Time's arrow

Not only vicious causal loops can create a problem for time travel. Suppose that we have a situation in which a later event A causes an earlier event B, whereas B causes A. This does not seem to lead to any logical contradiction, yet it gives rise to a serious conceptual problem. Suppose that in the year 2011 you are visited by yourself from the year 2020, and your older self hands you in a complete blueprint for a time machine. You then build it, and in 2020 you enter your machine to visit yourself in 2011 and deliver the plans. The question is: where did the blueprint come from, and who invented the time machine? Yet another paradox goes as follows. The traveller from the future hands you in an empty notebook and instructs you to enter the time machine, go to the future and write an entry about your journey in the notebook. Then you are supposed to give the notebook to the traveller in the future, whereupon he enters the time machine, goes back in time and gives you in the past the instruction to write another entry in the notebook while travelling to the future. This creates an immediate problem: how many entries are there in the notebook? It looks like the number of entries should correspond to the number of ‘loops’ made, but actually there is only one loop, not many repeated ones.

Apparently this last example is akin to the grandfather paradox, as the traveller attempts to do something in the past (i.e. convince you to enter the time machine and write something in the notebook) which would cause some future event (the traveller’s entering the time machine with an empty notebook) to disappear. So perhaps the solution would be, again, that something should prevent you from writing an entry in the notebook, and thus the situation in the future will be exactly as it should: the time traveller, despite his attempts, will enter the time machine in the future with an empty notebook. On the other hand, if the notebook you receive in the past already contains an entry, the problem looks more like the blueprint paradox we discussed above. You try to write another entry, but you fail, so you hand in the notebook with the old entry to the future traveller. But the question remains: who wrote the entry?

Even if the notion of time travel can escape some of the most threatening conceptual difficulties, the question remains whether it is nomically possible. We will not attempt to answer this question generally, but only with respect to one physical theory, namely the special theory of relativity. First let us notice that the conceptual framework of STR seems to leave room for a realisation of a journey to the past. Let us consider three events A, B and C such that C is in the absolute past of A, whereas B is space-like separated from both A and C. We know that there is a frame of reference f in which event B looks later than event A, so at least in principle it seems possible to send somebody from A to B along the ordinary direction of time from the past to the future. But there is also a different frame of reference f’ in which C is later than B, hence another trip from B to C seems feasible. Combining these two together, we can see that the journey took us from A to its absolute past. But this scenario is not physically realizable, due to the fact that in order to reach B from A (and C from B) the traveller would have to accelerate beyond the speed of light, and this is prohibited by the laws of relativistic dynamics. Let us note in passing that STR does not exclude the possibility of the existence of superluminal particles (called tachyons) – particles that travel faster than light. But if tachyons exist, they cannot cross the light barrier by slowing down to a subluminal speed.

How can we explain the fact that time has an objective direction, in contrast with space? And what does it mean precisely that time has a direction? The directionality (or asymmetry) of time can be defined as the property of being ordered by the asymmetric relation “being earlier than”. But points on a spatial line can be similarly ordered in one direction. To distinguish between the two cases let us observe that the temporal relation of being earlier than is an intrinsic relation of two moments which does not depend on anything extrinsic from these moments. On the other hand, for a spatial line to be ordered in a similar fashion we have to choose some external “point of reference”. For instance, we can order points from left to right, but this ordering depends on the position of the external observer. Analogously, it is possible to order points on a meridian, but again we have to choose the South Pole and the North Pole to do that.

Is the directionality a primitive and irreducible property of time, or can it be grounded in some more fundamental property? There are three standard ways of grounding time’s arrow: in the psychological arrow, causal arrow and thermodynamic arrow. The psychological arrow is based on the observation that perceptions always precede memories. My perception of a given occurrence is designated as being earlier than my memory of the same occurrence. But clearly this criterion can be applied to a very narrow category of events only: the mental ones. How, then, can we use the criterion of such a limited scope to order all the events? The answer is that in some circumstances it may be sufficient to define a direction for only two points in time, and the rest will be ordered accordingly. Let us suppose that the set of events is equipped with a symmetric structure defined by the three-place relation of being between. This means that we can say that an event x lies between events y and z (in short, B(x, y, z)) without deciding whether y is earlier than z or vice versa. Now the question is: What has to be added to the structure defined by the relation B in order to have a full linear order? It is easy to notice that if we decide for a pair of selected events a and b which of them is earlier and which later, the rest of the events will be ordered thanks to the relation B. For instance, let’s suppose that a is earlier than b (E(a, b)), and consider two events x, y lying between a and b (i.e. such that B(x, a, b) and B(y, a, b)). In this case, E(x, y) iff B(x, a, y) (or, equivalently, B(y, x, b)). This definition can be easily extended for all possible distributions of x and y with respect to a and b.
From this it follows that in principle we need only one instance of a perception and its memory to give time its direction (under the condition that the relation of betweenness is already given). But in practice we obviously need more such instances. The psychological arrow suffers from obvious shortcomings. It is strongly anthropocentric, as it requires the existence of humans (sentient beings) in order for time to have a direction. But surely there would be earlier and later events even if there were no humans. Another difficulty is that this criterion excludes a priori the possibility of clairvoyance (which may be impossible physically, but does not look like a contradictory concept). Knowing the future requires that a ‘memory’ of a given event is earlier than its perception (by ‘perception’ we mean the ordinary way of seeing things, and not the one claimed by the clairvoyants), but this contradicts the criterion.

The causal arrow assumes that if x is a cause of y, x is earlier than y. This reduction of the directionality of time encounters the following two objections. First, it excludes the possibility of backward causation (we considered this possibility when analysing time travel). Second, grounding the direction of time in the causal relation requires that we define causality without referring to temporal precedence. And yet some conceptions of causality, famously including Hume’s analysis, rely on just that. If we insist that causality grounds time’s arrow, we have to find an alternative way of making sure that the causal relation will be asymmetric.

The most commonly accepted of all three arrows is the thermodynamic arrow. It relies on the second law of thermodynamics which states that for isolated systems their entropy (a measure of disorder) never decreases. The second law explains why we observe so many irreversible processes (heat transfer, dissolution, etc.). Thus the thermodynamic criterion states that if x and y are two macrostates of a given isolated system, and the entropy of x is smaller than the entropy of y, then x is earlier than y. But there is a well-known foundational problem associated with this account of time’s arrow. The theory which describes the interactions that underlie all thermodynamic processes is just classical mechanics of many particles, and this theory is known to be time-reversible (if a process is admissible by the laws of classical mechanics, so is its reversed version). The standard attempt to explain the observed thermodynamic asymmetry is due to L. Boltzmann, and it is based on the fact that one macrostate of a system can be realized by a great number of various microstates (defined by the positions and velocities of all individual molecules of the system). Boltzmann proved that for a given macrostate the vast majority of corresponding microstates are such that their dynamic evolution leads to the increase of entropy. Thus the second law can be seen as probabilistic only, but the probability that a system will actually violate it is extremely small.

However, one of the main problems with Boltzmann’s argument is that it is essentially symmetric, i.e. it can be used in support of the claim that the system at a given moment evolved from a state of higher entropy. One way of avoiding this difficulty is to assume that the initial state of the universe had extremely low entropy, and hence the overwhelming tendency of the systems is to increase their entropy. This means that the thermodynamic arrow has to be grounded ultimately in a singular fact about the origin of the universe, together with the probabilistic laws of statistical mechanics. One interesting consequence of this fact is that it is theoretically possible that when the universe reaches the state of maximal entropy, the tendency can be reversed and the majority of systems will follow the entropy-decreasing evolution. It is not clear whether this would mean that the direction of time got reversed.

Reading:

E.J. Lowe, Chapter 18 "Causation and the direction of time", pp. 325-344, A Survey of Metaphysics.