# Does cosmological time expand like space does?

The FRW metric is given by: $$ds^2=-dt^2+a^2(t)\ dr^2$$ where $ds$ is an interval of proper length, $dt$ is an interval of cosmic time, $dr$ is an interval of co-moving co-ordinate distance and $a(t)$ is the scale factor (also $c=1$).

If I take $dt=0$ then I find that an interval of proper distance $ds$ is given by: $$ds = a(t)\ dr$$

Thus the proper distance between two nearby co-moving points is proportional to the scale factor - space expands.

If I take $ds=0$ then I obtain the null geodesic equation describing the path of a light ray: $$dt = a(t)\ dr$$

Thus the light travel time between two nearby co-moving points is also proportional to the scale factor.

Does this imply that intervals of cosmic time expand along with intervals of space?

The natural clocks in co-moving co-ordinates are expanding light-clocks.

Maybe in order to derive the constant units of the "atomic" time that we experience, $d\tau$, from expanding cosmic time intervals $dt$, we need to use the conformal time equation: $$d\tau = dt/a(t)$$

• possible duplicate of Time slowing down vs. universe expanding – rob Jun 4 '14 at 21:32
• I don't think it's a duplicate. This question is a technical one, whereas the other seems more of a random thought. – Nathaniel Jun 5 '14 at 0:25
• possible duplicate of Does time expand with space? (or contract) – John Rennie Jun 5 '14 at 9:07
• John, I've linked what seems to me a better duplicate than the one Rob suggests. – John Rennie Jun 5 '14 at 9:08
• What do you think of my idea that the natural clock in co-moving co-ordinates is an expanding light clock?When corrected by dividing by the scale factor you get our atomic time. – John Eastmond Jun 6 '14 at 14:52