# Wald测试粒子近似方法在弱的宇宙监督假设中的应用Application of Wald Test Particle Approximation Method in the Weak Cosmic Censorship Conjecture

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The naked singularities will break down all the laws of physics. Roger Penrose conceived the Cosmic Censorship Conjecture (CCC) to avoid the naked singularities. A gedanken experiment was given by Wald in 1974 to investigate the CCC. We reviewed the Wald’s gedanken experiment and introduced the research progress in recent years.

1. 引言

2. Wald测试粒子近似方法介绍

Kerr-Newman [12] 黑洞的线元为

$\begin{array}{c}\text{d}{s}^{2}=-\left(1-\frac{2Mr-{Q}^{2}}{{\rho }^{2}}\right)\text{d}{t}^{2}+\frac{{\rho }^{2}}{\Delta }\text{d}{r}^{2}+{\rho }^{2}\text{d}{\theta }^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[\left({r}^{2}+{a}^{2}\right){\mathrm{sin}}^{2}\theta +\frac{\left(2Mr-{Q}^{2}\right){a}^{2}{\mathrm{sin}}^{4}\theta }{{\rho }^{2}}\right]\text{d}{\varphi }^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{2\left(2Mr-{Q}^{2}\right)a{\mathrm{sin}}^{2}\theta }{{\rho }^{2}}\text{d}t\text{d}\varphi \end{array}$ (1)

${r}_{+}=M+\sqrt{{M}^{2}-{Q}^{2}-{a}^{2}}$ (2)

${M}^{2}={Q}^{2}+{a}^{2}$ (3)

Kerr-Newman黑洞拥有事件视界必须要满足条件

${M}^{2}\ge {Q}^{2}+{a}^{2}$ (4)

${M}^{2}<{Q}^{2}+{a}^{2}$ (5)

${\left(M+E\right)}^{2}<{\left(\frac{aM+L}{M+E}\right)}^{2}+{\left(q+Q\right)}^{2}$ (6)

$E<\frac{QqM+aL}{{M}^{2}+{a}^{2}}$ (7)

$E=\frac{{g}_{t\varphi }}{{g}_{\varphi \varphi }}\left(q{A}_{\varphi }-L\right)-q{A}_{t}+\sqrt{\left(\frac{{g}_{t\varphi }^{2}-{g}_{\varphi \varphi }{g}_{tt}}{{g}_{\varphi \varphi }^{2}}\right)\left({\left(L-q{A}_{\varphi }\right)}^{2}+{m}^{2}{g}_{\varphi \varphi }\left(1+{g}_{rr}{\stackrel{˙}{r}}^{2}+{g}_{\theta \theta }{\stackrel{˙}{\theta }}^{2}\right)\right)}\text{\hspace{0.17em}}$ (8)

$E>\frac{{g}_{t\varphi }}{{g}_{\varphi \varphi }}\left(q{A}_{\varphi }-L\right)-q{A}_{t}$ (9)

${E}_{\mathrm{min}} (10)

3. Wald测试粒子方法研究综述

3.1. 考虑高阶修正后极端Kerr-Newman黑洞的视界破坏

Wald假想试验发现极端的Kerr-Newman黑洞通过捕获一个测试粒子是无法出现裸奇点的，Wald在计算时对粒子一些参数作了线性近似。后来有研究者指出 [13] ，在不作线性近似的情况下，测试粒子是可以破坏一个极端的Kerr-Newman黑洞的，其所允许的破坏黑洞的粒子的能量范围很窄。定义新参量W

$W={\left(M+E\right)}^{2}$ (11)

${W}^{2}-{\left(Q+q\right)}^{2}-{\left(aM+L\right)}^{2}<0$ (12)

${W}_{1} (13)

${W}_{1,2}=\frac{{\left(Q+q\right)}^{2}±\sqrt{{\left(Q+q\right)}^{4}+4{\left(aM+L\right)}^{2}}}{2}$ (14)

$\begin{array}{r}\hfill W>{\left(\frac{aL+qQM}{{a}^{2}+{M}^{2}}+M\right)}^{2}\equiv {W}_{3}\end{array}$ (15)

$s\equiv {W}_{2}-{W}_{3}=\frac{{\left(Q+q\right)}^{2}+\sqrt{{\left(Q+q\right)}^{4}+4{\left(aM+L\right)}^{2}}}{2}-{\left(\frac{aL+qQM}{{a}^{2}+{M}^{2}}+M\right)}^{2}>0$ (16)

3.2. 近极端RN黑洞视界破坏

Hubeny指出 [14] ，通过向一个近极端的RN黑洞投入一个带电的粒子，黑洞的视界可以被破坏。其中粒子的最小能量E

$E>\frac{qQ}{{r}_{+}}$ (17)

${r}_{+}=M+\sqrt{{M}^{2}-{Q}^{2}}$ (18)

$\begin{array}{r}\hfill E (19)

$q>{r}_{+}\left(\frac{M-Q}{{r}_{+}-Q}\right)=\frac{{r}_{+}-Q}{2}$ (20)

$m (21)

3.3. 近极端Kerr黑洞视界面的破坏

Jacobson和Sotiriou研究发现 [15] ，一个近极端的Kerr黑洞视界可以通过捕获一个有角动量的粒子被破坏。文章中定义测试粒子带有 $\delta E$ 的能量和 $\delta J$ 的角动量。为了使测试粒子能够掉入黑洞，其角动量必须满足条件

$\begin{array}{r}\hfill \delta J>\delta {J}_{\mathrm{min}}=\left({M}^{2}-J\right)+2M\delta E+{\left(\delta E\right)}^{2}\end{array}$ (22)

$\begin{array}{r}\hfill \delta J<\delta {J}_{\mathrm{max}}=\frac{2M{r}_{+}}{a}\delta E\end{array}$ (23)

3.4. 近极端Kerr-Newman黑洞视界面的破坏

$E<{E}_{\mathrm{max}}=\frac{QeM+aL}{{M}^{2}+{a}^{2}}-\frac{{M}^{3}}{2\left({M}^{2}+{a}^{2}\right)}{\left(\frac{\delta }{M}\right)}^{2}$ (24)

$E\ge =A-B\left(\frac{\delta }{M}\right)-\left(\frac{\left(2+{\mathrm{sin}}^{2}\alpha \right)A-4B}{2+2{\mathrm{cos}}^{2}\alpha }\right){\left(\frac{\delta }{M}\right)}^{2}$ (25)

$A=\frac{\left(L/M\right)\mathrm{cos}\alpha +e\mathrm{sin}\alpha }{1+{\mathrm{cos}}^{2}\alpha }\ge 0$ (26)

$B=\frac{\left(L/M\right)\mathrm{cos}\alpha +e{\mathrm{sin}}^{3}\alpha }{{\left(1+{\mathrm{cos}}^{2}\alpha \right)}^{2}}\ge 0$ (27)

$a=\sqrt{{M}^{2}-{\delta }^{2}}\mathrm{cos}\alpha$ (28)

$Q=\sqrt{{M}^{2}-{\delta }^{2}}\mathrm{sin}\alpha$ (29)

3.5. 更多的研究进展

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