In this talk, I will present a series of results concerning general covariance in effective quantum black hole (BH) models. Within the framework of spherically symmetric models, general covariance is precisely formulated as a set of equations, leading to necessary and sufficient conditions for a generally covariant theory. Using these conditions, we derive equations for the effective Hamiltonian constraint and solve them to obtain the most general form of the Hamiltonian constraint describing generally covariant BH models.

Remarkably, the results show that any one-parameter family of static, spherically symmetric spacetime metrics can be embedded as the unique vacuum solution set of a generally covariant theory of pure gravity. This provides a complete resolution to the inverse problem of Birkhoff’s theorem and establishes a systematic method for reconstructing covariant dynamics from a given geometry. Finally, we apply this framework to loop quantum gravity-inspired spacetimes and derive the corresponding covariant effective Hamiltonian constraint.

Inviter: Jie-Qiang Wu