Relation of gauge formalisms for pulsations of general-relativistic stellar models

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Abstract
There have been two recent reformulations of the equations for even-parity perturbations of general-relativistic stellar models, in both of which fluid perturbation variables are absent in the final set of equations. The recent reformulation by Chandrasekhar and Ferrari uses the diagonal coordinate gauge and leads to a fifth-order system of differential equations; we have recently presented a reformulation, based on the Regge-Wheeler coordinate gauge, which leads to a fourth-order system. The difference in the orders is similar to that for perturbations of Schwarzschild and of Reissner-Nordström black holes; in both cases the diagonal-gauge formulation led to a system one degree higher than that for equations based on the Regge-Wheeler gauge. For perturbations of holes, however, the equations could be reduced by one degree. We show that this is the case also for the Chandrasekhar-Ferrari equations for stellar perturbations. More important, we show that the extra degree of freedom, in all descriptions based on the diagonal gauge, is due to the fact that the diagonal gauge is an incomplete constraint on coordinates; a one degree of freedom set of gauge transformations can be made within the diagonal gauge. This previously unnoticed degree of freedom is responsible for the extra degree of freedom in the Chandrasekhar-Ferrari equations, and the related black-hole equations. It also provides an a priori solution with which those equations can be reduced.

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