Length Contraction causes Time Dilation?

[Dale]

thinking about the hypothetical inertial frame is one thing

There is nothing hypothetical about the inertial reference frame.

thinking about the hypothetical inertial frame is one thing, but it is another to claim the particular details of measurement within said hypothetical inertial frame are the cause of the effects you experience in an accelerated frame.

Nobody claimed that. The explanations would be different in a non-inertial frame, but equally valid.

 

[Max™]

What has GR to do with this?

Well, working with just SR is kinda like me telling you calculate integrals and differentials, but not letting you use the fundamental theorem of calculus to flesh the structure out properly.

SR is great because the mathematical form doesn't require you to keep track of multi-dimensional tensor systems, but it's just not the same as the richer understanding you gain when you include GR into the structure.



As for the inertial frame, if you were flying from here to Betelgeuse you would need to accelerate until the halfway point and then decelerate the rest of the way in order to reach relativistic speeds such as we're discussing, presumably some sort of nuke-pusher drive or maybe an anti-matter pulse engine.


In such a scenario, yes, the inertial frame is a hypothetical, there wouldn't be a period where you would be able to claim you were at rest, much less that your frame hadn't been boosted compared to your origin frame.

 

[Passionflower]

Well, working with just SR is kinda like me telling you calculate integrals and differentials, but not letting you use the fundamental theorem of calculus to flesh the structure out properly.

SR is great because the mathematical form doesn't require you to keep track of multi-dimensional tensor systems, but it's just not the same as the richer understanding you gain when you include GR into the structure.



As for the inertial frame, if you were flying from here to Betelgeuse you would need to accelerate until the halfway point and then decelerate the rest of the way in order to reach relativistic speeds such as we're discussing, presumably some sort of nuke-pusher drive or maybe an anti-matter pulse engine.


In such a scenario, yes, the inertial frame is a hypothetical, there wouldn't be a period where you would be able to claim you were at rest, much less that your frame hadn't been boosted compared to your origin frame.

Sorry Max I lost you completely here.

 

[Dale]

In such a scenario, yes, the inertial frame is a hypothetical, there wouldn't be a period where you would be able to claim you were at rest, much less that your frame hadn't been boosted compared to your origin frame.

No, the inertial frame is not hypothetical in any way regardless of whether or not the ship is ever at rest in it.

In addition to the sloppy language I pointed out above you seem to have this mistaken notion that there is a requirement for some object to be at rest in order for a reference frame to be valid. That is simply not the case. We need not restrict our analysis to frames where some object is at rest and we need not restrict our analysis to objects which are at rest in our chosen frame.

 

[Max™]

Again, why would you think that is what I was saying?

I said an observer who is undergoing constant acceleration can't claim they are in an inertial frame, is this wrong now?

 

[Dale]

I said an observer who is undergoing constant acceleration can't claim they are in an inertial frame, is this wrong now?

Yes, this is wrong as written. It is not possible for something to be "in a frame" or "out of a frame". A frame is a coordinate system, it has infinite spatial extent. An object may be "at rest in a frame" or "moving in a frame", but it exists in every possible frame at all times regardless of its motion.

An observer who is undergoing constant proper acceleration can't claim that they are at rest in an inertial frame. But they may certainly use any arbitrary inertial frame to do physics calculations in anyway.

If you really understand this then why would you repeatedly use the word "hypothetical" to describe the inertial frame where the Earth-Betelgeuse distance is 2 ly?

 

[Max™]

I said they can't claim they are in an inertial frame, by which I would have assumed at this point in the discussion it would be implied that I meant "can't claim they are at rest in one", so it is a hypothetical scenario to argue that said observer would be unable to distinguish changes due to a boosted frame of reference from changes due to the universe being smushed up around them.


Yes, from a suitably chosen frame the distance is measured as 2 ly, and if you were at rest in such a frame you would not claim this distortion was due to time dilation.

The observer being described in this scenario can not make such a claim, and accordingly would conclude that the distortion of their measurements was due to time dilation/them experiencing lorentz contraction with their measuring rods.



Similarly, it is hard to ignore the variations in gravity between the Earth and Betelgeuse, though they may be exceptionally slight, nonetheless they break symmetry between all frames, any gravity is an acceleration, even if you were "coasting at rest" (free fall), a sufficiently accurate clock/ruler would display the effects compared to what you would expect when you compared a clock/ruler you know to be in a well at Earth or Betelgeuse to a hypothetical situation where you're coasting between them in perfectly flat spacetime at infinite distance from a gravity well such as those described in SR.

You can define a local frame in GR as an inertial one (assuming you're not spinning, ideally), and if you can get your acceleration components to vanish you could claim your hypothetical ideal observer in that frame is at rest in their locally flat spacetime (i.e. is following a path congruent to a geodesic in that spacetime), and that is about as close as GR gets to the ideal Lorentz frames from SR.

 

[Dale]

Yes, from a suitably chosen frame the distance is measured as 2 ly, and if you were at rest in such a frame you would not claim this distortion was due to time dilation.

The observer being described in this scenario can not make such a claim, and accordingly would conclude that the distortion of their measurements was due to time dilation/them experiencing lorentz contraction with their measuring rods.

Even if you are not at rest in such a frame you may still use it, and if you used that frame then your time dilation would only be due to your velocity in that frame and other frames (including the Earth frame) would be time dilated.

You seem to think that it is essential for an accelerating observer to refer their observations (including their acceleration) back to the Earth inertial frame and that any other view (either a non-inertial frame or a different inertial frame) represents a "distortion". That misses the point of relativity entirely. Any inertial frame is equally valid and is not considered distorted or invalid in any way. From the 2-ly frame it is the Earth rods and clocks that are so contracted and dilated that they measure 640 ly.

So why insist on referring the ship observations back to the Earth frame rather than some other perfectly valid frame?

 

[Max™]

Even if you are not at rest in such a frame you may still use it, and if you used that frame then your time dilation would only be due to your velocity in that frame and other frames (including the Earth frame) would be time dilated.

Yes, I am aware that this is the case in SR, this is what I was saying, SR is an idealized hypothetical case of GR where acceleration is less important, and where spacetime is a flat Minkowski spacetime.

You seem to think that it is essential for an accelerating observer to refer their observations (including their acceleration) back to the Earth inertial frame and that any other view (either a non-inertial frame or a different inertial frame) represents a "distortion". That misses the point of relativity entirely. Any inertial frame is equally valid and is not considered distorted or invalid in any way. From the 2-ly frame it is the Earth rods and clocks that are so contracted and dilated that they measure 640 ly.

No, I am pointing out that the only important things in General Relativity are paths and their lengths, and as such it is important to treat your choice of frames in a manner which best suits the state in your local region or the local region of another hypothetical observer.

Similarly, an arbitrary choice of Lorentz frames can only be inserted into GR as an approximation of the local spacetime for a particular ideal observer. GR is all about ideal observers, they're everydamnwhere, and it is convenient to assume ideal test particles following true geodesic paths.

In such a situation, those geodesics between two events define minimum possible lengths in time for timelike paths, or space for spacelike paths.


Any inertial frame is equally valid in SR, but not GR, at most they are locally valid.

So why insist on referring the ship observations back to the Earth frame rather than some other perfectly valid frame?

I am actually comparing the observations to an idealized geodesic path between Earth and Betelgeuse, perhaps I shouldn't assume everyone else will take GR into consideration, but there is no reason to NOT use GR to better explain some of the less intuitive aspects of SR, is there?

 

[Dale]

Yes, I am aware that this is the case in SR, this is what I was saying, SR is an idealized hypothetical case of GR where acceleration is less important, and where spacetime is a flat Minkowski spacetime. ...

perhaps I shouldn't assume everyone else will take GR into consideration, but there is no reason to NOT use GR to better explain some of the less intuitive aspects of SR, is there?

Sure, but at these scales both the FLRW metric and the Schwarzschild metric are essentially flat. So GR doesn't add anything significant here.

No, I am pointing out that the only important things in General Relativity are paths and their lengths, and as such it is important to treat your choice of frames in a manner which best suits the state in your local region or the local region of another hypothetical observer. ...

In such a situation, those geodesics between two events define minimum possible lengths in time for timelike paths, or space for spacelike paths. ...

I am actually comparing the observations to an idealized geodesic path between Earth and Betelgeuse

And in this case the length of the geodesic in question is ~2 ly, not 640 ly. So in terms of paths and lengths the 2 ly frame is the one which best suits the problem, not the Earth frame.

 

[Max™]

No, there are proper times and proper lengths, and there are coordinate choices which can lead to different results when those projections are applied to non-boosted frames.

The 2 ly frame is NOT the best solution for describing the problem, it should be nothing more than a sidenote, an "oh, and an observer using his position as the origin of an unaccelerated set of coordinates will project those measurements in such a manner that he will claim the distance is 2 ly", not "this is what happens, there are no invariant properties anywhere, take your beautiful hyperbolic geometry and flush it because we like the description of shrinking rulers and slowing clocks"...

I mean, I suppose your personal taste could lead you to prefer the "contraction/dilation is king" description, but it is not uniquely correct, and it is not the most useful explanation either.

If it had never been put forth in terms of shrinking rulers and slowing clocks we wouldn't have to deal with people getting stuck on silly things like the twin paradox or ladder and barn paradox.


We could have people discussing whether it is better to pronounce "cosh" as (kawsh) or (kose-aytch) instead, ah well.

 

[ghwellsjr]

No, there are proper times and proper lengths, and there are coordinate choices which can lead to different results when those projections are applied to non-boosted frames.

The 2 ly frame is NOT the best solution for describing the problem, it should be nothing more than a sidenote, an "oh, and an observer using his position as the origin of an unaccelerated set of coordinates will project those measurements in such a manner that he will claim the distance is 2 ly", not "this is what happens, there are no invariant properties anywhere, take your beautiful hyperbolic geometry and flush it because we like the description of shrinking rulers and slowing clocks"...

I mean, I suppose your personal taste could lead you to prefer the "contraction/dilation is king" description, but it is not uniquely correct, and it is not the most useful explanation either.

If it had never been put forth in terms of shrinking rulers and slowing clocks we wouldn't have to deal with people getting stuck on silly things like the twin paradox or ladder and barn paradox.


We could have people discussing whether it is better to pronounce "cosh" as (kawsh) or (kose-aytch) instead, ah well.

Max, proper times and proper distances are not effected by anyone's choice of reference frame or even if SR is used to analyze the problem. Co-ordinate times and co-ordinated distances are what are effected by the choice of reference frame. It doesn't matter how you analyze the problem, assuming, of course, that you do it in any correct way, the traveler will be able to survive the 2 year trip, his precision clock will advance only 2 years and any correct means that he uses to calculate his distance will determine that he has traveled just under 2 light-years (or whatever your specific scenario is). Again, it doesn't matter which frame of reference you choose to analyze the scenario in. You could even do it with LET believing in an absolute ether rest frame and get the same answers. Do you doubt this?

 

[soothsayer]

What Max is essentially asserting, I think, is that we cannot just pick EITHER time dilation OR length contraction to describe the journey at relativistic speeds to a far away star, we have to use both. Otherwise there is not point in describing a time dilation AND length contraction. If we say that the velocity is such that the length of the distance between the two bodies in the reference frame of the ship is 2ly, it's not enough to just say "Well, this gives us a shuttle velocity of about 0.9999c, so the trip will take just over 2 years." there is also time dilation in the reference frame of the ship, does it not factor in?

 

[Dale]

The 2 ly frame is NOT the best solution for describing the problem.

I agree, any inertial frame is just as good. But by your reasoning it is the frame where the coordinate interval is closest to the proper interval and it is the frame where the geodesic you were discussing is parallel to the time axis. The Earth frame certainly has no such features, so by your above comments I am surprised that you don't consider it the best solution.

 

[ghwellsjr]

What Max is essentially asserting, I think, is that we cannot just pick EITHER time dilation OR length contraction to describe the journey at relativistic speeds to a far away star, we have to use both. Otherwise there is not point in describing a time dilation AND length contraction. If we say that the velocity is such that the length of the distance between the two bodies in the reference frame of the ship is 2ly, it's not enough to just say "Well, this gives us a shuttle velocity of about 0.9999c, so the trip will take just over 2 years." there is also time dilation in the reference frame of the ship, does it not factor in?

Time dilation and length contraction are reciprocal. The traveler cannot measure or tell if he is experiencing them, he can measure and observe them in the Earth and the star and the space between them. Observers on Earth and the star cannot tell if they are experiencing time dilation and length contraction, they can measure and observe them in the traveler.

Max's problem appears to be that he thinks these measurements are frame dependent, in spite of so many people telling him otherwise, and he probably won't accept it from me either.

 

[Max™]

Note: I am not asserting that time dilation and length contraction both have to be considered, if anything I am asserting that they shouldn't be provided as primary explanations of something easily and correctly represented by hyperbolic trigonometry.

If I were to assert that time dilation and length contraction should be used, though, I would note that any situation where you do has to consider the relativity of simultaneity as well, since dilation/contraction alone are not enough.

Max, proper times and proper distances are not effected by anyone's choice of reference frame or even if SR is used to analyze the problem. Co-ordinate times and co-ordinated distances are what are effected by the choice of reference frame. It doesn't matter how you analyze the problem, assuming, of course, that you do it in any correct way, the traveler will be able to survive the 2 year trip, his precision clock will advance only 2 years and any correct means that he uses to calculate his distance will determine that he has traveled just over 2 light-years. Again, it doesn't matter which frame of reference you choose to analyze the scenario in. You could even do it with LET believing in an absolute ether rest frame and get the same answers. Do you doubt this?

No, I don't doubt that any method he uses to calculate his distance, if it is based on rulers, or time of flight for a photon, or judging the distance between endpoints based on their appearance, all will give a measurement of 2 light years.

This is because you're assuming the presence of an object sitting between Earth and Betelgeuse which has a length altered by length contraction.


It isn't really wrong, but it's just a description of a kinematic effect, not a dynamic alteration. More accurately you would measure the distance between Event A: the ship passes the Earth, and Event B: the ship passes Betelgeuse.

Alternatively, you would measure the duration/distance between Event A': Earth passes the ship, and Event B': Betelgeuse passes the ship, and apply the appropriate Lorentz transformations to get your result.


Here, found a better wording explaining why we're not getting each other:

According to the “geometric” approach, special
relativity primarily describes the geometry of spacetime. It does not depend on anything
funny happening to the vectors, clocks, rulers, odometers or other objects that inhabit
spacetime. Objects are neither Lorentz contracted nor time dilated; they are completely
unaffected by boosts. The components obtained by projecting a vector onto this-or-that
reference frame are affected, but that is a property of the projective geometry of the situation,
not a property of the vector itself.

http://www.av8n.com/physics/spacetime-trig.pdf


As I mentioned before, the way I learned relativity was the pure geometrical approach Wheeler had been pushing for years, so it is odd to see people arguing for the less useful description/explanation of events.


Oh, btw, I don't know how I missed this earlier, but no, a 2 ly path is NOT a timelike geodesic for this set of events, it is not remotely close, the geodesic between two points is the unaccelerated worldline which maximizes proper time, not one which pushes it close to zero.

 

[ghwellsjr]

Note: I am not asserting that time dilation and length contraction both have to be considered, if anything I am asserting that they shouldn't be provided as primary explanations of something easily and correctly represented by hyperbolic trigonometry.

Are you asserting that Einstein's explanation is not easy or not correct?

If I were to assert that time dilation and length contraction should be used, though, I would note that any situation where you do has to consider the relativity of simultaneity as well, since dilation/contraction alone are not enough.

Granted, but why after so many posts on your part, you decide now to bring this up for the first time?

Max, proper times and proper distances are not effected by anyone's choice of reference frame or even if SR is used to analyze the problem. Co-ordinate times and co-ordinated distances are what are effected by the choice of reference frame. It doesn't matter how you analyze the problem, assuming, of course, that you do it in any correct way, the traveler will be able to survive the 2 year trip, his precision clock will advance only 2 years and any correct means that he uses to calculate his distance will determine that he has traveled just under 2 light-years (or whatever your specific scenario is). Again, it doesn't matter which frame of reference you choose to analyze the scenario in. You could even do it with LET believing in an absolute ether rest frame and get the same answers. Do you doubt this?

No, I don't doubt that any method he uses to calculate his distance, if it is based on rulers, or time of flight for a photon, or judging the distance between endpoints based on their appearance, all will give a measurement of 2 light years.

But is there some other method that would give a different measurement?

This is because you're assuming the presence of an object sitting between Earth and Betelgeuse which has a length altered by length contraction.


It isn't really wrong, but it's just a description of a kinematic effect, not a dynamic alteration. More accurately you would measure the distance between Event A: the ship passes the Earth, and Event B: the ship passes Betelgeuse.

Alternatively, you would measure the duration/distance between Event A': Earth passes the ship, and Event B': Betelgeuse passes the ship, and apply the appropriate Lorentz transformations to get your result.

Well, as I said, you could even use LET and believe in an absolute ether rest frame (defined by you as one in which both Earth and Betelgeuse are at rest). Way to go! I didn't think anyone would go for that but here we have a true believer.

But even your preferred absolute ether rest frame still affirms that the the traveler is experiencing time dilation and length contraction, correct?

And it also affirms that the traveler will measure that Earth and Betelgeuse are the ones experiencing time dilation and length contraction and not himself, correct?

Here, found a better wording explaining why we're not getting each other:


http://www.av8n.com/physics/spacetime-trig.pdf


As I mentioned before, the way I learned relativity was the pure geometrical approach Wheeler had been pushing for years, so it is odd to see people arguing for the less useful description/explanation of events.

Is the pure geometric approach of Wheeler the same as the hyperbolic trigonometric explanation mentioned at the beginning of your post?

Does this approach (or any other you want to consider) explain what each observer measures and observes and does it give any different answers than Einstein's method?

Oh, btw, I don't know how I missed this earlier, but no, a 2 ly path is NOT a timelike geodesic for this set of events, it is not remotely close, the geodesic between two points is the unaccelerated worldline which maximizes proper time, not one which pushes it close to zero.

Was this meant for someone else?

 

[Max™]

Are you asserting that Einstein's explanation is not easy or not correct?

Uh, no, he is the one that said it's just a kinematic effect (i.e. description of motion without delving into the dynamics of the causes), last time I checked.

It definitely isn't as easy as the hyperbolic rotations are. If you understand circular trig (not the hardest subject), you can combine what you know of Euclidean rotations to pick up the geometrical explanation of SR easily.

Granted, but why after so many posts on your part, you decide now to bring this up for the first time?

Well, my main thrust is that presenting length contraction as a cause is incorrect, and that presenting lorentz contraction/dilations as explanations in general is a less elegant/useful formulation.

Pretty sure I questioned why anyone would choose that over a description of hyperbolic angle rotations in the first couple of pages, I do admit that I took for granted that everyone knows the importance of the relativity of simultaneity, but I did bring it up with the description of the scenario indirectly because of how it forces one to specify events.

But is there some other method that would give a different measurement?

Well, yes, if by different measurement you mean "not confusing", then if he treats his measurement of the distance as being passively transformed by the Lorentz effects within his chosen coordinate system, he will be working with the same vector as any other passively transformed set of coordinates would.

If he actively transforms the measurements then you're mapping everything back into your original coordinates in a different way, lending the impression that this is some physical alteration of what you're measuring, which is extremely misleading.

Well, as I said, you could even use LET and believe in an absolute ether rest frame (defined by you as one in which both Earth and Betelgeuse are at rest). Way to go! I didn't think anyone would go for that but here we have a true believer.

What? This has nothing to do with the quote you responded to, I said nothing at all about an absolute ether frame.

I said the mistake in applying a Lorentz contraction to the distance between Earth and Betelgeuse is that you're treating that distance as though it is a physical object, which will give a different result than you would get measuring the distance between two events which are co-local with Earth and Betelgeuse respectively.

But even your preferred absolute ether rest frame still affirms that the the traveler is experiencing time dilation and length contraction, correct?

And it also affirms that the traveler will measure that Earth and Betelgeuse are the ones experiencing time dilation and length contraction and not himself, correct?

Uh... I said nothing at all about a preferred absolute ether rest frame, seriously, where are you getting this?

Nothing in my post is in any way related to such a concept, at all, how did you misread it so completely?


I said using length contraction on the distance as though it is an object is incorrect, that you should apply lorentz transformations to the distance you measure between two events if you're going to use that approach.

Is the pure geometric approach of Wheeler the same as the hyperbolic trigonometric explanation mentioned at the beginning of your post?

Does this approach (or any other you want to consider) explain what each observer measures and observes and does it give any different answers than Einstein's method?

Well, it allows you to construct the Lorentz transformations from the hyperbolic trig functions:

⎡cosh, sinh⎤
⎣sinh, cosh⎦

Which then shows how different observers in different frames, whether they've been boosted or not, would decompose the components of any particular vector into spatial component and temporal component, which then explains the measurements you would make if you were using the contraction/dilation explanation.

Was this meant for someone else?

Yeah, the portion about the geodesic was a response to a comment about "the frame where it's 2 ly is just as good as any other to use as a geodesic", but it doesn't maximize the proper time of the path, so it by default can't be the timelike geodesic for a worldline between those two events.

 

[IsometricPion]

A test mass undergoing (non-gravitational) acceleration does not follow a geodesic. So, technically a test mass that takes a path starting at rest on Earth and ending at rest on Betelgeuse is not following a geodesic (i.e., there must be points on its path where (non-gravitational) acceleration occurs). The coordinate independent (space-time rather than just space based) length associated with a path is the proper time that elapses over the path (note that a path need not be a geodesic to have a well-defined length/proper time elapsed). In the case of the rocketship, it is clear that the elapsed proper time between the start and end of its journey is slightly more than two years (assuming the acceleration/decleration is large enough). It is certainly possible to specify coordinate systems in which the coordinate time and (spatial) distances the path covers are much larger than this, but all coordinate systems agree on the proper time elapsed on the path. All of this carries over in the SR approximation since different inertial frames are simply different coordinate systems covering minkowski space. In particular, the proper time elapsed over the path is still coordinate indepenent and therefore frame invarient. There are frames in which the time elapsed and (spatial) distance traveled are longer, but all agree on the proper time elapsed on the path. One's choice of inertial frame (or non-inertial for that matter) in SR does not matter any more than the choice of coordinates matters in any other exercise in geometry (it is only the coordinate invariant properties that matter).

P.S. Sorry for the "wall-of-text".

 

[ghwellsjr]

Well, yes, if by different measurement you mean "not confusing", then if he treats his measurement of the distance as being passively transformed by the Lorentz effects within his chosen coordinate system, he will be working with the same vector as any other passively transformed set of coordinates would.

If he actively transforms the measurements then you're mapping everything back into your original coordinates in a different way, lending the impression that this is some physical alteration of what you're measuring, which is extremely misleading.

You were the one who put the word "if" on your statement with regard to an observer making a measurement. I wouldn't put any if's on it. You want to tie measurements in with coordinate systems and transforms. I keep saying that measurements are made without regard to any coordinate system or any transformation (or any theory, for that matter).

What? This has nothing to do with the quote you responded to, I said nothing at all about an absolute ether frame.

I said the mistake in applying a Lorentz contraction to the distance between Earth and Betelgeuse is that you're treating that distance as though it is a physical object, which will give a different result than you would get measuring the distance between two events which are co-local with Earth and Betelgeuse respectively.

Uh... I said nothing at all about a preferred absolute ether rest frame, seriously, where are you getting this?

Nothing in my post is in any way related to such a concept, at all, how did you misread it so completely?

I said using length contraction on the distance as though it is an object is incorrect, that you should apply lorentz transformations to the distance you measure between two events if you're going to use that approach.

You prefer the Earth-Betelgeuse rest frame over the traveler's rest frame (after he gets inertial). You even admitted as much here:

There are two events at the start of the thought experiment, we'll just assume the ship was flying past the Earth to confirm that it is not simply a confusion on my part due to an admitted preference for background star reference frames, and to take acceleration out of the equation entirely.

You think it is important or necessary or better or more convenient or less misleading or more elegant or more useful or less confusing for the traveler to establish his rest frames before and after acceleration so that he can transform his measurements done in his traveling rest frame into his pre-acceleration rest frame. And that is the same as believing in an absolute ether rest frame. Now if you would do this in a reciprocal manner and always explain how the Earth should transform all of its measurements into the traveler's rest frame, then I would take back my characterization of your position.

 

[Dale]

we cannot just pick EITHER time dilation OR length contraction to describe the journey at relativistic speeds to a far away star, we have to use both.

Well, in the Earth frame the Earth-Betleguese distance is not length contracted, so that cannot be an explanation in that frame. In the ship frame the clock is not time dilated, so that cannot be an explanation in that frame. In all other frames there will be both length contraction and time dilation.

 

[Dale]

Note: I am not asserting that time dilation and length contraction both have to be considered, if anything I am asserting that they shouldn't be provided as primary explanations of something easily and correctly represented by hyperbolic trigonometry.

This makes sense to me, I do prefer nice consistent geometric explanations to "piecemeal" explanations.

This is because you're assuming the presence of an object sitting between Earth and Betelgeuse which has a length altered by length contraction.

I don't believe that anyone has assumed that. Length contraction is a property of the Lorentz transforms and does not require the presence of an object. Although without an object it should probably be called "distance contraction".

More accurately you would measure the distance between Event A: the ship passes the Earth, and Event B: the ship passes Betelgeuse...

Oh, btw, I don't know how I missed this earlier, but no, a 2 ly path is NOT a timelike geodesic for this set of events, it is not remotely close, the geodesic between two points is the unaccelerated worldline which maximizes proper time, not one which pushes it close to zero.

Yes, it is. Go ahead and work it out. Assuming that the ship is inertial between events A and B then that path is a geodesic by definition and all other paths from A to B will have less proper time.

 

[Dale]

Well, my main thrust is that presenting length contraction as a cause is incorrect, and that presenting lorentz contraction/dilations as explanations in general is a less elegant/useful formulation.

I agree completely. The Minkowski geometric explanation is much more elegant and simple.

I said the mistake in applying a Lorentz contraction to the distance between Earth and Betelgeuse is that you're treating that distance as though it is a physical object, which will give a different result than you would get measuring the distance between two events which are co-local with Earth and Betelgeuse respectively.

I don't think anyone is treating it as a physical object. But the distance between Earth and Betelgeuse is not the same as the distance between two events, one on the Earth worldline and one on the Betelgeuse worldline. Let's discuss this geometrically.

If we have two parallel timelike worldlines then the distance between those worldlines is defined as the spacetime interval between any two simultaneous events on those worldlines. Because this definition relies on simultaneity it is frame variant, and as you hyperbolic-rotate those worldlines into other frames you will get different pairs of events and therefore different distances. The distance is maximized in the frame where the worldlines are also parallel to the time axis. Note that there is no requirement that an object be present, only that there be two parallel worldlines.

Now, events A and B are timelike separated, so they are not simultaneous in any frame, which means that the interval between A and B is not the same thing as the distance between Earth and Betelgeuse in any frame. This should be clear from the geometric fact that distances are spacelike and the interval from A to B is timelike.

Yeah, the portion about the geodesic was a response to a comment about "the frame where it's 2 ly is just as good as any other to use as a geodesic", but it doesn't maximize the proper time of the path, so it by default can't be the timelike geodesic for a worldline between those two events.

It does maximize the proper time of the path between A and B, as long as the ship is inertial between A and B. Can you justify your assertion that it does not?

 

[Max™]

You were the one who put the word "if" on your statement with regard to an observer making a measurement. I wouldn't put any if's on it. You want to tie measurements in with coordinate systems and transforms. I keep saying that measurements are made without regard to any coordinate system or any transformation (or any theory, for that matter).

Wait how are you making a measurement without a coordinate system, and why would you ignore the transforms involved due to your choice of frame?

You prefer the Earth-Betelgeuse rest frame over the traveler's rest frame (after he gets inertial). You even admitted as much here:

No, I stated that it (over)simplifies the case if you pretend there is no acceleration involved, meaning you can declare the ship to be in inertial motion or Earth-Betelgeuse are in motion, either choice is fine, though an incomplete description of events.

You think it is important or necessary or better or more convenient or less misleading or more elegant or more useful or less confusing for the traveler to establish his rest frames before and after acceleration so that he can transform his measurements done in his traveling rest frame into his pre-acceleration rest frame. And that is the same as believing in an absolute ether rest frame. Now if you would do this in a reciprocal manner and always explain how the Earth should transform all of its measurements into the traveler's rest frame, then I would take back my characterization of your position.

No, I think it is important to consider the worldlines involved (thanks for reminding me Dale!), I was just using a strange way of saying it due to the odd mental contortions required to put myself into the contracting/dilating mode of thought.

Simply insisting on a reciprocal transformation back and forth says nothing interesting about the path itself, it just says you can apply a formula.

Similarly, conflating the measurement of the path from one frame with the worldline itself just confuses things.

It does maximize the proper time of the path between A and B, as long as the ship is inertial between A and B. Can you justify your assertion that it does not?

Doh, I see what I did, I forgot we were ignoring gravity.

The geodesic assuming purely inertial movement is different from the one where you consider the tradeoff between avoiding time dilation, and increasing your motion through time due to being in freefall.


Yes, if you assume gravity can be ignored it is a geodesic, otherwise there is a path with longer time given the proper acceleration profile.

I don't think anyone is treating it as a physical object. But the distance between Earth and Betelgeuse is not the same as the distance between two events, one on the Earth worldline and one on the Betelgeuse worldline. Let's discuss this geometrically.

This is something I've had in the back of my head, but kept forgetting to state deliberately, thank you for reminding me of that.


Yes, it could be described more cleanly as choosing a timelike slice that connects two events along the worldlines, which can be pictured easily by overlaying an x and t spacetime diagram so Earth (or betelgeuse, or the ship) starts at one origin, and the other worldline(s) starts out some distance across the x axis. Set it into motion and plot paths between the worldlines. The ship can then be treated as a point with no spatial motion in one coordinate system, where the Earth and then betelgeuse are on very steep diagonals which cross the ships worldline with two years between them. Switching to the other graph, the ship is then a line which crosses into the x-axis of the graph intersecting the Earth path at one t coordinate, then continues diagonally across it until it intersects the betelgeuse path at another t coordinate.


Both descriptions are correct in their frame, and both descriptions can be converted into the other frame readily enough without making any mention of rulers contracting or clocks slowing.

It is then easy enough to see that there is no path which connects the earth/betelgeuse worldlines that is shorter than 640 light years/zero duration (positive spacelike interval). Additionally there is no path that can be followed which is shorter than 640 light years/640 years (null interval).


You can determine that the angle which the earth/betelgeuse worldlines are inclined by in the ships coordinate system means they have significant rapidity, and then you can take the proper time between them in the ship frame, and with that you can explain the apparent 2 light year separation between them. If you say that the 2 light year separation is the proper length between them, that would imply you ignored your use of a lorentz contraction when working out the distance between them and behaved as though it was actually a 2 light year long measuring rod that flew past you.


I'm saying you could(should) be using a transformation into your coordinate frame to produce a coordinate length. Claiming that your 2 year proper time between them is due to the coordinate length would be silly, but if you were treating it as a proper length I can see why you might make that statement. Similarly you can passively apply a transformation from the Earth or betelgeuse frames to determine what an observer along that worldline will measure.

Doing this makes it easy to see that the worldline itself does not depend on any observer, while also explaining why different observers will construct their measurement of the path vector along the worldline with a particular choice of spacelike and timelike components in their coordinate system.

 

[Dale]

It is then easy enough to see that there is no path which connects the earth/betelgeuse worldlines that is shorter than 640 light years/zero duration (positive spacelike interval).

I think you meant longer. There is no spacelike geodesic which is longer than 640 light years.

You can determine that the angle which the earth/betelgeuse worldlines are inclined by in the ships coordinate system means they have significant rapidity, and then you can take the proper time between them in the ship frame, and with that you can explain the apparent 2 light year separation between them. If you say that the 2 light year separation is the proper length between them, that would imply you ignored your use of a lorentz contraction when working out the distance between them and behaved as though it was actually a 2 light year long measuring rod that flew past you.

I am not sure what you are talking about here. The proper time is the frame-invariant timelike interval from A to B and doesn't have much to do with the frame-variant 2 light year spatial distance between Earth and Betelgeuse in the ship frame.

 

[Max™]

I think you meant longer. There is no spacelike geodesic which is longer than 640 light years.

Whoops, yeah, I was thinking about the ∆τ² with no timelike portion being -(1/c²)*∆s², and thus less than 0, so I wanted to say -640 light years, got to love sign change mistakes, huh?

I am not sure what you are talking about here. The proper time is the frame-invariant timelike interval from A to B and doesn't have much to do with the frame-variant 2 light year spatial distance between Earth and Betelgeuse in the ship frame.

Exactly!

This is what the thread was started about, I had a guy tell me that the ship frame 2 light year coordinate distance was the reason the proper time is about 2 years.

 

[erics]

Correct me if I am wrong, but based on the Twin Paradox where one twin stays put on Earth (assume Earth is an object at rest here)... and the other one travels .866 of C (Lorentz Factor of 2) of for many light years and then makes a U-turn and comes back to Earth...

If the twin on the ship never made a uturn to turn around, and the twin on the ship attempted to calculate the age of his twin who is resting on Earth (NOT even counting for how long it takes the information to reach the ship from earth... he would conclude that the twin on Earth is actually the one experiencing Time Dilation...

he would determine that events happening on Earth are happening slower than in his frame of reference... eventhough this is the opposite of what the twin on Earth views of the twin on the ship...

If the transmission between signals sent from the Earth to the ship did reverberate back and forth at a speed faster than life, there would be timetravel between the two twins...telepathic communication going on... literally

What each could calculate they are experiencing simultaneously, though not through light... is a different bending of universal clocks and a combination of space and time segments scattered each his own... with only one thing in common... the observed speed of light

even the time calculated on Earth by the traveler on the ship would not appear to be smooth throughout the journey to and from earth... and the time differential between the twins accumulates to be half that of the other twin in both frames of reference only when the exercise is commenced.

is this right?

what actually happens is that there is no absolute way to determine what time on the Earth clock was it when the ship was experiencing time "t" along the journey... because of the relativity of simultaneity not even this can be observed in absolute terms

what can be only observed is each separate reference frame. For the twin on earth, the ship would be half the normal length and the twin would be calculated to be undergoing a smooth time dilation.

For the twin on the ship, the Earth would be going through dilation for the first leg of the trip and the distance from Earth would be chopped in half whenever the ship's velocity is at that .866 of c.
Then at the point of the U-turn... a massive amount of Earth time would appear to have been skipped over completely in the calculations of the traveling twin on the ship, so much so that the time dilations happening to Earth that's observed by the traveler on the ship on the return leg of his trip, would not be able to make up for that the skip in Earth time at the ship's U-turn de-acceleration and acceleration was so large that in the traveler's reference frame, the Earth STILL aged older than the age of the ship...which turns out to equate the total for the other twin's calculation of time in the end from his separate reference frame.

The appearance of time dilation and length contraction for the twin on the ship would be observed on Earth discounting for the lag of images it takes light to emit... on both legs of the journey... it's just during the de-acceleration and turn-around, the twin on the ship would jump a time stage and see the Earth's age shoot up a few months/years almost instantly

this in total brings both twins to the same age disparity in completely different ways in completely different reference frames. Did I get this right? Correct what I got wrong.



ON ANOTHER NOTE:
I think if a person were to travel at near light speed, but orbiting Earth in a circle, in this situtation there would be complete asymmetry. The traveling twin would in this case see the Earth speed up in aging at a constant rate due to contant inertia as velocities are linerar-based and not circular-based (and this case the acceleration and force would be applied persistently, unlike when traveling away from earth)... and the length contactions would somewhat spin around only to match the direction parallel to the ship at each segment of the orbit.
In circling the globe at near light speed, there would be a case where simultaneity is still in tact, so both the traveler and the person at rest could observe each other's reference frames in one piece.


The one piece that I am unable to get and probably never will is how I cannot intuitively apply length contraction to observe the relationship of speeds that U+V = 1 + (UtimesV) when length contraction is not proporational to the measurements of this equation.

For example if one is moving at .8606 the speed of light, where Gamma = 2...then you would not think intuitively that light would be passing with respect to Earth's velocity in the frame of reference of the traveler, only 1-.8606 the speed of light in the opposite direction to total with the .8606 to yield 1C.

 

[Passionflower]

This is what the thread was started about, I had a guy tell me that the ship frame 2 light year coordinate distance was the reason the proper time is about 2 years.

Well it is! It takes the ship a little over 2 years to get to the other side because it travels near light speed and the distance for the ship is 2 ly.

After 75 posting we have apparently not made any progress.

 

[erics]

Well it is! It takes the ship a little over 2 years to get to the other side because it travels near light speed and the distance for the ship is 2 ly.

After 75 posting we have apparently not made any progress.

Or you mean the ship perceives the length of everything in front and in back of it to be cut in half with respect to objects moving with respect to Earth.

Then since there is time dilation on top of this, then the perception of speed traveled by the guy in the ship is identical to the perception of speed calculated by the guy on Earth looking at the ship's speed from his frame of reference...because time dilation and length contraction have a reciprocal effect on speed calculation

if you fix the lengths and the times proportionally, then they have to agree on the speeds.

But then there's this other factor to the puzzle... time dilation of the people and happenings on Earth perceived by the traveler in the ship while moving linearly away from Earth. Again, correct me if I'm wrong on this part. THIS peculiarity is related to the relativity of simultaneity where over large distances, the order of events can be reversed in one's reference frame versus that of another observe... yet these separate observations cannot be communicated timely enough to disturb the space-time continuum because of the fact that the information of these observations between two potential observers that qualify for this, would transfer between the two observers at no faster than the speed of light in every possible instance of this...which makes the two observers' reference frames disconnected. So no reversal of time sequence can ever actually be observed or witnessed in any single reference frame until both events have passed in both frames of reference.

 

[Max™]

Well it is! It takes the ship a little over 2 years to get to the other side because it travels near light speed and the distance for the ship is 2 ly.

After 75 posting we have apparently not made any progress.

/sigh

No, the coordinate length measured from the ship is NOT the reason it takes 2 years of proper time... it is apparently the reason, yes, but that is a description of appearances, not what is actually happening.

If the distance for the ship was 2 light years and it was traveling near light speed, it would measure a different time. The distance the ship crosses is different from the measurement the ship makes of the distance it crosses.

 

[erics]

I substantially agree with you, Max, unless you make the case that the ship is resting relative to the objects you are using to define what "rest" means" and what velocity means...then you can argue the ship's readings of length are the proper lengths and the cosmos are all moving backwards, but this is silly

 

[Max™]

In the ship frame on the left where it is at rest, and the earth/betelgeuse are moving past it, the proper time is 2 years, the coordinate length is 2 light years.

Considering the worldlines of the earth, betelgeuse, and the ship on the right, ignoring the various motions of the earth/betelgeuse for this scenario (technically the Earth line would be a spiral around the sun, and deflected towards vega, while betelgeuse has a motion that I am not certain of, but which is definitely nonzero around the milky way.

Setting those aside, considering both of them to be in an inertial frame for simplicity, the proper length of the purely spacelike interval between them is 640 light years, and 640+ years after the ship crosses the Earth worldline, it's worldline crosses the betelgeuse worldline.

The timelike interval along the ships worldline is about 2 years, and thus the proper time is 2 years. The spacelike component is 640 light years, which would be measured in the coordinates of the ship frame as 2 light years~.


It is true that an observer on Earth or Betelgeuse would experience a proper time of 640+ years between the ship crossing Earth/crossing Betelgeuse, and it is true that the ship observer would experience a proper time of 2 years.

There is no conflict there, these frames have been rotated compared to each other, and they would transport a clock differently through time. This is a real dynamic effect, the duration between (S, E') and (S, B') really is 2 years, and there is nothing wrong with saying the distance between them is 640 light years, but it is measured as 2 light years in the ship coordinates.


There is something wrong with saying that your measurement of 2 light years in the ship coordinates (as Dale so nicely put it, a frame variant quantity) is why it took 2 years (a frame invariant quantity) to cross that distance.

 

[Passionflower]

The distance the ship crosses is different from the measurement the ship makes of the distance it crosses.

OK, this is completely fruitless, you are not here to learn but to argue that you are right.

There is something wrong with saying that your measurement of 2 light years in the ship coordinates (as Dale so nicely put it, a frame variant quantity) is why it took 2 years (a frame invariant quantity) to cross that distance.

Time = distance/velocity even in relativity!

 

[Max™]

OK, this is completely fruitless, you are not here to learn but to argue that you are right.

Well, no, I'm here trying to figure out why anyone would make that claim as I phrased it originally. I am not here specifically to learn, because that would imply that I don't spend my time learning elsewhere, it is what I do.

If you're assuming I'm just confused about some relativistic "paradox" and seeking clarification, you didn't read what I said, and are responding to something else.


What I'm talking about is not about me being right, it's about an odd claim I saw someone make which I can't resolve with anything I've learned about relativity over the last couple of decades. So far the biggest mistakes I've made were due to odd choices of language, or a simple sign change error... nothing of the scale which would lead me to so completely misinterpret the difference between an effect due to coordinate choices and an effect due to the geometry of spacetime itself.

Apparently Dale was able to understand me once I cleaned up my phrasing:

I am not sure what you are talking about here. The proper time is the frame-invariant timelike interval from A to B and doesn't have much to do with the frame-variant 2 light year spatial distance between Earth and Betelgeuse in the ship frame.

This is what I started out thinking, which led me to wonder why someone would claim the proper time is due to a coordinate distance measurement.

Time = distance/velocity even in relativity!

>.>

Uh, no, though it may technically be crudely true in a sense, time is not just bluntly defined as d/v, distance is measured in meters, which are defined as the distance light covers in a set amount of time.

If you're measuring anything but the path a beam of light covered, then you will need to consider the lorentz transformations.


In this case, if you're disagreeing with Dale and I, you're saying that since you measure the distance in your set of coordinates as 2 light years, that causes the proper time that elapses for you to be 2 years.


If you were saying "you measure the distance in your set of coordinates to be 2 light years BECAUSE your proper time is 2 years", that is different, but you don't seem to be saying that... and the distinction of cause and effect is rather important here... there is no way for your choice of a coordinate system to cause an effect which is due to your state of motion.

 

[harrylin]

[[..]
There is something wrong with saying that your measurement of 2 light years in the ship coordinates (as Dale so nicely put it, a frame variant quantity) is why it took 2 years (a frame invariant quantity) to cross that distance.

Indeed, and probably everyone agrees with that - your measurement cannot cause a duration, although it can predict it.
Therefore you also received mostly agreement about your main issue that Length Contraction does not physically cause Time Dilation - although most attention was on sayings of your own. As that and most other issues have been answered, I'm out of this thread.

Harald

 

[Dale]

If the distance for the ship was 2 light years and it was traveling near light speed, it would measure a different time. The distance the ship crosses is different from the measurement the ship makes of the distance it crosses.

Max, I want to be supportive here because I agree with your basic point that length contraction does not cause time dilation, but you keep making wrong comments like this in your arguments. In the ship frame the distance is 2 light years and Betelegeuse is traveling near light speed. The ship is at rest so it is not time dilated so it does not measure "a different time" as you say.

 

[Dale]

there is nothing wrong with saying the distance between them is 640 light years, but it is measured as 2 light years in the ship coordinates.

There is something wrong with saying that. You are stating a frame variant quantity (the distance between Earth and Betelgeuse) without specifying the frame. The correct way to say this is "there is nothing wrong with saying the distance between them is 640 light years in the Earth frame, but it is 2 light years in the ship coordinates" or even "there is nothing wrong with saying the proper distance between them is 640 light years, but it is 2 light years in the ship coordinates".

 

[Dale]

Indeed, and probably everyone agrees with that - your measurement cannot cause a duration, although it can predict it.
Therefore you also received mostly agreement about your main issue that Length Contraction does not physically cause Time Dilation - although most attention was on sayings of your own.

Exactly. I agree with Max's conclusion, but his arguments are wrong and/or sloppy.

 

[erics]

Yeah I'm not sure length contraction causes time dilation, but they are 100% correlated between the 2 reference frames. If the traveler parallel to the path of a resting observer experiences length contraction, and given the resting observer knows the speed of the traveler and can recalculate images discounting back from when the images are received on earth...

then the resting observer will determine that the traveler is experiencing time dilation and it would be proportional to the traveler's degree of length contraction experienced... EVERY TIME! You might as well put the word cause there. Makes no difference.

 

[Dale]

Yeah I don't think length contraction causes time dilation, but they are correlated.

Yes, and as my statistics professor continuously emphasized, correlation is not causation! They are correlated because they are both caused by the same thing, the Minkowski metric (or equivalently the Lorentz transform).

 

[Passionflower]

..you're saying that since you measure the distance in your set of coordinates as 2 light years, that causes the proper time that elapses for you to be 2 years.

No I am not saying that, I am saying:

The proper time it takes to go to the destination is slightly over 2 years because time = distance/velocity. In this case we have:

Distance is 2 and velocity being slightly under 1.

Thus time is slightly over 2.

And it is clear to me you do not understand SR, here are some quotes from you:

someone stated that "if I were to travel to Betelgeuse at a sufficient velocity I would reduce the distance between myself and Betelgeuse until it is say, 2 light years, which means I would only experience 2 years or so during my journey"

He is correct.

you can't get to Betelgeuse in 2 actual years, so you can't claim that the 2 years you observed was a proper time, or that the 2 light year distance was a proper distance.

Yes you can.

Yeah, I don't have a problem with it being 2 light years in that reference system, and yes I do tend to take for granted that such frames are not as... interesting as ones where the "background stars" are at rest.

Both frames are on an equal footing.

THAT is what sent me off in a tizzy, the way he was implying that moving really fast makes distances shorter, and crossing those shorter distances takes less time

He implied well.

No, the distance being contracted doesn't mean you only have to cross 2 light years which would take just over 2 years at your velocity, either that statement is not true, or I am quite mistaken about special relativity.

Well, then you must be mistaken.

I could go on with more quotes but I stop here.

 

[Max™]

As has been said, my choice of language is odd, but my conclusion is sound, I'm cool with that. You can stop with the "clearly you must not understand SR if you don't use the description which I personally prefer as opposed to another completely equivalent one you find more aesthetically satisfying", ok? I understand SR fine, thanks. I understand GR as well.

Yes, and as my statistics professor continuously emphasized, correlation is not causation! They are correlated because they are both caused by the same thing, the Minkowski metric (or equivalently the Lorentz transform).

Yeah, the correlation=causation argument is extremely irritating.



I didn't mean it experiences time dilation in it's frame, I meant the path the clock was transported along has a certain timelike length and a certain spacelike length.

I don't like the dilation/contraction descriptions at all, they are strange. When you consider the different geometries involved it is clean and pretty, when you force it into a "clocks and rulers changing" explanation it is ugly and distorted.

"there is nothing wrong with saying the proper distance between them is 640 light years, but it is 2 light years in the ship coordinates"

This is indeed more clearly phrased than I've been stating it, sorry, I have a terrible habit of assuming everyone uses the same interpretation of a word which I intended, generally when I refer to distance without tying it to a particular frame, I'm talking about a proper distance. When I talk about time without tying it to a particular frame, I'm talking about a proper time.

It's just a habit due to treating the lorentz transforms as a frame result for non-accelerated observers (and in general not dealing with purely inertial worldlines rather than freely falling ones) in different relative states of motion.

 

[Passionflower]

generally when I refer to distance without tying it to a particular frame, I'm talking about a proper distance.

He is another example where it is clear to me you do not understand it. In special relativity there is no point in talking about distance without mentioning the frame of reference.

 

[Dale]

I meant the path the clock was transported along has a certain timelike length and a certain spacelike length.

The path a clock is transported along has only one invariant length and it is always timelike, never spacelike. A clock's path may cross a certain spatial distance in some frame, but it never becomes spacelike.

 

[Max™]

Uh, there is a point in talking about a proper distance/proper time, they relate to the components of different types of intervals.


While you might not see a reason to talk about frame independent quantities, that isn't the same thing as there being no reason to do so.

The path a clock is transported along has only one invariant length and it is always timelike, never spacelike. A clock's path may cross a certain spatial distance in some frame, but it never becomes spacelike.

Again, whoops, I just meant "spatial component", I appreciate the tips for cleaning up my language though.

 

[Passionflower]

While you might not see a reason to talk about frame independent quantities, that isn't the same thing as there being no reason to do so.

Distance is not frame invariant in relativity. Proper time however is.

 

[Max™]

Distance is not frame invariant in relativity. Proper time however is.

There is actually a way to treat relativity in which proper lengths and proper times are invariant, and the geometry of the paths they follow causes the different measurements due to coordinate projections from one frame or another. This is kinda my point. You don't have to exclusively use the "rulers/clocks vary" instead of the "hyperbolic geometry of paths vary" description.

 

[IsometricPion]

There is actually a way to treat relativity in which proper lengths and proper times are invariant

Assuming a -+++ sign convention, a path is said to have a proper length if the interval along it has a positive value. Similarly, a path has a proper time if its invariant length is negative. So, if a path has a proper length it cannot also have a proper time and vice versa. I guess what you mean is that if you compare a spacelike path between Earth and Betelgeuse with a timelike path, the proper length of the (by "the" I mean the one you seem to have chosen, which connects the Earth with Betelgeuse and where simultaneity is defined according to the Earth being at rest) spacelike path is 640 light-years and the proper time of the timelike path (that we have been discussing) is ~2 years. However, an observer taking a timelike path (as all observers with mass must) does not measure the distance according to the spacelike path (if tachyons existed, I suppose I could say "and vice versa along the spacelike path"). The timelike (non-stationary with respect to the endpoints of the spacelike path) observer will agree as to the proper length of the spacelike path and the fact that his measurements indicate a distance that is shorter than said length. This does not allow the observer to make any conclusions about the "actual distance" because the proper length is only purely a (spatial) distance in one frame if the observer chooses to use a different frame the distance will be different than the proper length (and why shouldn't it be, they are not equivalent concepts, one is invariant the other is not and the frame variant version happens to correspond with what is defined as distance).

 

[Max™]

Alternatively, you can take the part where you say "frame invariant quantity, and frame variant quantity" and replace them with "path independent quantity and path dependent measurement of a coordinate quantity", last time I checked.

 

[IsometricPion]

True, I could have. Is there a significant difference between what I wrote and that with which you wanted to replace it?

Regardless, proper lengths are not the same as distances since the two transform differently. By definition, distance is a frame variant (or path dependent measure of a coordinate) quantity, so there can be no actual distance.