Noncommutative inspired Schwarzschild black hole, Voros product and Komar energy
aa r X i v : . [ g r- q c ] D ec Noncommutative inspired Schwarzschild black hole, Vorosproduct and Komar energy
Sunandan Gangopadhyay a,b ∗ a Department of Physics, West Bengal State University, Barasat, India b Visiting Associate in Inter University Centre for Astronomy & Astrophysics,Pune, India
Abstract
The importance of the Voros product in defining a noncommutative Schwarzschild blackhole is shown. The entropy is then computed and the area law is shown to hold upto order √ θ e − M /θ . The leading correction to the entropy (computed in the tunneling formalism) isshown to be logarithmic. The Komar energy E for these black holes is then obtained anda deformation from the conventional identity E = 2 ST H is found at the order √ θe − M /θ .This deformation leads to a nonvanishing Komar energy at the extremal point T H = 0 ofthese black holes. Finally, the Smarr formula is worked out. Similar features also exist for adeSitter–Schwarzschild geometry . In this presentation, we discuss some of the issues of noncommutative inspired Schwarzschildblack hole [3, 4] which has gained considerable interest recently. The primary point of interest isthat there is no clear cut connection of this type of noncommutativity with standard notions of anoncommutative (NC) spacetime where point-wise multiplications are replaced by appropriatestar multiplications. We shall point out that the Voros star product [5] plays an importantrole in obtaining the mass density of a static, spherically symmetric, smeared, particle-likegravitational source. Our second objective is to derive quantum corrections to the semiclassicalHawking temperature and entropy by the tunneling mechanism by going beyond the standardsemiclassical approximation [6] for this black hole spacetime. Finally, we would like to reviewthe status of the relation between the Komar energy [7], entropy and Hawking temperature( E = 2 ST H ) in the context of NC inspired Schwarzschild black holes.The first issue can be addressed by taking recourse to the formulation and interpretationalaspects of NC quantum mechanics [8, 9]. We observe that the inner product of the coherent states | z, ¯ z ) (used in the construction of the wave-function of a free point particle) can be computedby using a deformed completeness relation (involving the Voros product) among the coherentstates Z θdzd ¯ z π | z, ¯ z ) ⋆ ( z, ¯ z | = 1 Q (1)where the Voros star product between two functions f ( z, ¯ z ) and g ( z, ¯ z ) is defined as f ( z, ¯ z ) ⋆ g ( z, ¯ z ) = f ( z, ¯ z ) e ← ∂ ¯ z → ∂ z g ( z, ¯ z ) . (2) ∗ [email protected], [email protected], [email protected] This presentation is based on the work in references [1, 2]. ψ ~p = ( p | z, ¯ z ) = 1 √ π ¯ h e − θ h ¯ pp e i q θ h ( p ¯ z +¯ pz ) ; p = p x + ip y , z = 1 √ θ ( x + iy ) (3)where the momentum eigenstates are normalised such that ( p ′ | p ) = δ ( p ′ − p ) and satisfy thecompleteness relation Z d p | p )( p | = 1 Q . (4)It turns out that a consistent probabilistic interpretation of this wave-function can be given onlywhen the Voros product is incorporated. With these observations and interpretations in place,we now write down the overlap of two coherent states | ξ, ¯ ξ ) and | w, ¯ w ) using the completenessrelation for the position eigenstates in eq.(1)( w, ¯ w | ξ, ¯ ξ ) = Z θdzd ¯ z π ( w, ¯ w | z, ¯ z ) ⋆ ( z, ¯ z | ξ, ¯ ξ ) . (5)A simple inspection shows that ( w, ¯ w | z, ¯ z ) = θ e −| ω − z | satisfies the above equation. A straight-forward dimensional lift of this solution from two to three space dimensions immediately moti-vates one to write down the mass density of a static, spherically symmetric, smeared, particle-likegravitational source in three space dimensions as ρ θ ( r ) = M (4 πθ ) / exp − r θ ! . (6)The above discussion clearly brings out the important role played by the Voros product indefining the mass density of the NC Schwarzschild black hole. Solving Einstein’s equations withthe above mass density incorporated in the energy-momentum tensor leads to the following NCinspired Schwarzschild metric [3],[4] ds = − − Mr √ π γ ( 32 , r θ ) ! dt + − Mr √ π γ ( 32 , r θ ) ! − dr + r ( d ˜ θ + sin ˜ θdφ ) . (7)The event horizon of the black hole can be found by setting g tt ( r h ) = 0 in eq.(7), which yields r h = 4 M √ π γ ( 32 , r h θ ) . (8)The large radius regime ( r h θ >>
1) is taken where one can expand the incomplete gammafunction to solve r h by iteration. Keeping upto next to leading order √ θe − M /θ , we find r h ≃ M (cid:20) − M √ πθ (cid:18) θ M (cid:19) e − M /θ (cid:21) . (9)Now for a general stationary, static and spherically symmetric space time, the Hawking temper-ature ( T H ) is related to the surface gravity ( κ ) by the following relation [11] T H = κ π (10)where the surface gravity of the black hole is given by κ = 12 (cid:20) dg tt dr (cid:21) r = r h . (11)2ence the Hawking temperature for the NC inspired Schwarzschild black hole upto order √ θe − M /θ is given by T H = 18 πM " − M θ √ πθ − θ M − θ M ! e − M /θ . (12)The first law of black hole thermodynamics can now be used to work out the Bekenstein-Hawkingentropy. The law reads dS BH = dMT H . (13)Hence the Bekenstein-Hawking entropy in the next to leading order in θ is found to be S BH = Z dMT H = 4 πM − r πθ M (cid:18) θM (cid:19) e − M /θ . (14)To express the entropy in terms of the NC horizon area ( A θ ), we use eq.(9) to get A θ = 4 πr h = 16 πM − r πθ M (cid:18) θ M (cid:19) e − M /θ + O ( θ / e − M /θ ) . (15)Comparing equations (14) and (15), we find that at the leading order in θ (i.e. upto order √ θ e − M /θ ), the NC black hole entropy satisfies the area law (in the regime r h θ >> S BH = A θ . (16)We now look for corrections to the semiclassical area law upto leading order in θ .To do so, we first compute the corrected Hawking temperature ˜ T H . For that we use the tunnelingmethod by going beyond the semiclassical approximation [6]. Considering the massless scalarparticle tunneling under the background metric (7), the corrected Hawking temperature is givenby ˜ T H = T H " X i ˜ β i ¯ h i ( M r h ) i − . (17)Applying the first law of black hole thermodynamics once again with this corrected Hawkingtemperature, gives the following expression for the corrected entropy/area law : S = A θ h + 2 π ˜ β ln A θ − π ˜ β ¯ h A θ + O ( √ θe − M θ )= S BH + 2 π ˜ β ln S BH − π ˜ β ¯ hS BH + O ( √ θe − M θ ) . (18)Finally, we proceed to investigate the status of the relation between the Komar energy E , entropy S and Hawking temperature T H E = 2 ST H (19)in the case of these NC inspired black holes. The expression for the Komar energy E for theNC inspired Schwarzschild metric (7) is given by [1] E = 2 M √ π γ , r θ ! − M r θ √ πθ e − r / (4 θ ) . (20)3e therefore identify M as the mass of the black hole since E = M in the limit r → ∞ . Thisidentification plays an important role as we shall see below.The above expression computed near the event horizon of the black hole upto order √ θe − M /θ gives E = M − M √ πθ M θ + 1 ! e − M /θ − M s θπ e − M /θ . (21)Finally, using eqs.(12), (14) and (21), we obtain E = 2 ST H + 2 s θπ e − M /θ + O ( θ / e − M /θ )= 2 ST H + 2 s θπ e − S/ (4 πθ ) + O ( θ / e − S/ (4 πθ ) ) (22)where in the second line we have used eq.(14) to replace M by S/ (4 π ) in the exponent. In-terestingly, we find that the relation E = 2 ST H gets deformed upto order √ θe − M /θ which isconsistent with the fact that the area law also gets modified at this order. The deformation alsoyields a nonvanishing Komar energy at the extremal point T H = 0 of these black holes [2]. Also,we have once again managed to write down the deformed relation in terms of the Komar energy E , entropy S and the Hawking temperature T H . Similar features are also present for a de-SitterSchwarzschild geometry [12]. Eq.(22) can also be written with M being expressed in terms ofthe black hole parameters S and T H using eq.(21) M = 2 ST H + 12 π √ πθ S + S πθ + 6 πθ ! e − S/ (4 πθ ) + O ( θ / e − S/ (4 πθ ) ) . (23)We name eq.(23) as the Smarr formula [13] for NC inspired Schwarzschild black hole since M has been identified earlier to be the mass of the black hole. References [1] Banerjee R, Gangopadhyay S and Modak S K 2010 Phys. Lett. B
181 [arXiv:0911.2123[hep-th]].[2] Banerjee R and Gangopadhyay S 2011 Gen. Rel. Grav
547 [arXiv:gr-qc/0510112].[4] Nicolini P 2009 Int.J.Mod.Phys.A
095 [arXiv:0805.2220].[7] Komar A 1959 Phys. Rev. The Komar energy computed near the event horizon of the black hole plays an important role in obtainingthe coefficient of the logarithmic correction term (18) in the entropy [1]. L467.[11] Bardeen J M, Carter B and Hawking S W 1973 Comm.Math.Phys.30