Summary
With the aim of generalizing the relation between the fundamental tensor and the affine connexion of the real and the hermitian case of the unified field theory, a relation with eight arbitrary constants is contemplated; conditions for its general invariance and transposition invariance are imposed, and a relation with only two arbitrary coefficients is obtained; which, however, except in three particular cases, is substantially nothing else but the real or hermitian relation. One of the particular cases is noteworthy, as it imposes the first quadruplet of Maxwell's equations on the skew part of the fundamental tensor [relation (19) in the text].
Riassunto
Nell'intento di generalizzare le relazioni tra il tensore fondamentale e la connessione del caso reale e del caso hermifiano della teoria unitaria, si considera una relazione contenente otto costanti arbitrarie; imponendo ad essa le condizioni di invarianza tensoriale e di invarianza per trasposizione, il numero dei coefficienti arbitrari viene ridotto a due. Tuttavia, fatta eccezione per tre casi particolari, tale nuova relazione si riduce ancora, sostanzialmente, al caso reale o al caso hermitiano. Uno dei casi particolari è notevole, in quanto impone alla parte emisimmetrica del tensore fondamentale il primo quadrupletto delle equazioni di Maxwell.
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Footnotes
A. Einstein:Ann. of Math.,46, 578 (1945);A. Einstein andE. G. Straus:Ann. of Math.,47, 731 (1946). For the real one, see e.g. the three papers byE. Schrödinger which appeared in theProceedings of the Royal Irish Academy: I, A51, 163 (1947); II, A51, 205 (1948); III, A52, 1 (1948); and, mainly for what concerns the inadmissability of the «strong» form of the field equations:A. Einstein andB. Kaufman:Sur l'état actuel de la théorie générale de la gravitation, published in the volumeLouis de Broglie, physicien et penseur (Paris, 1953), page 321.
V. Hlavatý:Proc. of the Nat. Acad. of Sciences,39, 507 (1953).
See, e.g.,L. P. Eisenhart:Non-riemannian geometry (New York, 1927), p. 83 and following.
SeeE. Schrödinger:Space-Time Structure (Cambridge, 1950), p. 64 and following.
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Bertotti, B. On the relation between fundamental tensor and affinity in unified field theory. Nuovo Cim 11, 358–365 (1954). https://doi.org/10.1007/BF02783625
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DOI: https://doi.org/10.1007/BF02783625