Length Contraction causes Time Dilation?

[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.

 

[Max™]

Nope, that sums it up rather well, the 2 ly distance is a frame variant quantity, the 640 ly length component of the vector is an invariant quantity which can be measured as a 2 ly distance from the inertial frame of the ship.

Accordingly, claiming the 2 year long timelike component of the vector is due to the choice of coordinate system which gives a 2 ly distance isn't just silly, it's wrong.

 

[Rap]

Sorry, I have been away a few days, I am not sure where this thread has gone. I am responding to Max's response to my last post of a few days ago.

A lightlike path connecting Earth and Betelgeuse is 640 light years long and takes 640 years to travel.

You forgot to mention that distance is with respect to the Earth. It is not a frame invariant statement.

Is he doomed to watch the squished up universe hurtle past him, Unable to consider that perhaps he was in a boosted frame, and that just maybe his measurements were distorted by it?.

Yes, he is doomed, because he cannot, by any physical means, decide whether he is in a boosted frame and the "universe" (Earth and Betelguese) is not, or whether the squished up universe is in a boosted frame and he is not.

 

[Max™]

Thanks to Dale pointing out the source of confusion due to my odd wording, we've worked it out pretty well.


Incidentally, you wouldn't observe the universe being squashed, it would appear rotated, Penrose-Terrell Rotation, but yeah, if I was talking about distance it wouldn't be frame invariant. I was talking about the spatial length of particular component of the vector between those two events, which is frame invariant.


In the way I've learned SR (indeed, the way it has generally been "correctly" described since Minkowski formulated it as a hyperbolic geometry over a century ago), it isn't time dilation/length contraction, those are just a result of applying YOUR particular set of coordinates onto a measured quantity, due to the way varying paths through spacetime involve different rotations.

 

[Calrid]

It seems to me a lot of these misunderstanding wouldn't occur if people just paid attention by which I mean the two guys having a misunderstanding with the OP.

It isn't called space-time because its fun to use the - symbol, it's because you can't effect one movement wise without effecting the other relatively speaking, even if you are supposedly stationary or in a rest frame. There's no cause length to time, there's just the maths and space-time, c is the speed limit of the universe the maths hence comes from the relativistic transformation in 4 dimensions: get it? Makes you want to throw a rubber at their heads. Pay attention Smith, see me later Brown!

Looking at a graph of just two dimensions +1 this becomes obvious, it makes you wonder where they learned this, because they must of been off sick the day they taught the basics at least.

 

[ghwellsjr]

Nope, that sums it up rather well, the 2 ly distance is a frame variant quantity, the 640 ly length component of the vector is an invariant quantity which can be measured as a 2 ly distance from the inertial frame of the ship.

Accordingly, claiming the 2 year long timelike component of the vector is due to the choice of coordinate system which gives a 2 ly distance isn't just silly, it's wrong.

Max, consider a second traveler leaving a planet orbiting Betelgeuse who follows exactly the same acceleration profile as the one leaving Earth that has been considered in this thread. They leave at the same time according to a rest frame common to Earth and Betelgeuse. After achieving their final speed, they will be at rest in a different frame of reference.

What do you call the distance between them? Proper or coordinate? Variant or invariant? Do you use the same terminology that you use for the distance between Earth and Betelgeuse or something different?

 

[Max™]

I call the distance a coordinate measurement, frame variant.

I call the space component of the vector between them a proper length, frame invariant.

The time component of that separation between them at the end of their journeys is 0 by choice, so there is a spacelike interval between them, so they wouldn't even disagree about the coordinate length in this particular reciprocal crossing scenario, though each would claim the other had gone a different distance, of course.


A vector doesn't care what frame you're putting it in, the length is the same, only the direction it seems to point varies with different frames, because of the hyperbolic rotations which different frames undergo. The observed effect of that apparent change in the vector is that it would appear to be lorentz contracted by an amount based on the motion of the observer's frame relative to the vector.

 

[Dale]

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.

No, "frame invariant" is not the same as "path independent". The change in potential energy in a conservative field is path independent but frame variant, and the proper time is frame invariant but path dependent. They are most definitely not synonyms.

 

[Dale]

the 640 ly length component of the vector is an invariant quantity

No, components of a vector transform as components, not as a scalar. The proper distance between Earth and Betelgeuse is a scalar whose value is 640 ly. It is not a component of a vector. The coordinate distance between Earth and Betelgeuse is a component of a vector (the 4-vector displacement) and it is frame variant as all components of vectors must be.

 

[ghwellsjr]

I call the distance a coordinate measurement, frame variant.

I call the space component of the vector between them a proper length, frame invariant.

The time component of that separation between them at the end of their journeys is 0 by choice, so there is a spacelike interval between them, so they wouldn't even disagree about the coordinate length in this particular reciprocal crossing scenario, though each would claim the other had gone a different distance, of course.


A vector doesn't care what frame you're putting it in, the length is the same, only the direction it seems to point varies with different frames, because of the hyperbolic rotations which different frames undergo. The observed effect of that apparent change in the vector is that it would appear to be lorentz contracted by an amount based on the motion of the observer's frame relative to the vector.

I'm confused by your last statement. I have only described two frames: the initial one in which everyone and everything is at rest and a second one in which the two travelers finally find themselves at rest. Instead of speaking in generalities, where exactly is the length contraction that you spoke of?

But my main question is: do you consider the second frame that I defined just as valid as the first one? If the distance between the two travelers has changed in some way, is it just as valid, significant, preferred, analyzed, etc, etc, etc, as you afforded the distance between the two planets and therefore, the distance between the two travelers before they started out?

 

[IsometricPion]

No, "frame invariant" is not the same as "path independent".

I stand (or, rather, sit) corrected. Come to think of it, a simple calculation demonstrates that this cannot be the case (though I guess it should be obvious since the length can be extremized). The proper time of the rocketship between Earth and Betelgeuse is about 2 years, compare this with a path that stays at Earth for 2 years and then takes the spacelike path between the then simultaneous (according to Earth) points of the Earth's and Betelgeuse's worldlines. The latter path has a (proper) length of 640-2=638 light-years versus the -2 light-years for that of the rocketship. Even though these two paths connect the same two events, they do not have the same invariant length. (should have calculated before I wrote in my previous post)

 

[Dale]

No worries. That confusion is pretty common as is the confusion between invariant and conserved quantities.

 

[Max™]

No, "frame invariant" is not the same as "path independent". The change in potential energy in a conservative field is path independent but frame variant, and the proper time is frame invariant but path dependent. They are most definitely not synonyms.

Sorry, I meant the length of the vector doesn't depend on measurements made from a frame following another path through spacetime, my mistake.