Is Light Speed Consistent in Conformal Time?

[johne1618]

Let us assume a flat FRW metric
<br /> ds^2=-dt^2+a(t)^2(dx^2+dy^2+dz^2).<br />
where t is cosmological time, x,y,z are comoving space coordinates, the speed of light c=1 and a(t_0)=1 at the present cosmological time t_0.

Imagine a light beam traveling in the x-direction. It travels on a null geodesic ds=0 therefore its path obeys the relation
<br /> a(t)dx=dt<br />
Therefore at the present time t_0 during an interval of cosmological time dt the light beam travels a proper distance a(t_0)dx=dx.

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

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

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 t_0.

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

Let us assume that observers actually measure time in units of conformal time d\tau such that
<br /> dt = a(t) d\tau<br />
Then for the later observer at cosmological time t we have
<br /> \frac{a(t) dx}{dt} = \frac{a(t) dx}{a(t) d\tau} = \frac{dx}{d\tau} = 1<br />
This agrees with the speed of light measured by the present observer at cosmological time t_0
<br /> \frac{a(t_0)dx}{dt}=\frac{dx}{a(t_0)d\tau}=\frac{dx}{d\tau}=1<br />
Thus if we assume that both observers measure conformal time \tau rather than cosmological time t then both will agree with the other's measurement of the speed of light.

 

[marcus]

Here's a picture showing conformal time (in the bottom frame). You've probably seen it many times but's still worth glancing at now and then I find. Jorrie has it in his signature along with Lightcone calculator.