# Length Contraction causes Time Dilation?

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.

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.

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 learnt this, because they must of been off sick the day they taught the basics at least.

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ghwellsjr
Gold Member
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?

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.

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Dale
Mentor
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
Mentor
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
Gold Member
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?

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

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.