The initial-value formulation of the λ-R model

Abstract

We apply the conformal method to solve the initial-value formulation of general relativity to the λ-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. We obtain a modified Lichnerowicz–York equation for the conformal factor of the metric and derive its properties. We show that the behavior of the equation depends on the value of the coupling constant λ. In the absence of a cosmological constant, we recover the existence and uniqueness properties of the original equation when λ > 1/3 and the trace of the momentum of the metric, π, is non-vanishing. For π = 0, we recover the original Lichnerowicz equation regardless of the value of λ and must therefore restrict the metric to the positive Yamabe class. The same restriction holds for λ < 1/3, a case in which we show that if the norm of the transverse-traceless data is small enough, then there are two solutions. Taking the equations of motion into account, this allows us to prove that there is, in general, no way of matching both constraint-solving data and time evolution of phase-space variables between the λ-R model and general relativity, thereby proving the non-equivalence between the theories outside of the previously known cases λ = 1 and π = 0 and of the limiting case of λ → ∞, with a finite π, which we show to yield geometries corresponding to those of general relativity in the maximal slicing gauge.