流体とブラックホールの間に見られる類似性・双対性

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1822 2013 56-68 56 (MIYAMOTO, Umpei) Department of

1 (MIYAMOTO, Umpei) Department of Physics, Rikkyo University 1 : ( $)$ 1 [ 1: ( $BH$ )

2 $(r, \theta, \phi)$ $t$ 4 $(x^{a})_{a=0,1,2,3}:=$ $c$ $(ct, r, \theta, \phi)$ $x^{a}$ $x^{a}+dx^{a}$ $ds^{2}= \sum_{a,b=0}^{3}g_{ab}dx^{a}dx^{b}$ (1) $g_{ab}$ ( $\Sigma$ ) $G_{ab}+ \Lambda g_{ab}=\frac{8\pi G}{c^{4}}T_{ab}$ (2) $G_{ab}$ $A$ 2 $G$ $T_{ab}$ (2) (2) ( ) $T_{ab}=(\rho+P)u_{a}u_{b}+Pg_{ab}$ (3) $(u_{a}, \rho, P)$ $(T_{ab}=\Lambda=0)$ 4 ( ) 2.2 (2) $T_{ab}=\Lambda=0$ ( ) $d s^{2}=-f(r)c^{2}dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\omega_{2}^{2}, f(r)=1-\frac{r_{0}}{r}$ (4) $d\omega_{2}^{2}:=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ $M$ 2 $r_{0}$ $r_{0}= \frac{2gm}{c^{2}}$ (5)

3 $\kappa$ 58 $r_{0}=0$ $f(r)\equiv 1$ $(r,\theta, \phi)$ 3 $r_{0}>0$ $r<r_{0}$ $r=0$ ( $)$ $r=r_{0}$ $M$ $r$ $m$ $v$ ( $)$ $\frac{1}{2}mv^{2}-g\frac{mm}{r}=0$ (6) $v=v_{esc}:=\sqrt{\frac{2gm}{r}}$ (7) ( ) (7) $v_{e8c}arrow c$ $rarrow r_{0}=2gm/c^{2}$ $r$ $r_{0}$ $M$ $r_{0}$ $A=4\pi r_{0}^{2}$ $A$ $A+\delta A(\delta A\geq 0)$ $\delta(mc^{2})$ $\delta(mc^{2})=\frac{c^{2}}{8\pi G}\kappa\delta A$ (8) 2 $\kappa$ ( $\kappa=c^{2}/2r_{0}$ ) $Mc^{2}$ $U$ $T$ $A$ $S$ (8) $\delta U=T\delta S$ $i$ $r_{0}$ $\delta J$ 2 $J$ $Q$ (8) $\delta Q$

4 59 ( ) $T= \frac{\hslash}{2\pi ck_{b}}\kappa$ (9) ( ) $k_{b}$ 3 (8) ( ) $S= \frac{c^{3}k_{b}}{4g\hslash}a=\frac{k_{b}}{4\ell_{p}^{2}}a$ (10) $\ell_{p}:=\sqrt{g\hslash}/c^{3}$ $(\sim 10^{-33} cm)$ (10) 4 $(c, G, \hslash, k_{b})$ ( $e^{i\pi}=-1$ ) [ (4) $f(r) arrow 1-(\frac{r_{0}}{r})^{n},$ $d\omega_{2}^{2}arrow d\omega_{n+1}^{2},$ $n=1,2,3,$ $\ldots$ (11) 4 $r_{0}$ $(n+1)$ $S^{n+1}$ 5 $+dz^{2}$ $z$ $(n+4)$ $r_{0}$ $(n+1)$ $S^{n+1}$ $z$ ( $2[a]$ ) 3 $c=g=\hslash=k_{b}=1$ $c=g=\hslash=k_{b}=1$ 4 (2)

5 60 2: $[(a)arrow(b)arrow(c)arrow(d)$ $z$ $(r\cdot=0)$ $S^{n+1}$ $]$ $z=$ $(n+1)$ 3.2 (2) (Gregory-Laflamme) [1] $\delta g_{ab}\propto\exp(-i\omega t+ikz)$ (12) $\omega^{2}<0$ ( $\omega$ ) ( 3) $\lambda:=2\pi/k$ $L$ $L>\lambda_{c^{\backslash },}:=2\pi/k_{ }\sim r_{0}/\sqrt{n}$ $k(0<k<k_{c}\sim\sqrt{n}/r_{0})$ $L$ $(r_{0}\gg$ $r_{0}$ $L)$ $r_{0}\lessapprox L$ $r_{0}$ ( ) 5 6 $n\geq n_{c}:=10$ $[3]$ [5] (5 ) $[$ $2(c)]$ $[$ $2(d)]$ $[$ $2(b)]$ $r\cdot(t)\propto(t_{0}-t)$ ( $t_{0}$ ) 5 $-\cdot$ 6 [2]

6 61 3: $\delta g_{ab}\propto\exp(-i\omega t+ikz)$ $\omega(k)$ $0<k<k_{c}\sim\sqrt{n}/r_{0}$ $(-i\omega>0)$ 7 [ ( ) ( ) ( ) $[6]_{0}$ ( 2 ) $(n>n_{c})\sim$ [7] $P$ $v^{i}(i=$ $\rho$ $1,2,$ $\ldots,p)$ $\partial_{t}\rho+\partial_{i}(\rho v^{i})=0,$ $\partial_{t} (\rho v^{i})+\partial_{j}\pi^{ij}=0$ (13) 7 [1] ( )

7 $\nu$ $\zeta$ 62 ( ) $\Pi_{ij}=\Pi_{ij}^{(0)}+\Pi_{ij}^{(1)}$ $\Pi_{ij}^{(0)}=pv_{i}v_{j}+P\delta_{ij}$ $\Pi_{ij}^{(1)}=-\eta(\partial_{i}v_{j}+\partial_{j}v_{\dot{t}}-\frac{2}{p}\delta_{ij}\partial_{k}v^{k})-\zeta\delta_{ij}\partial_{k}v^{k}$ (14) $\eta$ $\eta$ $\Pi_{ij}^{(0)}$ $\Pi_{1j}^{(.1)}$ $x\vdash-$ (13) ( ) $T_{\mu\nu}$ $\partial_{\mu}t^{\mu\nu}=0, x^{l^{\iota}};=(ct, z^{i})$ (15) ) $\eta,$ (13) ( $P$ $\rho$ $\Pi_{ij}^{(0)}+\Pi_{ij}^{(1)}$ $\Pi_{ij}^{(m)}$ $\Pi_{ij}^{(m\geq 2)}$ $m$ $\partial_{\mu}t^{\mu\nu}=0, T^{\mu\nu}=\sum_{m=0}^{\infty}\epsilon^{m}T_{(m)}^{\mu\nu}$ (16) $\epsilon$ ; $T_{(m>1)}^{\mu\nu}$ $\epsilon(<1)$ $O(\epsilon)$ $T_{(0)}^{\mu\nu}=\rho u^{\mu}u^{\nu}+pp^{\mu\nu}$ (17) $T_{(1)}^{\mu\nu}=-2_{\mathcal{C}7 }\sigma^{\mu\nu}-c\zeta P^{\mu\nu}\partial_{\alpha}u^{\alpha}$ $T^{\mu\nu}$ $(0)$ (3) $P^{\mu\nu}:=u^{\mu}u^{\nu}+\eta^{\mu\nu}$ $\eta^{\mu\nu}=$ diag $(-1,1, \ldots, 1)$ 1 ) $\mu$ ( $\sigma^{\mu\nu}:=(1/2)p^{\mu a}p^{\nu\beta}[\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}-(2/p)p_{\alpha\beta}\partial_{\gamma}u^{\gamma}]$ 4.2 $\delta\rho\propto\exp(-i\omega t+ikz)$

8 $c_{s}^{2}$ 63 (13) ( ) $\omega^{2_{=c_{s}^{2}k^{2}}^{j}}, c_{s}^{2};=\frac{dp}{d\rho}$ (18) $P=P(\rho)$ $c_{\hslash}^{2}<0$ ( ) 5 [lo] 5.1 : $z$ $p(\geq 1)$ $(n+3)$ $+ \sum_{i=1}^{p}(dz^{i})^{2}$ $p$ $S^{n+1}$ $x^{a}:=(x^{\mu}, r);=(ct, z^{i},r)$ (19) $z^{i}$ $p$ ( ) $(p+1)$ $(u^{\mu}u_{\mu}=-1)$ $(r_{0}, u^{/1})$ $r_{0}$ $T= \frac{n\hslash c}{4\pi k_{b}r_{0}}$ (20) $p$ $p$ $z^{2}\underline{\nearrow\sim}\prime z^{1}$ 4: 2

9 64 $\delta g_{ab}$ (2) (16) $z^{i}$ $p$ $p(\geq 2)$ $p=1$ 2 4 $p$ $r_{0}$ $u^{\mu}$ $T(r_{0})$ $(r_{0}, u^{\mu})arrow(r_{0}(x^{\nu}), u^{\mu}(x^{\nu}))$ (21) $x^{\nu}=(t, z^{i})$ 4 ( $)$ $T$ $u^{\mu}$ ( $r_{0}$ ) (2) $g_{ab}^{(0)}$ $p$ (2) $g_{ab}^{(0)}+\delta g_{ab}$ $\delta g_{ab}$ $X;=(r_{0}(x^{\nu}), u^{\mu}(x^{\nu}), \delta g_{ab}(x^{a}))$ (22) $X^{\int\iota}$ $\lambda:= \frac{x}{\partial_{\mu}x} $ (23) $(\lambda\gg r_{0})$ ( 2(b) ) $r_{0}$ $r_{0}/\lambda$ $X= \sum_{m=0}^{\infty}\epsilon^{m}x_{(m)}, \epsilon:=\frac{r_{0}}{\lambda}\ll 1$ (24) (2) $\sum_{n=1}^{\infty}\epsilon^{m}g_{(m)}^{ab}=0$ (25) $m=1$ $r_{0}\partial_{\mu}\ln X=O(\epsilon)$ (2) $\partial_{\mu}arrow\epsilon\partial_{\mu}$ (2) (25) (derivative expansion) $r$ Kd ( $V$ $)$ [11]

10 $\rho$ 65 $G_{(m\geq 1)}^{ab}=0$ $(x^{\mu}, r)$ $r$ $(r=r_{0})$ $(r=\infty)$ $x^{\mu}$ (24) $J\triangleright$ $\epsilon=r_{0}/\lambda$ $\epsilon$ (16) ( ) ( ) (16) (17) $\partial_{\mu}t_{(,n-1)}^{\mu\nu}=0$ $(n+3+p)$ 1 $\epsilon G_{(1)}^{ab}=0$ $P=- \frac{\rho}{n+1}=-\frac{\omega_{n+1}c^{4}}{16\pi G}r_{0}^{n}$ (26) $P$ $(p+1)$ $\partial_{\mu}t_{(0)}^{\mu\nu}=0$ $\Omega_{n+1}$ $(n+1)$ 2 $\epsilon G_{(J)}^{ab}+\epsilon^{2}G_{(2)}^{ob}=0$ $\eta=\frac{\omega_{n+1}c^{3}}{16\pi G}r_{0}^{n+1},$ $\zeta=\frac{\omega_{n+1^{c^{3}}}}{8\pi G}r_{0}^{n+1}(\frac{1}{p}+\frac{1}{n+1})$ (27) 8 $\vdash-$ $\partial_{\mu}(t_{(0)}^{l^{\iota\nu}}+\epsilon T_{(1)}^{\mu.\nu})=0$ 4 (2) (16) (26) (27) $r_{0}$ (16) $T_{(m\geq 2)}^{\mu\nu}$ $G_{(m\geq 3)}^{ab}$ $=$ 0 ( $)$ 5.2 (26) $c_{s}^{2};= \frac{dp}{d\rho}=-\frac{1}{n+1}<0$ (28) $p$ $p$ [10]. (27) 3 $\epsilon\sim \partial_{z}r_{0} \ll 1$ $r_{0}$ $ u^{\mu} $ 8 $(r_{0} (t, z), u^{z}(t, z))$ $(p=1)\iota_{\llcorner}^{\sim}$ (2)

$66 ( ) [5] ( ) ro $\propto(t_{0}-t)$ 2 $n_{c}$ $\partial_{\mu}t_{(m\geq 2)}^{\mu\nu}=0$ $(r_{0}arrow 0)$ 6 $(T_{ab}=\Lambda=0)$ (2) $(A <0)$$

11 66 ( ) [5] ( ) ro $\propto(t_{0}-t)$ 2 $n_{c}$ $\partial_{\mu}t_{(m\geq 2)}^{\mu\nu}=0$ $(r_{0}arrow 0)$ 6 $(T_{ab}=\Lambda=0)$ (2) $(A <0)$ $AdS/$CFT(ailti-de Sitter/conformal field theory) [12] ( ) $[13]_{0}$ ( ) [14] $r(t)\propto(t_{0}-t)$ $[$15, $16]$ ($AdS/$CFT $(AdS/$CFT $l$ $)$ ) $AdS/$CFT [17,18] 5

12 $\eta$ 67 5: 5 RHIC (Relativistic Heavy Ion Collider) LHC (Large Hadron Collider) ( ) 5 ( ) RHIC $\eta$ ( ) $\frac{\eta}{s}=\frac{\hslash}{4\pi k_{b}}$ (29) [19] (29) 5 $p$ $p$ (10) $s= \frac{\omega_{n+1}c^{3}k_{b}}{4g\hslash}r_{0}^{n+1}$ (30) $\mathcal{s}$ (27) $r_{0}$ ( (29) [1] R. Gregory and R. Laflamme: Black strings and $p$-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837, [arxiv:hep-th/ ]. [2] G. T. Horowitz and K. Maeda: Fate of the black string instability, Phys. Rev. Lett. 87 (2001) , [arxiv:hep-th/ ]. [3] E. Sorkin: $A$ critical dimension in the black-string pha.se transition, Phys. Rev. Lett. 93 (2004) , [arxiv:hep-th/ ].

68 [4] H. Kudoh and U. Miyamoto: On non-uniform smeared black branes, Class. Quant. Grav. 22 (2005) 3853, [arxiv:hep-th/0506019]. [5] L. Lehner and F.

13 68 [4] H. Kudoh and U. Miyamoto: On non-uniform smeared black branes, Class. Quant. Grav. 22 (2005) 3853, [arxiv:hep-th/ ]. [5] L. Lehner and F. Pretorius: Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship, Phys. Rev. Lett. 105 (2010) , $[arxiv: $ [hep-th] $].$ [6] V. Cardoso and O. J. C. Dias: Gregory-Laflamme and Rayleigh-Plateau instabilities, Phys. Rev. Lett. 96 (2006) [arxiv:hep-th/ ]. $i$ [7] U. Miyamoto and $K.$. Maeda: Liquid bridges and black strings in higher dimensions, Phys. Lett. $B664$ (2008) 103, $[arxiv: $ [hep-th] $].$ [8] M. M. Caldarelli, O. J. C. Dias, R. Emparan and D. Klemm, Black holes as lumps of fluid, JHEP 0904, 024 (2009), $[arxiv: $ [hep-th] $].$ [9] U. Miyamoto: Curvature driven diffusion, Rayleigh-Plateau, and Gregory-Laflamme, Phys. Rev. $D78$ (2008) , $[arxiv: $ [hep-th] $].$ [10] J. Camps, R. Emparan and N. Haddad, Black Brane Viscosity and the Gregory- Laflammc $I_{Ilstab;]ity},$ JHEP 1005 (2010) 042, $[arxiv: $ [hep-th] $].$ [11] 1995 [12] S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 0802 (2008) 045, $[arxiv: $ [hep-th] $].$ [13] O. Aharony, S. Minwalla and T. Wiseman: Plasma-balls in large $N$ gauge theories and localized black holes, Class. Quant. Grav. 23 (2006) 2171, [arxiv:hep-th/ ]. $i$ [14] $K.$. Maeda and U. Miyamoto: Black hole-black string phase transitions from hydrodynamics, JHEP 0903 (2009) 066, $[arxiv: $ [hep-th] $].$ [15] J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Rev. Mod. Phys. 69, 865 (1997). [16] U. Miyamoto: One-Dimensional Approximation of Viscous Flows, JHEP 1010 (2010) 011, $[arxiv: $ [hep-th] $].$ [17] $AdS/CFT$ 54-3(2010)110 [18] $94-3(2010)350$ [19] P. Kovtun, D. T. Son and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole $p1_{1}$ysics, Phys. Rev. Lett. 94, (2005) [hep- $th/ ].$

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt