Gravity Clock: Pendulum Periods for Observers A & B

[edpell]

OK here is a pendulum:

A gravity clock consists of two spherical masses one large of rest mass M and one small of rest mass m. The smaller mass is suspended by a rigid frame, of negligible mass, at a height R above the center of the large mass. It is the bottom of a pendulum arm, of negligible mass, of length L. When displaced the pendulum has the period

$T \approx 2 {\pi}R{\sqrt{\frac{L}{GM}}}.$

Given two observers A who will travel with the gravity clock and B who will remain behind in the initial inertial frame. When the gravity clock and observer A are set in motion at velocity v with respect to observer B what period does A see? What period does B see?

 

[Dale]

The period according to B is time dilated relative to the period according to A.

 

[edpell]

Dale yes, I agree that the canonical answer is that B observes A's clock as running slower by a fact,or of gamma. Also B observes that the inertia mass of M increases by a factor of gamma still canonical physics. Does the increase in the mass M in anyway effect the period observed by B? The equation for the pendulum seems to say yes(?). How do you see it?

 

[bcrowell]

Dale yes, I agree that the canonical answer is that B observes A's clock as running slower by a fact,or of gamma. Also B observes that the inertia mass of M increases by a factor of gamma still canonical physics. Does the increase in the mass M in anyway effect the period observed by B? The equation for the pendulum seems to say yes(?). How do you see it?

Isn't it the gravitational mass of M that's relevant, rather than the inertial mass (assuming M>>m)?

Rather than talking about a planet of mass M and a pendulum of mass m, I think you might as well talk about bodies of mass M and m orbiting around their common center of mass. I don't think there's any important difference between the two experiments. However, the orbiting example may be simpler to analyze, since there are no nongravitational forces. If there are no nongravitational forces, then the system is simply a solution to the Einstein field equations. The Einstein field equations have general covariance, so a solution is still a solution if we switch to a different frame of reference. This allows you to sidestep all the nasty complications of trying to describe the transformation in terms of special-relativistic length contraction, time dilation, and inertia. So I think Dale is clearly correct, and it may just be difficult to verify that he's correct by a nasty, complicated method.

 

[edpell]

It is my understanding that inertial mass and mass and gravitational mass are all different names for the same thing from a GR point of view.

OK let say we have two masses M and m (with M>>m). The "test mass" m is in orbit around the mass M. When both the masses and B are in the same inertial frame B observes a period T. When M and m are in a frame moving way from B at a velocity v what period T does B observe? Is it just T times gamma? If the mass of M goes as M time gamma does that have any effect on the period observed? If I have done the math right the period is proportional to M to the -1/2 power. Does this in any way effect the period observed by B?