Faradave » 06 Nov 2019, 21:35 wrote:Jorrie wrote:I fail to spot any spacetime events indicated in either of your last two diagrams. …To make your depictions more understandable, please indicate the two events that you are referring to and in which inertial frames their coordinates are measured.

OK, the two events that I'm interested in is the origin (0,0) and the orange one (d,t') as per black frame, because they would correspond to e.g. the original colocation event and the turnaround event in the 'twins parable' respectively. To make it easier to grasp, let's put in typical "twins-values", e.g. v/c = 0.6 and t=1 yr. Now what is the spacetime interval between these two 'tangible' events?

The green twin is the only one present at both events and her clock reads 1 yr, so for her the spacetime interval is 1 yr (because in her inertial frame she is at dx=0, dt=1). There are no other related events happening in the black twin's inertial frame (after the origin where they have crossed paths). But we know that the spacetime interval in the black frame between the two relevant events must also be 1 yr, by the invariance of the interval between inertial frames.

So I repeat my statement that the black arrow d is simply the coordinate distance between events (0,0) and (d,t') according to the black frame (with the given values, it is 0.6 lyr).

The black arrow d would be the spacetime interval between some (unrelated) red event that happen simultaneously with the orange event as per the black frame, but it would not be simultaneous as per the green frame. I say again, label the horizontal axes "space" and the objection will disappear!

I agree that the your other diagram:

can be called an "interval-time" diagram.

But I fail to see the utility of it in terms of practical situations, where massive objects and photons crisscross spacetime, e.g. what would the world-lines of the twins look like? And of photons send between them?