Misc,
From the Lorentz Group to the Celestial Sphere
(Nov 26, 2016)
Abstract
In these lecture notes we review the isomorphism between the (connected)
Lorentz group and the set of conformal transformations of the sphere. More
precisely, after establishing the main properties of the Lorentz group, we show
that it is isomorphic to the group SL(2,C) of complex 2 by 2 matrices with unit
determinant. We then classify conformal transformations of the sphere, define
the notion of null infinity in Minkowski space-time, and show that the action
of Lorentz transformations on the celestial spheres at null infinity is
precisely that of conformal transformations. In particular, we discuss the
optical phenomena observed by the pilots of the "Millenium Falcon" during the
jump to lightspeed.
