基于空间—时间—物质理论和膜世界模型的黑洞研究
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摘要
在20世纪20年代,Kaluza与Klein利用高维空间成功地统一了引力场与电磁场。此后,额外维进入了人们的视野。进入新世纪以后,高维理论在实践与理论方面都有望取得巨大的进展。在实践方面,最令人振奋的是演微缩版的“宇宙大爆炸”大型强子对撞机Large Hadron Collider(LHC)的运行。若果真如人们所预料的那样,额外维在宇宙早期和黑洞视界附近最活跃的话,高维时空效应极有可能在Planck能标的试验中被探测到。一个例子就是,高能粒子碰撞就有可能产生额外维的原初黑洞。在理论方面,两个著名的Arkani-Hamed,Dimopoulos,Dvali(ADD)模型与Randall,Sundrum(RS)模型利用brane与bulk构建了一类新的高维模型——膜世界(brane world),成功地解决了粒子物理当中存在已久的等级问题(Hierarchy problem)。鉴于这些情况,额外维尤其是膜世界理论逐渐成为理论物理研究中的一大研究热点。
本学位论文研究的对象是Space-Time-Matter(STM)中(等价于膜世界理论)的一类黑洞。在该模型中,时空是由一个(1+3)维的3-brane嵌入到一个更高维的bulk当中,标准模型中的普通物质被限制在膜上,例如fermions、bosons等等。在bulk中只有引力场与不带电的标量场才可以自由传播。如果物质被限制在膜上,则物质由于引力的作用不可避免地遭遇引力塌缩,在膜上自然就形成了黑洞。它的视界由于额外维的作用会在额外维方向上有一个延伸,所以这种塌缩体在膜上看去是一个黑洞(球状),而在整个bulk中看去则为一黑弦(弦状)。此黑弦解为一精确的五维真空解,包含来自于第5维坐标的宇宙学常数∧。由于5维时空是Ricci平坦的,在bulk中是不包含任何的宇宙学常数,所以在膜上时空是呈现Schwarzschild-de Sitter几何性质的。
具体说来,论文分为七章来阐述的,其中前三章是论文的背景知识:黑洞和宇宙学常数(第一章)、高维理论(第二章)以及膜世界理论(第三章);我们的工作包含在最后四章中:黑洞的辐射(第四章)、黑洞的熵(第五章)、实标量场的演化(第六章)、黑洞的拟正则模式(第七章)。
在黑洞辐射方面,由于标量场Klein-Gordon方程关于时空的五个分量耦合在一起,我们发现第五维明显影响四维时空的有效辐射。通过研究发现辐射方程的势函数包含一个来自额外维的关键参数,该参数将整个辐射模式分成连续谱型和离散谱型两类。
在黑洞熵的方面,论文采取改进的薄层砖墙模型与广义测不准原理两种不同的方法来研究。在前一种方法中,通过计算量子场的波相而获得了5维黑弦时空的量子统计熵。沿着第5维的模式被Randall-Sundrum质量关系所半经典量子化。黑洞的熵用一个2维面积来描述,在小质量近似下该熵为内外视界面积的线性和。由于量子化的额外维,熵表达式中的比例系数也成为离散的。值得强调的是小质量近似可以通过将第二张膜外推而得到。在第二种广义测不准原理中,熵可以不通过对bulk的结构作任何的限制,而且没有任何的截断就可以很自然地得到。量子场的发散问题在这里得到了很好的解决,态密度与自由能在视界的附近是收敛的。同样小质量近似可以通过视界附近度规函数的渐进行为很自然地也能得到。
在实标量场的演化中,论文详细地分析了可解的离散谱情况。由于类Schr(o|¨)dinger方程中包含了两个不同的乌龟坐标与径向坐标,鉴于两者之间复杂的超越函数关系,我们采用切线近似来处理并得到了两种极端情况下(∧_1=0.11与∧_2=10~(-3))波函数的数值解。在内外视界相距很远的情况中(∧_2),由于穿透率变弱,相对于普通的4维情况,波函数在黑洞视界附近变得更加松散,而在宇宙学视界附近发生很明显的堆积;而在内外视界相距很近的情况中(∧_1),波函数则在内视界发生堆积,且堆积密度是很明显大于外视界的堆积。
在黑洞的拟正则模式方面,论文采用低频条件下的三阶WKB近似来求解,得到了低频下的不同拟正则模式下的频率。通过结果的分析发现:标量场在额外维与宇宙学常数的影响下衰减得更慢,共鸣效应的影响导致第二、三模式高频区的衰减变慢。
Since in 1920s Kaluza and Klein had unified successfully gravity field and electromagnetic field,the higher dimensional gravitational-field theory came within people's range of vision. Recently,with the small-scale Big Bang model—Large Hadron Collider(LHC) being in operation,the extra dimensional small black hole is very likely to be produced by the collision of particles.Meanwhile,there are two famous theoretical models—Arkani-Hamed, Dimopoylos,Dvali(ADD) and Randall-Sundrum(RS) are developed in the turn of the century.So the developments of theory and experiment set off a new wave of researching extra dimension.
In this thesis,the study object is a kind of extra dimensional black hole in which standard model fields(such as fermions,gauge bosons fields) are confined on a(3+1) dimensional hypersurface(3-brane) without accessing along the transverse dimensions.The branes are embedded in the higher dimensional space(bulk),in which only gravitons and scalar particles without charges could propagate under standard model gauge group.If matter trapped on the brane undergoes gravitational collapse,a black hole will form naturally and its horizons extends into the extra dimension which is transverse to brane.Such higher dimensional object looks like a black hole on the brane is actually a black string in the higher dimensional brane world.Hence a 5D Ricci-flat black string is obtained naturally.On the brane the space gives a SdS geometric construction.
Specifically,this thesis is organized as follows:the former three chapters are the background knowledge concluding black hole and cosmological constant(chapter 1),higher dimensional gravitational-field theory(chapter 2),brane world(chapter 3);the last four chapters are our work concluding black hole radiation(chapter 4),black hole entropy(chapter 5),real scalar field evolution(chapter 6) and quasi-normal modes(chapter 7).
In the respect of black hole radiation,as the equations are coupled,it is found that the structure of the fifth dimension(as for membrane and induced-matter theory) affects the nature of the radiation observed in four-dimensional space-time.From these study it is known that the potential function of radiation equation contains a key parameter from extra dimension.This parameter divides the whole radiation into two parts,one is continuous spectrum and the other is discrete one.
The respect of black hole entropy is studied from two sides:one is Thin-Layer Approach (TLA) and the other is Generalized Uncertainty Principle(GUP).In the TLA case,the statistical-mechanical entropies of 5D Ricci-flat black string is calculated through the wave modes of the quantum field with improved thin-layer brick-wall method.The modes along the fifth dimension are semi-classically quantized by Randall-Sundrum mass relationship.The two-dimensional area is used to describe this black string's entropy which,in the small-mass approximation,is a linear sum of the area of inner and outer horizons.The proportionality coefficients of entropy are discretized with quantized extra dimensional modes.The smallmass approximation is naturally justified by the assumption of far apart two branes.In the GUP case,entropy is obtained without any cutoff and any constraint on the bulk's configuration rather than the usual uncertainty principle.The system's density of state and free energy are convergent in the neighborhood of horizon.The small-mass approximation is determined by the asymptotic behavior of metric function near horizons.
In the respect of real scalar field evolvement,the tangent approximation is chosen to unite tortoise coordinate and radial coordinate,which have transcendental function relationship, in Schr(o|¨)dinger-type equation.The numerical solutions of two extreme cases(∧_1=10~(-3) and∧_2=0.11) are obtained.In the case of∧_1,the inner and outer horizons are widely separated. Due to the penetrating power is weak,waves are concentrated near cosmological horizon and become much sparser near even horizon than that of 4D case.In the case of∧_2,two horizons are close to each other.Waves are concentrated near black hole even horizon and the stacking density is larger than that of outer cosmological horizon.
In the respect of quasi-normal modes,by using the classical third-order WKB approximation, the evolution of frequencies is carefully analyzed in two aspects,one is the induced cosmological constant and the other is the quantum number n.The massless scalar field decays more slowly because of the existence of the fifth dimension and the induced cosmological constant.Especially,the effect of resonance slows down the decaying of high-frequency.
本学位论文研究的对象是Space-Time-Matter(STM)中(等价于膜世界理论)的一类黑洞。在该模型中,时空是由一个(1+3)维的3-brane嵌入到一个更高维的bulk当中,标准模型中的普通物质被限制在膜上,例如fermions、bosons等等。在bulk中只有引力场与不带电的标量场才可以自由传播。如果物质被限制在膜上,则物质由于引力的作用不可避免地遭遇引力塌缩,在膜上自然就形成了黑洞。它的视界由于额外维的作用会在额外维方向上有一个延伸,所以这种塌缩体在膜上看去是一个黑洞(球状),而在整个bulk中看去则为一黑弦(弦状)。此黑弦解为一精确的五维真空解,包含来自于第5维坐标的宇宙学常数∧。由于5维时空是Ricci平坦的,在bulk中是不包含任何的宇宙学常数,所以在膜上时空是呈现Schwarzschild-de Sitter几何性质的。
具体说来,论文分为七章来阐述的,其中前三章是论文的背景知识:黑洞和宇宙学常数(第一章)、高维理论(第二章)以及膜世界理论(第三章);我们的工作包含在最后四章中:黑洞的辐射(第四章)、黑洞的熵(第五章)、实标量场的演化(第六章)、黑洞的拟正则模式(第七章)。
在黑洞辐射方面,由于标量场Klein-Gordon方程关于时空的五个分量耦合在一起,我们发现第五维明显影响四维时空的有效辐射。通过研究发现辐射方程的势函数包含一个来自额外维的关键参数,该参数将整个辐射模式分成连续谱型和离散谱型两类。
在黑洞熵的方面,论文采取改进的薄层砖墙模型与广义测不准原理两种不同的方法来研究。在前一种方法中,通过计算量子场的波相而获得了5维黑弦时空的量子统计熵。沿着第5维的模式被Randall-Sundrum质量关系所半经典量子化。黑洞的熵用一个2维面积来描述,在小质量近似下该熵为内外视界面积的线性和。由于量子化的额外维,熵表达式中的比例系数也成为离散的。值得强调的是小质量近似可以通过将第二张膜外推而得到。在第二种广义测不准原理中,熵可以不通过对bulk的结构作任何的限制,而且没有任何的截断就可以很自然地得到。量子场的发散问题在这里得到了很好的解决,态密度与自由能在视界的附近是收敛的。同样小质量近似可以通过视界附近度规函数的渐进行为很自然地也能得到。
在实标量场的演化中,论文详细地分析了可解的离散谱情况。由于类Schr(o|¨)dinger方程中包含了两个不同的乌龟坐标与径向坐标,鉴于两者之间复杂的超越函数关系,我们采用切线近似来处理并得到了两种极端情况下(∧_1=0.11与∧_2=10~(-3))波函数的数值解。在内外视界相距很远的情况中(∧_2),由于穿透率变弱,相对于普通的4维情况,波函数在黑洞视界附近变得更加松散,而在宇宙学视界附近发生很明显的堆积;而在内外视界相距很近的情况中(∧_1),波函数则在内视界发生堆积,且堆积密度是很明显大于外视界的堆积。
在黑洞的拟正则模式方面,论文采用低频条件下的三阶WKB近似来求解,得到了低频下的不同拟正则模式下的频率。通过结果的分析发现:标量场在额外维与宇宙学常数的影响下衰减得更慢,共鸣效应的影响导致第二、三模式高频区的衰减变慢。
Since in 1920s Kaluza and Klein had unified successfully gravity field and electromagnetic field,the higher dimensional gravitational-field theory came within people's range of vision. Recently,with the small-scale Big Bang model—Large Hadron Collider(LHC) being in operation,the extra dimensional small black hole is very likely to be produced by the collision of particles.Meanwhile,there are two famous theoretical models—Arkani-Hamed, Dimopoylos,Dvali(ADD) and Randall-Sundrum(RS) are developed in the turn of the century.So the developments of theory and experiment set off a new wave of researching extra dimension.
In this thesis,the study object is a kind of extra dimensional black hole in which standard model fields(such as fermions,gauge bosons fields) are confined on a(3+1) dimensional hypersurface(3-brane) without accessing along the transverse dimensions.The branes are embedded in the higher dimensional space(bulk),in which only gravitons and scalar particles without charges could propagate under standard model gauge group.If matter trapped on the brane undergoes gravitational collapse,a black hole will form naturally and its horizons extends into the extra dimension which is transverse to brane.Such higher dimensional object looks like a black hole on the brane is actually a black string in the higher dimensional brane world.Hence a 5D Ricci-flat black string is obtained naturally.On the brane the space gives a SdS geometric construction.
Specifically,this thesis is organized as follows:the former three chapters are the background knowledge concluding black hole and cosmological constant(chapter 1),higher dimensional gravitational-field theory(chapter 2),brane world(chapter 3);the last four chapters are our work concluding black hole radiation(chapter 4),black hole entropy(chapter 5),real scalar field evolution(chapter 6) and quasi-normal modes(chapter 7).
In the respect of black hole radiation,as the equations are coupled,it is found that the structure of the fifth dimension(as for membrane and induced-matter theory) affects the nature of the radiation observed in four-dimensional space-time.From these study it is known that the potential function of radiation equation contains a key parameter from extra dimension.This parameter divides the whole radiation into two parts,one is continuous spectrum and the other is discrete one.
The respect of black hole entropy is studied from two sides:one is Thin-Layer Approach (TLA) and the other is Generalized Uncertainty Principle(GUP).In the TLA case,the statistical-mechanical entropies of 5D Ricci-flat black string is calculated through the wave modes of the quantum field with improved thin-layer brick-wall method.The modes along the fifth dimension are semi-classically quantized by Randall-Sundrum mass relationship.The two-dimensional area is used to describe this black string's entropy which,in the small-mass approximation,is a linear sum of the area of inner and outer horizons.The proportionality coefficients of entropy are discretized with quantized extra dimensional modes.The smallmass approximation is naturally justified by the assumption of far apart two branes.In the GUP case,entropy is obtained without any cutoff and any constraint on the bulk's configuration rather than the usual uncertainty principle.The system's density of state and free energy are convergent in the neighborhood of horizon.The small-mass approximation is determined by the asymptotic behavior of metric function near horizons.
In the respect of real scalar field evolvement,the tangent approximation is chosen to unite tortoise coordinate and radial coordinate,which have transcendental function relationship, in Schr(o|¨)dinger-type equation.The numerical solutions of two extreme cases(∧_1=10~(-3) and∧_2=0.11) are obtained.In the case of∧_1,the inner and outer horizons are widely separated. Due to the penetrating power is weak,waves are concentrated near cosmological horizon and become much sparser near even horizon than that of 4D case.In the case of∧_2,two horizons are close to each other.Waves are concentrated near black hole even horizon and the stacking density is larger than that of outer cosmological horizon.
In the respect of quasi-normal modes,by using the classical third-order WKB approximation, the evolution of frequencies is carefully analyzed in two aspects,one is the induced cosmological constant and the other is the quantum number n.The massless scalar field decays more slowly because of the existence of the fifth dimension and the induced cosmological constant.Especially,the effect of resonance slows down the decaying of high-frequency.
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