# 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.

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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