Let two observers B and T be at the base and top of a building respectively. Given gravitational time dilation, both observers measure T’s clock running faster than B’s clock.

Questions:
1. Does B measure the Moon orbiting the Earth faster than T measures?
2. Does B measure the Earth-Moon distance as less than T measures?

I think:
1. Yes, just like T’s clock runs faster.
2. Yes, in concordance with Kepler’s third law, which implies that objects orbiting faster orbit closer.

What do you think?

Originally posted by zanket
Given gravitational time dilation, both observers measure T’s clock running faster than B’s clock.
Correct. Each observer will measure his own clock running normally. B will measure T's clock as running fast. T will measure B's clock as running slow.
Questions:
1. Does B measure the Moon orbiting the Earth faster than T measures?
Yes, he will.
2. Does B measure the Earth-Moon distance as less than T measures?
Yes, time dilation and length contraction go hand-in-hand.

- Warren

Thanks. Does B measure the building’s height as less than T measures? Assume the measurement is done with laser ranging.

I think yes. If B measures the Earth-Moon distance as less than T measures, then B also measures the Earth-Top distance as less.

I'm not so sure. Kepler's law is derived from Newton's gravitational law, which does not hold in general relativity.

Here’s a relevant site: Orbits in Strongly Curved Spacetime (http://www.fourmilab.ch/gravitation/orbits/).

Been thinking about this. The implication of Kepler’s law that objects orbiting faster orbit closer must be true in GR as well, otherwise Kepler’s law would be useless even in slightly curved spacetime. The site linked to above graphically confirms that GR matches Kepler in this respect.

zanket

Keplers law does not hold if the gravitational fields are strong, as in the case of a black hole or neutron star, or in the case of our Sun and Mercury. It cannot explain precession.

From the site:

When the test mass reaches its greatest distance from the black hole, a yellow line is plotted from the centre of the black hole to that point, the apastron of the orbit. In Newtonian gravity, the apastron remains fixed in space. The effects of General Relativity cause it to precess.

True. To avoid confusion, I change #2 at the top of this thread to:

2. Yes, in concordance with GR, which implies that objects orbiting faster orbit closer.

I think I can tidy my logic a bit on #2 at the top of this thread.

In GR, circular orbits for a given orbital velocity have a specific radius. The radius decreases as the orbital velocity increases.

B measures a faster orbital velocity for the Moon than T measures, so B calculates the Moon’s orbital radius as less than T calculates. The proper Earth-Moon distance lessens as altitude lessens.