Cycles of Time: An Extraordinary New View of the Universe

by Roger Penrose

This illustrates the key fact about the metric of space-time, namely that it is really something that has much more directly to do with the measurement of time rather than distance. Instead of providing a length measurement for curves, it directly provides us with a time measurement. Moreover, it is not all curves that are assigned a time measure: it is for the curves referred to as causal, that could be the world-lines of particles, these curves being everywhere either timelike (with tangent vectors within the null cones, achieved by massive particles) or null (with tangent vectors along the null cones, achieved by massless particles). What the space-time metric g does is to assign a time measure to any finite segment of a causal curve (the contribution to the time measure being zero for any portion of the curve which is null). In this sense, the ‘geometry’ that the metric of space-time possesses should really be called ‘chronometry’, as the distinguished Irish relativity theorist John L. Synge has suggested.