4 - The Geometry of Flat Spacetime

Summary

In this chapter we shall consider the intrinsic geometric structure of flat spacetime M. In particular, we shall use the results of the previous chapter to show that M has a natural physically defined affine structure (i.e., the notion of a displacement vector makes sense) and, most importantly, that the space of displacement vectors possesses a natural, physically defined metric.

Spacetime Vectors

A spacetime displacement is simply an ordered pair of spacetime points. We write OP = - PO and OP + PQ = OQ. If P lies inside N(O), the null cone of O, we say that OP is timelike, in which case O and P lie on the world line of an observer or a massive particle. There are two types of timelike displacements: if P lies in the future of O (according to an observer whose world line passes through O and P), we say that OP is future-pointing otherwise, of course, we say it is past-pointing. If P lies on N(O), we say that OP is null, in which case O and P lie on a null ray or the world line of a massless particle. Again, there are two types of null vectors, future-pointing and past-pointing, depending on whether they lie on N⁺(O) or N⁻(O). If OP is neither timelike nor null, that is, if P lies outside N(O), we say that OP is spacelike (Fig. 4.1).

For a timelike displacement OP, the points O and P lie on a world line l with proper time t. If Q also lies on l and t (Q) – t (O) = a[t (P) – t (O)], we write OQ = α OP.