Understand Proper Distances in SR/GR: A Layman's Guide

[rede96]

I'm just an interested laymen with no background in physics trying to get a grip on some of the principles of SR and GR. I've got a grip of some of the basics but one thing I am struggling with is how distances in GR are coordinate dependent.

If I take a simple example using SR to start and imagine two objects A and B, which are at rest wrt each other and are separated by some distance. Anyone moving relative to these two objects would measure a different distance to the proper distance. However if they needed to use their respective measurements of distance in order to work out say the time to take a rocked ship to move between A and B or to calculate the length of an object that fits exactly between A and B, then I assume they would need to do some sort of transformation in order to make accurate predictions for the rocket ship or make the right size of object to fit exactly between A and B.

So using the examples above, my questions are:

1) Is SR, how would the proper distance be measured if A was moving wrt B?

2) How are proper distances worked out in GR, as I am told that distances are coordinate dependent in GR. Although I am not sure what that even means.

Thanks for any help

 

[PeterDonis]

Is SR, how would the proper distance be measured if A was moving wrt B?

If A and B are not at rest relative to each other, there is no such thing as the "proper distance" between them.

How are proper distances worked out in GR

If two objects in curved spacetime are at rest relative to each other, then the proper distance between them can be defined similarly to the way it is defined in SR. If they are not at rest relative to each other, there is no such thing as the proper distance between them, as above.

 

[rede96]

If A and B are not at rest relative to each other, there is no such thing as the "proper distance" between them.

So just sticking with the SR example for now, if object A had a really long pole attached to it of a known length (As measured in the rest frame of A) that reached out into space, if object B was to move past and touch the end of the pole, couldn't a person in the FOR of object A say at that instant the proper distance between A and B was the length of the pole?

 

[PeterDonis]

couldn't a person in the FOR of object A say at that instant the proper distance between A and B was the length of the pole?

No, because A and B are not at rest relative to each other.

 

[A.T.]

...a person in the FOR of object A say at that instant the proper distance...

The idea of a "proper distance" is that all FORs agree on it.

 

[rede96]

No, because A and B are not at rest relative to each other.

That's the bit I don't understand. At the point B touches the pole attached to A, for that instant it seems the whole system can be said to be at rest. Everything is touching everything else, so how can they not be at rest for that instant? I understand A might not know just when B touches the end of his pole as it might be too far away so there will be time lag. But B will know when he touches the pole and if he already knows the rest length of the pole, why can't he say I am the pole's length away from A? I just don't get it.

So rather than just stating I am not correct, which is fine, could you explain why?

 

[rede96]

The idea of a "proper distance" is that all FORs agree on it.

But if all FORs knew the length of the pole, as they had previously measured it when they were at rest wrt the pole, then if they saw B touch the pole (which is connected to A) then couldn't all FORs agree that the distance between A and B at that instance was the length of the pole?

Sorry if these are really stupid questions, but I'm just struggling to get this.

 

[PAllen]

If A and B are not at rest relative to each other, there is no such thing as the "proper distance" between them.

Well, you could define a distance per A using a natural simultaneity for A, as of some moment of A's history. Same for B. These could come out very different, of course, as they would involve completely different pairs of events on the respective histories. Proper distance is often used as geodesic distance between points on some simultaneity hypersurface. That's the most common use in cosmology, where the simultaneity is, just by convention, the surface of common proper time for the all comoving trajectories. Of course, different natural simultaneity definitions (e.g. radar, or surface of spacelike geodesics 4-orthogonal to a world line) yield different proper distances. Even more technical note: in using radar to define a proper distance, you are only using it as a simultaneity convention, not attaching any meaning to travel times per se. The proper distance would be geodesic length within a simultaneity surface.

If two objects in curved spacetime are at rest relative to each other, then the proper distance between them can be defined similarly to the way it is defined in SR. If they are not at rest relative to each other, there is no such thing as the proper distance between them, as above.

Well, in GR, there no general definition of mutual rest for distant objects. If you can define mutual rest unambiguously, you could define relative velocity unambiguously. Even for static spacetimes, while there is a natural choice for mutual rest between certain world lines, you have not removed the feature that parallel transport of 4-velocity along different paths would still makes this definition conventional; and for other world lines in static spacetime you don't even have a candidate unambiguous criterion for mutual rest.

Further, as noted above, cosmologists routinely talk about proper distance between objects clearly not at mutual rest by any natural criterion.

 

[PAllen]

But if all FORs knew the length of the pole, as they had previously measured it when they were at rest wrt the pole, then if they saw B touch the pole (which is connected to A) then couldn't all FORs agree that the distance between A and B at that instance was the length of the pole?

Sorry if these are really stupid questions, but I'm just struggling to get this.

Yes, but if the pole was at rest with respect to B instead, it would show different distances. And if a third observer moving differently from A and B used their own pole, they would get yet another set of distances. So using an idealized pole does nothing to establish a unique proper distance, as I explained to you ad-nauseum in the other thread.

To be clear, we have A, at some instant touching moving B with a pole at rest with respect A. We also have B using a pole at rest with respect to B, such that it touches A at the same moment as A's measurement. These poles will have different rest lengths. Note that per B, the event on A's history that A claims their pole was touching B is not correct; per B, A's pole was touching at a different time on A's history than A claims.

 

[rede96]

Yes, but if the pole was at rest with respect to B instead, it would show different distances.

Just so I am clear, are you saying if A detached the pole and moved away, B then came to rest wrt to the pole and attached it, then A moved past and touched the end of the pole, then this situation is NOT symmetrical to the first instance when the pole was attached to A and B moved past and touched the end of the pole?

If so then that I definitely don't understand!

Also, if a third observer managed to take a picture of the exact moment when B touched the end of the pole, and the picture showed all 3 objects connected, why can't the length of the pole represent the distance between the two objects in that case?

as I explained to you ad-nauseum in the other thread.

Sorry, I know I am persistent but this particular topic fascinates me and I'd really love to understand it. And although I appreciate the help, so far there have been no answers I could understand that have helped me to resolve this problem.

 

[Ibix]

Proper length of an object is its length in its rest frame. The proper distance between two objects at rest is the same as the proper length of a pole, also at rest with respect to the objects, fitting exactly between them.

When one of the objects is in motion then you can't find a rod at rest with respect to both of them. If you pick a rod at rest with respect to one it will be moving, and hence length contracted, with respect to the other, so they will never agree on the distance between them.

 

[PeterDonis]

At the point B touches the pole attached to A, for that instant it seems the whole system can be said to be at rest.

No, it can't. B is moving relative to A and the pole. There's no such thing as being at rest for an instant; the concept only makes sense over an extended period of time.

so far there have been no answers I could understand that have helped me to resolve this problem.

I think that is because you are refusing to accept the simple statements that have been given to you: there is no absolute notion of "distance", and the concept of "at rest" makes no sense for only a single instant. Whether or not you find those statements intuitively plausible, they are what relativity says. So if you want to understand relativity, you have to accept them.

I am closing this thread since, once again, I don't see the point of continuing to rehash the same statements. Either you accept them or you don't. If you don't, I'm sorry, but there's nothing more we can do to help you understand.