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John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
The supertasks we have seen involve some exotic processes. I have argued that one cannot dismiss these supertasks simply by asserting that completion of an infinity of acts is impossible. However it does not follow that all supertasks are immune from criticism. "The gods," for example, fails as a supertask since it is based on contradictory assumptions.
The Thomson lamp, however, does not involve a similar contradiction; it is just unexpected and perhaps odd.
If some supertasks prove ultimately to hide contradictions, how can we be sure that there is not a similar problem with the others, even if they seem innocent enough on a superficial scan? That is, might the assumptions in these other supertasks enable us to deduce a contradiction? That is the defining characteristic of an inconsistent set of assumptions.
The problem is more acute for the reversed supertasks such as the reversals of Black's transfer machine and the many accelerating spaceships. For both involve the sudden materialization of bodies in space. They might be admissible processes, but they surely strike us as strange, at least initially.
Proving the consistency of some set of propositions can be difficult and, in interesting cases, impossible. We have already seen earlier in the case of the Thomson lamp that we can give ourselves some assurance of the consistency of some set of propositions by means of a relative consistency proof. In such a proof, we realize the process in question within another systems whose consistency is not under challenge. If the realization succeeds then the supposition of consistency of that second system gives us some assurance of the consistency of the system in question.
In this chapter, we shall investigate whether can we provide something similar for the more puzzling of the supertasks, including the reversed supertasks. The strategy to be employed here is to realize the supertasks within what we otherwise take to be a consistent physical theory. That is, we assume it is a theory in which we cannot deduce a contradiction. The simpler is the physical theory, of course the better.
The physical theory we will use is the theory of collisions of Newtonian, elastic bodies. We will recreate several supertasks in the collisions of infinite sets of these bodies. As long as we are confident that this Newtonian theory is consistent, then we must in turn be confident of the consistency of the original supertask.
There can be no proof of the consistency of this physical theory. We shall see that there are systems in which consistency fails. They include the problem of accumulation points described below. I will suggest however that we can circumscribe the conditions under which these failures of consistency arise; and we can avoid them if we consider supertasks that obey a "decomposition" condition.
Otherwise, we will require some extreme idealizations. We will assume the possibility of bodies of arbitrarily small size, all of the same mass with arbitrarily large velocities. If someone wants to mount a further challenge to the consistency of the theory, these idealizations would be a natural starting point.
Before we turn to the mechanics of collisions, an alternative approach we might take is to posit a force field that would accelerate bodies as needed. For the accelerating spaceship, we might consider a force field that gets stronger as the spaceship comes to be more distant from us.
If the spaceship's position in space is given by "x," we might consider a force field that gives the spaceship an acceleration proportional to x. It is easy to show that this force field would yield an accelerating motion such that the distance x grows exponentially with time. While that is impressive, it is not fast enough. For in exponential growth there is a doubling time, such that the spaceship's speed doubles with the passing of each doubling time. Fast as it is, it is not fast enough. For no finite amount of doubling will send the spaceship off to infinity in finite time. At any finite time, the spaceship will always be somewhere in space, even if very far away.
While this proposal for a force field fails to yield the accelerating spaceship supertask, a related force field succeeds. Instead of a force field that accelerates the spaceship in proportion to x, consider one that accelerates it even more aggressively, say, with acceleration proportional to x3. In the figure below, the arrows indicate the force acting on a body at different x positions, when the acceleration due to the field is 2x3. (2x13= 2; 2x23= 16; 2x33= 54; etc.)
A few calculations show that such a field will send the spaceship to infinity in finite time. One such force field has the following relations between time t and position x.
Here is a plot of these positions x against the times t. The trajectory shows a rapid acceleration that continues such that the curve will extend over all values to x but never intersect the horizontal line marking t=1.
Such force fields give some sense that there is no contradiction in the motions to which they are adapted. However they open the lingering worry that there is something wrong with the force field itself. We can always posit a force field and stop there. But how do we know that the theory of force field is consistent? Just how are the fields produced? By masses as sources? Under which rules?
We can see the dangers of positing force fields indiscriminately in Berndadete's analysis of the gods reversed supertask. The posit is that the peculiar arrangement of unrealized obstructing walls creates an obstructing force field. To make the posit is to take only the first step. To have some assurance that this account is consistent, we need a fuller account of just what general circumstances lead to the appearance of these field. Without that account, all we have is a purely ad hoc proposal designed opportunistically to answer the pressing needs of the moment.
Perhaps the simplest of all theories of physics is the dynamics of elastic body collisions in Newtonian mechanics. In the rest of the chapter we shall investigate how supertasks may be realized in this dynamics. We shall have some success. If the supertasks can be realized in this simplest of dynamical theories, we should have some confidence in their consistency. However we shall have to proceed with caution, for we will eventually see there are some extreme circumstances in which the dynamics fails to be consistent (in association with accumulation points, to be explained below.)
We shall need only a few simple rules from Newtonian mechanics in order to recover the supertasks in the behavior of the elastic bodies. In all the cases below, we consider spherical bodies all of which have the same mass. They may need to be of different sizes. That means that smaller bodies are assumed to be made of denser material so that their masses remain the same. A familiar approximation to bodies behaving this way comes with balls colliding on a pool or billiard table.
The first rules concern bodies whose collisions are confined to one dimension of space. The simplest case is of two bodies approaching head on at the same speed v. When they collide, they rebound at the same speed but in the opposite directions. This is an immediate result of the symmetry of the initial state and the assumption that the collisions are elastic (so that no kinetic energy is lost in them). The signs on the speeds indicate direction. Plus is to the right; minus is to the left.
We recover the general one-dimensional case from this special case by noting that Newtonian mechanics obeys a principle of relativity of inertial motion. If we change our frame of reference to one that moves inertially at speed -u, that is, speed u to the left, we recover a new description of this same collision merely by adding u to each speed.
Writing W=u+v and w=-u+v we can relabel this last figure. Then we see that, in the general case, the effect of the collision is merely to swap speeds. The body that had speed W now takes speed w; and the body that had speed w now takes speed W.
Rule for collision of elastic bodies in
The two bodies exchange their velocities.
An important case arises when u=v and w=0. A body at rest is struck by a moving body and set into motion with the speed of striking body. The formerly moving body, however, comes to rest.
Collisions in two-dimensions are more complicated. However they can treated by a slightly more complicated version of the one-dimensional rule. The general principle for resolving these collisions is that we can treat the components of the motions in the two directions independently by the rules just determined for one-dimensional collisions. The one dimensional rule applies to the components of the velocities in the direction of the impact. The components perpendicular to the direction of the impact are unaffected.
For example, in the x-y plane shown, we have a body moving diagonally at speed w. Its motion has component speeds in the x and y direction of wx and wy as shown. Another body moving only in the x direction at speed W collides with it.
The result is shown in the figure. The y-speed of the original body is unaffected by the collision. The effect of the collision is merely to swap the speed W of the impinging body with x-speed wx of the original body.
Rule for collision of elastic bodies in
The two bodies exchange their velocity components in the direction of the impact. The components of their velocities perpendicular to the direct of impact remain unchanged.
When two spherical bodies collide, the direction of the impact is a line connecting their two centers. The direction is indicated by the shading:
The line of impact is in the x-direction. That means that the the x-components of the velocities of the two bodies are exchanged, but not the y-components.
These collision rules conform with a feature that will be important below. It is that the rules continue to hold if we time reverse all the processes. That is, we imagine that we take a video of some collision that conforms with the rules. We then run the video in reverse. What the video shows will then still conform with the above rules. That is, we cannot tell whether the video shown to us is the original video or the reversed video. Both depict admissible collision processes.
Rule for time reversal of elastic body
If any sequence of collisions is allowed by the dynamics, then a time reversed version of the collisions is also allowed.
Here is the case of the general one-dimensional collision run in reverse. First we have the forward collision described above:
Then we have its time reversal:
The reversed motion obeys the same rule of collision dynamics as the forward collision: the speeds of the two bodies are switched.
This condition of time reversibility is not a trivial condition. The more realistic case is of even slightly inelastic collisions. In them there are slight frictional losses in each collision. A time-reversed video of the collision would no longer obey the dynamics. It would show frictional losses converted back into motions.
A simple example of a process that is not time-reversible is a block sliding with friction on a table top. To begin, it is moving quickly. As friction impedes the motion, it comes to a halt. The reverse of this process is not permitted in ordinary mechanics. Blocks sitting on table tops with frictional connections do not set themselves into motion.
To realize Black's transfer machine in this simple mechanics, we start with a flat surface. We could imagine it to be a smooth table top, on which motions occur without friction. Let us call the two directions "up" corresponding to up in the figure; and "across" or "horizontal."
Black's marble is initially moving up at some small, constant speed. We should like the marble to transit to the left and right with increasing frequency so that an infinity of transits are completed prior to the marble reaching the top edge of the surface. To carry this out, the horizontal speed of the marble on each transit will need to increase; and it must do so without any upper limiting speed.
The scheme for realizing this in collision mechanics is straightforward. Each reversal of direction of the marble is brought about by a collision with another body moving horizontally across the surface. It is easy to ensure that each transit has the speed needed to complete the supertask motion. We just set up the colliding body to have the speed needed for the ensuing transit. The rules for elastic collisions then ensure that the marble and the colliding body simply switch their horizontal speeds and the marble acquires the speed needed.
The only complication is that we will need an
infinity of colliding bodies and we need them to fit into a finite
vertical space without interfering with each others motion. To meet this
condition, we must specify that the colliding bodies
become arbitrarily small for the later collisions.
We can now ask after the viability of the time reversal of Black's transfer machine. Recall that the oddity of time reversal is that the marble materializes instantly at the start of the reversed supertask. Is this really something that can happen in a consistent account?
The easy and obvious answer is yes! The mechanics for elastic collisions is time reversible. To realize the reversed transfer machine in collision mechanics, we merely take the time reverse of the process just described. What results is shown in the figure:
The marble does materialize in the course of the process. We might try to find something mysterious in its materialization. Obvious ways to do this fail. If we call the first moment of the reversed supertask time t=0, we might want to ask:
"Why does the marble materialize at time t=0?"
To answer, return to the original supertask. The marble's trajectory in the this supertask has no end. It just oscilates infinitely often between the two trays. Correspondingly, the marble's trajectory in the reversed supertask has no initial point at time t=0. Hence the question is based on a mistaken assumption. The marble doesn't exist at t=0. It doesn't materialize at t=0.
We might try to press the worry by asking:
"Why is there a marble at time t=0.01?"
That question has a benign answer:
"...because the marble already existed at earlier times, such as time t = 0.001."
We can continue these question for any time after t=0 and always be answered benignly. Why is there a marble at time t=0.000001? it is because there was a marble at t=0.0000001. And so on.
This time reversal does reveal something one might have suspected from the start. Given great liberties with idealizations, it does seem conceivable that we could arrange the bodies and their speeds so that the forward transfer machine process would occur.
Doing the same for the reversed supertask is not possible. That is, we cannot set up the bodies and their speeds so that the reversed supertask ensues. For that set up concerns the positions and speeds of the bodies at the initial time t=0. The marble does not exist at this initial time, so we cannot place it in a way that ensures that the reversed supertask ensues.
This does not mean that the reversed supertask is impossible. It just means that we cannot bring it about. We can set up as much as we can of the bodies we control at time t=0. All we can then say is that rules of collision mechanics allow several possible futures. In one of them, the marble materializes. We just cannot force it to happen. It may not material without there being any violation of rules of collision mechanics.
After time t=0, the marble does exist. So why not set up the colliding does and the marble with suitable motions at some moment very soon after time t=0? What about time t=0.0001? The trouble is once any non-zero time at all has passed, the bulk of the supertask is completed. Infinitely many collisions are to happen prior to time t=0.0001 and only finitely many afterwards. What we can set up at time t=0.0001 is just a finite tail of the supertask. It is not the infinite part.
This reversed supertask illustrates the failure of determinism in the mechanics of the collisions of infinitely many bodies. Determinism requires that, if we fix fully the conditions in the present, then future conditions are also fixed. That fails here. Take the case in which the marble is assured not to exist prior to the supertask. All we can set in the present are the positions and velocities of the impinging bodies. Then two things can happen: either the marble materializes and the collisions proceed as above; or it does not materialize and other collisions ensue. That is, fixing the present condition does not give us a unique future.
We can now return to the accelerating spaceship supertask and see how it can be implemented in a system of many colliding bodies. In place of the spaceship, we have a body that is to be accelerated off to infinity. Following the approach taken in the transfer machine, how this is to be done is now obvious. We have the schedule of speed increases in the first description of the spaceship supertask.
|Time interval||speed||distance covered
= speed x time
|0 to 1/2 = 1/2||2||1|
|1/2 to 3/4 = 1/4||4||1|
|3/4 to 7/8 = 1/8||8||1|
|7/8 to 15/16 = 1/16||16||1|
We choose one body to represent the spaceship. It is initially moving "up" in a two-dimensional space. It is then struck by horizontally moving bodies whose speeds are those of the schedule for the accelerating spaceship. That is, the horizontally moving bodies have speeds 2, 4, 8, 16, ... With each collision, the first mass acquires the horizontal speed of the body with which it collides. As long as the infinitely many colliding bodies are so set up that the totality of collisions is completed in a finite time, then the original mass is accelerated off to infinity, as in the original supertask. Here is a representation of the collisions:
At the conclusion of the supertask, the original body is no where in space.
We can now ask after the prospects of the reversed supertask. As before, since the collision processes are time reversible, the time reverse of this implementation of the supertask can also be realized in systems of colliding masses. Here is a representation of the collisions:
In this reversal, the body representing the spaceship materializes at the start of the supertask. The role of the colliding bodies is not to accelerate the materialized body. Rather it is to decelerate it, that is, to slow it down by braking its arbitrarily fast motion at the start of the reversed supertask. The body moving in from infinity successively encounters slower moving bodies, where the bodies are moving in the direction of the incoming body, with the schedule of speeds in the table above ... 16, 8, 4, 2, 0. As the incoming body collides with each, the two bodies switch horizontal velocity components and the the mass steps down through the speeds of the table above, but in the reverse order: ... 16, 8, 4, 2, 0.
We might say the body materializes "from infinity." That, of course, is loose talk. There is no place "infinity" where the body waited patiently for its moment to step onto the stage of space. The body simply did not exist in space at the first moment of the supertask. It does exist in space for all moments after.
While this materialization of the body seems strange, the arguments above in the transfer machine apply here as well. There is no contradiction with collision mechanics. It is just a strange and otherwise unexpected result.
Finally, with some idealizations, we may imagine it possible to set up bodies and their motions so that the original, forward supertask occurs. However, there is no corresponding possibility with the reversed supertask. We may have all the colliding bodies set up appropriately at the first moment of the supertask. However the materializing body does not yet exist and nothing we can do can assure us it will appear. It may appear and the reversed supertask may proceed as intended. Or it may not appear and the reversed supertask will not proceed.
Once again, we have a failure of determinism.
The treatment of the accelerating spaceship so far has dealt with only one spaceship. In earlier treatments of many accelerating spaceships, we saw a system of infinitely many spaceships such that all of them vanished from space at a single moment. We might try to implement this more complicated supertask here by considering infinitely many bodies each to be accelerated by their own suite of infinitely many colliding bodies. It turns out that there is a simpler and more elegant way to achieve the same result. In it, there is a space filled with bodies that suddenly empties at the end of the supertask. All the bodies disappear to infinity at the same moment.
The ingenious way to implement this new supertask was devised by Jon Pérez Laraudogoitia. The new supertask arises when we collapse all the masses in the accelerating spaceship supertask above into a single dimension of space. The collapse looks something like this:
All the bodies are now lined up horizontally and the initial vertical motion of the body to be accelerated is now removed. Otherwise, everything else stays the same as in the initial set up. The horizontal positions of all the bodies and their initial speeds are unchanged. All the bodies can now be the same size since there is no need to shrink them to fit into the space available. The result will look initially something like this. (The colliding masses have been numbered for ease of reference.)
This is the initial configuration. Once the supertask starts, all the bodies will collide with each other and there will be a dramatic confusion of bodies colliding with each other. One might despair of trying to figure out just what will ensue. However it turns out that it is surprisingly easy to figure out what will happen. The result will be the same: the body to be accelerated will be accelerated off to infinity on the same schedule as with the original supertask.
To see why, we need a simple result from collision mechanics, already implicit in the rules we have seen. Imagine that we have some collection of bodies lined up together and moving variously. Another body approaches it from the left with some speed higher than any of those in the collection. Call that speed "v." The approaching body will strike the first body of the collection. That first body will acquire the speed v of the approaching body. The process will then continue. This first body will strike the second body in the collection; and that body will acquire speed v. The process will continue in a cascade until the last body in the collection is ejected at the other end with speed v. That is, the speed v passes through the collection and emerges at the other side. The speed v is now carried by a different body.
Now return to the supertask. Consider, for example, the colliding body numbered three. Will its speed be communicated to the accelerated body? Body 3 moves faster than bodies 1 and 2. It approaches them, collides with them and the mechanism just described leads the speed of body 3 to be communicated to body 1, which then communicates it to the accelerated body.
The same is true for all other colliding bodies. The still higher speed of body 4 is communicated through bodies 1, 2 and 3 to the accelerated body.
This continues for all the colliding bodies. Their speeds are communicated successively to the accelerated body so that it is accelerated to arbitrarily high speeds. Since the communications are completed in finite time, it is accelerated off to infinity. The supertask is completed as before.
Now consider the fate of the colliding bodies. Since they are aligned in the same dimension of space, they cannot overtake each other. Each can collide only with same body immediately on either side. Body 3, for example, can only collide with bodies 2 and 4.
This means that all the speeds of the colliding bodies are transmitted to the accelerated body by repeated collisions with body 1. This means that, as the supertask proceeds, body 1 successively acquires all the speeds of the colliding bodies and is able to contact the accelerating body to pass them on. It follows that, as the supertask proceeds, body 1 itself must pass arbitrarily far away from the starting point and be, itself, accelerated to infinity.
Now consider body 2. This body 2 communicates all the speeds of bodies 3, 4, 5, ... to body 1. So by similar reasoning it too must be accelerated off to infinity.
This last reasoning can be continued for the remaining bodies 3, 4, 5, ... Each of them individually is accelerated to unlimited speed and off to infinity in the finite time of the supertask.
The combined result is that, in the course of the supertask, ALL the masses are accelerated to arbitrarily high speeds and off to infinity. What results is that, until the last moment of the supertask, space is filled with infinitely many masses. At then--POOF--at the last moment, all the bodies disappear!
Now finally we can consider the reversed supertask. In it, we have a completely empty space at the first moment of the reversed supertask. And then immediately after, space is filled with an infinity of masses pouring in from infinity. This striking result, we might recall, has been established within the mechanics of the collision of Newtonian bodies.
That is we first:
and then suddenly space fills:
As before, this pouring in from infinity of bodies can happen. But nothing we do in advance can assure it to happen.
All the supertasks considered in this chapter so far involve bodies moving at arbitrarily fast speeds. One might well ask whether a prohibition on arbitrarily fast speeds is all that is needed to eliminate supertasks in the simple mechanics of colliding bodies. It is a good question to ask since we know from relativity theory that arbitrarily high speeds cannot be realized for such bodies in the real world. It turns out that supertasks can still give unexpected results, even when all the speeds are the same and this speed can be quite small.
We will still need some odd assumptions, however, to recover an interesting supertask. In the case below, we will need to assume an infinite set of bodies in which the bodies become arbitrarily small so that they can fit into a finite space. That way, we can still complete an infinity of actions involving collisions between the bodies, without needing arbitrarily high speeds. While this escapes the violation of relativity, it violates another theory of modern physics. Quantum theory has placed a lower limit on the size of bodies like atoms and the exactness with which we can assign a position and speed at the same time to small bodies.
For this slow speed supertask, we consider an infinity of equally massive bodies. They are laid out at rest in a line such that all infinity of them fit in some finite interval of the line. If the interval extends from x=1 to x=0, we might locate the bodies at x coordinates:
1, 1/2, 1/3, 1/4, 1/5, ...
and so on. The particular set of locations used will not matter. All that matters is that there is an infinity of them in the interval.
A body approaches from the left as shown below and will collide with them.
A cascade of collisions will ensue. The moving body will collide with the first body. This moving body will transmit its speed to the first body in the line and come to rest.
This first body will continue the motion. It will collide with the second body, transmit the motion to it and come to rest.
This second body will continue the motion. It will collide with the third body, transmit the motion to it and come to rest.
The cascade will continue for all the remaining bodies. The speed of the initiating mass can be small--unit speed say. Then all the speeds in the process will be at that unit speed. Since the infinity of bodies fits into a finite interval of the line, the motion will propagate through the whole infinity of bodies in a finite time. An infinity of collisions will occur and they comprise the supertask.
The interesting question is what happens at the end of the supertask? A natural reaction is to expect that the last body in the infinite collection is ejected with unit speed as a result of the last collision. However, in an infinity of bodies and collisions, there is no last body and no last collision. To be concrete about it:
The 100th body is not ejected
since it collides with the 101st body and comes to rest.
The 1000th body is not ejected since it collides with the 1001st body and comes to rest.
The n-th body is not ejected since it collides with the (n+1)-th body and comes to rest.
That is, none of the bodies is ejected at the end of the supertask. Instead the supertask proceeds as follows. A moving body approaches the infinite collection of bodies. It collides with the first body and that triggers a cascade of collisions that ripple through the infinite collection. Then the ripple stops and all the masses are at rest. Nothing is moving any more.
We can now consider the reversed supertask. It is compatible with the collision mechanics since each collision is individually time reversible. In it the end state of the forward supertask becomes the initial state of the reversed supertask. After the initial moment, a reversed cascade of collisions spontaneously appears at the infinite end of the collected bodies. The motion propagates outward from the infinite end until the first body is ejected and all the remaining bodies return to rest.
In short, the system spontaneously sets all of its bodies into motion temporarily and ejects one of the bodies. The figure shows the last few collisions of the reversed supertask:
Once again, there is nothing we can do in advance to bring about this motion. We can set up the infinity of bodies in the right configuration. Then all we can do it wait. The dynamics assures us that the spontaneous motion may happen or it may not.
A further element is out of our control. The original supertask can proceed with any finite speed for the incoming body. It follows that the reversed supertask can eject a body spontaneously moving at any finite speed.
Both the forward and reversed supertasks have a feature that requires some thought. In the forward version, the kinetic energy and momentum of impinging body is lost. When the forward supertask is complete, nothing is moving. There is no momentum and no kinetic energy in the system. In the reversed super task, the opposite happens. We start with a system all of whose bodies are at rest and thus have no kinetic energy and no momentum. When the supertask is completed, we have an ejected body with some non-zero momentum and non-zero kinetic energy.
In short, kinetic energy and momentum are not conserved.
This may at first seem a problematic result. Are not these conservation laws the bedrock of physics? The answer is that these conservation laws are foundational for ordinary systems. An infinity of bodies in collisions is not an ordinary system. These conservation laws fail for such system. Indeed, we have just given the proof.
If this failure of conservation laws is still troubling you, this might help. Take the case of an infinite system of bodies in which all of them are moving at some non-zero speed. If there is a non-zero lower bound to all the speeds, then the total momentum and the total kinetic energy of the system is infinite. When these quantities are infinite, the conservation laws become degenerate. Adding or subtracting any finite amount of momentum or kinetic energy from them leaves them the same, that is, infinite. Is the concern is that the system of the superstask does not have all the infinity of bodies moving at the same time. That is answered by considering the infinity of bodies from another inertial frame of reference such that all the bodies are moving uniformly in the same direction. In that frame, the kinetic energy and momentum of the system is infinite.
This last slow speed supertask contains a lingering problem for collision mechanics as it is used here. The infinity of bodies, when fitted into a finite interval, must have an accumulation point. This is a point in space, such that regions that enclose it always have an infinity of bodies in it, no matter how small we make that region. The accumulation point of this supertask is at the infinite end of the row of bodies, at x=0 in the figure. We can draw a circle around this accumulation point. The circle will contain an infinity of bodies. No matter how small we make the circle, it will always contain an infinity of bodies.
This accumulation point creates a problem. It arises if we consider another body that approaches the infinite collection of bodies from the side of the accumulation point. We ask what will happen when the body reaches the accumulation point at x=0.
In answering, we arrive at a contradiction
with collision mechanics. Either the body stops (or bounces back)
at x=0 or it keeps moving at x=0.
• The body cannot stop (or bounce back) at x=0 since to do so it must collide with one of the bodies in the collection. As it approaches the accumulation point at x=0, it must collide with the first body it meets (if it is to collide with any). However the accumulation point is such that there is no first body for it to meet.
For example, the body at x=1/100, is not the first, since there are bodies closer to x=0 at x=1/101, x=102, ...
In collision mechanics, colliding with another body is the only way that a body changes its motion.
• The body cannot continue past x=0. For to advance to any position at all with x>0, it must pass through an infinity of bodies without being deflected by them.
For example to advance to x=0.001=1/1000, it must pass through the infinity of bodies at 1/1001, 1/1002, 1/1003, ...
In collision mechanics, the approaching body cannot pass through other bodies.
The great temptation in the literature is to assert that the approaching body does stop and that the infinite collection of bodies somehow creates a mysterious force field to halt it, without actually contacting it. We saw this type of analysis in Bernadete's analysis of the "gods" supertask. The mysterious force field is a speculative proposal for some new physics, not contained within the original collision mechanics. It lacks any coherent theory. It is an opportunistic attempt to escape an inescapable contradiction in collision mechanics.
We can sharpen the problem of the accumulation point by replacing the infinite bodies above by an infinite collection of disks. They diminish in size such that an infinity of them fit into a finite interval. They touch one another and are geared together such that a disk rotating clockwise engages with its neighbor that must rotate anticlockwise. (If we imagine the engagement of the neighboring disks to arise through tiny teeth in cogwheels, then the interactions of the disks are still governed by simple collision mechanics. The collisions are very simple--it is just one body pushing another.)
If one of the disks rotates, it follows that the entire collection of disks is in rotation. As we move along the disks, they alternate between clockwise and anticlockwise rotations.
When a flat plate encounters one of these discs, ordinary contact mechanics dictates that it will be deflected in the direction of the rotation of the side of the disc it contacts. As the figure shows, a plate contacting an anticlockwise rotating disk from the right is deflected upward. A plate contacting a clockwise rotating disk from the right is deflected downward.
We now ask, what happens when the plate approaches the accumulation point of the infinite collection of rotating disks. As before, simple collision mechanics can give no consistent account of the motion of the plate. In cannot stop since there is no first disk for it to collide with. But it cannot advance past the accumulation point since then it must pass through an infinity of disks if it is to advance any distance at all.
The novel addition provided addresses the possibility that somehow--we know not how--the plate does contact the disks and be halted in some way by the contact. The difficulty is immediate. Contact with any of the disks will not simply halt the plate. In accord with contact mechanics, the plate must be deflected upward or downward according to whether the disk contacted is rotating anticlockwise or clockwise. Yet were the plate to be deflected upward, we would have found rotation sense of the "last" disk of the infinite collection at the accumulation point. That is, we would have found the last member of the alternating infinite sequence:
clockwise, anticlockwise, clockwise, anticlockwise, clockwise, ...
That result would stand in contradiction with fact that this sequence has, by construction, no last member.
The analysis of these accumulation points (along with Bernadete's analysis of the "gods" supertask) employs a simple collision mechanics. The principle at issue is:
Rigid bodies in collision halt their relative motion at the first point of contact.
It is possible, these examples show us, to embed this sort of collision mechanics in a larger system such that the system overall is held to inconsistent requirements.
The correct analysis of these systems is to deduce results that are manifestly in contradiction and thus merely to display their contradictory character. Typically the first step is to show that that a body must be halted in its motion, even though it has not made first contact with another body. That conclusion contradicts the assumption elsewhere that there is no other force halting it.
The mistaken analysis is to halt the deduction of the inconsistency at an intermediate stage. One then suggests that the system displays some sort of mysterious power to halt motions. This strategy could be applied to any demonstration of an inconsistency with equally spurious results. Fortunately, we mostly do not fall into the trap of applying this mistaken analysis elsewhere.
That there exist both irresistible forces and immovable objects is contradictory. We show this by deducing that an irresistible force could move an immovable object. That shows the contradiction. We do not--I hope--thereby conclude that an irresistible force can nonetheless, mysteriously, move an immovable object.
A simple example of this mistaken analysis is provided by the paradox of Smullyan's rod. Here is Smullyan's account of it:
"A Curious Puzzle
When I ﬁrst studied high-school geometry, the following idea occurred to me. Imagine that you have an inﬁnite solid plane table with a ﬁnite rod bolted perpendicular to the table. To the top of this ﬁnite rod is hinged one end of an inﬁnite rod. The hinging allows the inﬁnite rod to move up and down, but the curious thing is that the rod cannot possibly move down because both it and the table are solid, and therefore the rod cannot pierce the table. And so, you have the curious phenomenon of the hinged rod being supported at only one end!"
The paradox considers a straight rod, attached to a high point above a table surface. It is free to swivel about a hinge.
The motion of the rod is governed by two principles:
1. Gravity pulls the rod downward so it will fall if the motion is unopposed. (More precisely, since the rod moves about the hinge as a fulcrum, the rod will undergo angular acceleration about the hinge due to the gravitationally induced turning couple, unless something opposes it.)
2. The rod's fall is halted at the first point of contact with the table surface.
Thus we infer without contradiction that a finite rod falls until its end strikes the table surface.
It further follows from 1. and 2. that the motion halts at whichever is the first point on the surface that the falling rod strikes. If the table surface is uneven, the motion would halt when the finite rod strikes the first point of the unevenness, in relation to its rotation:
Now comes the paradox. Assume that the rod is infinite in length and that the surface is also flat, infinite in length and parallel to the initial position of the rod.
Then we can infer that the rod cannot fall. If we imagine the rod to have fallen by even a slight rotational angle, then it intersects the table at some finite distance.
But it can never reach this intersection since it would have been halted already at some smaller rotational angle. Since this is true of any non-zero rotational angle, the only allowable position is that rod remains in its horizontal position, with support only at the hinge.
In short, there is no first point of contact, so the contact principle 2. cannot be invoked.
The mistaken analysis makes this allowable position the conclusion of the analysis. Somehow--mysteriously--the rod does not fall, even though it is not properly supported.
The correct analysis is that this allowable position is the one deduced by applying the contact principle 2. above. It stands in contradiction with the gravitational principle 1., that bodies fall under gravity if unobstructed. That is, the full analysis is simply that the assumptions 1. and 2., together with the assumptions specific to the system are contradictory. Nothing more can be inferred. There is no warrant for a mysterious levitation.
The goal of this chapter is to provide some assurance of the consistency of the supertasks considered. it does this by providing a surrogate in the collision mechanics that functions in the same way as the original supertask. This assurance depends upon the further assurance of the consistency of this collision mechanics. The problems above indicate that we cannot always assume consistency. In some cases, it fails. We can, I will now argue, localize with some confidence where the contradictions will arise and thus can know which systems avoid them.
All the contradictions we have seen above amount to a contradiction with the following expectation:
Continuity. At any given moment, there is a state of a system of colliding bodies that has developed from prior states under the rules of the collision mechanics without contradiction.
It is important to see that this continuity condition just says "there is a..." It does not require that this contradiction free description has to be unique. For that would be to require determinism; and we have seen that determinism fails. A failure of determinism is not a contradiction within the set or rules. It is merely the recognition that the set of rules allows systems of bodies to develop in time in multiple ways.
The problems of accumulation points, of the infinity of rotating disks and of Smullyan's rod all involve a contradiction with this proposition of Continuity. For in each case we can identify a moment at which we can find no present state that has evolved from past states without some contradiction with the rules of the collision mechanics.
When can we be assured that a system will evolve without contradiction? Alas, it is major result of 20th century logic that we cannot prove the consistency of complicated systems such as these. However all the failures of consistency just noted arise when the following condition is not met:
Decomposition. At any moment for a system of bodies undergoing collisions, the behavior of each individual body is determined by the rules of the collision mechanics; that is, the body either moves freely or is deflected by a collision with another body.
For example, this decomposition condition is violated when the incoming body arrives at the accumulation point:
At that moment, the motion of that one body cannot be decided without contradiction by the rules of collision mechanics
Loosely speaking, this decomposition condition just says that the behavior of the totality can be recovered as the sum of the behaviors of the individual bodies, when they individually follow the rules of collision mechanics.
Even though both allow surprising materializations of bodies, this decomposition condition is met by both reversed versions of the transfer machine and accelerating spaceship:
The motions of the materialized bodies at no moment proceed in contradiction with collision mechanics.
While there is no absolute assurance, it seems safe to assume consistency in a system of colliding bodies as long as this decomposition condition is met. For then the overall consistency of the system is based on the consistency of the behavior of each part. The condition is met by both forward and reversed versions of the supertask of the transfer machine, the accelerating spaceship and the slow speed supertask.
The configuration of bodies accelerating the
spaceship can be collapsed from a two-dimensional arrangement to a
one-dimensional arrangement. Can the same collapse be carried out for the
case of the forward transfer machine? If so, what happens to the colliding
bodies at the end of the supertask?
The physical assumptions used in the simple collision supertasks here are incompatible with quantum theory and relativity theory. What significance does that have for the analysis?
Energy and momentum conservation are violated in these supertasks. Does that mean that the relative consistency proofs fail?
Can you think of another case of a relative consistency proof?
Why is "clockwise" rotation in the familiar
June 20, July 23, September 22, 2021
Copyright, John D. Norton