SL2,C symmetry of the gravitational field
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Summary
In a previous paper the symmetry of the gravitational-field dynamical variables was discussed. A scheme was given according to which the spin coefficients and the Riemann tensor were represented in the form of linear combinations of the infinitesimal generators of the groupSL2,C, similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices, where the spin coefficients take the role of the Yang-Mills-like potentials and the Riemann tensor takes the role of the fields. In this paper we use this representation of the gravitational-field dynamical variables in order to write the field equations of general relativity, and obtain a simple and attractive representation for the field equations of Newman and Penrose. Comparison between the present method and that of Utiyama, based on the Yang-Mills theory, is discussed and the possibility of using it as a basis for quantization is briefly mentioned.
Riassunto
In un articolo precedente si è discussa la simmetria delle variabili dinamiche del campo gravitazionale. Si è dato uno schema secondo cui i coefficienti dello spin e il tensore di Riemann erano rappresentati nella forma di combinazioni di generatori infinitesimali del gruppoSL2,C in modo simile a quello con cui Yang e Mills scrivono le loro variabili dinamiche in termini delle matrici di spin di Pauli; in questo schema i coefficienti di spin hanno il ruolo di potenziali del tipo di Yang e Mills ed il tensore di Riemann ha il ruolo dei campi. In questo articolo si usa questa rappresentazione delle variabili dinamiche del campo gravitazionale per scrivere le equazioni di campo della relatività generale, e si ottiene una semplice ed attraente rappresentazione delle equazioni di campo di Newman e Penrose. Si discute il confronto tra questo metodo e quello di Utiyama, basato sulla teoria di Yang e Mills, e si menziona brevemente la possibilità di usarlo come base della quantizzazione.
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References
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