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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. ( m ) 

It might seem that the assumptions on the matter T 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 

<img src='content_image/379204.jpg'>

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 

<img src='content_image/379203.jpg'>

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 

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. 

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. 

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 vacuum case 

