# Time dilation and Transverse Doppler Effect – where does energy disappear?

Kinematic effects of Special Relativity like time dilation and length contraction are well known.

Article in Wikipedia makes it clear:

https://en.wikipedia.org/wiki/Time_dilation

“In the special theory of relativity, a moving clock is found to be ticking slowly with respect to the observer's clock. If Sam and Abigail are on different trains in near-lightspeed relative motion, Sam measures (by all methods of measurement) clocks on Abigail's train to be running slowly and similarly, Abigail measures clocks on Sam's train to be running slowly.”

One practical way to measure the amount of time dilation is measuring frequency of relatively moving source of radiation at points of closest approach – the Transverse Doppler Effect. That means, if Sam radiates with proper frequency f, Abigail measures frequency $$f_s/\gamma$$. If Sam measures frequency of Abigail, he measures $$f_a/\gamma$$. That means, the photon redshifts and redshifts again.

According to this logic, if Sam and Abigail exchange a photon, the photon will finally vanish. Or will Sam and Abigail will vanish themselves? Where does the energy disappear?

Please don't confuse it with relativistic Doppler redshift when emitter and absorber recede.

• According to this logic, if Sam and Abigail exchange a photon, the photon will finally vanish. Or will Sam and Abigail will vanish themselves? Where does the energy disappear? - I have read this a couple of times and have no idea what this means. Where did you get this from? Jul 7, 2017 at 6:39
• Let’s Sam and Abigail are very long relatively moving parallel perfect mirrors. Photon oscillates between these mirrors. Travel path (travelled distance) of photon is always the same. But each mirror measures redshift (time dilation). Thus, photon redshifts at each oscillation? Distance between mirrors doesn’t change. Where the "loss" of photon’s energy is going to? What is not clear here? Jul 7, 2017 at 12:43

Taken from the Wikipedia article on transverse Doppler shift:

Sometimes the question arises as to how the transverse Doppler effect can lead to a redshift as seen by the "observer" whilst another observer moving with the emitter would also see a redshift of light sent (perhaps accidentally) from the receiver.

It is essential to understand that the concept "transverse" is not reciprocal. Each participant understands that when the light reaches them transversely as measured in terms of that person's rest frame, the other had emitted the light afterward as measured in the other person's rest frame. In addition, each participant measures the other's frequency as reduced ("time dilation"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity.

• Thank you very much, but this chapter claims, that in certain frame photon was first absorbed and then emitted. I think it is absolutely impossible. In general it's just a mess of unrelated words. Emitter emits a photon, photon hits transversely moving mirror and comes back (transversely) to emitter. Once it was emitter, now it is observer. Jul 7, 2017 at 12:24

One of practical ways to measure amount of time dilation is measuring frequency of relatively moving source of radiation at points of closest approach – the Transverse Doppler Effect. That means, if Sam radiates with proper frequency f, Abigail measures frequency 𝑓𝑠/𝛾. If Sam measures frequency of Abigail, he measures 𝑓𝑎/𝛾. That means, the photon redshifts and redshifts again.

Actually, because of relativistic aberration and the relativity of simultaneity these are completely different photons. When in Abigail’s frame the moment that the transverse photon Sam is emitted is a specific event on Sam’s worldline. When that event is transformed to Sam’s frame it is actually emitted after the point of closest approach. And vice versa. So there is no redshifting and redshifting again, and hence no photon disappearing.

Please don't confuse it with relativistic Doppler redshift when emitter and absorber recede.

This is not possible since they are the same thing. What is transverse Doppler in one frame is redshift when emitter and absorber recede in the other frame. Since you are looking at this scenario in both directions it is unavoidable.

Let's say Sam has some light trapped in a box. According to Agibail that light in the box has momentum into the direction of motion of the box.

Transverse light according to Agibail is such that it has only transverse momentum.

If Sam opens the lid on the top of the box then according to Sam the out flowing light has no longitudinal momentum. According to Agibail it has longitudinal momentum. (Top is the side that points towards Agibail)

So Agibail must instruct Sam to point the box in the backwards direction. And now we can see where the energy "disappears".

It becomes kinetic energy of Sam, according to Agibail. It becomes kinetic energy of Agibail according to Sam.

• I mostly agree. So, energy of photon turns into kinetic and their relative velocity has to increase and photon fade away. But, to put a photon into box, Sam had to take it from somebody - only from Abigail. Abigail emits photon to Sam, Sam moves in Abigail's ref. frame and reflects photon back. Does that mean, that if you emit a photon towards transversely moving perfect mirror, photon will come back doubly- redshifted and the mirror will gain kinetic energy? Will mirror move relatively faster or slower after collision, if it cannot recede (Sam and Abigail are in the trains on rails?) Jul 8, 2017 at 11:03