Do clocks speed up in an expanding Universe?

[johne1618]

If on the other hand the physical distance is constant between two points in space, as would be the case for the two ends of a rigid rod, then the comoving (coordinate) separation between these two points can't possibly be the same as what it was in the beginning.

Ok - I accept your point that the co-ordinate separation between the ends of a rod with a fixed proper length must change. But I'm not using the rod's changing co-ordinate separation in my argument - I only use the rod's constant proper length.

At time $t_0$ I start with a rigid rod with proper length $l_0$ adjacent to an equal proper length of space $ds_0$.

As $a(t_0)=1$ we have:
<br /> ds_0 = a(t_0)\ dr_0 \\<br /> ds_0 = dr_0<br />
where $dr_0$ is the co-ordinate length of the space. Therefore at time $t_0$ we have:
<br /> ds_0 = dr_0 = l_0.<br />
Now at time $t$ the proper length of the space is given by:
<br /> ds = a(t)\ dr_0<br />
The rod is rigid so that its proper length is still $l_0$ (I don't care that its co-ordinate length has shrunk accordingly.)

The space is co-moving so that its co-ordinate length $dr_0$ has not changed.

Therefore at time $t$ we still have:
<br /> l_0 = dr_0<br />
and so at time $t$ we can assert:
<br /> ds = a(t)\ l_0.<br />
The cosmological time interval is given by $dt=a(t)dr_0/c=ds/c$ whereas the time interval measured by a rigid light clock, of length $l_0$, is given by $d\tau=l_0/c$ so that we have finally:
<br /> d\tau = \frac{dt}{a(t)}.<br />

 

[phsopher]

The cosmological time interval is given by $dt=a(t)dr_0/c=ds/c$ whereas the time interval measured by a rigid light clock, of length $l_0$, is given by $d\tau=l_0/c$ so that we have finally:
<br /> d\tau = \frac{dt}{a(t)}.<br />

The cosmological time interval of what?. What we want to consider is the time it takes for a light ray to travel the length of a rigid rod (that's the basis of our clock remember). You need the coordinate separation of the two ends of the rod at time $t$ in order to use the null-geodesic condition $\mathrm dt = a(t)\mathrm dr$. That thing you say you don't care about is precisely what you need in order to calculate the path of a light ray.

 

[johne1618]

The cosmological time interval of what?. What we want to consider is the time it takes for a light ray to travel the length of a rigid rod (that's the basis of our clock remember). You need the coordinate separation of the two ends of the rod at time $t$ in order to use the null-geodesic condition $\mathrm dt = a(t)\mathrm dr$. That thing you say you don't care about is precisely what you need in order to calculate the path of a light ray.

I am comparing a time interval, $dt$, measured by an expanding light clock with a time interval, $d\tau$, measured by a rigid light clock assuming that initially at time $t_0$ both light clocks have the same length.

Therefore we have:
<br /> dt = a(t)\ d\tau.<br />
As expected the time interval $dt$ measured by the expanding light clock scales with the scale factor $a(t)$.

I think the expanding light clock is the "natural" time keeper in a co-moving reference frame and measures intervals of cosmic time $t$.

One second measured by the expanding clock is equivalent to $1/a(t)$ seconds measured by the rigid clock.

Therefore from the perspective of rigid clocks and atomic systems using time $\tau$, that is our perspective, cosmic time $t$ is speeding up.

 

[phsopher]

Now I'm confused; when I asked you in the beginning of the thread if you were considering an expanding clock you answered no and said that the time would the same as that measured by rigid clocks.

Putting that aside, yes, the rate of an expanding clock will change; however, since the actual clocks we are using in reality do not expand with the universe, those will always tick at the same rate. The rigid clocks measure the cosmic time $t$ and continue to do so at the same rate.

 

[johne1618]

Now I'm confused; when I asked you in the beginning of the thread if you were considering an expanding clock you answered no and said that the time would the same as that measured by rigid clocks.

Putting that aside, yes, the rate of an expanding clock will change; however, since the actual clocks we are using in reality do not expand with the universe, those will always tick at the same rate. The rigid clocks measure the cosmic time $t$ and continue to do so at the same rate.

I admit that I've changed my mind.

I now think that a "natural" clock in co-moving co-ordinates is an expanding light clock just as a natural length measure is an expanding ruler. If those were the only measuring instruments we have then we would not notice that times/distances are increasing as the Universe expands.

However we have fixed rulers so that we can detect that co-moving distances are increasing. I would also say that we have fixed light clocks so that we can detect that time intervals are increasing. I think this implies that one second of cosmic time is equivalent to $1/a(t)$ seconds of fixed light clock time. As our time is fixed clock time then from our point of view one second of cosmic time is getting shorter and shorter.

 

[johne1618]

However, since the actual clocks we are using in reality do not expand with the universe, those will always tick at the same rate. The rigid clocks measure the cosmic time $t$ and continue to do so at the same rate.

Thinking about it again maybe you're right.

I've been barking up the wrong tree!

 

[johne1618]

Actually I now think there might be some truth in my argument: See my post about energy conservation in an expanding Universe:

https://www.physicsforums.com/showthread.php?p=4753126#post4753126

 

[Mordred]

conservation of entropy and the arrow of time, has been and still is a common argument for an evolving time. Or time reversal in some extreme cases. You didn't quite state that in the other post, however the similarity is there. The arrow of time argument in regards to the second law of thermodynamics has been around for as long as the concept of entropy was first introduced. To my knowledge though its never been universally accepted, nor proven. For one thing disorder to order does not mean time reverses, like the mathematics the model would have one believe. Entropy used as the measure of order.