Time in special relativity and time travel

We will now discuss the changes in the notion of time and space brought about by the development of the special theory of relativity. Let us start with a brief characteristic of the classical account of space and time as incorporated in the Galilean-invariant version of Newtonian mechanics (i.e. the version that dispenses with the concept of absolute motion and absolute position). The main assumption is that no inertial frame of reference is privileged, and uniform motion is relative. However, the notion of simultaneity remains absolute, i.e. frame-independent. For each moment of time the set of events occurring at that moment defines a three-dimensional space with the usual Euclidean metric (distance) attached to it. Hence, Galilean space-time foliates naturally into separate spaces defined at different times. However, there is no absolute connection between points in spaces at different times (no absolute co-location). The question of which point of space at t2 is a continuation of a point at t1 does not receive a frame-independent answer. If we think that spatial points (places) are objects which retain their identity over time, then no such objects are present in the Galilean version of classical mechanics.

A step towards special relativity is the realisation that the relation of simultaneity has no obvious empirical content, due to the fact that all signals (including light) travel at finite speeds. What we observe as our ‘present’ is actually already in the past (the farther, the more distant the observed event is). The standard, operationally defined notion of simultaneity is given as follows: two events x and y are simultaneous iff light signals sent from x and y meet exactly half way between x and y. But this definition is obviously not frame-independent. Suppose that the definition of simultaneity is satisfied in a frame f, and let us consider another frame f’ which moves with respect to f in the direction of the event y. The spatiotemporal point where the two beams of light meet will not be located in the middle of the distance between their locations x’ and y’, but rather closer to x', so from the perspective of f’ y happened earlier than x. We have to add that the signal definition of simultaneity presupposes that the speed of light is constant in all frames of reference.

In special relativity neither simultaneity nor co-location are invariant notions (independent of the frame of reference). Thus space-time cannot be absolutely divided into space and time. However, there is a relation between events (spatiotemporal points) which stays the same in all frames of reference. This relation is defined by the so-called spatiotemporal interval: cdt^2 – dx^2 – dy^2 – dz^2., where c – the speed of light, and dt, dx, dy and dz are temporal and spatial intervals between the two events. Two events for which the spatiotemporal interval is positive are called ‘time-like separated’. Such events can be connected by a signal travelling slower than light. If the interval equals zero, the events can be only connected by a beam of light. Events separated by a negative interval are called space-like separated. Such events cannot directly communicate by way of sending signals.

The basic structure of relativistic space-time (so-called Minkowski space-time) can be given with the help of light cones. For a given event x, its forward light cone consists of all events reachable from x by beams of light. Similarly, x’s backward light cone contains all events which can reach x using beams of light. The area within x’s backward light cone is called its absolute past, and within the forward light cone its absolute future. The events outside of both light cones are neither past nor future with respect to x, but they cannot be interpreted as being simultaneous with x. Their temporal relation with x is frame-dependent: for every event y space-like separated from x there is a frame of reference f in which x and y are simultaneous. But if we choose a different event y’ also space-like separated from x, the frame of reference in which y’ is simultaneous with x will be generally different from f.

Now we will discuss some issues related to the asymmetry of time. Let us start with the problem of time travel. Is time travel conceptually possible, or does it involve logical contradiction? First we have to decide what process can be called time travel. For a given object we can say that it travels in time if there is a difference between its own time and the external time of the world. If the interval measured with the object’s time is shorter than the external interval, we can speak about travel into the future. Such travel is not only possible but actually happens, according to the special theory of relativity. Due to the effect known as time dilation, if an object moves, its own time measures shorter intervals than the external time. If a traveller embarks on a journey and then comes back, his clock will show that his journey was shorter than when measured by the external clocks (this is the basis of the so-called twin paradox).

The most radical type of time travel is when the traveller goes into the past, i.e. the duration of his journey measured according to the external time is actually negative. Some philosophers claim that travel into the past involves contradiction, because a time traveller could change the past, and this is impossible. More specifically, the concept of changing the past is applied to states of affairs. In order to change a given past state of affairs – for instance, by scratching an inscription on a rock a thousand years ago – this state of affairs (the rock being unscratched) has to be both real (‘before’ the change) and unreal (‘after’ the change). But of course the time of the occurrence of these two contradictory states of affairs is the same, so the contradiction seems unavoidable. But it can be observed that the same problem applies to the apparently uncontroversial case of a change in the future. To literally change a state of affairs at a future time t requires that this state of affairs exist before but not after my action. But, again, this leads to logical contradiction. This difficulty can be avoided, though, when we apply the notion of change to things, not states of affairs. I certainly changed the past rock: before my intervention it wasn’t scratched, and afterwards it bore an inscription.

The most celebrated grandfather paradox exploits the possibility of vicious causal loops that seems to be opened by admitting time travel. The time traveller goes back to the times of his grandfather’s youth and kills him in the past. If his grandfather dies before he can have any children, the traveller will not be born in the future, and a contradiction ensues: the traveller both exists in the future (because he came from there to kill the grandfather) and does not exist (because his grandfather dies childless). This paradox has the following general form: there are two events A (the beginning of the journey to the past) and B (the killing of the grandfather) such that A is later than B, A causes B and B causes non-A. In order to avoid the problem, serious restrictions have to be imposed on the possible interactions of the time traveller with the past. Speaking loosely, each time the traveller tries to kill his grandfather something must get in the way to prevent him from accomplishing this task. Note that this extends to any action of the traveller (even seemingly innocent, such as leaving footprints on the grass) which might lead to the consequences threatening the entire future travel to the past as it precisely occurred. One general solution may be to assume that causal links directed from the future to the past do not ‘couple’ with the causal links leading in the ordinary temporal direction. But this would effectively imply that the traveller could only observe the past and not interact with it the normal way.

Reading:

B. Garrett, "Time travel", pp. 94-99, What is this thing called metaphysics?