Proving Reciprocity Of Time Dilation

[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.