Pendulum Clock vs GR: Time Discrepancy

[Patsy Hasty]

TL;DR

Imagine you are locked inside an elevator in space accelerating at 1g where a pendulum clock ticks normally. Then slow to 1/2g acceleration and the pendulum swings slower. Finally, at 0g the pendulum stops. Doesn't this conflict with GR and the Equivalence Principle?

According to general relativity (GR) time runs faster in a weak gravity field relative to a stronger one, for example: clocks run faster at the top of a tall building than at ground level. According to the principle of equivalence the accelerating space elevator should be analogous to gravity - but there seems to be a conflict, i.e., the pendulum clock is slower in weaker acceleration field.

 

[Orodruin]

That has nothing to do with how time is measured. The pendulum clock measures time correctly only if in the same gravitational field as where it has been calibrated. Furthermore, gravitational time dilation depends on gravitational potential, not gravitational acceleration.

 

[Ibix]

According to general relativity (GR) time runs faster in a weak gravity field relative to a stronger one,

This is not correct. The relative rates of clocks depend on the difference in gravitational potential between the clocks, not on the field strength. Although those things frequently change in the same direction, they don't always and they are very different things. An obvious case is if one were to drill a hole to the center of the Earth - clocks held lower in the hole would tick slow compared to clocks held higher in the tunnel all the way to the center, but the acceleration due to gravity decreases to zero at the center of the Earth. So the place you'd feel zero weight would be where you also measure the largest time dilation effect.

Furthermore, your experiment is misconceived. To make comparable measurements of time you need clocks that operate the same way under all conditions of interest. For example, a steel clock would operate strangely in a strong magnetic field, but that's not evidence of time acting funny in a magnetic field - it's just a bad experiment. As you yourself note, pendulum clocks don't tick at the same rate in different gravitational field strengths. So time measurements taken with identical pendulum clocks in different field strengths aren't comparable, just the same as the steel clock's measurements in a magnetic field would not compare to the measurements it makes normally.

Edit: you can, of course, calibrate pendulum clocks to some agreed standard at arbitrary local $g$ value, and you will (in principle) be able to detect time dilation between them, just as you would with atomic clocks. In practice I do not think a pendulum clock can be made precise enough to do this, but it's possible in principle.

 

[Dale]

Summary:: Imagine you are locked inside an elevator in space accelerating at 1g where a pendulum clock ticks normally. Then slow to 1/2g acceleration and the pendulum swings slower. Finally, at 0g the pendulum stops. Doesn't this conflict with GR and the Equivalence Principle?

According to the principle of equivalence the accelerating space elevator should be analogous to gravity

It is equivalent, you are just misapplying it. Suppose you have an atomic clock, a quartz clock, a mechanical clock, and a pendulum clock. Place all in a lab on Earth and normalize so that they all measure accurate time. Then place all in a lab on the moon, the pendulum clock will no longer keep accurate time, but the rest will. Now, place all in a lab on a rocket, far away from gravity. Have the rocket accelerate at earth’s gravitational acceleration, they will all keep accurate time again. Now, have the rocket accelerate at the moon’s gravitational acceleration, the pendulum clock will no longer keep accurate time but the rest will. The equivalence principle will thus hold.

 

[Ibix]

According to Einstein, "any physical process may be used to measure time".

Any process can be used to measure time, but if you want to compare measurements from different instruments, or from the same instrument in different conditions, you need to make sure that they're calibrated against some standard. Simply counting the ticks of a pendulum clock isn't enough. You seem aware that the ticks represent different lengths of time, but you are then trying to equate one tick with one standard unit of time, which is where you go wrong.

A spring-powered alarm clock measures time, and yet it is affected depending on the strength of the gravity field.

All mechanical clocks I can think of rely on a pendulum or balance wheel of some kind, which will be affected by gravitational field strength so won't accurately represent time anywhere $g$ isn't 9.81m/s² (or whatever value it was calibrated for). Only systems like quartz or atomic clocks, or impractical but easy to analyse arrangements like the light clock, will be unaffected by their proper acceleration. All will be affected by differences in gravitational potential.

 

[ergospherical]

In relativity an ideal clock is a point-particle which "ticks" every $\delta \tau$ of proper time along its wordline, but real clocks can of course only approximate this behaviour. Even atomic clocks are only good approximations when their accelerations are much less than the centripetal accelerations of electrons around the nuclei (still pretty high...).

 

[vanhees71]

For more precision and stablity the "nuclear clock" is around the corner!

https://en.wikipedia.org/wiki/Nuclear_clock

 

[Patsy Hasty]

This is not correct. The relative rates of clocks depend on the difference in gravitational potential between the clocks, not on the field strength. Although those things frequently change in the same direction, they don't always and they are very different things. An obvious case is if one were to drill a hole to the center of the Earth - clocks held lower in the hole would tick slow compared to clocks held higher in the tunnel all the way to the center, but the acceleration due to gravity decreases to zero at the center of the Earth. So the place you'd feel zero weight would be where you also measure the largest time dilation effect.

Furthermore, your experiment is misconceived. To make comparable measurements of time you need clocks that operate the same way under all conditions of interest. For example, a steel clock would operate strangely in a strong magnetic field, but that's not evidence of time acting funny in a magnetic field - it's just a bad experiment. As you yourself note, pendulum clocks don't tick at the same rate in different gravitational field strengths. So time measurements taken with identical pendulum clocks in different field strengths aren't comparable, just the same as the steel clock's measurements in a magnetic field would not compare to the measurements it makes normally.

Edit: you can, of course, calibrate pendulum clocks to some agreed standard at arbitrary local $g$ value, and you will (in principle) be able to detect time dilation between them, just as you would with atomic clocks. In practice I do not think a pendulum clock can be made precise enough to do this, but it's possible in principle.

Thankyou for your interesting and thoughtful reply. If I understand correctly, I see the error in my analogy. (1) There is no conflict with the Principle of Equivalence as the pendulum would behave exactly the same in a X gravitational field = accelerating at X. (2) Time dilation is a relative effect based on one particular observation frame, for example, the BASE of a tall tower on Earth's surface where the gravitational potential (gR) depends on the radius R. At other locations relative to the BASE, the gravitational potential (gR-gH) is LESS RELATIVE TO THE OBSERVATION FRAME, so time runs faster than at the BASE. Do I have that right?