# Length contraction in General Relativity

• I
• knowwhatyoudontknow
Before asking and answering such questions, shouldn't one first give a definition of what he means by length contraction in a general space-time?
Only if one is a mathematician. Now seriously. No, because it seemed that the source of disagreement between PeroK and me was not in different definitions.

Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.

Only if one is a mathematician. Now seriously. No, because it seemed that the source of disagreement between PeroK and me was not in different definitions.

Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.
I didn't mean a strict definition vs intuitive one. I meant that one needs to say what he means when there is more than one possibility even if it is vague and intuitive definition.

Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.
Anyone teaching SR would immediately be able to give you the precise definition of (coordinate) length as the distance between simultaneous measurements of the two ends of an object (in the given reference frame - usually assumed to be inertial).

And, the proper length is defined as the length in a reference frame in which the object is at rest.

That is the starting point. Without that, you cannot make any progress and conclude that length contraction is a thing.

• cianfa72 and martinbn
Anyone teaching SR would immediately be able to give you the precise definition of (coordinate) length ...
Sure, but here we are talking about GR, so precise definition is not so immediate.

Sure, but here we are talking about GR, so precise definition is not so immediate.
There is none. That's the point. It's not a core concept in GR. It only appears in more obscure scenarios.

Sure, but here we are talking about GR, so precise definition is not so immediate.
Can you give any definition, not necessarily precise?

There is none. That's the point. It's not a core concept in GR. It only appears in more obscure scenarios.
I think it's very subtle. Even in GR there is a well defined operational definition of spatial distance seen by a given observer . One can then compare distances as seen by different observers in relative motion with respect to each other, and this gives a GR generalization of length contraction. Sure, it's not a core concept in GR, but it does make some sense.

 Landau and Lifschitz, The Classical Theory of Fields, Sec. 84.

Can you give any definition, not necessarily precise?
See my post above.

I think it's very subtle. Even in GR there is a well defined operational definition of spatial distance seen by a given observer . One can then compare distances as seen by different observers in relative motion with respect to each other, and this gives a GR generalization of length contraction. Sure, it's not a core concept in GR, but it does make some sense.

 Landau and Lifschitz, The Classical Theory of Fields, Sec. 84.
No, that is a local situation.

No, that is a local situation.
I'm not sure what do you mean by that. If you mean that Eq. (84.6) in LL is local, one can always integrate it ##l=\int dl## to get a global distance. The integral is taken along a geodesic with respect to the spatial metric (84.7).

Another, possibly deeper, way to define proper spatial size of an object in GR is to define it with respect to simultaneity in which the spatial size is maximal. It was used to define the internal black hole volume in https://arxiv.org/abs/1411.2854 .

irst the ruler appears to not contract, because fast moving rulers do not appear to be contracted, for some optics related reason.
Where are you getting this from? Do you have a reference?

when the ruler gets close to the Rindler-horizon, then the ruler appears to contract, as the part of ruler closer to the horizon appear to move slower than the parts of ruler further away.
Reference?

If the rocket is not coordinate-accelerating, but hovering over a large black hole, the same thing is seen in that case too.
Reference?

Before asking and answering such questions, shouldn't one first give a definition of what he means by length contraction in a general space-time?
More than that, one should first read subsequent posts to see what the OP was actually asking about. See in particular post #25 and my responses to it.

• martinbn
I'm not sure what do you mean by that. If you mean that Eq. (84.6) in LL is local, one can always integrate it ##l=\int dl## to get a global distance. The integral is taken along a geodesic with respect to the spatial metric (84.7).
No, that is still local.

And, the proper length is defined as the length in a reference frame in which the object is at rest.
The object actually defines a worldtube in spacetime. The reference frame in which it is at rest should have as "grid lines of constant spatial coordinates" the worldlines in the congruence defined from the object's worldtube.

Last edited:
No, that is still local.
Is that a problem?

Is that a problem?
Yes, if the stick is not near you how do you assign length to it? But even if that worked, the main point since post #2 is that whatever conventions there are in the question, they need to be explicitly made. Otherwise the question is ill posed.

• PeroK
Yes, if the stick is not near you how do you assign length to it?
The reference in post #43 says how.

But even if that worked, the main point since post #2 is that whatever conventions there are in the question, they need to be explicitly made. Otherwise the question is ill posed.
It's explicit in the reference in post #43.

The reference in post #43 says how.

It's explicit in the reference in post #43.
No, it is not there. At least I cannot find it. Can you quote it or point out exactly where to look?

No, it is not there. At least I cannot find it. Can you quote it or point out exactly where to look?
Page 234, "Suppose a light signal is directed ...". It describes an experimental procedure for measuring distances.

@Demystifier - it's also pointed out in that section that integration of the spatial line element ##dl = \sqrt{\gamma_{\alpha \beta} dx^{\alpha} dx^{\beta}}## over a finite domain is only meaningful in static spacetimes, for otherwise the metric is time-dependent and the integral is worldline-dependent.

• Demystifier, cianfa72 and PeroK
for otherwise the metric is time-dependent and the integral is worldline-dependent.
Btw, even though the spacetime is static you must evaluate the integral in the appropriate coordinate chart -- i.e. in the adapted coordinate chart in which the metric does not depends on timelike coordinate.

• ergospherical
Page 234, "Suppose a light signal is directed ...". It describes an experimental procedure for measuring distances.
Which continues "...from some point B to an infinetesimally close point A..."

Which continues "...from some point B to an infinetesimally close point A..."
And later says that it can be integrated to finite distances (provided that metric is time-independent).

And later says that it can be integrated to finite distances (provided that metric is time-independent).
We are going in circles. This is local. An observer can do that and say what the length of a stick is, if the stick is right next to him. What about a second observe, who is far away? What is he going to do to measure the stick's length? All this requires some conventions to be specified. It seems that the OP is unaware of that.

What exactly is your position? It is unclear to me what your point is.

Where are you getting this from? Do you have a reference?

Terrell says "Lorentz contraction can't be seen". Well, maybe the important thing in the paper is not that thing, but the rotation that can be seen. Anyway that is were I got the idea that Lorentz contraction can't be seen.

Reference?
Well, my understanding of a Rindler-horizon is that it's a horizon were motion of stuff appears to freeze, as seen from far above the horizon. Now, if a ruler's one end near said horizon appears to be quite motionless, while the other end is still appears to be moving a little bit, then the length of the ruler appears to be decreasing.

And here's some kind of reference about the last thing, that a person hovering above a large black hole sees the same effects as the person on an accelerating spaceship.

https://www.physicsforums.com/threa...ple-and-rindler-horizons.1007879/post-6550886

Terrell says "Lorentz contraction can't be seen". Well, maybe the important thing in the paper is not that thing, but the rotation that can be seen. Anyway that is were I got the idea that Lorentz contraction can't be seen.
That depends a bit what you mean by "seen". Get a 1m long strip lamp and a 1m long piece of photo paper facing each other a centimetre or so apart and have a rod of rest length 1m pass between at 0.866c. Blink the light as the rod passes through and the shadow on the photo paper will be 0.5m long. However, if you stand and watch the rod or use a regular camera that we can model as pointlike then the light coming from the far end of the rod is older than the light from the near end, so you don't see the whole rod as it is at one time (per an Einstein frame) and this can sometimes counter the length contraction effect.

• PeroK