Hawking Radiation from General Kerr-(anti)de Sitter Black Holes Zhibo Xu∗ and Bin Chen† 7 School of Physics, Peking University, Beijing 100871, P. R. China 0 0 2 n a J Abstract 7 2 We calculate the total flux of Hawking radiation from Kerr-(anti)de Sitter black holes 3 v by using gravitational anomaly method developed in [6]. We consider the general Kerr- 1 6 (anti)de Sitter black holes in arbitrary D dimensions with the maximal number [D/2] 2 2 of independent rotating parameters. We find that the physics near the horizon can be 1 described by an infinite collection of (1 + 1)-dimensional quantum fields coupled to a 6 0 set of gauge fields with charges proportional to the azimuthal angular momentums m . / i h With the requirement of anomaly cancellation and regularity at the horizon, the Hawking t - p radiation is determined. e h : v i X r a ∗Email:[email protected] †Email:[email protected] 1 Introduction Hawkingradiationisoneofthemostimportantandintriguingeffectsinblackholephysics. It shows that black hole is not really black, it radiates thermally like black body. Precisely speaking, Hawkingradiationisthequantumeffectoffieldinabackgroundspace-timewith a future event horizon. It has a feature that the radiation is determined universally by the horizon properties. It has several derivations. The original one discovered by Hawking [2, 3] is by directly calculating the Bogoliubov coefficients between in and out states of fields in a black hole background. This approach relies on the fact that in a curved background the choice of vacuum for incoming and outgoing particle is not unique. Later on a derivation based on the path-integral quantum gravity was given in [4]. A few years ago, Parikh and Wilczek[5] proposed a tunneling picture in which particle pair production happens near the horizon and Hawking radiation could be obtained by calculating WKB amplitudes for classically forbidden paths. Very Recently, Robinson and Wilczek [6] have given a new derivation of Hawking ra- diation in the Schwarzschild black hole background through gravitational anomaly. This work is to some extent inspired by the work of Christensen and Fulling [7], in which the radiation created in the (1 + 1)-dimensional Schwarzschild black hole background was determined by the trace anomaly and the energy-momentum conservation law. In this approach, boundary conditions at the horizon and the infinity are required to specify the Unruh [8] vacuum. Moreover the method in [7] could not be applied to the cases in more than (2 + 1) dimensions. Robinson and Wilczek found that by dimension re- duction, the physics near the horizon can be described by an infinite collections of free (1 + 1)-dimensional fields because the mass and interaction terms of quantum fields in the background are suppressed. If one only consider the effective field theory outside the horizon, the theory become chiral since classically all ingoing modes can not affect physics outside the horizon. Quantum mechanically, the effective theory is anomalous with re- spect to gauge or general coordinate symmetries. The anomaly should be cancelled by the quantum effects of the classically irrelevant incoming modes. The condition for chiral and gravitational anomaly cancellation and regularity requirement at the horizon, com- bining with the energy-momentum conservation law, determines Hawking fluxes of the charge and energy-momentum. Robinson and Wilczek’s treatment once again shows that Hawking radiation (if we neglect the back-reaction on the background) is universal, it only depends on the property of the event horizon. In the further development, Iso et al. [9, 10] investigated the charged and rotating black hole. By using a dimensional reduction technique, they found each partial wave of 1 quantum fields in d = 4 rotating black hole background can be interpreted as a (1+1)- dimensional charged field with a charge proportional to the azimuthal angular momentum m. The totalflux ofHawking radiationcanbe determined by demanding gaugeinvariance and general coordinate covariance at the quantum level. And the boundary conditions are clarified. The results are consistent with the effective action approach. Murata et al.[11] extended the method to Myers-Perry black holes[14] with only one rotating axis andalsoclarifiedtheboundarycondition. TheHawking radiationfromgeneralspherically symmetric black holes[12] and BTZ black holes [13] have also been investigated. In this paper, we further extend Robinson and Wilczek’s derivation of Hawking ra- diation to general Kerr-(anti)de Sitter(K(A)dS) black holes [1] in D dimensions. For a general K(A)dS metric, there are at most N = [D−1] Killing symmetries corresponding to 2 the rotational symmetries in N orthogonal spatial 2-planes. For a quantum field in such backgrounds, the physics near the future event horizon could still be effectively described by an infinite collection of (1+1)-dimensional fields coupled to N U(1) gauge fields. We discuss such dimensional reduction in detail in sec. 2. In sec. 3, we obtain the Hawking fluxes by requiring anomaly cancellation and regularity condition. The final section is devoted to the conclusion. 2 Quantum Fields in general Kerr-(anti)de Sitter Black Holes In this section, we will discuss the quantum fields in general Kerr-(anti)de Sitter blacks holes and its effective dimensional reduction near the horizon. The Kerr-(anti)de Sitter metric in D-dimension has been studied carefully in [1]. Here we just give a brief review of its basic property. The metric takes the form in an Boyer-Lindquist coordinates 2 2M N a µ2dϕ N r2 +a2 ds2 = W(1 λr2)dt2 + Wdt i i i + i µ2dϕ2 − − VF − 1+λa2 1+λa2 i i i=1 i ! i=1 i X X 2 VFdr2 N+ǫ r2 +a2 λ N+ǫ r2 +a2 + + i dµ2 + i µ dµ (2.1) V 2M 1+λa2 i W(1 λr2) 1+λa2 i i − i=1 i − i=1 i ! X X where 1, D even ǫ =  0, D odd   2 N+ǫ µ2 W i ≡ 1+λa2 i=1 i X N 1 N+ǫ r2µ2 V rǫ−2(1 λr2) (r2 +a2), F i . ≡ − i ≡ 1 λr2 r2 +a2 i=1 − i=1 i Y X Here N is the integral part of (D 1)/2. There are N independent rotation parameters − a in N orthogonal spatial 2-planes. The ϕ ’s are azimuthal angular coordinates. And i i N+ǫ the µ ’s are the latitudinal coordinates obeying a constraint µ2 = 1, so only N +ε 1 i i − i=1 X latitudinal coordinates µ are independent. The λ is the cosmological constant. Up to the i sign of λ, the above metric describes different Kerr black holes in D dimensions: λ > 0, Kerr-de Sitter metric  λ = 0, Myers-Perry metric[14]     λ < 0, Kerr-Anti-de Sitter metric    Hawking fluxes of Myers-Perry black holes with only one azimuthal angular momentum has been discussed in [11]. In this paper, we will discuss Hawking radiation of the other two cases with any permissible angular momentums. Certainly our discussion apply to the Myers-Perry black holes with more than one angular momentums. It is remarkable that for the black holes in de Sitter spacetime, there exist a cosmo- logical event horizon. However, our motivation is to study the Hawking radiation of the black hole so we focus on the physics near the black hole event horizon. The metric (2.1) could be cast into a generalized Boyer-Lindquist form, 1 ds2 = Xdt2 +2Y dtdϕi+Z dϕidϕj +g dxadxb + dr2 (2.2) i ij ab B where ϕi,i = 1, N are periodic with period 2π and xa,a = 1, n with n = N +ǫ 1 ··· ··· − areindependent latitudinalcoordinates. Allthemetriccomponents onlydependonxa(µ ) i and the radial coordinate r. The Z and g are positive definite and their corresponding ij ab inverses are Zij and gab with ZijZ = δi, gabg = δa. jk k bc c From the discussion in [1], the angular velocities of the horizon are given by Ωi = Ωi H a (1 λr2 ) Ωi = Yi = i − H (2.3) H − |r=rH r2 +a2 H i where Yi = ZijY and r is the radius of the horizon. Using (2.1), r is just the largest j H H positive root of equation V 2M = 0 or in the metric (2.2) the largest positive root of − 3 equation B = 0. The angular velocities, relative to a non-rotating frame at infinity, is a little different from (4.4) in [1], which is defined relative to a rotating frame at infinity. The null generator l of the horizon is a linear combination of the Killing vector fields ∂ ∂ l = +Ωi (2.4) ∂t H∂ϕi The surface gravity on the horizon is κ2 = ( µL)( L) (2.5) ∇ ∇µ |r=rH where L2 = lµl = X +2Y Ωi +Z Ωi Ωj (2.6) − µ i H ij H H Note that L and B vanish on the horizon but ∂ L and ∂ B are non-zero. So near the r r horizon, we have L2 (∂ L2) (r r ), B (∂ B) (r r ) (2.7) ≈ r |r=rH − H ≈ r |r=rH − H Thus the surface gravity is: 1 1 V′(r ) κ = (∂ L2)(∂ B) = (1 λr2 ) H (2.8) 2 r r |r=rH 2 − H V(r ) H p The property that κ is a constant and YiY X = 0 on the horizon are very important i − to the following discussions. In order to do dimensional reduction, we need some other properties of the metric near the horizon. Define A = YiY X then i − L2 A = Z (Ωi +Yi)(Ωj +Yj) (2.9) − − ij H H From the definition (2.3) Ωi +Yi = 0, so near the horizon Ωi +Yi Ci(r r ), H |r=rH H ≈ − H then we have L2 A Z CiCj(r r )2, ∂ L2 = ∂ A (2.10) − ≈ − ij − H r r |r=rH When A = 0, the inverse of the metric (2.2) can be written as 6 1 1 gtt = , gij = YiYj +Zij −A −A 1 gti = git = Yi, grr = B, (2.11) A 4 Note that near the horizon A 0, Yi,Zij and gab are finite. This property is essential → to the dimensional reduction. The metric (2.2) can be written in another form 1 ds2 = Adt2 +Z (dϕi +Yidt)(dϕj +Yjdt)+ dr2 +g dxadxb (2.12) ij ab − B with 1 √ g = Ag g (2.13) 1 2 − B r where g = det(g ), g = det(Z ), g = det(g ) µν 1 ij 2 ab Now let’s consider a free complex scalar field for simplicity in the general Kerr-(anti)de Sitter black holebackground. Using the inverse ofmetric (2.11), the freepartof theaction is S = dDx√ ggµν∂ φ∗∂ φ free µ ν − − Z 1 1 1 = dtdrdNϕidnxa√ g ∂ φ∗∂ φ+ Yi∂ φ∂ φ∗ + Yi∂ φ∂ φ∗ t t t i i t − − −A A A Z (cid:18) 1 YiYj∂ φ∗∂ φ+Zij∂ φ∗∂ φ+B∂ φ∗∂ φ+gab∂ φ∗∂ φ (2.14) i j i j r r a b −A (cid:19) where ∂ denotes ∂ and ∂ denotes ∂ . i ∂ϕi a ∂xa Inorder to consider the physics near the horizon, we make acoordinatetransformation dr∗ = f(r)−1, where f(r) √A′B′ (r r ) = 2κ(r r ). In this frame, considering dr ≡ |r=rH − H − H the region near the outer horizon r , the finite terms Zij∂ φ∗∂ φ and gab∂ φ∗∂ φ are H i j a b suppressed by the factor f(r(r )), vanishing exponentially fast near the horizon. We can ∗ also substitute √AB, g g by f(r), g g because the omitting terms are suppressed 1 2 1 2|r=rH for the same reason. Similarly, one can redefine Yi by ai(1−λr2). So the action with − r2+a2i dominant terms is S = dtdrdNϕidnxa g (r )g (r ) f(r)−1 ∂ Yi∂ φ∗ ∂ Yi∂ φ 1 H 2 H t i t i − − − − Z +f(r)∂ φ∗∂ φ] p (cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (2.15) r r We can expand φ by a complete set of orthogonal functions of (ϕi, xa) with the measure g (r )g (r ). As we know, the angles ϕi are periodic with period 2π and 1 H 2 H coordinates xa come from µ which obey a constraint µ2 = 1. So (ϕi,xa) describe a p i i i compact manifold with a metric P ds2 = Z (r )dϕidϕj +g (r )dxadxb (2.16) ij H ab H 5 whose measure is g (r )g (r ). Then the eigenfunctions of the operator 2 of the 1 H 2 H ∇ compact manifold with the metric (2.16) comprise a complete orthogonal functions. Note p that there are N killing vectors ∂ which generate isometry. The eigenfunctions can be ∂ϕi given by N Y = expimjϕj f (xa) (2.17) m1···mNα α j=1 Y satisfying dxadϕi g (r )g (r )Y∗ Y = δ δ δ (2.18) 1 H 2 H m1···mNα n1···nNβ m1n1··· mNnN αβ Z p Performing the partial wave decomposition of φ in terms of these functions, φ = φ Y , (2.19) m1···mNα m1···mNα m1,·X··,mN,α the theory is reduced to a two-dimensional effective theory with an infinite collection of fields with quantum numbers(m , ,m ,α), simply denoted as φ . It is straightforward 1 N n ··· to show that the physics near the outer horizon can be effectively described by an infinite collection of massless (1+1)-dimensional fields with the following action S = dtdr f(r)−1 ∂ im Yj ∗φ∗ ∂ im Yj φ +f(r)∂ φ∗∂ φ (2.20) − − t − j n t − j n r n r n Z (cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3) 3 Anomalies and Hawking fluxes In this section, we will try to obtain the Hawking fluxes. We will follow the approach firstly proposed in [9, 10]. The basic point is that the Hawking fluxes can be determined by the anomaly cancellation of the effective chiral theory. From the effective action(2.20), near the horizon, each partial wave mode of the scalar field φ can be considered as (1+1)-dimensional complex scalar field in the backgrounds n of the metric g and N gauge potentials Ai µν µ g = f(r), g = f(r)−1, g = 0 tt rr rt − Ai = Yi, Ai = 0 (3.1) t r In this case, there are N U(1) gauge symmetries and N gauge currents which actually relate to angular momentum currents. Each gauge symmetry originates from the axial isometry along ϕi direction. With respect to gauge fields Ai, the field φ has charges m , µ n i 6 which is the azimuthal quantum number rotating along ϕi direction. The corresponding U(1) currents Jr can be defined from the D-dimensional energy-momentum tensor. i Jr = dnxadNϕi√ gTr . (3.2) i − − ϕi Z In effect, performing a partial wave decomposition and an integral, we find the result of right side of the above equation is just the current obtained from the two-dimensional ef- fective action. Similarly the energy-momentum tensor in two-dimensional effective action is the reduction of the one in D-dimension Tr = dnxadNϕi√ gTr. (3.3) t(2) − t Z Without bringing any confusion, from now on we denote Tr as Tr for simplicity. t(2) t As shown in [9], we can divide the region r [r , ] into two regions. One is H ∈ ∞ r [r +ε, ] which is apart from the horizon and the other is r [r ,r +ε] which is H H H ∈ ∞ ∈ near the horizon. In the region r [r +ε, ], each current is conserved. So we have H ∈ ∞ ∂ Jr = 0 (3.4) r i(o) Ontheotherhand,inthenearhorizonregion,theeffectivetwodimensionaltheorybecome chiralsince classically theingoing modes areirrelevant andthere areonlyoutgoingmodes. Inthiseffectivechiraltheory,thegaugesymmetries andgeneralcoordinatetransformation symmetries become anomalous quantum mechanically. The anomaly equation for each U(1) current near the horizon is[15, 16] m ∂ Jr = i∂ (3.5) r i(H) 4π rAt where = m Ai is the sum of N U(1)’s. At i t Actually one can take (r) more seriously. From effective action, we can take the A point of view that the scalar field coupled to a single gauge potential (r). There exist a A U(1) gauge symmetry associated with gauge potential (r). The corresponding current A denoted as (r) can be constructed from the original N U(1) gauge symmetries. Note J that each Jr is not independent for a fixed azimuthal angular momentum m . Their i i expectation values are related as 1 Jr = 1 Jr = r. The anomaly equation for (r) is mi i mj j J J 1 ∂ r = ∂ , (3.6) r r t J 4π A so we have anomaly equation (3.5). Solving the above equations in each region, we have, Jr = C i(o) i(o) m Jr = C + i ( (r) (r )) (3.7) i(H) i(H) 4π At −At H 7 where C and C are two integration constants. C is the value of the consistent i(o) i(H) i(H) current of the outgoing modes at the horizon and C is the value of the angular mo- i(o) mentum flux at infinity. It is our goal to determine C , which encodes the information i(o) of Hawking radiation. Under gauge transformations, the variation of the effective action is given by δW = g λ Jµ (3.8) − − (2) ∇µ i Z p where λ is a gauge parameter, and Jµ = Jµ θ (r)+Jµ H(r). (3.9) i i(o) + i(H) Here θ (r) = θ(r r ε) and H(r) = 1 θ (r), where θ(x) is a step function. Note that + H + − − − we have not take the contribution of the ingoing modes into account. Using equations (3.4)(3.5) we have m m δW = d2xλ δ(r r ǫ) Jµ Jµ + i +∂ i H(r) (3.10) − − H − i(o) − i(H) 4πAt r 4πAt Z h (cid:16) (cid:17) (cid:16) (cid:17)i Since the underlying theory must be gauge invariant, so δW = 0. Actually the last term is cancelled by quantum effects of the classically irrelevant ingoing modes [6]. Then the coefficient of the delta-function should vanish. With the results (3.7), we can obtain a relation between the two constants m i C = C (r ) (3.11) i(o) i(H) t H − 4πA In order to determine the value of C , one need to impose the regularity condition. i(o) As discussed in [9, 10], the regularity requires that the covariant current is zero on the horizon, m Jr = Jr + i H(r), Jr(r ) = 0 (3.12) i i 4πAt i H Then the flux of the anguflar momentum is obtainfed as m m N a (1 λr2 ) C = i (r ) = i m j − H (3.13) i(o) −2πAt H 2π j r2 +a2 j=1 H j X Similarly we can determine the flux of the energy-momentum tensor radiated from general Kerr-(anti)de Sitter black holes. In the presence of the effective gauge potentials Ai(r), the conservation equation outside the horizon is modified to be t ∂ Tr = r (3.14) r t(o) FrtJ(o) 8 where = ∂ . Note that the right hand side of the above relation depends simply on rt r t F A . With the definition of r, we have r = 1 (r ) C , where At J J(o) −2πAt H ≡ o N a (1 λr2 ) C = m j − H . (3.15) o j r2 +a2 j=1 H j X The solution of the above equation gives the value of the energy flux at spatial infinity Tr = a +C (r) (3.16) t(o) o oAt where a is an integration constant. Physically, it could be taken as the value of the total o energy flow of radiation measured at spatial infinity. On the other hand, there are gauge and gravitational anomalies near the horizon and the anomaly equation is now as ∂ Tr = r + µ +∂ Nr (3.17) r t(H) FrtJ(H) At∇µJ(H) r t where Nr = (f′2 + ff′′)/192π[9]. The second term indicates gauge anomaly while the t third term is gravitational anomaly[18] for the consistent energy-momentum tensor. From thedefinition of r and Eqs. (3.7)(3.11) we have r = C + 1 (r). Tr canbesolved J J(H) o 4πAt t(H) as r 1 Tr = a + dr∂ C + 2 +Nr (3.18) t(H) H r oAt 2πAt t ZrH (cid:18) (cid:19) where a is an integration constant. H Under the general coordinate transformation, the variation of the effective action is δW = d2x g ξt Tµ (2) µ t − − ∇ Z p 1 = d2xξt C ∂ (r)+∂ 2H(r)+NrH(r) 0 rAt r 4πAt t Z (cid:20) (cid:18) (cid:19) 1 + Tr Tr + 2 +Nr δ(r r ǫ) (3.19) t(o) − t(H) 4πAt t − H − (cid:18) (cid:19) (cid:21) where ξt is the transformation parameter and Tµ = Tµ θ (r)+Tµ H(r). The first term ν ν(o) + ν(H) is generated by classical current. The second term should be cancelled by the quantum effect of the ingoing modes. As we discussed before, the last term should vanish because the underlying theory is general coordinate transformation covariant. So we have: 1 a = a + 2(r ) Nr(r ) (3.20) o H 4πAt H − t H 9