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A simple and interesting classical mechanical supertask

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Abstract

This paper presents three interesting consequences that follow from admitting an ontology of rigid bodies in classical mechanics. First, it shows (in Sects. 4 and 5) that some of the most characteristic properties of supertasks based on binary collisions between particles, such as the possibility of indeterminism or the non-conservation of energy, persist in the presence of gravitational interaction. This makes them gravitational supertasks radically different from those that have appeared in the literature to date. Second, Sect. 6 proves that the role of gravitation in supertasks of this kind may be highly non-trivial. Third, (in Sects. 7 and 8), the gravitational supertasks found in the first part enable us to show that indeterminism of classical mechanics with gravitation is much more general than has been supposed until now. This result is especially interesting from the philosophical viewpoint, as it links the scope of indeterminism in a theory like classical mechanics directly with the nature of its ontological hypotheses.

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Notes

  1. I do not take account of “bifurcated supertasks” here, as they are not discussed in this paper.

  2. Indeed, note that, successively:

    $$\begin{aligned}&\hbox {v}_{2}^\prime = [(1 - {\upmu }_{2})/(1 + {\upmu }_{2})]\hbox {v}_{2}^{\prime \prime } = [(1 - {\upmu }_{2})/(1 + {\upmu }_{2})] [2/({\upmu }_{1} + 1)]\hbox {v}_{1}\\&\hbox {v}_{3}^{\prime \prime }= [2/({\upmu }_{2} + 1)]\hbox {v}_{2}^{\prime \prime } = [2/({\upmu }_{2} + 1)] [2/({\upmu }_{1} + 1)]\hbox {v}_{1}\\&\hbox {v}_{3}^\prime = [(1 - {\upmu }_{3})/(1 + {\upmu }_{3})]\hbox {v}_{3}^{\prime \prime } = [(1 - {\upmu }_{3})/(1 + {\upmu }_{3})][2/({\upmu }_{2} + 1)] [2/({\upmu }_{1} + 1)]\hbox {v}_{1}\\&\hbox {v}_{4}^{\prime \prime } = [2/({\upmu }_{3} + 1)]\hbox {v}_{3}^{\prime \prime } = [2/({\upmu }_{3} + 1)] [2/({\upmu }_{2} + 1)] [2/({\upmu }_{1}+ 1)]\hbox {v}_{1}\\ \end{aligned}$$

    .....................................

  3. Indeed, if we want the relation between the masses of the particles to be such that, except for \(\hbox {p}_{1}\), all of them will undergo two collisions (\(\hbox {p}_{1}\) only one) then it is sufficient that, after their last collision, all the particles move at the same velocity: \(\hbox {v}_{\mathrm{n}}^\prime = \hbox {v}_{\mathrm{n+1}}^\prime \) (what Atkinson and Johnson (2009) call the “constant recoil” case). Then

    $$\begin{aligned}{}[(1- {\upmu }_{\mathrm{n+1}})/(1+{\upmu }_{\mathrm{n+1}})][2/(1 + {\upmu }_{\mathrm{n}})]= [(1- {\upmu }_{\mathrm{n}})/(1+{\upmu }_{\mathrm{n}})] \end{aligned}$$

    so

    $$\begin{aligned} {\upmu }_{\mathrm{n+1}}(3 - {\upmu }_{\mathrm{n}}) = 1+{\upmu }_{\mathrm{n}}. \end{aligned}$$

  4. Indeed, when \(\hbox {p}_{\mathrm{n}}\) collides with \(\hbox {p}_{\mathrm{n+1}}\) it acquires a velocity \(\hbox {v}_{\mathrm{n}}^\prime = [(1 - {\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1}\) and it transmits to \(\hbox {p}_{\mathrm{n+1}}\) a velocity \(\hbox {v}_{\mathrm{n+1}}^{\prime \prime } = [2/({\upmu }_{\mathrm{n}} + 1)][2/({\upmu }_{\mathrm{n-1}} + 1)]..[2/({\upmu }_{2} + 1)] [2/({\upmu }_{1} + 1)]\hbox {v}_{1}\). At the end of a time \(\hbox {d}_{\mathrm{n+1}}/\hbox {v}_{\mathrm{n+1}}\)” after the collision of \(\hbox {p}_{\mathrm{n}}\) with \(\hbox {p}_{\mathrm{n+1}}\), \(\hbox {p}_{\mathrm{n+1}}\) collides with \(\hbox {p}_{\mathrm{n}+2}\) having covered the distance \(\hbox {d}_{\mathrm{n+1}}\), while in that time \(\hbox {p}_{\mathrm{n}}\) has advanced \([\hbox {d}_{\mathrm{n+1}}/\hbox {v}_{\mathrm{n+1}}^{\prime \prime }][(1 - {\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1}\). Therefore the final distance between \(\hbox {p}_{\mathrm{n}}\) and \(\hbox {p}_{\mathrm{n+1}}\) for \(\hbox {n} \ge 1\) (I call it final because hereafter it remains constant, owing to the fact that \(\hbox {p}_{\mathrm{n+1}}\) will go on to move at the same velocity as \(\hbox {p}_{\mathrm{n}}\), namely, \([(1 -{\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1})\) is

    $$\begin{aligned} \hbox {d}_{\mathrm{n}}^{*}= & {} \hbox {d}_{\mathrm{n+1}} - [\hbox {d}_{\mathrm{n+1}}/\hbox {v}_{\mathrm{n+1}}^{\prime \prime }][(1 - {\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1} \\= & {} \hbox {d}_{\mathrm{n+1}}(1 - [(1 - {\upmu }_{1})/(1 + {\upmu }_{1})][\hbox {v}_{1}/\hbox {v}_{\mathrm{n+1}}^{\prime \prime }]) \\= & {} \hbox {d}_{\mathrm{n+1}}(1 - [(1 - {\upmu }_{1})/(1 + {\upmu }_{1})][({\upmu }_{\mathrm{n}} + 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][({\upmu }_{1} + 1)/2]) \\= & {} \hbox {d}_{\mathrm{n+1}}(1 - [({\upmu }_{\mathrm{n}} + 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2]). \end{aligned}$$

  5. This number is clearly positive because \(\hbox {v}_{2}^{\prime \prime } - [(1 - {\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1}\) was and \(\hbox {v}_{\mathrm{n}}^{\prime \prime } - [(1 - {\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1} >\hbox {v}_{2}^{\prime \prime } - [(1 - {\upmu }_{1})/(1 + {\upmu }_{1})]\hbox {v}_{1}\), as \(\hbox {v}_{\mathrm{n}}^{\prime \prime } > \hbox {v}_{2}^{\prime \prime }\).

  6. Note however that, although by construction \(\hbox {d}_{\mathrm{n+1}} <\hbox {d}_{\mathrm{n}}\), it does not follow from here that \(\hbox {d}_{\mathrm{n+1}}^{*} <\hbox {d}_{\mathrm{n}}^{*}\). As \(\hbox {d}_{\mathrm{n}}^{*}= \hbox {d}_{\mathrm{n+1}}(1 - [({\upmu }_{\mathrm{n}}+ 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2])\) and \(\hbox {d}_{\mathrm{n+1}}^{*}= \hbox {d}_{\mathrm{n}+2}(1 - [({\upmu }_{\mathrm{n+1}} + 1)/2][({\upmu }_{\mathrm{n}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2])\), by being

    $$\begin{aligned}&[({\upmu }_{\mathrm{n+1}} + 1)/2][({\upmu }_{\mathrm{n}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2]) \\&\quad <[({\upmu }_{\mathrm{n}} + 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2] \end{aligned}$$

    is

    $$\begin{aligned}&(1 - [({\upmu }_{\mathrm{n+1}} + 1)/2][({\upmu }_{\mathrm{n}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2]) \\&\quad > (1 - [({\upmu }_{\mathrm{n}} + 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2]) \end{aligned}$$

    If we want \(\hbox {d}_{\mathrm{n+1}}^{*}< \hbox {d}_{\mathrm{n}}^{*}\) then

    $$\begin{aligned}&[\hbox {d}_{\mathrm{n}+2}/ \hbox {d}_{\mathrm{n+1}}] \\&\quad < [(1 - [({\upmu }_{\mathrm{n}} + 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2]\\&\qquad [(1 - {\upmu }_{1})/2])]/[ (1 - [({\upmu }_{\mathrm{n+1}} + 1)/2][({\upmu }_{\mathrm{n}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2])] \end{aligned}$$

    must be fulfilled, which is a more demanding condition than \((\hbox {d}_{\mathrm{n}+2}/\hbox {d}_{\mathrm{n+1}}) < 1\) because

    $$\begin{aligned}&[(1 - [({\upmu }_{\mathrm{n}} + 1)/2][({\upmu }_{\mathrm{n-1}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2])]\\&\qquad /[ (1 - [({\upmu }_{\mathrm{n+1}} + 1)/2][({\upmu }_{\mathrm{n}} + 1)/2]{\ldots } [({\upmu }_{2} + 1)/2][(1 - {\upmu }_{1})/2])] < 1 \end{aligned}$$

    With just this additional requirement we may say that \(\hbox {d}_{1}^{*} > \hbox {d}_{2}^{*} > \hbox {d}_{3}^{*} > \hbox {d}_{4}^{*} >\cdots > \hbox {d}_{\mathrm{n}}^{*} >\cdots \)

  7. Note that in this argument of the non-existence of singularities the total finite mass of the spheres is as essential as the fact that their radii \(\hbox {r}_{\mathrm{i}}\) are such that \(\hbox {r}_{\mathrm{i}} \ge \hbox {r}_{1} > 0\). If, on the other hand, it was the case that \(\lim _{\mathrm{i}\rightarrow \infty }\hbox {r}_{\mathrm{i}} = 0\) then (as I argue a little later in the main text of the paper) the existence of singularities may not be excluded a priori.

  8. As in point \(\wp \) the gravitational field is null but the potential V is \(-\infty \), an interesting question is posed in the case where a point particle \(\hbox {p}_{\wp }\) of mass m is precisely in \(\wp \). If one considers the gravitational potential V as merely a mathematical tool (not a physical thing after all) then the gravitational force on the mass m is null as the field in \(\wp \) is null. This means that \(\hbox {p}_{\wp }\) could remain at rest if all the other \(\hbox {p}_{\mathrm{i}}\) are at rest. In the contrary case, i.e. if V is considered to be a physical entity, the gravitational force on \(\hbox {p}_{\wp } = - \hbox {m}\nabla \hbox {V}\) will not have a defined value. In these circumstances the analysis of its evolution becomes much more problematic.

  9. If we permit the presence of central forces different from the gravitational, supertasks are also possible. Consider the case of a central potential not of a gravitational type (\(\hbox {V} = -\upalpha /\hbox {r}\)) but rather of squared inverse (\(\hbox {V} = -\upalpha /\hbox {r}^{2}\)) and a particle approaching the centre of the field. In the gravitational case, as we know, its spatial path is a conic section but in the case of squared inverse potential its path is a spiral. Furthermore, this spiral performs an infinite number of revolutions around the centre of the field while the particle takes a finite time to cover it, because its radial velocity grows on approaching the centre (Kotkin and Serbo 1971).

  10. The explanation of this difference obviously lies in the fact that I use the mechanical notion (idealization) of the rigid body, while Mather & McGehee, Gerver and Xia consider an alternative idealization: strictly point particles.

  11. This common characteristic is not casual and there is a historical explanation for it: the four authors mentioned were interested in finding solutions with non-collision singularities (pseudocollisions) to the Newtonian equations of motion. It was already well known that a necessary condition for having a non-collision singularity is that the motion become unbounded in finite time

  12. The point character is easily understood if, as is the case, the total energy is conserved at the same time as the paths of the particles are not bounded in a finite time. This non-bounding implies that the (positive) kinetic energy of the particles tends to infinite in a finite time. Then, the conservation of energy requires that the (negative) potential gravitational energy tends to less infinite in the same interval of time, which implies that the distance between at least some of the point particles arbitrarily approaches zero.

  13. It wouldn’t be difficult to show that any finite distribution of mass with the X axis as an axis of symmetry would be valid instead of point particle Q. So the argument I am going to put forward is very general.

  14. Note that a rough (but not perfectly rough) surface for slipping is characterized by having a non-null finite value of the coefficient of friction \({\upmu }\). Let A and B have rough (but not perfectly rough) surfaces for slipping. If, while remaining in contact with A, B exerts no pressure on A, then B may slide on A at constant velocity without needing any force at all. This, which is true for all \({\upmu }\ne 0\), also characterizes perfect roughness for slipping, the difference being when B exerts a non-null force N (usually called normal reaction) on A. If the surfaces in contact are just rough for slipping, the force required for sliding is \({\upmu }\hbox {N}\), but if they are perfectly rough for slipping then sliding is impossible whatever the force. To a certain extent one may consider perfect roughness for slipping as the limit of roughness when \({\upmu }\rightarrow \infty \).

  15. In this context, the coherence of any statement on the non-conservation of total mechanic energy requires proof that total mass and the total moment of inertia of the set of hollow spheres is finite. Only then can we truly talk about total translational kinetic energy and total kinetic energy of rotation. Remember that now we are not assuming that \({\upmu }_{\mathrm{n+1}} = (1+{\upmu }_{\mathrm{n}})/(3 - {\upmu }_{\mathrm{n}})\), which means that the theorem at the end of Sect. 3 is not directly applicable. We can get round this problem by once more making use of our assumption c): \({\uplambda }_{\mathrm{n+1}} = (1+{\uplambda }_{\mathrm{n}})/(3 - {\uplambda }_{\mathrm{n}})\), with \({\uplambda }_{\mathrm{i}} = \hbox {I}_{\mathrm{i}+1}/\hbox {I}_{\mathrm{i}} < 1 ~(1\le \hbox {i})\). This enables us to apply the theorem, with the uniform substitution of \(\hbox {m}_{\mathrm{i}}\) by \(\hbox {I}_{\mathrm{i}}\) and \({\upmu }_{\mathrm{i}}\) by \({\uplambda }_{\mathrm{i}}\), to conclude that the total moment of inertia of the set of hollow spheres is indeed finite. As \(\hbox {I}_{\mathrm{i}} = \hbox {m}_{\mathrm{i}}\cdot \hbox {r}_{\mathrm{i}}^{2}\), with \(\hbox {r}_{\mathrm{i}+1} > \hbox {r}_{\mathrm{i}}\), the convergence of \(\Sigma _{\mathrm{n}=1}\hbox { I}_{\mathrm{n}}\) implies that of \(\Sigma _{\mathrm{n}=1}\hbox {m}_{\mathrm{n}}\), so that the total mass of the hollow spheres is also finite.

  16. The definition of continuity of the world line of a particle given by Laraudogoitia is standard: if particle p is at point \(\hbox {x}_{0}\) at instant \(t_{0}\), then for every neighborhood V of \(\hbox {x}_{0}\) there exists a neighborhood U of \(t_{0}\) such that the particle is at V at every instant \(t \in U\). In our case the particles are not point particles, being instead hollow spheres and therefore the requirement of continuity must refer to the world lines of any points of the spheres. However this clarification it is not relevant in the problem under consideration here, which is why I do not take it into account in the main text: with regard to questions of continuity one may consider hollow spheres in the present context as point particles.

  17. Norton’s dome is impossible if we require that the condition of Lipschitz is fulfilled (a condition whose violation is at the root, as we know, of results of indeterminism like that particular one) and it is also impossible if we impede ontologically mixed circumstances. It is not true however that the strategy of not admitting ontologically mixed circumstances leads to the fulfilment of the condition of Lipschitz. This condition may always be violated by admitting sufficiently exotic forms of interaction. But I will go so far as to suggest that, if the active forces present in a situation (e.g. the force of gravity in the case of the dome) do not violate the condition of Lipschitz, then the total or net forces (the result of composing the active forces with the passive forces, i.e. with the reactions of the constraints) will not do so either if we do not admit ontologically mixed circumstances.

  18. They would be if, for example, besides gravitation they included an ontology of elastic bodies and not of rigid bodies.

  19. A more realistic treatment, that takes account of the temperature-dependence of the material properties, is the most ambitious theory of thermomechanics.

  20. Of this type, for instance, is Mindlin’s problem, in which the state of stress due to a force P applied at an internal point Q of the elastic half-space \(\hbox {z} \ge 0\) and normal to the boundary plane z = 0 is determined. Such a force can only be due to a point particle located in Q.

References

  • Atkinson, D. (2008). A relativistic Zeno effect. Synthese, 160, 5–12.

    Article  Google Scholar 

  • Atkinson, D., & Johnson, P. (2009). Nonconservation of energy and loss of determinism I. Infinitely many colliding balls. Foundations of Physics, 39, 937–957.

    Article  Google Scholar 

  • Earman, J. (1986). A primer on determinism. Dordrecht: Reidel.

    Book  Google Scholar 

  • Earman, J. (2004). Curie’s principle and spontaneous symmetry breaking. International Studies in the Philosophy of Science, 18, 173–199.

    Article  Google Scholar 

  • Gerver, J. L. (1991). The existence of pseudocollisions in the plane. Journal of Differential Equations, 89, 1–68.

    Article  Google Scholar 

  • Kotkin, G. L., & Serbo, V. G. (1971). Collection of problems in classical mechanics. Oxford: Pergamon Press.

    Google Scholar 

  • Laraudogoitia, J. P. (1998). Infinity machines and creation ex nihilo. Synthese, 115, 259–265.

    Article  Google Scholar 

  • Laraudogoitia, J. P. (2007). Supertasks, dynamical attractors and indeterminism. Studies in History and Philosophy of Modern Physics, 38, 724–731.

    Article  Google Scholar 

  • Mather, J., & McGehee, R. (1975). Solutions of the collinear four-body problem which become unbounded in finite time. In J. Moser (Ed.), Dynamical systems theory and applications (pp. 573–589). Berlin: Springer.

    Chapter  Google Scholar 

  • Norton, J. D. (1999). A quantum mechanical supertask. Foundations of Physics, 29, 1265–1302.

    Article  Google Scholar 

  • Norton, J. D. (2008). The dome: An unexpectedly simple failure of determinism. Proceedings of the 2006 Biennial meeting of the philosophy of science association. Philosophy of science (75): 786–798.

  • Papastavridis, J. G. (2002). Analytical mechanics. Oxford: Oxford University Press.

    Google Scholar 

  • Romero, G. E. (2014). The collapse of supertasks. Foundations of Science, 19, 209–216.

    Article  Google Scholar 

  • Wilson, M. (2013). What is “Classical Mechanics” anyway. In R. Batterman (Ed.), The Oxford handbook of philosophy of physics (pp. 43–106). Oxford: Oxford University Press.

    Google Scholar 

  • Xia, Z. (1992). The existence of non-collision singularities in newtonian systems. The Annals of Mathematics, 135, 411–468.

    Article  Google Scholar 

Download references

Acknowledgments

Research for this paper was partially supported by the projects FFI2012-33550 of the Minister of Economy and Competitiveness (Government of Spain), and IT644-13 of the Department of Education, Language Policy and Culture (Basque Government).

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  1. Departamento de Lógica y Filosofía de la Ciencia, Facultad de Letras, Universidad del País Vasco, c/ Paseo de la Universidad, 5, 01006, Vitoria-Gasteiz, Spain

    Jon Pérez Laraudogoitia

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Appendix: Proofs of theorems

Appendix: Proofs of theorems

Theorem

All solutions to the recurring equation \({\upmu }_{\mathrm{n+1}} = (1+{\upmu }_{\mathrm{n}})/(3 - {\upmu }_{\mathrm{n}})\) that satisfy the condition \({\upmu }_{\mathrm{i}} = {\mathrm{m}}_{\mathrm{i}+1}/{\mathrm{m}}_{\mathrm{i}} < 1~ (1 \le {\mathrm{i}})\) correspond to total finite mass.

Proof

One of these solutions is, as is immediately verifiable, \({\upmu }_{\mathrm{n}}={\upmu }_{\mathrm{n}}^{*} = \hbox {n}/(\hbox {n} + 2)\). Then, taking for instance \(\hbox {m}_{1} = 1\), we have successively \(\hbox {m}_{2} = 2/(2\cdot 3)\), \(\hbox {m}_{3} = 2/(3\cdot 4),\ldots , \hbox {m}_{\mathrm{n}} = 2/(\hbox {n}\cdot (\hbox {n}+1)),\ldots \). As \(\hbox {m}_{\mathrm{n}} \le 2/\hbox {n}^{2}\) and the series \(\Sigma _{\mathrm{n} = 1}^{\mathrm{n} = \infty }~ (2/\hbox {n}^{2})\) is convergent, it follows that \(\Sigma _{\mathrm{n} = 1}^{\mathrm{n} = \infty }\hbox { m}_{\mathrm{n}}\) is also convergent. Given that for the function y = (1 + x)/(3 \(-\) x) it always holds that \(\hbox {x} < \hbox {y} < 1\) when \(0 < \hbox {x} < 1\), it also follows that, in our conditions of \({\upmu }_{\mathrm{i}} = \hbox {m}_{\mathrm{i}+1}/\hbox {m}_{\mathrm{i}} < 1 (1 \le \hbox {i})\), the succession of values \({\upmu }_{1}\), \({\upmu }_{2}\), \({\upmu }_{3}\), \({\upmu }_{4}\),...is strictly increasing, besides being bounded by 1. This succession therefore has a limit, which I call L. As

$$\begin{aligned} {\lim }_{\mathrm{n}\rightarrow \infty } {\upmu }_{\mathrm{n+1}} = (1+ {\lim }_{\mathrm{n}\rightarrow \infty } {\upmu }_{\mathrm{n}})/(3 - {\lim }_{\mathrm{n}\rightarrow \infty } {\upmu }_{\mathrm{n}}) \end{aligned}$$

then inevitably L = (1+L)/(3 \(-\) L), from where \(\hbox {L} = \lim _{\mathrm{n}\rightarrow \infty } {\upmu }_{\mathrm{n}} = 1\). The masses of the particles are successively: \(\hbox {m}_{1}, \hbox {m}_{1}{\upmu }_{1}, \hbox {m}_{1}{\upmu }_{1}{\upmu }_{2}\), \(\hbox {m}_{1}{\upmu }_{1}{\upmu }_{2}{\upmu }_{3},\ldots , \hbox {m}_{1}{\upmu }_{1}{\upmu }_{2}{\upmu }_{3}\ldots {\upmu }_{\mathrm{n-1}}{\upmu }_{\mathrm{n}},\ldots \) and the total mass is the sum of the series \(\hbox {m}_{1} + \hbox {m}_{1}{\upmu }_{1} + \hbox {m}_{1}{\upmu }_{1}{\upmu }_{2} + \hbox {m}_{1}{\upmu }_{1}{\upmu }_{2}{\upmu }_{3} +\cdots + \hbox {m}_{1}{\upmu }_{1}{\upmu }_{2}{\upmu }_{3}\ldots {\upmu }_{\mathrm{n-1}}{\upmu }_{\mathrm{n}} +\cdots \). Certainly, with \({\upmu }_{\mathrm{n}}={\upmu }_{\mathrm{n}}^{*} = \hbox {n}/(\hbox {n} + 2)\) we have already seen that

$$\begin{aligned} \hbox {m}_{1} + \hbox {m}_{1}{\upmu }_{1}^{*} + \hbox {m}_{1}{\upmu }_{1}^{*}{\upmu }_{2}^{*} + \hbox {m}_{1}{\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*} +\cdots + \hbox {m}_{1}{\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*}\ldots {\upmu }_{\mathrm{n-1}}^{*}{\upmu }_{\mathrm{n}}^{*} +\cdots \end{aligned}$$

is convergent and that is why

$$\begin{aligned} 1 + {\upmu }_{1}^{*} + {\upmu }_{1}^{*}{\upmu }_{2}^{*} + {\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*} +\cdots + {\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*}\ldots {\upmu }_{\mathrm{n-1}}^{*}{\upmu }_{\mathrm{n}}^{*} +\cdots \end{aligned}$$

also is. Let us take an arbitrary positive integer k. \({\upmu }_{\mathrm{k}}^{*}\) = k/(k + 2). The infinite series

$$\begin{aligned}&{\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*}\ldots {\upmu }_{\mathrm{k-1}}^{*}{\upmu }_{\mathrm{k}}^{*} + {\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*}\ldots {\upmu }_{\mathrm{k-1}}^{*}{\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*} +{\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*}\\&\quad \ldots {\upmu }_{\mathrm{k-1}}^{*}{\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*}{\upmu }_{\mathrm{k}+2}^{*} + \cdots \end{aligned}$$

obtained by removing the k first terms from the previous one also converges. Therefore the series obtained from this latter by multiplying it by (\({\upmu }_{1}^{*}{\upmu }_{2}^{*}{\upmu }_{3}^{*}\ldots {\upmu }_{\mathrm{k}-1}^{*})^{-1}\) also converges. In other words, the infinite series \({\upmu }_{\mathrm{k}}^{*} + {\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*} + {\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*}{\upmu }_{\mathrm{k}+2}^{*} +\cdots \) converges. Now let us take as \({\upmu }_{1}\) any number \({\uplambda }_{1}\) between 0 and 1 and calculate the rest according to \({\uplambda }_{\mathrm{n+1}} = (1+{\uplambda }_{\mathrm{n}})/(3 - {\uplambda }_{\mathrm{n}})\). Given that \(\lim _{\mathrm{n}\rightarrow \infty } {\upmu }_{\mathrm{n}}^{*} = 1\) there will be a positive integer k such that \(0 < {\uplambda }_{1} < {\upmu }_{\mathrm{k}}^{*}\). As the function y = (1 + x)/(3 \(-\) x) strictly increases for \(0 < \hbox {x} < 1\) it follows that \(0 < {\uplambda }_{2} < {\upmu }_{\mathrm{k}+1}^{*}, 0 <{\uplambda }_{3} < {\upmu }_{\mathrm{k}+2}^{*}, 0 < {\uplambda }_{4} < {\upmu }_{\mathrm{k}+3}^{*},..\) and so on successively. In particular, \(0 < {\uplambda }_{1} < {\upmu }_{\mathrm{k}}^{*}, 0 < {\uplambda }_{1}{\uplambda }_{2} < {\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*}\), \(0 < {\uplambda }_{1}{\uplambda }_{2}{\uplambda }_{3} < {\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*}{\upmu }_{\mathrm{k}+2}^{*},\ldots \) and so on successively. As \({\upmu }_{\mathrm{k}}^{*} + {\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*} + {\upmu }_{\mathrm{k}}^{*}{\upmu }_{\mathrm{k}+1}^{*}{\upmu }_{\mathrm{k}+2}^{*} +\cdots \) is convergent, \({\uplambda }_{1} + {\uplambda }_{1}{\uplambda }_{2} + {\uplambda }_{1}{\uplambda }_{2}{\uplambda }_{3} +\cdots \) and \(1 + {\uplambda }_{1} + {\uplambda }_{1}{\uplambda }_{2} + {\uplambda }_{1}{\uplambda }_{2}{\uplambda }_{3} +\cdots \) are convergent too, as is, finally, \(\hbox {m}_{1} + \hbox {m}_{1}{\uplambda }_{1} + \hbox {m}_{1}{\uplambda }_{1}{\uplambda }_{2} + \hbox {m}_{1}{\uplambda }_{1}{\uplambda }_{2}{\uplambda }_{3} +\cdots \) \(\square \)

(GROI) There are no spatially bounded systems in classical mechanics with gravitation whose evolution is deterministic.

Proof of GROI: Let us consider a dynamic system \(\Sigma \) which evolves between t = 0 and t = 8T freely in a finite region of space V. Throughout this demonstration we are situated in an inertial frame of reference R in which the centre of mass of \(\Sigma \) moves at a uniform (non-null) velocity U such that it travels a distance U8T between t = 0 and t = 8T. In this frame R the system \(\Sigma \) evolves freely between t = 0 and t = 8T in a finite region of space W. The free evolution of \(\Sigma \) from t = 0 to t = 8T may occur in at least two ways:

  1. (a)

    without any other material body existing in space

  2. (b)

    a system of rigid spheres having appeared (spontaneously, from the self-excitation of the empty space) from t = 0 that collide with each other in a complex way and such that in t = 8T they are at rest in the state \(\hbox {I}^{*}{}^{0}_{\mathrm{left}}\). For this possibility to occur it is sufficient for the finite region W in which \(\Sigma \) evolves freely between t = 0 and t = 8T to fit in the interior of a spherical region of volume \(\hbox {V}_{\mathrm{p}1} = (4/3){\uppi }[\hbox {r}_{1} - [2{\upmu }_{1}\hbox {d}_{1}/({\upmu }_{1} + 1)]]^{3}\).

The evolution of \(\Sigma \) for t \(> 8\hbox {T}\) will differ greatly depending on whether the possibility described in a) or the one described in b) arises and this is the cause of indeterminism. In case a), the system \(\Sigma \) “as a whole” (to put it more precisely, its centre of mass) can continue moving at uniform velocity U for all eternity. In case b), the system \(\Sigma \) moving at velocity U will end up colliding with the innermost sphere \((\hbox {p}_{1})\) belonging to the infinite set of enclosed spheres at rest in the state \(\hbox {I}^{*}{}^{0}_{\mathrm{left}}\). Now the indeterminism is clear. Before t = 0, \(\Sigma \) (whose centre of mass moves in the frame of reference R at velocity U) was the only material system existing in space. After t = 0 the evolution of \(\Sigma \) may continue being the same (velocity U for its centre of masses), but it may also change from a certain moment as a consequence of its collision with the innermost sphere (at rest) of an infinite set of spheres that appear for \(\hbox {t} > 0\) as a consequence of the self-excitation of the empty space.

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Laraudogoitia, J.P. A simple and interesting classical mechanical supertask. Synthese 194, 545–570 (2017). https://doi.org/10.1007/s11229-015-0958-z

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