Lorentz invariance (LI) is a cornerstone of modern physics, and strongly supported by observations.
In fact, all the experiments carried out so far are consistent with it, and no evidence to show that such a symmetry needs to be broken at a certain energy scale. Nevertheless, there are various reasons to construct gravitational theories with broken LI. In particular, our understanding of space-times at Plank scale is still highly limited, and the renomalizability and unitarity of gravity often lead to the violation of LI.
One concrete example is the Horava theory of quantum gravity, in which the LI is broken in the ultraviolet (UV), and the theory can include higher-dimensional spatial derivative operators, so that the UV behavior is dramatically improved and can be made (power-counting) renormalizable.
On the other hand, the exclusion of high-dimensional time derivative operators prevents the ghost instability, whereby the unitarity of the theory—a problem that has been faced since 1977 [ K.S. Stelle, Phys. Rev. D 16, 953 (1977)]—is assured. In the infrared (IR) the lower dimensional operators take over, whereby a healthy low-energy limit is presumably resulted.
However, once LI is broken different species of particles can travel with different velocities, and in certain theories , such as the Horava theory mentioned above, they can be even arbitrarily large. This suggests that black holes may not exist at all in such theories, as any signal initially trapped inside a horizon can penetrate it and propagate to infinity, as long as the signal has sufficiently large velocity (or energy). This seems in a sharp conflict with current observations, which strongly suggest that black holes exist in our universe [R. Narayan and J.E. MacClintock, Mon. Not. R. Astron. Soc., 419, L69 (2012)].
A potential breakthrough was made recently by Blas and Sibiryakov [D. Blas and S. Sibiryakov, Phys. Rev. D84, 124043 (2011)], who found that there still exist absolute causal boundaries, the so-called universal horizons, and particles even with infinitely large velocities would just move around on these boundaries and cannot escape to infinity.
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