Conformal Compactifaction

Space is big. Really big.You just won't believe how vastly, hugely, mind-bogglingly big it is; I mean you may think it's a long way down the street to the chemist, but that's just peanuts to space.

--- "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.

Some points of interest:

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.

Conformal compactification is useful in dealing with the global behaviour of non-linear wave equations. A typical application is in

"Global well-posedness for large data of one-dimensional wave maps below the energy norm", with Mark Keel, submitted to IMRN. [dvi*] [Figure 1] [ps].

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