No book on time would be complete without a discussion of time travel (TT). In this chapter we review selected aspects of that concept in some detail: there are many discussions in the literature on TT, exploring the paradoxes and the constraints. Ours is not a review of those discussions simply because many of them are contextually incomplete. Our interest is in those aspects that may have some empirical validity.
Although TT is a common motif in science fiction, it has been and remains the focus of many serious studies in general relativity (GR). The fact is, TT is mathematically admissible in classical GR, as we shall show, but seems logically inadmissible in any context involving irreversibility, such as quantum mechanical (QM) or thermodynamics. The current failure of the quantum gravity programme to reach any conclusions may be related to this fact.
We do not need to discuss GR and QM to see what problems could arise were TT possible: temporal paradoxes arise in all approaches to the subject. Some reflection on TT soon leads to the conclusion that these paradoxes are all to do with information.
To see this, suppose we wanted to win a lot of money on the National Lottery. Do we actually need to construct a time machine at almost certainly enormous cost, time travel physically into the day after tomorrow, read in that day’s newspapers the numbers that have/will have been selected ‘at random’ in the previous day’s National Lottery draw, travel back to today, and then place an enormous bet on that outcome, before tomorrow’s Lottery Draw? Not really. All we would need to achieve the same result would be to obtain that information from the future. But of course, that is the rub: we can no more get this information by wishful thinking than we can actually travel in time and get it.
The flow of information is therefore the central issue in TT. This can be illustrated by differential equations (DEs), of the sort discussed in Chapter 12.
Images of Time. First Edition. George Jaroszkiewicz.
© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.
Conventionally, these are local in time, meaning that they involve functions and their derivatives, all of which are evaluated at the same instant of the observer’s time. Consider the simple first order DE
The solution is easily found: x(t) = e‘x(0). Suppose now that the equation was modified ‘very slightly’ to
where T is a positive constant representing a delay by a time T in the information about the current position of the system under observation (SUO) reaching the observer/mathematician. Although physically realistic, this equation is surprisingly subtle. To see this, suppose that the solution exists and is an analytic function of time. This means that we may make a Taylor expansion of x about t. Then (20.2) becomes
This is an infinite-order differential equation, a very different proposition to the original first-order differential equation (20.1). Equation (20.2) is an example of a difference-differential equation, higher order versions of which require sophisticated mathematical techniques [Pinney, 1958]. Such equations are usually not studied by physicists, who tend to represent delays (or retardations) via additional dynamical variables such as electromagnetic fields.
Equation (20.2) is physically realistic in that it reflects the fact that, in the real world, many processes are affected by delays. Indeed, before our age of near instant communications, significant delays (retardations) of information were a constant fact of life: letters would arrive days or weeks after posting. Even a Roman emperor had to wait many days before news from the frontier could reach him in Rome: the average courier speed over land at that time was about 50 miles a day [Ramsey, 1925].
Naval architects ran head-on into difference-differential equations when they started to use Newtonian mechanics to study the stability of ships. Consider a traditional masted sailing boat with its mast making an instantaneous angle в (t) with the vertical, as in Figure 20.1. The forces destabilizing the ship depend not only on the instantaneous position of the centre of mass, but also on the amount of water in the bilges swirling around in the lower decks. Forces due to such water would depend on the orientation of the boat a few seconds earlier, due to retardation effects. Much the same effect is experienced when a large ship passes a small boat: the boat starts to feel the passage of the ship after the ship has passed by.
Fig. 20.1 Boat oscillating at sea: g represents gravitational forces, b represents bilge water forces.