Emergent modified gravity: The perfect fluid and gravitational collapse

Canonical gravity is a formulation of general relativity where an evolving space reproduces the usual spacetime theory, and, unlike other formulations, it requires fewer assumptions---in particular, in the relationship between the gravitational field and the metric field. This freedom in the assumptions is the key to emergent modified gravity, a canonical theory in which the spacetime metric is not fundamental, but rather it is an emergent object composed of the truly fundamental fields. This peculiar feature of emergent modified gravity compared to other approaches can be used to model new effects, such as those provided by quantum gravity in an effective description. We discuss how matter fields can be coupled to emergent modified gravity, we realize the coupling of the (isentropic) perfect fluid, and we explicitly obtain the Hamiltonian of spherically symmetric systems and identify the symmetries of the coupled system. We formulate the Oppenheimer-Snyder collapse model in canonical terms as an adaptation of the fluid frame to the canonical foliation, permitting us to easily extend the conditions of the model to emergent modified gravity and obtain an exact solution to the dust collapsing from spatial infinity, including some effects suggested by quantum gravity in spherical symmetry. In this solution, the collapsing dust forms a black hole, and then reaches a minimum radius inside the black hole. While the geometry on this minimum-radius surface is regular in the vacuum, it turns out to be singular in the presence of dust. However, the fact that the geometry is merely emergent in this picture, and all the fundamental fields that compose the phase space are regular on such a surface, allows us to continue the canonical solution past it in a meaningful way, obtaining the global structure for the interior of the star. Thus, the matter reaches the minimum-radius surface with vanishing velocity and finite positive acceleration, and it proceeds to emerge outward, now behaving as a white hole. This star-interior solution can then be complemented by the vacuum solution describing the star-exterior region by a continuous junction at the star's radius. This gluing process can be viewed as the imposition of boundary conditions, which is nonunique and does not follow from solving the equations of motion alone. This ambiguity in the gluing of the two regions can give rise to different physical outcomes of the collapse compatible with the canonical dynamics. We discuss two such phenomena: the formation of a wormhole characterized by two distinct asymptotic regions, and the transition from a black hole to a white hole characterized by a single asymptotic region.