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.
REFERENCES
In what follows, we will have coordinates xi on the hypersurfaces Σt and use to denote the time coordinate.
This is only true for λ ≠ 1/3, as the same condition is a primary constraint of the model in that case.
The formulation is named after its authors: Richard Arnowitt, Stanley Deser, and Charles Misner.
The equations of motion for the lapse N and the shift Ni.
A metric Gijkl is said to be ultralocal if it does not depend on spatial derivatives of the (inverse) metric gij.
In Ref. 5, we imposed for simplicity. We are not performing the same simplification in the current paper.
The set of restrictions associated with the “almost always” was discussed in Subsection I B and holds in the present case.
Recall that in Sec. II, we discussed some cases for which there are two solutions.
Recall that, unlike in general relativity, it is not possible to cast all spherically symmetric spacetimes as spacetimes with spherically symmetric spatial hypersurfaces when the symmetry group is that of foliation-preserving diffeomorphisms.
