Sommario
Si vuole ottenere una classe di teorie gravitazionali deducibili da un principio variazionale, nell'ambito della teoria dei campi e nello spazio-tempo pseudoeuclideo non-rinormalizzato. Si richiede che tali teorie coincidano, al primo ordine nella costante di accoppiamento f, con la teoria di Einstein. Si assume inoltre la “gauge” di Hilbert al fine di escludere la presenza della componente vettoriale del potenziale ψαβ. Per ottenere la consistenza al secondo ordine delle equazioni di campo, si sostituisce, in queste ultime, al tensore della particellaT (p) αβ il più generale tensore energia-quantità-di-moto\(\bar T_{\alpha \beta }^{(tot)}\). Imponendo alle equazioni di campo di essere deducibili mediante un principio variazionale ove si varino i potenziali ψαβ, si ottiene una lagrangiana che, ove si varino le coordinate della particella di prova, dà le equazioni di moto. In tal modo si ottiene una classe di teorie dipendenti da 5 parametri arbitrari. Per un confronto con i dati sperimentali è necessario rinormalizzare, onde esprimere quantità osservabili. Si dimostra così che per soddisfare il principio di equivalenza al secondo ordine è necessario porre uno dei 5 parametri uguale a zero e che, con tale scelta, l'intera classe di teorie coincide, al secondo ordine, con la relatività generale.
Summary
We consider, in the field-theoretical approach, a class of gravitational theories deducible by a variational principle in the “unrenormalized” pseudo-Euclidean space-time. At first order in the coupling constant f we require the theories to coincide with the Einstein one. Moreover we assume the Hilbert gauge which assure the exclusion of the vector component of the gravitational potential ψαβ. To get the higher order consistency we substitute the most general energy-momentum tensor\(\bar T_{\alpha \beta }^{(tot)}\) for the particle tensorT (p) αβ in the field equations. Requiring the latter to be deducible by a variational principle varying the potentials ψαβ, we get a Lagrangian which, varying the particle coordinates, gives the equations of motion. So we get a class of theories depending on 5 arbitrary parameters. To have observable quantities we have to “renormalize”. So we realize that, to satisfy the equivalence principle, we have to put one of the arbitrary parameters equal to zero. With this choice the class of theories coincides at second order with general relativity.
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Research sponsored by the CNR, Gruppi di ricerca Matematica
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Spinelli, G. Second order gravitational field theories with hilbert gauge. Meccanica 10, 32–41 (1975). https://doi.org/10.1007/BF02148283
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DOI: https://doi.org/10.1007/BF02148283