Gravitational Length Contraction

Question

General Relativity predicts that a clock at rest in a gravitational field will run slower than a clock in free fall. Similarly, will a vertical ruler on the earth's surface be shorter than a ruler in free fall? Why or why not and by how much?

Answer 1

The reason it makes sense to talk about gravitational time dilation is that the gravitational field solution (Schwarzschild geometry) has a time-translation symmetry. If you're hovering at a certain altitude (all your Schwarzschild coordinates are constant except for $t$) and emit two light pulses at times separated by $\delta t$, it follows immediately from symmetry that any hovering receiver that detects those signals will detect them at a separation of $\delta t$. But that's coordinate time in both cases, which is related to proper time by the local metric. So actually the receiver will see a redshift or blueshift given by the ratio of (the square root of) the appropriate component of the metric at each location. Thus you can consistently think of this metric component as a "time dilation factor" and get the right answer.

There's nothing analogous for length contraction. This metric doesn't have a spatial translation symmetry. Even if it did, the idea of two light beams being emitted from different coordinate positions, and received with the same coordinate separation but a different proper separation, doesn't seem as deserving of the name "length contraction" as the former case seems to deserve the name "time dilation". FLRW cosmology is symmetric under spatial translations (and not time translations), and you can make exactly the above argument with $\delta r$ instead of $\delta t$. In fact this is the easiest (and usual) way of deriving the cosmological redshift formula. But no one calls it "length contraction". Maybe they would if the universe were contracting instead of expanding.

(edit: Since the Schwarzschild metric does have rotational symmetry, you can make this argument with an angular displacement $\delta\theta$, showing that the emitted and received physical distances are related by the ratio $r_\mathrm{emitter}/r_\mathrm{receiver}$. But it would be silly to call that length contraction—for starters, it's true in Newtonian physics also. It's kind of silly to call the $\delta t$ case time dilation too, since it's geometrically similar to this case, and as far as the universe is concerned, that intrinsic geometry is all that matters. The laws of physics, being local, can't even see the global time-translation symmetry that leads us to introduce the Schwarzschild coordinates that make the time-dilation picture seem sensible.)

Answer 2

Yes. We can figure out

$dr = ds \sqrt{1-\frac{2GM}{rc^2}}$.

Here is a straightforward derivation: http://www.physicsforums.com/showthread.php?t=404153

Answer 3

Here's a non GR explanation. Those who prefer to stick to majority views may wish to ignore it.

Mass can be thought of as a collection of waves, each with the potential to do work. The total potential of all the waves to do work is the total potential of the mass to do work.

If we describe this in terms of the distance over which it could potentially apply a force, we get the well known mass-energy equivalence equation:

$E=mc^2$

If we describe it in terms of the time over which it could potentially apply a force, we get the less familiar equation:

$\rho=mc$

Whichever way we choose to describe it, we are describing the same potential to do work.

When a mass falls freely through a gravitational field, it neither does work nor has work done on it. It's potential to do work therefore remains constant as it falls. However, we know that it gains momentum [aka KE] and loses energy [aka GPE] on the way down (what we are seeing is a result of the changing ratio of total energy to total momentum). From this, and the above equations it can be logically deduced that the unit of time is dilating while the unit of length is contracting.

During free fall $ |dE/ds| = |d\rho/dt| = mg$

Which leads to the conclusion that the rate of length contraction is the same as the rate of time dilation, but we don't know how to measure it.