On uniqueness of static black hole with conformal scalar hair

We discuss the uniqueness of the static black hole in the Einstein gravity with a conformally coupled scalar field. In particular, we prove the uniqueness of the region outside of the photon surface, not event horizon.


I. INTRODUCTION
It is well-known that static black holes do not have the scalar hair with non-negative potential in asymptotically flat spacetimes [1] (see Ref. [2] for a review). However, there is an exception, that is, we can have the Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) solution [3,4]. This is because this solution does not satisfy the regularity condition for the scalar field at the event horizon.
In this paper, we shall address the uniqueness issue of the BBMB black hole solution. In particular, we will examine if the static black hole spacetime with conformal scalar hair is spherically symmetric as the Schwarzschild spacetime [5]. As a result, we could prove the uniqueness of the photon surface of the BBMB solution, that is, the outside region of the photon surface is unique to be the BBMB solution in the Einstein gravity with a conformally coupled scalar field. In the BBMB solution, the photon surface corresponds to the unstable circular orbit of null geodesics (see Ref. [6] for the definition of the photon surface). In the Einstein frame, the system is reduced to the Einstein-massless scalar field system. In this system, the uniqueness of the outiside region of the photon surface has been proven in Ref. [7]. However, the proof in Ref. [7] cannot be applied to the current situation. This is because the photon surface of the BBMB black hole in the Jordan frame is singular in the Einstein frame.
Note that it will be difficult to prove the black hole uniqueness for cases except for vacuum/electrovacuum or well-motivated systems by string theory or so [8]. In this sense, the current result may encourage us to try to prove the uniqueness of static black holes with a hair although we know that the BBMB black hole itself is not stable [2].
The rest of this paper is organized as follows. In Sec. 2, we briefly review the BBMB black hole. In Sec. 3, we describe the basic equations for static spacetimes in the Einstein gravity with the conformally coupled scalar field. In Sec. 4, we show that the certain relation between the scalar field and the time lapse function holds once the scalar field is turned on. Then, in Sec. 5, we will present the proof of the uniqueness of the photon surface in the current system. Finally, we will give the summary and discussion in Sec. 6.

II. BBMB BLACK HOLE
Let us consider the Einstein equation with the conformally coupled scalar field [3,4], where φ is the scalar field and R is the Ricci scalar. The field equations are and where G µν is the Einstein tensor and The trace for Eq. (2) and Eq. (3) show us and For the current purpose, it is better to rearrange the Einstein equation as where From this, we can see that one needs a careful treatment at φ = ± 6/κ =: φ p to have the regular spacetime. The metric of the BBMB black hole is given by [3,4] where f (r) = (1 − m/r) 2 , m is the mass of black hole and dΩ 2 2 is the metric of the unit 2-sphere. The configuration of the scalar field is The metric itself is exactly the same with the extreme Reissner-Nordström black hole spacetime. The event horizon is located at r = m and the scalar field diverges at there. Here we note that the factor 1 − κφ 2 /6 in the left-hand side of Eq. (7) vanishes at r = 2m where there is the unstable circular orbit of null geodesics.

III. BASIC EQUATIONS IN STATIC SPACETIMES
Let us focus on static spacetimes from now on. Then the metric is written as The Latin indices indicate the spatial components. The locus of the event horizon is V = 0 and we assume that the spacetime is regular in the outside of the event horizon, that is, the domain of the outer communication.
In static spacetimes, the non-trivial parts of the Ricci tensor are and where D i and (3) R ij are the covariant derivative and the Ricci tensor on the t =constant hypersurface Σ, respectively. S µν defined in Eq. (8) is decomposed into The equation for the scalar field (6) becomes Since we focus on asymptotically flat cases, the asymptotic behaviors of metric are given by and For the scalar field, we impose From Eqs. (5), (12) and (13), we can see IV. SCALAR FIELD AND TIME LAPSE FUNCTION In this section, we will show that the scalar field φ is written by the time lapse function V uniquely if there is non-trivial scalar field exists in static spacetimes.
From the (0, 0)-component of the Einstein equation and Eq. (16), we have where Φ := (1 + ϕ)V and ϕ := ± κ/6φ. Now, in Σ, we focus on the region Ω which has the two boundaries; the surface S p specified by φ = φ p and the 2-sphere S ∞ at the spatial infinity.
In Ω, we assume that there are no event horizons, that is, V is strictly positive (V > 0). Since ϕ or φ follows Eq. (16), ϕ is a monotonic function which has the maximum value 1 at S p and minimum value 0 at the spatial infinity, that is, 0 ≤ ϕ ≤ 1 in Ω. Let us consider the conformally transformed spaceΩ(⊂Σ) with the metricg ij = (1 − ϕ) 2 g ij . Then Eq. (21) becomesD The volume integration of the above overΩ and the Gauss theorem give us We can show that the second term of the right-hand side vanishes as For the last equality in the above, we used the fact of ϕ| Sp = 1. Then, Eq. (23) tells us Next we consider the volume integration of ΦD 2 Φ = 0 overΩ and then In the second equality, we used the Gauss theorem. As Eq. (24), it is easy to show that the last term vanishes. Using Eq. (25) and the fact of Φ ∞ = 1, we can see Thus, Eq. (26) implies Therefore, D i Φ = 0 has to be satisfied everywhere on Ω. Together with the boundary condition at the spatial infinity (Φ ∞ = 1), this means that Φ = 1 holds everywhere. Thus we have the following relation between the time lapse function V and the scalar field φ; Of course, the BBMB solution satisfies this relation. Note that φ = φ p corresponds to V = 1/2 =: V p and the V = V p surface in the BBMB solution is composed of the closed circular orbit of photon (null geodesics). Now, both Eq. (16) and the (0, 0)-component of the Einstein equation give us the same equation where v := ln V . Then, we can use v as a "radial" coordinate later. The (i, j)-component of the Einstein equation becomes The trace part of the above and Eq. (30) (or Eqs. (20) and (30)) imply

V. UNIQUENESS OF THE BBMB PHOTON SURFACE
In this section, we will employ Israel's way to address the uniqueness of the BBMB black hole. First, we consider the foliation by v =constant 2-surfaces {S v } on t =constant hypersurfaces Σ. We write the induced metric on 2-surfaces as h ij := g ij − n i n j , where n i := ρD i v is the unit normal vector on the surface and Then we can derive the following equations; and where ξ := (2V − 1)ρ −1/2 , η := 2(2V + 1)ρ −3/2 , k ij is the extrinsic curvature of S v in Σ, k is its trace part and k ij := k ij − (1/2)kh ij is its traceless part. D i is the covariant derivative with respect to h ij . In the above, we used the following equations; and For the derivation of the second equation, we used Eq. (30). For the third one, we used Eq. (31) and the formula This comes from the definition of the extrinsic curvature and Eq. (30). Now we compute the curvature invariant R µν R µν to check the regularity of spacetime. It is written as In the above, we used the Einstein equation and Eq. (39). As commented below Eq. (8), one needs a careful treatment at the surface of V = 1/2 (φ = φ p ). For the regularity of spacetimes at the surface of V = 1/2, we require the conditions and Note that the index "p" indicates the evaluation at S p , e.g. ρ p = ρ| Sp . These features tell us that S p is totally umbilic and the photon surface defined in Ref. [6] 1 or photon sphere [9]. From Eqs. (32) and (39), the Kretschmann invariant can be written as where we also used and Thus, we can see that the Kretschmann invariant is finite in Ω when Eqs. (41) and (42) hold on S p . From now on, we focus on the region Ω in Σ which has the two boundaries, that is, S p and the spatial infinity S ∞ . The volume integrations of Eqs. (33), (34) and (35) over Ω give us and respectively. A p and (2) R are the area of S p and the Ricci scalar of S v , respectively. Equation (46) tells us that m is positive. In the left-hand side of Eq. (47), we used the fact of In particular, on S p , we have Then Eq. (47) shows us Because of m > 0, inequality (51) and the Gauss-Bonnet theorem tell us that the topology of S p is restricted to S 2 (that is, Sp (2) RdS = 8π) and then we have the following inequality with the help of Eq. (46); Using Eq. (46), Eq. (48) gives us the inequality This may be regarded as a mimic of the Penrose inequality [10] for the photon surface. Thus, we can conclude that the equality holds in inequalities (52) and (53), and theñ Finally, we use the Codazzi equation, to show that k is constant on each S v . From Eq. (45), we see that the right-hand side vanishes. Using Eq. (54), we can have Because of Eq. (49), we can see that (2) R is also constant on each S v . In addition, inequality (51) tells us that (2) R is positive. Thus, we can conclude that the region of Ω in the spacetime is maximally symmetric with the positive curvature and then spherically symmetric. Thus, we could see that the spacetime in Ω is unique to be the BBMB solution because the regular spherically symmetric solutions have been shown to be unique in Ref. [11]. Note that what we could prove is only the uniqueness for outside region of the photon surface and we did not discuss the inside of the photon surface. Therefore, this proof does not mean the uniqueness of the whole region of the BBMB black hole spacetime.

VI. SUMMARY AND DISCUSSION
In this paper, we proved that, in the Einstein gravity with a conformally coupled scalar field, if a single closed surface S p satisfying (1 − κφ 2 /6) = 0 exists and if there is no horizon in the region Ω whose boundaries are only S p and the spatial infinity, the geometry in Ω is the same as that outside of the BBMB photon surface. Since we employed Israel's way which can work for the proof of the uniqueness of single object, we cannot exclude the existence of multi-photon surface system. And we did not prove the uniqueness of the whole region of the BBMB black hole spacetime. We have no definite answer for the inside region of the photon surface. Accidentally, the uniqueness of the photon surfaces have been recently discussed for vacuum, electrovacuum and so on [7,9,[12][13][14]. Therein, the existence of photon surface or photon sphere was assumed by hand. On the other hand, in our current study on the Einstein gravity with a conformally coupled scalar field, the existence of the photon surface is automatically required to make the spacetime regular.
There are many remaining works. If one may examine the possibility of cases having multi-photon surface, one must employ the another proof developed by Bunting and Masood-ul-Alam [15]. One may be also interested in the uniqueness issue for the inside region of the photon surface, that is, the region between the photon surface and the event horizon. Some extension of our proof into other non-vacuum cases should be addressed. Finally, we may have the Penrose inequality for a kind of the photon surface for dynamical systems in the Einstein gravity with conformally coupled scalar fields. This is because we could have a mimic of the Penrose inequality in our study.