I Length contraction in General Relativity

[PeroK]

The original intent of my question was to reconcile the idea of time dilation and length contraction near a black hole.

As I pointed out above, if you study Sean Carroll's book Spacetime and Geometry in order to answer this question, then you'll be disappointed. Length contraction is not mentioned in the entire book; not even in the review of SR.

 

[Demystifier]

As I pointed out above, if you study Sean Carroll's book Spacetime and Geometry in order to answer this question, then you'll be disappointed. Length contraction is not mentioned in the entire book; not even in the review of SR.

What is really your point? Please choose one of the suggested answers that describes your opinion best.

a) Length contraction is entirely unphysical.
b) Length contraction makes sense only for flat spacetime.
c) Length contraction makes sense only when curvature is sufficiently small.
d) Length contraction makes sense only in the absence of horizons and/or coordinate singularities.

 

[PeroK]

What is really your point? Please choose one of the suggested answers that describes your opinion best.

a) Length contraction is entirely unphysical.
b) Length contraction makes sense only for flat spacetime.
c) Length contraction makes sense only when curvature is sufficiently small.
d) Length contraction makes sense only in the absence of horizons and/or coordinate singularities.

Length contraction is a useful concept in flat spacetime. Although, beyond introductory SR it becomes less useful. In particle physics, we use the energy-momentum relations and I don't recall length contraction ever being relevant to high-energy particle physics. In GR, the concept is of no practical use. There may be exceptions, but I suggest you are unikely to find discussion of length contraction in a GR text.

 

[Demystifier]

Length contraction is a useful concept in flat spacetime. Although, beyond introductory SR it becomes less useful. In particle physics, we use the energy-momentum relations and I don't recall length contraction ever being relevant to high-energy particle physics. In GR, the concept is of no practical use. There may be exceptions, but I suggest you are unikely to find discussion of length contraction in a GR text.

OK, I can agree that length contraction is not very useful in GR. But it's not totally useless. An interesting example is the submarine paradox [1]. In its standard formulation it combines length contraction and gravity, while recently I found a way to simplify it by considering a version of the paradox which does not use length contraction [2]. But just because the concept is not so much useful, it does not mean that the concept is totally wrong. I think it is perfectly legitimate to ask conceptual questions that involve length contraction in GR, even if GR is more naturally formulated without it.

[1] https://en.wikipedia.org/wiki/Supplee's_paradox
[2] https://arxiv.org/abs/2112.11162

 

[martinbn]

What is really your point? Please choose one of the suggested answers that describes your opinion best.

a) Length contraction is entirely unphysical.
b) Length contraction makes sense only for flat spacetime.
c) Length contraction makes sense only when curvature is sufficiently small.
d) Length contraction makes sense only in the absence of horizons and/or coordinate singularities.

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?

 

[Demystifier]

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.

 

[martinbn]

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.

 

[PeroK]

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.

 

[Demystifier]

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.

 

[PeroK]

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.

 

[PeroK]

There's a relevant discussion here:

https://physics.stackexchange.com/questions/168379/proper-length-in-gr

 

[martinbn]

Sure, but here we are talking about GR, so precise definition is not so immediate.

Can you give any definition, not necessarily precise?

 

[Demystifier]

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 [1]. 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.

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

 

[Demystifier]

Can you give any definition, not necessarily precise?

See my post above.

 

[martinbn]

I think it's very subtle. Even in GR there is a well defined operational definition of spatial distance seen by a given observer [1]. 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.

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

No, that is a local situation.

 

[Demystifier]

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).

 

[Demystifier]

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 .

 

[PeterDonis]

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?

 

[PeterDonis]

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.

 

[knowwhatyoudontknow]

I found this: https://www.physicsforums.com/threads/gravitational-length-contraction-and-dilation.404153/. Post# 9. What bearing does this have on the current discussion?

 

[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.

 

[cianfa72]

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.

 

[PeterDonis]

I found this: https://www.physicsforums.com/threads/gravitational-length-contraction-and-dilation.404153/. Post# 9. What bearing does this have on the current discussion?

It shows some of the issues involved if you try to give a well-defined meaning to "gravitational length contraction".

 

[Demystifier]

No, that is still local.

Is that a problem?

 

[martinbn]

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.

 

[Demystifier]

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.

 

[martinbn]

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?

 

[Demystifier]

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.

 

[ergospherical]

@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.

 

[cianfa72]

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.