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Lorentz-violating theory and tunneling radiation characteristics of Dirac particles in curved spacetime of Vaidya black hole

Pu Jin Yang Shu-Zheng Lin Kai

Lorentz-violating theory and tunneling radiation characteristics of Dirac particles in curved spacetime of Vaidya black hole

Pu Jin, Yang Shu-Zheng, Lin Kai
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  • In this paper, the modified Hawking radiation for Dirac particles via tunneling from the apparent horizon of Vaidya black hole is studied by using the Lorentz-violating Dirac field theory. We first extend the gamma matric from flat spacetime to the curved spacetime in the Lorentz-violating Dirac field theory, and generalize the general derivative to the covariant derivative. Then, by considering the commutative relation of the gamma matric, the Dirac equation in the Lorentz-violating Dirac field theory is obtained, which contains three correction terms related to the Lorentz-symmetry violation. In the semiclassical approximation, the modified Hamilton-Jacobi equation is obtained by using the commutative relation of gamma matric and treating the aether-like vector in the Lorentz-violating theory as a constant. We find that the modified Hamilton-Jacobi equation contains only two correction terms based on the Lorentz-symmetry violation, i.e. the corrected term containing the parameter a affects the mass term of the Dirac field, and the aether-like term containing the parameter c modifies the coefficient term of the action S of the separating variable. According to the modified Hamilton-Jacobi equation, we study the effect of Lorentz-symmetry violation on the characteristics of Hawking radiation for Dirac particles via tunneling from the apparent horizon ra = 2M(v) of Vaidya black hole (the apparent horizon of Vaidya black hole coincides with the timelike limit surface, so the apparent horizon can be regarded as the boundary of Vaidya black hole). Since the Hawking tunneling radiation of black holes is the radial property at the horizon of black holes, we finally find that only the aether-like term containing the parameter c can modify the characteristics of Dirac particles’ tunneling radiation from the black hole. In addition, the corrected Hawking temperature of the black hole caused by considering the effect on the Lorentz-violating Dirac field theory has a small correction related to the aether-like term, which is consistent with the results obtained by studying the characteristics of Hawking tunneling radiation for scalar particles in the Lorentz-violating scalar field theory. The results suggest that the Lorentz-symmetry violation theory may provide a new method to further study the information loss paradox of black holes.
      Corresponding author: Yang Shu-Zheng, szyangphys@126.com
    [1]

    Horava P 2009 Phys. Rev. D 79 084008

    [2]

    Jacobson T, Mattingly D 2001 Phys. Rev. D 64 024028

    [3]

    Lin K, Mukohyama K, Wang A, Zhu T 2014 Phys. Rev. D 89 084022

    [4]

    Mukohyama S 2010 Class. Quant. Grav. 27 223101

    [5]

    Kostelecky V A, Samuel S 1989 Phys. Rev. Lett. 63 224

    [6]

    Jackiw R, Kostelecky V A 1999 Phys. Rev. Lett. 82 3572

    [7]

    Colladay D, McDonald P 2007 Phys. Rev. D 75 105002

    [8]

    Nascimento J R, Petrov A Yu, Reyes C M 2015 Phys. Rev. D 92 045030

    [9]

    Casana R, Ferreira M M, Jr, Moreira R P M 2011 Phys. Rev. D 84 125014

    [10]

    Hawking S W 1974 Nature 248 30

    [11]

    Hawking S W 1975 Commun. Math. Phys. 43 199

    [12]

    Robinson S P, Wilczek F 2005 Phys. Rev. Lett. 95 011303

    [13]

    Damoar T, Ruffini R 1976 Phys. Rev. D 14 332

    [14]

    Sannan S 1988 Gen. Relativ. Gravit. 20 239

    [15]

    Kraus P, Wilczek F 1995 Nucl. Phys. B 433 403

    [16]

    Parikh M K, Wilczek F 2000 Phys. Rev. Lett. 85 5042

    [17]

    Hemming S, Keski-Vakkuri E 2001 Phys. Rev. D 64 044006

    [18]

    Jiang Q Q, Wu S Q, Cai X 2007 Phys. Rev. D 75 064029

    [19]

    Iso S, Umetsu H, Wilczek F 2006 Phys. Rev. D 74 044017

    [20]

    Medved A J M 2002 Phys. Rev. D 66 124009

    [21]

    Parikh M K 2006 The Tenth Marcel Grossmann Meeting Rio de Janeiro, Brazil, February, 2006 pp1585-1590 [arXiv: hep-th/0402166]

    [22]

    Zhang J Y, Zhao Z 2006 Phys. Lett. B 638 110

    [23]

    Akhmedov E T, Akhmedova V, Singleton D 2006 Phys. Lett. B 642 124

    [24]

    Srinivasan K, Padmanabhan T 1999 Phys. Rev. D 60 24007

    [25]

    Shankaranarayanan S, Padmanabhan T, Srinivasan K 2002 Class. Quantum Grav. 19 2671

    [26]

    Kerner R, Mann R B 2008 Class. Quantum Grav. 25 095014

    [27]

    Kerner R, Mann R B 2008 Phys. Lett. B 665 277

    [28]

    Li R, Ren J R, Wei S W 2008 Class. Quantum Grav. 25 125016

    [29]

    Chen D Y, Jiang Q Q, Zu X T 2008 Class. Quantum Grav. 25 205022

    [30]

    Criscienzo R D, Vanzo L 2008 Europhys. Lett. 82 60001

    [31]

    Li H L, Yang S Z, Zhou T J, Lin R 2008 Europhys. Lett. 84 20003

    [32]

    Jiang Q Q 2008 Phys. Lett. B 666 517

    [33]

    Lin K, Yang S Z 2009 Int. J. Theor. Phys. 48 2061

    [34]

    Lin K, Yang S Z 2009 Phys. Rev. D 79 064035

    [35]

    Lin K, Yang S Z 2009 Phys. Lett. B 674 127

    [36]

    Lin K, Yang S Z 2011 Chin. Phys. B 20 110403

    [37]

    Yang S Z, Lin K 2019 Acta Phys. Sin. 68 060401

    [38]

    Criscienzo R D, Nadalini M, Vanzo L, Zernini S, Zoccatelli G 2007 Phys. Lett. B 657 107

    [39]

    Hayward S A 1998 Class. Quantum Grav. 15 3147

    [40]

    Kim S W 2014 Crav. & Cosm. 20 247

    [41]

    Kodama H 1980 Prog. Theor. Phys. 63 1217

  • [1]

    Horava P 2009 Phys. Rev. D 79 084008

    [2]

    Jacobson T, Mattingly D 2001 Phys. Rev. D 64 024028

    [3]

    Lin K, Mukohyama K, Wang A, Zhu T 2014 Phys. Rev. D 89 084022

    [4]

    Mukohyama S 2010 Class. Quant. Grav. 27 223101

    [5]

    Kostelecky V A, Samuel S 1989 Phys. Rev. Lett. 63 224

    [6]

    Jackiw R, Kostelecky V A 1999 Phys. Rev. Lett. 82 3572

    [7]

    Colladay D, McDonald P 2007 Phys. Rev. D 75 105002

    [8]

    Nascimento J R, Petrov A Yu, Reyes C M 2015 Phys. Rev. D 92 045030

    [9]

    Casana R, Ferreira M M, Jr, Moreira R P M 2011 Phys. Rev. D 84 125014

    [10]

    Hawking S W 1974 Nature 248 30

    [11]

    Hawking S W 1975 Commun. Math. Phys. 43 199

    [12]

    Robinson S P, Wilczek F 2005 Phys. Rev. Lett. 95 011303

    [13]

    Damoar T, Ruffini R 1976 Phys. Rev. D 14 332

    [14]

    Sannan S 1988 Gen. Relativ. Gravit. 20 239

    [15]

    Kraus P, Wilczek F 1995 Nucl. Phys. B 433 403

    [16]

    Parikh M K, Wilczek F 2000 Phys. Rev. Lett. 85 5042

    [17]

    Hemming S, Keski-Vakkuri E 2001 Phys. Rev. D 64 044006

    [18]

    Jiang Q Q, Wu S Q, Cai X 2007 Phys. Rev. D 75 064029

    [19]

    Iso S, Umetsu H, Wilczek F 2006 Phys. Rev. D 74 044017

    [20]

    Medved A J M 2002 Phys. Rev. D 66 124009

    [21]

    Parikh M K 2006 The Tenth Marcel Grossmann Meeting Rio de Janeiro, Brazil, February, 2006 pp1585-1590 [arXiv: hep-th/0402166]

    [22]

    Zhang J Y, Zhao Z 2006 Phys. Lett. B 638 110

    [23]

    Akhmedov E T, Akhmedova V, Singleton D 2006 Phys. Lett. B 642 124

    [24]

    Srinivasan K, Padmanabhan T 1999 Phys. Rev. D 60 24007

    [25]

    Shankaranarayanan S, Padmanabhan T, Srinivasan K 2002 Class. Quantum Grav. 19 2671

    [26]

    Kerner R, Mann R B 2008 Class. Quantum Grav. 25 095014

    [27]

    Kerner R, Mann R B 2008 Phys. Lett. B 665 277

    [28]

    Li R, Ren J R, Wei S W 2008 Class. Quantum Grav. 25 125016

    [29]

    Chen D Y, Jiang Q Q, Zu X T 2008 Class. Quantum Grav. 25 205022

    [30]

    Criscienzo R D, Vanzo L 2008 Europhys. Lett. 82 60001

    [31]

    Li H L, Yang S Z, Zhou T J, Lin R 2008 Europhys. Lett. 84 20003

    [32]

    Jiang Q Q 2008 Phys. Lett. B 666 517

    [33]

    Lin K, Yang S Z 2009 Int. J. Theor. Phys. 48 2061

    [34]

    Lin K, Yang S Z 2009 Phys. Rev. D 79 064035

    [35]

    Lin K, Yang S Z 2009 Phys. Lett. B 674 127

    [36]

    Lin K, Yang S Z 2011 Chin. Phys. B 20 110403

    [37]

    Yang S Z, Lin K 2019 Acta Phys. Sin. 68 060401

    [38]

    Criscienzo R D, Nadalini M, Vanzo L, Zernini S, Zoccatelli G 2007 Phys. Lett. B 657 107

    [39]

    Hayward S A 1998 Class. Quantum Grav. 15 3147

    [40]

    Kim S W 2014 Crav. & Cosm. 20 247

    [41]

    Kodama H 1980 Prog. Theor. Phys. 63 1217

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  • Received Date:  28 March 2019
  • Accepted Date:  25 June 2019
  • Available Online:  01 October 2019
  • Published Online:  05 October 2019

Lorentz-violating theory and tunneling radiation characteristics of Dirac particles in curved spacetime of Vaidya black hole

    Corresponding author: Yang Shu-Zheng, szyangphys@126.com
  • 1. College of Physics and Space Science, China West Normal University, Nanchong 637002, China
  • 2. School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
  • 3. Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics,China University of Geosciences, Wuhan 430074, China
  • 4. Escola de Engenharia de Lorena, Universidade de Sao Paulo, 12602-810, Lorena, Sao Paulo, Brazil

Abstract: In this paper, the modified Hawking radiation for Dirac particles via tunneling from the apparent horizon of Vaidya black hole is studied by using the Lorentz-violating Dirac field theory. We first extend the gamma matric from flat spacetime to the curved spacetime in the Lorentz-violating Dirac field theory, and generalize the general derivative to the covariant derivative. Then, by considering the commutative relation of the gamma matric, the Dirac equation in the Lorentz-violating Dirac field theory is obtained, which contains three correction terms related to the Lorentz-symmetry violation. In the semiclassical approximation, the modified Hamilton-Jacobi equation is obtained by using the commutative relation of gamma matric and treating the aether-like vector in the Lorentz-violating theory as a constant. We find that the modified Hamilton-Jacobi equation contains only two correction terms based on the Lorentz-symmetry violation, i.e. the corrected term containing the parameter a affects the mass term of the Dirac field, and the aether-like term containing the parameter c modifies the coefficient term of the action S of the separating variable. According to the modified Hamilton-Jacobi equation, we study the effect of Lorentz-symmetry violation on the characteristics of Hawking radiation for Dirac particles via tunneling from the apparent horizon ra = 2M(v) of Vaidya black hole (the apparent horizon of Vaidya black hole coincides with the timelike limit surface, so the apparent horizon can be regarded as the boundary of Vaidya black hole). Since the Hawking tunneling radiation of black holes is the radial property at the horizon of black holes, we finally find that only the aether-like term containing the parameter c can modify the characteristics of Dirac particles’ tunneling radiation from the black hole. In addition, the corrected Hawking temperature of the black hole caused by considering the effect on the Lorentz-violating Dirac field theory has a small correction related to the aether-like term, which is consistent with the results obtained by studying the characteristics of Hawking tunneling radiation for scalar particles in the Lorentz-violating scalar field theory. The results suggest that the Lorentz-symmetry violation theory may provide a new method to further study the information loss paradox of black holes.

    • 物理学发展至今, 几乎完美地解释了所有物理实验中发现的各种物理现象. 在天体物理和宇宙尺度下, 广义相对论可以很好地解释“我们所处的宇宙为什么会是如此”的问题; 在微观领域, 量子场论则完美地阐述了各种微观物理现象. 随着科学研究的不断深入, 当今的物理学依然面临着一些前沿热点问题的挑战, 在天文观测上, 人们发现星系周围存在着大量的暗物质, 同时宇宙还在某种暗能量的驱动下加速膨胀. 在微观高能领域中, 量子引力理论则迟迟无法完美地建立起来. 这些未解之谜深深地困惑着当今的理论物理学和天文学的研究者. 由于广义相对论是一个不可重整化的引力理论, 因此几种修正的引力理论被提出了, 有人猜测, 构成广义相对论基石的“洛伦兹对称性”在高能下可能会产生破缺. 在这个基本假设下, 人们提出了各种洛伦兹对称性破缺的引力模型[1-3]. 洛伦兹对称性破缺的理论原则上可以解决引力不可重整化的难题, 另外关于这些洛伦兹对称性破缺的模型的一些研究显示暗物质可能只是洛伦兹对称性破缺的理论模型的一个效应[4]. 在弦论[5]、电动力学[6]、非阿贝尔理论[7]等方面的研究中, 洛伦兹对称性的破缺也得到了广泛地关注. 近年来, 人们通过引入类以太场项, 给出了平直时空中洛伦兹对称性破缺的Dirac方程, 并对这个理论中的以太场项的量子修正进行了研究[8,9]. 在这个理论中, 类以太场的存在导致时空的洛伦兹对称性的消失, 可能会在高能情况下表现出和洛伦兹对称性的理论不同的性质来.

      另一方面, 在弯曲时空中, 霍金证明黑洞视界附近由于剧烈的量子效应, 可能辐射出霍金辐射[10,11]. 霍金辐射的存在巧妙地把引力论、量子论、热力学有机地联系到了一起[12-14], 因此研究者期望通过对黑洞霍金辐射的研究来理解量子引力理论的动力学行为. 我们可以用隧穿的观点来解释黑洞的霍金辐射效应: 黑洞视界可以被视为一个势垒, 黑洞视界内的虚粒子有概率通过量子效应穿过此势垒, 并在视界外实化成实粒子形成霍金辐射. 研究人员利用隧穿理论提出了计算黑洞霍金温度和黑洞熵的方法[15-23], 随后通过把标量场方程化为Hamilton-Jacobi方程的方法简化了隧穿辐射的研究[24,25]. 2008年开始, 研究人员逐渐对黑洞的费米子隧穿辐射感兴趣[26-33], 2009年我们提出了用半经典近似求得费米子的Hamilton-Jacobi方程, 并进一步研究了高维和旋转时空的霍金辐射特征[34-36].

      目前, 在利用洛伦兹破缺理论研究黑洞的隧穿辐射特征方面, 研究人员只对静态情况下标量粒子的Hawking隧穿辐射进行了研究[37], 而对于在动态弯曲时空中洛伦兹破缺理论对旋量粒子Hawking隧穿辐射的修正的研究还未有报道. 因此, 在动态弯曲时空中研究洛伦兹破缺理论对Dirac粒子量子隧穿辐射特征的修正, 是一个有意义的前沿研究课题. 本文针对这一课题, 研究动态黑洞时空中洛伦兹对称性破缺的Dirac隧穿辐射. 由于黑洞时空是极端引力的时空, 在黑洞视界附近各种引力和量子效应更加明显, 因此对黑洞视界处黑洞热力学的研究将有助于人们更深刻地理解洛伦兹对称性破缺带来的各种量子效应. 在第2节中, 我们将根据文献[8]的思路, 构造一个弯曲时空中的洛伦兹对称性破缺的Dirac方程, 然后我们运用半经典近似把这个Dirac方程进行化简, 得到一个变形的Dirac-Hamilton-Jacobi方程.

    2.   洛伦兹对称性破缺的Dirac方程和修正的Hamilton-Jacobi方程
    • 文献[8]中给出了平直时空中的洛伦兹对称性破缺的作用量, 我们将平直时空中的${\bar \gamma ^\mu }$矩阵推广到弯曲时空中的${\gamma ^\mu }$矩阵, 考虑到${\gamma ^\mu }$矩阵的对易关系, 并应用普通微商到协变微商的推广, 即可得到弯曲时空中洛伦兹对称性破缺的Dirac方程为

      这里的${{\varPsi }}$是Dirac方程的函数, 而${{\slashed{D}}} \equiv {{{\gamma }}^\mu }{{{D}}_\mu } \equiv $$ {{{\gamma }}^\mu }\left( {{\partial _\mu } + \dfrac{{\rm{i}}}{2}\varGamma _\mu ^{\alpha \beta }{\varPi _{\alpha \beta }}} \right)$, 其中最后一项是旋联络项, 体现了弯曲时空中的旋量协变导数的性质. abc项是很小的量, 因此这几项满足关系: $a,b,c \ll m$. 在平直时空中的类以太矢量${u^\alpha }$是一个常量, 自然满足条件

      在弯曲时空中, 上述条件作为类以太矢量${u^\alpha }$需要满足的条件出现, ${u^\alpha }$不再一定是一个常量了. 在Vaidya时空中, 我们定义${{{\gamma }}^5} \equiv - \dfrac{{\rm{i}}}{{{r^2}\sin \theta }}{{{\bar \gamma }}^0}{{{\bar \gamma }}^1}{{{\bar \gamma }}^2}{{{\bar \gamma }}^3}$, 其中平直时空中的gamma矩阵${{{\bar \gamma }}^\alpha }$和弯曲时空中的gamma矩阵${{{\gamma }}^\mu }$分别满足条件

      为了研究Vaidya黑洞表观视界处的隧穿辐射, 我们利用半经典近似对弯曲时空中洛伦兹对称性破缺的Dirac方程进行处理, 因此可以假设

      注意到${{\varPsi }}$在矩阵方程中是一个列矩阵函数, 因此按照(4)式, ${{\psi }}$也是一个列矩阵函数. 把(4)式代入变形的Dirac方程(1)式, 并考虑到$\hbar $是一个小量, 因此把$\hbar $的所有高阶项都略去, 最后得到一个半经典的矩阵关系:

      然后, 考虑到gamma矩阵的反对易关系(3)式, 我们有

      于是(5)式变成

      (7)式的计算中, 我们已经用到了a是小量的条件, 我们把a进行展开并忽略掉了高阶项, 在后面的运算中, 我们始终把abc视为小量, 在展开式中忽略掉高阶项. 接下来, 我们可以把(7)式写成矩阵方程形式, 然后令矩阵的行列式为零即可得到修正的Dirac-Hamilton-Jacobi方程. 但是根据(3)式, 我们可以用一个更简单的方法推导出这一修正的Dirac-Hamilton-Jacobi方程. 通过把上式两边乘以${\rm{i}}{{{\gamma }}^\upsilon }{S_{,\upsilon }}$, 即

      于是我们得到

      我们发现由洛伦兹对称性破缺的Dirac方程在半经典近似下得到的变形Dirac-Hamilton-Jacobi方程仅受ac两个修正项的修正, 其中a项仅仅对质量项修正, 而c项则对S项前的函数进行了修正. 这暗示着, 只有c项修正会影响到霍金辐射的性质. 我们将在第3节中利用这个Dirac-Hamilton-Jacobi方程对Vaidya黑洞表观视界处的热力学性质进行研究.

    3.   Vaidya黑洞的隧穿辐射及其修正
    • 在宇宙空间中, 由于黑洞可以不断吸积物质或者由霍金辐射蒸发质量, 因此真实的黑洞一定是动态黑洞. 最简单的动态黑洞是球对称不带电的Vaidya黑洞, 其线元可以写为

      这里

      由于动态Vaidya黑洞的表观视界${r_{\rm{a}}} = 2M\left( v \right)$与类时极限面(无限红移面)重合, 因此可以把表观视界${r_{\rm{a}}}$视为Vaidya黑洞的边界, 根据霍金辐射量子效应理论, 穿越这一边界的粒子到达类时区. 所以, 我们研究了Vaidya黑洞表观视界处的霍金隧穿辐射修正. Vaidya黑洞的逆变度规张量为

      这里$M\left( v \right)$是黑洞的质量, 在真实宇宙中由于黑洞对周围物质的吸积作用, 动态黑洞的质量可能增加, 也可能由于量子隧穿辐射效应的蒸发而导致黑洞质量的减少. 另一方面, 静态球对称史瓦西黑洞的表观视界、事件视界和无限红移面是重合在一起的, 然而在Vaidya黑洞中, 表观视界和事件视界不再重合. 近来的一些研究显示, 把动态黑洞表观视界视为霍金辐射的源区更加合理[38-40]. 接下来, 我们在表观视界处研究黑洞的霍金辐射.

      这里我们假设${u^\alpha } = \dfrac{{{c_v}}}{{\sqrt F }}\delta _v^\alpha + \sqrt F {c_r}\delta _r^\alpha $, 其中的${c_v}$${c_r}$是常数, 因此${u_\alpha } = \sqrt F \left( {{c_r} - {c_v}} \right)\delta _\alpha ^v + \dfrac{{{c_v}}}{{\sqrt F }}\delta _\alpha ^r$. 很明显, 这个形式的类以太矢量是满足条件(2)式的. 代入变形的Dirac-Hamilton-Jacobi方程, 并且假设$S = R\left( {v,r} \right) + Y\left( {\theta,\varphi } \right)$我们得到径向的Dirac-Hamilton-Jacobi方程

      这里的${C_L}$是分离变量常数. 于是我们有

      (14)式中, 我们已经在表观视界处用到了Kodama矢量条件$\dot R = - \omega $[41]. 同时在表观视界附近, 我们有$F\left( {r \to {r_a}} \right) \to 0$, 所以

      我们有

      因此, 在表观视界处的黑洞隧穿率是

      其中, 霍金温度是

      这里的${T_{\rm{h}}}$是未修正的史瓦西黑洞事件视界处的霍金温度, 而修正后的霍金温度${T_{\rm{H}}}$则与$\left( 2c_r^2 + c_v^2 - \right. $$\left. 2{c_r}{c_v} \right)cm$相关. 因此我们发现类以太矢量项只有c项可以修正霍金温度, 但是零静止质量粒子时, 修正项消失. 同时, 我们发现$2c_r^2 + c_v^2 - 2{c_r}{c_v}$总是大于零, 这意味着当c大于零时候, 修正霍金温度变高; 而当c小于零时候, 修正霍金温度降低. 也就是说, 修正霍金温度与类以太矢量修正项系数的正负有关. 而我们之前的研究中也发现, 应用洛伦兹破缺理论研究标量粒子的修正霍金温度也是与类以太矢量修正项系数的正负有关的[37].

      另一方面, 我们也可以研究了${u^\alpha } = \left( {0,0,{u^\theta },0} \right)$${u^\alpha } = \left( {0,0,0,{u^\varphi }} \right)$的情况, 其中前一种情况要求${u^\theta } \propto {r^{ - 1}}$, 而后一种情况要求${u^\varphi } \propto {r^{ - 1}}{\sin ^{ - 1}}\theta $, 所以${u^\theta }$${u^\varphi }$的修正只能影响角向部分, 而无法影响到作为黑洞径向行为的霍金辐射的性质.

    4.   结 论
    • 本文把洛伦兹对称性破缺的Dirac方程推广到弯曲时空中, 我们发现Dirac方程的洛伦兹对称性被三项类以太矢量项破缺. 然而当我们用半经典近似得到Dirac方程的时候, 我们发现只有ac项会影响Dirac-Hamilton-Jacobi方程的一阶修正, 其中的a项影响方程的质量项, c项则修正S的系数项. 由于黑洞霍金辐射是黑洞时空的径向性质, 因此我们发现只有c项会影响到黑洞的Dirac粒子隧穿辐射特征.

      本文的研究只涉及到不带电的动态黑洞情况, 宇宙中真实的黑洞通常是旋转的, 在下一步的工作中我们将继续研究旋转动态的黑洞时空中Dirac粒子的隧穿理论. 另一方面, 2018年双星合并所产生的引力波被首次直接观测到, 这一成果不但证明了引力波的存在, 同时也是黑洞存在的直接证明. 源于黑洞时空视界附近的引力辐射将携带许多极端引力场的引力性质. 因此可以预期, 今后人们对黑洞性质的研究将越来越有兴趣. 我们也将在今后的工作中对自旋为2的引力子霍金辐射以及自旋为3/2的引力费米子霍金辐射进行深入的研究.

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