Noether charge formalism for Weyl invariant theories of gravity

Gravitational theories invariant under transverse diffeomorphisms and Weyl transformations have the same classical solutions as the corresponding fully diffeomorphism invariant theories. However, they solve some of the problems related to the cosmological constant and in principle allow local energy non-conservation. In the present work, we obtain the Noether charge formalism for these theories. We first derive expressions for the Noether currents and charges corresponding to transverse diffeomorphisms and Weyl transformations, showing that the latter vanish identically. We then use these results to obtain an expression for a perturbation of a Hamiltonian corresponding to evolution along a transverse diffeomorphism generator. From this expression, we derive the first law of black hole mechanics, identifying the total energy, the total angular momentum, the Wald entropy, and the contributions of the cosmological constant perturbations and energy non-conservation. Lastly, we extend our formalism to derive the first law of causal diamonds.