# Length contraction in GR

In general relativity, $\sqrt{1-v_{e}^2}$ (with $v_e$ the escape velocity as a fraction of the speed of light) is the factor by which time is dilated in a gravitational field. Is it also the factor by which objects get contracted?

• Gravitational length contraction is not a thing. See John Rennie's answer here for why such a thing is not defined. – J. Murray Feb 27 '18 at 1:21
• @JMurray: My question does not involve black holes or event horizons. Your reference does mention radial lengths expansions. I'm asking: what is the factor as a function of $v_e$. – Pierre Berrigan Feb 27 '18 at 2:50
• The presence of black holes is irrelevant. The point is that the idea of gravitational length contraction does not make sense. – J. Murray Feb 27 '18 at 3:34
• I fail to understand why you would insist on the question not making sense. John Rennie's answer that you refer me to, says (I quote:) "...there is a sense in which radial distances are stretched not contracted..." and even gives the Schwarzschild metric deriviate to measure it. – Pierre Berrigan Feb 27 '18 at 13:17

As mentioned in the answer linked in the comments, the notion of gravitational length contraction doesn't make sense.

Special Relativity

Imagine the two of us standing next to one another, holding identical, synchronized clocks. You then start sprinting around in circles while I sit still. When you are finished, you come back and compare your clock to mine, and find that they no longer agree. This is time dilation.

Now imagine the two of us standing next to one another, holding identical meter sticks. You start sprinting around again. This time, as you pass me, we each hold out our meter sticks, and when we line them up next to each other, we each observe the other's meter stick to be a bit shorter. This is length contraction.

General Relativity

Imagine the two of us standing next to each other, holding identical, synchronized clocks. Now you take your clock to a location of different gravitational potential - maybe to the top of a mountain - and sit there for a while. When you come back down and we compare our clocks, we see that the clocks no longer agree. This is gravitational time dilation.

Now, I'm sure you knew all of that. What I'm trying to make clear is that there is no analogous way to talk about gravitational length contraction. We might start off as before, but then it breaks down:

Imagine the two of us standing next to one another, holding identical meter sticks. You then travel to a region of different gravitational potential. When you come back ... the gravitational potential is now the same as it was before, so nothing has happened, and your stick is the same length as mine.

The trouble is that distances need to be compared locally - but if two meter sticks are in the same region of specetime, then they have the same gravitational potential, so there is no way for any gravitational length contraction effect to cause a discrepancy between them.

In the linked question, John Rennie makes the statement

[...] there is a sense in which radial distances are stretched not contracted [...]

but this is not the effect you are talking about. This refers to the fact that if you draw a circular loop around the Earth, then you could measure its circumference $C$ and divide by $2\pi$ to get the distance to the Earth's core.

However, the proper distance to the Earth's core is not the same the number you just calculated. It would be equal if the space enclosed by the loop were flat, but because of the curvature, the proper distance to the core is a bit larger than $C/2\pi$.

This does not refer to length contraction in any meaningful way - rather, it expresses the fact geometry in curved space is not the same as it is in flat space.