Let us assume that an observer is stationary at the origin in expanding space. We assume the FRW metric near the origin is given by:(adsbygoogle = window.adsbygoogle || []).push({});

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

Let us assume that the observer measures time by bouncing a light beam at a mirror that is at a constant proper unit distance away from him.

By substituting ##dt=0## in the metric we find that an interval of proper distance ##ds## is given by:

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

Integrating we find the proper distance ##S## to an object at co-moving co-ordinate ##r## is given by:

$$S=a(t)\ r$$

Thus if the mirror has constant proper distance ##S=1## then it follows a path in co-moving co-ordinates given by:

$$r=\frac{1}{a(t)}$$

A light beam is described by substituting ##ds=0## in the metric to get the null geodesic:

$$dr=\frac{dt}{a(t)}$$

Integrating we get the path of a light beam:

$$r = \int \frac{dt}{a(t)}$$

The spacetime diagram below shows the observer attempting to measure cosmological time ##t## using the light clock.

One can see that as the mirror gets closer in co-moving co-ordinate ##r## the period of the light clock gets smaller and smaller. Thus the light clock is getting faster relative to cosmological time ##t##.

Now let us make a transformation to conformal time ##\tau## given by the relationship:

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

The transformed radial co-ordinate ##\rho## is given by:

$$\rho\ d\tau = r\ dt$$

Therefore the path of the mirror in conformal co-ordinates is given by:

$$\rho = r\ \frac{dt}{d\tau}$$

$$\rho = \frac{1}{a(t)} \cdot a(t) = 1$$

The path of a light beam in co-moving co-ordinates is given by:

$$dr=\frac{dt}{a(t)}$$

Therefore in conformal co-ordinates we have:

$$d\rho=d\tau$$

On integrating:

$$\rho = \tau$$

Thus in conformal co-ordinates the light clock is described by the diagram below:

One can now see that the light clock period is constant - the clock is working properly.

Thus a light clock measures conformal time ##\tau## rather than cosmological time ##t##.

Is this argument correct?

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# Light clocks measure conformal time - detailed argument

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