Lorentz-violating graviton masses: getting around ghosts, low strong coupling scale and VDVZ discontinuity

A theory with the action combining the Einstein--Hilbert term and graviton mass terms violating Lorentz invariance is considered at linearized level about Minkowskian background. It is shown that with one of the masses set equal to zero, the theory has the following properties: (i) there is a gap of order $m$ in the spectrum, where $m$ is the graviton mass scale; (ii) the dispersion relations at ${\bf p}^2 \gg m^2$ are $\omega^2 \propto {\bf p}^2$, the spectrum of tensor modes being relativistic, while other modes having unconventional maximum velocity; (iii) the VDVZ discontinuity is absent; (iv) the strong coupling scale is $(mM_{Pl})^{1/2}$. The latter two properties are in sharp contrast to the Lorentz-invariant gravity with the Pauli--Fierz mass term.