Role of the extra coupling in the kinetic term in Ho<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ř</mml:mi></mml:math>ava-Lifshitz gravity

Abstract

We analyze different claims on the role of the coupling constant $\ensuremath{\lambda}$ in so-called $\ensuremath{\lambda}$-$R$ models, a minimal generalization of general relativity inspired by Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity. The dimensionless parameter $\ensuremath{\lambda}$ 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 Bellor\'{\i}n and Restuccia, who showed that all nonprojectable $\ensuremath{\lambda}$-$R$ 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 $\ensuremath{\lambda}$-$R$ models by a physical, cosmological argument. We show that their analysis can be interpreted consistently within the projectable sector of Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity, thus resolving the apparent contradiction.