# 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*:

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
by , and likewise omit &ip.eop; from the notation for other matrices. Corresponding to (4.4.1) and (4.4.2), we have

and

respectively, where

and

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