Einstein's Concept of Individual time

[vla]

How is Einstein's concept of Individual time different from a second dimension of time?

I am thinking of the example of the twins; one twin goes into space at a velocity approaching the speed of light and the other remains on Earth. When the traveler returns he is actually younger than his twin depending on his speed and distance.

If I were to graph the timeline from point A (departure) to point B (return) for both twins, the traveller's timeline would be shorter. If I were graphing these as two separate line segments it suggests that there would be points on one line that would not exist on the traveller's line. Why doesn't this suggest a plane, or two dimensional view of time?

Please forgive me if this seems too elementary a question, but I am not a scientist or even a science student - I am just a curious layperson.

 

[Nugatory]

If I were to graph the timeline from point A (departure) to point B (return) for both twins, the traveller's timeline would be shorter. If I were graphing these as two separate line segments it suggests that there would be points on one line that would not exist on the traveller's line. Why doesn't this suggest a plane, or two dimensional view of time?

You can graph them as two separate line segments. Imagine that you have a graph with the horizontal axis being distance and the vertical axis being time. The departure point is (0,0), so both timelines start there. The point of return is (0,T) where T is the time that the at-home twin waited for the traveller's return. Now you can draw the at-home twin's timeline as a vertical line from (0,0) to (0,T), and the travelers timeline sloping out to (D,T/2) where D is the distance travelled, and then another line sloping from there back to (0,T).

(If you already know this, I apologize for restating the elementary; you've used just enough non-standard terminology that I'm not sure if you're already familiar with these diagrams).

This is all very well, except that now you'll be looking at this graph and thinking that the traveller's timeline is longer, not shorter - it's two sides of a triangle. The trick is that in space-time the distance between two points is calculated as the square root of t^2-x^2, not the square root of t^2+x^2 as you expect from the ordinary Pythagorean theorem.

(Disclaimer: I've played fast and loose with several conventions here, bashed over some subtleties, and been completely sloppy about who is measuring D and T. You'll also have noticed that you can easily find the square root of a negative number popping up in some distance calculations... Well, whaddaya expect for just two paragraphs of answer?)

 

[A.T.]

How is Einstein's concept of Individual time different from a second dimension of time?

The usual geometrical interpretation (Minkowski's) is that coordinate time is a dimension. But you can draw diagrams where the Individual time of each object (it's proper time) is a dimension. Here is a comparison of the two approaches for the twin case that you mentioned:
http://www.adamtoons.de/physics/twins.swf