Here is some motivation for <a href=PS.html>this work</a>.
This is intended to be a modest generalization of <a href=../../Isometry>these maps</a> where we consider how a quadratic form with signature (1, 2) elucidates a 2D space with uniform negative curvature.
I want to see what happens when we choose a quadratic form with signature (2, 1).
The ‘indefinite sphere’ S in that 3 space is a hyperboloid of one sheet:
x<sup>2</sup> + y<sup>2</sup> − z<sup>2</sup> = 1.
That form provides our metric for both the 3 space and S.
I consider the metric at point (1, 0, 0) which is in S.
In hopeful mimicking of the classic pattern of hyperbolic models we project from point (−1, 0, 0) thru a point in S onto the plane x=1.
That plane is tangent to S at (1, 0, 0) for what its worth.
Note too that the line (1, t, t) is both the plane and in S.
Ditto the line (1, t, −t).
These two lines are null geodesics in S.
<p>We carry a hidden weapon: the indefinite orthogonal transformations on the 3 space, based on the quadratic form, which isometrically map S onto S.
This gives us a cheap argument of uniformly of the curvature tensor in S under transformations including the “<a href=http://en.wikipedia.org/wiki/Lorentz_transformation>Lorentz boost</a>”.
<p>Note first that the signature of the metric on S is (1, 1).