Models of Space - Time

2. Abstract space-time. We need first an axiomatic foundation strong enough to support both our mathematical considerations and their applications to physics. DEFINITION. An n+1 dimensional space-time (n*tl) consists of (A) An n+1 dimensional vector space V over the real numbers plus a symmetric bilinear real form A »B (inner product) such that: (1) There exists a vector A with A -A <0 . (2) Any 2-dimensional sub space of V contains a vector A with A-A>0. (B) A set x of objects p,q, • • • (points or "events") plus a mapping {pi <Ù~*P~Q. °f x X x into V such that: (1) (p-a) + <St-r)=p-r. (2) p—q = 0 implies p = q. (3) Given any point q and any vector A there exists a point p with p-q = A. Any V satisfying (A) yields a model of space-time (vector spacetime) on setting x = V. The Minkowski model V—X^^M consists of all w + 1-tuples of real numbers x = (xiy • • • , xn, xn+i) with x-y =#iyi+ • • • +%nyn—xn+iyn+i. (When n = 3, X4 = ct, where t is time and c is the velocity of light.) Every n+l dimensional vector spacetime is isomorphic to R?M, but this result is physically misleading. Eventually we set n = 3, x = the physical space-time continuum, and F=(§4, the spin model of (vector) space-time we shall construct.