Topology and Constraints

Wheeler and Feynman (1949) were the first to claim that the fact that nature is continuous could be used to argue that causal influences from later events to earlier events, as are made possible by time travel, will not lead to paradox without the need for any constraints. Maudlin (1990) showed how to make their argument precise and more general, and argued that nonetheless it was not completely general. Imagine the following set-up. We start off having a camera with a black and white film ready to take a picture of whatever comes out of the time machine. An object, in fact a developed film, comes out of the time machine. 

We photograph it, and develop the film. The developed film is subsequently put in the time machine, and set to come out of the time machine at the time the picture is taken. This surely will create a paradox: the developed film will have the opposite distribution of black, white, and shades of gray, from the object that comes out of the time machine. For developed black and white films (i.e. negatives) have the opposite shades of gray from the objects they are pictures of. But since the object that comes out of the time machine is the developed film itself it we surely have a paradox. However, it does not take much thought to realize that there is no paradox here. What will happen is that a uniformly gray picture will emerge, which produces a developed film that has exactly the same uniform shade of gray.

 No matter what the sensitivity of the film is, as long as the dependence of the brightness of the developed film depends in a continuous manner on the brightness of the object being photographed, there will be a shade of gray that, when photographed, will produce exactly the same shade of gray on the developed film. 

This is the essence of Wheeler and Feynman's idea. Let us first be a bit more precise and then a bit more general. For simplicity let us suppose that the film is always a uniform shade of gray (i.e. at any time the shade of gray does not vary by location on the film) The possible shades of gray of the film can then be represented by the (real) numbers from 0, representing pure black, to 1, representing pure white. Figure 0 Let us now distinguish various stages in the chronogical order of the life of the film. In stage S1 the film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage S2 of the life of the film, it has been developed and is about to enter the time machine. Stage S3 occurs just after it exits the time machine and just before it is photographed. Stage S4 occurs after it has been photographed and before it starts fading away. Let us assume that the film starts out in stage S1 in some uniform shade of gray, and that the only significant change in the shade of gray of the film occurs between stages S1 and S2. During that period it acquires a shade of gray that depends on the shade of gray of the object that was photographed. I.e. the shade of gray that the film acquires at stage S2 depends on the shade of gray it has at stage S3. 

The influence of the shade of gray of the film at stage S3, on the shade of gray of the film at stage S2, can be represented as a mapping, or function, from the real numbers between 0 and 1 (inclusive), to the real numbers between 0 and 1 (inclusive). Let us suppose that the process of photography is such that if one imagines varying the shade of gray of an object in a smooth, continuous manner then the shade of gray of the developed picture of that object will also vary in a smooth, continuous manner. This implies that the function in question will be a continuous function. Now any continuous function from the real numbers between 0 and 1 (inclusive) to the real numbers between 0 and 1 (inclusive) must map at least one number to itself. One can quickly convince oneself of this by graphing such functions. For one will quickly see that any continuous function f from [0,1] to [0,1] must intersect the line x=y somewhere, and thus there must be at least one point x such that f(x)=x. Such points are called fixed points of the function. Now let us think about what such a fixed point represents. It represents a shade of gray such that, when photographed, it will produce a developed film with exactly that same shade of gray. The existence of such a fixed point implies a solution to the apparent paradox. Let us now be more general and allow color photography. One can represent each possible color of an object (of uniform color) by the proportions of blue, green and red that make up that color. (This is why television screens can produce all possible colors.) Thus one can represent all possible colors of an object by three points on three orthogonal lines x, y and z, that is to say, by a point in a three-dimensional cube. 

This cube is also known as the ‘Cartesian product’ of the three line segments. Now, one can also show that any continuous map from such a cube to itself must have at least one fixed point. So color photography can not be used to create time travel paradoxes either! Even more generally, consider some system P which, as in the above example, has the following life. It starts in some state S1, it interacts with an object that comes out of a time machine (which happens to be its older self), it travels back in time, it interacts with some object (which happens to be its younger self), and finally it grows old and dies. Let us assume that the set of possible states of P can be represented by a Cartesian product of n closed intervals of the reals, i.e. let us assume that the topology of the state-space of P is isomorphic to a finite Cartesian product of closed intervals of the reals. Let us further assume that the development of P in time, and the dependence of that development on the state of objects that it interacts with, is continuous. Then, by a well-known fixed point theorem in topology (see e.g. 

Hocking and Young 1961, p 273), no matter what the nature of the interaction is, and no matter what the initial state of the object is, there will be at least one state S3 of the older system (as it emerges from the time travel machine) that will influence the initial state S1 of the younger system (when it encounters the older system) so that, as the younger system becomes older, it develops exactly into state S3. Thus without imposing any constraints on the initial state S1 of the system P, we have shown that there will always be perfectly ordinary, non-paradoxical, solutions, in which everything that happens, happens according to the usual laws of development. Of course, there is looped causation, hence presumably also looped explanation, but what do you expect if there is looped time? Unfortunately, for the fan of time travel, a little reflection suggests that there are systems for which the needed fixed point theorem does not hold. Imagine, for instance, that we have a dial that can only rotate in a plane. We are going to put the dial in the time machine. Indeed we have decided that if we see the later stage of the dial come out of the time machine set at angle x, then we will set the dial to x+90, and throw it into the time machine. Now it seems we have a paradox, since the mapping that consists of a rotation of all points in a circular state-space by 90 degrees does not have a fixed point. 

And why wouldn't some state-spaces have the topology of a circle? However, we have so far not used another continuity assumption which is also a reasonable assumption. So far we have only made the following demand: the state the dial is in at stage S2 must be a continuous function of the state of the dial at stage S3. But, the state of the dial at stage S2 is arrived at by taking the state of the dial at stage S1, and rotating it over some angle. It is not merely the case that the effect of the interaction, namely the state of the dial at stage S2, should be a continuous function of the cause, namely the state of the dial at stage S3. It is additionally the case that path taken to get there, the way the dial is rotated between stages S1 and S2 must be a continuous function of the state at stage S3. And, rather surprisingly, it turns out that this can not be done. Let us illustrate what the problem is before going to a more general demonstration that there must be a fixed point solution in the dial case. Forget time travel for the moment. 

Suppose that you and I each have a watch with a single dial neither of which is running. My watch is set at 12. You are going to announce what your watch is set at. My task is going to be to adjust my watch to yours no matter what announcement you make. And my actions should have a continuous (single valued) dependence on the time that you announce. Surprisingly, this is not possible! For instance, suppose that if you announce "12", then I achieve that setting on my watch by doing nothing. Now imagine slowly and continuously increasing the announced times, starting at 12. By continuity, I must achieve each of those settings by rotating my dial to the right. If at some point I switch and achieve the announced goal by a rotation of my dial to the left, I will have introduced a discontinuity in my actions, a discontinuity in the actions that I take as a function of the announced angle. So I will be forced, by continuity, to achieve every announcement by rotating the dial to the right. But, this rotation to the right will have to be abruptly discontinued as the announcements grow larger and I eventually approach 12 again, since I achieved 12 by not rotating the dial at all. 

So, there will be a discontinuity at 12 at the latest. In general, continuity of my actions as a function of announced times can not be maintained throughout if I am to be able to replicate all possible settings. Another way to see the problem is that one can similarly reason that, as one starts with 12, and imagines continuously making the announced times earlier, one will be forced, by continuity, to achieve the announced times by rotating the dial to the left. But the conclusions drawn from the assumption of continuous increases and the assumption of continuous decreases are inconsistent. So we have an inconsistency following from the assumption of continuity and the assumption that I always manage to set my watch to your watch. So, a dial developing according to a continuous dynamics from a given initial state, can not be set up so as to react to a second dial, with which it interacts, in such a way that it is guaranteed to always end up set at the same angle as the second dial. 

Similarly, it can not be set up so that it is guaranteed to always end up set at 90 degrees to the setting of the second dial. All of this has nothing to do with time travel. However, the impossibility of such set ups is what prevents us from enacting the rotation by 90 degrees that would create paradox in the time travel setting. Let us now give the positive result that with such dials there will always be fixed point solutions, as long as the dynamics is continuous. Let us call the state of the dial before it interacts with its older self the initial state of the dial. And let us call the state of the dial after it emerges from the time machine the final state of the dial. We can represent the possible initial and final states of the dial by the angles x and y that the dial can point at initially and finally. The set of possible initial plus final states thus forms a torus. (See figure 1.) 

Figure 1 Suppose that the dial starts at angle I. The initial angle I that the dial is at before it encounters its older self, and the set of all possible final angles that the dial can have when it emerges from the time machine is represented by the circle I on the torus (see figure 1). Given any possible angle of the emerging dial the dial initially at angle I will develop to some other angle. One can picture this development by rotating each point on I in the horizontal direction by the relevant amount. Since the rotation has to depend continuously on the angle of the emerging dial, ring I during this development will deform into some loop L on the torus. Loop L thus represents the angle x that the dial is at when it is thrown into the time machine, given that it started at angle I and then encountered a dial (its older self) which was at angle y when it emerged from the time machine. 

We therefore have consistency if x=y for some x and y on loop L. Now, let loop C be the loop which consists of all the points on the torus for which x=y. Ring I intersects C at point . Obviously any continuous deformation of I must still intersect C somewhere. So L must intersect C somewhere, say at . But that means that no matter how the development of the dial starting at I depends on the angle of the emerging dial, there will be some angle for the emerging dial such that the dial will develop exactly into that angle (by the time it enters the time machine) under the influence of that emerging dial. This is so no matter what angle one starts with, and no matter how the development depends on the angle of the emerging dial. Thus even for a circular state-space there are no constraints needed other than continuity. Unfortunately there are state-spaces that escape even this argument. Consider for instance a pointer that can be set to all values between 0 and 1, where 0 and 1 are not possible values. I.e. suppose that we have a state-space that is isomorphic to an open set of real numbers. Now suppose that we have a machine that sets the pointer to half the value that the pointer is set at when it emerges from the time machine. Figure 2 Suppose the pointer starts at value I. 

As before we can represent the combination of this initial position and all possible final positions by the line I. Under the influence of the pointer coming out of the time machine the pointer value will develop to a value that equals half the value of the final value that it encountered. We can represent this development as the continuous deformation of line I into line L, which is indicated by the arrows in Figure 2. This development is fully continuous. Points on line I represent the initial position x=I of the (young) pointer, and the position y of the older pointer as it emerges from the time machine. Points on line L represent the position x that the younger pointer should develop into, given that it encountered the older pointer emerging from the time machine set at position y. Since the pointer is designed to develop to half the value of the pointer that it encounters, the line L corresponds to x=1/2y. We have consistency if there is some point such that it develops into that point, if it encounters that point. Thus, we have consistency if there is some point on line L such that x=y.

However, there is no such point: lines L and C do not intersect. Thus there is no consistent solution, despite the fact that the dynamics is fully continuous. Of course if 0 were a possible value L and C would intersect at 0. This is surprising and strange: adding one point to the set of possible values of a quantity here makes the difference between paradox and peace. One might be tempted to just add the extra point to the state-space in order to avoid problems. After all, one might say, surely no measurements could ever tell us whether the set of possible values includes that exact point or not. Unfortunately there can be good theoretical reasons for supposing that some quantity has a state-space that is open: the set of all possible speeds of massive objects in special relativity surely is an open set, since it includes all speeds up to, but not including, the speed of light. Quantities that have possible values that are not bounded also lead to counter examples to the presented fixed point argument. And it is not obvious to us why one should exclude such possibilities. So the argument that no constraints are needed is not fully general. An interesting question of course is: exactly for which state-spaces must there be such fixed points. We do not know the general answer.

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