Do we also travel at a fixed velocity?

[A.T.]

I'm talking about a situation involving multiple worldlines that don't depart from a common point (in Minkowski spacetime) at all, like objects that have been drifting towards each other from infinity until they finally meet, or multiple worldlines that never cross at all, or three worldlines that each cross the others but all three never cross at a single point.

Ok, but where is the problem in a space-propertime-diagram with this scenarios? You have their spatial position, and proper-time interval since the begin of the observation. You just plot these two in a diagram, for each object.

 

[robphy]

2) T1 after O, B sends signal to A, which A mirrors, so it returns to B T2 after O

That one.
So, T1 and T2 mark off on B's worldline the sending and reception events of the radar experiment performed by B.

You can count off the age difference in a Minkowski Diagram:
physics.syr.edu/courses/modules/LIGHTCONE/LightClock/#twins[/url]
and you can relate it to the geometry of Minkowski spacetime.

I like this too. It's a bit more complicated tough, and involves an understanding of pseudo Euclidean geometry.

Actually it is derived using the standard physics-textbook thought-experiments...

The result I've shown and emphasized is a picture of the ticks of each clock [rather than equations of a pseudo-Euclidean geometry] to directly address how the clocks tick off time.

From this point on, one could address the pseudo-Euclidean aspects, if desired. In other words, after the physics is demonstrated, the underlying mathematics can be discussed... rather than the other way around.

 

[JesseM]

Ok, but where is the problem in a space-propertime-diagram with this scenarios? You have their spatial position, and proper-time interval since the begin of the observation. You just plot these two in a diagram, for each object.

OK, I guess since you're using a particular inertial frame's definition of spatial position for the horizontal axis, you can also use that frame's definition of simultaneity to define the moment that is the "beginning of observation"--I had been thinking about the relativity of simultaneity without realizing that the Epstein diagram already required you to specify which frame you're using.

 

[A.T.]

Sorry for my late reply. About Epstein(space-propertime) diagrams:

So, T1 and T2 mark off on B's worldline the sending and reception events of the radar experiment performed by B.

Since objects at c don't experience proper time, their world lines in an Epstein diagram are horizontal. While you could use these diagrams to display a radar experiment, it wouldn't provide much insight, unless you animate them.