Question

Bob is in a gravitational potential well, he moves a long vertical stick up and down a distance of 1 meters. Alice observes the upper end of the stick, at upper location.

There is the phenomenon of redshift of energy, so either Alice measures a shorter motion than 1 meters, or alternatively Alice measures a weakened force.

So I'm asking: is Bob weak, or is Bob short?

Answer 1

Bob is weak. Bob will find it harder to do the work of raising the meterstick, because Bob is slowed down by gravitational time dilation.

The proper way to answer this is to consider Rindler coordinates for an accelerating frame, where the metric is

The "dr" coordinate is radial distance, and integrating this gives the meterstick length. The r^2 factor in front of d? is the square of the redshift factor. The length of the meterstick is unchanged for all values of r, assuming it is neglibibly elastic in the range of gravitational fields considered, so that r is varying between two large values. The Rindler approximation is a valid local description of the gravitational field, in a region small enough so that the curvature is not important.

If you place two mirrors at two values of r, and let a photon bounce back and forth between the two mirrors, the answer becomes obvious. When the photon hits the lower mirror, it is more energetic, and it pushes the meterstick down by a large amount, then when it hits the upper mirror, it is less energetic, and it pushes the meterstick up by a smaller amount, but the center of mass of the stick over one photon cycle doesn't move.

Further, the back and forth bounces of the photon give an equal amount of pushes per unit tau at both locations (this is clear from the fact that tau is a killing vector for the metric, so that the process is tau stationary when averaged over many cycles). But a unit of tau at Bob's position is shorter than a unit of ? at Alice's position, so there are more strong pushes per unit proper time at Bob's position, balancing out the fewer weak pushes at Alice's position.

Each of these effects goes as the square root of the time component of the metric, so that Bob is weaker by the ratio of his to Alice's value of .

In the Minkowski point of view, in order to keep the meterstick accelerating, you have to dump a certain amount of momentum per unit proper time into the stick. But you don't need to push as hard at the top to do this, both because the proper time is longer there so you don't need to put as much energy per second (because your second counts for more), and because the energy you put in gets blueshifted when it gets to the center of mass, so that it is giving you more oomph per kick. The effect is as the square of the time-dilation factor, from two cooperating effects