Cycles of Time: An Extraordinary New View of the Universe

by Roger Penrose

the signature of the metric. In algebraic terms, this simply refers to a few + signs being changed to – signs, and it basically tells us how many of a set of n mutually orthogonal directions, for an n-dimensional space, are to be considered as ‘timelike’ (within the null cone) and how many ‘spacelike’ (outside the null cone). In Euclidean geometry, and its curved version known as Riemannian geometry, we think of all directions as being spacelike. The usual idea of ‘space-time’ involves only 1 of the directions being timelike, in such an orthogonal set, the rest being spacelike. We call it Minkowskian if it is flat and Lorentzian if it is curved. In the ordinary type of (Lorentzian) space-time that we are considering here, n=4, and the signature is ‘1+3’ separating our 4 mutually orthogonal directions into 1 timelike direction and 3 spacelike ones. ‘Orthogonality’ between spacelike directions (and between timelike ones, had we had more than 1 of them) means simply ‘at right angles’, whereas between a spacelike and a timelike direction it looks geometrically more like the situation depicted in Fig. 2.18, the orthogonal directions being symmetrically related to the null direction between them. Physically, an observer whose world-line is in the timelike direction regards events in an orthogonal spacelike direction to be simultaneous.