3 discussing the spherically symmetric solutions of scalar- tensor gravity because the Brans-Dicke-like scalar field present in these theories acts as an effective form of mat- ter. goes through zero to change sign in the interior [62]. The theorem does not make statements valid on a black hole horizon surface which may be present and it does not imply that the solution of the Einstein equations is the Schwarzschild metric. A solution with a matter distribu- tion diverging on a horizon is possible; this happens, for example, for the Brans class I solutions of scalar-tensor gravity [19] (in which the Brans-Dicke-like scalar field can be regarded as a form of effective matter) which are static and spherically symmetric but have a scalar field diverging at the horizon—see Sec. IV. By regarding vacuum ( T ( m ) ab = 0) as a form of static matter, the Jebsen-Birkhoff theorem assumes its most familiar form. Jebsen-Birkhoff theorem (version 2): a spher- ically symmetric solution of the vacuum Einstein equations is necessarily static in a region in which t remains timelike and ( r, θ, ϕ ) stay spacelike. It might seem that the assumptions on the matter T ( m ) ab are too strong because this tensor contains the metric g$_{ab}$ and, therefore, any assumption on T ( m ) ab means to require already, in some way, that the metric g$_{ab}$ is static. Indeed, this is true but the assumption of a static matter distribu- tion leaves room for physically relevant situations. The first is vacuum, in which no-matter is necessarily (and trivially) of the static form and gives rise to the more fa- miliar version of the Jebsen-Birkhoff theorem upon which we comment in the next subsection. Another non-trivial situation is that of a cosmological constant seen as a form of effective matter described by T (Λ) ab , which is spherically symmetric and static. In this case the solution is not the Schwarzschild but the Schwarzschild-(anti)de Sitter one. It is well known that, for example, the Schwarschild-de Sitter (Kottler) line element can be put in the locally static form The vacuum assumption rules out the possibility of a cosmological constant as well as electro-vacuum, and the Schwarzschild-(anti)de Sitter and Reissner- Nordstrom solutions. The solution is then forced to be Schwarzschild. Note that the vacuum as defined by T$_{ab}$ = 0 is not necessarily a trivial configuration in alternative theories of gravity. In scalar-tensor gravity the Brans-Dicke-like scalar field φ describing the gravitational field together with the metric g$_{ab}$ may be non-constant and still give a vanishing effective stress-energy tensor T ( φ ) ab ; this is re- ferred to as a “non-gravitating” or “stealth” scalar field. Two examples of massive waves of a φ -field coupled non- minimally to the Ricci curvature and with a potential are given in [29]; they achieve T ( φ ) ab = 0 and no other form of matter is present. As a result, the spacetime is Minkowskian (spherically symmetric and static), provid- ing a non-trivial realization of “vacuum” and of version 2 of the Jebsen-Birkhoff theorem. Other examples of non- gravitating matter distributions are given in Refs. [30]. In the language of field theory, the field content of Gen- eral Relativity consists only of a massless spin two field and gravitational radiation is quadrupole to lowest order. Hence spherically symmetric sources cannot excite grav- itational radiation and the gravitational charge ( i.e. , the mass-energy) of the source is conserved—the spacetime around a spherically symmetric source must be static. It is usually remarked that the Jebsen-Birkhoff theo- rem allows, as a corollary, an extension to General Rel- ativity of the iron sphere theorem of Newtonian gravity stating that the gravitational field of a spherically sym- metric distribution of mass inside a spherical cavity is zero. In spherical symmetry, if the energy distribution inside a cavity is static, the solution of the Einstein equa- tions will be static. In vacuo, it is Minkowski space, i.e. , the Schwarzschild solution corresponding to zero mass. If matter inside the cavity consists only of a cosmological constant the interior solution is spatially homogeneous The absence of a radial energy current, J$_{(}$$_{r}$$_{)}$ = 0, with- out the assumption of staticity of ρ and P$_{(}$$_{r}$$_{)}$ is not suf- ficient to guarantee that the metric is static. A coun- terexample is the McVittie solution describing a spher- ical object embedded in a cosmological background [28] for which J$_{(}$$_{r}$$_{)}$ = 0 but the metric is time-dependent except for the special case in which it reduces to the Schwarzschild-(anti)de Sitter one. Although less familiar than the vacuum version, ver- sion 1 of the Jebsen-Birkhoff theorem is more suitable for in the region of spacetime manifold between the black hole and the cosmological horizons. Another physical situation allowed by the assumptions of the theorem is electro-vacuum [27]. If the mass distribution carries a static electric charge Q with no radial current the solution is the static Reissner-Nordstrom metric given by The vacuum case ds 2 = − ( 1 − 2 GM r + Q 2 r 2 ) dt 2 + ( 1 − 2 GM r + Q 2 r 2 ) − 1 dr 2 + r $^{2}$d Ω 2 $_{2}$. (15) ds 2 = − ( 1 − 2 GM r − Λ r 2 3 ) dt 2 + ( 1 − 2 GM r − Λ r 2 3 ) − 1 dr 2 + r $^{2}$d Ω 2 (14)