This is a question following on from my previous post Time-like Killing vector in FRW metric?

For simplicity I take the spatially flat FRW metric in cartesian co-ordinates given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$

I transform the time co-ordinate $t$ to conformal time $\tau$ using the relationship: $$d\tau = \frac{dt}{a(t)}$$ so that the metric now becomes: $$ds^2 = a^2(t)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ We can introduce a function $\alpha(\tau)$ giving $$ds^2 = \alpha^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$

One can now see that the FRW metric has a time-like conformal Killing vector field..

Now electromagnetic waves are conformally (or scale) invariant. If one makes the transformation: $$x \rightarrow \lambda x$$ $$t \rightarrow \lambda t$$ the electromagnetic wave equations are invariant.

Thus it seems to me that there is a symmetry here that Nature could obey. If it did so then the implication would be that the energy of electromagnetic waves is constant as the Universe expands.Therefore wave lengths that scale with the Universe are associated with constant energy.

However atoms have a fixed wavelength so that their energy would increase as the Universe expands.

Thus the cosmological redshift would be explained not by electromagnetic waves losing energy in an expanding Universe but rather by atomic energies increasing.

In conventional cosmology one tacitly assumes that atomic energies are constant as the Universe expands. But what is the basis for such an assumption? At least with photons one can offer a symmetry argument, consistent with general relativity, as to why their energy should stay constant.

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    $\begingroup$ Isn't this just a suggestion to use a energy unit which decreases in time relative to the ones commonly used? $\endgroup$ – fqq Jul 14 '14 at 10:24
  • $\begingroup$ @fqq Well put. He's just offsetting the density evolution of the different $\Omega$ by a factor a through what boils down to a time varying redefinition of the Joule. Now all he needs to do is convince himself that this doesn't affect any observable whatsoever. It just makes everything needlessly more convoluted. $\endgroup$ – ticster Jul 14 '14 at 11:10
  • $\begingroup$ I think the implication of my hypothesis is that the masses of all particles, including the Plank mass, increase with the cosmological scale factor. $\endgroup$ – John Eastmond Jul 14 '14 at 11:11
  • $\begingroup$ I think the mass density now goes like $a/a^3 = 1/a^2$ leading to a linearly expanding cosmology ($a(t) \propto t$) which is very close to what is observed. $\endgroup$ – John Eastmond Jul 14 '14 at 11:14
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    $\begingroup$ Actually I think ticster is right. I now think what I'm hypothesising is that the definition of energy scales with the scale factor. $\endgroup$ – John Eastmond Jul 15 '14 at 10:01

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