Hyperbolicity of scalar-tensor theories of gravity

Abstract

Two first order strongly hyperbolic formulations of scalar-tensor theories of gravity allowing non- minimal couplings (Jordan frame) are presented along the lines of the 3 � 1 decomposition of spacetime. One is based on the Bona-Massoformulation, while the other one employs a conformal decomposition similar to that of Baumgarte-Shapiro-Shibata-Nakamura. A modified Bona-Massoslicing condition adapted to the scalar-tensor theory is proposed for the analysis. This study confirms that the scalar-tensor theory has a well-posed Cauchy problem even when formulated in the Jordan frame. Scalar-tensor theories of gravity (STT) are alternative theories of gravitation where a scalar field is coupled non- minimally to the curvature associated with the physical metric (this is the so-called Jordan frame representation). The term ''physical metric'' refers to a situation where test particles follow the geodesics of that metric. The variation of the action of the STT with respect to the physical metric gives rise to field equations which contain an effective energy-momentum tensor (EMT) involving second order derivatives in time and space of the scalar field. Such EMT has the property that ''ordinary matter,'' i.e., matter asso- ciated with fields other that the scalar field, obeys the (weak) equivalence principle which mathematically trans- lates into a conserved EMT for ordinary matter alone. Since a priori, it was not clear how such second order derivatives could be eliminated (in terms of lower order derivatives), or managed so as to obtain a quasilinear system of hyperbolic equations for which the Cauchy problem was well-posed (in the Hadamard sense), many people decided to abandon this approach in favor of the so- called Einstein frame representation where the nonminimal coupling is absorbed into the curvature by means of a conformal transformation of the metric. The new confor- mal metric is unphysical in the sense that (non-null) test particles do not follow the geodesics of that metric. However, the mathematical advantage is that the field equations for the nonphysical metric resemble the standard Einstein field equations with an unphysical effective EMT which involves at most first order derivatives of a suitable transformed scalar field (this EMT is unphysical because the ''ordinary matter'' part is not separately conserved). In the Einstein frame one can show that by using standard gauges (e.g., harmonic gauge) the field equations acquire the form required in the application of theorems that establish the well-posedness of the Cauchy problem. In view of the apparent mathematical advantages and disadvantages of the Jordan and Einstein frames, several