5 time-independent ( φ = φ ( r )), eq. (28) yields Static (but non-constant) Brans-Dicke-like field Let us assume that the spacetime metric which solves the field equations (17) and (18) is spherically symmetric with line element (1). Then so that either ˙ B = 0 or φ ( r ) = C/r $^{2}$, where C > 0 is a constant. If φ = C/r 2 then eq. (29) yields the only non-vanishing Christoffel symbols are in vacuo and B = B ( r ), therefore we focus on the ˙ B = 0 situation. Eq. (30) reduces to and the d’Alembertian of φ is If ∂T ( m ) 11 /∂t = 0 (in particular in vacuo), the right hand side of this equation is t -independent, implying that A can depend from t at most through a multiplicative factor as in A ( t, r ) = f ( t ) a ( r ). In this case a redefinition of the time coordinate d ¯ t = f ( t ) dt absorbs the time-dependent factor into ¯ t and the metric can be cast in locally static form. Then, the other field equations imply that also the radial pressures T ( m ) ii ( i = 1 , 2 , 3) are time-independent. The fact that the scalar field φ needs to be static or non-gravitating in order to rescue the Jebsen-Birkhoff theorem was established on several occasions for partic- ular scalar-tensor theories. Sch¨ ucking [34] derived the result for Jordan’s theory, a precursor of Brans-Dicke the- ory [31]; Reddy [37] studied the electro-vacuum case of the Sen-Dunn theory [63] and of the conformally cou- pled scalar field [45] (the former was revisited in [41] and the latter in [42]; the Sen-Dunn theory was reconsidered, and some errors of [37] corrected, in [38]). Venkateswarlu and Reddy studied electrovacuum in more general scalar- tensor theories [46]. To summarize, when the Brans-Dicke-like scalar is static but non-constant, the solution of the field equa- tions can be different from Schwarzschild-(anti)de Sitter. If this field is constant the theory reduces to General Rel- ativity, for which the Jebsen-Birkhoff theorem holds. As a corollary, the spherically symmetric solution inside an empty cavity is static only if the scalar field φ is assumed to be static or non-gravitating (Minkowski space if there is no potential and (anti-)de Sitter space if the potential is non-zero). By performing the conformal transformation of the metric and redefining non-linearly the Brans-Dicke-like If there are no radial energy currents ( T ( m ) 01 = 0) and, in particular, in vacuo ( T ( m ) ab = 0) and assuming that φ is THE JEBSEN-BIRKHOFF THEOREM IN THE EINSTEIN FRAME REPRESENTATION OF SCALAR-TENSOR GRAVITY 2 A ′ Ar − B 2 r 2 + 1 r 2 = 8 π φ T ( m ) 11 + ω φ 2 ( φ ′ 2 + B 2 A 2 ˙ φ 2 ) + B 2 A $^{2}$φ ( ¨ φ − ˙ A A ˙ φ − AA ′ B 2 φ ′ − 2 A 2 B $^{2}$r φ ′ ) − V B 2 2 φ . (30) A 2 ( 1 r 2 + 2 B ′ B $^{3}$r − 1 B $^{2}$r2 ) = 8 π φ T ( m ) 00 + ω 2 φ 2 ( ˙ φ 2 + A 2 B 2 φ ′ 2 ) + A 2 B $^{2}$φ ( φ ′′ − B ˙ B A 2 ˙ φ − B ′ B φ ′ + 2 φ ′ r ) + V A 2 2 φ , (29) 2 ˙ B Br = 8 π φ T ( m ) 01 + ω ˙ φφ ′ φ 2 + 1 φ ( ˙ φ ′ − A ′ A ˙ φ − ˙ B B φ ′ ) , (28) □ φ = − 1 A 2 ( ¨ φ − ˙ A A ˙ φ − AA ′ B 2 φ ′ ) + 1 B 2 ( φ ′′ − B ˙ B A 2 ˙ φ − B ′ B φ ′ ) + 2 φ ′ rB 2 . (27) The (0 , 1) , (0 , 0), and (1 , 1) components of the field equa- tions (17) yield Γ 3 13 = Γ 3 31 = 1 r , Γ 3 23 = Γ 3 32 = cos θ sin θ , (26) Γ 2 12 = Γ 2 21 = 1 r , Γ 2 33 = − sin θ cos θ , (25) Γ 1 22 = − r B 2 , Γ 1 33 = − r B 2 sin $^{2}$θ , (24) Γ 1 00 = AA ′ B 2 , Γ 1 01 = Γ 1 10 = ˙ B B , Γ 1 11 = B ′ B , (23) Γ 0 00 = ˙ A A , Γ 0 01 = Γ 0 10 = A ′ A , Γ 0 11 = B ˙ B A 2 (22) A ′ A ( 2 r + φ ′ φ ) = 8 π φ T ( m ) 11 + B 2 − 1 r 2 + ω ( φ ′ φ ) 2 − 2 r φ ′ φ − B $^{2}$V 2 φ . (33) B 2 = 2 C (2 ω + 3) 2 C + V r 4 (32) ˙ B B ( 2 r + φ ′ φ ) = 0 , (31) ∇ $^{c}$φ ∇ $_{c}$φ = − ˙ φ 2 A 2 + φ ′ 2 B 2 , (21)