Mapping Hawking temperature in the spinning constant curvature black hole spaces into Unruh temperature
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Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
Mapping Hawking temperature in the spinning constant curvatureblack hole spaces into Unruh temperature
Huaifan Li
Department of Physics and Institute of Theoretical Physics, Shanxi Datong UniversityDatong 037009, ChinaKey Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, P.O. Box 2735,Beijing 100190, ChinaDepartment of Applied Physics, Xi’ an Jiaotong UniversityXi’ an 710049, [email protected]
Bin Hu
Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, P.O. Box 2735,Beijing 100190, [email protected]
Received (Day Month Year)Revised (Day Month Year)We established the equivalence between the local Hawking temperature measured bythe time-like Killing observer located at some positions r with finite distances from theouter horizon r + in the 5-dimensional spinning black hole space with both negativeand positive constant curvature, and the Unruh temperature measured by the Rindlerobserver with constant acceleration in the 6-dimensional flat space by employing theglobally embedding approach. Keywords : Black hole; Hawking temperature; Unruh temperature; Globally embeddingapproach.PACS Nos.: 04.70.Dy; 04.50.Gh
1. Introduction
In the past decade it was shown that, for both Hawking 1 , , , a in de Sitter space will detect a temperaturegiven by p a + 1 /l / π , where l is the de Sitter radius. This result was soon gener-alized by Deser and Levin 7 , p a ± /l / π measured by accelerated detectors in (anti-)de Sitter (AdS/dS) geometries can be eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Huaifan li obtained from their corresponding Rindler observors with constant accelerations byusing the Global Embedding Approach (GEA). For more examples on the equiva-lence, such as Banados-Teitelboim-Zanelli, Schwarzschild, Schwarzschild-AdS (dS),and Reissner-Nordstr¨om solutions into higher dimensional Minkowskian spaces etc.,see 9 , , ,
12 and references therein.On the other hand, the so-called BTZ (Banados-Teitelboim-Zanelli) black holesolutions 13 ,
14 have played the important role in understanding microscopic degreesof freedom of black hole. The BTZ black hole, which is constructed by identifyingpoints along the orbit of a Killing vector in a three dimensional anti-de Sitter space,is an exact solution of Einstein field equations with a negative cosmological constantin three dimensions. This kind of black hole has a topology of M × S , where M denotes a conformal Minkowski space in two dimensions. Following the same logic,one can construct analogues of the BTZ solution, the so-called constant curvature(CC) black holes in higher n ≥ , ,
17. Because n -dimensional anti-de Sitter space has the topology R n − × S , thel CC black holeshave topology of M n − × S , which is quite different from the known topologyof M × S n − for the usual black holes in n dimensions. In addition, the exteriorregion of these CC black holes is time-dependent and thus, there is no global time-like Killing vector 15. Because of this feature, it is difficult to discuss Hawkingradiation and thermodynamics associated with these black holes. For example, see18 , ,
20 and references therein.Comparing with the anti-de Sitter case, the n -dimensional de Sitter space hasthe different topology R × S n − . Similar to the negative constant curvature case,a positive constant curvature spacetime was constructed in 21 by identifying pointsalong a rotation Killing vector in a de Sitter space. Such solutions turn out tobe counterparts of the three-dimensional Schwarzschild de Sitter solution in higherdimensions, and have an associated cosmological horizon. There is a parameter inthe solution, which can be explained as the size of cosmological horizon.The equivalence between the lower dimensional Hawking temperature and higherdimensional Unruh temperature has been well established for the general spinningBTZ black hole in 9 ,
26 and the spinnless negative/positive CC black hole in 22.It should be noted that the Hawking temperature obtained by Cai and Myung 22is the same as that obtained from semi-classical tunneling method 23. Recently, areduced approach to the study of Hawking/Unruh effects including their unificationwas proposed by authors of 25. The primary goal of this work is to generalize theequivalence for the spinnless CC black hole established in 22 to the spinning one.The rest of this paper is organized as follows. In section II, we briefly review theconstruction of the spinless negative/positive CC black hole from the (anti)-de Sitterspace by identifying the points along a boost direction. In section III and IV, wefirstly construct the 5-dimensional spinning black hole with negative and positiveconstant curvature, respectively, and then establish the equivalence between theHawking temperature in 5-dimension and the Unruh temperature in 6-dimension.Finally, we conclude in section V.eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH
Mapping Hawking temperature in the spinning constant curvature black hole spaces into Unruh temperature
2. review on the spinless constant curvature black hole
In this section, we begin with a brief review the construction of the spinless CCblack hole from the (anti-)de Sitter space by identifying the points along one boostdirection. The n -dimensional (anti-)de Sitter space is defined as (the universal cov-ering) of the hypersurface embedded into a ( n + 1)-dimensional Minkowskian space,satisfying − x + x + · · · + x n − ∓ x n = ∓ l , (1)here and in the following paragraph of this section, we denote the upper and lowersign for anti-de Sitter and de Sitter space, respectively. The (anti-)de Sitter spaceadimits the boost along Killing vector ξ = ( r + /l )( x n − ∂ n ± x n ∂ n − ) with norm ξ = ( r /l )( ∓ x n − + x n ).In what follows, we will firstly discuss the anti-de Sitter case, in which the( n + 1)-dimensional flat space has two timelike coordinates ds = − dx + dx + · · · + dx n − − dx n . (2)To go further in the discussion, let us introduce the local Kruskal coordinates( y α , φ ) on anti-de Sitter space (in the region ξ > x α = 2 ly α − y , α = 0 , · · · , n − , (3) x n − = lrr + sinh (cid:18) r + φl (cid:19) , (4) x n = lrr + cosh (cid:18) r + φl (cid:19) (5)where r and y are defined as r = r + y − y , (6)and y = η αβ y α y β [ η αβ = diag( − , , · · · , −∞ < φ < ∞ and −∞ < y α < ∞ with the restriction − < y <
1. By usingKruskal coordinate, the induced metric on the anti-de Sitter space reads ds = l ( r + r + ) r dy α dy β η αβ + r dφ . (7)Without lossing any information, we shall restrict the discussion to the five dimen-sional case. The negative CC black hole can be easily obtained by introducing thelocal “spherical” coordinates ( t, r, θ, χ ) y = f cos θ sinh( r + t/l ) , y = f sin θ sin χ , (8) y = f cos θ cosh( r + t/l ) , y = f sin θ cos χ , (9)with f ( r ) = [( r − r + ) / ( r + r + )] / . (Note that these coordinates, with ranges 0 <θ < π/
2, 0 ≤ χ < π , −∞ < t < ∞ , and r + < r < ∞ , do not cover the wholeeptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Huaifan li region r > r + but only − < y , y < ds = l N d Ω + N − dr + r dφ , (10)with N ( r ) = ( r − r ) /l and d Ω = − cos θdt + l r ( dθ + sin θdχ ) . (11)In this coordinate frame, the Killing vector, generated a boost, becomes into ξ = ∂ φ with norm ξ = r , and the spinless negative CC black hole can be obtainedby identifying the points along the Killing vector ξφ ∼ φ + 2 nπ , n ∈ Z . (12)The horizon for the negative CC black hole in these coordinates is located at r = r + ,the point where N vanishes.The positive CC black hole can be constructed analogously, by identifying thepoints along the same Killing vector ξ = ∂ φ with norm ξ = r . In details, we firstlyembed the n -dimensional de Sitter space into a ( n + 1)-dimensional Minkowskianspace ds = − dx + dx + · · · + dx n − + dx n . (13)Furthermore, we define the Kruskal coordinates ( y α , φ ) on the n -dimensional deSitter space in the region (0 ≤ ξ ≤ r ), x α = 2 ly α − y , α = 0 , · · · , n − , (14) x n − = lrr + sin (cid:18) r + φl (cid:19) , (15) x n = lrr + cos (cid:18) r + φl (cid:19) , (16)with r = r + − y y , y = η αβ y α y β ,η αβ = diag( − , , · · · , . (17)Here the coordinate range is −∞ < y α < + ∞ , and −∞ < φ < + ∞ with therestriction − < y < r positive. Under the Kruskal coordinateframe, the induced metric takes the same from with (7). We can also introduceSchwarzschild coordinates to describe the solution. Using local spherical coordinates( t, r, θ, ξ ) defined as (8) with f ( r ) = [( r + − r ) / ( r + r + )] / , and the coordinate rangesare 0 < θ < π/
2, 0 ≤ χ < π , −∞ < t < ∞ , and 0 < r < r + , we find that thesolution can be expressed as ds = l N d Ω + N − dr + r dφ , (18)eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Mapping Hawking temperature in the spinning constant curvature black hole spaces into Unruh temperature with N ( r ) = ( r − r ) /l and d Ω = − cos θdt + l r ( dθ + sin θdχ ) . (19)Like what happens for the negative constant curvature case, the spinless positiveCC black hole can be constructed by identifying the points along the Killing vector ξ = ∂ φ with norm ξ = r φ ∼ φ + 2 nπ , n ∈ Z . (20)In these coordinates r = r + is the cosmological horizon. The only difference isthat N = ( r − r ) /l in the negative CC black hole space there is replaced by N = ( r − r ) /l in the positive one.
3. spinning negative constant curvature black hole
In this section, we firstly construct the 5-dimensional spinning negative CC blackhole by virtue of the GEA, then we calculate both the local Hawking temper-ature observed by a time-like Killing observer in the 5-dimensional hypersurfaceand the corresponding Unruh temperature observed by a Rindler observer in the6-dimensional flat space.By using the “spherical” coordinate ( t, ρ, θ, φ, χ ) x = l sinh ρ cos θ sinh (cid:16) r + l t − r − l φ (cid:17) ,x = l sinh ρ cos θ cosh (cid:16) r + l t − r − l φ (cid:17) ,x = l sinh ρ sin θ sin χ , (21) x = l sinh ρ sin θ cos χ ,x = l cosh ρ sinh (cid:16) r + l φ − r − l t (cid:17) ,x = l cosh ρ cosh (cid:16) r + l φ − r − l t (cid:17) , with r + and r − ( r + > r − ) two arbitrary real constants with dimensions of length,the 5-dimensional spinning negative CC black hole can be obtained ds = cos θ (cid:2) − N l dt + r ( dφ + N φ dt ) (cid:3) + N − dr + l r − r r − r − (cid:0) dθ + sin θdχ (cid:1) (22)+ l r − r − r − r − sin θ (cid:16) r + l dφ − r − l dt (cid:17) , by identifying φ ∼ φ + 2 nπ , n ∈ Z , (23)eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Huaifan li where N = ( r − r )( r − r − ) l r , (24) N φ = − r + r − r , (25)and r = r cosh ρ − r − sinh ρ . (26)In these coordinates, −∞ < t < ∞ , 0 < θ < π/
2, 0 ≤ χ < π and r + < r < ∞ .In the 5-dimensional spinning space, the local observer is chosen as the onewhose trajectory follows the time-like Killing vector ζ = ∂ t + r − /r + ∂ φ , namely the“time-like Killing observer” ( O K ) φ = r − r + t , r, θ, χ = const . , (27)then the proper velocity of ( O K ) is u µ = ( t, r, θ, φ, χ ) = r + cos θ q ( r − r )( r − r − ) , , , r − cos θ q ( r − r )( r − r − ) , , (28)and the proper acceleration for ( O K ) reads a µ = (cid:18) , r − r − l r , − ( r − r − ) tan θl ( r − r ) , , (cid:19) . (29)Thus the a equals a = q ( r − r − ) + tan θ ( r − r − ) l q r − r . (30)The Killing vector tangent to the worldline of the time-like observer ( O K ) is ζ = ∂ t + r − /r + ∂ φ , consequently, the Hawking temperature which enters thethermodynamical relations can be obtained through the surface gravity [ κ = − ( ∇ µ ξ ν )( ∇ µ ξ ν )] 2 πT HK = κ = r − r − r + l . (31)This result is consistent with the one which is obtained by embedding the spinningCC black hole into a Chern-Simons supergravity theory 16. However, as pointedout in 22 the method used in 16 has two drawbacks, one is that the result cannotbe degenerated to the spinnless case, the other is that it cannot be generalized toother dimensions. Along the integral curves generated by the Killing vector ζ theline element becomes ds = cos θ (cid:20) − N l dt + r ( r − r + dt + N φ dt ) (cid:21) + · · · , = − cos θ ( r − r )( r − r − ) r dt + · · · . (32)eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Mapping Hawking temperature in the spinning constant curvature black hole spaces into Unruh temperature Armed with the above results, we can define the Tolman temperature which is thelocal Hawking temperature observed by the time-like Killing observer ( O K ) locatedat r πT = κ √− ˆ g = q r − r − l cos θ q r − r , (33)where ˆ g is the Tolman redshift factor which can be read off from (32).On the other hand, the worldline of the time-like Killing observer in the 5-dimensional spinning black hole space coincides with the trajectory of Rindler ob-server with the constant acceleration a − = x − x = l r − r r − r − cos θ , (34)thus, the corresponding Unruh temperature reads2 πT DU = a = q r − r − l cos θ q r − r = r l + a . (35)From (33) and (35) we can easily see that the local Hawking temperature measuredby a time-like Killing observer in the 5-dimensional space is nothing but the Unruhtemperature observed by a Rindler observer in the 6-dimensional flat space.
4. spinning positive constant curvature black hole
Analogously, the 5-dimensional spinning positive CC black hole can also be con-structed by using the “spherical” coordinate ( t, ρ, θ, φ, χ ) x = l sin ρ cos θ sinh (cid:16) r + l t − r − l φ (cid:17) ,x = l sin ρ cos θ cosh (cid:16) r + l t − r − l φ (cid:17) ,x = l sin ρ sin θ sin χ , (36) x = l sin ρ sin θ cos χ ,x = l cos ρ sin (cid:16) r + l φ + r − l t (cid:17) ,x = l cos ρ cos (cid:16) r + l φ + r − l t (cid:17) . Plugging (36) into the 6-dimensional Minkowskian metric and identifying φ ∼ φ +2 nπ with n ∈ Z , one can derive the induced metric which describes a 5-dimensionalspinning black hole with positive constant curvature ds = cos θ (cid:2) − N l dt + r ( dφ + N φ dt ) (cid:3) + N − dr + l r − r r + r − (cid:0) dθ + sin θdχ (cid:1) + l r + r − r + r − sin θ (cid:16) r + l dφ + r − l dt (cid:17) , (37)eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Huaifan li where N = ( r − r )( r + r − ) l r , (38) N φ = r + r − r , (0 < r − < r + ) , (39)and r = r cos ρ − r − sin ρ , < r < r + . (40)The ranges of other “spherical” coordinates ( t, θ, φ, χ ) are the same as the case withnegative constant curvature. In these coordinates, the cosmological horizon locatesat r + and the ranges 0 < r < r + represents the interior of the horizon 24.The 5D CC black hole is constructed by indentifying φ ∼ φ + 2 nπ , n ∈ Z (41)The time-like Killing observer ( O K ) is chosen as φ = − r − r + t , r, θ, χ = const . , (42)then the corresponding proper velocity is u µ = ( t, r, θ, φ, χ ) = r + cos θ q ( r − r )( r + r − ) , , , − r − cos θ q ( r − r )( r + r − ) , . (43)The proper acceleration for observer ( O K ) can be derived by differentiating theproper velocity (43) with respect to the proper time a µ = (cid:18) , − r + r − l r , − ( r + r − ) tan θl ( r − r ) , , (cid:19) , (44)thus the a equals a = q ( r + r − ) + tan θ ( r + r − ) l q r − r . (45)The Killing vector tangent to the worldline of the time-like observer ( O K ) reads ζ = ∂ t − r − /r + ∂ φ , so the Hawking temperature reads2 πT HK = κ = r + r − r + l . (46)This result is consistent with those obtained in 24. Along the integral curves gen-erated by the Killing vector ζ , the line element becomes ds = cos θ (cid:20) − N l dt + r ( r − r + dt + N φ dt ) (cid:21) + · · · , = − cos θ ( r − r )( r + r − ) r dt + · · · . (47)eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Mapping Hawking temperature in the spinning constant curvature black hole spaces into Unruh temperature The local Hawking temperature measured by the time-like Killing observer ( O K )located at r equals the Hawking temperature in (46) rescaled by a redshift factor2 πT = κ √− ˆ g = q r + r − l cos θ q r − r , (48)where the redshift factor are derived in (47).The corresponding Unruh acceleration and temperature can also be obtained byusing the same logic in the negative case a − = x − x = l r − r r + r − cos θ , (49)and 2 πT DU = a = q r + r − l cos θ q r − r = r l + a . (50)As we expect that the tempratures (48) and (50) coincide exactly.
5. Conclusion
In this work, we established the equivalence between the local Hawking temperaturemeasured by the time-like Killing observer located at some positions r with finitedistances from the outer horizon r + in the 5-dimensional spinning CC black holespace, and the Unruh temperature measured by the Rindler observer with constantacceleration in the 6-dimensional flat space by employing the globally embeddingapproach. For the spinning black hole with negative constant curvature, the localHawking or Unruh temperature equals to q r − r − / πl cos θ q r − r , while forthe postive case, the temperature reads q r + r − / πl cos θ q r − r , where l is theradius of (anti-)de Sitter spaces and θ is the spherical coordinate of time-like Killingobserver. Our results can recover the previous one in both the general spinning BTZblack hole 17 when one sets θ = 0 and the spinnless black hole with negative/positiveconstant curvature 22 when one sets both θ = 0 and r − = 0.
6. Acknowledgments
We are grateful to Hai-Qing Zhang, Zhang-Yu Nie and Yun-Long Zhang for theirvarious discussions. We especially thank Rong-Gen Cai for the useful commentson draft and the persistent encouragements during all stages of this work. BHthanks the nice accommodation during “the 5th Asian Winter School on Strings,Particles and Cosmology”, Jeju, Korea. This work was supported in part by a grantfrom Chinese Academy of Sciences and in part by the National Natural ScienceFoundation of China under Grant Nos. 10821504, 10975168, 11035008, 11075098eptember 26, 2018 23:38 WSPC/INSTRUCTION FILE MHTCCBH Huaifan li and 11175109 and by the Ministry of Science and Technology of China under GrantNo. 2010CB833004.
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