More on spacetime thermodynamics in the light of Weyl transformations

It was recently found that, after performing a Weyl conformal transformation, the familiar analogy between black hole mechanics and black hole thermodynamics becomes ambiguous. It was argued that this fact can be traced back to the fundamental dichotomy between matter and geometry, which is at the heart of Einstein's field equations. As a further study of this issue, we investigate here the general link between spacetime thermodynamics and Weyl transformations from two other angles. We first examine the conformal behaviour of the horizon entropy within Wald's approach based on the fundamental diffeomorphism symmetry of $pure$ geometry. We then revisit $-$ using Weyl transformations $-$ Jacobson's derivation of Einstein's field equations, the starting point of which is precisely built on the fundamental dichotomy between matter and geometry. As a result, we show that in order for Jacobson's approach to be able to yield the right Einstein field equations in the conformal frame, a specific conformal behavior of the horizon temperature and its entropy $-$ different from what Wald's approach implies $-$ is required. The two approaches to spacetime thermodynamics become thus incompatible in the conformal frame. In addition, we show $-$ in greater detail in the conformal frame $-$ that in the presence of a null dust the thermodynamics approach for extracting Einstein's field equations necessarily fails. An extensive discussion of the whole issue is given.