general relativity - length contraction in a gravitational field

As space-time is distorted in a gravitational field, relativistic effects such as time dilation and length contraction take effect.

Time dilation is explained simply enough: closer to the source of gravity, slower the time passage.

However, space contraction gives no such clear answer. for example, a single thread on the matter contains the following:

• gravitational field would produce a transverse expansion of distances


• gravitational fields produce increased distances in the direction perpendicular to the field


• [gravitational fields] produce decreased distances in the radial direction

So, to pose the question as bluntly and directly as I can:

Ignoring all other effects (relativistic or otherwise), which is shorter: a measuring stick closer to the source of gravity or a farther one?

If the orientation of the stick plays a part, try to answer for both the case where it's pointing towards the source of gravity, as well as perpendicular.

Similarly, if the position of the observer relative to source of gravity plays a part try to take that into account (but it doesn't seem to in the case of time dilation so I don't expect it here either)

Answer

Your question isn't well defined, because there is no way to compare the two measuring sticks.

I can take two clocks, put them at different heights, then bring them back together and I will find that they now show different times due to time dilation. However if I take two rulers, put them different distances from a black hole then bring them back together I will find that they are still the same length. The reasons for this are covered in the answer Ben Crowell linked above.

However there is a sense in which radial distances are stretched not contracted (as rockandAir8747 says in their answer). But to understand what this means you need to understand what we mean by the Schwarzschild radial distance. The Schwarzschild coordinates use a radial coordinate $r$ that we naively interpret as the distance from the centre of the black hole. However we cannot measure the distance to the centre of the black hole because there's an event horizon in the way. So what we do is lay out our measuring tape in a circle centred on the black hole and measure the circumference of this circle. Because we know that in flat space the circumference is $2\pi r$ we can take our measured circumference and divide by $2\pi$ to get the Schwarzschild radial distance $r$.

But this $r$ coordinate is calculated assuming space is flat, so it shouldn't be any surprise that it is different from the distance you'd measure if you let down a measuring tape towards the event horizon. There's a nice image showing this difference in the question Difference between coordinate and proper distance in Schwarzschild geometry:

The $dr$ in the bottom diagram is the distance in Schwarzschild coordinates while the $dr$ in the top diagram is the distance you'd measure with a ruler, and it's obviously bigger than the Schwarzschild $dr$.

I suppose in this sense you could claim that the ruler is length contracted near to the black hole, because it takes more rulers than you expect to cover the distance $dr$.

If you want the gory details, including an equation for how to calculate lengths measured towards the event horizon then see the questions:

The measured distance, s, between the two Schwarzschild radii $r_1$ and $r_2$ is given by:

$$ s = \left[ r\sqrt{1-\frac{r_s}{r}} + \frac{r_s}{2} log \left( 2r \left( \sqrt{1-\frac{r_s}{r}} + 1 \right) - r_s \right) \right]_{r_1}^{r_2} $$