We analyze different claims on the role of the coupling constant λ in so-called λ-R models, a minimal generalization of general relativity inspired by Hořava-Lifshitz gravity. The dimensionless parameter λ appears in the kinetic term of the Einstein-Hilbert action, leading to a one-parameter family of classical theories. Performing a canonical constraint analysis for closed spatial hypersurfaces, we obtain a result analogous to that of Bellorín and Restuccia, who showed that all nonprojectable λ-R models are equivalent to general relativity in the asymptotically flat case. However, the tertiary constraint present for closed boundary conditions assumes a more general form. We juxtapose this with an earlier finding by Giulini and Kiefer, who ruled out a range of λ-R models by a physical, cosmological argument. We show that their analysis can be interpreted consistently within the projectable sector of Hořava-Lifshitz gravity, thus resolving the apparent contradiction.

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We derive spherically symmetric solutions of the classical λ-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. Starting from a 3 þ 1 decomposition of the four-metric in a general spherically symmetric ansatz, we perform a phase space analysis of the reduced model. We show that its constraint algebra is consistent with that of the full λ-R model, and also yields a constant mean curvature or maximal slicing condition as a tertiary constraint. Although the solutions contain the standard Schwarzschild geometry for the general relativistic value λ ¼ 1 or for vanishing mean extrinsic curvature K, they are in general nonstatic, incompatible with asymptotic flatness, and parametrized not only by a conserved mass. We show by explicit computation that the four-dimensional Ricci scalar of the solutions is in general time dependent and nonvanishing.

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