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Abstract

Weyl transverse gravity is a gravitational theory that is invariant under
transverse diffeomorphisms and Weyl transformations. It is characterised by
having the same classical solutions as general relativity while solving some of
its issues with the cosmological constant. In this work, we first find the
Noether currents and charges corresponding to local symmetries of Weyl
transverse gravity as well as a prescription for the symplectic form. We then
employ these results to derive the first law of black hole mechanics in Weyl
transverse gravity (both in vacuum and in the presence of a perfect fluid),
identifying the total energy, the total angular momentum, and the Wald entropy
of black holes. We further obtain the first law and Smarr formula for
Schwarzschild-anti-de Sitter and pure de Sitter spacetimes, discussing the
contributions of the varying cosmological constant, which naturally appear in
Weyl transverse gravity. Lastly, we derive the first law of causal diamonds in
vacuum.