Near-horizon symmetries of the Schwarzschild black hole with supertranslation field

Using the exact solution to Einstein equations of Compere and Long for the Schwarzschild metric containing supertranslation field, we study the near-horizon symmetries of the metric. We consider class of metrics with supertranslation field depending only on a spherical angle $\theta $. After reviewing the action of supertranslations preserving the static gauge of the metric, we study general transformations preserving the near-horizon form of the metric. We determine the asymptotic Killing vectors and calculate the charge corresponding to the asymptotic symmetries preserving the form of the metric at the horizon.


Introduction
The study of asymptotically flat spacetimes initiated by Bondi, Metzner and Sachs (BMS) [1,2] has attracted much attention recently (for a review and references [3]). The symmetry group of the asymptotically flat gravity extends the Poincare group and contains supertranslations, the angulardependent translations at null infinity, which generalize translations. The vacua with different supertranslation fields are physically different in a sense that their (superrotation) charges are different [4,5,6,7,8,9]. Different vacua differ one from another by the cloud of massless particles they contain [10,12].
Of particular interest is the study of black holes containing supertranslation field. It was suggested that evaporation of black holes with inclusion of soft quanta may be unitary [10,11,12]. Finite supertranslation diffeomorphisms map physical states to inequivalent physical states having different charges. Stationary black holes resulting from the collapse are diffeomorphic to the Kerr metric [13,14] and have the same ADM charges. Because diffeomorphisms contain supertranslations, the metric of a black hole, in general, contains supertranslation field. It was discussed the possibility that the account of supertranslation (and superrotation) hair may contribute to solving the information problem [15,16,17,18].
The BMS transformations are naturally formulated at the null infinity, and there is a complicated problem of extension of asymptotically defined metric with supertranslation field in a closed form to the bulk. In paper [8] a family of vacua containing supertranslation field was constructed in the bulk. In paper [9] a solution-generated technique was developed and applied to construction of solutions to the Einstein equations containing supertranslation field generalizing the Schwarzschild metric.
Because many properties of black holes are associated with physics near horizon, and having at hand the Schwarzschild metric containing supertranslation field, it is natural to study its near-horizon symmetries.
We consider the class of metrics containing supertranslation field depending only on a spherical angle θ. The metrics are solutions of the Einstein equations and are generalization of the Schwarzschild solution. We transform the metric to a coordinate system in which horizon of the metric is located at r = 2M. A specific property of this metric is that, in distinction to the metrics without supertranslation field, the expansion of the metric near horizon is in powers of (r − 2M) 1/2 and not in powers of r − 2M [19,20].
We review the action of supertranslations preserving the gauge of the metric [9]. We find sufficient condition on supertranslations T (θ) preserving the gauge of the metric to preserve the near-horizon form of the metric.
We perturbatively solve equations for the asymptotic Killing vectors preserving the functional form of the metric at the horizon (cf. [21,22,23,24,25,26,27,28] and many other papers). We calculate the variation of the surface charge corresponding to the asymptotic horizon symmetries. We find that in the case of the black hole with supertranslation field C(θ) depending only on spherical angle θ the variation can be integrated to a charge in a closed form.

Schwarzshild metric containing supertranslation field
In this section we briefly review the exact solution of the Einstein equations generalizing the Schwarzschild metric and containing supertranslation field constructed in paper [9]. Next, we transform the metric of [9] to a form with the horizon located at the surface r = 2M, where M is interpreted as a mass of black hole.
Vacuum solution of the Einstein equations containing supertranslation field diffeomorphic to the Schwarzschild metric is C(θ, ϕ) is a supertranslation field, C 00 is the lowest spherical harmonic of C(θ, ϕ). In the following we do not write C 00 explicitly understanding C → C − C 00 . Here Covariant derivatives D A are defined with respect to the metric on the sphere ds 2 = dθ 2 + sin 2 θdϕ 2 . The tensor C AB and the functions U and E are In the following we consider the case of metrics with C depending only on angle θ. In this case the components C AB are Here prime is differentiation over θ. Horizon of the metric (1) is located at the surface ( see [9], Eq.(45)) To obtain another form of the metric with horizon located at r = 2M, we introduce a new variable r = r(θ, ρ) defining The inverse transformation ρ = ρ(θ, r) is obtained from the relation Introducing the functions Expressing ρ as a function of r and C, we obtain we have Because K(r) is an increasing function of r having its minimum at r = 2M, and because the cosmic censorship conjecture implies (see Eq.(47) of [9]) that We obtain the line element (1) in a form [29] Horizon of the metric (14) is located at the surface r H = 2M. In Appendix A we consider solution of the geodesic equations for null geodesics in the metric (14). We find the asymptotic of the solution in the limit V (r) → 0. In this limit dr dt = constV (r), i.e. the same relation as in the Schwarzschild metric. The surface r = 2M is the surface of the infinite redshift, i.e. horizon [30]. In the following, we set M = 1.
In this section, first, we review the supertranslations preserving the gauge of the metric (1). Next, we write the generator of supertranslations preserving the gauge of the metric (14). Imposing the requirement that supertranslations do not alter the functional form of the metric at the horizon, we determine necessary conditions on supertranslations to preserve the near-horizon form of the metric. We begin with the metric component g tt and find conditions on supertranslations under which the functional form of g tt at the horizon is preserved. Considering other components of the metric, we show that the condition obtained for g tt is sufficient to preserve the near-horizon form of all components. The metric (1) is written in the static gaugeg tρ =g tθ = 0 andg ρa = 0. Generator of supertranslations preserving this gauge was obtained in [9], and in the case C = C(θ) is In the following we include T 00 in T writing simply T . The metric components transform as where δ T C = T and δ T D k C = D k T . The action of the transformation (16) on the componentg tt is where Horizon of the metric (1) is located at the surface ρ s = 1/2. Transformations (16) form a commutative algebra w.r.t. the modified bracket [4,5,9] [ξ 1 , where giving for [ξ 1 , ξ 2 ] θ mod zero result. Next, we consider supertranslations acting on the metric (14). The generator of supertranslations in variables (r, θ) acting on the metric (14) is obtained from the generator (16) by transformation Using (6) to calculate ∂r/∂ρ s = 1 − 1/(4ρ 2 s ), and expressing χ i T through r, θ, we obtain In the near-horizon region, denoting r = 2 + x, |x| ≪ 1, we have Acting by the generator of supertranslations on the component g tt of the metric (14), we obtain In the near-horizon region To preserve the functional form (26) and The action of the term χ θ T ∂ θ on the metric components do not change their behavior as x → 0. Let us consider the action of χ r T ∂ r . When acting onḡ rr ,ḡ rθ ,ḡ θθ ,ḡ ϕϕ the operator χ r T ∂ r produces the terms proportional to V −1/2 . When acting on the terms V −1 and V −1/2 the operator χ r T ∂ r produces an extra factor V −1 . However, because of condition (28), the total singularity of each term remains the same or reduces.
Condition (27) gives a restriction on supertranslation T (θ) Using (29) we obtain To obtain the relation (30) we have used (29). Eq. (29) is the first-order equation on T (θ). T (θ) is expressed through C(θ) and is determined up to an arbitrary constant.

Asymptotic horizon symmetries of the metric
In this section we consider general diffeomorpisms which do not change the form of the metric components in the near-horizon region. We find transformations of the leading-order parts of the metric components. At x = r − 2 ≪ 1 the metric (14) takes the form The coefficients of expansions of the metric components in series in x 1/2 are functions of θ. Here κ is introduced to treat the component g tt on the same footing as other components. In the final expressions κ is set to unity. We consider transformations generated by vector fields χ i which do not change the form of the metric components in the leading orders. The vector fields generating the near-horizon transformations have the following structure Under the diffeomorphisms generated by vector fields χ k the metric components are transformed as Transformations preserving the gauge conditions g rt = g θt = 0 are L χ g rt = ∂ r χ t g tt + ∂ t χ r g rr + ∂ t χ θ g θr = 0, (34) L χ g θt = ∂ θ χ t g tt + ∂ t χ r g rθ + ∂ t χ θ g θθ = 0.
It is seen that the leading-order parts of the metric components are transformed through the functions The vector fields in the generator of supertranslations (23) satisfying condition (27) form a subset of the vector fields (32) The Lie brackets of the vector fields in which we retain the components (43) are [χ (1) , [χ (1) , In the next section we use transformations (42) to calculate the surface charge of the asymptotic horizon symmetries.

The surface charge of asymptotic horizon symmetries
In this section we calculate the variation of the surface charge corresponding to the asymptotic Killing vector field χ i . The metric variations needed to obtain the variation of the charge were obtained in (42). Details of the calculation are presented in Appendix B. In our calculations we used the relations specific to the case of C depending only on θ.
Substituting the above expressions, we obtain the variation of the surface charge Here integration is over the surface r = 2. Setting κ = 1 in correspondence with the metric (14) and integrating the variation (60), we obtain If χ t 0 is independent of θ, the surface charge is proportional to the surface of the horizon, i.e. to the geometric entropy of the black hole.
The Lie bracket of the charges is where χ t (12),0 is defined in(46).
In this note we studied the horizon symmetries of the metric containing supertranslation field depending only on spherical angle θ. We transformed the metric obtained in [9] to the form with horizon located at the surface r = 2M. After reviewing the action of supertranslations preserving the form of the metric [9], we have determined the requirements on the generator of supertranslations to preserve the near-horizon form of the metric. It was found that to preserve the functional form of the metric in the near-horizon region, at r − 2M ≪ 1, the r-component of the generator of supertranslations should be of order O(r − 2M). From this requirement it follows that parameter of supertranslations, T (θ), must satisfy a condition Next, we studied the general transformations preserving the near-horizon form of the metric and calculated the charge corresponding to the asymptotic horizon Killing symmetries. In calculation of the variation of the charge we used the relations between the metric components specific to the case of the supertranslation field depending only on angle θ. In the case of supertranslation field depending only on θ the variation of the charge can be integrated to a charge in a closed form. The charge is proportional to the area of horizon surface i.e. to the geometric entropy of a black hole.
The metric is written in the gauge g ρρ = g vρ = g ρa = 0 with accuracy O(ρ 2 ). Horizon of the metric is located at ρ = 0. In the near-horizon region the metric components are expanded in power series in ρ . The charge of the asymptotic near-horizon symmetries for the metric (63) was obtained in [25,26] in a form dzdz |γ|(2T κΩ − y a θ a Ω).
Here T is a part of the asymptotic Killing vector χ v . The volume √ γΩdzdz, where Ω zz = Ωγ zz , is an analog of the volume (ḡ θθḡϕϕ ) 1/2 sin θdθdϕ in (61), and T is an analog of χ t . It is seen that structures of charges in both cases are similar. In distinction to the metric (63), the components of the metric (14) are expanded in powers of x 1/2 = (r − 2M) 1/2 . In the limit of zero supertranslation field the terms with fractional powers of (r − 2M) vanish.

Appendix A
In this Appendix we consider a solution of the geodesic equations in the metric (14). We find an asymptotic of a solution for null geodesic in the limit V (r) → 0. Following the treatment of [33] we start from the Lagrangian corresponding to the metric (15) 2L = −Vṫ 2 +ṙ 2 Vḡ rr + 2ṙθ V 1/2ḡ rθ +θ 2ḡ θθ +φ 2 sin 2 θḡ ϕϕ .
The derivatives are taken with respect to an affine parameter on geodesic τ . The function ϕ is the cyclic variable which it is possible to set equal to zero [33]. We look for solution in the limit V (r) → 0 in a forṁ In the limit V = 0 we have r = 2M = 2, V ,r = 2M/r 2 = 1/2. Substituting the Ansatz in the Lagrange equations and retaining the leading in V → 0 terms, we have From the system (A9)-(A10) it follows that Cḡ rθ + 2Aḡ θθ = 0, (A11) Substituting the Ansatz in relation (A5), we reduce it to Using the relation (52),ḡ rrḡθθ −ḡ 2 rθ =ḡ θθ , from the Eq. (A12) we find that C 2 = E 2 .
From this relation it follows that the surface r = 2M is the surface of infinite redshift [30].