关于相对论性紧致天体的若干研究
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摘要
在天体物理学和宇宙学中,黑洞、拓扑缺陷和孤子星是被经常讨论的三类相对论性紧致天体。黑洞是爱因斯坦方程的解,且是一种简单而优美的相对论性天体,它仅仅需要用几个参量如质量,电荷和角动量就能确定。研究黑洞拟正则模的动因之一就是确定黑洞的特征参量。拟正则模是指黑洞受到外界扰动后的后期出现的一类不断振荡衰减的特征信号,这些特征信号由黑洞的特征参量决定的。拓扑缺陷是早期宇宙相变而产生的相对论性紧致天体。自发对称破缺为拓扑缺陷提供了一个形成机制,在相对论性场论中得到了稳定拓扑缺陷的解。如果对称性是规范对称性,称拓扑缺陷为局部的;如果对称性是整体的,则称其为整体拓扑缺陷。这些拓扑缺陷能解释宇宙标准模型未能解决的很多问题,如宇宙结构的起源,重子不对称,星系的形成和演化等。孤子星是相对论性非线性场论中的拓扑缺陷,这类拓扑缺陷具有恒星到星系的质量。孤子星很可能在早期宇宙中就已形成并可能是暗物质的候选者。我们在这篇论文中分别研究了Schwarzschild-AdS小黑洞,弦黑洞和声学黑洞的拟正则模与它们的特征参量之间的关系,还研究了暗能量对整体单极子演化的影响。
本文共由四个部分组成:首先我们对黑洞、拓扑缺陷和孤子星三类相对论性紧致天体作一个简明的,且与全文自洽的评述,作为论述我们研究结果的准备知识。在第二部分中,我们分别用有限差分法和谱方法计算了Schwarzschild黑洞的拟正则模。我们发现当黑洞处于基态(n=0)时,黑洞拟正则模的振动周期随着小黑洞的视界r_+的变小而减小,而衰减频率缓慢地减小;拟正则模的振荡周期和衰减周期都随着角量子数l的增加而变小。此外,黑洞拟正则频率的实部和虚部都随着n的增加而增大,这里的n＞0表示黑洞处在激发态。再用有限差分法研究了弦黑洞的拟正则模和弦黑洞的特征参量之间的关系。在1+1维弦黑洞时空中拟正则模的振荡周期随着黑洞质量的增加而变小,而其衰减周期稍微有所增大;拟正则模的振荡频率和衰减周期都随着参量Q的增加而增大。在1+3维弦黑洞时空中拟正则模的振荡周期和衰减周期都随着视界的增大而变大。在第三部分中,我们研究了声学黑洞的拟正则模和拖尾信号。声学黑洞是根据超声速流体理论在实验室里构造的一种类黑洞。我们用有限差分法研究了1+3维声学黑洞的拟正则模和它的特征参量的关系,同时也计算了它的晚期拖尾信号。对于l≥2,我们的数值结果与用WKB法的一阶,三阶和六阶近似得到的结果一致,它还表明声学黑洞的晚期拖尾信号是以幂律形式衰减的,其衰减表达式为φ≈7.36×10~(-32)t~(-10)。声学黑洞的拟正则模振荡频率和衰减周期都随着角动量l的增加而增大。当l趋于很大时,拟正则频率的实部随着角动量l的增加而线性增大而虚部则趋向于某个数。在第四部分中,我们研究了在quintessence中整体单极子的演化。讨论了在球对称quintessence中整体单极子的爱因斯坦方程解。这个新的爱因斯坦方程解跟状态方程参量w_q有关。我们发现整体单极子在单极子核外产生一个类de Sitter的引力场以及一个立体欠缺角。当w_q→-1/3时quintessence的密度趋于零,所以w_q=-1/3的整体单极子解不存在。quintessence中整体单极子的外视界半径随着quintessence的密度减小而增大。
Black hole,topological defect and soliton star are three kinds of relativistic compact objects,which is a hotpoint of investigation in cosmology and astrophysics. A black hole is not only a solution for the Einstein's equation,but also it is a simple and elegant celestial body,in the sense that only a few parameters,like mass,charge and angular momentum are enough to describe it.One of major motivations to study quasinormal modes of black hole is to estimate black hole parameters.After perturbing a black hole,there is a quasinormal ringing outside the black hole.The frequency and damping of the 'ring' depend only on the structure of the background spacetime.Topological defects originated from phase transitions in the early universe. Of particular interest for cosmology is the theoretical expectation that at high temperature,symmetries that are spontaneous symmetry breaking(SSB) today were restored,and that during the evolution of universe there were phase transitions.On the other hand,there are the topological defect solutions in the SSB relativistic field theories.Topological defects are invoked to explain a variety of cosmic enigmas, including the large-scale structure of cosmic,the baryon asymmetry,etc.In relativistic non-linear field theory,topological defects are a kind of topological soliton.Soliton star is a soliton with star mass and star size.It maybe formed in the early universe and is a candidate for dark matter.Here,the relations between quasinormal modes of small AdS-Schwarzschild black hole,stringy black hole and acoustic black hole and their parameters are discussed,respectively,and the effect of quintessence on the global monopole is also considered.
In this paper,there are four sections.In section one,we have a brief review of black hole,topological defect,soliton star.In section two,we introduce two numerical methods to calculate quasinormal modes of a small Schwarzschild black hole.One is the ordinary finite element method.The other is Frobenius method.We find that the oscillating quasi-period and the imaginary part of the fundamental quasinormal modes (n=0) decrease with a decrease of the event horizon;the oscillating quasi-period and the damping time scale decrease as the multipole index increases.Furthermore,the real part and the imaginary part of the quasinormal frequency increase with an increase of n.Then,using the ordinary finite element method,the relation between quasinormal modes of stringy black hole and its parameters is discussed.In 1+1 dimensional stringy black hole spacetimes,the oscillating quasi-period decreases as the mass increases,but the damping time scale slightly increases with the mass.The oscillating frequency and the damping time scale are both increasing as the parameter Q increases.In 1+3 dimensional stringy black hole spacetimes,the oscillating time and the damping time increase with an increase of the event horizon.In section three, we study the quasinormal modes and late-time tails of a canonical acoustic black hole. Acoustic black hole is an analogue black hole made in the laboratory.We show the relations between quasinormal modes of the canonical acoustic black hole and its parameters.And we also study its late-time tails.For l≥2,the numerical results are consistent with the first,third and sixth orders WKB method,and show that the late-time tail decays as the power-law falloff of the formΦ≈7.36×10~(-32)t~(-10).The oscillating quasi-period and damping time scale are both increase with the angular momentum l.In the limit of large l,the real part of fundamental quasinormal frequency increases linearly and imaginary part tends to a constant with the angular momentum.In section four,we investigate new static spherically-symmetric solutions of Einstein equations with a global monopole surrounded by quintessence.The new solutions are more complicated,which depend on the parameter of equation of state Wq.We show that the gravitating global monopole produces a gravitational field of de Sitter kind outside the core in addition to a solid angular deficit.Furthermore,W_q= -1/3 solution cannot exist because the density of quintessence tends to zero as W_q→-1/3.The new feature is the appearance of outer horizon for the case of a global monopole surrounded by quintessence.
本文共由四个部分组成:首先我们对黑洞、拓扑缺陷和孤子星三类相对论性紧致天体作一个简明的,且与全文自洽的评述,作为论述我们研究结果的准备知识。在第二部分中,我们分别用有限差分法和谱方法计算了Schwarzschild黑洞的拟正则模。我们发现当黑洞处于基态(n=0)时,黑洞拟正则模的振动周期随着小黑洞的视界r_+的变小而减小,而衰减频率缓慢地减小;拟正则模的振荡周期和衰减周期都随着角量子数l的增加而变小。此外,黑洞拟正则频率的实部和虚部都随着n的增加而增大,这里的n＞0表示黑洞处在激发态。再用有限差分法研究了弦黑洞的拟正则模和弦黑洞的特征参量之间的关系。在1+1维弦黑洞时空中拟正则模的振荡周期随着黑洞质量的增加而变小,而其衰减周期稍微有所增大;拟正则模的振荡频率和衰减周期都随着参量Q的增加而增大。在1+3维弦黑洞时空中拟正则模的振荡周期和衰减周期都随着视界的增大而变大。在第三部分中,我们研究了声学黑洞的拟正则模和拖尾信号。声学黑洞是根据超声速流体理论在实验室里构造的一种类黑洞。我们用有限差分法研究了1+3维声学黑洞的拟正则模和它的特征参量的关系,同时也计算了它的晚期拖尾信号。对于l≥2,我们的数值结果与用WKB法的一阶,三阶和六阶近似得到的结果一致,它还表明声学黑洞的晚期拖尾信号是以幂律形式衰减的,其衰减表达式为φ≈7.36×10~(-32)t~(-10)。声学黑洞的拟正则模振荡频率和衰减周期都随着角动量l的增加而增大。当l趋于很大时,拟正则频率的实部随着角动量l的增加而线性增大而虚部则趋向于某个数。在第四部分中,我们研究了在quintessence中整体单极子的演化。讨论了在球对称quintessence中整体单极子的爱因斯坦方程解。这个新的爱因斯坦方程解跟状态方程参量w_q有关。我们发现整体单极子在单极子核外产生一个类de Sitter的引力场以及一个立体欠缺角。当w_q→-1/3时quintessence的密度趋于零,所以w_q=-1/3的整体单极子解不存在。quintessence中整体单极子的外视界半径随着quintessence的密度减小而增大。
Black hole,topological defect and soliton star are three kinds of relativistic compact objects,which is a hotpoint of investigation in cosmology and astrophysics. A black hole is not only a solution for the Einstein's equation,but also it is a simple and elegant celestial body,in the sense that only a few parameters,like mass,charge and angular momentum are enough to describe it.One of major motivations to study quasinormal modes of black hole is to estimate black hole parameters.After perturbing a black hole,there is a quasinormal ringing outside the black hole.The frequency and damping of the 'ring' depend only on the structure of the background spacetime.Topological defects originated from phase transitions in the early universe. Of particular interest for cosmology is the theoretical expectation that at high temperature,symmetries that are spontaneous symmetry breaking(SSB) today were restored,and that during the evolution of universe there were phase transitions.On the other hand,there are the topological defect solutions in the SSB relativistic field theories.Topological defects are invoked to explain a variety of cosmic enigmas, including the large-scale structure of cosmic,the baryon asymmetry,etc.In relativistic non-linear field theory,topological defects are a kind of topological soliton.Soliton star is a soliton with star mass and star size.It maybe formed in the early universe and is a candidate for dark matter.Here,the relations between quasinormal modes of small AdS-Schwarzschild black hole,stringy black hole and acoustic black hole and their parameters are discussed,respectively,and the effect of quintessence on the global monopole is also considered.
In this paper,there are four sections.In section one,we have a brief review of black hole,topological defect,soliton star.In section two,we introduce two numerical methods to calculate quasinormal modes of a small Schwarzschild black hole.One is the ordinary finite element method.The other is Frobenius method.We find that the oscillating quasi-period and the imaginary part of the fundamental quasinormal modes (n=0) decrease with a decrease of the event horizon;the oscillating quasi-period and the damping time scale decrease as the multipole index increases.Furthermore,the real part and the imaginary part of the quasinormal frequency increase with an increase of n.Then,using the ordinary finite element method,the relation between quasinormal modes of stringy black hole and its parameters is discussed.In 1+1 dimensional stringy black hole spacetimes,the oscillating quasi-period decreases as the mass increases,but the damping time scale slightly increases with the mass.The oscillating frequency and the damping time scale are both increasing as the parameter Q increases.In 1+3 dimensional stringy black hole spacetimes,the oscillating time and the damping time increase with an increase of the event horizon.In section three, we study the quasinormal modes and late-time tails of a canonical acoustic black hole. Acoustic black hole is an analogue black hole made in the laboratory.We show the relations between quasinormal modes of the canonical acoustic black hole and its parameters.And we also study its late-time tails.For l≥2,the numerical results are consistent with the first,third and sixth orders WKB method,and show that the late-time tail decays as the power-law falloff of the formΦ≈7.36×10~(-32)t~(-10).The oscillating quasi-period and damping time scale are both increase with the angular momentum l.In the limit of large l,the real part of fundamental quasinormal frequency increases linearly and imaginary part tends to a constant with the angular momentum.In section four,we investigate new static spherically-symmetric solutions of Einstein equations with a global monopole surrounded by quintessence.The new solutions are more complicated,which depend on the parameter of equation of state Wq.We show that the gravitating global monopole produces a gravitational field of de Sitter kind outside the core in addition to a solid angular deficit.Furthermore,W_q= -1/3 solution cannot exist because the density of quintessence tends to zero as W_q→-1/3.The new feature is the appearance of outer horizon for the case of a global monopole surrounded by quintessence.
引文
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