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## Hawking’s theorem and the Jebsen-Birkhoff theorem in Brans-Dicke gravity 

Hawking’s theorem [22] states that a stationary space- time containing a black hole is a solution of the Brans- Dicke field equations (with V = 0) if and only if it is a solution of the Einstein field equations, and therefore it must be axially symmetric or static. The theorem is usually taken to mean that Brans-Dicke black holes are exactly the same as those of General Relativity: this is 

an overstatement and in fact many solutions of scalar- tensor theories including, but not necessarily limited to, Brans-Dicke gravity are known which describe black holes with a static scalar field and do not coincide with the Schwarzschild metric. For these solutions the scalar field either goes to zero or diverges on an event or apparent horizon and this feature invalidates the proof of Hawk- ing’s theorem, as we shall see below. The most well known example is probably that of Brans’ class I solu- tions given by [19] 

<img src='content_image/382473.jpg'>

<img src='content_image/382474.jpg'>

where µ, C, φ$_{0}$ and λ are constants with 

Clearly, this metric is static and not the Schwarzschild solution while it satisfies the vacuum Brans-Dicke field equations with V = 0 for r > µ (three other classes of spherically symmetric solutions were found by Brans [19], although they are not all independent from each other [51]). Note that for the positive values of C and λ usually considered, the Brans-Dicke scalar field goes to zero on the horizon. 

Let us consider now the proof of Hawking’s theorem, which is performed in the Einstein frame [22]. Another theorem [52] states that a stationary black hole in Gen- eral Relativity must be axisymmetric and have spherical topology and relies on the weak (or the null) energy con- dition being satisfied. Hawking’s theorem extends this previous theorem to Brans-Dicke theory [22], aiming to prove that the scalar field is static. It is pointed out in [22] that the advantage of going to the Einstein frame is that the rescaled Brans-Dicke scalar 

(obtained by integrating eq. (35), where φ ∗ is a constant) has canonical kinetic energy density and obeys the weak 

<img src='content_image/382472.jpg'>

<img src='content_image/382471.jpg'>

and null energy conditions. The assumption that space- time is stationary then implies that it is also axially sym- metric [52] and, therefore, there exist a timelike Killing field t a and a spacelike Killing field ψ a (outside the hori- zon) and the Einstein frame scalar ˜ φ must necessarily be constant along the orbits of t a and ψ a in order to re- spect these symmetries, hence ∂ a ˜ φ can only be spacelike or zero outside the horizon. 

Consider now a 4-dimensional volume V bounded by two Cauchy hypersurfaces S and S ′ at two consecutive instants of time, a portion of the black hole event hori- zon, and spatial infinity [22]: the Einstein frame equa- tion of motion (39) in vacuo and with V = 0 becomes ˜ □ ˜ φ = 0, where ˜ □ ≡ ˜ g ab ˜ ∇ $_{a}$˜ ∇ b is the Einstein frame d’Alembertian. Multiplying this equation by ˜ φ , integrat- ing over V , and using the Gauss theorem and the identity ˜ φ ˜ □ ˜ φ = ˜ ∇ c ( ˜ φ ˜ ∇ $_{c}$˜ φ ) − ˜ ∇ c ˜ φ ˜ ∇ $_{c}$˜ φ , one obtains [22] 

The integral over the boundary ∂ V on the right hand side is split into four contributions: 

<img src='content_image/382470.jpg'>

<img src='content_image/382469.jpg'>

