Based on semiclassical tunneling method, we focus on charged fermions tunneling from higher-dimensional Reissner-Nordström black hole. We first simplify the Dirac equation by semiclassical approximation, and then a semiclassical Hamilton-Jacobi equation is obtained. Using the Hamilton-Jacobi equation, we study the Hawking temperature and fermions tunneling rate at the event horizon of the higher-dimensional Reissner-Nordström black hole space-time. Finally, the correct entropy is calculation by the method beyond semiclassical approximation.

Fundação de Amparo à Pesquisa do Estado de São Paulo2012/08934-0Conselho Nacional de Desenvolvimento Científico e TecnológicoCoordenação de Aperfeiçoamento de Pessoal de Nível SuperiorNational Natural Science Foundation of China1157302211375279

Hawking radiation is an important prediction in modern gravitation theory [1–5]. Recently, Kraus et al. proposed quantum tunneling theory to explain and study Hawking radiation [6–31], and then semiclassical Hamilton-Jacobi method is put forward to research the properties of scalar particles’ tunnels [32–36]. In 2007, Kerner and Mann investigated the 1/2 spin fermion tunneling from static black holes [37]. In their work, the spin up and spin down cases are researched, respectively, and the radial equations are obtained, so that they can finally determine the Hawking temperature and tunneling rate at the event horizon. Subsequently, Kerr and Kerr-Newman black holes cases, the charged dilatonic black hole case, the de Sitter horizon case, the BTZ black hole case, 5-dimensional space-time cases, and several nonstationary black hole cases were all researched, respectively [38–48], and we used Hamilton-Jacobi method to study the fermion tunneling from higher-dimensional uncharged black holes [49, 50]. However, up to now, no one has studied higher-dimensional charged black holes cases, so we set out to research that case. In our work, we developed the Kerner and Mann method and proved that the semiclassical Hamilton-Jacobi equation can be obtained not only with the Klein-Gordon equation of curved space-time, but also with the Dirac equation in curved space-time. Applying the Hamilton-Jacobi equation, we can then obtain semiclassical Hawking temperature and tunneling rate at the event horizon of higher-dimensional Reissner-Nordström black hole.

In modern physics theory, the concept of an extra dimension can help to solve some theoretical issues, so several higher-dimensional metrics of curved space-time were investigated. The metric of static charged (n+2)-dimensional Reissner-Nordström black hole is given by [10, 51–55] (1)ds2=-frdt2+f-1rdr2+r2dΩn2,where dΩn2 is the metric of n-dimensional sphere(2)dΩn2=∑i=1nhiidθi2=dθ12+sin2θ1dθ22+sin2θ1sin2θ2dθ32+⋯+∏i=1n-1sin2θidθn2,fr=1-ωnMrn-1+ωnQ22n-1Vnr2n-2,ωn=16πnVn,where M and Q are mass and electric charge of black hole, and the electromagnetic potential is (3)Aμ=Qn-1Vnrn-1,0,0,0,…,where Vn is volume of unit n-sphere (we can adopt the units G=c=ħ=1). The outer/inner horizon located at (4)r±n-2=ωn2M±M2-nQ28πn-1.Obviously, at the horizons, the equation f(r±)=0 should be satisfied. However, the physical property near the inner horizon cannot be researched, so we just study the fermion tunneling at the outer event horizon of this black hole. The charged Dirac equation in curved space-time is (5)γμDμΨ+mħΨ=0,μ=t,r,θ1,…,θn,where (6)Dμ=∂μ+Γμ+iqAμħΓμ=18γ~a,γ~beaνebν;μ,where m and q are mass and electric charge of the particles, and ebν;μ=∂μebν-Γμναeab is the covariant derivative of tetrad ebν. The gamma matrices in curved space-time need to be satisfied (7)γμ,γν=2gμνI.After the gamma matrices are defined, we choose the gamma matrices in (n+2)-dimensional flat space-time as (8)γ~m×m1=Im/2×m/200-Im/2×m/2γ~m×m2=0Im/2×m/2Im/2×m/20γ~m×mη=0iγ~m/2×m/2η-2-iγ~m/2×m/2η-20,η=3,4,5,…,n+2,where Im/2×m/2 and γ~m/2×m/2η are unit matrices and flat gamma matrices with m/2×m/2 order and m=2(n+2)/2 is the order of the matrices in even (odd) dimensional space-time. Corresponding to the flat case, the gamma matrices can be chosen as (9)γm×mt=ifγ~m×m1γm×mr=fγ~m×m2γm×mη=r-1hηηγ~m×mη,η=3,4,5,…,n+2.

Now, let us simplify the Dirac equation via semiclassical approximation, and rewrite the spinor function as(10)Ψ=Am/2×1t,r,…,xη,…Bm/2×1t,r,…,xη,…ei/ħSt,r,…,xη,…,where Am/2×1(t,r,…,xη,…) and Bm/2×1(t,r,…,xη,…) are column matrices with m/2×1 order and S is classical action. Via semiclassical approximation method, Substituting (10) into (5) and dividing the exponential term and multiplying by ħ, we can get (11)CDEFAm/2×1Bm/2×1=0(12)C=-1f∂S∂t+qAtIm/2×m/2+mIm/2×m/2(13)D=if∂S∂rIm/2×m/2-∑ηr-1hηη∂S∂xηγ~m/2×m/2η-2(14)E=if∂S∂rIm/2×m/2+∑ηr-1hηη∂S∂xηγ~m/2×m/2η-2(15)F=1f∂S∂t+qAtIm/2×m/2+mIm/2×m/2.Solving (11), we have (16)E-FD-1CAm/2×1=0F-EC-1DBm/2×1=0.It is evident that the coefficient matrices of (16) must vanish, when Am/2×1 and Bm/2×1 have nontrivial solutions. Due to the fact that CD=DC, we can write the condition that determinant of coefficient vanish as (17)detED-FC=0.From the relation of flat gamma matrices {γ~μ,γ~ν}=2δμν, we can obtain the semiclassical Hamilton-Jacobi equation in (n+2)-dimensional Reissner-Nordström space-time (18)-1f∂S∂t+qAt2+f∂S∂r2+⋯+gηη∂S∂xη+⋯+m2=0.Using the Hamilton-Jacobi equation, in charged static space-time, we can separate the variables for the action as (19)S=-ωt+Rr+Y…,xη,…+K,K is a constantand the Hamilton-Jacobi equation is broken up as (20)-1fω-qAt2+fdRdr2+m2=λr2(21)∑ηhηη∂Y∂xη2+λ=0,where (20) and (21) are radial and nonradial equations, respectively, and λ is a constant. However, we only research on the radial equation, because the tunnel at the event horizon is radial. From (20), we can get (22)dRrdr=±ω-qAt2r2+fλ-m2r2fr.Near the event horizon, the radial action is given by (23)ImR±=±πω-ω0f′r++ImC,where R+ is part of the outgoing solution, while R- is the part of incoming solution, and (24)ω0=qQn-1Vnr+n-1.So the tunneling rate is (25)Γ=ProboutProbin=exp-2ImS+exp-2ImS-=exp-2ImR++ImKexp-2ImR-+ImK=exp-4πω-ω0f′r+,where Im represents the imaginary part of the function, and the Hawking temperature is (26)TH=f′r+4π.

However, above calculation is worked on semiclassical approximation, because we ignored all higher order terms of O(ħ). Recently, Banerjee and Majhi proposed a new method beyond semiclassical approximation to research the quantum tunneling, and their works show that the conclusion should be corrected [56–65], and this correct entropy may be applied in quantum gravity theory. Now let us generalize this work in higher-dimensional Reissner-Nordström black hole space-time.

Because the tetrad eμa in the space-time are given by (27)eμa=diagf,1f,r,rsinθ1,…,r∏i=1n-1sinθi,so that Γμ is (28)γ~aeaμΓμ=γ~1fn2r+f′4f+12r∑k=1n-1γ~k+1n-kcotθk∏i=1k-1sinθi.It means the Dirac equation becomes (29)iγ~0f∂∂t+iqAtħΨ+γ~1f∂∂r+n2r+f′4fΨ+∑k=1n-1γ~k+1r∏i=1k-1sinθi∂∂θk+n-kcotθk2Ψ+γ~n+1r∏i=1n-1sinθi∂Ψ∂θn+mħΨ=0,and this equation can be simplified at event horizon (30)iγ~0∂∂t+iqA0ħΨ+γ~1∂∂r∗+f′4Ψ=0,because f→0 at event horizon, and dr∗=dr/f is tortoise coordinate. On the other hand, the space-time background is static, and Ψ can be rewritten as (31)Ψ=ArBre-i/ħωt,where A(r) and B(r) are matrices with m/2×1 and ω is frequency or energy of Dirac particle. Finally, applying the definitions of γ~0 and γ~1, we get (32)ω-ω0ħf∂∂r-f′4fħf∂∂r-f′4f-ω-ω0AqBq=0;here Aq and Bq are qth elements of matrices A(r) and B(r), respectively. Above equation becomes (33)∂Bq/∂r∂Aq/∂r=ω-ω0Aq-ħf′Bq/4-ω-ω0Bq-ħf′Aq/4,so (34)ω-ω02∂∂rAq2+Bq2-ħf′4Bq∂Aq∂r-Aq∂Bq∂r=0.At event horizon f′(r0)≠0 and depends on the position r0, so above equation implies (35)∂∂rAq2+Bq2=0,Bq∂Aq∂r-Aq∂Bq∂r=0,and the solution is (36)Aq2+Bq2=0.Above relation means Aq and Bq can be rewritten as (37)Aq=Cqei/ħRqr,Bq=Fqei/ħRqr,and Cq=±iFq are constants.

Next, let us use the method beyond semiclassical approximation to expand Rq(r) and K=ω-ω0 as (38)Rq=Rq0+∑i=1∞ħiRqir,K=ω-ω0=K0+∑i=1∞ħiKi,so we get (39)ħ0: -iK0f∂Rq0∂r∂Rq0∂r-iK0fCqFq=0,ħ1: -iK1f∂Rq1∂r+if′4f∂Rq1∂r+if′4f-iK1fCqFq=0,ħk: -iKkf∂Rqk∂r∂Rqk∂r-iKkfCqFq=0,k≥2.Therefore, the determinants of matrices vanish: (40)ħ0: Rq0±=±∫K0fdr,ħ1: Rq1±=±∫K1-if′r+/4fdr,ħk: Rqk±=±∫Kkfdr,k≥2,ImRqi=ImRqi+-ImRqi-=2πKif′r+.In order to calculate the tunneling rate and Hawking temperature, we rewrite ImRqi=βi/AhiImRq0 (i≥0, Ah is area of black hole, and βi are dimensionless constant parameters) since the forms of Rqj are the same, so the total radial action is (41)ImRq=ImRq0r+∑i=1∞ħiRqir=1+∑i=1∞βiħiAhiRq0.The tunneling rate of Dirac particle at event horizon is given by (42)Γ-h=exp-2ħ1+∑i=1∞βiħiAhiRq0=exp-4πK0ħf′r+1+∑i=1∞βiħiAhi.From the relation between tunneling rate and Hawking radiation, we get the temperature of black holes (43)Th=1+∑i=1∞βiħiAhiTH.Finally, the laws of black hole thermodynamics request (44)Sh=∫dSh=∫dM-AdQThr=r+=Ah4π+πβ1lnAh+⋯=SH+πβ1lnSH+⋯and SH=Ah/4π is entropy of semiclassical approximation, and this result shows the correction of entropy is logarithmic correction.

In this paper, we studied fermions tunneling from higher-dimensional Reissner-Nordström black holes and obtained the Hamilton-Jacobi equation from charged Dirac equation. This work shows that semiclassical Hamilton-Jacobi equation can describe the property of both 0 spin scalar particles and 1/2 spin fermions. In this work, we did not emphasize dimensions of space-time larger than (3+1) dimensions, so the method also can be used in the research of (3+1)-dimensions and lower cases.

As we all know, the information loss is an open problem in black hole physics, and the information of particles maybe vanishes at the singularity. In order to solve this difficulty, Horowitz and Maldacena proposed a boundary condition, which is called the black hole final state, at singularity of black hole to perfectly entangle the incoming Hawking radiation particles and the collapsing matter [66, 67]. Due to the boundary condition, any particle which is falling into the black holes completely annihilates. It is a new and interesting idea to investigate the black hole physics and quantum gravity, so we also will work on this area in the future.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported in part by FAPESP no. 2012/08934-0, CNPq, CAPES, and National Natural Science Foundation of China (no. 11573022 and no. 11375279).

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