一般相対性理論に関するリーマン計量の変形について

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1 ( ) 1 $(N^{4}, g)$ $N$ 4 $g$ $(3, 1)$ $R_{ab}- \frac{1}{2}rg_{ab}=t_{ab}$ (1) $R_{ab}$ $g$ $R$ $g$ ( ) $T_{ab}$ $T$ $R_{ab}- \frac{1}{2}rg_{ab}=0$ 4 $R_{ab}=0$ $\mathbb{r}^{3,1}$ ( ) (trivial) $T_{ab}=-(F_{ac}F_{b}^{c}+ \frac{1}{4}f_{cd}f^{cd}g_{ab})$ $F$ 2 $divf=\mathcal{j}, df=0$ $\mathcal{j}$ 4 $(\rho,j)$

2 138 Hilbert-Einstein $\mathcal{h}(g)=\int_{n}\{r_{g}+l\}d\mu_{g}$ $L$ ( ) $L$ $F^{ab}F_{ab}$ 2 $g=-dt^{2}+\delta, N^{4}=\mathbb{R}\cross \mathbb{r}^{3}$ $\delta$ $\mathbb{r}^{3}$ $dx^{2}+dy^{2}+dz^{2}$ Schwarzschild (1916) $g=-v^{2}dt^{2}+u^{4}\delta, N^{4}=\mathbb{R}\cross(\mathbb{R}^{3}\backslash B)$ $B=B_{m/2}(0)$ $v= \frac{1-m/2r}{1+m/2r},$ $u=1$ $\frac{m}{2r}$ $m$ $m\geq 0$ (Causality) $r=m/2$ 4 ( ) Reissner-Nordstr\"om ( )(1918) $g=-v^{2}dt^{2}+u^{4}\delta, N^{4}=\mathbb{R}x(\mathbb{R}^{3}\backslash B)$

3 $\mathbb{r}$ 139 $B=B\sqrt{m^{2}-q^{2}}/2(0)$ $v= \frac{1-(m^{2}-q^{2})/4r^{2}}{1+m/r+(m^{2}-q^{2})/4r^{2}}, u=\sqrt{1+\frac{m}{r}+\frac{m^{2}-q^{2}}{4r^{2}}}$ $E=u^{-6} \nabla(\frac{q}{r}), B\equiv 0.$ $\nabla$ gradient (Causality) $m\geq$ $ q $ $q=0$ Reissner-Nordstr\"om Schwarzschild Maj umdar-papapetrou (1947) $g=-u^{2}dt^{2}+u^{-2}\delta N^{4}=\mathbb{R}\cross(\mathbb{R}^{3}\backslash \bigcup_{i=1}^{n}\{p_{i}\})$, $u=(1+ \sum_{i=1}^{n}m_{i}/r_{i})^{-1} E=\nabla\log u, B\equiv 0.$ $m_{i}>0$ $r_{i}$ $N=1$ (extremal Reissner-Nordstr\"om solution) $m= q $ Schwarzschild Reissner-Nordstr\"om Majumdar-Papapetrou $(N^{4}, g)$ $(\mathbb{r}\cross\sigma^{3}, -dt^{2}+g^{(3)})$ $(\Sigma^{3}, g^{(3)})$ 3 Theorem 1 (Chrusciel, Heusler, Bunting, Masood-ul-Alam, Tod, [4] ). Reissner-Nordstr\"om Majumdar-Papapetrou $ds^{2}=-v^{2}dt^{2}+g, A=\phi dt$

4 140 $R_{ij} = V^{-1}\nabla_{i}\nabla_{j}V-2V^{2}\nabla_{a}\phi\nabla_{b}\phi+V^{-2} \nabla\phi ^{2}g_{ij}$ $\triangle_{g}v = V^{-1} \phi ^{2}$ $\triangle_{g}\phi = V^{-1}\nabla_{a}\nabla^{a}\phi$ $\Sigma^{(3)}=$ { $t=$ const. } $(i=$ $1,2,3)$ $V>0$ $Varrow 1$ as $xarrow\infty$ and $V _{\partial\sigma}=0$ $V$ 3 $(M, g, k, E, B)$ $R_{ab}- \frac{1}{2}rg_{ab}=t_{ab}$ $\mu =T(N, N)=\frac{1}{2}(R^{(3)}+(trk)^{2}- k ^{2})$ $J =T(N, \cdot)=div(k-(trk)g)$, $dive=o$, div$b=0$ $N$ $M$ $J$ $\mu$ $M$ 2 $E$ $B$ $\mu=0tr^{2}$ (Gauss eqn), $J=0$ tr(codazzi $eqn$) $\mu\geq J+2E\cross B + E ^{2}+ B ^{2}$ $M$ $N^{4}$ $k=0$ $B\equiv 0$ $R^{(3)}\geq 2\Vert E\Vert_{g}^{2}.$

5 141 $M^{3}$ $R^{(3)}$ 3 $g$ $(M, g, E)$ Theorem 2 (Choquet-Bruhat Geroch [12] ). $(M, g, k)$ $(M^{3},g)$ $k$ $(N^{4}, g)$ 2 $N^{4}\cong^{diff}$ $M^{3}\cross(-\epsilon, \epsilon)$ Corollary 3. $(M, g, k, E, B)$ $(M^{3}, g)$ $k$ $(N^{4}, g)$ $N$ $M$ $E$ $B$ $F$ Theorem 4 ( (Bonnet)). $n$ $g$ $k$ $(M, g, k)$ $\{\begin{array}{ll}r_{ijkl}=k_{ij}k_{kl}-k_{il}k_{jk} (Gauss equation)d_{i}k_{jk}-d_{j}k_{ik}=0 (Codazzi equation).\end{array}$ $g$ $k$ $\iota$ 1 2 $M^{n}arrow \mathbb{r}^{n+1}$ ( )

6 142 ( ) Gauss Codazzi $(M, g, E)$ $K\subset M$ $M\backslash K$ $\mathbb{r}^{3}\backslash B_{1}(0)$ $\mathbb{r}^{3}$ $g$ $E$ $g_{ij}-\delta_{ij}=o_{1}(r^{-1}), E=O(r^{-2})$. ( ) ADM $m= \frac{1}{16\pi}\lim_{rarrow\infty}\int_{s_{r}}(g_{ij,i}-g_{ii,j})v^{j}$ $q_{e}= \frac{1}{4\pi}\lim_{rarrow\infty}\int_{s_{r}}e\cdot\nu, q_{b}=\frac{1}{4\pi}\lim_{rarrow\infty}\int_{s_{r}}b\cdot\nu.$ 4 $N^{4}=\mathbb{R}\cross(\mathbb{R}^{3}\backslash \{O\})$ Schwarzschild $g=-v^{2}dt^{2}+u^{4}\delta, v=\frac{1-m/2r}{1+m/2r}, u=1+\frac{m}{2r}$ $($ $r=\sqrt{x^{2}+y^{2}+z^{2}})$ $( \mathbb{r}^{3}\backslash \{r=0\}, (1+\frac{m}{2r})^{4}\delta)$ $\{r=$ $\}$2 $A(r)$ $A(r)=4 \pi r^{2}(1+\frac{m}{2r})^{4}$

7 143 $A(r)$ $r=m/2$ Schwarzschild $\{t=$ $\mathbb{r}^{4}=\{(x, y, z, w)\}$ $\}$ Flamm Paraboloid 4 3 $r= \frac{1}{2m}w^{2}+\frac{m}{2} (r=\sqrt{x^{2}+y^{2}+z^{2}})$, $w=0$ $m/2$ ( 2 ) $(r, w)\mapsto(r, -w)$ $\}$ $\{t=$ $\mathbb{z}_{2}$ Schwarzschild $(M, g)$ ( ) ( $=$ ) 3 $M$ $M$ $(N$ $)$ ( 1 ) $\mathbb{r}^{3}\backslash \bigcup_{i=1}^{n}b^{3}$ $M^{3}$ (outermost) ( ) (cf. Flamm Paraboloid) $\Sigma^{2}$ $\Sigma$ $(M, g)$ $A$ $r_{0}=\sqrt{\frac{a}{4\pi}}$ $\Sigma$ (area radius) 5 Penrose ( ) ADM ([8] ) ADM Theorem 5 $(Schoen-Yau[10,11] ,$ Witten $[14] 1981)$. $(M, g)$ $(R_{g}\geq 0)$ $ADM$ $m$ \delta)$ $m=0$ $(M, g)$ $(\mathbb{r}^{3},

$$(M, g, E, B)$ $(R_{g}\geq 2\Vert E\Vert^{2})$ Dominant Energy Condition $m\geq\sqrt{q_{e}^{2}+q_{b}^{2}}$ 144 $(M,g, E, B)$ Majumdar-Papapetrou $\{t=$ $\}$ $ADM$ / Penrose $ADM$ Theorem 7 (Bray [1],$

8 Theorem 6 $($Gibbons, Hawking, Horowitz, Perry, $[3]1983)$. $(M, g, E, B)$ $(R_{g}\geq 2\Vert E\Vert^{2})$ Dominant Energy Condition $m\geq\sqrt{q_{e}^{2}+q_{b}^{2}}$ 144 $(M,g, E, B)$ Majumdar-Papapetrou $\{t=$ $\}$ $ADM$ / Penrose $ADM$ Theorem 7 (Bray [1], Huisken/Ilmanen [5] 2001). $(M, g)$ Dominant Energy Condition $(R_{g}\geq 0)$ $M$ $r_{0}$ $ADM$ m $m \geq\frac{r_{0}}{2},$ $(M,g)$ Schwarzschild $\{t=$ $\}$ Schwarzschild $(B\equiv 0)_{\backslash }$ Jang [6] (1979) Gibbons [2] (1983) pure electric Penrose Reissner$-$Nordstr \"om $m$ $r_{0}$ $q$ $\Sigma$ 1 $M$ Theorem 8 $(Jang [6], Huisken/$Ilmanen [$5], Khuri-$Disconzi). $(M, g, E)$ $(R_{g}\geq 2\Vert E\Vert^{2})$ $hg\not\in$ $M$ 1

9 145 m ro $q$ $(M, g, E)$ $ADM$ $m \geq\frac{1}{2}(r_{0}+\frac{q^{2}}{r_{0}})$ $(M, g, E)$ Reissner-Nordstr\"om $\{t=$ $\}$ Majumdar-Papapetrou Theorem 9 (Weinstein-Yamada [13] 2005). $(M, g, E)$ $m< \frac{1}{2}(r_{0}+\frac{q^{2}}{r_{0}})$ $q^{2}/r_{0})$ Jang [6] Gibbons [2] $m\geq 1/2(r_{0}+$ $m-\sqrt{m^{2}-q^{2}}\leq r_{0}\leq m+\sqrt{m^{2}-q^{2}}.$ Penrose $r_{0}\leq m+\sqrt{m^{2}-q^{2}}$ Weinstein-Yamada [13] $m-\sqrt{m^{2}-q^{2}}\leq r_{0}$ Theorem 10 (Khuri-Weinstein-Yamada [7]). $(M, g, E)$ $(R_{g}\geq 2\Vert E\Vert_{g}^{2})$ $(M, g, E)$ $ADM$ m $r_{0}$ $q$ $r_{0}\leq m+\sqrt{m^{2}-q^{2}}$ $(M, g, E)$ Reissner-Nordstr\"om $\{t=$ $\}$

10 146 $r_{0}\leq m+\sqrt{m^{2}-q^{2}}$ 2 $m\geq q $ if $r_{0}\leq q$ $m \geq\frac{1}{2}(r_{0}+\frac{q^{2}}{r_{0}})$ if $q<r_{0}$ Gibbons Hawking Horowitz Perry [3] $m\geq q $ $q$ $r_{0}$ $m \geq\frac{1}{2}(r_{0}+_{r_{0}}l^{2})$ $q<r_{0}$ $q<r_{0}$ $m \geq 0$ $m \geq \frac{1}{2}r_{0}$ $m$ $\geq$ $\frac{1}{2}(r_{0}+\frac{q^{2}}{r_{0}})$ if $r_{0}\geq q $ $m$ $\geq$ $ q $ if $r_{0}< q $ ( ) 5.1 $R^{3}\backslash \{0\}$ $g(t)=\mathcal{u}(t)^{4}(dr^{2}\cdot+r^{2}d\omega^{2})$. $\mathcal{u}(t)=e^{-t}+\frac{m}{2re^{-t}}.$

11 147 Schwarzschild $g(t)$ $\phi_{t}$ $r\mapsto r\exp 2t$ $\phi_{t}^{*}g(o)$ $u(t)=$ $\mathcal{u}(t)/\mathcal{u}(0)$ $g(t)=u(t)^{4}g(0)$ $xarrow\infty$ $v(r_{t})=0$ Dirichlet $varrow-1$ $\triangle_{g(t)}v(t)=0$ $t$ $r_{0}=m/2$ $u(t)= \exp\int_{0}^{t}v(\tau)d\tau$ $r_{t}=r_{0}e^{2t}$ Schwarzschild $\mathbb{z}_{2}$ Flamm $r=r_{0}$ Dirichlet $v$ $r=r_{0}$ $R^{3}\backslash \{0\}$ $R^{3}\backslash \{0\}$ $g(t)=\mathcal{u}(t)^{4}(dr^{2}+r^{2}d\omega^{2})$ $\mathcal{u}(t)=(e^{-t}+\frac{m+q}{2re^{-t}})^{1/2}(e^{-t}+\frac{m-q}{2re^{-t}})^{1/2}$ Reissner-Nordstr\"om $g(t)$ $\phi_{t}$ $r\mapsto r\exp 2t$ $\phi_{t}^{*}g(o)$ $u(t)=\mathcal{u}(t)/\mathcal{u}(o)$ $t$ $g(t)=u(t)^{4}g(0)$ $xarrow\infty$ $varrow-1$ $v(r_{t})=0$ Dirichlet $r_{t}$ $r_{0}e^{2t}$ $r_{0}=\sqrt{m^{2}-q^{2}}/2$ $\triangle_{g(t)}v(t)=\vert E\Vert_{g(t)}^{2}v(t)$ $u(t)= \exp\int_{0}^{t}v(\tau)d\tau.$ $u$ $v$ Schwarzschild Schwarzschild Reissner-Nordstr\"om $(M,g, E)$ Dirichlet $t$ 3 Reissner-Nordstr\"om Penrose

12 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$ 148 $\bullet$ $\partial M_{t}$ $ \partial M_{t} = \partial M $ $q_{t}=q$ $\frac{d}{dt}m_{t}\leq 0$ ( ) Reissner-Nordst\"om Maxwell $div_{g_{t}}e_{t}=0$, $\geq 2 E_{t} ^{2}$ $E=0$ HBray ( $\phi_{t}$ ) $m_{t}=m_{0}.$ $Reissner-Nordstr\ddot{o}m$ $r\mapsto r\exp 2t$ References [1] H. Bray. Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differential Geometry, , [2] G. Gibbons. The isoperimetric and Bogomolny inequalities for black holes. In T.J. Willmore and N. Hitchin, editors, Global Riemannian Geometry, pages John Wiley & Sons, New York, [3] G. W. Gibbons, S. W. Hawking, Gary T. Horowitz, and Malcolm J. Perry. Positive mass theorems for black holes. Comm. Math. Phys. 88(1983), [4] M. Heusler. Black Hole Uniqueness Theorems Cambridge Lecture Notes in Physics, [5] G. Huisken and T. Ilmanen. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geometw, , [6] P. Jang. Note on cosmic censorship. Phys. Rev. $D,$ $20(4) $, [7] M. Khuri, G. Weinstein and S. Yamada, The Riemannian Penrose Inequality with Charge for Multiple Black Holes, $\cdot 2013.$ arxiv ,

13 149 [8] M. Mars. Present status of the Penrose inequality. Class. Quant. Grav., , [9] R. Penrose. Naked singularities. Ann. New York Acad. Sci., , [10] R. Schoen and $S$.-T. Yau. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys., $65(1)45-76$, [11] R. Schoen and S-T. Yau. Proof of the positive mass theorem. II. Comm. Math. Phys., $79(2) $, [12] R. Wald. General Relativity, University of Chicago Press, [13] G. Weinstein and S. Yamada. On a Penrose inequahty with charge. Commun. Math. Phys., $257(3) $, [14] E. Witten. $A$ new proof of the positive energy theorem. Communications in Mathematical Physics, , $1007/BF $

ADM-Hamiltonian Cheeger-Gromov 3. Penrose

ADM-Hamiltonian Cheeger-Gromov 3. Penrose ADM-Hamiltonian 1. 2. Cheeger-Gromov 3. Penrose 0. ADM-Hamiltonian (M 4, h) Einstein-Hilbert M 4 R h hdx L h = R h h δl h = 0 (Ric h ) αβ 1 2 R hg αβ = 0 (Σ 3, g ij ) (M 4, h ij ) g ij, k ij Σ π ij = g(k