8 in itself. The formal equivalence of f ( R ) theories with scalar-tensor gravity has beeen discovered and rediscov- ered many times [18]. Metric f ( R ) gravity is equiva- lent to a Brans-Dicke theory with Brans-Dicke parame- ter ω = 0 and a non-trivial potential, while Palatini f ( R ) gravity is equivalent to an ω = − 3 / 2 Brans-Dicke theory with potential (see [7] for details). In the Palatini version, the scalar φ is non-dynamical, as has been pointed out in various works [7, 54, 56, 57]. The metric-affine version of f ( R ) gravity [6] is not equivalent to a scalar-tensor theory and will not be considered here. The contributions from the portions of the Cauchy hyper- surfaces S and S ′ cancel out because they have the same absolute value due to the time symmetry but opposite signs because of the opposite directions of the outgoing unit normal on these hypersurfaces. The contribution from spatial infinity vanishes because ˜ φ vanishes there, whereas the contribution from the integral over the por- tion of the horizon is supposed to vanish because the pro- jection of ∂ a ˜ φ along the null vector tangent to the hori- zon, which is a linear combination of t a and ψ $^{a}$, vanishes due to the symmetries [22]. This point is crucial: the ar- gument is not valid in general because the Einstein frame scalar ˜ φ may not be defined at the horizon and, indeed, this is the case for the Brans class I solutions (45) and (46) in which the Einstein frame scalar ˜ φ ∝ ln φ diverges on the horizon because the Jordan frame φ → 0 + there. The conformal transformation to the Einstein frame be- comes ill-defined at the horizon and its variables ( ˜ g$_{ab}$, ˜ φ ) cannot be used on this surface. Of course, nothing forbids to use the Jordan frame instead of the Einstein one, but then the scalar φ violates the weak and null energy con- ditions because its stress-energy tensor given by eq. (17) has a non-canonical structure containing second deriva- tives of φ instead of being quadratic in the first deriva- tives (indeed, it is now well known that nonminimally coupled scalar fields can violate all the energy conditions [53]). Let us consider now the ω = − 3 / 2 , V = 0 Brans-Dicke equivalent of Palatini f ( R ) gravity. In vacuo, electro- vacuo, or in any region in which the trace of the matter energy-momentum T ( m ) is constant, the d’Alembertian disappears from the field equation (18), which reduces to no longer a differential but an algebraic or trascenden- tal equation. If eq. (51) has solutions they are of the form φ =const. ≡ φ$_{0}$ . In this case the field equation (17) reduces to which describes General Relativity with a cosmological constant Λ = V ( φ$_{0}$ ) 2 φ$_{0}$ , for which Birkhoff’s theorem holds if the matter distribution described by T ( m ) ab is static (in- cluding the vacuum case T ( m ) ab = 0) which is consistent, of course, with the previous assumption T ( m ) =const. To summarize, the Jebsen-Birkhoff theorem holds in Palatini f ( R ) gravity with a static matter distribution; the fact that the Jebsen-Birkhoff theorem holds when T ( m ) ab = 0 is well known [7] and is due to the non- dynamical nature of the Brans-Dicke scalar present in this class of theories, which acts as an effective form of matter without dynamics (this feature has been studied in detail, see the discussion and references in [7]). Metric f ( R ) gravity is equivalent to an ω = 0 Brans- Dicke theory with a non-trivial potential. This time, in vacuo, the field equation (18) reduces to which is now a true dynamical equation. Since φ is dynamical and, in general, time-dependent the Jebsen- Birkhoff theorem is not valid, which has been noted on We are now ready to come back to f ( R ) gravity, which is the original motivation for our work, even though the understanding of the Jebsen-Birkhoff theorem and spherical symmetry in scalar-tensor gravity has merit Campanelli and Lousto [21] report static solutions of Brans-Dicke theory which possess a static but radially- dependent scalar. These authors comment on the vio- lation of the null energy condition in the Jordan frame being the cause of the violation of the no-hair theorem. However, this comment (echoed in [51]) appears to be a bit misleading because Hawking’s theorem discusses the Einstein frame scalar ˜ φ which does satisfy the energy con- dition, while Campanelli and Lousto refer to the Jordan frame φ which doesn’t. Their comment is technically cor- rect but, per se , does not help explaining the violation of Hawking’s theorem. Note that, if the Brans-Dicke scalar φ does not go to zero or diverges on the horizon, Hawk- ing’s theorem applies and the solution is forced to be Schwarzschild. The scalar field in the known solutions violating the Hawking theorem does turn out to be static (which is what [22] intended to prove), but it has a radial de- pendence and these solutions do not coincide with the Schwarzschild metric. R$_{ab}$ − 1 2 g$_{ab}$R = 8 π φ$_{0}$ T ( m ) ab − V ( φ$_{0}$ ) 2 φ$_{0}$ g$_{ab}$ , (52) □ φ = 1 3 [ φ dV dφ − 2 V ( φ ) ] , (53) Metric f ( R ) gravity THE JEBSEN-BIRKHOFF THEOREM IN f ( R ) GRAVITY 8 πGT ( m ) + φ dV dφ − 2 V ( φ ) = 0 , (51) Palatini f ( R ) gravity