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Abstract

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.