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Amateur here, so go easy.

Given that cosmological time-dilation is the only GR-consistent reading of cosmological redshifting due to the accelerated expansion of space (due to the constancy of the speed of light, and thus ‘distance’- since speed is distance/ time, time is the only variable, and thus must be dilated across expanding space; a dilation that is responsible for the red shifting of photons relative to the expansion of space).

AND

Given that this redshift is itself a measurable energy shift (energy is frequency) that is correlated to relativistic transformations related to that time dilation (the degree of redshift is equal to the degree of time-dilation), in turn, seems to show that energy, as a measurable value, or objective quantity (at least of massless particles like photons), is relativistic.

*I know this would not be true from a single reference frame. That would be accounted for due to special relativity: According to special relativity, the photon’s frequency would be reduced by the Lorentz factor, so that the received frequency would be redshifted by the same factor. However, cosmological time- dilation is frame- invariant, as it is a universal (it is itself an effect of the universal of accelerated cosmic expansion), and thus all observers measure the same redshift/ distance. We can look in any direction, at any distance travelled and measure the same relative redshift of eg type 1a supernovae photons with known initial energies (say, n oscillations per megaparsec). Not surprising, due to general relativity in an acceleratedly expanding universe.

This seems to suggest that cosmological time- dilation CAUSES energy shifts in photons, and thus that energy (of massless particles like photons, at least) is transformed relativistically.

If yes, I wonder how this relates (if at all) to $E=mc^2$?

Am I wrong? Am I missing something?

Thoughts?

Note: This is a genuine question, so please don’t reply with obscurantist, facetious, or dismissive answers. Also, if you use equations, please explain them conceptually. I am but an amateur trying to make sense of things and their interrelationships.

Many thanks in advance for your thoughts!

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  • $\begingroup$ Photons don't have mass, so the rest energy, $E=mc^2$, of a photon is not defined. $\endgroup$
    – Paul T.
    Commented Feb 23, 2022 at 16:32
  • $\begingroup$ Not sure what the question is. Are you asking for confirmation that redshifted photons have lower energy? The energy of a photon is not a relativistic invariant. $\endgroup$
    – ProfRob
    Commented Feb 23, 2022 at 20:42
  • $\begingroup$ Ok, then what is the relationship between cosmological time- dilation and measurable photon energy loss (redshift), given that there seems to be a correlation (causation?!), and cosmological time-dilation IS a relativistic invariant in this instance (I mean, is a result of relativistic invariance)? $\endgroup$ Commented Feb 23, 2022 at 21:27
  • $\begingroup$ Thanks Paul. I thought about that. That does go some way towards answering my question, at least negatively. I’m probably barking up the wrong tree, but my substantive question is still open, as rephrased above in comment. Thanks a lot. $\endgroup$ Commented Feb 23, 2022 at 21:35

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Given that cosmological time-dilation is the only GR-consistent reading of cosmological redshifting due to the accelerated expansion of space

That is not a given, or at least it isn’t something that I would accept as a given.

energy, as a measurable value, or objective quantity (at least of massless particles like photons), is relativistic

Yes, energy is relative. Energy has the same relationship to momentum as time has to space. Energy and momentum transform together in what is called the four-momentum.

However, cosmological time- dilation is frame- invariant

Time dilation (cosmological or not) is not frame-invariant. Cosmologists just have a standard reference frame that they assume by convention. So they often speak as though it were frame-invariant instead of as though they were describing a frame-varying quantity in an understood frame.

It is like, if I say “I was traveling 100 kph”, you know from understood context that I meant in the ground’s reference frame. It is not that my speed is invariant, just that the frame that defines the speed is understood.

If yes, I wonder how this relates (if at all) to 𝐸=𝑚𝑐2

The full expression is $m^2 c^2=E^2/c^2-p^2$ where $p$ is the momentum. When $p=0$ this simplifies to the famous $E=mc^2$ and when $m=0$ this same formula simplifies to the photon’s $E=pc$, but the full expression works for all particles regardless of their mass or speed.

In this formula the mass $m$ is a relativistic invariant. Different frames will disagree about both $p$ and $E$, but they will all agree about $m$.

This is independent of any of the above regarding cosmological expansion.

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