content_image/398906.jpg content_image/398905.jpg content_image/398904.jpg content_image/398903.jpg content_image/398902.jpg content_image/398900.jpg content_image/398899.jpg content_image/398898.jpg content_image/398897.jpg content_image/398910.jpg content_image/398907.jpg content_image/398909.jpg content_image/398908.jpg
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 

the only non-vanishing Christoffel symbols are 

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and the d’Alembertian of φ is 

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The (0 , 1) , (0 , 0), and (1 , 1) components of the field equa- tions (17) yield 

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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 

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time-independent ( φ = φ ( r )), eq. (28) yields 

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 

in vacuo and B = B ( r ), therefore we focus on the ˙ B = 0 situation. Eq. (30) reduces to 

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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). 

## THE JEBSEN-BIRKHOFF THEOREM IN THE EINSTEIN FRAME REPRESENTATION OF SCALAR-TENSOR GRAVITY 

By performing the conformal transformation of the metric and redefining non-linearly the Brans-Dicke-like 

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