Covariant Collapse

Gravitational collapse to black holes is one of the main examples of physical phenomena that rely on properties of space-time. It leads to large densities and curvature and is therefore expected to be sensitive to quantum effects at least in its final stages, but consistently combining quantum physics with space-time structure is a notoriously difficult problem. Nevertheless, the approach to high curvature may be studied by effective or modified theories of gravity that aim to implement consistent space-time descriptions of dynamical equations different from the classical ones of general relativity. An important set of consistency conditions is implied by the requirement that modified dynamics of space-time should, at least in an effective description, maintain a valid notion of general covariance in the sense that solutions of the theory can still be analyzed by suitable line elements. If this condition were violated, several concepts relevant for gravitational collapse and black holes, such as horizons or curvature singularities, would no longer be available in the usual form. A detailed analysis shows that the covariance condition, applied in a canonical formulation, is restrictive and rules out several proposals based only on an analysis of spatially homogeneous models for the Schwarzschild interior. At the same time, the condition does allow new and previously unrecognized features of gravitational collapse in modified or quantum gravity.