Dirac particles' tunnelling from 5-dimensional rotating black strings influenced by the generalized uncertainty principle
aa r X i v : . [ h e p - t h ] D ec Dirac particles’ tunnelling from 5-dimensional rotating blackstrings influenced by the generalized uncertainty principle
Deyou Chen Institute of Theoretical Physics, China West Normal University,Nanchong 637009, China
Abstract:
The standard Hawking formula predicts the complete evap-oration of black holes. Taking into account effects of quantum gravity, weinvestigate fermions’ tunnelling from a 5-dimensional rotating black string.The temperature is determined not only by the string, but also affectedby the quantum number of the emitted fermion and the effect of the extraspatial dimension. The quantum correction slows down the increase of thetemperature, which naturally leads to the remnant in the evaporation.
The semi-classical tunnelling method is an effective way to describe theHawking radiation [1]. Using this method, the tunnelling behavior of mass-less particles across the horizon was veritably described in [2]. In the re-search, the varied background spacetime was taken into account. The tun-nelling rate was related to the change of the Bekenstein-Hawking entropy andthe temperature was higher than the standard Hawking temperature. In theformer researches, the standard temperatures were derived [3, 4, 5, 6, 7, 8],which imply the complete evaporation of black holes. Thus the varied back-ground spacetime accelerates the black holes’ evaporation. This result wasalso demonstrated in other complicated spacetimes [9, 10, 11, 12, 13]. Ex-tended this work to massive particles, the tunnelling radiation of generalspacetimes was investigated in [14, 15]. The same result was derived by therelation between the phase velocity and the group velocity.In [16], the standard Hawking temperatures were recovered by fermionstunnelling across the horizons. In the derivation, the action of the emittedparticle was derived by the Hamilton-Jacobi equation [17]. This derivationis based on the complex path analysis [18]. In this method, we don’t needthe consideration of that the particle moves along the radial direction [19,20, 21, 22]. This is a difference from the work of Parikh and Wilczek [2].The tunnelling radiation beyond the semi-classical approximation wasdiscussed in [23, 24, 25]. Their ansatz is also based on the Hamilton-Jacobi E-mail: [email protected] h . Using theexpansion, one can get the quantum corrections over the semiclassical value.The corrected temperature is lower than the standard Hawking temperature.The higher order correction entropies were derived by the first law of blackhole thermodynamics.Taking into account effects of quantum gravity, the semi-classical tun-nelling method was reviewed in the recent work [26, 27]. In [26], the tun-nelling of massless particles through quantum horizon of a Schwarzschildblack hole was investigated by the influence of the generalized uncertaintyprinciple (GUP). Through the modified commutation relation between theradial coordinate and the conjugate momentum and the deformed Hamil-tonian equation, the radiation spectrum with the quantum correction wasderived. The thermodynamic quantities were discussed. In the fermionicfields, with the consideration of effects of quantum gravity, the generalizedDirac equation in curved spacetime was derived by the modified fundamentalcommutation relations [28], which is [27] (cid:20) iγ ∂ + iγ i ∂ i (cid:16) − βm (cid:17) + iγ i β ¯ h (cid:16) ∂ j ∂ j (cid:17) ∂ i + m ¯ h (cid:16) β ¯ h ∂ j ∂ j − βm (cid:17) + iγ µ Ω µ (cid:16) β ¯ h ∂ j ∂ j − βm (cid:17)i ψ = 0 . (1)This derivation is based on the existence of a minimum measurable length.The length can be realized in a model of GUP∆ x ∆ p ≥ ¯ h h β (∆ p ) + β < p > i , (2)where β = β l p ¯ h is a small value, β < is a dimensionless parameter and l p is the Planck length. Eq. (2) was derived by the modified Heisenberg al-gebra [ x i , p j ] = i ¯ hδ ij (cid:2) βp (cid:3) , where x i and p i are position and momentumoperators defined respectively as [28, 29] x i = x i ,p i = p i (1 + βp ) , (3) p = P p j p j , x i and p j satisfy the canonical commutation relations[ x i , p j ] = i ¯ hδ ij . Thus the minimal position uncertainty is gotten as∆ x = ¯ h p β q β < p > , (4)2hich means that the minimum measurable length is ∆ x = ¯ h √ β [28]. Tolet ∆ x have a physical meaning, the condition β > The Kerr metric describes a rotating black hole solution of the Einstein equa-tions in four dimensions. When we add an extra compact spatial dimensionto it, the metric becomes ds = − ∆ ρ (cid:16) dt − a sin θdϕ (cid:17) + sin θρ h adt − ( r + a ) dϕ i + ρ ∆ dr + ρ dθ + g zz dz , (5)where ∆ = r − M r + a = ( r − r + )( r − r − ), ρ = r + a cos θ , g zz is usually set to 1. The above metric describes a rotating uniform blackstring. r ± = M ± √ M − a are the locations of the event (inner) horizons. M and a are the mass and angular momentum unit mass of the string,3espectively. A fermion’s motion satisfies the generalized Dirac equation (1).To investigate the tunnelling behavior of the fermion, it can directly choosea tetrad and construct gamma matrices from the metric (5). The metric(5) describes a rotating spacetime. The energy and mass near the horizonsare dragged by the rotating spacetime. It is not convenient to discuss thefermion’s tunnelling behavior. For the convenience of constructing the tetradand gamma matrices, we perform the dragging coordinate transformation dφ = dϕ − Ω dt , where Ω = (cid:0) r + a − ∆ (cid:1) a ( r + a ) − ∆ a sin θ , (6)on the metric (5). Then the metric (5) takes on the form as ds = − F ( r ) dt + 1 G ( r ) dr + g θθ dθ + g φφ dφ + g zz dz = − ∆ ρ ( r + a ) − ∆ a sin θ dt + ρ ∆ dr + g zz dz + ρ dθ + sin θρ h ( r + a ) − ∆ a sin θ i dφ . (7)Now the tetrad is directly constructed from the above metric. It is e aµ = diag ( √ F , / √ G, √ g θθ , √ g φφ , √ g zz ) . (8)Then gamma matrices are easily constructed as follows γ t = 1 √ F I − I ! , γ θ = q g θθ σ σ ! ,γ r = √ G σ σ ! , γ φ = q g φφ σ σ ! ,γ z = √ g zz − I I ! . (9)When measure the quantum property of a spin-1/2 fermion, we can get twovalues. They correspond to two states with spin up and spin down. Thewave functions of two states of a fermion in the metric (7) spacetime takeon the form as 4 ( ↑ ) = A B exp (cid:18) i ¯ h I ↑ ( t, r, θ, φ, z ) (cid:19) , (10) ψ ( ↓ ) = C D exp (cid:18) i ¯ h I ↓ ( t, r, θ, φ, z ) (cid:19) , (11)where A, B, C, D are functions of ( t, r, θ, φ, z ), and I is the action of thefermion, ↑ and ↓ denote the spin up and spin down, respectively. In thispaper, we only investigate the state with spin up. The analysis of the statewith spin down is parallel. To use the WKB approximation, we insert thewave function (10) and the gamma matrices into the generalized Dirac equa-tion (1). Dividing by the exponential term and considering the leading termsyield four equations. They are − B √ F ∂ t I − B √ G (1 − βm ) ∂ r I + A √ g zz (1 − βm ) ∂ z I − Am (1 − βm − βQ ) + Bβ √ GQ∂ r I − Aβ √ g zz Q∂ z I = 0 , (12) A √ F ∂ t I − A √ G (1 − βm ) ∂ r I − B √ g zz (1 − βm ) ∂ z I − Bm (1 − βm − βQ ) + Aβ √ GQ∂ r I + Bβ √ g zz Q∂ z I = 0 , (13) − B (cid:18) i q g θθ ∂ θ I + q g φφ ∂ φ I (cid:19) (1 − βm − βQ ) = 0 , (14) − A (cid:18) i q g θθ ∂ θ I + q g φφ ∂ φ I (cid:19) (1 − βm − βQ ) = 0 , (15)where Q = g rr ( ∂ r I ) + g θθ ( ∂ θ I ) + g φφ ( ∂ φ I ) + g zz ( ∂ z I ) . It is difficult toget the expression of the action from the above equations. Considering theproperty of the spacetime, we carry out separation of variables as5 = − ( ω − j Ω) t + W ( r ) + Θ( θ, φ ) + J z, (16)where ω is the energy of the emitted fermion, j is the angular momentumand J is a conserved momentum corresponding to the compact dimension.Eqs. (14) and (15) are irrelevant to A, B . Inserting Eq. (16) into themyields i q g θθ ∂ θ Θ + q g φφ ∂ φ Θ = 0 , (17)which implies that Θ is a complex function other than the constant solution.In the former research, it was found that the contribution of Θ could be can-celed in the derivation of the tunnelling rate. Using Eq. (17), an importantrelation is easily gotten as g θθ ( ∂ θ Θ) + g φφ ( ∂ φ Θ) = 0 . (18)Now our interest is the first two equations. Inserting Eq. (16) into Eqs. (12)and (13), canceling A and B and neglecting the higher order terms of β , weget A ( ∂ r W ) + B ( ∂ r W ) + C = 0 , (19)where A = 2 βG F,B = − [1 − βg zz ( ∂ z I ) ] GF,C = [1 − βm − βg zz ( ∂ z I ) ]( m − g zz ( ∂ z I ) ) F + ( ∂ t I ) . (20)Solving the above equation at the event horizon yields the imaginary part ofthe radial action. Based on the invariance under canonical transformations,we adopt the method developed in [32]. The tunnelling rate isΓ ∝ exp [ − h Im I p r dr ] = exp (cid:20) − h Im (cid:18)Z p outr dr − Z p inr dr (cid:19)(cid:21) = exp (cid:20) ∓ h Im Z p out,inr dr (cid:21) . (21)6n the above equation, H p r dr is invariant under canonical transformations.Here let p r = ∂ r W . Thus the solutions of Im R p out,inr dr are determined byEq. (19), which is Im I p r dr = 2 ImW out = 2 Im Z dr s ( E − j Ω) + (1 − βm − βg zz J )( m − g zz J ) FGF (1 − βg zz J ) × " β ( E − j Ω) F + m − g zz J ! = 2 π ( ω − j Ω + )( r + a ) r + − r − [1 + β Ξ( J, θ, r + , j )] , (22)where g zz = 1, Ω + = ar + a is the angular velocity at the event horizon.Ξ( J, θ, r + , j ) is a complicated function of J, θ, r + , j , therefore, we don’t writedown here. It should be that Ξ( J, θ, r + , j ) >
0. If adopt Eq. (22) tocalculate the tunnelling rate, we will derive two times Hawking temperature,which was showed in [31]. This is not in consistence with the standardtemperature. With careful observations, Akhmedova et. al. found that thecontribution coming from the temporal part of the action was ignored [32].When they took into account the temporal contribution, the factor of twoin the temperature was resolved.To find the temporal contribution, we use the Kruskal coordinates (
T, R ).The region exterior to the string ( r > r + ) is described by T = e κ + r ∗ sinh ( κ + t ) ,R = e κ + r ∗ cosh ( κ + t ) , (23)where r ∗ = r + κ + ln r − r + r + − κ − ln r − r − r − is the tortoise coordinate, and κ ± = r + − r − r ± + a ) denote the surface gravity at the outer (inner) horizons.The description of the interior region is given by T = e κ + r ∗ cosh ( κ + t ) ,R = e κ + r ∗ sinh ( κ + t ) . (24)To connect these two patches across the horizon, we need to rotate thetime t as t → t − iκ + π . As pointed in [32], this rotation would lead to an7dditional imaginary contribution coming from the temporal part, namely, Im ( E ∆ t out,in ) = πEκ + , where E = ω − j Ω + . Thus the total temporalcontribution is Im ( E ∆ t ) = πEκ + . Therefore, the tunnelling rate isΓ ∝ exp (cid:20) − h (cid:18) Im ( E ∆ t ) + Im I p r dr (cid:19)(cid:21) = − π ( ω − j Ω + )( r + a )¯ h ( r + − r − ) (cid:20) β Ξ( J, θ, r + , j ) (cid:21) . (25)This is the Boltzman factor expression and implies the temperature T = ¯ h ( r + − r − )4 π ( r + a ) h β Ξ( J, θ, r + , j ) i = T (cid:20) − β Ξ( J, θ, r + , j ) (cid:21) , (26)where T = ¯ h ( r + − r − )4 π ( r + a ) is the standard Hawking temperature of the Kerrstring and shares the same expression of the temperature of the 4-dimensionalKerr black hole. It shows that the corrected temperature is determined bythe mass, angular momentum and extra dimension of the string, but alsoaffected by the quantum number (energy, mass and angular momentum)of the fermion. Therefore, the property of the emitted fermion affects thetemperature when the effects of quantum gravity are taken into account.When a = 0, the metric (5) is reduced to the Schwarzschild string metric.Then the imaginary part of the radial action (22) is reduced to Im I p r dr = 2 πωr + h β (cid:16) ω + 3 m / J / (cid:17)i . (27)Adopting the same process, we get the temperature of the Schwarzschildstring as T = ¯ h πr + [1 + β ( ω + 3 m / J / h πM h − β (cid:16) ω + 3 m / J / (cid:17)i . (28)It shows that the effect of the extra dimension and the quantum number(energy, mass and angular momentum) of the fermion affect the temperature8f the Schwarzschild string. It is obviously that the quantum correction slowsdown the crease of the temperature. Finally, the string can not evaporatecompletely and there is a blanched state. At the this state, the remnant isleft. The effect of the extra dimension plays an role of impediment duringthe evaporation. When J = 0, Eq. (28) describes the temperature of the4-dimensional Schwarzschild black hole. The remnant was derived as ≥ M p /β , where M p is the Planck mass and β is a dimensionless parameteraccounting for quantum gravity effects [27]. In this paper, we investigated the fermion’s tunnelling from the 5-dimensionalKerr string spacetime. To incorporate the influence of quantum gravity, weadopted the generalized Dirac equation derived in [27]. The corrected tem-perature is not only determined by the mass, angular momentum and extradimension, but also affected by the quantum number of the emitted fermion.The quantum correction slows down the increase of the temperature. Fi-nally, the balance state appears. At this state, the string can not evaporatecompletely and the remnant is left. This can be seen as the direct conse-quence of the generalized uncertainty principle.
Acknowledgements
This work is supported by the National Natural Science Foundation ofChina with Grant No. 11205125.
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