Chapter 22 Applications to Physics

22.1 Linear Isometries on Lorentz Vector Spaces

Let (V, g, D) be an oriented Lorentz vector space, and let ε = (e 1, …, e m ) be an orthonormal basis for V that is positively oriented with respect to D. We denote by L m the set of linear isometries on V:

equation

where we recall that Lin(V, V) is the vector space of linear maps from V to V.

Let A be a map in Lin(V, V). For brevity, we denote images by , and likewise omit &ip.eop; from the notation for other matrices. Corresponding to (4.4.1) and (4.4.2), we have

equation

and

equation

respectively, where

and

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