On the Linearity of Form Isometries

Let G and H be symmetric, bilinear forms on vector spaces V and W respectively. A form isometry is a function f from V into W which satisfies $G( {x - y,x - y} ) = H( {f( x ) - f( y ),f( x ) - f( y )} )$ for all x and y in V.Form isometrics are necessarily affine-linear in the following situations:(1) when H is nonsingular and f surjective (see Ratz [6]); (2) when G is nonsingular and f surjective; (3) when G is nonsingular and the dimension of W is finite and less than or equal to the dimension of V; (4) when H is positive definite on W; or (5) when $G = H$, $V = W$, and V has an orthogonal decomposition as the sum of a positive definite subspace and a negative definite subspace, one of which is finite-dimensional.The original motivation for this research was to derive the Lorentz transformations of special relativity theory from the Minkowski metric in $R^4 $ without the presupposition of linearity. Various proposals in the literature which achieve this objective are discussed here.