霍金辐射和黑洞熵
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摘要
从上世纪70年代发现霍金辐射至今,对黑洞的研究已经取得了长足的进展。虽然尚未确证黑洞的存在,但对黑洞的理论研究已经揭示出一些涉及物理学和哲学基本观念的问题,具有划时代的意义。
论文首先介绍了霍金辐射出现的的历史背景及其所带来的信息疑难问题,即量子纯态演化为热混合态,违背了量子力学中的幺正性原理。接着介绍了解决信息疑难问题的一种比较好的方法-派瑞克方法。其出发点是将黑洞的霍金辐射理解成一种量子隧穿,势垒强弱取决于出射粒子自身能量的大小。用这种方法,派瑞克等人计算了零质量粒子穿过Schwarschild黑洞和Reissner-Nordstrom黑洞的出射修正谱。结果显示粒子的隧穿几率和贝根斯坦-霍金熵的变化量有关,出射谱线偏离纯热谱,满足幺正性原理,支持信息守恒。接下来把派瑞克方法的推广到了更一般的稳态黑洞。此外哈密顿-雅克比方法揭示了派瑞克方法的一些不足之处,即用雅克比方法得到的黑洞温度与霍金得到的并不一致,是后者的两倍。为了解决这一难题,论文引入了各项同性坐标系。考虑了入射粒子的贡献,得到了正确的霍金温度,并证明了在派瑞克方法使用的坐标系(Painleve-Gullstrand坐标系)中入射粒子的贡献为零。
接下来介绍了黑洞熵的一些性质。根据黑洞热力学第二定律,黑洞熵在数值上等于黑洞表面积的四分之一,且在顺时方向上永不减少。黑洞外部并不是平静的,充满了扰动场。一般来说,扰动场对黑洞熵的影响表现为黑洞熵公式中增加的一个对数修正项。修正项中的系数和扰动场的模型有关,是待定的。第三章在正则系统中引入拉普拉斯方法计算了热场扰动对黑洞熵的修正,因为正则系统允许黑洞和外界热场有能量的交换。论文还介绍了另外一套方法,这种方法的主要特点是将黑洞看成一个处在热场中的盒子,那么黑洞就会和它周围的物质达到热平衡状态。论文将这种方法应用到了不同维度的各种黑洞时空当中,得到了一些有用的结果。最后引入了随动坐标系的观测者,得出了Schwarzschild黑洞在热场扰动下,随动观测者观测到的黑洞熵和他所处的位置有关。
最后,介绍了广义不确定原理。经研究证实,当引力效果很弱的时候,广义不确定原理可以退化成海森堡不确定关系；当引力场强度不能忽略的时候,海森堡不确定关系不能替代广义不确定原理。根据广义不确定原理,位置的不确定度存在一个最小值。论文用广义不确定原理研究了量子场扰动对Schwarzschild黑洞熵和温度的影响。接下来论文引入了推广的不确定原理(EUP、GEUP)和修正的不确定原理(MDR),并在各种黑洞时空中进行了推广,得到了一些比较有用的结果并进行了分析。
The black hole theory have achieved remarkable success since Hawking radiation was discovered in 1970s. Although the existence of black hole is not confirmed now, the property of black hole fling down a challenge to the basic concept of physics and philosophy, which will surely become a landmark in the history of science。
The thesis introduced the background of Hawking radiation and its twin brothers-information puzzle, namely, the pure quantum state is translated into mixed thermal state, which violates the underlying unitary theory. The method of Per Kraus, which is regarded as an candidate to resolve information puzzle, shows that Hawking radiation can be consid-ered as a tunneling process and the barrier is defined by the tunneling particle itself. Per Kraus investigated Hawking radiation as massless particles tunneling in Schwarzschild and Reissner-Nordstrom black holes. The result shows that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum, but is consistent with an underlying unitary theory. The thesis will introduce many excellent work to extend the method of Per Kraus to the other black hole spacetime. The Hamilton Jacobi approach shows that the value of black hole temperature is twice as before with the method of Per Kraus. In order to deal with this problem, the thesis introduce the Isotropic coordinates and take into account the contribution of ingoing solution to the action, then the standard Hawking temperature is achieved. At the same time, the thesis point out that the imaginary part of action for ingoing particles is zero in the Painleve-Gullstrand coordinates.
The property of black hole entropy is introduced in chapter two. Based on the second law of thermodynamics, black hole entropy, which will never decrease with time, equal to a quarter of black hole area. There are disturbing field around black hole, so it is hard for black hole to achieve equilibrium with itself. Generally speaking, a logarithmic correction term will be exist in the black hole entropy formula based on the effect of disturbing field. The coefficient of correction term can be regarded as model-dependent parameters. the thesis use the Laplace method to calculate black hole entropy corrections with thermal fluctuations in canonical ensemble, which allows black hole to exchange energy with surrounds. Another method shows that if we put black hole in a cavity of finite radius by immersing it in an isothermal bath, then the thermal equilibrium will be achieved between the black hole and surroundings. We extend this method to various black hole spacetime, then some interesting result is obtained. Finally, the thesis show that the observer in servo-coordinates will find that the value of black hole entropy is related to the position himself under the thermal fluctuations in Schwarzschild black holes.
In the end of thesis, the thesis introduced the generalized uncertainty principle(GUP). It is shown that the GUP reduces to the conventional Heisenberg relation in situations of weak gravity but transcends it when gravitational effects can no longer be ignored. The GUP implies that there are a lower bound limit on the uncertainty in position. Based on GUP, the thesis investigate the quantum fluctuations on black hole entropy and tempera-ture in Schwarzschild black holes. Next, the thesis will introduce the extended uncertainty principle(EUP), the generalized extended uncertainty principle(GEUP) and the modified dis-persion relations(MDR), then the thesis extend these theory to investigate the property of black holes. Many excellent results are obtained.
论文首先介绍了霍金辐射出现的的历史背景及其所带来的信息疑难问题,即量子纯态演化为热混合态,违背了量子力学中的幺正性原理。接着介绍了解决信息疑难问题的一种比较好的方法-派瑞克方法。其出发点是将黑洞的霍金辐射理解成一种量子隧穿,势垒强弱取决于出射粒子自身能量的大小。用这种方法,派瑞克等人计算了零质量粒子穿过Schwarschild黑洞和Reissner-Nordstrom黑洞的出射修正谱。结果显示粒子的隧穿几率和贝根斯坦-霍金熵的变化量有关,出射谱线偏离纯热谱,满足幺正性原理,支持信息守恒。接下来把派瑞克方法的推广到了更一般的稳态黑洞。此外哈密顿-雅克比方法揭示了派瑞克方法的一些不足之处,即用雅克比方法得到的黑洞温度与霍金得到的并不一致,是后者的两倍。为了解决这一难题,论文引入了各项同性坐标系。考虑了入射粒子的贡献,得到了正确的霍金温度,并证明了在派瑞克方法使用的坐标系(Painleve-Gullstrand坐标系)中入射粒子的贡献为零。
接下来介绍了黑洞熵的一些性质。根据黑洞热力学第二定律,黑洞熵在数值上等于黑洞表面积的四分之一,且在顺时方向上永不减少。黑洞外部并不是平静的,充满了扰动场。一般来说,扰动场对黑洞熵的影响表现为黑洞熵公式中增加的一个对数修正项。修正项中的系数和扰动场的模型有关,是待定的。第三章在正则系统中引入拉普拉斯方法计算了热场扰动对黑洞熵的修正,因为正则系统允许黑洞和外界热场有能量的交换。论文还介绍了另外一套方法,这种方法的主要特点是将黑洞看成一个处在热场中的盒子,那么黑洞就会和它周围的物质达到热平衡状态。论文将这种方法应用到了不同维度的各种黑洞时空当中,得到了一些有用的结果。最后引入了随动坐标系的观测者,得出了Schwarzschild黑洞在热场扰动下,随动观测者观测到的黑洞熵和他所处的位置有关。
最后,介绍了广义不确定原理。经研究证实,当引力效果很弱的时候,广义不确定原理可以退化成海森堡不确定关系；当引力场强度不能忽略的时候,海森堡不确定关系不能替代广义不确定原理。根据广义不确定原理,位置的不确定度存在一个最小值。论文用广义不确定原理研究了量子场扰动对Schwarzschild黑洞熵和温度的影响。接下来论文引入了推广的不确定原理(EUP、GEUP)和修正的不确定原理(MDR),并在各种黑洞时空中进行了推广,得到了一些比较有用的结果并进行了分析。
The black hole theory have achieved remarkable success since Hawking radiation was discovered in 1970s. Although the existence of black hole is not confirmed now, the property of black hole fling down a challenge to the basic concept of physics and philosophy, which will surely become a landmark in the history of science。
The thesis introduced the background of Hawking radiation and its twin brothers-information puzzle, namely, the pure quantum state is translated into mixed thermal state, which violates the underlying unitary theory. The method of Per Kraus, which is regarded as an candidate to resolve information puzzle, shows that Hawking radiation can be consid-ered as a tunneling process and the barrier is defined by the tunneling particle itself. Per Kraus investigated Hawking radiation as massless particles tunneling in Schwarzschild and Reissner-Nordstrom black holes. The result shows that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum, but is consistent with an underlying unitary theory. The thesis will introduce many excellent work to extend the method of Per Kraus to the other black hole spacetime. The Hamilton Jacobi approach shows that the value of black hole temperature is twice as before with the method of Per Kraus. In order to deal with this problem, the thesis introduce the Isotropic coordinates and take into account the contribution of ingoing solution to the action, then the standard Hawking temperature is achieved. At the same time, the thesis point out that the imaginary part of action for ingoing particles is zero in the Painleve-Gullstrand coordinates.
The property of black hole entropy is introduced in chapter two. Based on the second law of thermodynamics, black hole entropy, which will never decrease with time, equal to a quarter of black hole area. There are disturbing field around black hole, so it is hard for black hole to achieve equilibrium with itself. Generally speaking, a logarithmic correction term will be exist in the black hole entropy formula based on the effect of disturbing field. The coefficient of correction term can be regarded as model-dependent parameters. the thesis use the Laplace method to calculate black hole entropy corrections with thermal fluctuations in canonical ensemble, which allows black hole to exchange energy with surrounds. Another method shows that if we put black hole in a cavity of finite radius by immersing it in an isothermal bath, then the thermal equilibrium will be achieved between the black hole and surroundings. We extend this method to various black hole spacetime, then some interesting result is obtained. Finally, the thesis show that the observer in servo-coordinates will find that the value of black hole entropy is related to the position himself under the thermal fluctuations in Schwarzschild black holes.
In the end of thesis, the thesis introduced the generalized uncertainty principle(GUP). It is shown that the GUP reduces to the conventional Heisenberg relation in situations of weak gravity but transcends it when gravitational effects can no longer be ignored. The GUP implies that there are a lower bound limit on the uncertainty in position. Based on GUP, the thesis investigate the quantum fluctuations on black hole entropy and tempera-ture in Schwarzschild black holes. Next, the thesis will introduce the extended uncertainty principle(EUP), the generalized extended uncertainty principle(GEUP) and the modified dis-persion relations(MDR), then the thesis extend these theory to investigate the property of black holes. Many excellent results are obtained.
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