What happens to time on the hands of a clock?

[Mattergauge]

TL;DR Summary

The effect of SR and GR on the hands of a clock and its indication of time.

I ask this question as a follow-up of an already answered one (about the effect of velocity on time). I'm looking for the effects of SR and GR on a clock's seconds hand. The effect is very tiny though, as you can imagine. Instead of making the size of the clock vary, we can vary c, the speed of light. Imagine that the hand moves continuously and that it's a massive Born-rigid stick with a mass of one kilogram and length of one meter. It can be treated like 1d. When will effect be noticed? For which c will this be? Will it affect the timekeeping of the clock?

 

[PeterDonis]

I'm looking for the effects of SR and GR on a clock's seconds hand.

Unless the clock is large enough that tidal gravity is significant from one side of it to the other, we can ignore GR effects.

As for SR effects, first you need to get clear about what a "tick" of the clock means. It has nothing to do with any "rate of time flow" of the hand vs. the clock face. The "ticks" of the clock are the events at which the tip of the hand coincides with markings on the clock face. Those events are invariant and define the "time flow" of the clock itself. The fact that, if you were to place a second clock (perhaps a quartz digital clock so no effects of "motion of the hands" are present at all for it, to avoid confusion) at the tip of the hand, the fact that this clock's "rate of time flow" might be different from that of the original clock is irrelevant to the behavior of the original clock.

Instead of making the size of the clock vary, we can vary c, the speed of light.

No, you can't, because in SR, the speed of light is invariant.

 

[PeterDonis]

perhaps a quartz digital clock so no effects of "motion of the hands" are present at all for it, to avoid confusion

This comment, btw, shows that if we want to consider the simplest idealized "clock" in relativity, we should not be considering an analog clock with hands. We should be considering a clock with no moving parts, which registers time using something like motionless LED digits. We should also ensure that the physical size of the clock is small enough that (a) tidal gravity is negligible across the clock, so we can put it in a curved spacetime without affecting its function, and (b) light travel time across the clock is small compared with the smallest time interval the clock needs to register (for example, if the clock is to register time at the nanosecond level, it needs to be much smaller than one foot across).

 

[Mattergauge]

Of course the speed of light is invariant. Maybe I should have introduced a clock that's scalable in size. If the seconds hand grows larger it will affects the metric on the coordinate system co-rotating with the hand. The hand will off course move in the spacetime of the clock's body, being Minkowskian, in which it stands still and the body only moving through time. Clocks put on the hand will show a difference with a clock placed on the body though.

 

[PeterDonis]

Maybe I should have introduced a clock that's scalable in size.

I don't see the point of this, since the whole purpose of the idealization of a "clock" is to avoid any complications due to the size of the clock. See my post #3.

 

[PeterDonis]

If the seconds hand grows larger it will affects the metric on the coordinate system co-rotating with the hand.

The metric, and any other invariant, is unaffected by the state of motion of any particular object or any particular choice of coordinates you make.

The hand will off course move in the spacetime of the clock's body

There is no such thing as "the spacetime of the clock's body". There is just spacetime, which in the case we are discussing is flat Minkowski spacetime.

I think you need to work on your understanding of the basics of relativity.

Clocks put on the hand will show a difference with a clock placed on the body though.

I already addressed this in post #2.

 

[Dale]

Clocks put on the hand will show a difference with a clock placed on the body though.

Wait, are you putting clocks on clocks? If so, then it will be impossible to consider the big clock to be a clock. At least I can’t see how.

 

[PeroK]

Summary:: The effect of SR and GR on the hands of a clock and its indication of time.

In general, we consider an event as a point in spacetime. Yet, any specific physical event - a changing LED display, for example - must have some spatial and temporal extent. Einstein mentions this in the 1905 paper, where he says in a footnote:

We shall not here discuss the inexactitude which lurks in the concept of simultaneity of
two events at approximately the same place, which can only be removed by an abstraction

 

[PeterDonis]

If so, then it will be impossible to consider the big clock to be a clock. At least I can’t see how.

That's the whole point! :)

Just to clarify this issue for future readers of this thread:

You can in fact still consider the big clock to be a clock; I described how in post #2 (the events where the tip of the hand coincides with markings on the clock's face define the "ticks"). I also noted in that post that the fact that the "tick rate" of a second (obviously much smaller) clock at the tip of the big clock's hand will, in general, be different from the "tick rate" of the big clock (I am glossing over a number of technicalities here) does not prevent the big clock from functioning as a clock.

However, of course this is not a very good way to define a clock, for various reasons. A much better way is the way I described in post #3 (make the clock small enough that any possible effects that could cause difficulties are negligible).

 

[Dale]

That's the whole point! :)

Well if the point is to not consider the big clock as a clock then don’t call it a clock. Call it a centrifuge or a carousel. Confusing language will lead to confusing responses.