Motivated by statements in the literature which contradict two general theorems, the static and spherically symmetric Brans solutions of scalar-tensor gravity are analyzed explicitly in both the Jordan and the Einstein conformal frames. Depending on the parameter range, these solutions describe wormholes or naked singularities but not black holes.

The Noether charge associated to diffeomorphism invariance in teleparallel gravity is derived. It is shown that the latter yields the Arnovitt-Deser-Misner mass of an asymptotically flat spacetime. The black hole entropy is then investigated based on Wald’s prescription that relies on the Noether charge. It is shown that, like in general relativity, the surface gravity can be factored out from such a charge. Consequently, the similarity with the first law of thermodynamics implied by such an approach in general relativity does show up also in teleparallel gravity. It is found that, based on the expression of the first law of black hole mechanics, which is preserved in teleparallel gravity, entropy can thus be extracted from such a Noether charge. The resulting entropy can very naturally be expressed as a volume integral, though. As such, it is shown that the conformal issue that plagues the entropy-area law within general relativity does not arise in teleparallel gravity based on Wald’s approach. The physics behind these features is discussed.

We report the explicit form of the general static, spherically symmetric, and asymptotically flat solution of vacuum Brans-Dicke gravity in the Jordan frame, assuming that the Brans-Dicke scalar field has no singularities or zeros (except possibly for a central singularity). This general solution is conformal to the Fisher-Wyman geometry of Einstein theory and its nature depends on a scalar charge parameter. Apart from the Schwarzschild black hole, only wormhole throats and central naked singularities are possible.

The conformal transformation of the Hawking–Hayward quasi-local mass is re-examined. It has been found recently that the conformal transformation of the latter exhibits the 'wrong' conformal factor compared to the way usual masses transform under conformal transformations of spacetime. We show, in analogy with what was found recently for the Misner–Sharp mass, that unlike the purely geometric definition of the Hawking–Hayward mass, the latter exhibits the 'right' conformal factor whenever expressed in terms of its material content via the field equations. The case of conformally invariant scalar–tensor theories of gravity is also examined. The equivalence between the Misner–Sharp mass and the Hawking–Hayward mass for spherically symmetric spacetimes manifests itself by giving identical peculiar behaviors under conformal transformations.