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M. Bonetti and E. Barausse, “Post-Newtonian constraints on Lorentz-violating gravity theories with a MOND phenomenology,” Phys. Rev
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M. Crisostomi, M. Hull, K. Koyama, and G. Tasinato, “Horndeski: beyond, or not beyond?,” JCAP
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2021

Degenerate Horava gravity

Barausse, Enrico, Crisostomi, Marco, Liberati, Stefano et al.

Understand

Horava gravity breaks Lorentz symmetry by introducing a dynamical timelike scalar field (the khronon), which can be used as a preferred time coordinate (thus selecting a preferred space-time foliation).

Adopting the khronon as the time coordinate, the theory is invariant only under time reparametrizations and spatial diffeomorphisms.
In the infrared limit, this theory is sometimes referred to as khronometric theory.
Here, we explicitly construct a generalization of khronometric theory, which avoids the propagation of Ostrogradski modes as a result of a suitable degeneracy condition (although stability of the latter under radiative corrections remains an open question).