Master field equations for spherically symmetric gravitational fields beyond general relativity

I construct a set of master field equations encompassing all possible second-order field equations for spherically symmetric spacetimes beyond general relativity. The structure of 2-dimensional Horndeski theory is leveraged to define a spherically symmetric gravitational tensor that is identically conserved and can play an equivalent role as the Einstein tensor in the coupling to spherically symmetric stress-energy tensors. The master field equations provide a framework in which to study a wide variety of previously intractable problems and prove general results across different theories, as illustrated with the example of the Jebsen--Birkhoff theorem that holds generally. This work sets up the stage for the study of the dynamics of spherically symmetric spacetimes beyond general relativity and provides powerful tools for the exploration of a new paradigm of black hole physics.