0

A classical pendulum clock is powered by gravitational potential energy by weights. While a hybrid pendulum clock is somehow propelled by electric current. Both have the same pendulum swing as the regulator.

A pendulum's frequency is dependent both on its length as on the value of small g according to Huygens formula. Considering the length stays the same, but only small g varies due to height differences between locations: let's say first position is at sea level, and the next position is at 20 km above sea level. Considering the value of g drops significantly there, the frequency of ticking should become lower at that altitude. Time dilation should also be taken into account, but this is only a very small effect.

However: one of the pendulums is powered by a weight (which is also dependent on the value of g), while the other is powered by an electric battery (which is sensitive to time dilation). What effect would this difference have on their mutual frequencies? Would they start to run out of phase, as one is dependent on a gravitational potential energy power source and the other on a quantum-mechanical potential energy power source. Of which only the latter is intrinsically sensitive to time dilation!

Would love to hear your thoughts on this conundrum. Do the clocks tick in phase with eachother at both locations, or will they start to differ due to the time dilation in the electric power source? Is there some way to do this calculation?

Qmechanic
  • 201,751
Apsteronaldo
  • 109
  • Why would they run out of phase? "Both have the same pendulum swing as the regulator." Also, gravitational time dilation affects everything, both the electronics and the falling weight. – PM 2Ring Jan 29 '24 at 02:28
  • 1
    IDK if there is any difference, but if there is any difference, I bet the difference is too small to be measured by any practical pendulum clock. – Solomon Slow Jan 29 '24 at 02:28
  • FWIW, the time dilation due to Earth's gravity (at the surface) is ~21.9 milliseconds per year. Good luck measuring at that precision using a pendulum. – PM 2Ring Jan 29 '24 at 02:38
  • @PM2Ring That's actually an interesting question for an experimentalist. LIGO reports measured violin mode Q factors of 1e9 (!) for their mirror suspensions (https://repository.lsu.edu/cgi/viewcontent.cgi?article=2187&context=physics_astronomy_pubs). If I am not mistaken that just about gets us to the required level of phase noise. Maybe I am overlooking something, but I wouldn't be completely dismayed about making relativistic measurements with a "grandfather clock". – FlatterMann Jan 29 '24 at 05:20
  • @FlatterMann Sure, but those LIGO mechanical oscillations aren't quite the same as what we normally call a pendulum. I mentioned in a comment on a previous question by the OP that a precision (evacuated) pendulum clock can be used to detect the tidal variation in g. See Precision Pendulum Clocks, Gravity and Tides, but much higher precision is required for time dilation measurements, eg Clocks, Kids, and General Relativity on Mt Rainier – PM 2Ring Jan 29 '24 at 11:58
  • @PM2Ring I take your word for it. My first impression was that LIGO's technology, when applied in a slightly different way, might actually work. It's not many orders of magnitude off, as far as I can tell. The kid's project is really cool. – FlatterMann Jan 29 '24 at 14:54
  • @PM2Ring Thank you for your insightful comments, I am trying to find a way to play around with this behaviour of mass compared to electromagnetic energy in response to time dilation, and see if I can construct somehow a way to make it count, useful or measurable and help us perhaps say something about that relatively unknown factor of the gravitational constant G, or perhaps a difference between quantum mechanical potential energy and gravitational potential energy; a.k.a. the difference between quantum and spacetime. I feel like there is still something to be delved here. Don Quichot I know – Apsteronaldo Feb 02 '24 at 20:37
  • I will keep thinking about your comments which are clearly valid, and whether I can improve this setup in some way to show some effect. – Apsteronaldo Feb 02 '24 at 20:44
  • "this behaviour of mass compared to electromagnetic energy in response to time dilation" What behaviour are you talking about? If you have a collection of clocks based on various mechanisms, they all respond to time dilation the same way (assuming the clocks are in proper working order). – PM 2Ring Feb 02 '24 at 23:27
  • @PM2Ring Right you are, both gravitational potential as electromagnetic potential energy decrease at altitude while time dilation diminishes; one as a byproduct of gravity diminishing and the other literally because time dilation is diminishing. My phrasing was sloppy, and I didn't realise both potential energies decrease at altitude: I was only thinking in the slowing down and running faster of clocks versus pendulums, somehow automatically thinking the same would be true for their energies. As you can tell I visualize everything, and it sometimes is hard to add the correct variables to it. – Apsteronaldo Feb 03 '24 at 20:07
  • @PM2Ring To try to get out of this mess of definitions we are using, with the most problematic one both calling the measurement of the pendulum and the electronic clock 'time', and both being dependent on 'time dilation' except one directly in its slowing of clocktime behaviour, and the other in the effect of gravity (in the weak field limit of general relativity), this makes me think. To get things straight for myself, I will ask another question on the forum. Maybe you can help me there as well. It is much appreciated while I try to wrestle with these definitions to get a clear picture – Apsteronaldo Feb 03 '24 at 20:13
  • @Apsteronaldo It's a very important principle in relativity that all clocks respond the same way to time dilation. If some type of clock behaved differently, that would break relativity. In SR, that would let us detect an absolute rest frame (and according to relativity there's no such thing). In GR, it would break the equivalence principle. See https://physics.stackexchange.com/q/570966/123208 which focuses on SR, but similar reasoning applies to GR. – PM 2Ring Feb 03 '24 at 20:18
  • In fact, we call it "time dilation" because all clocks respond the same way. Otherwise, we'd call it "clock malfunction". ;) Of course, this assumes that all the clocks are in good working order, and that their mechanism is unaffected by the extreme conditions of our relativity experiment. So you can't smash the clock with a brick and then claim that bricks stop time. :D Similarly, you can't use a gravity-based clock like a pendulum or hourglass under conditions where they are experiencing variable acceleration. But if the acceleration is a non-zero constant you can calibrate such clocks. – PM 2Ring Feb 03 '24 at 20:26
  • @PM2Ring To avoid turning this into a chat, I like to ask my last question for today. Is the time dimension a real thing to you (like Newton and Einstein), or artificial (like Leibniz, and perhaps Boltzmann)? Because I sympathize with the latter, and currently feel I can't say whether a clock with variable acceleration is less 'real' than a battery clock...which is completely artificial time to me. Just a projection onto reality. Why would that be the correct time, and the pendulum time not? As the last measures the time of the movement of the planets (namely gravity), it's more real to me! – Apsteronaldo Feb 03 '24 at 20:37
  • 1
    @Apsteronaldo Yes, we're definitely getting into Chat territory. ;) See John Rennie's excellent post on time: https://physics.stackexchange.com/q/235511/123208 I agree with Einstein: time is fundamentally geometrical: it's a component of spacetime geometry. So time is like a spatial direction in some respects, but there are also hugely important differences. – PM 2Ring Feb 03 '24 at 22:06

1 Answers1

2

Clocks are designed with some sort of a frequency standard. Often the frequency standard requires energy input, but the frequency standard is designed to be rather insensitive to the energy input. The energy input typically does not need to be carefully regulated and, by design, even fairly large variations will cause little variation in the frequency.

In your case, the frequency standard is the pendulum. Whether the power is provided by a battery or by weights is not particularly important. The gravitational potential and gravitational acceleration affect the frequency, but the details of the power source do not. So they tick in phase, within the limits of their stability.

Dale
  • 99,825
  • Thank you for your comment. I agree with your analysis, as indeed a battery powered quartz watch in its ticking doesn't depend on how full its battery is charged. I wonder though if there might be an edge case, with very little energy, when the potential energy of the hybrid becomes determined by quantum mechanics and the behaviour starts to change though compared to the classical pendulum clock with the gravitational potential as the energy source. So on the micro energy scales gravity might differ from quantum mechanics as the energy source. Just thinking out loud here, wonder what you think – Apsteronaldo Feb 02 '24 at 20:32