energy conservation and supertasks

ARTICLE IN PRESSJ. Perez Laraudogoitia / Studies in History and Philosophy of Modern Physics 39 (2008) 364–379 365

the energy in a supertask involving infinitely many identical particles with non-uniformlybounded velocities is not well defined, as the total energy would be ‘‘infinite’’). Recently,Atkinson (2007) and Perez Laraudogoitia (2007a, 2007b) have proposed examples ofsupertasks involving an infinite number of particles with finite total mass. These examplesare basically of two kinds, which I shall call ‘‘deterministic supertasks’’ and‘‘indeterministic supertasks’’.

By ‘‘deterministic supertask’’, I mean a deterministic process that lasts a finite time andwhich takes place exclusively by means of an infinite number of elastic, deterministiccollisions not globally independent; and by ‘‘indeterministic supertask’’ I mean anindeterministic process that lasts a finite time and which takes place exclusively by meansof an infinite number of elastic, deterministic collisions not globally independent.

The condition that collisions be elastic means that the laws of conservation of particledynamics are complied within them. The condition of the collisions not being globallyindependent is not essential for the developments that follow and we could actually dowithout it here. It has been included with the secondary object of avoiding an overly broadnotion of supertask (whether deterministic or indeterministic) that would allow some fairlyuninteresting processes, involving infinite collisions ‘‘too disconnected’’ from each other, tobe classified as such. To be more precise, suppose that the particles of the set P evolve(during a finite period of time) exclusively by means of an infinite number of elastic,deterministic collisions between them following the process PR from the initial condition Cgiven at the instant t0. For each qAP, let Pq be the set of particles of P causally connectedwith q (obviously through elastic collisions) during the process PR from the initialcondition C. We shall say that the collisions involved in PR (from C) are globallyindependent when, for each qAP, Pq is finite and there is one sole evolution possible for q

from C. The intuitive idea underlying this is perfectly clear: the elastic collisions involved inPR are globally independent when, for each qAP, q is causally connected only with a finiteset of particles of P and its (q’s) evolution is unique, thus not depending in the slightest onthe particles that are not connected causally with it in PR. This way, the idea of ‘‘globalindependence’’ indeed adds rigor to the intuition expressed above that the infinite collisionsare ‘‘too disconnected’’ from each other. Obviously the processes analyzed, for example, inAtkinson (2007) and Perez Laraudogoita (2007a, 2007b) are supertasks (somedeterministic, others indeterministic) because the collisions involved in them are notglobally independent (each particle there is causally connected with infinitely manyparticles by means of collisions). In contrast, let us suppose a process Q in which, for eachnA{1,2,3,y}, the particles P2n and P2n�1 are small spheres the centers of which movepermanently in the coordinate plane Xn ¼ n and that they collide head-on in tn ¼ 1/n at thepoint in space Xn ¼ n, Yn ¼ 0, Zn ¼ 0. Although the infinite collisions take place in a unitinterval of time we would be hard put to call this a supertask, since the collisions areglobally independent in an evident sense (as reflected in my technical characterization ofglobal independence): they take place in arbitrarily remote spatial regions with no possibleconnection between them. It is interesting to note that, despite appearances, what we arelooking at here is rather more than just a question of definition: in a relativistic contextthere are inertial systems of reference in which the set of collision events at points Xn ¼ n,Yn ¼ 0, Zn ¼ 0 at the instants of time tn ¼ 1/n becomes a set of collision events that takeplace in an infinite interval of time. Therefore to acknowledge that the sequence ofcollisions at the points Xn ¼ n, Yn ¼ 0, Zn ¼ 0 in the instants tn ¼ 1/n makes a supertaskimplies acknowledging that process Q is a supertask or not depending on the inertial

ARTICLE IN PRESSJ. Perez Laraudogoitia / Studies in History and Philosophy of Modern Physics 39 (2008) 364–379366

system of reference considered. If the supertaskhood of a process has to be relativisticallyinvariant in special relativity, as would seem reasonable, then processes like Q should notbe considered supertasks.

2. The question

As opposed to what we know in the sphere of supertasks of infinite total mass (aparadigmatic example of which is to be found in Perez Laraudogoitia, 1996), new exampleswith finite total mass have clearly revealed that there is no intrinsic connection betweendeterministic supertasks and the non-conservation of energy: there are cases ofdeterministic supertasks where energy is conserved and cases where it is not. The point Iwant to deal with here is the possible connection between indeterministic supertasks andthe non-conservation of energy. This point is interesting and not obvious. In the case oftotal infinite mass, all the known indeterministic supertasks (with defined energy, ofcourse) lead to the non-conservation of energy and this likewise occurs in the case of finitetotal mass. The question is: Is there any intrinsic connection between indeterministicsupertasks and the non-conservation of energy? In other words, is there something in thevery notion of an indeterministic supertask (as always, with defined energy) that impliesthe non-conservation of energy? In this paper I show that the answer is no. However, thepath to this answer is very different from the one that has enabled us to answer ‘‘no’’ to thecorresponding question for the case of deterministic supertasks. Again, the answer involvesproviding an example of an indeterministic supertask with well-defined energy in whichenergy is conserved; what changes radically is the nature of the example required and themethod to be followed.All this can be seen from a complementary, but equally important perspective. Some

time ago, Alper and Bridger proposed a criterion for excluding certain types of supertasksfrom the realm of physics (at least from legitimate physics), namely, the non-conservationof energy. In the conclusion of their joint paper with Earman and Norton, they state:

For an isolated dynamical system to earn the designation ‘Newtonian’ is it necessarythat it conserve energy? Alper and Bridger answer yes, arguing that the First Law ofThermodynamics trumps Newton’s laws of motion. (Alper, Bridger, Earman, &Norton, 2000, p. 291)

reiterating this idea in their joint paper with Perez Laraudogoitia:

[Alper & Bridger] maintain that ‘Newtonian’ should mean ‘Global Newtonian’because the global condition maintains the primacy of the laws of conservation ofmomentum and energy. In the global analysis, Newton’s laws break down for thosesystems y in which conservation laws fail. (Perez Laraudogoitia, Alper, & Bridger,2002, p. 177)

From what we know about supertasks with finite mass it is clear that Alper and Bridger’scriterion does not enable us to exclude deterministic supertasks as non-physical (i.e. it doesnot enable us to exclude all of them), but the question of whether the requisite of theconservation of energy permits the exclusion of indeterministic supertasks as non-physicalremains open. In the present paper we shall see this is not so, because, even when admittingthe condition of the conservation of energy, we can find examples of indeterministic


supertasks. This is another way of establishing (as I mentioned above) that there is nointrinsic connection between indeterministic supertasks and non-conservation of energy.

The example I give below is also general in an especially interesting sense: it willdemonstrate that there are indeterministic supertasks compatible with the conservation ofenergy, whether the laws of Newtonian dynamics or those of relativistic dynamics areadopted. Although the model’s initial conditions are fixed and precise, they have beendesigned to illustrate the essential point independently of the precise details of the type ofunderlying dynamical laws.

Some final words are required on the originality of my result. In dynamics (whetherNewtonian or relativistic) it is well known that there is no intrinsic connection between theindeterministic nature of a process and the non-conservation of energy in that process, asshown by the fact that there are trivial indeterministic processes where the energy isconserved (for example, elastic binary collisions between point particles in more than onespatial dimension). But rather than studying indeterministic dynamical processes ingeneral, in this paper I am interested in a particular type of such processes, i.e.indeterministic supertasks (as defined above) and with this restriction the question is notanswered because, as I stated above, all indeterministic supertasks known to date lead tothe non-conservation of energy. The existence or otherwise of an intrinsic connection inthis case is an open question. Although answering it does not teach us anything new aboutdynamic processes in general, it does teach us something new about dynamic supertasks.

3. A two-dimensional space with point particles

The supertasks to be introduced will be executed by point particles moving in two-dimensional space and subject to mutual interactions (not at a distance) as a consequenceof collisions. The initial condition (let us say, at t ¼ 0) common to them all can be specifiedvery simply in the following terms. Beginning with the origin of coordinates in the XY

plane, trace a segment s1 of length O2 that bisects the first quadrant. At the lowest point ofs1, i.e. at the origin of the coordinates, place a point particle P1 of mass 1/21 moving at unitvelocity towards the upper semi-plane in the direction of s1. Beginning at the mid-point ofs1, trace a segment s2 perpendicular to s1 and downwards to arrive at X-axis. At the lowestpoint of s2, i.e. on X-axis, place a point particle P2 of mass 1/22 moving at unit velocitytowards the upper semi-plane in the direction of s2. Beginning at the mid-point of s2, tracea segment s3 perpendicular to s2 and downwards to arrive at X-axis. At the lowest point ofs3, i.e. on X-axis, place a point particle P3 of mass 1/23 moving at unit velocity towards theupper semi-plane in the direction of s3. In general, then, beginning with the mid point of sn,trace a segment sn+1 perpendicular to sn down to X-axis. At the lowest point of sn+1, i.e. onX-axis, place a point particle Pn+1 of mass 1/2n+1 moving at unit velocity towards theupper semi-plane in the direction of sn+1 (nX1). This elaborate description in factcorresponds to the simple initial configuration that appears in Fig. 1.

It is not difficult to accept that the initial coordinate x of the particles P2n+1 of oddsubscript is

Xo2nþ1 ¼ ð2=3Þð4

n � 1Þ=4n, (1)

while the initial coordinate x for the particles P2n of even subscript is

Xo2n ¼ 1� ð1=3Þð4n�1 � 1Þ=4n�1. (2)

ARTICLE IN PRESS

Y

Xp1 p2p3 p4p5 p6……

••••••

Fig. 1. Initial state of an infinite system of particles that performs a supertask.

J. Perez Laraudogoitia / Studies in History and Philosophy of Modern Physics 39 (2008) 364–379368

Before seeing how all these might evolve, we should remember we are interested in forms ofevolution that satisfy the laws of dynamics. This means that we can use a dynamic theoryof which the usual laws of dynamics are only a part, i.e. a dynamic theory that contains(perhaps among other things) the usual laws of dynamics.1 It is natural that this should beso because we will be trying to model indeterministic supertasks and yet the binarycollisions between point particles Pi (which, as we shall see, will be involved) are alreadydynamically indeterminate in two spatial dimensions. It is precisely to undo thisindetermination, by choosing only one of the multiple alternatives open after a binarycollision of the type described, that we need to endow the underlying dynamic theory withmore structure. This we do by constructing a dynamic theory of hidden variables.Specifically, we introduce a hidden variable that can take positive entire numbers as values,such that each point particle, besides being characterized by its mass, is also characterizedby the value that the ‘‘hidden variable’’ takes in it (which we admit is constant in time). Tosimplify, let us suppose that i is the value of the hidden variable of the particle Pi. In fact,the specific values of the hidden variable are not important because its scale of values ismerely ordinal. That is sufficient to undo the indetermination in the final state after thebinary collision of two particles Pi and Pj (i6¼j) in our model, which will be done in thefollowing way:(VO) Consider an inertial reference system in which Pi and Pj (i6¼j) collide following

mutually perpendicular spatial paths (if two particles collide there is always such a system).Immediately after the collision, Pi will not abandon the straight-line path along which itwas moving immediately before the collision if the value of its hidden variable is greater

1Obviously, in the context of our objective this strategy will only be of value if in the amplified dynamical theory

we do not permit new forms of interaction that are not present in the initial dynamical theory. Otherwise, the

possibility of indeterministic supertasks in which energy is conserved in the amplified dynamics may not tell us

anything of interest about the initial dynamics.


than that of Pj. If Pi does not abandon its straight-line path then we shall say that itdisperses Pj and also that Pj is dispersed by Pi.

Although (VO) still leaves as indeterminate what happens after the binary collision oftwo point particles with the same hidden variable value, we do not need to concernourselves with this because it is irrelevant for our model: we have simply undone to acertain extent the indeterminism of particle dynamics. Further, (VO)’s compatibility withthis dynamics is evident.

It also needs to be noted that the idea of a dynamic theory (non-quantum, as is the onewe are dealing with) of hidden variables is neither exotic nor revolutionary. The namemight sound a little extravagant but it corresponds to a practice that is not unknown, forexample, in the study in engineering of the mechanics of rigid bodies. Consider the obliqueimpact between two rigid bodies. If the surfaces of these bodies are ‘‘smooth’’, the finalstate after the collision is dynamically determined; it is not so determined, however, if thesurfaces are ‘‘rough’’. To achieve dynamic determination, engineers introduce (in the frameof what they call Coulomb’s hypothesis) a friction coefficient between the surfaces incontact (Brach, 1991). This friction coefficient is in fact a hidden variable associated withthe pair of rigid solids that collide, because the surfaces of both bodies (even when‘‘rough’’) are taken as locally flat, so that the origin of the coefficient cannot be explainedby means of any surface structure: locally flat rigid solids all have the same surfacestructure. So, if engineers find it useful to use dynamic theories of hidden variables fortechnical purposes, why should not they be useful for philosophers of science forconceptual purposes?

4. Two indeterministic supertasks

(VO) implies that the binary collision between two of our particles, Pi and Pj, may bedealt with as regards energy and momentum exchange as a one-dimensional problembecause the velocity component of the dispersed particle perpendicular to the path of thedisperser particle does not change as a result of the collision. From now on, andthroughout the rest of this section, I shall be using as dynamic laws the laws of Newtoniandynamics. The reason is that they are simpler, while our significant results may be easilygeneralized to relativistic dynamics. One fact of Newtonian dynamics is essential to ouranalysis:

(I) After the one-dimensional elastic binary collision between a point particle ofmass m at rest and another point particle of mass m/2 moving at velocity v ¼ 1, the moremassive particle acquires velocity n00 ¼ 2

3 while the less massive particle stays at velocityn0 ¼ �1

3.

The system of particles the initial state of which was described in the previous sectionis capable of executing a denumerable infinity of different supertasks (all, of course,starting from the same initial state) in which both linear momentum and (kinetic)energy are conserved. To demonstrate that, in our Newtonian dynamics amplified with(VO), indeterministic supertasks in which the energy is conserved are possible, it isenough to state just two of that denumerable infinity of different supertasks. As weshall see, each one comprises an infinite sequence (different in either case) of deter-ministic elastic binary collisions not globally independent and developed fromexactly the same initial state (which explains the ‘‘indeterministic’’ label for bothsupertasks).


SUPERTASK A

—Each particle P2n+1 collides once and only once with P2(n+1) (n ¼ 0,1,2,3,y), andthese are the only collisions that occur.That the collision between P2n+1 and P2(n+1) is kinematically possible obviously follows

from the initial condition stated in the previous section (referred to as IC from now on).That the collision between P2n+1 and P2(n+1) is dynamically possible (i.e. it is a de facto

possibility and not just ‘‘in principle’’), follows from the self-consistent nature of supertaskA. To prove this self-consistency, imagine a hypothetical situation in which the onlyparticles present are P2n+1, P2(n+1), P2m+1 and P2(m+1), with n, m 2 0; 1; 2; 3; . . . and m4n

without loss of generality. The Fig. 2 represents extensions of the initial paths along whicheach one began moving: the extension of the initial path of Pi we denote as tPi.From the construction we know that tP2(n+1) and tP2(m+1) are straight lines forming an

angle (p/2)+(p/4) with the X-axis positive. Therefore the distance between them will havethe value

A ¼ ðXo2ðnþ1Þ � Xo

2ðmþ1ÞÞcosðp=4Þ ¼ ð1=p2ÞðX o

2ðnþ1Þ � Xo2ðmþ1ÞÞ. (3)

But we said before that

X o2n ¼ 1� ð1=3Þð4n�1 � 1Þ=4n�1, (2)

from which we have

A ¼ ð1=p2Þf½1� ð1=3Þð4n � 1Þ=4n� � ½1� ð1=3Þð4m � 1Þ=4m�g

¼ ð1=3p2Þððð4m � 1Þ=4mÞ � ðð4n � 1Þ=4nÞÞ. (4)

Analogously we know from the construction that tP2n+1 and tP2m+1 are straight lines thatform an angle p/4 with the X-axis positive. Therefore the distance between them will have

B

A

tp2m+1 tp2(m+1)

tp2(n+1)tp2n+1

Fig. 2. A subsystem of four particles to analyze supertask A.


the value

B ¼ ðXo2mþ1 � X o

2nþ1ÞCosðp=4Þ ¼ ð1=p2ÞðXo

2mþ1 � Xo2nþ1Þ. (5)


Xo2nþ1 ¼ ð2=3Þð4

n � 1Þ=4n, (1)

from which we have

B ¼ ð1=p2Þf½ð2=3Þð4m � 1Þ=4m� � ½ð2=3Þð4n � 1Þ=4n�g

¼ ðp2=3Þððð4m � 1Þ=4mÞ � ðð4n � 1Þ=4nÞÞ, (6)

which means that B ¼ 2A is satisfied. Likewise tP2n+1 and tP2(n+1) are mutuallyperpendicular, as are tP2m+1 and tP2(m+1), and the first collision that takes place is thedispersal of P2m+1 by P2(m+1) (remember (VO)). Let us see now that there will only be onemore collision, namely, the subsequent dispersal of P2n+1 by P2(n+1). This follows fromthree easy-to-establish facts:

(a)
After dispersing P2m+1, P2(m+1) does not collide with either P2n+1or P2(n+1). The firstfollows because, after colliding with P2m+1, the velocity of P2(m+1) changes sign(although not direction, acquiring under (I) the value �1
3) so that it begins to movefurther and further away from tP2n+1. The second follows because, after colliding withP2m+1, the velocity of P2(m+1) remains parallel to tP2(n+1). Both things continue to betrue after the dispersal of P2n+1 by P2(n+1) , because, under (VO), P2(n+1) does notabandon the path tP2(n+1) and, under (I), P2n+1 acquires a positive component of thevelocity parallel to tP2(n+1) , of value 2

3, which helps send it away even faster from

P2(m+1) . For this argument to be correct, it presupposes that P2m+1 shall never collideeither with P2(n+1) or with P2n+1. This is established in (b) and (c).

(b)
After being dispersed by P2(m+1), P2m+1 does not collide with P2(n+1). Indeed, ifP2(m+1) did not exist then P2m+1 would cross the path tP2(n+1) at a certain moment t*(in which, in fact, it would coincide with P2(n+1) and collide with it). The existence ofP2(m+1) does not change the value t* of the moment of crossing but as, after beingdispersed by it, P2m+1 acquires a positive component of the velocity parallel to tP2(n+1)
(of value 23, which follows from (I)), it is clear that P2m+1 will cross the path tP2(n+1) in

t* at a point at which it will not coincide (and therefore will not collide) with P2(n+1).For this argument to be correct, it presupposes that P2m+1 will never collide withP2n+1. This is established in (c).

(c)
After being dispersed by P2(m+1), P2m+1 does not collide with P2n+1. Indeed, from (I)it follows that, after being dispersed by P2(m+1), P2m+1 maintains its positivecomponent of the velocity parallel to tP2m+1 and acquires a positive component of thevelocity parallel to tP2(n+1) (although this latter component is smaller than theprevious one, its value is 2
3). Given that in the rectangle in the figure above B ¼ 2A it

follows that P2m+1 crosses the path tP2(n+1) before crossing tP2n+1, which means thatP2m+1 cannot collide with P2n+1 before the latter collides with P2(n+1). After P2n+1 hasbeen dispersed by P2(n+1) it acquires a velocity parallel to the one acquired P2m+1 afterbeing dispersed by P2(m+1) (a consequence of (I)) although along a different straightline (an obvious consequence of the geometry of the figure above) which means it willstill be impossible for P2m+1 to collide with P2n+1.


From (a)–(c) it does indeed follow that with the four particles P2n+1, P2(n+1), P2m+1 andP2(m+1) alone there will only be two collisions: the dispersal of P2m+1 by P2(m+1) and the
dispersal of P2n+1 by P2(n+1). If we now consider an arbitrary non-negative integer a andtake all the possible sets of four particles P2a+1, P2(a+1), P2m+1, P2(m+1), with m a non-negative integer different from a, it so happens that in any of those sets the only collisionP2a+1 and P2(a+1) undergo is the one that takes place between the two of them. This meansthat, when the infinite particles are present, it is a self-consistent possibility that the onlycollision P2a+1 and P2(a+1) undergo is the one that takes place between them. As a may beany non-negative integer we may deduce that supertask A, as described above, is self-consistent.The fact that energy (and linear momentum) is conserved in supertask A may be proved
numerically, although it is not necessary to do so. The conservation follows directly fromthe fact that each particle present undergoes precisely one collision, in which the energy(and linear momentum) is conserved. Indeed, P2n+1 collides with, and only with, P2(n+1)

(n ¼ 0,1,2,3,y), so we mentally divide the system of infinite particles P1, P2, P3,y into an

infinity of exhaustive and exclusive binary subsystems (i.e. they consist of only twoparticles): subsystem S1 (with P1 and P2), subsystem S3 (with P3 and P4),y in general thesubsystem S2n+1 (with P2n+1 and P2(n+1)). In supertask A these subsystems do notmutually exchange either energy or momentum, remaining isolated from each other. Theenergy and the momentum will therefore be conserved in supertask A if they are conservedin each subsystem. But this is obviously true because the only dynamic process that takesplace in subsystem S2n+1 is the collision between P2n+1 and P2(n+1), in which both energyand linear momentum are conserved.2 Note the contrast with situations like the classic

2One might imagine, for the purpose of calculating energies and linear momenta, that the different pairs P2n+1,

P2(n+1) of particles that collide with each other move in different, but parallel planes. In any case, for a strict

demonstration of the conservation of kinetic energy and linear momentum in supertask A, an additional condition

still has to be fulfilled: both the initial energy and momentum must be finite. If they were infinite, then, by the

reasoning conducted in the main text, the final energy and momentum should also be infinite and then to talk of

conservation in a global sense (i.e. for supertask A) would be pointless. Since particle Pn has mass 1/2n

(n ¼ 1,2,3,y) and moves initially at unit velocity forming an angle p/4 with the positive direction of X-axis, for

odd n, and with the negative direction of X-axis, for even n, it is clear that the horizontal component of the initial

total linear momentum is

ð1=2Þð1=p2Þ � ð1=4Þð1=

p2Þ þ ð1=8Þð1=

p2Þ � ð1=16Þð1=

p2Þ þ � � � ¼ ð1=3Þð1=

p2Þ.

Analogously, the vertical component of the initial total linear momentum is

ð1=2Þð1=p2Þ þ ð1=4Þð1=

p2Þ þ ð1=8Þð1=

p2Þ þ ð1=16Þð1=

p2Þ þ � � � ¼ ð1=

p2Þ.

The initial total linear momentum is therefore finite. Likewise, the initial total kinetic energy is finite and has the

value, as is immediately clear, of

ð1=2Þð1=2Þ þ ð1=2Þð1=4Þ þ ð1=2Þð1=8Þ þ � � � ¼ ð1=2Þ.

Incidentally, these calculations will be used to corroborate numerically the conservation of total kinetic energy

and linear momentum in the case of supertask A. From (VO) and (I) it follows that, after the supertask has been

performed, the horizontal component of the velocity of Pn is (1/O2)–(2/3O2) for odd n and (1/3O2) for even n.


example in ‘‘A Beautiful Supertask’’ (Perez Laraudogoitia, 1996), where exclusive andexhaustive decomposition into non-mutually interacting subsystems in which energy andmomentum are conserved is not possible. To assure that supertask A is an indeterministicsupertask in accordance with the definition we gave in Introduction; all that remains now isto check that the elastic binary and deterministic collisions of which it consists are notglobally independent. This is easier to see than might be thought because it followsimmediately from the self-consistency of another indeterministic supertask (in which theenergy is also conserved) possible from the same initial condition IC, and which I shall call

SUPERTASK B

—Each particle P2n+1 collides once and only once with P2n (n ¼ 1,2,3,y), and these arethe only collisions that occur.

We shall prove the self-consistency of supertask B in a way similar to the one used insupertask A. Imagine once again a hypothetical situation in which the only particlespresent are P2n+1, P2n, P2m+1 and P2m, with n, m 2 1; 2; 3; . . . and m4n. The Fig. 3represents the extensions of the initial paths along which each one started moving.

From the construction we know that tP2n and tP2m are straight lines that form an angle (p/2)+(p/4) with the X-axis positive. Therefore the distance between them will have the value

A ¼ ðX o2n � X o

2mÞCosðp=4Þ ¼ ð1=p2ÞðXo

2n � X o2mÞ. (7)


Xo2n ¼ 1� ð1=3Þð4n�1 � 1Þ=4n�1, (2)

from which we have

A ¼ ð1=p2Þf½1� ð1=3Þð4n�1 � 1Þ=4n�1� � ½1� ð1=3Þð4m�1 � 1Þ=4m�1�g

¼ ð1=3p2Þððð4m�1 � 1Þ=4m�1Þ � ðð4n�1 � 1Þ=4n�1ÞÞ ¼ ð2=3Þð

p2Þðð1=4nÞ � ð1=4mÞÞ.

(8)

(footnote continued)

The horizontal component of the final total linear momentum is therefore

ð1=2Þðð1=p2Þ � ð2=3

p2ÞÞ þ ð1=4Þð1=3

p2Þ þ ð1=8Þðð1=

p2Þ � ð2=3

p2ÞÞ

þ ð1=16Þð1=3p2Þ þ � � � ¼ ð1=3

p2Þ,

which coincides with the horizontal component of the initial total linear momentum. Analogously, the vertical

component of the velocity of Pn after the performance of supertask A is (1/O2)+(2/3O2) for odd n and (�1/3)(1/

O2) for even n. The vertical component of the final total linear momentum is therefore

ð1=2Þðð1=p2Þ þ ð2=3

p2ÞÞ � ð1=4Þð1=3

p2Þ þ ð1=8Þðð1=

p2Þ

þ ð2=3p2ÞÞ � ð1=16Þð1=3

p2Þ þ � � � ¼ ð1=

p2Þ,

which coincides with the vertical component of the initial total linear momentum. Finally, the square of the

velocity of Pn after the performance of supertask A is 12+(2/3)2 ¼ 1+(4/9) for odd n and (1/3)2 ¼ (1/9) for even

n. So the final total kinetic energy is

ð1=2Þð1=2Þð1þ ð4=9ÞÞ þ ð1=2Þð1=4Þð1=9Þ þ ð1=2Þð1=8Þð1þ ð4=9ÞÞ

þ ð1=2Þð1=16Þð1=9Þ þ � � � ¼ ð1=2Þ,

which coincides with the initial total kinetic energy.

Of course, the general argument for conservation given in the main text of the paper makes all these numerical

corroborations strictly unnecessary.

ARTICLE IN PRESS

B

AA

tp2m+1 tp2m

tp2n

tp2n+1

Fig. 3. A subsystem of four particles to analyze supertask B.

J. Perez Laraudogoitia / Studies in History and Philosophy of Modern Physics 39 (2008) 364–379374

Analogously we know from the construction tP2n+1 and tP2m+1 are straight lines thatform an angle p/4 with the X-axis positive and we saw on dealing with supertask A that thedistance between them is

B ¼ ðp2=3Þððð4m � 1Þ=4mÞ � ðð4n � 1Þ=4nÞÞ ¼ ð

p2=3Þðð1=4nÞ � ð1=4mÞÞ, (9)

which means that A ¼ 2B is satisfied. Likewise tP2n+1 and tP2n are mutually perpendi-cular, as are tP2m+1 and tP2m, and the first collision that takes place is the dispersal of P2m

by P2m+1. Let us see now that there will only be one more collision, namely, thesubsequent dispersal of P2n by P2n+1. This follows from three easy-to-establish facts:

(a*)
After dispersing P2m, P2m+1 does not collide with either P2n+1or P2n. The secondfollows because, after colliding with P2m, the velocity of P2m+1 changes sign(although not direction, acquiring under (I) the value �1
3) so that it begins to move

further and further away from tP2n. The first follows because, after colliding with P2m,the velocity of P2m+1 remains parallel to tP2n+1. Both things continue to be true afterthe dispersal of P2n by P2n+1, because, under (VO), P2n+1 does not abandon the pathtP2n+1 and, under (I), P2n acquires a positive component of the velocity parallel totP2n+1, of value 2

3, which helps send it away even faster from P2m+1. For this

argument to be correct, it presupposes that P2m shall never collide either with P2n+1

or with P2n. This is established in (b*) and (c*).
(b*) After being dispersed by P2m+1, P2m does not collide with P2n+1. Indeed, if P2m+1 did
not exist then P2m would cross the path of tP2n+1 at a certain moment t* (in which, infact, it would coincide with P2n+1 and collide with it). The existence of P2m+1 doesnot change the value t* of the moment of crossing but as, after being dispersed by it,P2m acquires a positive component of the velocity parallel to tP2n+1 (of value

23, which

follows from (I)), it is clear that P2m will cross the path tP2n+1 in t* at a point at which

ARTICLE IN PRESS

3No

supert

2n+1

J. Perez Laraudogoitia / Studies in History and Philosophy of Modern Physics 39 (2008) 364–379 375

it will not coincide (and therefore will not collide) with P2n+1. For this argument to becorrect, it presupposes that P2m will never collide with P2n. This is established in (c*).

(c*)
After being dispersed by P2m+1, P2m does not collide with P2n. Indeed, from (I) itfollows that, after being dispersed by P2m+1, P2m maintains its positive component ofthe velocity parallel to tP2m and acquires a positive component of the velocity parallelto tP2n+1 (although this latter component is smaller than the previous one, its value is23). Given that in the rectangle in the figure above is A ¼ 2B it follows that P2m crossesthe path tP2n+1 before crossing tP2n , which means that P2m cannot collide with P2n
before the latter collides with P2n+1. After P2n has been dispersed by P2n+1 it acquiresa velocity parallel to the one acquired by P2m after being dispersed by P2m+1 (aconsequence of (I)) although along a different straight line (a consequence of thegeometry of the figure above) which means it will still be impossible for P2m to collidewith P2n.

From (a*), (b*) and (c*) it does indeed follow that with the four particles P2n+1, P2n,P2m+1 and P2m alone there will only be two collisions: the dispersal of P2m by P2m+1 andthe dispersal of P2n by P2n+1.

3 If we now consider an arbitrary positive integer a and takeall the possible sets of five particles P1, P2a, P2a+1, P2m, P2m+1, with m a positive integerdifferent from a, it so happens that in any of these sets the only collision P2a and P2a+1

undergo is the one that takes place between the two of them and P1 does not undergo anycollision at all (the argument for this last being identical to the one in (a*), (b*), (c*) abovewhich showed that P2n+1 does not collide with P2m+1 or P2m, one may take n ¼ 0 with thereservation that then particle P2n does not exist). This means that, when the infiniteparticles are present, it is a self-consistent possibility that the only collision P2a and P2a+1

undergo is the one that takes place between them, while P1 does not undergo any collisionat all. As a may be any positive integer we may deduce that supertask B, as describedabove, is self-consistent.

Now it is clear that the binary collisions of which supertask A consists are not globallyindependent (and that, therefore, A is an indeterministic supertask). Indeed, although, forexample, P1 is causally connected only with P2 in supertask A, its evolution is not uniquegiven the self-consistency of supertask B. By symmetrical reasoning, and given that it hasalready been shown that supertask A is self-consistent, it follows that the binary collisionsof which supertask B consists are not globally independent either and, consequently, thatsupertask B is an indeterministic supertask.

Finally, the explanation of why energy (and linear momentum) is conserved in supertaskB is essentially identical to the one given in the case of supertask A. With this we havedemonstrated that, in our Newtonian dynamics amplified with (VO), indeterministicsupertasks in which the energy is conserved are possible.

At this point it might be worth considering a point of view often found amongst manycritics of the notion of supertask, although I have never actually seen it defended in theliterature. Given that from the initial condition IC different forms of evolution are possible,might this indeterminism not be a consequence of IC not in fact being well defined? If thatwere the case, supposed indeterministic supertasks like the one in the present paper would

te that the argument to establish this is identical to the analogous argument presented with regard to

ask A when there the subscripts 2(m+1), 2m+1, 2(n+1) and 2n+1 are uniformly replaced by 2m+1, 2m,

and 2n, respectively.


not enable us to say anything of interest about dynamics, and should in fact be excluded asnot well defined. My main criticism of this approach is that it is arbitrary, for two reasons. Inthe first place, because, with no other justification, it considers that the indeterminism thatarises in systems with infinite particles is a sign that the problem is not well defined while theindeterminism that emerges in systems with a finite number of particles is accepted as a‘‘real’’ limitation on the predictive capability of dynamics. In the second place, because manyinitial conditions for indeterministic supertasks may be altered trivially to avoidindeterminism and even the actual performance of a supertask. For example, in IC thevalue of the velocity of any of the particles Pn is Vn ¼ 1. If we define the new IC* as identicalto IC except that for even n it is Vn ¼ 1+e41 then the evolution from IC* does not generateindeterminism or even a supertask: after IC* each particle continues moving eternally at itsconstant velocity. Any of the critics referred to above will then have to say either that IC* iswell defined although IC is not, a move that could be classified as ‘‘ad hoc by default’’, orthat neither IC* nor IC are well defined, a move that is ‘‘ad hoc by excess’’ as IC* creates nodifficulties of any kind in dynamics.

5. The relativistic point of view

Now we shall see that the arguments contained in the previous section remain validwhen Newtonian dynamics is replaced by relativistic dynamics. It will then be clear thatindeterministic supertasks in which the energy is conserved are possible in relativisticdynamics amplified with (VO).(I) was essential in our prior demonstration that, in Newtonian dynamics amplified with

(VO), indeterministic supertasks in which the energy is conserved are possible. Itsrelativistic analog, which we shall call (IR), differs from it only in the numerical values of v0

(which now is not �13) and v00 (which will not now be 2

3). But the specific numerical values

n0 ¼ �13and n00 ¼ 2

3were not necessary in the previous section. There, of v0 we use only (in

(a) and (a*)) the fact that it is negative, of v00 we use only (in (b) and (b*)) the fact that it ispositive and (in (c) and (c*)) the fact that it is positive but smaller than 2 (although I didnot explicitly mention this last part). What is essential therefore are the bounds v0o0 and0ov00o2; and these bounds are also valid in (IR)! 0ov00o2 is necessary because in the one-dimensional binary collision of (IR) the momentum would not be conserved if v00p0 andthe energy would not be conserved if v00X2. v0o0 is necessary because, in the relativisticone-dimensional collision of an incident particle against a target particle at rest of greatermass (at rest), the incident particle always retreats.4 So in our relativistic dynamicsamplified with (VO), indeterministic supertasks in which energy is conserved are possible.

4The equations of the conservation of energy and momentum applied to the collision between particle 1 and

particle 2 (the target) give (taking the velocity of light as unit)

m1gðv1Þ þm2 ¼ m1gðv01Þ þm2gðv002Þ; m1v1gðv1Þ ¼ m1v01gðv01Þ þm2v002gðv

002Þ,

where g (v) ¼ (1�v2)�1/2.

A direct calculation enables us to verify that (see Atkinson, 2007), taking g1 ¼ m2/m1 and with e(v) ¼ O((1�v)/

(1+v)),

�ðv01Þ ¼ ðg1 þ �ðv1ÞÞ=ð1þ g1�ðv1ÞÞ.

As particle 1 is the incident we take 14v140 and thus 0oe (v1)o1. As the mass of particle 1 is smaller than the

mass of particle 2 we have g141 and then e (v1)o1 implies e(v1)(g1�1)o(g1�1), i.e. (g1+e(v1))/(1+g1e(v1))) ¼ e(v10)41. But then we must have v1

0o0, i.e. the incident particle retreats.


6. Conclusion

We have demonstrated that, as stated in section two of the paper, there areindeterministic supertasks compatible with the conservation of energy, as they arepossible when the laws of dynamics (whether Newtonian or relativistic) are implementedwith (VO) (and, of course, the result of that implementation is an internally consistentdynamical theory because it is only occupied with undoing some indeterminations leftby the initial dynamical laws). In other words, there is no intrinsic connection betweenindeterministic supertasks and the non-conservation of energy. This annuls any possibilityof excluding indeterministic supertasks as ‘‘non-physical’’ by resorting to the conserva-tion of energy,5 because in the A and B supertasks introduced in this paper we maysubstitute trivially the role of the hidden variables by mere external restraint conditionswhich constrain the possible movements of the particles after each collision (withoutinteracting with them, of course, as in each collision the energy and momentum areconserved, so that such indeterministic supertasks consist exclusively in elastic collisions,as the definition given in Introduction demands). Another original feature of the presentpaper is the path taken to prove the non-existence of this kind of intrinsic connection;conceptually speaking, it is very different from the one that facilitates proof of thehomologous result for deterministic supertasks. With deterministic supertasks one simplyprovides (without hidden variables) an example of a deterministic supertask withconservation of energy in Newtonian dynamics (see Section 2 of Perez Laraudogoitia2007b) and another example in relativistic dynamics (see Section 4 of Perez Laraudogoitia2007b). Such a direct path to proof does not seem easy (if it is possible, after all)with regard to indeterministic supertasks and, besides being more elegant, resorting tohidden variables may also be necessary: in particular, one may use them to deal withthe Newtonian case and the relativistic case by means of a single identical example forboth.6 It is important at this point to pinpoint the role of the hidden variables in this paperto avoid possible misunderstandings. To this end, we shall begin by noting that theexistence of an intrinsic connection in dynamics (whether Newtonian or relativistic)between indeterministic supertasks and the non-conservation of energy may be expressedsymbolically thus:

Dynamics ‘ Int, (10)

where

Int: for any process x, that x is an indeterministic supertask implies that in x the energy isnot conserved.

From (10)

Dynamics ‘Mat, (11)

5The impossibility of excluding deterministic supertasks by this path was mentioned in Introduction.6At first sight it would seem possible to avoid having to resort to hidden variables if, instead of point particles,

we were to use rigid, ‘‘smooth’’ bodies in our example, as their binary collisions are dynamically determined. But

even if this worked for Newtonian dynamics, it would not serve for relativistic dynamics, because in relativistic

dynamics rigid bodies are not possible. A sufficiently general proof therefore seems to need the use of hidden

variables.


where

Mat: for any process x, if x is an indeterministic supertask, then in x the energy is notconserved (the term ‘‘Mat’’ suggests that a material conditional is involved, althoughuniversally quantified).The most difficult part of my argument consisted in establishing, in the previous sections

of the paper, that

Dynamicsþ ðVOÞ ‘ there exists at least one process x such that x is an

indeterministic supertask and in x the energy is conserved: (12)

In other words, that

Dynamicsþ ðVOÞ ‘ no Mat: (13)

Then, from (11) and (13)

Dynamicsþ ðVOÞ ‘Mat and no Mat: (14)

As Dynamics+(VO) is a consistent theory

NOTfDynamicsþ ðVOÞ ‘Mat and no Matg. (15)

Therefore, by a Reductio ad Absurdum from (10) to (14), (15)

NOTfDynamics ‘ Intg. (16)

And Eq. (16) expresses this paper’s ultimate target conclusion: in dynamics there is nointrinsic connection between indeterministic supertasks and the non-conservation ofenergy. Note that the role of the hidden variables (i.e. of (VO)) in my argument is merelyformal and in this sense very different from the role they have in quantum mechanics (or inthe mechanics of rigid bodies with friction mentioned above). For example, rather thanjust a formal recourse, quantum mechanic hidden variables aim to say something aboutreality, which means that the hypotheses that introduce them always run the risk of beingcriticized as arbitrary and/or suspect, when not dismissed as false. In contrast, (VO) neitherhas, nor needs, such pretensions, making its arbitrariness and possible empirical falsityirrelevant.Finally, I would also like to take advantage of this paper to propose the method of

hidden variables as an appropriate means of demonstrating general results of compatibilityin classical dynamics and relativistic dynamics. Here I have given an example of their useto respond to certain issues concerning supertasks, as from (16) it follows that not Int isconsistent (compatible) with dynamics. One anonymous referee has complained that theliterature on supertasks depends too much on the introduction of particular examples tothe detriment of more powerful general theorems. I think the method of hidden variables,as a general method for proof of compatibility as I have already said, may help to invertthis trend. In any case, one should not lose sight of the fact that dynamic supertaskstypically take place in systems of infinite particles, where the possibilities of evolution (and,in particular, of pathological evolution) are much greater than in finite systems. To developa suitable intuition on infinite systems one must therefore become familiar with manyparticular examples. And this sort of good intuition is essential if, as mathematiciansprescribe, one has to ‘‘see’’ first that a general theorem is correct before trying todemonstrate it.


References

Alper, J. S., Bridger, M., Earman, J., & Norton, J. D. (2000). What is a Newtonian system? The failure of energy

conservation and determinism in supertasks. Synthese, 124, 281–293.

Atkinson, D. (2007). Losing energy in classical, relativistic and quantum mechanics. Studies in History and

Philosophy of Modern Physics, 38, 170–180.

Brach, R. M. (1991). Mechanical impact dynamics. Rigid body collisions. New York: Wiley.

Perez Laraudogoitia, J. (1996). A beautiful supertask. Mind, 105, 81–83.

Perez Laraudogoitia, J. (2007a). Avoiding infinite masses. Synthese, 156, 21–31.

Perez Laraudogoitia, J. (2007b). Supertasks, dynamical attractors and indeterminism. Studies in History and

Philosophy of Modern Physics, 38, 724–731.

Perez Laraudogoitia, J., Bridger, M., & Alper, J. S. (2002). Two ways of looking at a Newtonian supertask.

Synthese, 131, 173–189.

energy conservation and supertasks

Documents