Rest length in general relativity

[jostpuur]

I would be interested to know how the author thinks that all observers agree on rest length in general relativity.

Is it the case, that in general relativity the rest length is not absolute?

How is the rest length defined?

Is it the problem, that it is not clear how one should choose the rest frame of some object, if the object is large?

 

[gendou2]

In general relativity, is the rest length absolute?

What do you mean "absolute"?
Assuming only intelligent observers, all observers in all reference frames will agree on what the rest length of an object is.
Is that what you were wondering?

How is the rest length defined?

The rest length of an object is how long it is when you measure it in it's reference frame.

How should one choose the rest frame of a large object?

No different from choosing the rest frame of a small object?
Take the Earth, for example.
Ignoring it's rotation, when you stand still on the surface, you're in it's rest frame.

 

[MeJennifer]

The rest length of an object is how long it is when you measure it in it's reference frame.

How do you define a reference frame in a non-stationary spacetime, which is the spacetime of our universe?

 

[gendou2]

How do you define a reference frame in a non-stationary spacetime, which is the spacetime of our universe?

This is how I define a reference frame:
http://www.google.com/search?q=define:+reference+frame

I don't know about stationary versus non-stationary spacetime.
Please explain how it follows that one cannot define reference frames within a non-stationary spacetime.

 

[Fredrik]

The problem isn't to define what a coordinate system is, it's to choose which one to use. It's definitely not a trivial question.

Proper length is the integral of $\sqrt{dx^2+dy^2+dz^2-dt^2}$ along a space-like curve, but which one? If the object is a spinning rod for example, one of the endpoints of the curve is an event on the world line of one of the endpoints of the rod, but what's the other endpoint of the curve? It's an event on the world line of the other endpoint of the rod, but which one? And even if you have an answer to that, do you know which curve connecting the two events you should use?

 

[Mentz114]

The problem isn't to define what a coordinate system is, it's to choose which one to use. It's definitely not a trivial question.

Proper length is the integral of $\sqrt{dx^2+dy^2+dz^2-dt^2}$ along a space-like curve, but which one? If the object is a spinning rod for example, one of the endpoints of the curve is an event on the world line of one of the endpoints of the rod, but what's the other endpoint of the curve? It's an event on the world line of the other endpoint of the rod, but which one? And even if you have an answer to that, do you know which curve connecting the two events you should use?

Observables in GR are coordinate independent because they are scalars obtained by contracting tensors. The rest length of a rod is such an observable, so it doesn't matter what coordinate system is chosen.

You have given the SR definition of proper length, incidentally. As you know it is different in GR.

 

[jostpuur]

Observables in GR are coordinate independent because they are scalars obtained by contracting tensors. The rest length of a rod is such an observable, so it doesn't matter what coordinate system is chosen.

Not precisely. The expression for length also involves an integral

$$\int \sqrt{g_{ij}\frac{dx^i}{d\alpha}\frac{dx^j}{d\alpha}} d\alpha$$

which is supposed to be carried out on a given instant, so it depends on the chosen frame. The rest length is supposed to be invariant, which could be achieved if there existed a unique frame where the integration is supposed to be carried out. If I understood correctly what MeJennifer and Fredrik were saying, then this unique frame doesn't exist, although I don't yet understand why this is the case.

 

[JesseM]

This is how I define a reference frame:
http://www.google.com/search?q=define:+reference+frame

I don't know about stationary versus non-stationary spacetime.
Please explain how it follows that one cannot define reference frames within a non-stationary spacetime.

You can certainly define a coordinate system where an object is at rest in general relativity, the problem is that you can construct all different kinds of coordinate systems where this is true and which nevertheless disagree about the object's coordinate length, and all these coordinate systems are equally valid. This is different from SR, where only inertial coordinate systems are equally valid, and there is a set procedure for constructing inertial coordinate systems which guarantees that if two inertial coordinate systems agree an object is at rest, they will agree on its coordinate length.

 

[jostpuur]

I just realized the integral might have as well be written as

$$\int\sqrt{g_{\mu\nu}\frac{d x^{\mu}}{d\alpha}\frac{d x^{\nu}}{d\alpha}}d\alpha,$$

and then you don't need to be in the rest frame.

So... hmhmh... even this integral is not coordinate independent? Even though the integration path in the physical space would be fixed? hmhmh... the integral looks invariant to me The rest length is not unique despite that this integral is invariant?

 

[pmb_phy]

This is how I define a reference frame:
http://www.google.com/search?q=define:+reference+frame

I don't know about stationary versus non-stationary spacetime.
Please explain how it follows that one cannot define reference frames within a non-stationary spacetime.

A stationary spacetime is a spacetime for which there exists a coordinate system in which the components of the metric tensor are not explicit functions of time.

..which is supposed to be carried out on a given instant, so it depends on the chosen frame.

The quantity that you just defined is an invariant. It is independant of the coordinate system used. The value of this quantity, known as the proper distance, depends on the worldline, not on the coordinate system. However for small spatial distances this quantity is unique.

Pete

 

[jostpuur]

The quantity that you just defined is an invariant. It is independant of the coordinate system used. The value of this quantity, known as the proper distance, depends on the worldline, not on the coordinate system. However for small spatial distances this quantity is unique.

I noticed I made a mistake in the post #7, and continued about it in the post #9, but I got new questions there, which still puzzle me.

 

[jostpuur]

Is it so that the integration path cannot be defined uniquely when determining the rest length of an object?

 

[pmb_phy]

Is it so that the integration path cannot be defined uniquely when determining the rest length of an object?

No. While the proper length of a path is path dependant the length is path dependant and its that path that you choose and that should be unique. I can't think of a counter example.

Pete

 

[jostpuur]

the matter clear?

In order to find the rest length of an object, we first choose a rest frame of the object, and set the length to be

$$\int \sqrt{g_{ij}\frac{dx^i}{d\alpha}\frac{dx^j}{d\alpha}} d\alpha,$$

where the integration is carried out only in the spatial space, with fixed time. Since $$dx^0/d\alpha=0$$, the integral can as well be written as

$$\int\sqrt{g_{\mu\nu}\frac{d x^{\mu}}{d\alpha}\frac{d x^{\nu}}{d\alpha}}d\alpha,$$

and then the integral does not depend on the chosen frame anymore, and is invariant.

The problem with uniqueness rises from the fact, that if the rest frame cannot be chosen uniquely in the beginning, then the integration path is not unique either. So this is the reason why there is not unique rest length in general relativity?

 

[jostpuur]

No. While the proper length of a path is path dependant the length is path dependant and its that path that you choose and that should be unique. I can't think of a counter example.

Pete

It could be I don't want to see the counter example. I'm not devoting my life for the general relativity

MeJennifer and JesseM seemed to be confident that the rest frame cannot be chosen uniquely, so I think I'm satisfied with it, assuming that they have learned this from some reliable source.

 

[Mentz114]

The rest length of an object is its length as measured and reported by an observer at rest wrt to the measured object. That's it. There is no ambiguity in this. It is not observer dependent because we chose the observer.

What are we all talking about ?

 

[jostpuur]

The rest length of an object is its length as measured and reported by an observer at rest wrt to the measured object. That's it. There is no ambiguity in this. It is not observer dependent because we chose the observer.

What are we all talking about ?

It is not possible to be intuitively capable of telling what is happening if there is a large object in space with non-trivial time-depending metric. The observer might have to travel along the object in order to measure its length. Or if the observer has installed measuring devices all over the object before the measurement, he will have to be able to synchronize them somehow, and then be able to interpret the data. It's not like the observer who is in rest with the object merely measures the distance, and that's it.

 

[pmb_phy]

It could be I don't want to see the counter example. I'm not devoting my life for the general relativity

MeJennifer and JesseM seemed to be confident that the rest frame cannot be chosen uniquely, so I think I'm satisfied with it, assuming that they have learned this from some reliable source.

MeJennifer was referring to non-stationary spacetimes. A body in such a spacetime would, in general, have its rest length a function of time because the spatial contraction would be a function of time. In that sense it is not unique, but (I think) its still invariant.

Hurkyl pointed out something else that might be bothering MeJennifer. The fact that the proper length of a body would depend on the bodies orientation in a gravitational field

However the FAQ that started this inquiry was about SR although the author didn't say that, unfortunately.

Pete

 

[Hurkyl]

The rest length of an object is its length as measured and reported by an observer at rest wrt to the measured object. That's it. There is no ambiguity in this. It is not observer dependent because we chose the observer.

What are we all talking about ?

Objects have spatial extent.

In special relativity, there's no reason different parts of the object must be at rest with each other, and therefore there's no reason to expect an observer can be at rest with all parts of the object.

In general relativity, we cannot even define what it means for things at two different locations in space-time to be at rest with each other, so "observer at rest with the mesaured object" is, strictly speaking, nonsense.


AFAIK, the best we can do is assume some sort of smallness condition that ensures all reasonable methods all give approximately the same answer.

 

[jostpuur]

MeJennifer was referring to non-stationary spacetimes. A body in such a spacetime would, in general, have its rest length a function of time because the spatial contraction would be a function of time. In that sense it is not unique, but (I think) its still invariant.

Hurkyl pointed out something else that might be bothering MeJennifer. The fact that the proper length of a body would depend on the bodies orientation in a gravitational field

However the FAQ that started this inquiry was about SR although the author didn't say that, unfortunately.

Pete

MeJennifer's comment caught my attention, but I thought continuing it in the FAQ thread would have been a little bit off topic. But with new thread, it doesn't matter anymore how off topic this is!

"Rest length of a spacelike curve" is not ambiguous. "Rest length of an object" is ambiguous, because there is no canonical way to choose which spacelike curve expresses the rest length of the object.

You get the same problem in special relativity too -- there are only a few special cases which determine 'obvious' sorts of spacelike curve to measure. e.g. an inertially traveling object, or one undergoing uniform constant acceleration.

Well this is a new twist in the story. Is this because the velocity of the object is not unique even in some fixed frame, necessarily? Some parts of the object are moving with different velocities, making the rest frame difficult concept?

So this thing never had much to do with the general relativity?

 

[jostpuur]

Yeah. I wasn't seeing the Hurkyl's post #19 when writing my previous post.

 

[pmb_phy]

In general relativity, we cannot even define what it means for things at two different locations in space-time to be at rest with each other, so "observer at rest with the mesaured object" is, strictly speaking, nonsense.

What justification do you have for this assertion?

Pete

 

[Mentz114]

Hurkyl,

In general relativity, we cannot even define what it means for things at two different locations in space-time to be at rest with each other, so "observer at rest with the mesaured object" is, strictly speaking, nonsense.

Exactly. So the question is nonsense too. Or at least ill-defined. The use of 'rest' in GR does not have the meaning it does in SR, and can only be defined over a small region. But, my definition could be said to be valid because it's operational.

M

 

[jostpuur]

What justification do you have for this assertion?

I suppose it's the usual: You need to do parallel transporting to compare velocities at different space time points. The relative velocity depends on the chosen path of parallel transport.

Exactly. So the question is nonsense too. The use of 'rest' in GR does not have the meaning it does in SR. But, my definition is valid because it's operational.

Your definition attempt about rest frame in SR is not valid, because some parts of the object are not necessarily in rest with respect to other parts of the object.

 

[pmb_phy]

I suppose it's the usual: You need to do parallel transporting to compare velocities at different space time points. The relative velocity depends on the chosen path of parallel transport.

This is altogether different. Parallel transport has nothing to do with proper distance.

Pete

 

[MeJennifer]

proper distance.

Note that term 'proper distance' in cosmology has a particular meaning for FRW spacetimes.

 

[Mentz114]

jost,

Your definition attempt about rest frame in SR is not valid, because some parts of the object are not necessarily in rest with respect to other parts of the object.

This is getting circular - the object I'm measuring in my rest frame has ALL its parts at rest by definition ! If not, it is not in my rest frame. For heavens sake.

The concept you use of 'rest length' in your question suffers the same flaw. What do you mean by it ? Are you saying that some of your 'rest' frame is not at rest ? That is contradictory.

I just think you are using the word 'rest' incorrectly. Are you asking about the possibility that the measured length changes if the orientation to the gravitation changes ?

M

PS it's good to see you here in the relativity section - I enjoyed your many contributions in QM.

 

[pmb_phy]

Note that term 'proper distance' in cosmology has a particular meaning for FRW spacetimes.

This is news to be. Can you be more specific?

Pete

 

[MeJennifer]

See for instance:

Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe
Authors: Tamara M. Davis, Charles H. Lineweaver
http://arxiv.org/abs/astro-ph/0310808

 

[robphy]

This is altogether different. Parallel transport has nothing to do with proper distance.

Pete

So, given two 4-velocities at different events in a general spacetime,
how do you compare those unit-vectors to see if they are parallel
(which is presumably one way to characterize that those particles are at rest with respect to each other)?

 

[jostpuur]

The concept you use of 'rest length' in your question suffers the same flaw. What do you mean by it ?

I'm not sure. Actually I asked in the opening post, that what are we supposed to mean by it

I didn't know what MeJennifer was talking about in her post, so I decided to ask something.

 

[pmb_phy]

MeJennifer was referring to non-stationary spacetimes. A body in such a spacetime would, in general, have its rest length a function of time because the spatial contraction would be a function of time.

Ummmm ... at least this is what I think she was referring to. Is this correct MJ?

Pete

 

[JesseM]

This is getting circular - the object I'm measuring in my rest frame has ALL its parts at rest by definition ! If not, it is not in my rest frame. For heavens sake.

What is "your rest frame" in general relativity? One can use arbitrary coordinate systems, so you could find one coordinate system where you were at rest but the object (or one part of the object) was not, and another where you were both at rest. There isn't a specific procedure for constructing your frame as there is in special relativity where you use inertial rulers and clocks and synchronize clocks using the Einstein synchronization convention.

 

[pmb_phy]

What is "your rest frame" in general relativity? One can use arbitrary coordinate systems, so you could find one coordinate system where you were at rest but the object (or one part of the object) was not, ...

All that would mean is that the rest length is time dependant as mentioned above.

Pete

 

[JesseM]

All that would mean is that the rest length is time dependant as mentioned above.

Does "time dependent" mean it depends on a particular coordinate system's definition of simultaneity? Of course in GR surfaces of simultaneity can bend and wave pretty much arbitrarily (as long as the surface is spacelike) from one end of the object to another depending on what coordinate system you choose, they aren't neat planes as in SR, so if this is true it seems like you could make the "rest length" of an object pretty much anything you wanted depending on your choice of coordinate system.