Here's lookin at you, Steve and Victor
So I'm trying to make a Minkowski-like space-time diagram with a proper time axis. This obviously presents some conceptual challenges when dealing with multiple frames of reference, and a hard to shake Minkowski diagram intuition.
By a proper time axis, I mean that as a particle passes by a given point in space, you ask the particle what time its watch says it is, and there's your space-time coordinate. Right off the bat it sets any null geodesics to flat, horizontal lines (which is actually kind of a nice consequence. You'd like something called a null vector to be 0 categorically in some sense). It also obviates time-dilational paradoxes like the twin paradox (when they reunite, the traveling twin's proper time will be smaller than the other twin's proper time, end of story).
When thinking about the twin paradox, however, the very strength of this kind of diagram reveals its more unintuitive side, that of simultaneity. Points on this diagram do not represent events in space-time, simply because the time is subjective to the path. I'd really like to make a diagram which is isomorphic to Minkowski's, but I'm not sure just yet if that's possible. I don't care how many other axis need to be shoved in there to keep enough information, but there must be, of course, the one axis which is the proper time axis.
With a Minkowski diagram, the coordinates are space-time from one inertial reference frame, and all other inertial reference frames are embedded within their own skewed axes. In this new kind of diagram, the proper time is a universal axis, so to communicate relative issues like simultaneity there will need to be a way to relate the different paths to one another.
One way to do all this would be to make a 3D Minkowski diagram, space, time, and proper time. If the 2D version of the diagram is truly not isomorphic then there would seem to be little recourse (other than just showing both 2D diagrams next to each other). The 3D approach will clearly never catch on though, so the trick is to either prove it is isomorphic, or to find enough properties where it reveals interesting properties of scenarios to be worth having.
It's flagrant downside is that it simply won't contain more information than Minkowski's, although it will make some properties explicit whereas they may be more hidden within Minkowski's. It's purpose is pedagogical, so it had better be worth the additional effort. It's hard to tell if it would be worth it for an audience just learning the subject. On the one hand it'll help visualize relative proper time, on the other hand it could be confusing for someone struggling to make the traditional diagram intuitive.
By a proper time axis, I mean that as a particle passes by a given point in space, you ask the particle what time its watch says it is, and there's your space-time coordinate. Right off the bat it sets any null geodesics to flat, horizontal lines (which is actually kind of a nice consequence. You'd like something called a null vector to be 0 categorically in some sense). It also obviates time-dilational paradoxes like the twin paradox (when they reunite, the traveling twin's proper time will be smaller than the other twin's proper time, end of story).
When thinking about the twin paradox, however, the very strength of this kind of diagram reveals its more unintuitive side, that of simultaneity. Points on this diagram do not represent events in space-time, simply because the time is subjective to the path. I'd really like to make a diagram which is isomorphic to Minkowski's, but I'm not sure just yet if that's possible. I don't care how many other axis need to be shoved in there to keep enough information, but there must be, of course, the one axis which is the proper time axis.
With a Minkowski diagram, the coordinates are space-time from one inertial reference frame, and all other inertial reference frames are embedded within their own skewed axes. In this new kind of diagram, the proper time is a universal axis, so to communicate relative issues like simultaneity there will need to be a way to relate the different paths to one another.
One way to do all this would be to make a 3D Minkowski diagram, space, time, and proper time. If the 2D version of the diagram is truly not isomorphic then there would seem to be little recourse (other than just showing both 2D diagrams next to each other). The 3D approach will clearly never catch on though, so the trick is to either prove it is isomorphic, or to find enough properties where it reveals interesting properties of scenarios to be worth having.
It's flagrant downside is that it simply won't contain more information than Minkowski's, although it will make some properties explicit whereas they may be more hidden within Minkowski's. It's purpose is pedagogical, so it had better be worth the additional effort. It's hard to tell if it would be worth it for an audience just learning the subject. On the one hand it'll help visualize relative proper time, on the other hand it could be confusing for someone struggling to make the traditional diagram intuitive.