How's energy stated in special relativity.?

Another take: Speed of light is the same in any frame of reference. The light beam will cover the same amount of ground in the same time (doing work) with the same end result. Energy is conserved.

 

regarding

What work is done transporting a photon from one point to another?? none. Once light is emitted in a vacuum it does not change color...it does not change energy. Furthermore in an expanding universe such as ours, cosmological distance [expansion] also changes over time.

"We get different time periods..."

What about the 'Doppler shift' of electromagnetic waves, as in a radar speed trap? We infer speed from observed frequency shift. But photon signal frequency and wavelength can be viewed as fixed... Doppler shift is a particular explanation of redshift, based on the relative velocity of source and detector, with a particular relativistic formula. The frequency of the signal itself remains fixed coming and going; It's the the relative motion between the emitter and absorber [the Doppler effect] that causes the effect, not anything to do with photons themselves.

 

Energy is capability to do work. Measuring is doing some work. For the measuring of the light source in another frame moving along another geodesic line, the ray of light has to cross. That will give doppler shifts doppler broadening of the lines. Total energy content as measured will be the same.

From the book:

4. Tests of Time Dilation and Transverse Doppler Effect

The Doppler effect is the observed variation in frequency of a source when it is observed by a detector that is moving relative to the source. This effect is most pronounced when the source is moving directly toward or away from the detector, and in pre-relativity physics its value was zero for transverse motion (motion perpendicular to the source-detector line). In SR there is a non-zero Doppler effect for transverse motion, due to the relative time dilation of the source as seen by the detector. Measurements of Doppler shifts for sources moving with velocities approaching c can test the validity of SR's prediction for such observations, which differs significantly from classical predictions; the experiments support SR and are in complete disagreement with non-relativistic predictions.

 

What is all this supposed to show??

 

It was only the middle sentence that we are taking issue with; we agree energy IS conserved in special relativity. The middle sentence is wrong on two counts: no work is done; and over vast [cosmological] distances, light does NOT cover the same 'ground' [distance] in the same time due to varying rates of cosmological distance.

For adit's benefit, a way to think about a simple case: think of an emitter and receiver of electromagnetic radiation, perhaps a few miles apart: Say one is moving with respect to the other; the measured frequency at each will be different. You can tell this is a measurement issue by then having the emitter and receiver stationary with respect to each other and taking another set of measurements: voila, the emitted and received frequency are now measured to be identical.

 

A massive object as seen from different frames has different energies due to different speeds, a massless photon has different energies as seen from different frames not due to different speeds, but different frequencies.