions so time dilation by itself is a bit of a red herring. What matters in the end are invariant measurements, such as differential aging.

 

[Dale]

Except that it has nothing to do with actually measuring time dilation as what you measure are physical invariants and then is based on calculations with underlying conventions.

This is an example of the kind of scenario I described in the last part of post #8: the "moving emitter" and the "lab" are using different simultaneity conventions to make their conclusions about who is time dilated relative to who.

Sure. There is no problem stipulating to the fact that there are standard conventions and that time dilation itself is intrinsically related to the standard simultaneity convention.

Nevertheless, in the scientific literature there are many experiments claiming to test time dilation. Several of them do so using the Doppler effect. So showing that the same relativistic Doppler effect applies for a moving transmitter and a moving receiver is reasonably portrayed as an experimental test of the reciprocity of time dilation.

 

[Orodruin]

Sure. There is no problem stipulating to the fact that there are standard conventions and that time dilation itself is intrinsically related to the standard simultaneity convention.

Nevertheless, in the scientific literature there are many experiments claiming to test time dilation. Several of them do so using the Doppler effect. So showing that the same relativistic Doppler effect applies for a moving transmitter and a moving receiver is reasonably portrayed as an experimental test of the reciprocity of time dilation.

Sure, but then what they are really testing is that the Doppler effect does not depend on whether receiver or emitter is moving in the lab frame.

 

[PeterDonis]

it has nothing to do with actually measuring time dilation

Yes, fair point: "time dilation" itself is not something that can be directly measured since it always depends on a choice of simultaneity convention.

 

[Sagittarius A-Star]

What matters in the end are invariant measurements, such as differential aging.

The measurement of differential aging proofs also, that time-dilation of the "moving" clock with reference to the rest-frame of the "stationary" clock is a real effect.

This 'time dilation', like length contraction, is no accident of convention but a real effect. Moving clocks really do go slow. If a standard clock is taken at uniform speed $v$ through an inertial frame $S$ along a straight line from point $A$ to point $B$ and back again to $A$, the elapsed time $T_0$ indicated on the moving clock will be related to the elapsed time $T$ indicated on the clock fixed at $A$ by the Eq. (21) ...

Source:
http://www.scholarpedia.org/article/Special_relativity:_kinematics#Special_relativistic_kinematics

 

[Dale]

Sure, but then what they are really testing is that the Doppler effect does not depend on whether receiver or emitter is moving in the lab frame.

I agree completely, but if you want experimental evidence for the reciprocity of time dilation, that is a reasonable thing to really test. It is consistent with the existing literature testing the Doppler effect and describing the results in terms of time dilation (under the standard conventions).

 

[Orodruin]

The measurement of differential aging proofs also, that time-dilation of the "moving" clock with reference to the rest-frame of the "stationary" clock is a real effect.

No, it shows that differential aging is a real effect. How you accounted for that differential aging in terms of coordinates is less relevant than the actual invariant measurement.

I agree completely, but if you want experimental evidence for the reciprocity of time dilation, that is a reasonable thing to really test. It is consistent with the existing literature testing the Doppler effect and describing the results in terms of time dilation (under the standard conventions).

If you want experimental evidence for time dilation you’re out of luck (unless you want to do Einstein synchronization -or similar- and work with that because that physically realizes the coordinate time measurement in the rest frame of the synchronized clocks). I agree with what @PeterDonis said:

Yes, fair point: "time dilation" itself is not something that can be directly measured since it always depends on a choice of simultaneity convention.

If it is a simultaneity convention that can be physically realized then you can measure ”time dilation” and its reciprocity. You cannot really measure it unless you physically construct the coordinate time of each frame.

 

[vanhees71]

Sure. There is no problem stipulating to the fact that there are standard conventions and that time dilation itself is intrinsically related to the standard simultaneity convention.

Nevertheless, in the scientific literature there are many experiments claiming to test time dilation. Several of them do so using the Doppler effect. So showing that the same relativistic Doppler effect applies for a moving transmitter and a moving receiver is reasonably portrayed as an experimental test of the reciprocity of time dilation.

That's of course only right for the Doppler effect of light in vacuum. There the effect depends only on the relative velocity between source and receiver. For other kinds of waves, where a medium is involved (like sound waves, or em. waves in a dielectric), you have the (local) rest frame of the medium as an additional ingredient in the physical description, i.e., in some sense a "preferred reference frame". Here's a review on the general relativistic Doppler effect (ignoring the case of "anomalous dispersion" though):

https://itp.uni-frankfurt.de/~hees/pf-faq/rela-waves.pdf

 

[Sagittarius A-Star]

time dilation is reciprocal by definition of the standard simultaneity convention.

To my understanding, reciprocity of time dilation follows from the principle of relativity (SR postulate 1). Therefore, it is sometimes part of the derivation of LT.

Of course, SR postulate 1 refers to standard inertial coordinate systems, and those are based on Einstein-synchronization. The reason for using Einstein-synchronization:

The basic principle of clock synchronization is to ensure that the coordinate description of physics is as symmetric as the physics itself.

Source:
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations

 

[Orodruin]

To my understanding, reciprocity of time dilation follows from the principle of relativity (SR postulate 1). Therefore, it is sometimes part of the derivation of LT.

Of course, SR postulate 1 refers to standard inertial coordinate systems, and those are based on Einstein-synchronization. The reason for using Einstein-synchronization:

Source:
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations

Well … yes. I don’t see how any of that would contradict what I said in what you quoted.

The ”world’s fastest LT derivation” to me sounds a bit pretentious. According to what measure? Starting from where exactly? He also seems to make it ”faster” by leaving most of the algebra to the reader.

 

[Sagittarius A-Star]

He also seems to make it ”faster” by leaving most of the algebra to the reader.

Yes. I addition, @robphy found an interesting pattern in this derivation:

T+X and T-X are light-cone coordinates

 

[Orodruin]

Yes. I addition, @robphy found an interesting pattern in this derivation:

Yes, I kind of noted that as well. It makes me wonder if it would not be ”faster” to utilize this from the beginning, but I did not have time to think much more about it.

Generally, I feel that much of how relativity is typically presented is based on historical development rather than focusing on a modern geometrical approach.

 

[robphy]

Generally, I feel that much of how relativity is typically presented is based on historical development rather than focusing on a modern geometrical approach.

In my experience, typical physicists follow the “physicsy” textbook MichelsonMorley/Einsteinian storyline, stopping short of Minkowskian spacetime…. probably because of Einstein’s initial reaction to the mathematicians.
In other words, “math is hard”.

 

[DanAil]

This is a very helpful discussion. Thank you!

Summarizing the comments so far on how to experimentally prove reciprocity in time dilation:

1. GPS position fix (gathering more information on that)
2. Doppler experiments (very interesting):

a. Pound-Rebka experiment​

b. Relativistic Doppler experiments​

c. Transverse Doppler effect​

3. Time dilation itself can not be directly measured since it depends on a choice of simultaneity convention..

 

[Dale]

If you want experimental evidence for time dilation you’re out of luck (unless you want to do Einstein synchronization -or similar- and work with that because that physically realizes the coordinate time measurement in the rest frame of the synchronized clocks)

I understand what you are saying, but I think that this claim is too strong.

In a standard Einstein synchronized inertial reference frame the difference between the Newtonian Doppler shift and the relativistic Doppler shift is the relativistic time dilation. Therefore, a measurement of the relativistic Doppler shift itself is sufficient to establish that there is time dilation in a standard inertial frame. It is not necessary to fully physically realize the coordinate time in each frame.

Certainly, the Ives and Stilwell experiment is often considered a test of time dilation, and it was a Doppler measurement. And Einstein also proposed using the transverse Doppler shift to test relativistic time dilation. Neither asserted that a full physical realization of the moving coordinates was needed.

 

[Orodruin]

In a standard Einstein synchronized inertial reference frame

The key is that you have to refer to a reference frame at all to get at your conclusion that time dilation is the difference between the relativistic and classical Doppler shift. You basically define something in a particular way and then should not be surprised to find that what you defined has the defined property. By adopting Einstein synchronization you are making time dilation reciprocal by default.

It is worth noting that, like differential aging, Doppler type measurements involve closed curves in spacetime. In this case the world lines of emitter and receiver and (at least) two null geodesics.

 

[vanhees71]

I understand what you are saying, but I think that this claim is too strong.

In a standard Einstein synchronized inertial reference frame the difference between the Newtonian Doppler shift and the relativistic Doppler shift is the relativistic time dilation. Therefore, a measurement of the relativistic Doppler shift itself is sufficient to establish that there is time dilation in a standard inertial frame. It is not necessary to fully physically realize the coordinate time in each frame.

Most directly is the "extreme case" of the transverse Doppler effect, i.e., the frequency shift in observation perpendicular to the direction of the wave propagation, which is purely due to time dilation. An observer at rest with the source moving perpendicular to the direction of wave propagation you get a redshift. For an observer moving perpendicular to the direction of wave propgation (from his point of view) in the rest frame of the source you get a blue shift.

Certainly, the Ives and Stilwell experiment is often considered a test of time dilation, and it was a Doppler measurement. And Einstein also proposed using the transverse Doppler shift to test relativistic time dilation. Neither asserted that a full physical realization of the moving coordinates was needed.

 

[Sagittarius A-Star]

the frequency shift in observation perpendicular to the direction of the wave propagation, which is purely due to time dilation.

From this follows:

This 'time dilation', like length contraction, is no accident of convention but a real effect.

Source:
http://www.scholarpedia.org/article/Special_relativity:_kinematics#Special_relativistic_kinematics

 

[Sagittarius A-Star]

It is worth noting that, like differential aging, Doppler type measurements involve closed curves in spacetime. In this case the world lines of emitter and receiver and (at least) two null geodesics.

SR, as a classical theory, works equally good with the wave model and particle model of light.

• In case of the wave model, you can make the spatial distance and angle (in the frame of the receiver) between the two light paths arbitrarily small, by choosing an arbitrarily high frequency in the transverse Doppler effect experiment. So the delta between the two delays by the light transmission can be made arbitrarily small.
• In case of the particle model (photon), you need only one null geodesic. Because of $E=h\nu$, the 4-momentum of the photon transforms according the the same aberration/Doppler formula as the 4-frequency.

 

[Orodruin]

SR, as a classical theory, works equally good with the wave model and particle model of light.

• In case of the wave model, you can make the spatial distance and angle (in the frame of the receiver) between the two light paths arbitrarily small, by choosing an arbitrarily high frequency in the transverse Doppler effect experiment. So the delta between the two delays by the light transmission can be made arbitrarily small.
• In case of the particle model (photon), you need only one null geodesic. Because of $E=h\nu$, the 4-momentum of the photon transforms according the the same aberration/Doppler formula as the 4-frequency.

The wave model is based on derivatives, which implicitly assume two nearby geodesics.

The particle model assumes parallel transport, which assumes a vector field in a neighbourhood around the geodesic which coincidences with the 4-momentum at the point of origin.

 

[PAllen]

From @Sagittarius A-Star Rindler link:

"Another striking instance of time dilation is provided by the 'relativistic focusing' of beams of charged particles emerging from high-energy particle accelerators: they are surprisingly coherent. If we regard a stationary cloud of such particles, expanding under its own electrostatic repulsion, as a kind of clock, then the slow spread of beams at high velocity is an almost visible manifestation of the slowing down of a series of such clocks moving at high speed."

This is a really interesting example, but has the same "refutation" as atmospheric muons reaching the ground: The reason is not time dilation but distance contraction in the COM frame of particle cluster. Thus, it remains correct to state that pure time dilation (rather than differential aging or Doppler) is a coordinate dependent description of some some measurement. That is, the measurements are invariant, but whether time dilation is the reason is frame dependent.

 

[Sagittarius A-Star]

That is, the measurements are invariant, but whether time dilation is the reason is frame dependent.

Time-dilation with reference to the rest frame of the receiver is the reason, if the angle is 90° in the receiver frame.

Then, because of aberration, the reason can be also described as a combination of time-dilation of the receiver and a longitudinal Doppler effect component, both with reference to the rest frame of the sender.

 

[vanhees71]

SR, as a classical theory, works equally good with the wave model and particle model of light.

• In case of the wave model, you can make the spatial distance and angle (in the frame of the receiver) between the two light paths arbitrarily small, by choosing an arbitrarily high frequency in the transverse Doppler effect experiment. So the delta between the two delays by the light transmission can be made arbitrarily small.
• In case of the particle model (photon), you need only one null geodesic. Because of $E=h\nu$, the 4-momentum of the photon transforms according the the same aberration/Doppler formula as the 4-frequency.

The wave model is based on derivatives, which implicitly assume two nearby geodesics.

The particle model assumes parallel transport, which assumes a vector field in a neighbourhood around the geodesic which coincidences with the 4-momentum at the point of origin.

The "particle model" is derived from the more comprehensive "wave model" via the eikonal approximation. It's also highly misleading to think about photons in terms of localized zero-mass particles. Photons are described by quantum field theory and defined as asymptotic free Fock states.