Bi-conformal symmetry and static Green functions in the higher-dimensional Reissner-Nordstrom spacetimes
BBi-conformal symmetry and static Green functions in the higher-dimensionalReissner-Nordstr¨om spacetimes
Valeri P. Frolov and Andrei ZelnikovTheoretical Physics Institute, Department of PhysicsUniversity of Alberta, Edmonton, AB, Canada T6G 2E1
We study a static scalar massless field created by a source located near an electrically chargedhigher dimensional spherically symmetric black hole. We demonstrated that there exist bi-conformaltransformations relating static field solutions in the metric with different parameters of the mass M and charge Q . Using this symmetry we obtain the static scalar Green function in the higherdimensional Reissner–Nordstr¨om spacetimes. PACS numbers: 04.50.Gh, 04.40.Nr, 11.10.Kk
I. INTRODUCTION
In this paper we continue studying minimally coupledmassless scalar fields created by static sources placed inthe vicinity of a higher dimensional static black holes.For this purpose we use the method of bi-conformal trans-formations, which was developed and applied to the caseof the Schwarzschild-Tangherlini metrics in our previouspaper [1]. This method is based on the following obser-vations.A scalar massless field Φ in a D -dimensional spacetimewith metric g µν ( µ, ν = 0 , . . . , D −
1) obeys the equation (cid:3) Φ = − πJ . (1.1)Let us consider a static scalar field Φ ( X ) created by asource J ( X ) in the static spacetime with the metric ds = − α dt + g ab dx a dx b ,X = ( t, x a ) , α = α ( x ) , g ab = g ab ( x ) . (1.2)Then the equation (1.1) is reduced and takes the formˆ F Φ = − πJ , ˆ F = 1 α √ g ∂ a (cid:0) α √ gg ab ∂ b (cid:1) . (1.3)Here g = det( g ab ). The red-shift factor α is connectedwith the norm of the static Killing vector ξ as follows: α = (cid:112) − ξ = √− g tt . The equation (1.3) is invariantunder the following bi-conformal transformations Φ = ¯ Φ , g ab = Ω ¯ g ab , α = Ω − n ¯ α , J = Ω ¯ J , (1.4)where n ≡ D − x a .This transformation consists of a bi-conformal map [2,3] of the original background D -dimensional metric g µν Ψ Ω : g → ¯ g , (1.5)accompanied by a properly chosen rescaling of the chargedensity J . If one starts with a solution of the Einsteinequations, a new metric, obtained as a result of this trans-formation, is not necessarily a solution of the Einstein equations with a physically meaningful stress-energy ten-sor. However, it may happen that for a specially cho-sen transformation this new metric has enhanced sym-metries.An interesting example is a Majumdar-Papapetroumetric, describing the gravitational field of a set of higherdimensional extremely charged black holes in equilib-rium. Under properly chosen bi-conformal map this met-ric reduces to the higher dimensional Minkowski metric.This allows one to solve the static scalar field equationin the Majumdar-Papapertou exactly (see [4]).In the paper [1] we demonstrated that the methodof bi-conformal transformations can be used for solv-ing static equations in spacetimes of static sphericallysymmetric black holes. The enhanced symmetry of thebi-conformal metric ¯ g was used in that paper to obtainthe static Green functions for the equation (1.3) in ahigher dimensional Schwarzschild-Tangherlini spacetime.In this paper we demonstrate how this method works forthe case a charged higher dimensional black hole.There are many possible applications of the proposedresult. One of them is an old problem of finding a self-energy and a self-force of charged particles near blackholes [5–8]. In four dimension the closed form of theexact solution for the field of a point charges in the blackhole geometry was obtained earlier [7–12].The recent interest to the problem of a self-force isstimulated by a study of the back-reaction of the fieldon the particle moving near black holes [13] in connec-tion with the gravitational wave emission by such parti-cles. More recently, several publications discussed higher-dimensional aspects of this problem (see, e.g., [14, 15]).This study was stimulated by general interest to space-times and brane models with large extra dimensions.The paper is organized as follows. In Section II we dis-cuss bi-conformal transformations of higher dimensionalspherically symmetric metrics and demonstrate that theReissner-Nordstr¨om metrics are bi-conformally related tothe higher dimensional Bertotti-Robinson metric. Thelatter is a product of 2D anti-de Sitter space and a sphere.Using this result we construct a bi-conformal map of theReissner-Nordstr¨om metrics with different parameters ofmass and charge. In Section III we obtain useful rep- a r X i v : . [ h e p - t h ] D ec resentations for static Green functions in a spacetime ofstatic spherically symmetric higher dimensional chargedblack holes. Section IV contains example of calculationsof the static Green functions for 4,5 and 6 dimensionalblack holes. Section V contains discussion of the obtainedresults and their possible generalizations. II. BI-CONFORMAL MAP AND SYMMETRYENHANCEMENT OF STATIC SPHERICALLYSYMMETRIC SPACETIMESA. Symmetry enhancement condition
Let us consider an application of the method of thebi-conformal maps to the case of a general static spheri-cally symmetric D -dimensional metric. The correspond-ing metric is ds = − f ( r ) dt + w − ( r ) dr + r dω n +1 , (2.1)where n = D − dω n +1 is the line element on a( n + 1)-dimensional unit sphere dω n +1 = dθ n + sin θ n dω n , dω = dφ . (2.2)We denote θ ≡ φ ∈ [0 , π ]. All other coordinates θ i> ∈ [0 , π ]. This metric is invariant under time trans-lations and spatial rotations. Since ds , t and r have thesame dimensionality of the length, the metric (2.1) canbe presented in the form ds = a dS , where the dimen-sionless metric dS is obtained from (2.1) by substituting t → t/a and r → r/a , where a is an arbitrary constantparameter with the dimensionality of length.Let us apply a bi-conformal transformation (1.4) tothis metric with Ω = r/a . This choice guarantees that Ωis dimensionless. After this bi-conformal transformationone has d ¯ s = dh + a dω n +1 ,dh = − (cid:16) ra (cid:17) n f ( r ) dt + a r w ( r ) dr . (2.3)The scalar curvature of the two-dimensional metric dh is R = − a f { rf w (cid:48) (2 nf + rf (cid:48) )+ w [2 r f (cid:48)(cid:48) − r ( f (cid:48) ) + 2 r (2 n + 1) f f (cid:48) + 4 n f ] (cid:9) . (2.4)Here and later ( . . . ) (cid:48) = d ( . . . ) /dr . The metric dh pos-sesses an enhanced symmetry if its 2D curvature R isconstant. We denote its value by R = − b , (2.5)where b is a constant of the dimensionality of the length.The equations (2.4) and (2.5) can be solved to determine the function w . The result is w = (cid:18) a n b + Cr n f (cid:19) (cid:18) rf (cid:48) nf (cid:19) − . (2.6)Here C is an integration constant. Let us suppose thatfunction f has the following asymptotic at infinity f = f + f r − γ + . . . , γ ≥ . (2.7)Then (2.6) shows that asymptotic value of w at the in-finity is a / ( n b ). The spacetime does not have a solidangle deficit and is asymptotically flat only if anb = 1 . (2.8)In what follows we always assume that this conditions issatisfied.By using the relation (2.6) one finds such functions { f ( r ) , w ( r ) } , for which the bi-conformal transformationof the metric (2.1) has an enhanced symmetry. The cor-responding metric d ¯ s = dh + a dω n +1 (2.9)is a direct sum of the two dimensional anti-de Sitter met-ric dh and the metric a dω n +1 on ( n + 1)-dimensionalsphere. The ratio of the curvature radii for these twometrics is fixed by the condition (2.8). This metric de-scribes a particular Bertotti-Robinson spacetime and canbe written in the following canonical form d ¯ s = a (cid:20) n (cid:18) − ( ρ − d ¯ σ + 1 ρ − dρ (cid:19) + dω n +1 (cid:21) . (2.10)Let us emphasize that the parameter a has dimensional-ity of the length and it is arbitrary. B. Bi-conformal map of Reissner-Nordstr¨ommetric to the Bertotti-Robinson space
Let us consider a special case of the metric (2.1) withan extra condition w = f . (2.11)For this choice the relation (2.6) becomes an equationwhich allows one to obtain the function f . The ordinarydifferential equation (2.6) with w = f is of the first or-der. Hence its solution besides the constant C containsanother arbitrary integration constants C . It is possibleto show that one can choose these constants so that thesolution takes the form f = 1 − Mr n + Q r n . (2.12)For real positive M and real Q , which satisfies the condi-tion | Q | ≤ M , the metric (2.1) with (2.11) and (2.12) isthe metric of a higher dimensional spherically symmetricelectrically charged black hole with M and Q being itsmass and charge, respectively.In order to rewrite the metric d ¯ s , obtained as a resultof the bi-conformal map (2.3), in the standard (canon-ical) form (2.10) it is sufficient to make the followingcoordinate transformations r n = M + µρ , t = a n +1 nµ ¯ σ , µ = (cid:112) M − Q . (2.13)We denote this bi-conformal map as followsΨ Ω : g M,Q → ¯ g BR , Ω = r/a . (2.14)
C. Bi-conformal transformations within theReissner-Nordstr¨om family of solutions
The method of bi-conformal maps was used in thepaper [1] to obtain static Green functions in thebackground of the higher dimensional Schwarzschild-Tangherlini spacetimes. For this purpose, one usesat first the enhanced symmetry of a related Bertotti-Robinson space to find the D -dimensional Green func-tion in this space, and after this one obtains the staticGreen function by means of the dimensional reduction.One can apply the same method for finding static Greenfunctions in the Reissner-Nordstr¨om geometry. However,there exist another much simper way. One can generatethe corresponding static Green function in the spacetimeof charged black holes by using the already known Greenfunction for the Schwarzschild-Tangherlini spacetime.For this purpose let us notice that the canonical form(2.10) is universal in the following sense: It is the same forany Reissner-Nordstr¨om metric and it does not dependon its parameters M and Q . This observation opens aninteresting possibility to relate metrics with different pa-rameters. Let us introduce new coordinates ˆ t and ˆ r ˆ r n = ˆ M + ˆ µρ , ˆ t = a n +1 n ˆ µ ¯ σ , ˆ µ = (cid:113) ˆ M − ˆ Q , (2.15)and denote ˆΩ = ˆ r/a . (2.16)Then one has the following bi-conformal map of theReissner-Nordstr¨om metric with parameters ˆ M and ˆ Q to the canonical Bertotti-Robinson metricΨ ˆΩ : g ˆ M, ˆ Q → ¯ g BR . (2.17)Combining the direct bi-conformal map (2.14) with thebi-conformal map, inverse to (2.17), one obtain a bi-conformal mapΨ = Ψ − ◦ Ψ Ω : g M,Q → g ˆ M, ˆ Q . (2.18)This bi-conformal map is a transformation of the originalReissner-Nordstr¨om metric with parameters M and Q to a similar metric with different parameters ˆ M and ˆ Q . Thestatic equation (1.3) is invariant under such a transfor-mation provided one in addition properly transforms thesource term J → ˆ J .In other words the solutions for the static field Φ in theoriginal background space are simply related to solutionsin a spacetime with modified parameters of the mass andthe charge. In particular, if one knows the static Greenfunction in the spacetime of uncharged black hole, onecan obtain the static Green function for the charged blackhole by using the above described transformations. In thenext section we demonstrate how this method works inmore detail. III. STATIC GREEN FUNCTIONSA. Bi-conformal map of static Green functions
Following the paper [1] we define a static Green func-tion G ( x, x (cid:48) ) as follows G ( x, x (cid:48) ) = (cid:90) ∞−∞ dt G Ret ( t, x ; 0 , x (cid:48) ) . (3.1)Here G Ret ( t, x ; 0 , x (cid:48) ) is a retarded Green function in theD-dimensional spacetime. This static Green function sat-isfies the the equationˆ F G ( x, x (cid:48) ) = − δ ( x − x (cid:48) ) α √ g . (3.2)In what follows, we assume that this Green function isdecreasing when one of its parameters x tends to infinityand remains regular at the horizon (for more details see[1]).The static Green function is simply related to the ex-pression for a scalar field created by a point charge. Thecurrent of a static point charge q positioned at the point y reads J ( x ) = q δ ( x − y ) √ g . (3.3)In this case the scalar field at the point x takes theform[20] Φ ( x ) = 4 πq α ( y ) G ( x, y ) . (3.4)The field of a distributed source q ( y ) can be easily ob-tained by integration over y of the right-hand side of thisrelation.It is convenient to introduce a new radial variable ρ related to the radial coordinate r as follows (2.13) ρ = r n − Mµ . (3.5)The Reissner-Nordstr¨om metric takes the form ds = − µ ( ρ − M + µρ ) dt + ( M + µρ ) /n (cid:20) n ( ρ − dρ + dω n +1 (cid:21) . (3.6)The horizon corresponds to ρ = 1 and the gravitationalradius r g is given by the expression r n g = M + µ . Thesurface gravity at the horizon is κ = nµr n +1 g . (3.7)In these coordinates the equation for the static Greenfunction takes the form (cid:2) n ( ρ − ∂ ρ + 2 n ρ ∂ ρ + (cid:52) n +1 ω (cid:3) G ( x, x (cid:48) )= − nµ δ ( ρ − ρ (cid:48) ) δ ( ω, ω (cid:48) ) . (3.8)Here (cid:52) n +1 ω and δ ( ω, ω (cid:48) ) are the Laplace operator and acovariant delta-function on the unit ( n + 1)-dimensionalsphere, respectively, (cid:52) n +1 ω = ∂ θ n + n cos θ n sin θ n ∂ θ n + 1sin θ n (cid:52) nω , (cid:52) ω = ∂ φ ,δ n +1 ( ω, ω (cid:48) = δ ( θ n − θ (cid:48) n )sin n θ n δ n ( ω, ω (cid:48) ) ,δ ( ω, ω (cid:48) ) = δ ( φ − φ (cid:48) ) . (3.9)Because of the spherical symmetry of background ge-ometry, the resulting static Green functions are the func-tions of radial coordinates of the observer ρ , the source ρ (cid:48) , as well as the angular distance γ ≡ γ n +1 between thesource and the observational point. G ( x, x (cid:48) ) = G ( ρ, ρ (cid:48) ; γ ) . (3.10)The angular distance on the ( n + 1) dimensional spherecan be written explicitly in terms of the angular coordi-nates (2.2)cos γ n +1 = cos θ n cos θ (cid:48) n + sin θ n sin θ (cid:48) n cos γ n ,γ = φ − φ (cid:48) . (3.11)The canonical Bertotti-Robinson spacetime (2.10) ishomogeneous. In the paper [1] we have used the knowl-edge of heat kernels on homogeneous spaces to derive thestatic Green functions. Now, using the bi-conformal sym-metry of the static operator (1.3), we can use these re-sults to derive the static Green functions in the Reissner-Nordstr¨om spacetime with arbitrary parameters of themass M and the charge Q .One can see that the bi-conformal transformation (1.4)with Ω = ra = ( M + µρ ) /n a , (3.12) leads to the Bertotti-Robinson canonical metric (2.10), if¯ σ is identified with the rescaled Reissner-Nordstr¨om timecoordinate t ¯ σ = ¯ κt , ¯ κ = nµa n +1 = (cid:16) r g a (cid:17) n +1 κ . (3.13)Here κ is given by (3.7) and ¯ κ is the surface gravity ofthe horizon ρ = 1 in the Bertotti-Robinson spacetime,normalized according to the Killing vector ¯ ξ µ = δ µt ¯ κ = −
12 ¯ ξ α ; β ¯ ξ α ; β (cid:12)(cid:12)(cid:12) ρ =1 . (3.14)One can define the static Green function in the canonicalmetric (2.10) as the integral over the dimensionless timecoordinate ¯ σ ¯ G ( x, x (cid:48) ) = (cid:90) ∞−∞ d ¯ σ ¯ G Ret (¯ σ, x ; 0 , x (cid:48) ) , (3.15)where ¯ G Ret (¯ σ, x ; ¯ σ (cid:48) , x (cid:48) ) is a retarded Green function in thecanonical Bertotti-Robinson spacetime (2.10). It satisfiesthe the equation (cid:2) n ( ρ − ∂ ρ + 2 n ρ ∂ ρ + (cid:52) n +1 ω (cid:3) ¯ G ( x, x (cid:48) )= − n a n +1 δ ( ρ − ρ (cid:48) ) δ ( ω, ω (cid:48) ) . (3.16)The left-hand side of this equation coincides with thatof (3.8). The right-hand sides of these equations differonly by a constant factor related to the rescaling of thetime coordinate (3.13). Thus, the static Green functionsin these spaces also differ only by a constant factor G ( ρ, ρ (cid:48) ; γ ) = 1¯ κ ¯ G ( ρ, ρ (cid:48) ; γ ) . (3.17)To construct the bi-conformal map (2.18) relatingReissner-Nordstr¨om with different parameters M and Q one proceeds as follows. Let us define two radial coordi-nates r and ˆ r by the relation r n − Mµ = ˆ r n − ˆ M ˆ µ ≡ ρ . (3.18)Here µ = (cid:112) M − Q , ˆ µ = (cid:113) ˆ M − ˆ Q . (3.19)This allows one to express the new coordinate ˆ r in termsof the original radial coordinate r . The time coordinatesare related as follows ˆ t = µ ˆ µ t . (3.20)The the bi-conformal transformation withΩ = (cid:20) M + µρ ˆ M + ˆ µρ (cid:21) /n (3.21)relates two arbitrary Reissner-Nordstr¨om metrics (3.6)characterized by the parameters M, Q and ˆ
M , ˆ Q , corre-spondingly.Because of the time rescaling the relation betweenthe static Green functions in these Reissner-Nordstr¨omspacetimes becomes µ G ( r, r (cid:48) ; γ ) = ˆ µ ˆ G (ˆ r, ˆ r (cid:48) ; γ ) . (3.22)Note that though the static Green function depends onthe time rescaling, this dependence is dropped out of theexpression for the scalar field Φ . The resulting Φ is invari-ant with respect to the time rescaling. One can say thatthe static scalar potentials Φ for all Reissner-Nordstr¨omgeometries are given by the same function of ρ . In termsof the radial coordinates r and ˆ r they are related by thecoordinate transformation (3.18). Therefore, as soon aswe know the static scalar Green function for a particularchoice of the charge of a black hole, for example, for aneutral one, the identity (3.22) makes it possible to gen-erate the solution for the scalar field near the Reissner-Nordstr¨om black hole with an arbitrary mass and charge.Since the cases of even and odd-dimensional spacetimesdiffer, so that we shall treat them separately. B. Even-dimensions
In even dimensions the exact static Green function canbe represented in the form of the integral G ( x, x (cid:48) ) = 1 nµ
12 (2 π ) n +32 (cid:18) ∂∂ cos γ (cid:19) ( n +1) / (cid:90) π dσ A n . (3.23)Here n = D − χ ) = ρρ (cid:48) − (cid:112) ρ − (cid:112) ρ (cid:48) − σ . (3.24)When n ≥
2, the functions A n ( σ, ρ, ρ (cid:48) ; γ ) are given bythe integral A n = (cid:90) ∞ χ dy (cid:112) cosh ( y ) − cosh ( χ ) sinh (cid:0) yn (cid:1)(cid:113) cosh (cid:0) yn (cid:1) − cos ( γ ) . (3.25)At large y the integrand in (3.25) behaves like exp[ − y ( n − / (2 n )]. Therefore, (3.25) is convergent for any n ≥ n = 1) theintegrand has to be modified to guarantee convergence ofthe integral. For example, one can subtract the asymp-totic of the integrand, which does not depend on γ . Since(3.23) contains the derivative of A n over γ , the resultingGreen function does not depend on the particular formof the subtracted γ -independent asymptotic. Thus, for n = 1 one can choose A = (cid:90) ∞ χ dy sinh ( y ) (cid:112) cosh ( y ) − cosh ( χ ) × (cid:34) (cid:112) cosh ( y ) − cos ( γ ) − (cid:112) cosh ( y ) + 1 (cid:35) . (3.26)Substitution ρ = r n − Mµ = r n − M (cid:112) M − Q (3.27)into (3.23) gives the static Green function of a scalarcharge near the Reissner-Nordstr¨om black hole (3.6) interms of the radial coordinate r . C. Odd-dimensions
In odd-dimensional spacetimes we have G ( x, x (cid:48) ) = 1 nµ √ π ) n +42 (cid:18) ∂∂ cos γ (cid:19) n/ (cid:90) π dσ B n , (3.28)where n = D − χ is given by (3.24) and B n = (cid:90) ∞ χ dy √ cosh y − cosh χ sinh (cid:0) yn (cid:1) cosh (cid:0) yn (cid:1) − cos γ . (3.29)Similarly to the even dimensions, the Green functionin the radial coordinates r can be obtained after the sub-stitution (3.27). IV. CLOSED FORM OF THE GREENFUNCTION: EXAMPLESA. Four dimensions. D = 4 In four dimensions ( n = 1) the integral (3.26) can bedone and one obtains A = ln (cid:18) cosh ( χ ) + 1cosh ( χ ) − cos ( γ ) (cid:19) . (4.1)The integral over σ can be taken explicitly and we obtainthe closed form for the static Green function G ( x, x (cid:48) ) = 14 πµ (cid:112) ρ + ρ (cid:48) − ρρ (cid:48) cos γ − γ ,ρ = r − Mµ = r − M (cid:112) M − Q . (4.2)When written in terms of the radial coordinate r it reads G ( x, x (cid:48) ) = 14 π R , (4.3)where R = ( r − M ) + ( r (cid:48) − M ) − r − M )( r (cid:48) − M ) cos γ − ( M − Q ) sin γ . (4.4)This formula exactly reproduces the closed form of thewell known result for the scalar Green function in four-dimensional Schwarzschild geometry [8, 11, 16]. It iseasy to check that using the bi-conformal symmetry(3.22) this solution could be generated from that of theSchwarzschild case ( Q = 0).In the limit of the extremally charged black hole Q = M the obtained solution (4.2) reproduces the result [4] forthe four-dimensional Majumdar-Papapetrou geometry. B. Five dimensions. D = 5 The other case, when there exists a closed form forthe static Green function is five-dimensional ( n = 2)Reissner-Nordstr¨om black hole. One can generate thissolution using the bi-conformal symmetry (3.22) fromthat of the Tangherlini black hole [1], or, equivalently,just make the substitution (3.27) in the expression forthe five-dimensional Green function (see eq.(6.14) of [1]). G ( x, x (cid:48) ) = 18 π µ ρ − / ( ρ (cid:48) − / × ∂∂ cos γ { κ [ F ( ψ, κ ) + K ( κ )] } , (4.5)where F and K are the elliptic functions ρ = r − Mµ , (4.6)andsin ψ = cos γ √ (cid:113) ρρ (cid:48) − (cid:112) ρ − (cid:112) ρ (cid:48) − , κ = √ ρ − / ( ρ (cid:48) − / (cid:113) ρρ (cid:48) + (cid:112) ρ − (cid:112) ρ (cid:48) − − γ . (4.7)To the best of our knowledge, this closed form forthe static Green function in five-dimensional Reissner-Nordstr¨om black hole is new.In the limit of the extremally charged black hole, when Q = M , the expression (4.5) leads to G ( x, x (cid:48) ) = 14 π R , (4.8)where R = ( r − M ) + ( r (cid:48) − M ) − γ (cid:112) r − M (cid:112) r (cid:48) − M .
It exactly reproduces the result [4] for the five-dimensional Majumdar-Papapetrou geometry in the caseof a single extremal black hole of the mass M . C. Six dimensions. D = 6 Application of the (3.23) to six-dimensional ( n = 3)Reissner-Nordstr¨om black hole leads to A = (cid:90) ∞ χ dy y − cosh χ ) / sinh (cid:0) y (cid:1)(cid:113) cosh (cid:0) y (cid:1) − cos ( γ )= 3 (cid:90) ∞ cosh( χ/ dz (cid:112) z − z − cosh χ √ z − cos γ . (4.9)This integral can be expressed in terms of the ellipticfunction F A = 6 (cid:112) v ( w − u ) F (cid:32) arcsin (cid:114) w − uw , w ( v − u ) v ( w − u ) (cid:33) , (4.10)where p = cosh( χ/ , w = 2( p − cos γ ) ,u = 3 p − i (cid:112) p − , v = 3 p + i (cid:112) p − . (4.11)Note that A is real in spite of the complexity of thefunctions u and v . Thus the static Green function inthe six-dimensional Schwarzschild-Tangherlini spacetimeis given by the integral G ( x, x (cid:48) ) = 148 π µ (cid:18) ∂∂ cos γ (cid:19) (cid:90) π dσ A . (4.12)It is problematic to obtain an answer for the Greenfunctions in a closed form for D ≥
6. However, a rathersimple integral representation is possible in all higher di-mensions. For some applications, like computing of theself-force and self-energy of scalar charges this integralrepresentations is sufficient to obtain the final results ina closed form.
V. DISCUSSION
In this paper we demonstrated that there exist bi-conformal transformations relating static solutions ofthe minimally coupled massless field equation in theReissner-Nordstr¨om spacetimes with different values ofthe parameters of the mass M and the charge Q . Weused this symmetry to generate expressions for the staticGreen functions in such space starting from similar Greenfunctions for the neutral (uncharged) higher dimensionalblack holes, that have been obtained earlier [1]. To checkthe obtained results, we considered limit of higher di-mensional extreme black holes with | Q | = M . This is aspecial case of the Majumdar-Papapetrou metrics relatedby means of a bi-conformal map to the flat spacetime. Itis possible to show that the obtained static Green func-tions in a generic Reissner-Nordstr¨om spacetime obey acorrect flat spacetime limit.Natural applications of the results obtained in our ear-lier publication [1] and in this paper is study of theproblem of the self-energy and self force of point scalarcharged in the background of higher dimensional staticblack holes. Especially interesting is the origin of nearhorizon logarithmic terms in these expressions in odd di-mensional black holes [14, 15] and the relation of theseterms with the bi-conformal anomalies (see discussion in[17–19]). It is interesting also to test the method of the bi-conformal transformations in application to electric fieldsof static sources in the static black hole backgrounds. An-other interesting question is: Is it possible to generalizethe method of bi-conformal to the case of fields from sta- tionary sources in a spacetime of rotating black holes.We are going to address these questions in our furtherwork. Acknowledgments
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D88 , 024032 (2013), [1303.1816].[20] Note that rescaling of the time variable by a constant fac-tor t = c ˜ t leads to ˜ α = cα , ˜ J = J , ˜ G ( x, x (cid:48) ) = c − G ( x, x (cid:48) ),˜ Φ = Φ .