Vacuum polarization of massive spinor and vector fields in the spacetime of a nonlinear black hole
aa r X i v : . [ h e p - t h ] F e b Vacuum polarization of massive spinor and vector fields in the spacetime of anonlinear black hole
Jerzy Matyjasek ∗ Institute of Physics, Maria Curie-Sk lodowska Universitypl. Marii Curie-Sk lodowskiej 1, 20-031 Lublin, Poland
Building on general formulas obtained from the approximate renormalized effective action, thestress-energy tensor of the quantized massive spinor and vector fields in the spacetime of the regularblack hole is constructed. Such a black hole is the solution to the coupled system of nonlinearelectrodynamics and general relativity. A detailed analytical and numerical analysis of the stress-energy tensor in the exterior region is presented. It is shown that for small values of the charge aswell as large distances from the black hole the leading behavior of the stress-energy tensor is similarto that in the Reissner-Nordstr¨om geometry. Important differences appear when the inner horizonbecomes close to the event horizon. A special emphasis is put on the extremal configuration and itis shown that the stress-energy tensor is regular inside the event horizon of the extremal black hole.
PACS numbers: 04.62.+v,04.70.Dy
I. INTRODUCTION
One the most important and intriguing open questions in the physics of compact objects is the issue of the final stageof black hole evolution and the problem of singularities residing in the internal region of the black holes. Althoughthe definitive answer to these questions would require the application of the full machinery of the (unknown as yet)quantum gravity or even more sophisticated approach, one can still obtain valuable results within the semiclassicalframework. Of course, the equations of the semiclassical gravity cannot be used to describe the evolution of thesystem completely: such equations are expected to break down in the Planck regime. On the other hand, however,having established the domain of applicability of the theory precisely one can obtain interesting and important results.Moreover, a careful analysis of the solutions to the semiclassical equations can show us tendencies in the evolution ofthe system, indicating its possible continuation.It is well known that the physical content of quantum field theory in spacetimes describing black holes is carried bythe renormalized stress-energy tensor (SET) evaluated in a suitable state [1]. Treating the renormalized stress-energytensor as the source term of the semiclassical Einstein field equations, one can, in principle, determine the backreaction of the quantized fields upon the spacetime geometry of black holes unless the (expected) quantum gravityeffects become dominant. Therefore, form the point of view of the semiclassical approach, it is crucial to have at one’sdisposal a general formula describing functional dependence of the renormalized stress-energy tensor on a wide classof metrics.In the semi-classical approach we are confronted with two major problems: construction of the renormalized stress-energy tensor on the one hand, and studying its influence via semi-classical equations on the system on the other.Unfortunately, even such a simplified approach leads to the equations that are still far too complicated to be solvedexactly and it is natural that much effort has been concentrated on developing approximate methods, referring tonumerical calculations or both.It seems that for the massive fields, the approximation based on the Schwinger-DeWitt expansion [2, 3, 4] is ofrequired generality. Indeed, it has been shown that for sufficiently massive fields, the renormalized effective action, W (1) ren , can be expanded in powers of m − . It is because the nonlocal contribution to the total effective action can beneglected, and, consequently, there remains only the vacuum polarization part which is local and determined by thegeometry of the spacetime. The stress-energy tensor can, therefore, be obtained by functional differentiation of theeffective action with respect to the metric tensor:2 g / δδg µν W (1) ren = h T µν i ren . (1)Such a tensor describing the vacuum polarization effects of the quantized massive fields in the vacuum type-D geome-tries has been constructed and subsequently applied in a series of papers by Frolov and Zel’nikov [5, 6, 7]. They used ∗ Electronic address: [email protected], [email protected] the Schwinger - DeWitt method [2, 3] and constructed the first order approximation of the effective action, omittingthe terms which do not contribute to the final results in the Ricci-flat spaces. These results have subsequently beenextended in Refs. [8] and [9], where the most general formulas describing the renormalized stress-energy tensor ofthe massive scalar, spinor and vector fields have been calculated. As the effective action consists of 10 (integrated)purely geometric terms constructed from the curvature tensor and multiplied by the spin-dependent coefficients, itsuffices to calculate their functional derivatives with respect to metric tensor only once. The stress-energy tensorof the scalar, spinor and vector fields can easily be obtained by taking the linear combination of the thus obtainedfunctional derivatives with the spin-dependent coefficients. Interested reader is referred to [8] and [9]. (Especially seeEqs. 7-18 and Table I of Ref [9]).The range of applicability of such a stress-energy tensor is dictated by the limitations of the validity of the renor-malized effective action: it can be used in any spacetime provided the mass of the quantized field is sufficiently great,i.e., when the Compton length, λ c , is much smaller than the characteristic radius of curvature, L, where the lattermeans any characteristic length scale associated with the geometry in question. Assuming, for example, that L isrelated to the Kretschmann scalar as K = R µνστ R µνστ ∼ L − , (2)one has a simple criterion for the validity of the Schwinger-DeWitt expansion. Typically, L ∼ M H , where M H isthe black hole mass, and, therefore, one expects that the approximation would be accurate provided m and L satisfy mL ∼ M H m ≫ . Using a different method, Anderson, Hiscock, and Samuel [10, 11] evaluated h T µν i ren of the massive scalar field witharbitrary curvature coupling for a general static, spherically symmetric spacetime and applied the obtained formulasto the Reissner-Nordstr¨om (RN) spacetime. (See also Ref. [12].) Their approximation is equivalent to the Schwinger- DeWitt expansion; to obtain the lowest (i. e. m − ) terms, one has to use sixth-order WKB expansion of the modefunctions. Numerical calculations reported in Ref. [10, 11] indicate that the Schwinger-DeWitt method always providea good approximation of the renormalized stress energy tensor of the massive scalar field with arbitrary curvaturecoupling as long as the mass of the field remains sufficiently large. The techniques presented in refs. [10, 11] havebeen successfully applied in a number of cases. Specifically, the vacuum polarization in the electrically charged blackholes have been studied in Refs. [10, 11], important issue of the black hole interiors in [13], the stress-energy tensorin the spacetimes of various wormhole types in [14] and the back reaction calculations in [15].On the other hand, the general formulas of Refs. [8, 9] have been applied in the spactimes of the Reissner-Nordstr¨om [8] and dilaton black holes [16]. Various aspects of the back reaction problem have been studied inRefs. [17, 18, 19]. Especially interesting in this regard are the regular black hole geometries, being the solutions ofthe coupled system of equations of the nonlinear electrodynamics and gravitation. The stress-energy tensor of themassive scalar fields (with an arbitrary curvature coupling) in such spaces have been studied in [9, 20].The issue of the regular black holes in general relativity has a long and interesting history. For example, one of themethods that can be used in construction of such configurations consits in replacing the singular black hole interiorby a regular core. This idea appeared in mid sixties [21, 22, 23] and its various realizations have been and still areinvestigated. For example, in the models considered in Refs. [24, 25] part of the region inside the event horizonis joined through a thin boundary layer to de Sitter geometry. Similar idea has been applied in the calculationsreported in Ref. [26], where the singular interior of the extreme Reissner-Nordstr¨om black hole has been replaced bythe Bertotti-Robinson geometry. Of course, such a geometric surgery does not exhaust all interesting possibilities.The regular geometries constructed with the aid of suitably chosen profile functions, or, better, the exact solutionsconstructed for specific, physically reasonable sources are of equal importance [27, 28, 29, 30, 31, 32].Recent interest in the nonlinear electrodynamics is partially motivated (beside a natural curiosity) by the fact thatthe theories of this type frequently arise in modern theoretical physics. For example, they appear as effective theoriesof string/M-theory. Moreover, on general grounds one expects that it should be possible to construct the regularblack hole solutions to the coupled system of equations of the nonlinear electrodynamics and gravity.One of the most interesting and intriguing regular solutions has been constructed by Ay´on-Beato and Garc´ıa [33]. Ithas subsequently been reinterpreted by Bronnikov in Refs. [34, 35]. The former solution describes a regular static andspherically symmetric configuration parametrized by the mass and the electric charge whereas the latter describes aformally similar geometry characterized by M and the magnetic charge Q . We shall refer to the solutions of this kindas ABGB geometries. It should be noted that the electric solution does not contradict the no-go theorem, which statesthat if the Lagrangian of the matter fields, L , is an arbitrary function of F = F µν F µν with the Maxwell asymptoticsin a weak field limit ( F µν is the electromagnetic tensor), then it cannot have a regular center. This is because theformulation of the nonlinear electrodynamics employed by Ay´on-Beato and Garc´ıa ( P framework in the nomenclatureof Refs. [34, 35]) is not the one to which one refers in the assumptions of the theorem. Specifically, the solution ofAy´on-Beato and Garc´ıa has been constructed in a formulation of the nonlinear electrodynamics obtained from theoriginal one ( F framework) by means of a Legendre transformation (see Ref. [35] for details).For certain values of the parameters the ABGB line element describes a black hole and an attractive feature of thissolutions that simplifies calculations is the possibility to express the location of the horizons in terms of the Lambertspecial functions [9, 20]. Moreover, as the function L ( F ) coincides with the Lagrangian density of the Maxwelltheory in the weak field limit, one expects that at large distances the static and spherically symmetric solution shouldapproach the Reissner-Nordstr¨om solution. A similar behavior should occur outside the event horizon for | e | /M ≪ , where e denotes either the electric or magnetic charge and M is the mass.The objective of this paper is to construct the renormalized stress-energy tensor of the quantized neutral massivespinor and vector fields in the spacetime of the regular ABGB black hole. The stress-energy tensor of the scalar fieldswith arbitrary curvature coupling has been constructed and discussed in Refs. [9, 20]. The results presented here arethe basic ingredients of the first-order back reaction calculations. They can also be used in the analysis of the variousenergy conditions and quantum inequalities. II. THE RENORMALIZED EFFECTIVE ACTION
The source term of the semi-classical Einstein field equations is given by the stress-energy tensor. Ideally, sucha tensor should be constructed from the renormalized effective action, W eff , in a standard way, i.e., by functionaldifferentiation of W eff , with respect to the metric. Unfortunately, neither the exact nor the approximate form of W eff is known in general. However, in a large mass limit of the quantized fields one can construct its local approximationsatisfactorily describing the vacuum polarization effects.The massive scalar, spinor and vector fields in curved spacetime satisfy the differential equations( − ✷ + ξR + m ) φ (0) = 0 , (3)( γ µ ∇ µ + m ) φ (1 / = 0 (4)and ( δ µν ✷ − ∇ ν ∇ µ − R µν − δ µν m ) φ (1) = 0 , (5)respectively, where ξ is the curvature coupling constant, and γ µ are the Dirac matrices obeying standard relations γ α γ β + γ β γ α = 2ˆ1 g αβ . The lowest-order approximation of the renormalized effective action, W (1) ren , of the quantizedmassive fields satisfying equations (3-5) is given by a remarkably simple expression W (1) ren = 132 π m Z g / d x [ a (0)3 ] − tr [ a (1 / ] tr [ a (1)3 ] − [ a (0)3 | ξ =0 ] (6)Here [ a ( s )3 ] is the coincidence limit of the fourth Hadamard-DeWitt-Minakshisundaram-Seeley [36] coefficient of thescalars ( s = 0) , spinors ( s = 1 /
2) and vectors ( s = 1). Making use of elementary properties of the Dirac matrices andthe Riemann tensor, after simple calculations, one obtains the first term of the asymptotic expansion of the effectiveaction in the form [37, 38] W (1) ren = 1192 π m X i =1 α ( s ) i W i = 1192 π m Z d xg / (cid:16) α ( s )1 R ✷ R + α ( s )2 R µν ✷ R µν + α ( s )3 R + α ( s )4 RR µν R µν + α ( s )5 RR µνρσ R µνρσ + α ( s )6 R µν R νρ R ρµ + α ( s )7 R µν R ρσ R ρ σµ ν + α ( s )8 R µν R µλρσ R νλρσ + α ( s )9 R ρσµν R µνλγ R λγρσ + α ( s )10 R ρ σµ ν R µ νλ γ R λ γρ σ (cid:17) (7)where the numerical coefficients α ( s ) i depending on the spin of the field are given in a Table I.Up to now, we have not specified the quantum state of the field. However, the construction of the effective actionhas been carried out with the assumption that the state in question may be identified with the Hartle-Hawking state.A closer examination of the problem indicates that outside the narrow strip in the closest vicinity of the event horizon,the results obtained in the Hartle-Hawking as well as the Unruh and the Boulware states are almost indistinguishable TABLE I: The coefficients α ( s ) i for the massive scalar, spinor, and vector fields = 0 s = 1/2 s = 1 α ( s )1 12 ξ − ξ + − − α ( s )2 1140 128 928 α ( s )3 ` − ξ ´ − α ( s )4 − ` − ξ ´ − α ( s )5 130 ` − ξ ´ − − α ( s )6 − − − α ( s )7 2315 471260 − α ( s )8 11260 191260 61140 α ( s )9 177560 297560 − α ( s )10 − − as they differ by the contributions of the real particles. On the other hand, inside that region the stress-energy tensorstrongly depends on the chosen state and may diverge at the event horizon. On general grounds, one expects that forregular geometries the Schwinger-DeWitt approximation yields a regular stress-energy tensor at the event horizon.It should be stressed that although the effective action W (1) ren can, in principle, be calculated for any line element, itsphysical applications are limited to the quantized fields in the large mass limit. Moreover, the technical difficulties onemay encounter in the process of calculation may prevent direct application of the effective action and the stress-energytensor. Finally, observe that the effective action approach employed in this paper requires the metric of the spacetimeto be positively defined. Hence, to obtain the physical stress-energy tensor one has to analitically continue the resultsconstructed for the Euclidean metric. III. THE REGULAR ABGB BLACK HOLE
An interesting solution to the coupled system of nonlinear electrodynamics and gravity representing a class of theblack holes parametrized by a mass and a charge has been constructed recently by Ay´on-Beato and Garc´ıa [33] andby Bronnikov [34, 35]. The former describes electrically charged configuration in the P -framework whereas the latterdescribes geometry of the magnetically charged solution in the F -framework. Both line elements are formally identicaland can be written in the form ds = − f ( r ) dt + f − ( r ) dr + r (cid:0) dθ + sin θdφ (cid:1) , (8)where f ( r ) = 1 − M H r (cid:20) − tanh (cid:18) e M H r (cid:19)(cid:21) , (9) M H is the black hole mass and e is either the magnetic or the electric charge. For small values of the charge it differsoutside the event horizon from the Reissner-Nordstr¨om solution by terms of order O ( e ) . Similarly, at large distancesthe the function f ( r ) also closely resembles that of the RN solution. Indeed, expanding metric potentials in a powerseries one concludes that the ABGB solution behaves asymptotically as f ( r ) = 1 − M H r + e r − e M H r + O ( 1 r ) . (10)On the other hand, and this is even more interesting and has profound consequences, the interior of the ABGBsolution is regular. This can be demonstrated by studying behavior of various curvature invariants. It can be shownthat curvature invariants factorize in such a way that there is a common multiplicative factor, which for r → (cid:0) − e /M H r (cid:1) . For e = 0 the ABGB solution reduces to the Schwarzschild line element and it isthe nonlinear charge, no matter how small, that leads to the dramatic changes of the geometry.The spacetime described by the line element (8) with (9) has been extensively studied in [9, 20, 33]. Specifically, ithas been shown that although the metric coefficient f ( r ) is a complicated function of r , the location of the horizonsmay be elegantly expressed in terms of the Lambert functions [9]. Since these results are, apparently, not widelyknown we shall summarize a few basic facts. For a short description of the Lambert functions the reader is referredto [39].Making use of the substitution r = M x and e = q M , and subsequently introducing a new unknown function W by means of the relation x = − q W − q , (11)one arrives at exp( W ) W = − q q / . (12)Since the Lambert function is defined as exp( W ( s )) W ( s ) = s, (13)one concludes that the location of the horizons as a function of q = | e | /M, is given by the real branches of the Lambertfunctions x + = − q W (0 , − q exp( q / − q , (14)and x − = − q W ( − , − q exp( q / − q . (15)The functions W (0 , s ) and W ( − , s ) are the only real branches of the Lambert function with the branch point at s = − / e , where e is the base of natural logarithms. Finally, observe, that simple manipulations of Eqs. (14) and(15) indicate that for q extr = 2 w / = 1 . , (16)the horizons r + and r − merge at x extr = 4 w w =0 . , (17)where w = W (1 / e) and W ( s ) is a principal branch of the Lambert function W (0 , s ) . Inspection of (16) reveals another interesting feature of the ABGB geometries: the black hole solution exists for q greater than the analogous ratio of the parameters of the RN solution. The three types of the ABGB solutionstherefore are: the regular black hole with the inner and event horizons for q < q extr , the extremal black hole for q = q extr , and the regular configuration for q > q extr . As have been observed earlier at large distances as well as forsmall charges the geometry of the ABGB solution resembles that of the Reissner-Nordstr¨om. There is, however, onenotable distinction: for q > , the Reissner-Nordstr¨om solution describes unphysical naked singularity whereas theregular geometry for q > q extr could be interpreted as a particle like solution. IV. STRESS-ENERGY TENSOR
The renormalized stress-energy tensor of the quantized massive scalar (with an arbitrary curvature coupling), spinorand vector fields in a large mass limit has a general form h T µν i ren = 196 π m g / X i =1 α ( s ) i δδg µν W i , (18)where W i can be obtained from Eq. (7) and the spin dependent coefficients are listed in Table I. The purely geometricobjects δW i /δg µν have been calculated in Refs. [8, 9]. It has been shown that the thus obtained renormalized stress-energy tensor consists of approximately 100 terms (constructed from the Riemann tensor, its covariant derivativesand contractions) combined with the numerical coefficients depending on the spin of the quantized field. However,such a local geometric structure of the stress-energy tensor has its price: h T µν i ren does not describe the process ofparticle creation which is a nonlocal phenomenon. Fortunately, for sufficiently massive fields, the contribution of thereal particles can be neglected and the Schwinger-DeWitt action satisfactorily approximates the total effective action.The general expression describing the renormalized stress-energy tensor of the quantized fields in a large mass limitis rather complicated and to avoid unnecessary proliferation of lengthy formulas it will be not presented here. Forits full form as well as the technical details the interested reader is referred to [9] (Especially see Eqs.7-18) and [8].Because of numerous identities that hold for the Riemann tensor, the final form of the stress-energy tensor is notunique and obviously depends on adapted simplification strategies. It should be noted, however, that any othercalculation based on the effective action (7) with the numerical coefficients α ( s ) i for scalar, spinor and vector fieldmust yield results identical to those of Refs. [8, 9]. Recently, an equivalent form of the renormalized stress-energytensor of the quantized massive fields has been constructed by Folacci and Decanini [40].As has been stated earlier, the general formulas of Refs. [8, 9] have been successfully applied in a number cases, suchas Reissner-Nordstr¨om spacetime [8], dilatonic black holes [16], and various back reaction calculations [17, 18, 19].Moreover, the renormalized stress-energy tensor of the quantized massive scalar field with the arbitrary curvaturecoupling in the ABGB geometry has been calculated and exhaustively discussed in Refs. [8, 9]. In this section weshall extend these calculations to the massive spinor and vector fields.Since the general form of the stress-energy tensor is rather complicated, one expects that its components evaluatedfor the specific line element are, except simple geometries, formidable. Our calculations in the ABGB backgroundclearly shows that this is indeed the case, and once again, to avoid unnecessary proliferation of long formulas we shallnot display them here . On the other hand, one can obtain a great deal of information studying the behaviour of thecomponents of the stress-energy tensor in some physically important regimes. Below we shall consider expansions ofthe stress-energy tensor for small q, large x, and study the configuration near and at the extremality limit. Specialattention will be put on the regularity issues and the interior of the extreme ABGB black hole. A. General features of the stress-energy tensor in the ABGB spacetime
Each component of h T νµ i ( s ) ren in the ABGB spacetime has a general form h T νµ i ( s ) ren = 196 π m M H (cid:18) − tanh q x (cid:19) X i,j,k α ( s ) ijk q i x j tanh k q x , (19)where 0 ≤ i ≤ , ≤ j ≤
15, 0 ≤ k ≤ , and α ( s ) ijk are numerical coefficients depending on the spin of the field. Forsimplicity, we have omitted tensor indices in right hand side of the above equation. This result can be contrastedwith the analogous expression obtained for the Reissner-Nordstr¨om black hole h T νµ i ( s ) RN = 196 π m M H X i,j β ( s ) ij q i x j , (20)where 0 ≤ i ≤ , ≤ j ≤
12 and β ( s ) ij are, as before, the numerical coefficients depending on s. In both cases thestress-energy tensor is covariantly conserved and falls as r − as r → ∞ . The latter behavior indicates that there isno need to impose spherical boxes in the back reaction calculations. Moreover, both tensors are regular at the eventhorizon.Since the Lagrangian density of the classical (nonlinear) field considered in this paper tends to its Maxwell analogueas F (= F µν F µν ) → , one expects that in this limit, regardless of the spin of the quantized field, the leading behaviorof the renormalized stress-energy tensor of the quantized massive fields is similar to the analogous terms constructed inthe Reissner-Nordstr¨om geometry. On the other hand, for the configurations near the extremality limit the differencesbetween the tensors outside the event horizon should be more prominent.It can be demonstrated that the difference between the radial and time components of the stress-energy tensorfactorizes: h T rr i ( s ) ren − h T tt i ( s ) ren = (cid:20) − x (cid:18) − tanh q x (cid:19)(cid:21) F ( x ) , (21) The complete results in various formats can be obtained from the author where F ( x ) is a regular function. Now, let us consider a freely falling frame. A simple calculation shows that theframe components of the tensor T νµ are T (0)(0) = γ (cid:0) T − T (cid:1) f − T , (22) T (1)(1) = γ (cid:0) T − T (cid:1) f + T , (23) T (0)(1) = − γ p γ − f (cid:0) T − T (cid:1) f , (24)where γ is the energy per unit mass along the geodesic. One concludes, therefore, that since all components of h T νµ i ( s ) ren are regular and (cid:16) h T rr i ( s ) ren − h T tt i ( s ) ren (cid:17) /f is by (21) finite, the stress-energy tensor of the quantize massive fields isregular in freely falling frame. B. Stress-energy tensor on AdS × S spacetime Let us postpone the detailed analysis of the stress-energy tensor in the ABGB spacetime for a while and considera far more simple case of the
AdS × S geometry. Such geometries are closely related to the extremal black holes.Indeed, AdS × S can be obtained by expanding the geometry of the vicinity of the event horizon into a wholemanifold. Various aspects of the geometries of this type have been discussed, for example, in [8, 17, 41, 42, 43, 44,45, 46, 47, 48, 49, 50, 51].The extremal ABGB black hole is described by a line element (8) with f ( r ) = 1 − M H r (cid:20) − tanh (cid:18) M H wr (cid:19)(cid:21) . (25)Now, in order to investigate the geometry in the vicinity of the event horizon, x extr and to obtain uniform approxi-mation we introduce new coordinates ˜ t = t/ε and r = r + ε/ ( h y ) , (26)where h = (1 + w ) / (32 M H w ) (27)and r = r extr . Expanding the function f ( r ) in powers of ε, retaining quadratic terms and subsequently taking thelimit ε = 0 we obtain ds = 1 hy (cid:0) − dt + dy (cid:1) + r d Ω . (28)Since h − > r , the line element does not belong to the Bertotti-Robinson class, contrary to the near-horizon geometryof the Reissner-Nordstr¨om solution. Alternatively, this can easily be demonstrated making use of the relation f ′′ ( r + ) = 2 r + 8 πT µµ , (29)where prime denotes differentiation with respect to the radial coordinate, as the stress-energy tensor of the nonlinearelectromagnetic field, T νµ , has nonvanishing trace at the event horizon.Other frequently used representations of the line element (28) can be obtained through the change of coordinatesystem. Using, for example, h / t = e τ coth χ, h / y = e τ sinh − χ (30)and sinh χ = R h − , τ h = T (31)one obtains ds = 1 h (cid:0) − sinh χdt + dχ (cid:1) + r d Ω (32)and ds = − (cid:0) R h − (cid:1) d T + d R R h − r d Ω , (33)respectively. Topologically the geometry described by the line element (28) is a direct product of the two-dimensionalanti-de Sitter geometry and the two-sphere of constant curvature; its curvature scalar is simply a sum of the curvaturesof the subspaces AdS and S : R = K AdS + K S , (34)where K AdS = − h and K S = 2 /r . Now, let us return to the stress-energy tensor of the massive fields. Simple calculations yield h T νµ i ( s ) ren = 196 π m diag h A ( s ) , A ( s ) , B ( s ) , B ( s ) i νµ , (35)where A (1 / = 121 h + 160 r h + 142 r , (36) B (1 / = − h − r h − r (37)and A (1) = 835 h + 15 r h + 435 r , (38) B (1) = − h − r h − r (39)for spinor and vector fields, respectively. In view of our earlier discussion we expect that the results (35-39) coincidewith the components of the stress-energy tensor calculated at the event horizon of the extremal ABGB black hole. C. Stress-energy tensor of massive spinor and vector fields in the spacetime Reissner-Nordstr¨om black hole
The renormalized stress-energy tensor of the quantized massive spinor and vector fields in the Reissner-Nordstr¨omspacetime has been constructed in Ref. [8]. It turns out that although the general formulas describing h T νµ i ( s ) ren arerather complicated, its components calculated in the Reissner-Nordstr¨om spacetime are simple functions of the radialcoordinate due to the spherical symmetry and the form of the metric potentials. These results will be used for thecomparison with the analogous results obtained in the ABGB spacetime and we reproduce them here for the reader’sconvenience.The components of the spinor field read h T tt i (1 / RN = 140320 π m x M H (cid:0) x + 10544 x q − x q + 21832 x q − x − xq + 4917 q + 5400 x q (cid:1) , (40) h T rr i (1 / RN = 140320 π m x M H (cid:0) x + 1080 x q − x − x q + 3560 x q + 8440 x q − xq + 2253 q (cid:1) , (41)and h T θθ i (1 / RN = − π m x M H (cid:0) − x + 12080 x q − x q + 30808 x q + 1512 x − xq + 9933 q + 3240 x q (cid:1) . (42)Similarly, for the massive vector field one has h T tt i (1) RN = 110080 π m x M H (cid:0) q + 1665 x + 41854 x q + 93537 x q − xq − x − x q + 12150 q x (cid:1) , (43) h T rr i (1) RN = 110080 π m x M H (cid:0) x − x + 12907 x q − x q − xq + 2430 q x + 6442 x q + 5365 q (cid:1) , (44)and h T θθ i (1) RN = − π m x M H (cid:0) q − x + 20908 x q + 30881 x q − xq + 4854 x − x q + 7290 q x (cid:1) . (45)Although there are no numeric calculations of the stress-energy tensor of the quantized massive spinor and vectorfields against which one could test the results (40- 45), we expect that the approximation is reasonable so long themass of the field is sufficiently large. Thanks to the detailed analytical and numerical calculations carried out inRefs. [10, 11] we know that this is indeed the case for the massive scalar field. It is a very important result, indicatingthat that the exact stress-energy tensor of the scalar field may satisfactorily be approximated with the accuracy withina few percent provided M H m ≥
2. Further, as the sixth-order WKB-approach employed in [10, 11] is equivalent tothe Schwinger-DeWitt expansion in inverse powers of m , this affirmative result yields a positive verification of thelatter approach.Finally, let us consider the stress-energy tensor of the massive fields in the spacetime of the extreme Reissner-Nordstr¨om black hole. Its horizon value is given by h T νµ i ( s ) ren = β ( s ) π m M H diag[1 , , − , − , (46)where β (1 / = 37 /
12 and β (1) = 19 . It can be easily demonstrated that it coincides with the stress-energy tensor ofthe massive field in the Bertotti-Robinson geometry.
D. Massive spinor fields in ABGB spacetime
Now, let us return to ABGB geometry and consider h T νµ i ( s ) ren near the event horizon of the extremal black hole.It can be shown that the renormalized stress-energy tensor of the massive spinor field for x close to x extr may beapproximated by h T νµ i (1 / ren = (1 + w ) × π m M H w (cid:20) A (1 / νµ + 12 w B (1 / νµ ( x − x extr ) (cid:21) + O ( x − x extr ) , (47)where A (1 / tt = A (1 / rr = 57 + 44 w + 37 w + 10 w , (48) A (1 / θθ = A (1 / φφ = − (99 + 29 w + 15 w + 5 w ) (49)0and B (1 / tt = B (1 / rr = − ( w + 1) ( w + 3)(52 + 7 w + 15 w ) , (50) B (1 / θθ = B (1 / φφ = ( w + 1) (cid:0)
355 + 156 w + 58 w + 17 w (cid:1) . (51)Numerically, one has h T νµ i (1 / ren = 1 m M H − diag [1 . , . , − . , − . νµ − m M H − diag [5 . , . , . , . νµ ( x − x extr ) + O ( x − x extr ) . (52)To this end, observe that x → x extr limit of (47) coincides with the stress-energy tensor of the massive spinor fieldin AdS × S spacetime. To demonstrate this, it suffices to substitute into Eqs. (36) and (37) the explicit forms of r and h as given by Eqs (17) and (27), respectively.Having established the expansion of the components of the stress-energy tensor for the extremal configuration letus analyze their leading behavior for q ≪ . It can be shown that expanding the stress-energy tensor in powers of q one obtains h T νµ i (1 / ren = h T νµ i (1 / RN + q π m M H t (1 / νµ + O (cid:0) q (cid:1) (53)where t (1 / tt = − − x + 945 x x , (54) t (1 / rr = − − x + 35 x x , (55)and t (1 / θθ = t (1 / φφ = 1775 − x + 280 x x , (56)where h T νµ i (1 / RN is given by Eqs. (40-42). Inspection of (40-42) and (54-56) indicates that for q ≪ O (cid:0) q (cid:1) terms.Now, let us consider the leading behavior of the stress-energy tensor at large distances ( r/r + ≫ q may be written as h T νµ i (1 / ren = 1 π m M H ˜ t (1 / νµ + O (cid:0) x − (cid:1) (57)where ˜ t (1 / tt = 3 (cid:0) q − (cid:1) x + 149 − q x + q (cid:0) q + 5458 − q (cid:1) x , (58)˜ t (1 / rr = 15 q + 7560 x − q + 492520 x + q (cid:0) q + 422 − q (cid:1) x π , (59)and ˜ t (1 / θθ = ˜ t (1 / φφ = − (cid:0) q + 7 (cid:1) x + 221 + 1251 q x − q (cid:0) q − q (cid:1) x . (60)Once again, the leading behavior of h T νµ i (1 / ren as r → ∞ , (which is governed by the first term in the above equationsand strongly depends on q ) is identical to the analogous behavior in the Reissner-Nordstr¨om case. On the other hand,substituting q = 2 w / into Eqs. (58-60) one obtains the expansion of the stress-energy tensor at large distances fromthe event horizon of the extreme black holes. It should be noted, however, that any comparison of the extremal ABGBand Reissner-Nordstr¨om black holes should be interpreted with care as the extremality limit occurs for different valuesof q. E. Massive vector fields in ABGB spacetime
The calculations of the renormalized stress-energy tensor of the quantized massive vector fields proceed along thesame lines as for the spinor case. Repeating the steps necessary to calculate the SET of the massive spinor field andfocusing attention on the narrow strip near the degenerate event horizon of the extremal black hole, one has h T νµ i (1) ren = (1 + w ) × π m M H w (cid:20) A (1) νµ + 12 w B (1) νµ ( x − x extr ) (cid:21) + O ( x − x extr ) , (61)where A (1) tt = A (1) rr = 27 + 26 w + 19 w + 4 w , (62) A (1) θθ = A (1) φφ = − w + 3 w + w ) (63)and B (1 / tt = B (1 / rr = − ( w + 1) ( w + 3)(25 + 7 w + 6 w ) , (64) B (1 / θθ = B (1 / φφ = 16 ( w + 1) (cid:0) w + 485 w + 66 w (cid:1) . (65)Since the location of the event horizon as well as the value of q extr depend on the particular value of the Lambertfunction one can easily determine numerical value of the components of the stress-energy tensor on the event horizon.Making use of (61) one obtains h T νµ i (1) ren = 1 m M H − diag [6 . , . , − . , − . νµ − m M H − diag [35 . , . , . , . νµ ( x − x extr ) + O ( x − x extr ) . (66)Using, once again, Eqs.(17) and (27) one can easily demonstrate that the horizon value of the stress-energy tensor(61) reduces to that calculated in AdS × S , geometry.For any value of the radial coordinate and small q, the stress-energy tensor may be approximated by h T νµ i (1) ren = h T νµ i (1) RN + q π m M H t (1) ba + O (cid:0) q (cid:1) (67)where t (1) tt = − (212249 − x + 34020 x )13440 x , (68) t (1) rr = − − x + 3780 x x (69)and t (1) θθ = t (1) φφ = 82501 − x + 15120 x x . (70)Now, expanding the general stress-energy tensor of the vector field for r/r + ≫ q ) in the form h T νµ i (1) ren = 1 π m M H ˜ t (1) νµ + O (cid:0) x − (cid:1) , (71)where ˜ t (1) tt = 270 q + 37224 x − q + 6111680 x − q (cid:0) − q + 25515 q − (cid:1) x , (72)2˜ t (1) rr = − − q x − q − x − q (cid:0) − − q + 2835 q (cid:1) x , (73)and ˜ t (1) θθ = ˜ t (1) φφ = 3 (cid:0) − q (cid:1) x + 15854 q − x + q (cid:0) − q + 11340 q − (cid:1) x . (74)Finally observe that the q = 2 w / limit taken in Eqs. (71-74) supplements the discussion of the extremal black holes.The results presented in Sec. IV D and IV E can be applied in further calculations. In the proofs of various theoremsin General Relativity, for example, the stress-energy tensor is expected to satisfy some restrictions usually addressedto as the energy conditions. Their detailed studies are worthwhile as the violation of the energy conditions frequentlyleads to exotic, yet physically interesting situations. Of course, the main role played by the renormalized stress-energytensor is to serve as the source term of the semi-classical Einstein field equations. For the problem on hand one cancalculate the back reaction on the metric in the first-order approximation. Unfortunately, the components of themetric tensor of the quantum-corrected spacetime are rather complicated functions of the radial coordinate, eachconsisting of several hundred terms [52]. Therefore, to analyze the quantum-corrected spacetime it is necessary torefer to approximations or even to numerical calculations. F. Inside the event horizon of the extremal ABGB black hole
In this subsection we shall analyze the stress-energy tensor inside the extremal ABGB black hole. The line elementinside the degenerate horizon is regular, and, for r → f ∼ − x exp ( − w/x ) . (75)This may be contrasted with the analogous behavior of the Reissner-Nordstr¨om solution f ∼ x . (76)Even without detailed calculations certain qualitative features of h T νµ i ( s ) ren can be deduced from this formulas. Indeed,since the stress-energy tensor is constructed form the Riemann tensor, its covariant derivatives up to certain orderand contractions, the result of all this operations, in view of the asymptotic relation (75), should be regular. This canalso be demonstrated using Eq. (19), which, in the case in hand, can be written in the form h T νµ i ( s ) ren = 196 π m M H (cid:18) − tanh 2 wx (cid:19) X i,j,k ˜ α ( s ) ijk w i x j tanh k wx , (77)where for each component ˜ α ( s ) ijk are numerical coefficient depending on the spin of the massive field (we have omittedtensor indices to make the formulas more transparent). Alternatively, one can utilize approximation of the componentsof the stress-energy tensor valid small r h T νµ i ( s ) ren ∼ π m M H exp ( − w/x ) X i,j ˜ β ( s ) ij w i x j . (78)Inspection of Eqs. (77) or (78) shows that h T νµ i ( s ) ren → r → . This is simply because the Schwinger-DeWittapproximation is local and depends on the geometric terms constructed from the curvature. Since the line elementhas the Euclidean asymptotic as r → , then, regardless of the spin of the field, the renormalized stress-energy tensormust vanish in that limit.It should be noted, however, that the regularity of the source term does not necessarily leads to the regularity ofthe quantum corrected geometry. Indeed, the latter requires that various curvature invariants of the self-consistentsolution of the semi-classical equations with the total source term given by the sum of classical stress-energy tensorof the nonlinear electrodynamics and of the quantized massive fields be regular. However, since the resulting semi-classical equations comprise a very complicated system of sixth-order differential equations, there are no simple wayto construct the appropriate solutions. A comprehensive discussion of the analogous situation in the quadratic gravityhas been carried out in [53].3 G. Numerical results
Considerations of the previous sections concentrated on the approximate analytical results valid in a few importantregimes: q ≪ x ≫ r and q, one has to refer to numerical calculations, as our complete but rathercomplicated results are, unfortunately, not very illuminating. Below we describe the main features of the constructedtensors and present them graphically. Related discussion of the spin 0 field has been carried out in Refs.[9, 20].First, let us consider the horizon values of the components of h T νµ i ( s ) ren . Spherical symmetry and regularity imposesevere constrains on the structure of the stress-energy tensor at the event horizon. It suffices, therefore, to consideronly its two independent components, say, h T tt i ( s ) ren and h T θθ i ( s ) ren . The run of this components as functions of q isexhibited in Figs. 1 and 2, for spinor and vector fields, respectively.The run of the stress-energy tensor for a several exemplary values of q is exhibited in Figs. 3-13. Each curverepresents the radial dependence of the rescaled component of h T νµ i ( s ) ren for a given q. We shall start our discussion ofthe numerical results with the spin-1 / ρ (1 / (cid:16) ρ ( s ) = −h T tt i ( s ) ren (cid:17) is alwaysnegative at the event horizon, and, thus, by continuity, it is negative in its vicinity. Further ρ (1 / attains a positivelocal maximum as can be clearly seen in Fig 3. For q > / √ − as r → ∞ . Further, inspection of the leading behavior of Eq. (58) shows that ρ (1 / is positive atlarge distances for q < / √ . The radial pressure p (1 / r ( p ( s ) r = h T rr i ( s ) ren ) is positive at the event horizon and p (1 / r ( r + ) = − ρ (1 / ( r + ); subse-quently, it decreases monotonically to 0 + with r . The behavior of p (1 / r is plotted in Fig. 5.The tangential pressure, p (1 / θ (cid:16) p ( s ) θ = h T θθ i ( s ) ren (cid:17) is plotted in Figs. 6 and 7. At the event horizon it is positive for q < . r/r + ≈ . − as r → ∞ . A closer examination indicatesthat for q > . , it develops a local maximum, which disappears near the extremality limit.In general, there are no qualitative similarities between components of the renormalized stress-enegy tensor of themassive spinor and vector fields, as can be easily seen in Figs. 8-13. In the vicinity of the event horizon the energydensity of the massive vector field is positive for q < .
581 and negative otherwise. For q > .
465 the energy densityapproaches a maximum, and, subsequently, regardless of q it has a minimum. As the leading behavior as r → ∞ isgoverned by the first term in rhs of (72), p (1) t → − . Other qualitative and quantitative features of the energy densitycan easily be inferred from Figs. 8 and 9.Numerical calculations indicate that for q < .
387 the radial pressure, p (1) r , is negative and monotonically increasesto 0 − as r → ∞ . For 0 . < q < .
919 there appears a local minimum in the closest neighborhood of the eventhorizon, and, for 0 . < q < .
919 , the radial pressure approaches a local maximum. Finally, for q > . + . The run of p (1) r for a few exemplary values of q is plotted in Figs. 10 and 11.The tangential pressure of the vector field is negative on the event horizon and increases to a global maximum.Subsequent behavior of p (1) θ depends on q : it decreases monotonically to 0 + for q < .
534 whereas for q > .
534 thetangential pressure has a local minimum and increases to 0 − . Some other qualitative and quantitative features, as forexample the numerical values of p (1) θ at the maxima and minima can easily be inferred from Figs. 12 and 13.The numerical calculations carried out in the external region of the extremal configuration shows that the run ofthe stress-energy tensor qualitatively follows the analogous behavior for q = 1 case, and, consequently, it will not bediscussed separately.Now, let us consider the vacuum polarization effects inside the event horizon of the extremal configuration. Therun of the rescaled components of stress-energy tensor of the massive spinor field is exhibited in Figs 14-16. All thecomponents display oscillatory behavior for r/r + > . , indicating that the back reaction effects would be especiallyinteresting there. Such a behavior can easily be understood in relation with the behavior of the line element. Indeed,for small r the line element closely resembles that of a flat spacetime, and, consequently, the vacuum polarizationeffects are small. On the other hand, for r/r + > .
05 the function f ( r ) changes noticeably leading to the changes ofthe stress-energy tensor. The competition of the local geometric terms δW i /δg µν lead to its oscillatory-like behavior.Numerical calculations indicate that the stress-energy tensor of the quantized vector field is qualitatively similar tothat of the spinor field and approximately one has h T νµ i (1) ren ≈ × h T νµ i (1 / ren (79)The basic features of the stress-energy tensor of the quantized massive vector field can easily be infrred form Figs.14-16 and the above relation.4 V. FINAL REMARKS
In this paper we have constructed the renormalized stress-energy tensor of the massive spinor and vector fields inthe spacetime of ABGB black hole. The scalar case has been analyzed extensively in our two previous papers. Themethod employed here is based on the observation that the first-order effective action could be expressed in terms ofthe (traced) coincidence limit of the coefficient a . Functional differentiation of this action with respect to the metrictensor yields the most general first-order (i.e. proportional to m − ) stress-energy tensor. Such a generic tensors ofthe quantized massive scalar, spinor and vector fields have been constructed for the first time in [8, 9].Application of our general formulas, although conceptually straightforward, is technically rather intricate, andproduces quite complex results. Therefore, for clarity, we have analyzed the leading behavior of h T νµ i ( s ) ren in somephysically important regimes. This discussion has been supplemented with detailed numerical calculations. Theresults have also been used to construct and analyze the stress-energy tensor in AdS × S , spaces, which are naturallyrelated to the near horizon geometry of the extremal ABGB black hole.A special emphasis in this paper has been put on the extremal configurations. Specifically, it has been shownthat the stress-energy tensor of the massive fields is regular inside the degenerate event horizon. This result raisesimportant question of the nature of the black hole interior in the back-reaction problem. Preliminary calculationscarried out in [53] for the quadratic gravity, which, for certain calculational purposes, may be considered as some sortof a toy model of the semi-classical theory, indicate that at least for the first-order calculations it is possible to obtainregular solution, at the expense of a small modification of the classical nonlinear action. Of course, the stress-energytensor of the quantized massive fields constructed in the general static, spherically symmetric and asymptoticallyflat spacetime is far more complicated than quadratic terms [53], however, the general pattern that lies behind thecalculations should be, in general, the same. The calculations carried out so far indicate that this is indeed the case,although lengthy and complicated results expressed in term of the polylogarithms are rather hard to analyze andmanipulate. Moreover, it would be interesting to investigate the back reaction problem for any q outside the eventhorizon. Finally, observe that the ABGB solution with the cosmological constant may provide an interesting settingfor studying the influence of the quantized fields upon ultraextremal horizons. These problems are being studied andthe results will be reported elsewhere. FIG. 1: This graph shows behavior of the rescaled components of h T tt i (1 / ren and h T θθ i (1 / ren [ λ = 5760 π m M H ] of the renormalizedstress- energy tensor of the quantized massive spinor field at the event horizon. The time component is always positive andincreases with q, whereas the angular component is positive for q < . . For the extremal configuration λ h T tt i (1 / ren = 5 . λ h T θθ i (1 / ren = − . . FIG. 2: This graph shows behavior of the rescaled components of h T tt i (1) ren and h T θθ i (1) ren [ λ = 5760 π m M H ]of the renormalizedstress- energy tensor of the quantized massive vector field at the event horizon. The time component increases with q and isnegative for q < . . For the extremal configuration λ h T tt i (1) ren = 35 .
080 and λ h T θθ i (1) ren = − . h T tt i (1 / ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive spinor field in the spacetime of the ABGB black hole. From top to bottom at the event horizonthe curves are plotted for q = 1 − i/ , ( i = 0 , , ..., . Each curve attains a negative minimum in the vicinity of r + . .FIG. 4: This graph shows the radial dependence of the rescaled component h T tt i (1 / ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive spinor field in the spacetime of the ABGB black hole for 1 . < r/r + < .
5. From top to bottomthe curves are plotted for q = 1 − i/ , ( i = 0 , ..., h T rr i (1 / ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive spinor field in the spacetime of the ABGB black hole. From top to bottom the curves areplotted for q = 1 − i/ , ( i = 0 , ..., . Each curve decreases monotonically to 0 + with r. FIG. 6: This graph shows the radial dependence of the rescaled component h T θθ i (1 / ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive spinor field in the spacetime of the ABGB black hole. From top to bottom at the event horizonthe curves are plotted for q = i/ , ( i = 1 , ..., . For q < .
823 the component h T θθ i (1 / ren is positive in the vicinity of the eventhorizon.FIG. 7: This graph shows the radial dependence of the rescaled component h T θθ i (1 / ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive spinor field in the spacetime of the ABGB black hole for 1 . < r/r + < . . From top to bottom(in the minima) the functions are plotted for q = i/ , ( i = 1 , ..., . FIG. 8: This graph shows the radial dependence of the rescaled component h T tt i (1) ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive vector field in the spacetime of the ABGB black hole. At the event horizon h T tt i (1) ren is positivefor q > . . For q < .
465 the curves reach minimum.FIG. 9: This graph shows the radial dependence of the rescaled component h T tt i (1) ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive vector field in the spacetime of the ABGB black hole. From top to bottom (at r = 2 . r + )horizon the curves are plotted for q = 1 − i/ , ( i = 0 , ..., . FIG. 10: This graph shows the radial dependence of the rescaled component h T rr i (1) ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive vector field in the spacetime of the ABGB black hole. From top to bottom at the event horizonthe curves are plotted for q = 1 − i/ , ( i = 0 , ..., . For q < . h T rr i (1) ren increases with r to 0 − whereas for q > .
919 it is amonotonically decreasing function.FIG. 11: This graph shows the radial dependence of the rescaled component h T rr i (1) ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive vector field in the spacetime of the ABGB black hole for 1 . < r/r + < . FIG. 12: This graph shows the radial dependence of the rescaled component h T θθ i (1) ren [ λ = 5760 π m M H ] of the stress-energytensor of the quantized massive vector field in the spacetime of the ABGB black hole. It is always negative at the event horizonand increases to a local maximum. From top to bottom the curves are for q = 1 − i/ , ( i = 1 , ...,
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