Length Contraction causes Time Dilation?

[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.