If we believe in reincarnation, time travel, or a series of Big Bang expansions followed by Big Crunch contractions, we may have to drop the notion that time is linear and use periodic functions. The big danger here is that such discussions can easily be contextually incomplete: we should always ask who is monitoring cyclic time and establishing that it is it indeed cyclic. Our view is that this would require an exophysical observer based on a linear time perceiving an SUO with periodic behaviour.
Signature and multi-dimensional time
The advent of special relativity (SR) led to the idea that time and space were part of spacetime, a four-dimensional continuum with a novel distance structure related to Pythagoras’s theorem in standard plane geometry. Recall that in the geometry of the Euclidean plane, a right-angled triangle with sides of length x, y, and я satisfies Pythagoras’ ‘theorem’, я2 = x2 + y2, if я is the length of the hypotenuse, the longest side. In spacetime, suppose we have two events with relative coordinates x, y, я, and t, as measured by observers in some inertial frame. Then the SR ‘distance’s, corresponding to the hypotenuse in Pythagoras’ theorem, is given by Minkowski’s ‘line-element’ [Minkowski, 1908]
where c is the speed of light. An excellent account of Minkowski’s ideas about this concept is given in [Petkov, 2012]. The importance here to us is the signature, the pattern of pluses and minuses in (7.5). In our convention, we have the pattern (+,-,-,-). We shall discuss the physical implication of having different signatures, such as (+, +, +, +) or (+, +,-,-) in Chapter 12. In the latter case, the spacetime appears to have two time dimensions, something that plays havoc with all the standard notions of ordering and causality that the one-dimensional image of time based on R embodies. Signature is a fundamental concept in the study of time and is discussed further in the Appendix.