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The infinitesimal holonomy group structure of EEnstein-Maxwell space-times

Структура бесконечно малой голономной группы пространства-времени Эйнштейна-Максвелла

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Il Nuovo Cimento B (1971-1996)

Summary

This paper further investigates and interprets the structure of Einstein-Maxwell space-times in terms of the infinitesimal holonomy groups (IHG) of theC ∞ Riemannian connection. In particular, this investigation provides a fundamental physical classification of Einstein-Maxwell space-times in terms of the IHG group structure of the Riemann curvature tensor. It will be shown that the Maxwell fieldsF μλ and *F μλ of a given Einstein-Maxwell space-time define a representation of a subalgebra of the Lie algebra of the IHG. The main results of this paper are stated in the form of two theorems and a corollary. Also the results obtained here indicate that physical insight could be gained by studying more general gauge fields in terms of the IHG of the relevant Einstein-Yang-Mills space-times.

Riassunto

Questo lavoro studia ulteriormente e interpreta la struttura degli spazi-tempo di Einstein e Maxwell in termini dei gruppi infinitesimali di olonomia (IHG) della connessione riemannianaC ∞. In particolare, questo studio fornisce una classificazione fisica fondamentale degli spazi-tempo di Einstein e Maxwell sulla base della struttura del gruppo IHG del tensore di curvatura di Riemann. Si mostra che i campi di MaxwellF μλ e *F μλ di un dato spazio-tempo di Einstein e Maxwell definiscono una rappresentazione di una subalgebra dell'algebra di Lie dell'IHG. I più importanti risultati di questo lavoro sono forniti sotto forma di due teoremi e un corollario. Anche il risultati ottenuti qui indicano che l'indagine fisica potrebbe essere migliorata studiando i campi di gauge più generali sulla base dell'IHG degli importanti spazi-tempo di Einstein e Yang-Mills.

Резюме

В этой статье исследуется и интерпретируется структура пространства-времени Эйнштейна-Максвелла в терминах бесконечно малых голономных групп. В частности, это исследование обеспечивает фундаментальную физическую классификацию пространства-времени Эйнштейна-Максвелла в терминах структуры бесконечно малой голономной группы для риманова тензора кривизны. Показывается, что Максвелловские поляF μλ и *F μλ для заданного пространства-времени Эйнштейна-Максвелла определяют представление субалгебры алгебры Ли для бесконечно малой голономной группы. Основные результаты этой статьи состоят в формулировке двух теорем и следствия. Полученные результаты указывают, что физический смысл может быть получен посредством исследования более общих калибровочных полей в терминах бесконечно малых голономых групп для соответствующего пространства-времени Эйнштейна-Янга-Миллса.

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References

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  12. These necessary conditions (3.1), (3.2) and (3.3) are given, for example, in (9).. For simplicity we use condition (3.3) rather thanR 00≥0. In addition to these algebraic conditions there is also a differential condition (which is a consequence of requiring the Maxwell equations to be satisfied) that must hold if the space-time is to be a source-free Einstein-Maxwell space-time (9).G. Y. Rainich:Trans. Am. Math. Soc.,27, 106 (1925);C. W. Misner andJ. A. Wheeler:Ann. Phys. (N. Y.),2, 525 (1957). Since we are not concerned here with the differential condition, theorem 3.1 could equally as well apply to the more general class of space-times defined by the algebraic conditions alone.

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  14. This simple descriptive statement does not hold when the Maxwell fields are null fields and the IHG is of rotation classR 3,R 6, orR 10. For these three null-field cases corollary 3.1 shows that only the linear combination (3.11) can be chosen as a generator of the group.

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  16. In connection with these three cases see the remarks made in footnotes (14,16). These necessary conditions (3.1), (3.2) and (3.3) are given, for example, in (9) ;C. W. Misner andJ. A. Wheeler:Ann. Phys. (N. Y.),2, 525 (1957). For simplicity we use condition (3.3) rather thanR 00≥0. In addition to these algebraic conditions there is also a differential condition (which is a consequence of requiring the Maxwell equations to be satisfied that must hold if the space-time is to be a source-free Einstein-Maxwell space-time (9).G. Y. Rainich:Trans. Am. Math. Soc.,27, 106 (1925);C. W. Misner andJ. A. Wheeler:Ann. Phys. (N. Y.),2, 525 (1957)., Since we are not concerned here with the differential condition, theorem 3.1 could equally as well apply to the more general class of space-times defined by the algebraic conditions alone. This simple descriptive statement does not hold when the Maxwell fields are null fields and the IHG is of rotation classR 3,R 6, orR 10. For these three null-field cases corollary 3.1 shows that only the linear combination (3.11) can be chosen as a generator of the group.

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  17. In this connection the classification of unquantized Yang-Mills fields into radiative and nonradiative cases byEguchi (7). would be useful.

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  18. Clearly such space-times must have IHG generated by a set of bivector generators transforming into itself under the dual operation.

  19. For comparison we note that, for nonnull Einstein-Maxwell space-times with IHG in rotation classR 15,P ab (cf. equation (5.6)) takes the form (5.10) with β=γ=0.

  20. See, for example,C. W. Misner, K. S. Thorne andJ. A. Wheeler:Gravitation (San Francisco, Cal., 1970), p. 360, 361.

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  1. Department of Physics, North Carolina State University, 27650, Raleigh, N. C., U.S.A.

    L. K. Norris & W. R. Davis

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  1. L. K. Norris
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  2. W. R. Davis
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Norris, L.K., Davis, W.R. The infinitesimal holonomy group structure of EEnstein-Maxwell space-times. Nuov Cim B 53, 209–232 (1979). https://doi.org/10.1007/BF02739891

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