The generalized first law for more general matter

A bstract In previous work, a first law of generalized entropy was derived from semiclassical gravitational dynamics around thermal setups using an assumed relation between the matter modular Hamiltonian and stress tensor. Allowing for non-minimal coupling curvature any tensor fields, we show however, that states is given by integrated bulk Noether current associated to time translation plus spacetime boundary term. One generally cannot express this in terms components. Still, working with correct expression Hamiltonian, are able recover entropy, added benefits over result. Firstly, Wald-Dong contributions resulting included. Secondly, equations motion, allow non-vanishing expectation value unperturbed background state, account field perturbations as part its variation. Finally, quantum allowed contribute nontrivially asymptotic energy, e.g. necessary, even minimally coupled Maxwell field, expected thermodynamic charged black holes.