4 and isotropic about every point and, therefore, (anti-)de Sitter. Einstein equations as With the advent of the Jordan [31] and Brans-Dicke [32] theories first and of scalar-tensor theories later [33], the validity of the Jebsen-Birkhoff theorem was investi- gated in alternative gravity [34–39, 41–46]. where T ( m ) ab is the matter energy-momentum tensor and we assume that φ > 0 in conjunction with ω > − 3 / 2 to guarantee the positivity of the effective gravitational coupling [47] In general, the theorem does not hold in scalar-tensor gravity: one needs to impose that the effective stress- energy tensor T ( φ ) ab of the Brans-Dicke-like scalar field of the theory is time-independent in order for the metric to be static (this requirement is usually achieved by impos- ing that φ is time-independent, but is also obtained by a stealth field φ [29, 30]). The failure of the theorem in the presence of time-dependent scalars opens the door for new phenomenology in scalar-tensor gravity which is un- known in General Relativity. The failure of the Jebsen- Birkhoff theorem is to be expected: since scalar-tensor gravity has a new spin zero degree of freedom in compar- ison with General Relativity, scalar monopole radiation can occur. In Einstein’s theory monopole radiation is for- bidden by the Jebsen-Birkhoff theorem, which is a con- sequence of the fact that the gravitational field is repre- sented only by a spin two field. Since in Einstein’s theory gravitational radiation is necessarily quadrupole to low- est order, spherically symmetric pulsating sources cannot generate gravitational radiation and the metric must be static. This is no longer true in scalar-tensor gravity, in which the time-varying monopole moment of a radially pulsating spherical source generates propagating scalar radiation which makes also the metric non-static. Let us consider a scalar-tensor theory of gravity de- scribed by the action In this form, it is easy to see that the scalar field φ acts as an effective form of matter in the field equa- tions (17) and therefore, by imposing that the matter stress-energy tensor T ( m ) ab vanishes, one is left with an effective stress-energy tensor T ( φ ) ab such that T ( φ ) 00 could be time-dependent and T ( φ ) 0 i = 0 if φ depends on time. In other words, a time-dependent Brans-Dicke-like field φ spoils the validity of the Jebsen-Birkhoff theorem and only the assumption that φ is time-independent (or that T ( φ ) ab is static, as for a time-dependent stealth field φ ) re- stores the staticity of a spherically symmetric solution. This can be checked explicitly using the field equations, which we do in the following. The case φ =const. ≡ φ$_{0}$ > 0 is trivial and eq. (17) reduces to where V$_{0}$ ≡ V ( φ$_{0}$ ), so that the theory degenerates to General Relativity with the cosmological constant Λ ≡ V$_{0}$/ (2 φ$_{0}$ ). If T ( m ) ab is such that the energy distribution is static (including the case T ( m ) ab = 0), version 1 of the Jebsen-Birkhoff theorem holds and the metric is static in the region in which the coordinate gradients preserve their causal character. The same happens if T ( φ ) ab vanishes with φ =const. The field equations can be written in the form of effective R$_{ab}$ − 1 2 g$_{ab}$R = 8 π φ$_{0}$ T ( m ) ab − V$_{0}$ 2 φ$_{0}$ g$_{ab}$ (20) S$_{ST}$ = 1 16 π ∫ d 4 $_{x}$√ − g [ φR − ω ( φ ) φ g $^{ab}$∇ $_{a}$φ ∇ $_{b}$φ − V ( φ ) ] + S ( m ) . (16) G$_{eff}$ = 2 ( ω + 2) 2 ω + 3 1 φ . (19) (2 ω + 3) □ φ = 8 πT ( m ) − dω dφ ∇ $^{c}$φ ∇ $_{c}$φ + φ dV dφ − 2 V , (18) R$_{ab}$ − 1 2 g$_{ab}$R = 8 π φ T ( m ) ab + ω ( φ ) φ 2 ( ∇ $_{a}$φ ∇ $_{b}$φ − 1 2 g$_{ab}$ ∇ $^{c}$φ ∇ $_{c}$φ ) + 1 φ ( ∇ a ∇ $_{b}$φ − g$_{ab}$ □ φ ) − V ( φ ) 2 φ g$_{ab}$ ≡ 8 π φ ( T ( m ) ab + T ( φ ) ab ) , (17) THE JEBSEN-BIRKHOFF THEOREM IN SCALAR-TENSOR GRAVITY The trivial case φ = constant