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By George D. Birkhoff, Ralph Beatley

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S;;; -r+. 1. is regular, then -r is regular; If tW, . ; If-r is regular and tW, . 1. l. ) = . , . 1.. 1. 1. 1. , and (tW. 1. 1. 1.. l w. Generally, a two-dimensional quadratic space is hyperbolic precisely when it is regular and isotropic. Every regular isotropic CAUSAL MODELS AND SPACE-TIME GEOMETRIES 31 metric space "Y is split by a hyperbolic plane: "Y = (~vEe ~w) Til'"' for some iI'"' ~ "Y. This theorem will be useful for clarifying the nature of subspaces of the Minkowski space. Now let us turn briefly to the external properties of.

The set of all isometries on "Y, o ("Y) = {cp: "Y -+ => qJL::: il'"'L, I "Y cp is an isometry}, is called the orthogonal group of isometries of "Y. It is indeed a group with respect to functional composition and, in fact, a subgroup of the group of all invertible linear transforms on "Y. It is trivial to check that v 1. W iff cp(v) 1. cp(w) for any cp E O("Y). The group O("Y) is decomposable into two disjoint sets: O("Y) = 0+ ("Y) u 0- ("Y), where 0+ ("Y) is called the group of rotations (it is indeed a group and, in fact, a normal subgroup of O("Y», and O-("Y) is called the set of reflections (it is not a subgroup of o ("Y».

1. 1.. l w. Generally, a two-dimensional quadratic space is hyperbolic precisely when it is regular and isotropic. Every regular isotropic CAUSAL MODELS AND SPACE-TIME GEOMETRIES 31 metric space "Y is split by a hyperbolic plane: "Y = (~vEe ~w) Til'"' for some iI'"' ~ "Y. This theorem will be useful for clarifying the nature of subspaces of the Minkowski space. Now let us turn briefly to the external properties of. quadratic spaces. ') be two quadratic spaces. Then by an isometry from "Y onto "Y' we mean a linear one-to-one and onto mapping cp: "Y -+ "Y' such that vow = cp(v)-' cp(w) for all v, WE "Y, and we write "Y :::= "Y'.

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