The initial value formulation of the $\lambda$-R model

Abstract

We apply the conformal method to solve the initial value formulation of general relativity to the $\lambda$-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. We obtain a generalised Lichnerowicz-York equation for the conformal factor of the metric and derive its properties. We show that the behaviour of the equation depends on the value of the coupling $\lambda$. In the absence of a cosmological constant, we recover the existence and uniqueness properties of the original equation when $\lambda>1/3$ and the trace of the momentum of the metric, $\pi$, is non-vanishing. For $\pi=0$, we recover the original Lichnerowicz equation regardless of the value of $\lambda$ and must therefore restrict the metric to the positive Yamabe class. The same restriction holds for $\lambda<1/3$, a case in which we show that the spatial Ricci scalar must also be large enough to guarantee the existence of at least one solution. 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 $\lambda$-R model and general relativity, thereby proving the the non-equivalence between the theories outside of the previously known cases $\lambda=1$ and $\pi=0$.