- #1

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$$ds^2=-c^2dt^2+a(t)^2dr^2$$

If we take [itex]dt=0[/itex] then the proper distance [itex]ds(t)[/itex] between two spatially separated points at cosmological time [itex]t[/itex] is given by:

$$ds(t)=a(t)dr$$

Now at the present time [itex]t_0[/itex] we can define [itex]a(t_0)=1[/itex] so that we also have:

$$ds(t_0)=dr$$

Therefore by eliminating [itex]dr[/itex] in the above equations we find:

$$ds(t)=a(t)\ ds(t_0)$$

If we define [itex]ds(t)=1[/itex] so that a hydrogen atom has a unit proper diameter, at any time [itex]t[/itex], then the

*equivalent*diameter at the present time [itex]t_0[/itex] is given by:

$$ds(t_0)=\frac{1}{a(t)}$$

According to quantum mechanics the mass/energy of a quantum system is inversely proportional to its size.

Therefore if the mass/energy of the hydrogen atom at time [itex]t[/itex] is one unit then the mass/energy of an

*equivalent*atomic system at the present time [itex]t_0[/itex] is [itex]a(t)[/itex] units.

Thus can one infer that hydrogen atoms at time [itex]t[/itex] in the future have an energy that is a factor [itex]a(t)[/itex] higher relative to the energy of hydrogen atoms today?