1 Introduction

CPHT-RR021.042018

Black hole memory effect

Laura Donnay, Gaston Giribet, Hernán A. González, Andrea Puhm

Center for the Fundamental Laws of Nature, Harvard University

17 Oxford Street, Cambridge, MA 02138, USA.

Black Hole Initiative, Harvard University

20 Garden Street, Cambridge, MA 02138, USA.

Physics Department, University of Buenos Aires and IFIBA-CONICET

Ciudad Universitaria, Pabellón 1, Buenos Aires, 1428, Argentina.

Center for Cosmology and Particle Physics, New York University,

726 Broadway, New York, NY 10003, USA.

Institute for Theoretical Physics, TU Wien,

Wiedner Hauptstr. 8-10, 1040 Vienna, Austria.

CPHT, Ecole Polytechnique, CNRS,

91128, Palaiseau, France.

We compute the memory effect produced at the black hole horizon by a transient gravitational shockwave. As shown by Hawking, Perry, and Strominger (HPS) such a gravitational wave produces a deformation of the black hole geometry which from future null infinity is seen as a Bondi-Metzner-Sachs (BMS) supertranslation. This results in a diffeomorphic but physically distinct geometry which differs from the original black hole by their charges at infinity. Here we give the complementary description of this physical process in the near-horizon region as seen by an observer hovering just outside the event horizon. From this perspective, in addition to a supertranslation the shockwave also induces a horizon superrotation. We compute the associated superrotation charge and show that its form agrees with the one obtained by HPS at infinity. In addition, there is a supertranslation contribution to the horizon charge, which measures the entropy change in the process. We then turn to electrically and magnetically charged black holes and generalize the near-horizon asymptotic symmetry analysis to Einstein-Maxwell theory. This reveals an additional infinite-dimensional current algebra that acts non-trivially on the horizon superrotations. Finally, we generalize the black hole memory effect to Reissner-Nordström black holes.

## 1 Introduction

Over the last few years we have learned that gravitational and gauge field dynamics in asymptotically Minkowski spacetime entails a rich mathematical structure whose relevance for physics had been largely overlooked. This observation led to a revision of the notion of vacua in gravity and gauge theories in asymptotically flat spacetimes, which is of crucial importance for the scattering problem. This mathematical structure, expressed in the emergence of an infinite set of symmetries, unveils a surprising connection among three previously known but seemingly disconnected topics: a) the soft theorems for the S-matrix of gravity and gauge theories in asymptotically flat spacetimes [1], b) the enhanced symmetry group that governs the dynamics in the asymptotic region [2, 3, 4], and c) the memory effect produced by transient gravitational waves [5, 6].

Recently, there has been substantial progress in understanding the three components of this triangle and how they are interconnected. The new insights raise the hope for a better understanding of scattering processes, especially when gravitons, or even black holes are involved. Besides, they could lead to a better comprehension of the physical meaning of the infinite-dimensional symmetries exhibited by Minkowski spacetime in its asymptotic domain.

The existence of a set of infinite-dimensional asymptotic symmetries in the future (and past) null infinity region(s) has been known for a long time [2, 3, 4]. However, only recently the importance of these symmetries has been understood and significant advances have been made [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The algebra generated by these symmetries, known as Bondi-Metzner-Sachs (BMS) algebra, originally appeared in the study of classical gravitational radiation in asymptotically flat spacetimes. The application of the BMS symmetry algebra to study the S-matrix in flat spacetime has been proposed recently [11] and, since then, several physical systems have been studied within this framework; see [21] and references therein and thereof. In particular, a revision of the problem of formation and evaporation of black holes has recently been initiated in [22, 23, 24], where it was suggested that the BMS symmetry would be of importance for the information loss problem.

Here, we will not address the information loss puzzle, but rather another problem which is related to the black hole memory: To understand how the BMS symmetry that underlies scattering processes involving a black hole can be measured by an observer hovering just outside the event horizon. We will establish a connection between the description of the black hole geometry in terms of the BMS symmetries in the asymptotic region at null infinity and its description in terms of the symmetries that emerge in the near-horizon region by computing the gravitational memory effect in the vicinity of the black hole horizon produced by an incoming shockwave. We will refer to it as the black hole memory effect [25]. A similar process has recently been studied by Hawking, Perry, and Strominger (HPS) [25], who showed that a transient shockwave produces a disturbance in the spacetime corresponding to a BMS supertranslation at null infinity. This provides a concrete example of a physical process that endows a black hole with BMS hair of the type suggested in [23]. However, it remained an open question how this phenomenon is seen from the point of view of an observer close to the horizon. Here, we will show that the BMS supertranslation hair of [25] can be understood as a supertranslation111Supertranslations on the future horizon of Schwarzschild black hole have been also studied in [23], where the canonical construction of the Bondi-gauge preserving supertranslations was given. composed with a superrotation from the point of view of the near-horizon geometry. We compute the conserved charge associated to this superrotation and find its form to agree with the one of HPS.

This is relevant for several reasons: First, it describes a physical process whose effect on the horizon geometry can be captured by the symmetries discovered in [26]. Second, this shows that, in addition to horizon supertranslations [22], horizon superrotations are crucial to describe the physics in the vicinity of the black hole. Third, this gives a bulk complementary description of the process studied in [25], which sheds light on the connection between the symmetries emerging in different regions of the spacetime.

We begin in Section 2 with a review of the results of [25] of how the action of the BMS symmetries on the Schwarzschild geometry can be understood as a perturbation produced by a transient gravitational wave. In Section 3, we describe the same physical process from the point of view of an observer in the near-horizon region using the symmetry analysis of [26, 27]. We compute the superrotation and supertranslation charges associated to the asymptotic symmetries, and we show how the charges found in the near-horizon region relate to those computed at null infinity. In Section 4, we extend the analysis to the case of electrically and magnetically charged black holes. As a prerequisite, we first generalize the near-horizon asymptotic symmetry analysis of [26, 27] to Einstein-Maxwell theory. This is shown to yield an additional infinite-dimensional current algebra on which the horizon charges of the gravity sector act non-trivially. We discuss the physical interpretation of the extended symmetry and of their associated charges by analyzing the particular case of non-extremal Reissner-Nordström black holes. We evaluate the zero-modes of the charges on the dyonic solution and discuss their interpretation. The extremal case, which exhibits qualitatively different features, is analyzed separately. In Section 5, we generalize the black hole memory effect to the Reissner-Nordström black hole. Section 6 contains our conclusions.

## 2 Gravitational shockwaves and BMS hair

We begin by reviewing the salient features of the BMS analysis of [25] and their proposal for a dynamical mechanism for generating BMS hair on black holes.

Consider a static Schwarzschild black hole whose line element in advanced Bondi coordinates is given by

where is the advanced time and () represents the angular position on the 2-sphere with unit metric . The horizon is located at . At past null infinity , which is defined as the null surface obtained by taking the limit while keeping fixed, the only non-vanishing conserved charge is the mass and hence (1) represents a bald Schwarzschild black hole. In [25], HPS constructed BMS supertranslation hair at null infinity and showed that a physical process for the bald Schwarzschild black hole to acquire such hair is given by perturbing the geometry (1) with a linearized gravitational shockwave prepared at advanced time whose energy density to leading order in large radial distance is given by

 Tvv=μ+T(z)4πr2δ(v−v0). (2)

The function characterizes the angular profile of the shockwave and, following [25] we explicitly write its monopole contribution separately. Solving the conservation equation for the stress-tensor in the background (1) yields the subleading contributions to (2) which break spherical symmetry, namely

 (3)

Here, obeys the equation , with and being the covariant derivative and the Laplacian on the 2-sphere, respectively. As explained in [25], the solution to these equations can be conveniently expressed in terms of the Green function connecting two different angular positions and as defined by

 D2(D2+2)G(z,w)=4√γ δ(2)(z−w), (4)

where denotes the determinant of the metric on the sphere. Defining

 C(z)=∫d2w G(z,w) T(w), (5)

the stress-tensor components (3) become

 Tvv=14πr2(μ+14D2(D2+2)C−3M2rD2C)δ(v−v0), (6) TvA=−3M8πr2DAC δ(v−v0).

This represents the energy-momentum contribution of the linearized shockwave. Its effect on the background is to produce a perturbed metric with the perturbation given by

 (7)

This perturbation was shown in [25] to be equivalent to acting on the Schwarzschild geometry (1) with a large diffeomorphism generated by the asymptotic BMS Killing vector

 ζ=ζv∂v+ζA∂A+ζr∂r, (8)

with components

 ζv=f,ζA=1rDAf∂A,ζr=−12D2f, (9)

and where ; see [25] for more details. This large diffeomorphism corresponds to a supertranslation that changes the BMS (superrotation) charges at null infinity. The resulting supertranslated black hole metric takes the form222The extension of the supertranslated geometry into the bulk is gauge dependent. Here it is done by requiring the Bondi gauge to be preserved [25].

The location of the supertranslated event horizon is

 (r+)f=r++12D2f, (11)

and thus depends on the angular variables through . Note that the solution (10) is exact in but only linear333See [28] for finite BMS transformations at null infinity. in and therefore has to be understood up to order . The supertranslated Schwarzschild black hole (10) is a different physical configuration than the unperturbed bald geometry (1) as it carries non-vanishing superrotation charge [23, 25]

 QHPSY=18π∫I−+d2z√γ YANA=−3M8π∫I−+d2z√γ YA∂Af, (12)

where is any smooth vector field on the sphere, is the angular momentum aspect, and is the 2-sphere that represents the remote future of past null infinity444The antipodal matching condition [14] relates the field configurations at to those at , the latter corresponding to the 2-sphere in the remote past of future null infinity . We refer to [25] for the details about the prescription for the matching conditions and integration. .

For the spacetime is described by the bald Schwarzschild black hole (1). The perturbation by the shockwave at turns on non-vanishing superrotation charge (12) and for the spacetime is described by the supertranslated Schwarzschild geometry (10). See figure 1.

The action of large diffeomorphisms corresponding to supertranslations on the Schwarzschild geometry at null infinity can thus be understood as the physical process of sending in a gravitational shockwave with an asymmetric angular profile. This concludes our review of [25] and their interpretation of the action of BMS transformations at null infinity. We now turn to the horizon.

## 3 Soft hair on Schwarzschild horizons

We now discuss how the process of deforming the black hole geometry by an incoming shockwave is seen by an observer located close to the horizon and, moreover, how supertranslations at null infinity get encoded in the symmetry transformations at the horizon. Since the supertranslated black hole solution (10) is valid for finite values of the radial distance , we can investigate this question using the near-horizon analysis of [26, 27]. There, it was shown that if one starts from the general form of a near-horizon metric

where the horizon is located at , the ellipsis stand for terms, are in principle arbitrary functions555The function does not ultimately appear in the conserved charges [27] and we will omit it in the following. of the advanced time and the angles , and assuming the gauge fixing conditions

 gρρ=0,gvρ=1,gAρ=0, (14)

then there exists a set of asymptotic diffeomorphisms preserving (13) generated by an infinite-dimensional algebra that includes both supertranslations and superrotations. These are diffeomorphisms

 χ=χv∂v+χA∂A+χρ∂ρ, (15)

of the form

 χv=f,χA=YA−∂Bf∫ρdρ′gAB,χρ=−ρ∂vf+∂Af∫ρdρ′gABgvB, (16)

where and are -independent functions whose -dependence is constrained by

 ∂vYA=0,κ∂vf+∂2vf=0. (17)

These last two equations follow from demanding the leading terms of not to depend on the fields, and from taking the surface gravity, , to be constant. The diffeomorphisms (16) subject to (17) have been shown to give rise to an infinite-dimensional algebra consisting of two copies of the Virasoro algebra generated by (superrotations) and two Abelian current algebras generated by (supertranslations). For non-extremal black holes (), the time-independent part of can be interpreted as a supertranslation in the retarded time of the future horizon . Its time-dependent part can be thought of as a superdilation in or, alternatively, as a supertranslation in the affine parameter along the event horizon. For extremal black holes (), the roles of supertranslations and superdilations are interchanged.

The diffeomorphisms (16) subject to the constraints (17) preserve the generic form of the metric (13), but change the functions as follows

 δχκ=0, (18) δχθA=LχθA+f∂vθA−2κ∂Af−2∂v∂Af+ΩBC∂vΩABDCf, δχΩAB=f∂vΩAB+LχΩAB.

We will now discuss how the transient gravitational shockwave of HPS [25] and its deformation of the horizon can be interpreted as the change (18) in the near-horizon metric from the bald Schwarzschild black hole (1) to the supertranslated one (10). This relates the BMS supertranslation at null infinity to the horizon supertranslations and superroations (16)-(17).

To make contact with the asymptotic symmetry analysis of [26, 27]666The infinite-dimensional symmetries at the horizon have been also discussed in references [29, 30, 31, 32, 33, 34, 35, 36, 37]; see also references therein and thereof. we can write the near-horizon metric of the supertranslated (10) Schwarzschild black holes in the form (13) by changing coordinates and expanding (10) near the horizon. To leading order in , this gives

where we used . From this one can read off the and contributions to the metric components induced at the horizon of the supertranslated Schwarzschild black hole, namely

The corresponding metric functions for the bald Schwarzschild geometry (1) are obtained by setting in (20). Asking (20) to be generated by acting with (18) on the geometry of the unperturbed Schwarzschild horizon, we find

which corresponds to a horizon supertranslation composed with a horizon superrotation, the latter given by777Notice that we are not assuming the vector to be a holomorphic function here.

 YA=1r+DAf. (22)

Here, is the HPS supertranslation at null infinity which turns out to coincide with the horizon supertranslation at . This hence shows that the disturbance produced by the gravitational shockwave, which from null infinity is seen as the action of a pure BMS supertranslation on the Schwarzschild geometry, is seen by the near-horizon observer as a supertranslation together with an induced superrotation given by (22).

We can now compute the conserved charges at the horizon associated to the horizon supertranslation and superrotation symmetries. For spacetimes of the form (13) these charges have been constructed in [26, 27] using the covariant formalism [38]. The horizon superrotation charge for the perturbed Schwarzschild black hole (19) is

 QY=18π∫d2z√γ YAθAΩ=M8π∫d2z√γ YA∂Af, (23)

where the integration is over the constant section of the horizon and so that . Note that the functional form of (23) is the same888There is an extra overall factor when directly comparing with the expression for . These two charges are quantities defined by integrating over different 2-surfaces. as the one of HPS given in (12). The additional horizon superrotations induce a supertranslation charge contribution

 δQT=κ8π∫d2z f δ(√detΩAB), (24)

which is absent at null infinity . The physical interpretation of the zero-mode of (24) is clear: it encodes the variation of the entropy (times the temperature) due to the transient shockwave, namely

 δQT|f=1=κ2πδA4, (25)

where is the variation of the horizon area in Planck units999We use units where Newton’s constant .. That is, , with being the Bekenstein-Hawking entropy and being the Hawking temperature.

This concludes our discussion of the black hole horizon memory effect for Schwarzschild black holes. In the following, we turn on gauge fields which will turn out to act non-trivially on the the horizon superrotation charge. To do so, we first need to extend the asymptotic symmetry analysis of [26, 27] to the Einstein-Maxwell theory and then generalize the discussion of the previous sections to Reissner-Nordström black holes.

## 4 Horizon symmetries for Einstein-Maxwell

The near-horizon geometry of a four-dimensional charged black hole takes the same convenient form in Gaussian null coordinates as that of its uncharged counterpart, namely (13), which we repeat here for convenience:

 gvv =−2ρκ+O(ρ2), (26) gvA =ρθA(zB)+O(ρ2), gAB =ΩAB(zC)+ρλAB(zC)+O(ρ2),

and we assume the following gauge fixing conditions for the metric

 gρρ=0,gvρ=1,gAρ=0. (27)

As in [26], the functions and depend on but are taken to be independent of the advanced time ; this accommodates the case of isolated horizons studied here101010One may relax this assumption, but then one has to treat subtleties regarding the integrability of the charges; see [27].. The asymptotic boundary conditions for the Maxwell field are

 Av=A(0)v+ρA(1)v(v,zA)+O(ρ2), (28) AB=A(0)B(zA)+ρA(1)B(v,zA)+O(ρ2),

and we choose the radial gauge condition

 Aρ=0. (29)

The Coulombian potential at the horizon, , is taken to be a fixed constant, is assumed to depend only on , while and are arbitrary functions of and . These conditions are analogous to those considered for Einstein-Maxwell theory at null [39, 40] and spatial [41] infinity. They are slightly more general than the horizon conditions considered in [31], and suffice to discuss physically interesting solutions such as Kerr-Newman black holes. We now study the horizon symmetries for metrics and gauge fields obeying (26)-(29). Depending on what is more convenient in specific examples, we will either consider the angular coordinates to be parametrized by complex variables or by standard polar variables .

A set of field transformations at the horizon is given by

 δ(χ,ϵ)gμν=Lχgμν,δ(χ,ϵ)Aμ=LχAμ+∂μϵ, (30)

where the vector field generates diffeomorphisms while is the gauge parameter. These transformations represent symmetries at the horizon if they respect the asymptotic form (26)-(28). The gauge fixing conditions (27) and (29) imply

 Lχgρρ=0,Lχgvρ=0,LχgρA=0,LχAρ+∂ρϵ=0, (31)

which yield the following form for and

 χv=f(v,zA), (32) χρ=Z(v,zA)−ρ∂vf+∂Af∫ρ0dρ′gABgvB, χA=YA(v,zA)−∂Bf∫ρ0dρ′gAB, ϵ=ϵ(0)(v,zA)−∫ρ0dρ′AB∂ρχB,

where , , and are arbitrary functions of and that do not depend on . Demanding that the leading piece of the vector field only depends on the coordinates but not on the fields (i.e. the arbitrary functions appearing in the metric and gauge field) leads to and  [27]. Implementing the boundary conditions for the remaining metric components, namely

 Lχgvv=O(ρ2),LχgvA=O(ρ2),LχgAB=O(ρ), (33)

yields the following components of the diffeomorphism generating vector field

 χv=f(v,zA), (34) χρ=−∂vf(v,zA)ρ+ρ22ΩθA(zB)∂Af(v,zA)+O(ρ3), χA=YA(zB)−ρΩ∂Af(v,zA)+ρ22Ω2λAB(zC)∂Bf(v,zA)+O(ρ3),

where we used the conformal gauge , which implies that the vector is a conformal Killing vector on the 2-sphere. For the gauge field, the boundary conditions (28) imply

 LχAv+∂vϵ=O(ρ),LχAB+∂Bϵ=O(1), (35)

yielding

 ϵ(0)(v,zA)=U(zA)−f(v,zA)A(0)v, (36)

where is an arbitrary function of the angular coordinates . This yields the gauge parameter

 ϵ=U(zA)−f(v,zA)A(0)v+ρ Ω−1∂Bf(v,zA)A(0)B(zA)+O(ρ2). (37)

Thus we find that the transformations (30) for the diffeomorphism vector field (34) and the gauge parameter (37) generate the horizon symmetries. The variations of the functions of the metric and the angular part of the gauge field are

 δ(χ,ϵ)κ=0=κ∂vf+∂2vf, (38) δ(χ,ϵ)θA=LYθA+f∂vθA−2κ∂Af−2∂v∂Af+ΩBC∂vΩAB∂Cf, δ(χ,ϵ)ΩAB=f∂vΩAB+LYΩAB, δ(χ,ϵ)A(0)v=0, δ(χ,ϵ)A(0)B=YC∂CA(0)B+A(0)C∂BYC+∂BU.

These variations generate a Lie algebra. If the gauge parameters and depended only on the spacetime coordinates but not on the fields, the Lie product

 [δ(χ1,ϵ1),δ(χ2,ϵ2)](gμν,Aμ)=δ(^χ,^ϵ)(gμν,Aμ), (39)

would take a simple form with and ; that is, the Lie bracket would be

 [(χ1,ϵ1),(χ2,ϵ2)]=(^χ,^ϵ). (40)

However, when the gauge parameters do depend on the fields, as in (34)-(37), one needs to resort to the modified Lie bracket [9]

 [(χ1,ϵ1),(χ2,ϵ2)]M=(^χ,^ϵ), (41)

where now

 ^χ=[χ1,χ2]+δ(χ1,ϵ1)χ2−δ(χ2,ϵ2)χ1, (42) ^ϵ=χμ1∂μϵ2−χμ2∂μϵ1+δ(χ1,ϵ1)ϵ2−δ(χ2,ϵ2)ϵ1.

With this modified bracket, one finds that the parameters (34) and (37) of the residual gauge symmetries form a representation of the infinite-dimensional Lie algebra which can be expressed as

 [(f1,YA1,U1),(f2,YA2,U2)]=(^f,^YA,^U), (43)

with

 ^f=f1∂vf2+YA1∂Af2−(1↔2), (44) ^YA=YB1∂BYA2−(1↔2), ^U=YA1∂AU2−(1↔2).

From this point on, the asymptotic symmetry analysis has to be treated separately for non-extremal and extremal horizons.

### 4.1 Non-extremal horizons

For isolated non-extremal horizons (), the first equation in (38) yields the linear equation

 0=κ∂vf+∂2vf, (45)

which has a solution of the form

 f(v,zA)=T(zA)+e−κv X(zA). (46)

We see from this that there are two distinct contributions to the supertranslation generator which have different physical interpretations. The first term in (46) generates supertranslation charge at the horizon. The exponential decay in advanced time in the second term in (46) resembles the so-called horizon redshift effect [42], where the energy of a photon moving tangential to the horizon undergoes a redshift proportional to . The wave analog of this effect is important in proving the linear stability of Schwarzschild and non-extreme Kerr spacetimes under scalar perturbations [43, 44]. In terms of these two different contributions the algebra (43) closes with

 ^T=YA1∂AT2−(1↔2), (47) ^X=YA1∂AX2−κT1X2−(1↔2).

Expanding the superrotation, supertranslation and electromagnetic charge generators in modes

 T(z,¯z)=∑m,nT(m,n)zm¯zn,X(z,¯z)=∑m,nX(m,n)zm¯zn, (48) Yz(z)=∑nznYn,Y¯z(¯z)=∑n¯zn¯Yn, U(z,¯z)=∑m,nU(m,n)zm¯zn,

where we used complex coordinates for the angular variables and , the algebra (43) becomes

 [Ym,Yn]=(m−n)Ym+n, (49) [¯Ym,¯Yn]=(m−n)¯Ym+n, [Yk,T(m,n)]=−mT(m+k,n), [¯Yk,T(m,n)]=−nT(m,n+k), [Yk,X(m,n)]=−mX(m+k,n), [¯Yk,X(m,n)]=−nX(m,n+k), [X(k,l),T(m,n)]=κX(m+k,n+l), [Yk,U(m,n)]=−mU(m+k,n), [¯Yk,U(m,n)]=−nU(m+k,n),

with the remaining commutators being zero. This algebra contains three sets of supertranslations currents, generated by , and , and two sets of Virasoro (Witt) modes which are in semi-direct sum with the supertranslations. The algebra contains ideals, generated by and . The supertranslation charge generator does not commute with but does commute with the generator of electromagnetic charge . The supertranslation zero-mode corresponds to the Killing vector associated to rigid translations in the advanced time , and consequently is associated to a notion of energy. A large set of generators , , , and commutes with this energy operator; it is thus natural to refer to them as soft horizon hairs. The generators , in contrast, behave under the action of as an expansion: . Hence, one may wonder about the existence of an additional conformal symmetry. However, as we will show next, does not appear in the conserved charge; and so we can conclude that it is pure gauge.

Diffeomorphisms and gauge symmetry transformations generated by the modes (49) have an associated set of conserved charges. The latter can be computed using again the method of [38], including now the gauge field contribution. We find111111The convention for the action used here is given in (52) below.

 Q[T,YA,U]=116π∫d2z√γΩ(2Tκ−YAθA−4UA(1)v−4A(0)BYBA(1)v), (50)

where there is indeed no contribution from . The first three terms in (50) correspond to the horizon charges computed in [27], while the fourth term is purely of electric origin, and the last term mixes the electromagnetic field with the superrotation vector field contribution. The charge (50) evaluated on the modes (49) obeys the algebra

 {Ym,Yn}=(m−n)Ym+n, (51) {¯Ym,¯Yn}=(m−n)¯Ym+n, {Yk,T(m,n)}=−mT(m+k,n), {¯Yk,T(m,n)}=−nT(m,n+k), {Yk,U(m,n)}=−mU(m+k,n), {¯Yk,U(m,n)}=−nU(m,n+k),

where , , , and are the charges associated to the modes , , , and , respectively. The brackets in (51) are defined as in [27]. The charge (50) and the algebra (51) generalize the results of [26] to include gauge fields. The generalization to the case of Abelian gauge fields is straightforward by considering with with .

In order to investigate the physical meaning of the charge (50), we can consider the static Reissner-Nordström solution to Einstein-Maxwell theory. The action of the theory is

 S=116π∫d4x√g(R−FμνFμν). (52)

The dyonic Reissner-Nordström solution in advanced Eddington-Finkelstein coordinates and a suitable gauge121212This amounts to performing a gauge transformation with parameter on the standard form of the gauge field. is given by the following metric and gauge field

 ds2 =−Δr2dv2+2dvdr+r2(dθ2+sin2θdϕ2), (53) A =−qrdv−p(cosθ−k)dϕ,

where with , where and are the electric and magnetic charges, respectively. The outer horizon is located at . The constant appearing in the gauge field is in principle arbitrary and can be changed by a gauge transformation. However, for the solution exhibits a special property: Provided , the gauge field configuration above is singular on the axis , where a Dirac string exists. Then, following standard practice, one can choose different gauges in each hemisphere, in such a way that either the north pole or the south pole is singularity free. This is achieved by choosing or , respectively.

After the coordinate transformation , the (outer) horizon is located at and the metric and the gauge field take a form suitable for comparison with the asymptotic symmetry analysis of the previous section. Expanding the metric near the horizon and comparing to (26), we can read off:

 κ=(r+−M)r2+,θA=0,Ωθθ=r2+,Ωϕϕ=r2+sin2θ,Ωθϕ=0. (54)

The expansion of the gauge field near yields

 Aρ=0,A(0)v=−qr+,A(1)v=qr2+,A(0)B=−p(cosθ−k)δϕB. (55)

The surface charge at the horizon for this static configuration with (54) and (55) yields

 Q[T,Yϕ,U]=116π∫dθdϕsinθ(2(r+−M)T−4qU+4pq(cosθ−k)Yϕ), (56)

where the range of integration over the constant section of the horizon has to be chosen such that the singularities of at and are avoided. This yields the zero-modes131313The charge , associated to the rigid translations , in terms of the complex variables is given by the charge .

 T(0,0) ≡ Q[1,0,0]=κr2+2, (57) U(0,0) ≡ Q[0,0,1]=−q, Y(0,0) ≡ Q[0,1,0]=0.

These three different contributions have the following interpretation: The first one, , has a simple interpretation in the context of black hole thermodynamics, as it gives the product between the Hawking temperature and the Bekenstein-Hawking entropy . The second contribution, , corresponds to the electric charge of the black hole. Finally, the third contribution, , gives the angular momentum of the black hole [26]. In the case of the static Reissner-Nordström solution, this gives zero141414The computation of this charge yields . Notice that had we performed the integral over the range without changing the gauge in each hemisphere, we would have rather obtained the result . This non-vanishing result comes from integrating a singular configuration of : it can be interpreted as the contribution from the Dirac string. which is consistent with the fact that the contribution of the electromagnetic field of the dyonic black hole to the total angular momentum is zero [45, 46].

### 4.2 Extremal horizons

In the above discussion of the horizon symmetries of the Reissner-Nordström black hole we assumed . The extremal limit, corresponding to , has to be treated separately. In particular, in this case equation (45) becomes

 ∂2vf=0, (58)

whose solution

 f=T(z,¯z)+v X(z,¯z), (59)

contains a linearly growing term in advanced time rather than an exponentially decaying one as in (46). This modifies the condition for closure of the algebra from (47) to

 ^T=T1X2+YA1∂AT2−(1↔2), (60) ^X=YA1∂AX2−(1↔2).

Expanding in modes we find the following algebra

 [Ym,Yn]=(m−n)Ym+n, (61) [¯Ym,¯Yn]=(m−n)¯Ym+n, [Yk,T(m,n)]=−mT(m+k,n), [¯Yk,T(m,n)]=−nT(m,n+k), [Yk,X(m,n)]=−mX(m+k,n), [¯Yk,X(m,n)]=−nX(m,n+k), [X(k,l),T(m,n)]=T(m+k,n+l), [Yk,U(m,n)]=−mU(m+k,n), [¯Yk,U(m,n)]=−nU(m+k,n),

with the other commutators being zero. It is interesting to compare (61) with (49). From the non-extremal algebra (49) one would naively expect and to commute when . However, (61) shows that this is clearly not the case; the limit is not continuous. Further comparison of the commutator reveals that the roles of and are interchanged between non-extremal and extremal horizons.

The set of conserved charges associated to (61) is

 Q[X,YA,U]=116π∫dzd¯z√γΩ(2X−YAθA−4UA(1)v−4A(0)BYBA(1)v). (62)

Notice that in contrast to (56) there is no dependence on in (62). From this we conclude that the modes associated to become pure gauge in the extremal case.

The algebra generated by these charges is the same as the one obeyed by the vector fields (61), namely

 {Ym,Yn}=(m−n)Ym+n, (63) {¯Ym,¯Yn}=(m−n)¯Ym+n, {Yk,X(m,n)}=−mX(m+k,n), {¯Yk,X