--- "The Hitchhiker's Guide to the Galaxy"
I could be bounded in a nutshell, and count myself king of infinite space, were it not that I have bad dreams.
--- Hamlet
This applet demonstrates the Penrose map which conformally maps spacetime (which is of course infinite) to a pre-compact set, which is commonly known as the "Einstein diamond".
The left-hand grid represents (one quadrant of) 1+1 Minkowski spacetime, with the horizontal axis denoting space x and the vertical axis denoting time t. Each grid spacing is 1/5 of a unit. The Penrose map
X = arctan(x+t) + arctan(x-t)
T = arctan(x+t) - arctan(x-t)
maps this quadrant conformally to the triangle (which is one quarter of the Einstein diamond) displayed on the right.
The same map works in higher dimensions, except that the position x is replaced by the radial variable r. (The angular variables are not affected by the Penrose compactification).
To use the applet, move the mouse around on the left-hand grid; a corresponding pointer will appear on the Einstein diamond. Curves can be drawn by dragging the mouse around. Conversely, if one moves or drags the mouse on the right-hand grid, then a pointer in Minkowski space will appear as given by the inverse Penrose map. Press any key to clear the screen.
 | Null rays (i.e. lines with slope +-1) get mapped to null rays.
| The entire line at infinity gets mapped to the diagonal border
of the Einstein diamond. Spacelike points at infinity get mapped
to the right-hand corner, timelike points to the upper corner, and
the null point gets mapped to the entire open line segment in between.
| Horizontal and vertical lines on the diamond get
inverse-mapped to hyperbolae on Minkowski space.
| |
