Time Dilation and The Twin Paradox?

JDoolin

Ah, I see. Someone has since added captions to the image since I last saw it on Wikipedia. I did not use that diagram, myself, because the fourth image shows non-orthogonal space and time axes, which struck me as deeply misleading. Now someone (Loedel?) has put a caption under it, which clears up at least what was intended, if not what was actually conveyed. The time axis should, of course, be vertical in every picture.

Thanks, bobc for the note on my user-page.

bobc2

Yes, I touched up the graphics from the Wiki sketches (boxes and captions). I thought the Loedel diagram for the twin paradox was interesting. But you are right, they should have put in the rest frame time axis for the Loedel diagram to make it a little more clear. Here are more graphics to help those who may not be familiar with Loedel diagrams.

Again, the Loedel diagram is constructed by finding a rest system (our black perpendicular coordinates below) in which two observers (blue and red twins) are moving in opposite directions at the same speed. The thing that makes it nice is that line distances on the screen have the same actual distance (or time) scaling for the two Lorentz boosted frames (the blue and red frames for the twins). Thus, in the rest frame corresponding to the black perpendicular reference coordinates (again, the vertical axis should have been shown), both of the twins travel the same distance (time) through the 4-D universe for the outgoing trip of the "traveling twin" (of course both twins are traveling with respect to the black perpendicular coordinates). To compare the distances between the twins for the return trip (the blue twin is doing the turn-around and return), you need to use hyberbolic calibration curves. We are using the black rest frame for making distance comparisons.

I have added the sketches below to help visualize the space-time scaling that results in the "traveling twin" taking a shorter path on the return trip. The hyperbolic curves needed to see the scaling of space-time distances are (hopefully) developed. The sketch on the left is a Loedel diagram. Two observers (the twins) are moving in opposite directions with the same speed with respect to rest frame represented by the perpendicular coordinates. Thus, the actual time and distance scaling on the diagram are the same for each twin during the out-going trip of the "traveling twin."

Again, for the return trip we must use hyperbolic calibration curves for distance (time) comparisons. But, the Loedel diagram allows us to use the Pythagorean theorem directly as shown, from which the metric and Lorentz transformations result. This result also gives us the hyperbolic curves that must be used for calibrating distances (times) in the black rest frame (perpendicular coordinate system). The upper right sketch below illustrates how a hyperbolic curve is used to calibrate points in the black frame located ten years from the origin. The locus of points, all at ten years distance, form a hyperbolic curve in accordance with the derived metric equation. The lower right sketch shows a collection of calibration curves for locating space-time distances (times).

Note that a photon worldline always bisects the time and space coordinate axes for all Lorentz coordinate systems (green lines rotated at a 45 degree angle in the rest frame of the black perpendicular coordinates).

bobc2

Here is an example of using the hyperbolic calibration curves. We have perpendicular black coordinates for the rest frame. The stay-at-home twin's worldline is along the black vertical time axis--the path is 13 years long (measured by stay-at-home's clock). The traveling twin's path through space-time is shown with the blue lines--the path is 10 years long (measured by traveling twin's clock).

JDoolin

The term "hyperbolic calibration curves" is apt. I kind of think of them being concentric. Just like concentric circles could be made as

[tex]x^2+y^2=r^2[/tex] as r={0,1,2,3,4,...} etc, all representing circles; the locus of positions equidistant from the origin.

you can have "concentric" hyperbolic curves, going

[tex]x^2-c^2 t^2=s^2[/tex] as s={0,1,2,3,4...}, all representing "concentric" space-like hyperbolas, the locus of events which, in some reference frame, are simultaneous with the origin event, all the same distance to the left or right of the origin in those respective frames,

and
[tex]c^2 t^2-x^2=\tau^2[/tex] as τ={0,1,2,3,4...} for the "concentric" time-like hyperbolas, the locus of events, which, in some reference frame, are all in the same position as the origin, all the same time since or before the origin in those respective frames.

In any case, whether you call it a "hyperbolic calibration curve" or a "concentric hyperbola" it's a helpful mental construct.

I especially like that you re-centered your hyperbolic calibration curve at the turn-around event.

The analogy with circles works so long as you re-center the circle with each measurement. You can't measure the path distance just by looking at the distance to the center to the end-point. You have to move the center of the circle every time the path turns. I'm probably just explaining something really obvious in a complicated way, but I remember myself thinking about it for a few days before it occurred to me, how to perfect the analogy; measuring distances with concentric circles vs measuring space-time intervals with "hyperbolic calibration curves."

bobc2

Excellent points, JDoolin. I'll adapt your "concentric hyperbola" teminology from now on.