Jose Natario, Rodrigo Vicente
We determine the timelike Killing vector field that gives the correct definition of energy for test fields propagating in a Kerr-Newman-de Sitter spacetime, and use this result to prove that test fields cannot destroy extremal Kerr-Newman-de Sitter black holes.
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The gedanken experiment by Wald to destroy a black hole using a test particle in the equatorial plane is adopted to the case of extremal magnetized black holes. We find the presence of external magnetic fields resulting from the "Ernst magnetization" permits a test particle to have strong enough energy to destroy the black hole. However, the corresponding effective potentials show that such particles would never reach the horizon.
If radiative and self-force effects are neglected, we find that feeding a test particle into a near-extremal Kerr-Newman-AdS black hole could lead to destroy their event horizon, giving rise to naked singularities. Hence radiative and self-force effects must be taken into account to further test cosmic censorship. Moreover, the allowed parameter range for this test particle is very narrow, this leaves the possibility of radiative and self-force effects considerations to cure the problem of WCC violation in Kerr-Newman-AdS spacetime.
We prove that (possibly charged) test fields satisfying the null energy condition at the event horizon cannot overspin/overcharge extremal Kerr-Newman or Kerr-Newman-anti de Sitter black holes, that is, the weak cosmic censorship conjecture cannot be violated in the test field approximation. The argument relies on black hole thermodynamics (without assuming cosmic censorship), and does not depend on the precise nature of the fields. We also discuss generalizations of this result to other extremal black holes.
We obtain an analytical expression for the electromagnetic quasinormal spectrum of the higher-dimensional nearly-extremal Schwarzschild-de Sitter black hole. The WKB method is used to verify the results, and a comparison with known results from previous works is briefly made as well.
It has been shown that a nearly extremal black hole can be overcharged or overspun by a test particle if radiative and self-force effects are neglected, indicating that the cosmic censorship might fail. In contrast, the existing evidence in literature suggests that an extremal black hole cannot be overcharged or overspun in a similar process. In this paper, we show explicitly that even an exactly extremal black hole can be destroyed by a test particle, leading to a possible violation of the cosmic censorship. By considering higher order terms, which were neglected in previous analysis, we show…
By neglecting the self-force, self-energy, and radiative effects, it has been shown that an extremal or near-extremal Kerr-Newman black hole can turn into a naked singularity when it captures charged and spinning massive particles. A straightforward question then arises: do charged and rotating black holes in string theory possess the same property? In this paper we apply the Wald's gedanken experiment, in his study on the possibility of destroying extremal Kerr-Newman black holes, to the case of (near-)extremal Kerr-Sen black holes. We find that feeding a test particle into a (near-)extremal …
We revisit here a previous argument due to Wald showing the impossibility of turning an extremal Kerr-Newman black hole into a naked singularity by plunging test particles across the black hole event horizon. We extend Wald's analysis to the case of near-extremal black holes and show that it is indeed possible to destroy their event horizon, giving rise to naked singularities, by pushing test particles towards the black hole as, in fact, it has been demonstrated explicitly by several recent works. Our analysis allows us to go a step further and to determine the optimal values, in the sense of …
We analyze in detail the highly damped quasinormal modes of $D$-dimensional extremal Reissner-Nordstr$\ddot{\rm{o}}$m and Reissner-Nordstr$\ddot{\rm{o}}$m-de Sitter black holes. We only consider the extremal case where the event horizon and the Cauchy inner horizon coincide. We show that, even though the topology of the Stokes/anti-Stokes lines in the extremal case is different than the non-extremal case, the highly damped quasinormal mode frequencies of extremal black holes match exactly with the extremal limit of the non-extremal black hole quasinormal mode frequencies.
In this work we investigate if the interaction of the Kerr-Taub-NUT spacetime with test scalar and neutrino fields can lead to the destruction of the event horizon. It turns out that both extremal and nearly extremal black holes can be destroyed by scalar and neutrino fields if the initial angular momentum of the spacetime is sufficiently large relative to its mass and NUT charge. This is the first example in which a classical field satisfying the null energy condition can actually destroy an extremal black hole. For scalar fields, the modes that can lead to the destruction of the horizon are …
We study the thermodynamics and the different thermodynamic geometric methods of Reissener-Nordström-de Sitter black hole and its extremal case, which is similar to the de Sitter black hole coupled to a scalar field, rather called an MTZ black hole. While studying the thermodynamics of the systems, we could find some abnormalities. In both cases, the thermodynamic geometric methods could give the correct explanation for the all abnormal thermodynamic behaviors in the system.
Destroying the Kerr Newman MOG Black hole with Scalar Test Field
本文通过标量测试场与 KN-MOG-BH 相互作用检验弱宇宙监督猜想，发现忽略反作用时标量场可使极端和近极端黑洞过度旋转形成裸奇点，而纳入反作用后仅能阻止极端黑洞的过度旋转。
The quasinormal-mode spectrum of the Schwarzschild-de Sitter black hole is studied in the limit of near-equal black-hole and cosmological radii. It is found that the mode_frequencies_ agree with the P"oschl-Teller approximation to one more order than previously realized, even though the effective_potential_ does not. Whether the spectrum approaches the limiting one uniformly in the mode index is seen to depend on the chosen units (to the order investigated). A perturbation framework is set up, in which these issues can be studied to higher order in future.
Unlike the extremal Reissner - Nordström black hole in ordinary spacetime, the one in anti-de Sitter spacetime is a minimum of action and has zero entropy if quantization is carried out after extremalization. However, if extremalization is carried out after quantization, then the entropy is a quarter of the area as in the usual case.
In this paper we have studied a black hole in de Sitter space which has a conformally coupled scalar field in the background. This black hole is also known as the $MTZ$ black hole. We have obtained exact values for the quasi normal mode frequencies under massless scalar field perturbations. We have demonstrated that when the black hole is near-extremal, that the wave equation for the massless scalar field simplifies to a Schr$\ddot{o}$dinger type equation with the well known P$\ddot{o}$shler-Teller potential. We have also used 6th order WKB approximation to compute quasinormal mode frequencies…
It is shown that the Kerr black hole could act as accelerators for spinning particles. In principle, it could obtain arbitrarily high energy for extremal Kerr black hole. In this paper, we extend the previous research to the Kerr-(anti-)de Sitter background and find that the cosmological constant plays an important role in the result. For the case of Kerr-anti-de Sitter black holes, like the Kerr background, only the extremal Kerr-anti-de Sitter black holes can have a divergent center-of-mass energy of collision. While, for the case of Kerr-de Sitter black holes, the collision of two spinning …
As in the grand canonical treatment of Reissner - Nordström black holes in anti-de Sitter spacetime, the canonical ensemble formulation also shows that non-extremal black holes tend to have lower action than extremal ones. However, some small non-extremal black holes have higher action, leading to the possibility of transitions between non-extremal and extremal black holes.
The destruction of the event horizon of a nonsingular black hole, which is not prevented by the weak cosmic censorship conjecture, might provide us the possibility to access quantum regime of gravity inside black hole. We investigate the possibility of overspinning a nonsingular rotating quantum-corrected black hole by a test particle and a scalar field in this paper, and analyse the effect of the quantum parameter on the destruction of the event horizon. For the test particle injection, both extremal and near-extremal black holes cannot be overspun due to the existence of the quantum paramete…
We first calculate equations of motion for particles in the Kerr-Sen-de Sitter black hole spacetime. Then in the eikonal regime, we analytically obtain the quasi-normal resonant modes of massless neutral scalar field perturbation and find its imaginary part to be characterized by the surface gravity of the near-extremal Kerr-Sen-de Sitter black hole with Cauchy horizon approaching the event horizon. We further show that the Penrose strong cosmic censorship conjecture is thus respected in this spacetime with dilaton scalar field and axion pseudoscalar field.
We calculate high-order quasinormal modes with large imaginary frequencies for electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, the analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1κ_1$ is small enough, where $r_1$ and $κ_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, to which corresponds quasin…
It has been shown previously that an extremal Reissner-Nordström or an extremal Kerr black hole cannot be overcharged or overspun by a test particle if radiative and self-force effects are neglected. In this paper, we consider extremal charged and rotating black holes with cosmological constants. By studying the motion of test particles, we find the following results: An extremal Reissner-Nordström anti-de Sitter (RN-AdS) black hole can be overcharged by a test particle but an extremal Reissner-Nordström de Sitter (RN-dS) black hole cannot be overcharged. We also show that both extrmal Kerr-de…
Gedanken Experiments to Destroy a Black Hole by a Test Particle: Multiply Charged Black Hole with Higher Derivative Corrections
本文通过思想实验证明，即使存在高阶导数修正，多电荷极端黑洞也不会被测试粒子摧毁，从而验证了弱宇宙监督猜想。
Motivated by Maggiore's new interpretation of quasinormal modes, starting from the first law of thermodynamics of black holes, we investigate area spectra of a near extremal Schwarzschild de sitter black hole and a higher dimensional near extremal Reissner-Nordstrom de sitter black hole. We show that the area spectra of all these black holes are equally spaced and irrelevant to the parameters of black holes.
Horizon structure and extremal configurations of Kerr-Newman-anti-de Sitter black holes in f(R) gravity
利用广义相对论中Kerr-Newman-(A)dS黑洞解与f(R)引力中常曲率对应解之间的对应关系，采用统一框架研究这些黑洞的视界结构和极端构型，主要关注反德西特情形，通过求解关于旋转参数平方a^2和逆曲率尺度l^{-2}的极端条件，得到闭合解析关系，确定了物理允许的极端区域并导出了电荷上限q_max=M/2，同时发现无电荷时存在最小旋转a_min=3√3M/8，而电荷的引入降低了最小旋转并扩展了极端分支。
Quasinormal modes of near-extremal Reissner-Nordström-de Sitter spacetimes
本文研究了近极端Reissner-Nordström-de Sitter黑洞上Klein-Gordon方程的准正规模式，定位了所有大小为$\mathcal{O}(κ_{\rm C})$的QNMs，并发现它们可由极端极限的近视界几何的QNMs近似。
We present an exact expression for the quasinormal modes of scalar, electromagnetic and gravitational perturbations of a near extremal Scwarzschild-de Sitter black hole and we show why a previous approximation holds exactly in this near extremal regime. In particular, our results give the asymptotic behavior of the quasinormal frequencies for highly damped modes, which as recently attracted much attention due to the proposed identification of its real part with the Barbero-Immirzi parameter.
We study extremal surfaces in de Sitter space in the Poincare slicing in the upper patch, anchored on spatial subregions at the future boundary ${\cal I}^+$, restricted to constant boundary Euclidean time slices (focussing on strip subregions). We find real extremal surfaces of minimal area as the boundaries of past lightcone wedges of the subregions in question: these are null surfaces with vanishing area. We also find complex extremal surfaces as complex extrema of the area functional, and the area is not always real-valued. In $dS_4$ the area is real. The area has structural resemblance wit…
Overcharging a nonsingular black hole in general relativity: the nonlinear electrodynamic field effects
本研究探讨ABG非奇异黑洞在非线性电动力学效应下可被带电粒子或标量场摧毁事件视界，与弱宇宙监督猜想一致，并可能用于探索黑洞内部。
The goal of this paper is to extend the particle-like quantization scheme presented in Pacheco and Silk (2020), to extremal Kerr-de Sitter black holes in four spacetime dimensions, thereby obtaining various quantized parameters, like the black hole mass and angular momentum, consistent with existing results, in the proper limits. Moreover, we show numerically, that for such extremal quantum black holes, there is a root mean square deviation from the First Law of black hole thermodynamics, of the order $\mathcal{O}(Λ^{0.45 \pm 0.01})$, where $Λ$ denotes the Cosmological Constant.
We investigate effect of a test magnetic field on the process of destroying near-extremal Kerr black hole with a charged test particle. It has been shown that it would be possible to throw a charged test particle into the near extremal rotating black hole and make it go past the extremality i.e. turn Kerr black hole into the Kerr-Newmann naked singularity. Typically in an astrophysical scenario black holes are believed to be surrounded by a magnetic field. Magnetic field although small, affects motion of charged particles drastically due to the large Lorentz force, as the electromagnetic force…
We study the instability of near extremal and extremal four-dimensional AdS charged hairy black hole to radial neutral massive and charged massless scalar field perturbations. We solve the scalar field equation by using the improved asymptotic iteration method and the time domain analysis and we find the quasinormal frequencies. For the charged scalar perturbations, we find the superradiance condition by computing the reflection coefficient in the low-frequency limit and we show that in the superradiance regime, which depends on the scalar hair charge, all modes of radial charged massless pert…
It has been shown that the quasinormal modes of perturebated fields can be used to investigate the validity of strong cosmic censorship (SCC). Relevant issues for Reissner-Nordstrom-de Sitter (RNdS) black holes and Born-Infeld-de Sitter (BI-dS) black holes have been discussed. In this paper, we investigate SCC in an asymptotic RN-dS black hole with logarithmic nonlinear electromagnetic field perturbed by massless scalar fields. It has been argued that SCC can be violated in a near-extremal RN-dS black hole. However, we find that the NLED effect can rescue SCC for a near-extremal logarithmic-de…
Near-extremal asymptotics and strong cosmic censorship for black holes immersed in a Chaplygin-like dark fluid
本文研究类恰普林 dark 流体中黑洞的无质量标量扰动的强宇宙审查，推导相关视界边界与近极值渐近行为，确定谱隙并分析强宇宙审查的潜在违反区域。
We show that near-horizon geometries in the presence of a positive cosmological constant cannot exist with ring topology. In particular, de Sitter black rings with vanishing surface gravity do not exist. Our result relies on a known mathematical theorem which is a straightforward consequence of a type of energy condition for a modified Ricci tensor, similar to the curvature-dimension conditions for the $m$-Bakry-Émery-Ricci tensor.
Uniqueness of extremal charged black holes in de Sitter
本文证明了四维带正宇宙常数的Einstein-Maxwell理论中极值带电黑洞（如Nariai和超冷黑洞）的唯一性定理，表明满足特定条件的解析解必局部等距于极值Reissner-Nordström-de Sitter黑洞或其近视界几何。
Overcharging an accelerating Reissner-Nordström-Anti-de Sitter black hole with test field and particle
本文通过带电场散射和带电粒子吸收，研究加速RN-AdS黑洞的弱宇宙监督猜想，发现极端黑洞不可过度充电，近极端黑洞在特定条件下可被过度充电，并指出需考虑二阶效应。
We consider gedanken experiments to destroy an extremal or near-extremal BTZ black hole by throwing matter into the horizon. These black holes are vacuum solutions to (2+1)-dimensional gravity theories, and are asymptotically $\mathrm{AdS}_3$. Provided the null energy condition for the falling matter, we prove the following---(i) in a Mielke-Baekler model without ghost fields, when torsion is present, an extremal BTZ black hole can be overspun and becomes a naked conical singularity; (ii) in 3-dimensional Einstein gravity and chiral gravity, which both live in torsionless limits of Mielke-Baek…
We present the Fermi story of strong cosmic censorship in the near-extremal Reissner-Nordstrom-de Sitter black hole. To this end, we first derive from scratch the criterion for the quasi-normal modes of Dirac field to violate strong cosmic censorship in such a background, which turns out to be exactly the same as those for Bose fields, although the involved energy momentum tensor is qualitatively different from that for Bose fields. Then to extract the low-lying quasi-normal modes by Prony method, we apply Crank-Nicolson method to evolve our Dirac field in the double null coordinates. As a res…
Quantum evolution of de Sitter black holes near extremality
本文利用耦合 de Sitter 量子场论的一维 Schwarzian 理论，计算近冷极端带电 de Sitter 黑洞经无质量标量粒子的量子能量转移，揭示其趋向宇宙学视界热平衡且偏离 Hawking 预测。
In this paper, we investigate minimal length effects on the thermodynamics and weak cosmic censorship conjecture in a RN-AdS black bole via charged particle absorption. We first use the generalized uncertainty principle (GUP) to investigate the minimal length effect on the Hamilton-Jacobi equation. After the deformed Hamilton-Jacobi equation is derived, we use it to study the variations of the thermodynamic quantities of a RN-Ads black hole via absorbing a charged practice. Furthermore, we check the second law of thermodynamics and the weak cosmic censorship conjecture in two phase spaces. In …
We rederive and extend the amplitude for charged spherical shells tunneling through the outer horizon of charged black holes. In particular, we explicitly confirm that an effective action approach with natural initial conditions for a spherical shell, including backreaction, reduces to the tunneling integral. Consequently, we establish a universal expression for the probability of emission in terms of the change in the horizon entropy. Notably, the result for the charged black hole also captures the superradiant regime of charged particle decay at low energies. We then explore an appropriately…
Logarithmic corrections to near-extremal entropy of charged de Sitter black holes
研究了带电德西特黑洞极值熵的对数修正，通过计算路径积分框架中的量子修正，揭示了冷和Nariai极限下熵修正的普遍性。
Challenging event horizons with spin (3/2) fields
本文研究用无质量自旋(3/2)场扰动克尔黑洞，发现小幅扰动可摧毁极端黑洞事件视界并将近极端黑洞推向超越极端状态，从而违反宇宙监督猜想和黑洞动力学第三定律，原因在于费米子场不满足零能量条件。
Uniqueness of the extremal Schwarzschild de Sitter spacetime
本文证明四维及更高维中含最大对称紧致截面的静态极值 Killing 视界的正宇宙学常数解析真空时空必为极值 Schwarzschild de Sitter 或 Nariai 解，并讨论极值双曲 Schwarzschild anti-de Sitter 的谱条件唯一性。
The nature of scalar wave functions near the horizon of Reissner Nordstrom (RN) extremal and Schwarzschild-de Sitter (SdS) extremal black holes are found using WKB approximation and the effect of reflection of waves from the horizon. The absorption cross section $σ_{abs}$ when RN extremal and SdS extremal black holes placed in a Klein-Gordon field is calculated.
A possible process to destroy a black hole consists on throwing point particles with sufficiently large angular momentum into the black hole. In the case of Kerr black holes, it was shown by Wald that particles with dangerously large angular momentum are simply not captured by the hole, and thus the event horizon is not destroyed. Here we reconsider this gedanken experiment for a variety of black hole geometries, from black holes in higher dimensions to black rings. We show that this particular way of destroying a black hole does not succeed and that Cosmic Censorship is preserved.
The quasinormal resonant modes of massless neutral fields in near-extremal Kerr-Newman-de Sitter black-hole spacetimes are calculated in the eikonal regime. It is explicitly proved that, in the angular momentum regime ${\bar a}> \sqrt{{1-2{\barΛ}\over{4+{\barΛ}/3}}}$, the black-hole spacetimes are characterized by slowly decaying resonant modes which are described by the compact formula $\Imω(n)=κ_+\cdot(n+{1\over2})$ [here the physical parameters $\{{\bar a},κ_+,{\barΛ},n\}$ are respectively the dimensionless angular momentum of the black hole, its characteristic surface gravity, the dimensio…
It has recently been shown that the strong cosmic censorship conjecture can be violated by the massless neutral scalar field in the nearly extremal Reissner-Nordstrom-de Sitter black hole. However, the formation of such a black hole by gravitational collapse necessitates the presence of the charged sector on top of the Einstein-Maxwell system. Thus we numerically calculate the quasi-normal modes for a massless charged scalar field in the Reissner-Nordstrom-de Sitter spacetime by generalizing the characteristic formulation to the charged case. As a result, the strong cosmic censorship turns out…
Non-extremal Island in de Sitter Gravity
针对传统岛公式在dS引力中无法给出合理极值岛的问题，本文构建dS$_2$膜嵌入AdS$_3$体的双全息模型，提出边界位于引力区边缘的非极值岛方案。
We investigate the weak cosmic censorship conjecture in extremal and near-extremal Kerr-Newman-(anti-)de Sitter black holes by the scattering of a massive scalar field with an electric charge. Under this scattering, the scalar field fluxes change the black hole state, as determined by the mass, angular momentum, and electric charge. The black hole may exceed its extremal condition because of these changes. However, we find that the black hole cannot be overcharged or overspun by the scattering. In particular, although the fluxes are closely associated with the asymptotic boundary conditions al…
We investigate the weak cosmic censorship conjecture in Kerr-(anti-)de Sitter black holes under the scattering of a scalar field. We test the conjecture in terms of whether the black hole can exceed the extremal condition with respect to its change caused by the energy and angular momentum fluxes of the scalar field. Without imposing the laws of thermodynamics, we prove that the conjecture is valid in all the initial states of the black hole (non-extremal, near-extremal, and extremal black holes). The validity in the case of the near-extremal black hole is different from the results of similar…
Extremal Black Hole Decay in de Sitter Space
通过隧穿方法研究de Sitter时空中带电黑洞衰变，发现衰变概率与黑洞熵变相关，并指出考虑反作用后存在违反Festina Lente界限但不导致大挤压的衰变通道。
Near--extremal gravitational collapse in 4+1 dimensions: Schwarzschild--de--Sitter space
研究四维加一维空间中引力波坍缩形成近极值视界的过程，发现黑洞质量接近极值，并探讨极值黑洞热力学第三定律在宇宙学背景下的有效性。
The linearized dynamics of charged massive scalar fields in the near-extremal charged Reissner-Nordström-de Sitter (RNdS) black-hole spacetime is studied analytically. Interestingly, it is proved that the non-asymptotically flat charged black-hole-field system is characterized by {\it unstable} (exponentially growing in time) complex resonant modes. Using a WKB analysis in the eikonal large-mass regime $Mμ\gg1$, we provide a remarkably compact analytical formula for the quasinormal resonant spectrum $\{ω_n(M,Q,Λ,μ,q)\}_{n=0}^{n=\infty}$ which characterizes the unstable modes of the composed RN…
Thermodynamic relation of accelerating black holes in anti-de Sitter spacetime
本文研究四维反德西特时空中的加速黑洞，验证了Goon和Penco普适极端性关系对其成立，并指出需在微扰参数与锥形缺陷/弦张力参数间建立匹配条件，其行为类似弱引力猜想。
In our earlier work (PRL 103 (2009) 161101) it was shown that nonextremal highly charged Reissner-Nordstrom-de Sitter black holes are gravitationally unstable in D>6-dimensional space-times. Here, we find accurate threshold values of the $Λ$-term at which the instability of the extremally charged black holes starts. The larger $D$ is, the smaller is the threshold value of $Λ$. We have shown that the ratio $ρ= r_{h}/r_{cos}$ (where $r_{cos}$ and $r_{h}$ are the cosmological and event horizons) is proportional to $e^{-(D-4)/2}$ at the onset of instability for D=7,8,...11, implying that the same …
Destroying the Event Horizon of Cold Dark Matter-Black Hole System
该研究针对被暗物质包裹的旋转类Kerr黑洞，通过引入测试粒子和标量场验证弱宇宙监督猜想，发现极端与近极端情形下二者均能破坏事件视界，且无冷暗物质时结论与前人研究一致。
We investigate the weak cosmic censorship conjecture in charged anti-de Sitter black holes with thermodynamic pressure and volume by the scattering of the charged scalar field in four and higher dimensions. We assume that the internal energy and electric charge of the black hole changes infinitesimally according to the energy and charge fluxes of the scalar field. Then, found to be an isobaric process, the changes in the black hole can be well reproduced to the first law of thermodynamics, although we find that the second law of thermodynamics is violated in extremal and near-extremal black ho…
It is known that the extremal Reissner-Nordström black hole possesses a discrete conformal isometry that exchanges the black hole horizon with infinity. It is also known that the Reissner-Nordström-de Sitter spacetime posseses a similar discrete conformal isometry which exchanges the event horizon of the black hole with the cosmological horizon. In this short paper we will continue this line of thought and extend the conformal isometry of the latter spacetime and give an unphysical interpretation of the negative root.
It was recently shown that Strong Cosmic Censorship might be violated for near-extremally-charged black holes in 4-dimensional de Sitter space under scalar perturbations. Here, we extend the study of neutral massless scalar perturbations in higher dimensions and discuss the dimensional influence on the validity of Strong Cosmic Censorship hypothesis. By giving an elaborate description of neutral massless scalar perturbations of Reissner-Nordström-de Sitter black holes in $d=4,5$ and $6$ dimensions we conclude that Strong Cosmic Censorship is violated near extremality.
Overspinning problem in Kerr black holes: second order corrections and self-energy
针对通过吸收高能高角动量物质摧毁克尔黑洞的思想实验，研究发现考虑对最终质量的二阶贡献且物质满足零能量条件时，极端和近极端克尔黑洞无法被摧毁，该二阶贡献经事件视界超曲面积分计算，与过往提出的相关作用项和自能下界相似。
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