Abstract
We present a systematic study of the geometric structure of non-singular spacetimes describing black holes in Lorentz-violating gravity. We start with a review of the definition of trapping horizons, and the associated notions of trapped and marginally trapped surfaces, and then study their significance in frameworks with modified dispersion relations. This leads us to introduce the notion of universally marginally trapped surfaces, as the direct generalization of marginally trapped surfaces for frameworks with infinite signal velocities (Hořava-like frameworks), which then allows us to define universal trapping horizons. We find that trapped surfaces cannot be generalized in the same way, and discuss in detail why this does not prevent using universal trapping horizons to define black holes in Hořava-like frameworks. We then explore the interplay between the kinematical part of Penrose’s singularity theorem, which implies the existence of incomplete null geodesics in the presence of a focusing point, and the existence of multiple different metrics. This allows us to present a complete classification of all possible geometries that neither display incomplete physical trajectories nor curvature singularities. Our main result is that not all classes that exist in frameworks in which all signal velocities are realized in Hořava-like frameworks. However, the taxonomy of geodesically complete black holes in Hořava-like frameworks includes diverse scenarios such as evaporating regular black holes, regular black holes bouncing into regular white holes, and hidden wormholes.
Highlights
The classification in [14, 15] describes alternative classes of metrics which fail to satisfy some additional, reasonable, physical criteria, e.g. due to not being globally hyperbolic, or due to describing black holes incompatible with semiclassical physics
The unit timelike vector field defines a preferred frame throughout spacetime, and all the degrees of freedom of the theory propagate with finite speeds with respect to this preferred frame
A particular way to construct a geometry in this class is taking one point in the universally marginally outer trapped surface (UMOTS)/universally marginally inner trapped surface (UMITS) and push it to infinite affine distance, which would be the equivalent of the behavior of marginally outer trapped surface (MOTS)/marginally inner trapped surface (MITS) in the everlasting horizons class discussed in [15]
Summary
The study of the causal structure of modified gravity theories [36] shows that one must distinguish between frameworks in which all propagating modes have finite signal velocities and frameworks in which at least one propagating mode has an infinite signal velocity. We start by analyzing the former framework which, since all its signal velocities are finite, is closer to general relativity. The spacetimes of interest describe the collapse of a regular distribution of matter from a given initial Cauchy surface with topology R3. On top of this structure we will define a metric gab and a preferred vector field na, both assumed to satisfy suitable regularity conditions
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