Null rotations

[Greg Egan]

Among Lorentz transformations, ordinary rotations and boosts are fairly
easy to understand, but "null rotations", the transformations that
preserve null vectors, are a little obscure.

I've written a small web page that explains some properties of null
rotations, specifically in the context of 2+1 dimensions:

http://gregegan.customer.netspace.net.au/GR2plus1/NullRotations.htm

This gives four ways to build a null rotation that preserves a given null
vector, and a movie of the action. The orbits are parabolas, except on
the plane that contains the preserved null ray, where they are straight
lines.

[Hans Aberg]

In article <20060702013519.5596D7B07C@mail.netspace.net.au>, Greg Egan
<gregegan@netspace.net.au> wrote:

> Among Lorentz transformations, ordinary rotations and boosts are fairly
> easy to understand, but "null rotations", the transformations that
> preserve null vectors, are a little obscure.

In the ordinary four-dimensional Lorentz space, this leads to a spin
structure via the Clifford algebra: In the complexified Lorentz space, one
chooses a direct sum of two maximal isotropic vectorspaces, i.e., each of
complex dimension two, consisting only of null vectors, and with
intersection 0 (and such a choice is equivalent to the choice of a spin
structure). The Clifford algebra can then be made acting on each of
these*maximal isotropic vectorspaces, and it is possible to extract the
chiral representation used in one form of the Dirac equation from this by
the choice of a suitable basis; the Feynman slash is in fact the Clifford
algebra action on such a spin representation.

--
Hans Aberg