The Jebsen-Birkhoff theorem in alternative gravity Valerio Faraoni ∗ Physics Department, Bishop’s University Sherbrooke, Qu´ ebec, Canada J1M 1Z7 We discuss the validity, or lack thereof, of the Jebsen-Birkhoff theorem in scalar-tensor theories by generalizing it and regarding the Brans-Dicke-like scalar as effective matter. Both the Jordan and Einstein frames are discussed and an apparent contradiction between static spherical solutions of scalar-tensor gravity and Hawking’s theorem on Brans-Dicke black holes is clarified. The results are applied to metric and Palatini f ( R ) gravity. PACS numbers: 04.50.Kd, 04.20.Cv, 04.70.Bw understanding in general scalar-tensor gravity, which is the goal of the present paper. Beginning with the realiza- tion that the Brans-Dicke-like scalar field of scalar-tensor gravity can, and most often is, regarded as an effective form of matter for effective Einstein equations, one is led to formulate a generalized Jebsen-Birkhoff theorem in the presence of matter in General Relativity, which is convenient for the study of spherical symmetry in scalar- tensor gravity (this is done in Secs. II and III using the Jordan frame). An alternative approach to scalar-tensor gravity consists of using its Einstein frame formulation, which is employed in Sec. IV. In this regard, an apparent contradiction between certain long-known exact spherical solutions of scalar-tensor theory [19, 21] and Hawking’s theorem for Brans-Dicke black holes [22] is elucidated. Almost as a byproduct of this work, light is shed on the validity, or lack thereof, of the Jebsen-Birkhoff theorem of Palatini and metric f ( R ) gravity in Sec. V. We adopt the notations of Ref. [9]. The cosmologi- cal constant Λ, if present, is seen as an effective form of matter described by the formal stress-energy tensor T (Λ) ab = − Λ 8 πG g$_{ab}$ (that is, in the following, “vacuum” implies that Λ = 0). The most general spherically symmetric line element can be written as where d Ω 2 = dθ 2 + sin $^{2}$θ dϕ 2 is the line element on the unit 2-sphere and r is an areal radius. The metric com- ponents g$_{0}$$_{i}$ ( i = 1 , 2 , 3), if present, can be eliminated by redefining the coordinates t and r ([17], see also [23]). The (0 , 1) , (0 , 0) , (1 , 1), (2 , 2), and (3 , 3) components of the Einstein field equations The recent attempts to explain the acceleration of the universe discovered with type Ia supernovae [1] have led to the introduction of an ad hoc dark energy comprising 76% of the energy content of the universe and with an extreme (possibly phantom) equation of state P ≃ − ρ , where ρ and P are the effective energy density and pres- sure of the dark energy [2]. Displeased with this ex- planation, many authors have turned to the possibility that dark energy does not exist and, instead, gravity must be modified in the infrared sector [3, 4] and the so-called f ( R ) gravity theories have been resurrected for this purpose. f ( R ) gravity theories in use in cosmol- ogy come in three versions, called the metric formalism [3, 4], Palatini formalism [5], and metric-affine gravity [6] (see [7] for a recent review and [8] for shorter introduc- tions). The study of spherically symmetric solutions has been crucial to understand the weak-field limit of these theories and confront them with Solar System observa- tions (remember that the three classical tests of Gen- eral Relativity are based on the spherically symmetric Schwarzschild solution [9, 10]). Morever, the study of strong field instabilities potentially fatal for the theory is carried out in spherical symmetry [11]. It is therefore im- portant to understand how the Jebsen-Birkhoff theorem well known from spherical symmetry in General Relativ- ity generalizes, or fails, in these extensions of Einstein’s theory. Students beginning in General Relativity are fa- miliar with Birkhoff’s theorem stating that a spherically symmetric solution of the vacuum Einstein equations is static [13] (it appears [14–16] that this result had actu- ally been discovered by Jebsen two years before Birkhoff [17]). It is now well-known that the Jebsen-Birkhoff the- orem does not hold in metric f ( R ) gravity while it is valid in Palatini f ( R ) gravity [7]. Since both metric and Palatini f ( R ) gravity admit a description as special scalar-tensor theories (Brans-Dicke theories with values of the Brans-Dicke parameter ω = 0 and ω = − 3 / 2, re- spectively, and endowed with a special potential for the Brans-Dicke scalar [7, 18]), a complete understanding of the Jebsen-Birkhoff theorem in f ( R ) gravity requires its arXiv:1001.2287v1 [gr-qc] 13 Jan 2010 INTRODUCTION THE JEBSEN-BIRKHOFF THEOREM IN GENERAL RELATIVITY ds 2 = − A $^{2}$( t, r ) dt 2 + B $^{2}$( t, r ) dr 2 + r $^{2}$d Ω 2 $_{2}$, (1) R$_{ab}$ − 1 2 g$_{ab}$R = 8 πG T ( m ) ab , (2)