Let us assume a flat FRW metric(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

ds^2=-dt^2+a(t)^2(dx^2+dy^2+dz^2).

[/tex]

where [itex]t[/itex] is cosmological time, [itex]x,y,z[/itex] are comoving space coordinates, the speed of light [itex]c=1[/itex] and [itex]a(t_0)=1[/itex] at the present cosmological time [itex]t_0[/itex].

Imagine a light beam travelling in the x-direction. It travels on a null geodesic [itex]ds=0[/itex] therefore its path obeys the relation

[tex]

a(t)dx=dt

[/tex]

Therefore at the present time [itex]t_0[/itex] during an interval of cosmological time [itex]dt[/itex] the light beam travels a proper distance [itex]a(t_0)dx=dx[/itex].

Now imagine a time [itex]t[/itex] in the future when the Universe has expanded by a factor [itex]a(t)[/itex].

During the same interval of cosmological time [itex]dt[/itex] the light beam now travels a proper distance [itex]a(t)dx[/itex].

Thus, in the future, the light beam travels further in the same interval of cosmological time and therefore its speed seems to have increased according to an observer at the present time [itex]t_0[/itex].

I think this paradox is resolved if the time interval the later observer at time [itex]t[/itex] measures expands by the same factor of [itex]a(t)[/itex] according to the present observer.

Let us assume that observers actually measure time in units of conformal time [itex]d\tau[/itex] such that

[tex]

dt = a(t) d\tau

[/tex]

Then for the later observer at cosmological time [itex]t[/itex] we have

[tex]

\frac{a(t) dx}{dt} = \frac{a(t) dx}{a(t) d\tau} = \frac{dx}{d\tau} = 1

[/tex]

This agrees with the speed of light measured by the present observer at cosmological time [itex]t_0[/itex]

[tex]

\frac{a(t_0)dx}{dt}=\frac{dx}{a(t_0)d\tau}=\frac{dx}{d\tau}=1

[/tex]

Thus if we assume that both observers measure conformal time [itex]\tau[/itex] rather than cosmological time [itex]t[/itex] then both will agree with the other's measurement of the speed of light.

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# Conformal or Cosmological Time?

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