Do we also travel at a fixed velocity?

[Dale]

Which is great because it allows to visualize the age difference geometrically. In a Minkowski diagram you don't see the age difference. In this visualization you see the standard twins paradox in both types of diagrams (space-propertime diagram is called Epstein diagram):
http://www.adamtoons.de/physics/twins.swf

Oh, I don't like that diagram at all! The end of the green line corresponds not to the point where it touches the yellow line but to the end of the yellow line. That is very confusing to me.

 

[robphy]

That's true in the sense that Epstein diagram doesn't contain the space like part of the Minkowski diagram. So you can "only" visualize world lines of massive objects and light.

A.T.,
How would one draw an Epstein diagram for a radar experiment with light rays?

More specifically, consider two inertial observers with different velocities that met at event O. One observer, after a short [proper-]time interval T1, sends a light-signal to the other. (The ratio of the time-intervals from O is the Doppler Factor.) After a [proper-]time interval T2, the first observer receives the radar echo.
How are these features shown on an Epstein diagram?

 

[robphy]

Which is great because it allows to visualize the age difference geometrically. In a Minkowski diagram you don't see the age difference.

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

 

[JesseM]

One remaining problem I have with this is that it doesn't really generalize to arbitrary sets of worldlines that may not have set their proper times to zero at a common origin point.

I see no problem there. They don't have to start at zero proper time, but it is convenient to use the starting proper times as zero. What you want to visualize, are different rates of proper time. An initial proper time offset is not really relevant.

I think you misunderstand--I'm not talking about the fact that two observers in a twin paradox situation might not set their proper times to zero at the moment they depart from one another, since as you say we can always imagine resetting their clocks to zero at that moment for convenience. 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.

 

[A.T.]

A.T.,
How would one draw an Epstein diagram for a radar experiment with light rays?

More specifically, consider two inertial observers with different velocities that met at event O. One observer, after a short [proper-]time interval T1, sends a light-signal to the other. (The ratio of the time-intervals from O is the Doppler Factor.) After a [proper-]time interval T2, the first observer receives the radar echo.

Lets call the guys A and B. Which scenario do you mean:
1) T1 after O, B sends signal to A, which A receives T2 after O
2) T1 after O, B sends signal to A, which A mirrors, so it returns to B T2 after O

You can count off the age difference in a Minkowski Diagram:
http://physics.syr.edu/courses/modules/LIGHTCONE/LightClock/#twins
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.

 

[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.