Abstract
There is broad consensus (both scientific and philosophical) as to what a rigid body is in classical mechanics. The idea is that a rigid body is an undeformable body (in such a way that all undeformable bodies are rigid bodies). In this paper I show that, if this identification is accepted, there are therefore rigid bodies which are unstable. Instability here means that the evolution of certain rigid bodies, even when isolated from all external influence, may be such that their identity is not preserved over time. The result is followed by analyzing supertasks that are possible in infinite systems of rigid bodies. I propose that, if we wish to preserve our original intuitions regarding the necessary stability of rigid bodies, then the concept of rigid body must be clearly distinguished from that of undeformable body. I therefore put forward a new definition of rigid body. Only the concept of undeformable body is holistic (every connected part of an undeformable body is not always an undeformable body) and every connected part of a rigid body in this new sense is always a rigid body in this new sense. Finally, I briefly discuss the connection between this conceptual distinction and the dimensionality of space, thereby enabling it to be supported from a new and interesting perspective.
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Notes
"A continuous body is a set B of points called material points" (Silhavý 1997, p. 29).
Consider P's interior topological boundary, which completely overlaps Q's topological boundary. Let us call this common boundary BPQ. Given the fact that the (constant) density of P and Q is the same (= ρ), it follows that P + Q is a homogeneous rigid body of density ρ and, therefore, dynamically identical to P*. In particular, as is evident, the density at any one point p BPQ is ρ, identical to the density of the corresponding point of P*. The same consideration applies, of course, to rigid bodies R and S discussed later in the text and also to all the cases of rigid bodies in contact in the subsequent sections of the paper.
Earman describes this principle thus: "particle world lines do not have beginning or end points and mass is constant along a world line" (p. 38). Strictly speaking he formulates it for point particles (with non-null mass). However extending it to the material points of a rigid solid (with non-null mass density, although their mass may be null) is natural. In this paper I assume this extension which is, understandably, also reasonable because a rigid body is partially specified by the material points that belong to it. The principle of conservation of mass also explains why an isolated unstable rigid body cannot instantly cease to exist and simply leave empty space in its place.
Each of the multiple collisions is dynamically undetermined. From the infinite solutions possible compatible with the laws of conservation, I have chosen a particularly interesting one. The same occurs in the other examples of collisions analyzed subsequently. The important thing in all this is not indeterminism as such; it is rather the type of forms of evolution that turn out to be compatible with dynamics even without being required by it. In the literature on Newtonian supertasks little or no philosophical advantage has yet been taken of indeterministic collisions (even in the non-philosophical technical literature, these types of situations are only considered in engineering-oriented publications such as, e.g. Brogliato 2016). The present paper also aims to put this to rights by showing a significant result that could not be reached by the usual path of analyzing deterministic binary collisions. Note that, since multiple collisions involving block d2i-1 and block d2i are dynamically undetermined, post-collision states other than those assumed in the basic supertask execution are possible. For example, both block d2i-1 and block d2i could rebound from each collision they undergo without losing any of their parts as a consequence of the collision. This would correspond to a form of evolution devoid of interest, whereby no supertask would take place. The form of evolution assumed in the basic supertask is therefore just one among an infinite number of other possible forms, most of which are of no interest. This is a possible, but not necessary, supertask. Nevertheless, such a possibility is all that is needed. As mentioned earlier, it corresponds to a form of evolution that is compatible with dynamics even though the later does not require it.
As a time [2/(√2 − 1)](L/v1) goes by from the de-blocking of the pair (d1, d2) to the completion of the type-ω supertask (which corresponds to the situation in Fig. 2), the same occurs from the beginning of the type-ω* supertask (which corresponds to the situation in Fig. 2) to the blocking of the pair (d1, d2). Finally, a time (2L/v1) evidently passes from the blocking of (d1, d2) to their de-blocking. In all, 2[2/(√2 − 1)](L/v1) + (2L/v1) = (2 + √2)2(L/v1).
The idea of extending the sphere of possible solutions is not uncommon and is useful in mathematical physics involving, for instance, differential equations. In this case one goes beyond classical solutions to generalized solutions (distributional and weak solutions). Kanwal (2004, pp. 228–229).
In other respects, the strategy of rejecting the possibility of certain supertasks on the sole basis of the kind of consequences they have sheds little light on the problem.
One might consider describing the situation, in more detail, in this way: all parts of C* are rigidly joined together, but not all parts of C + D are rigidly joined together. However, this only makes "empirical" sense, as noted above, if the space has at least four dimensions.
In particular, we saw that if the space is three-dimensional, then unstable three-dimensional undeformable non-rigid solids exist. The surprising thing was the form in which the new degrees of freedom of translation appeared, which were not dependent on increasing the dimensionality of space.
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Research for this work is part of the research project GIU19/051 (funded by the University of the Basque Country UPV/EHU)
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Laraudogoitia, J.P. Undeformable Bodies that are Not Rigid Bodies: A Philosophical Journey Through Some (Unexpected) Supertasks. Axiomathes 32, 605–625 (2022). https://doi.org/10.1007/s10516-021-09543-w
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DOI: https://doi.org/10.1007/s10516-021-09543-w