Note 9. Orthnormal hyperbolic coordinates

Express all coordinates in the reference frame of . Suppose that has coordinates in this frame with > 1. Then since belongs to the hyperboloid , we have , and so . Now define the "speed" of (as measured by ) to be .

Denote this number , and note that . I have assumed for this experiment that . We cannot establish these coordinates in the (trivial) case where is 0, so I handle that case separately. Then , and therefore

.

 

With these choices, let

(N9.1)

.

 

 Now select vectors to be

(N9.2)

 

 and

(N9.3)

.

 

It is easy to see that, with our definition of the inner product given in (6.1) of Hyperbolic Geometry of 2+1 Spacetime,

.

 

Therefore is an orthonormal system of vectors for observer with respect to the hyperbolic metric. The unique linear transformation carrying unprimed basis vectors to primed basis vectors is orthonormal and is sometimes called a "hyperbolic rotation" or a "boost transformation". It is also called a Lorentz Transformation. We saw this system of vectors in a Euclidean context in Planes Intersecting Cones.