黑洞时空中微扰的演化与相对论框架下的量子信息
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摘要
在广义相对论、量子力学、弦理论、热力学和统计物理等诸多学科的交叉领域——黑洞物理中，本文对当前国际物理学界极为关注的黑洞似正模、幂率拖尾、量子纠缠和量子隐形传态进行了研究。
利用Leaver的连续分数法精确求解了Schwarzschild黑洞、整体单极子黑洞和稳态轴对称爱因斯坦-麦克斯韦伸缩子黑洞三种时空中不同微扰场的似正模，主要结论如下：(1)当角量子数l(玻色场)或者j(费米场)很大时，Schwarzschild黑洞时空中任意自旋场似正模频谱的间距相等，即△_ω=2/3(?)-0．0000i，不依赖于自旋参数s和模量子数n。当模量子数n很大时，似正模是高度衰减的，其频谱虚部的间距也相等，始终为-1／(4M)，与角量子数l(或者j)以及自旋参数s无关。(2)整体单极子黑洞时空中任意自旋场的似正模依赖于对称性破缺参数H，其中似正模频谱的实部随着H的增加而减小，但虚部随着H的增加而增加。当模量子数n很大时，似正模频谱虚部的间距为-(1-H)~(3/2)/(4M)，依赖于对称性破缺参数H但与角量子数l和自旋参数s无关。(3)稳态轴对称爱因斯坦-麦克斯韦伸缩子黑洞时空中标量场的似正模频谱会随着单位质量角动量a的增加或者伸缩子参数D的减小在复平面内逆时针旋转并成螺旋状。但是，对于静态Garfinkle-Horowitz-Strominger(GHS)伸缩子黑洞，与该黑洞电荷相关的伸缩子参数D却无法使得似正模频谱成逆时针的螺旋状。这与荷电Reissner-Nordstr(o|¨)m(RN)黑洞中电荷能使似正模频谱成逆时针螺旋状的特性完全不同。提出了所谓的“类螺旋判据”：如果所考虑黑洞的热容始终为负值，那么该黑洞时空中似正模频谱在复平面内将无法出现螺旋状特性；反之，热容只要出现正值，该黑洞似正模频谱在复平面内必定可以在某一时空参数的连续变化下呈现螺旋状特性。这暗示着黑洞动力学演化与黑洞热力学不稳定性之间可能存在着某些联系。
利用黑洞Green函数法解析地研究了整体单极子黑洞时空中无质量扰动任意自旋场的晚期拖尾，结果表明：对于每一给定的角量子数l，该黑洞背景下无质量扰动任意自旋场的晚期衰减行为将由负幂率拖尾t~(-2[1+(?)])主导，依赖于对称性破缺参数H和自旋参数s。当H→0时，该结果退化为Schwarzschild黑洞时空中的幂率拖尾t~(-(2l+3))，显然与微扰场自旋参数s无关。
讨论了非惯性系下两个自由标量粒子和Dirac粒子非最大纠缠的衰减行为，分析了一般静态球对称渐近平直黑洞时空中Hawking温度对量子纠缠和量子隐形传态的影响，主要结论如下：(1)在非惯性系下，由于Unruh效应和不同坐标系下场量子化的不对等性，具有参数α和相应“归一化伙伴”(?)的两个不同初始态的相同非最大初始纠缠将沿着两条不同的轨迹衰减。在加速度无限大情况下，标量场双模态对于任意α值都不再是可提纯纠缠的；但Dirac场总是存在纠缠，其纠缠程度依赖于α。有趣的是，在此极限下的互信息恰好只是其对应初始值的一半，该结论不依赖于初始态参数α和场的种类。(2)在一般静态球对称渐近平直黑洞时空中，由于Hawking效应，具有参数α和相应“归一化伙伴”(?)的两个不同初始态的相同非最大初始纠缠将沿着两条不同的路径衰减。当Hawking温度无限大即黑洞完全蒸发以后，标量场双模态对于任意α值都不再是可提纯纠缠的，但此时系统的互信息恰好只是其对应初始值的一半。这类时空中量子隐形传态方案的保真度随着Hawking温度的增加而减小，恰好说明了量子纠缠的衰减性。
This thesis is devoted to the investigation of the quasinormal modes(QNMs),power-law tail,quantum entanglement and quantum teleportation in the black holephysics which is an intersectional field of general relativity,quantum mechanics,string theory,thermodynamics,statistics,and so on.
The QNMs of the Schwarzschild black hole,Schwarzschild black hole with aglobal monopole(SBHGM)and stationary axisymmetric Einstein-Maxwell dilatonaxion(EMDA)black hole are investigated by using the continued fraction method proposed by Leaver.Our main conclusions are as follows:(1)The QNMs associatedwith the decay of massless arbitrary spin fields around a Schwarzschildblack hole become evenly spaced for large angular quantum number l(for the boson perturbations)and j(for the fermion perturbations),and the spacing isgiven by△_ω=2/3(?)-0.0000i which is independent of the spin number s andovertone number n.It is also shown that the spacing for imaginary part of theQNMs at high overtones is equidistant and equals -1/(4M),which is independentof l(or j)and s.(2)The real part of the QNMs for arbitrary spin fields in thebackground of a SBHGM decreases as the symmetry breaking scale parameter Hincreases but the imaginary part increases instead.For the large overtone numbern,these QNMs become evenly spaced and the spacing for the imaginary partequals -(1-H)~(3/2)/(4M)which is dependent on H but independent on the quantum number l and spin number s.(3)The massless scalar quasinormal frequenciesof a stationary axisymmetric EMDA black hole move counterclockwise and get a spiral-like shape in the complex plane as the angular momentum per unit massa increases to its extremal value or the dilaton D decreases to its extremal value for the rotating black hole.However,for the non-rotating Garfinkle-Horowitz-Strominger(GHS)dilaton black hole,the dilaton parameter D,which is related tothe electric charge of this EMDA black hole,cannot make the frequencies spire inthe complex plane,which is qualitatively different from the charge of the Reissner-Nordstrom(RN)black hole.The so-call "Spiral-like Criterion" is obtained and itpoints out that the frequencies won't spire in the complex plane if the heat capacity for the considered black hole is always negative and vice versa.It seems to implythat there is some relation between the dynamical evolution and thermodynamicinstabilities for the black hole.
The late-time behavior of arbitrary spin fields in the background of a SB-HGM is studied by using the black-hole Green's function method.It is surprisingly found that this late-time behavior is dominated by an inverse power-law tail t~(-2[1+(?)]) which is dependent on the symmetry breaking scale parameter H and the spin number s for each quantum number l,and asH→0 it reduces to the Schwarzschild case t~(-(2l+3)) which is independent on s.
The entanglement between two modes of free scalar and Dirac fields as seenby two relatively accelerated observers is investigated and the effect of the Hawking temperature on the entanglement and teleportation for the scalar field in amost general,static and asymptotically flat black hole with spherical symmetry isalso analyzed.Our main conclusions are as follows:(1)It is found that the sameinitial entanglement for an initial state parameterαand its "normalized partner"(?) will be degraded by the Unruh effect along two different trajectoriesexcept for the maximally entangled state,which just shows the inequivalence ofthe quantization for a free field in Minkowski and Rindler coordinates.In the infinite-acceleration limit,the state does not have distillable entanglement for anyαfor the scalar field,but always remains entangled to a degree that is dependentonαfor the Dirac field.It is also interesting to note that in this limit the mutualinformation equals just half of the initial mutual information; this result isindependent ofαand the type of field.(2)It is shown that the same initial entanglement for the state parameterαand its "normalized partners"(?) willbe degraded by the Hawking effect with increasing Hawking temperature alongtwo different trajectories except for the maximally entangled state.In the infiniteHawking temperature limit,corresponding to the case of the black hole evaporating completely,the state has no longer distillable entanglement for anyα.It isinteresting to note that the mutual information in this limit equals just half ofthe initial mutual information.It has also been demonstrated that the fidelity ofteleportation decreases as the Hawking temperature increases,which just indicatesthe degradation of entanglement.
利用Leaver的连续分数法精确求解了Schwarzschild黑洞、整体单极子黑洞和稳态轴对称爱因斯坦-麦克斯韦伸缩子黑洞三种时空中不同微扰场的似正模，主要结论如下：(1)当角量子数l(玻色场)或者j(费米场)很大时，Schwarzschild黑洞时空中任意自旋场似正模频谱的间距相等，即△_ω=2/3(?)-0．0000i，不依赖于自旋参数s和模量子数n。当模量子数n很大时，似正模是高度衰减的，其频谱虚部的间距也相等，始终为-1／(4M)，与角量子数l(或者j)以及自旋参数s无关。(2)整体单极子黑洞时空中任意自旋场的似正模依赖于对称性破缺参数H，其中似正模频谱的实部随着H的增加而减小，但虚部随着H的增加而增加。当模量子数n很大时，似正模频谱虚部的间距为-(1-H)~(3/2)/(4M)，依赖于对称性破缺参数H但与角量子数l和自旋参数s无关。(3)稳态轴对称爱因斯坦-麦克斯韦伸缩子黑洞时空中标量场的似正模频谱会随着单位质量角动量a的增加或者伸缩子参数D的减小在复平面内逆时针旋转并成螺旋状。但是，对于静态Garfinkle-Horowitz-Strominger(GHS)伸缩子黑洞，与该黑洞电荷相关的伸缩子参数D却无法使得似正模频谱成逆时针的螺旋状。这与荷电Reissner-Nordstr(o|¨)m(RN)黑洞中电荷能使似正模频谱成逆时针螺旋状的特性完全不同。提出了所谓的“类螺旋判据”：如果所考虑黑洞的热容始终为负值，那么该黑洞时空中似正模频谱在复平面内将无法出现螺旋状特性；反之，热容只要出现正值，该黑洞似正模频谱在复平面内必定可以在某一时空参数的连续变化下呈现螺旋状特性。这暗示着黑洞动力学演化与黑洞热力学不稳定性之间可能存在着某些联系。
利用黑洞Green函数法解析地研究了整体单极子黑洞时空中无质量扰动任意自旋场的晚期拖尾，结果表明：对于每一给定的角量子数l，该黑洞背景下无质量扰动任意自旋场的晚期衰减行为将由负幂率拖尾t~(-2[1+(?)])主导，依赖于对称性破缺参数H和自旋参数s。当H→0时，该结果退化为Schwarzschild黑洞时空中的幂率拖尾t~(-(2l+3))，显然与微扰场自旋参数s无关。
讨论了非惯性系下两个自由标量粒子和Dirac粒子非最大纠缠的衰减行为，分析了一般静态球对称渐近平直黑洞时空中Hawking温度对量子纠缠和量子隐形传态的影响，主要结论如下：(1)在非惯性系下，由于Unruh效应和不同坐标系下场量子化的不对等性，具有参数α和相应“归一化伙伴”(?)的两个不同初始态的相同非最大初始纠缠将沿着两条不同的轨迹衰减。在加速度无限大情况下，标量场双模态对于任意α值都不再是可提纯纠缠的；但Dirac场总是存在纠缠，其纠缠程度依赖于α。有趣的是，在此极限下的互信息恰好只是其对应初始值的一半，该结论不依赖于初始态参数α和场的种类。(2)在一般静态球对称渐近平直黑洞时空中，由于Hawking效应，具有参数α和相应“归一化伙伴”(?)的两个不同初始态的相同非最大初始纠缠将沿着两条不同的路径衰减。当Hawking温度无限大即黑洞完全蒸发以后，标量场双模态对于任意α值都不再是可提纯纠缠的，但此时系统的互信息恰好只是其对应初始值的一半。这类时空中量子隐形传态方案的保真度随着Hawking温度的增加而减小，恰好说明了量子纠缠的衰减性。
This thesis is devoted to the investigation of the quasinormal modes(QNMs),power-law tail,quantum entanglement and quantum teleportation in the black holephysics which is an intersectional field of general relativity,quantum mechanics,string theory,thermodynamics,statistics,and so on.
The QNMs of the Schwarzschild black hole,Schwarzschild black hole with aglobal monopole(SBHGM)and stationary axisymmetric Einstein-Maxwell dilatonaxion(EMDA)black hole are investigated by using the continued fraction method proposed by Leaver.Our main conclusions are as follows:(1)The QNMs associatedwith the decay of massless arbitrary spin fields around a Schwarzschildblack hole become evenly spaced for large angular quantum number l(for the boson perturbations)and j(for the fermion perturbations),and the spacing isgiven by△_ω=2/3(?)-0.0000i which is independent of the spin number s andovertone number n.It is also shown that the spacing for imaginary part of theQNMs at high overtones is equidistant and equals -1/(4M),which is independentof l(or j)and s.(2)The real part of the QNMs for arbitrary spin fields in thebackground of a SBHGM decreases as the symmetry breaking scale parameter Hincreases but the imaginary part increases instead.For the large overtone numbern,these QNMs become evenly spaced and the spacing for the imaginary partequals -(1-H)~(3/2)/(4M)which is dependent on H but independent on the quantum number l and spin number s.(3)The massless scalar quasinormal frequenciesof a stationary axisymmetric EMDA black hole move counterclockwise and get a spiral-like shape in the complex plane as the angular momentum per unit massa increases to its extremal value or the dilaton D decreases to its extremal value for the rotating black hole.However,for the non-rotating Garfinkle-Horowitz-Strominger(GHS)dilaton black hole,the dilaton parameter D,which is related tothe electric charge of this EMDA black hole,cannot make the frequencies spire inthe complex plane,which is qualitatively different from the charge of the Reissner-Nordstrom(RN)black hole.The so-call "Spiral-like Criterion" is obtained and itpoints out that the frequencies won't spire in the complex plane if the heat capacity for the considered black hole is always negative and vice versa.It seems to implythat there is some relation between the dynamical evolution and thermodynamicinstabilities for the black hole.
The late-time behavior of arbitrary spin fields in the background of a SB-HGM is studied by using the black-hole Green's function method.It is surprisingly found that this late-time behavior is dominated by an inverse power-law tail t~(-2[1+(?)]) which is dependent on the symmetry breaking scale parameter H and the spin number s for each quantum number l,and asH→0 it reduces to the Schwarzschild case t~(-(2l+3)) which is independent on s.
The entanglement between two modes of free scalar and Dirac fields as seenby two relatively accelerated observers is investigated and the effect of the Hawking temperature on the entanglement and teleportation for the scalar field in amost general,static and asymptotically flat black hole with spherical symmetry isalso analyzed.Our main conclusions are as follows:(1)It is found that the sameinitial entanglement for an initial state parameterαand its "normalized partner"(?) will be degraded by the Unruh effect along two different trajectoriesexcept for the maximally entangled state,which just shows the inequivalence ofthe quantization for a free field in Minkowski and Rindler coordinates.In the infinite-acceleration limit,the state does not have distillable entanglement for anyαfor the scalar field,but always remains entangled to a degree that is dependentonαfor the Dirac field.It is also interesting to note that in this limit the mutualinformation equals just half of the initial mutual information; this result isindependent ofαand the type of field.(2)It is shown that the same initial entanglement for the state parameterαand its "normalized partners"(?) willbe degraded by the Hawking effect with increasing Hawking temperature alongtwo different trajectories except for the maximally entangled state.In the infiniteHawking temperature limit,corresponding to the case of the black hole evaporating completely,the state has no longer distillable entanglement for anyα.It isinteresting to note that the mutual information in this limit equals just half ofthe initial mutual information.It has also been demonstrated that the fidelity ofteleportation decreases as the Hawking temperature increases,which just indicatesthe degradation of entanglement.
引文
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