Abstract
It is shown that a notion of natural place is possible within modern physics. For Aristotle, the elements—the primary components of the world—follow to their natural places in the absence of forces. On the other hand, in general relativity, the so-called Carter–Penrose diagrams offer a notion of end for objects along the geodesics. Then, the notion of natural place in Aristotelian physics has an analog in the notion of conformal infinities in general relativity.
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Notes
Schummer (2008) presents a brief introduction to the main points of Aristotle’s physics.
However “the religious Newton” assumed God’s final causes: “We know him only by his most wise and excellent contrivances of things, and final causes” (Newton 1846, Book III, p. 506).
Apparently, because there are vestiges of teleology in the principle of least action, for example. See Stöltzner (1994).
See Physics III.
Nietzsche (2002, §22).
See, for example, Jammer (1954, chapter 1).
One may consider Einstein’s attempts of adopting Mach’s principle in general relativity as a modern form of an integrated cosmos.
Aristotle says that the center of the universe and the planet earth’s place coincide accidentally. “It happens, however, that the centre of the earth and of the whole is the same. Thus they [the elements earth] do move to the centre of the earth, but accidentally, in virtue of the fact that the earth’s centre lies at the centre of the whole” (Aristotle 1922a, 296b15).
At least for test particles. In this case, the tiny mass of that particles will not modify the metric, or spacetime geometry. On the other hand, massive bodies in motion modify the spacetime geometry. In this case, it is necessary to use backreaction effects to evaluate the body trajectory.
In this article, one adopts geometric unities. Then the speed of light in vacuum is set equal to 1.
Einstein static universe (Einstein 1917) is a special cosmological solution of Einstein’s field equations. It describes a non-expanding universe with dust, or pressureless matter. Moreover, Einstein adopted the cosmological constant to achieve such a solution of his field equations. It is well-known that Einstein static universe is unstable.
A diffeomorphism is special type of map between two manifolds, which may be represented by sets. In general relativity, a spacetime is a manifold \({\mathcal {M}}\) endowed with a metric \(g_{\mu \nu }\). It is also indicated by \(({\mathcal {M}}, g_{\mu \nu })\). A spacetime is a set of events. Two manifolds are diffeomorphic if there exists a differentiable and invertible map between them. See (Wald 1984, Appendix C).
In cosmology, matter means a pressureless perfect fluid.
Today it is possible to construct cosmological models without the initial singularity in the general relativity context. Such regular models are the so-called bouncing cosmologies (Novello 2008; Neves 2017a). Contrary to Minkowski spacetime, the cosmological metric in the standard model is not geodesically complete. In the past, the initial singularity is the end of trajectories. In Minkowski spacetime, as we have already seen, we have a structure of past infinities. This is the reason why Minkowski spactime is geodesically complete and the Friedmann–Lemaître–Robertson–Walker metric is not.
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I would like to thank IMECC-UNICAMP for the kind hospitality and referees for comments and suggestions.
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Neves, J.C.S. Infinities as Natural Places. Found Sci 24, 39–49 (2019). https://doi.org/10.1007/s10699-018-9556-0
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DOI: https://doi.org/10.1007/s10699-018-9556-0