Length Contraction causes Time Dilation?

Max™

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

Sorry Max I lost you completely here.

DaleSpam

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?

DaleSpam

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.

DaleSpam

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 refering the ship observations back to the Earth frame rather than some other perfectly valid frame?

Max™

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.

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.
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?

DaleSpam

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

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

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?

DaleSpam

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

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.

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:
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

Are you asserting that Einstein's explanation is not easy or not correct?
Granted, but why after so many posts on your part, you decide now to bring this up for the first time?
But is there some other method that would give a different measurement?
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?
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?
Was this meant for someone else?

Max™

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.

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.

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.

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.
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.
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.