Invariance of the Massless Field Equations under Changes of the Metric

Details Invariance of the Massless Field Equations under Changes of the Metric Invariance of the Massless Field Equations under Changes of the Metric

Mar 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998gregr..30..379t&link_type=abstract

General Relativity and Gravitation, Volume 30, Issue 3, p.379-387

Physics

4

Scientific paper

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field $Fij$ and the current density four-vector $Ji$, are invariant under the change of the metric $gij$ by $g'ij = gij + li lj$ if $li$ is a principal null direction of $Fij$ and that an analogous result holds in the case of the massless Klein-Gordon equation if $li$ is null and orthogonal to the gradient of the field and in the case of the null dust equations if $li$ is parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let $(gij, Fij)$ be an exact solution of the Einstein-Maxwell equations and let $li$ be a principal null direction of $Fij$, then $(gij + li lj, Fij)$ is also an exact solution of the Einstein-Maxwell equations if and only if $(li lj, 0)$ satisfies the Einstein-Maxwell equations linearized about the background solution $(gij, Fij)$. Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.

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