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Schwarzschild field inn dimensions and the dimensionality of space problem

Summary

The fact that our present laws of physics admit of a formal extension to spaces of an arbitrary number of dimensions suggests that there must be some principle (or principles) operative which in conjunction with these laws entails the observed specificity of spatial dimensionality,n=3. Generalizing from an approach suggested by the work ofEhrenfest (and independently byG. J. Whitrow) on the Newtonian keplerian problem inn dimensions, it is proposed that this principle may be tentatively summarized in the postulate that there shall be stable bound orbits or «states» for the equations of motion governing the interaction of bodies (considered as «material points’). This postulate is applied to the geodesic equations of motion obtained from a generalization of the Schwarzschild field to static systems with hyper-spherical symmetry, and it is shown that the bound state postulate uniquely entails the spatial dimensionality. This result is not entirely peculiar, to general relativity because it also holds for Newtonian theory (Ehrenfest-Whitrow) if one also introduces an asymptotic condition to exclude casesn<3. The Schrödinger hydrogen atom inn dimensions, is also briefly considered for which the postulate also excludesn>3, and in conjunction with the asymptotic conditionn<3. An attempt is made to understand the logical origin of this postulate and it is argued that if one assumes the basic representatives of a dynamics with a metric to be material points, one needs such a postulate to construct Einstein’s «practically rigid rods», since point bodies in themselves do not provide us with a measure of distance. Some brief qualitative applications of these ideas are, made to quantum electrodynamics.

Riassunto

Il fatto che le attuali leggi fisiche ammettono una estensione formale a spazi con un numero arbitrario di dimensioni, suggerisce che deve esistere qualche principio (o alcuni principi) operativi che in unione con queste leggi implichino l’osservata specificità della dimensionalità spaziale,n=3. Generalizzando uno spunto suggerito dal lavoro diEhrenfest (ed indipendentemente daG. J. Whitrow) sul problema di Kepleron dimensionale, si propone di riassumere in via di tentativo, questo principio nel postulato che devono esistere delleorbite o «stati» legati stabili nelle equazioni del moto che governano l’interazione dei corpi considerati come punti materiali. Si applica questo postulato alle equazioni geodetiche del moto ottenute da una generalizzazione del campo di Schwarzschild a sistemi statici con simmetria ipersferica e si dimostra che il postulato degli stati legati implica unicamente la dimensionalità spaziale. Questo risultato non è affatto peculiare della relatività generale perchè vale anche che escluda i casi conn<3. Si considera brevemente anche l’atomo di idrogeno di Schrödinger inn dimensioni, per cui il postulato escluden>3 e, in unione con la condizione asintotica,n<3. Si cerca di comprendere l’origine logica di questo postulato e si suggerisce che, se si suppone che i rappresentanti fondamentali di una dinamica avente una metrica siano punti materiali, si ha bisogno di un tale postulato per costruire le «sbarre praticamente rigide» di Einstein, in quanto i corpi puntiformi di per sè non ci forniscono una misura della distanza. Si fanno alcune brevi applicazioni qualitative di queste idee alla elettrodinamica quantistica.

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Correspondence to F. R. Tangherlini.

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Tangherlini, F.R. Schwarzschild field inn dimensions and the dimensionality of space problem. Nuovo Cim 27, 636–651 (1963). https://doi.org/10.1007/BF02784569

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