http://w.atwiki.jp/exactsolutions/ Spacetime Solutions at Glance Wiki ja 2013-08-01T11:10:35+09:00 1375323035 https://w.atwiki.jp/exactsolutions/pages/16.html *Einstein metrics $$S_D=\frac{1}{16\pi G}\int d^Dx \sqrt{-g} (R + \Lambda)$$ &aname(id_1){} **Black holes with spherical horizon topology in higher dimensions &aname(id_1_1){Asymptotically flat} |Dim|Metric|Authors (Age)|Symmetry|Parameters|Other features| |D≥4|Schwarzschild-Tangherlini|Tangherlini (1963)|static, spherically symmetric|mass|| |D≥4|Myers-Perry|Myers, Perry (1986)|stationary, axially symmetric|mass, rotations|| -&bold(){Schwarzschild-Tangherlini metric} [F. R. Tangherlini, "Schwarzschild field in n dimensions and the dimensionality of space problem", Nuovo Cim. 27 (1963) 636-651] The metric in $$D$$ dimensions is #center(){{{ $$ds^2=-\left(1-\frac{\mu}{r^{D-3}}\right)dt^2+\left(1-\frac{\mu}{r^{D-3}}\right)^{-1}dr^2+r^2 d\Omega_{D-2}^2$$ }}} where $$d\Omega_{D-2}^2$$ is the standard metric on $$S^{D-2}$$, given by #center(){{ $$d\Omega_{D-2}^2=d\theta_1^2+\sin^2\theta_1 d\theta_2^2+\cdots +(\sin^2\theta_1\cdots\sin^2\theta_{D-3})d\theta_{D-2}^2$$ }} The mass is #center(){{{ $$M = \frac{(D-2)\Omega_{D-2}}{16\pi G}\mu \,.$$ }}} -&bold(){Myers-Perry metric} [R. C. Myers, M. J. Perry, Ann. of Phys. 172 (1986) 304] The metric forms in odd and even dimensions are different. In odd dimensions $$D=2n+1$$, the metric is #center(){{{ $$ds^2=-dt^2+\frac{\Pi F}{\Pi-\mu r^2}dr^2+\sum_{i=1}^n(r^2+a_i^2)(d\mu_i^2+\mu_i^2d\phi_i^2)+\frac{\mu r^2}{\Pi F}\sum_{i=1}^n(dt-a_i\mu_i^2d\phi_i)^2$$ }}} where $$\mu_i$$ are constrained coordinates, that is, #center(){{ $$\sum_{i=1}^n\mu_i^2 = 1 \,.$$ }} In even dimensions $$D=2n$$, the metric is #center(){{{ $$ds^2=-dt^2+\frac{\Pi F}{\Pi-\mu r}dr^2+\sum_{i=1}^n(r^2+a_i^2)(d\mu_i^2+\mu_i^2d\phi_i^2)+\frac{\mu r}{\Pi F}\sum_{i=1}^n(dt-a_i\mu_i^2d\phi_i)^2$$ }}} where $$\mu_i$$ satisfy #center(){{ $$\sum_{i=1}^n\mu_i^2 + \alpha^2 = 1 \qquad (-1\leq\alpha\leq 1) \,.$$ }} In both cases, the functions contained in the metrics are given by #center(){{ $$F = 1-\sum_{i=1}^n\frac{a_i^2\mu_i^2}{r^2+a_i^2}\,, \qquad \Pi =\prod_{i=1}^n(r^2+a_i^2)\,.$$ }} The mass and angular momenta for the i-th rotational plane are #center(){{{ $$M = \frac{(D-2)\Omega_{D-2}}{16\pi G}\mu \,, \qquad J_i = \frac{\Omega_{D-2}}{16\pi G}\mu a_i\,.$$ }}} &aname(id_1_2){w/ cosmological constant} |Dim|Metric|Authors (Age)|Symmetry|Parameters|Other features| |5|Myers-Perry-(A)dS|Hawking, Hunter, Taylor-Robinson (1999)|stationary, axially symmetric|mass, rotations|| |D≥4|Myers-Perry-(A)dS|Gibbons, Lu, Page, Pope (2004)|stationary, axially symmetric|mass, rotations|| |D≥4|Myers-Perry-NUT-(A)dS|Chen, Lu, Pope (2006)|stationary, axially symmetric|mass, rotations, NUTs|| -&bold(){Myers-Perry-NUT-(A)dS metric} (Higher-dimensional Kerr-NUT-(A)dS metric) [W. Chen, H. Lu, C. N. Pope, Class. Quant. Grav. 23 (2006) 5323 ([[hep-th/0604125>http://arxiv.org/abs/hep-th/0604125]])] In even dimensions $$D=2n$$, the metric is #center(){{ $$ds^2=\frac{U}{X}dr^2+\sum_{\alpha=1}^{n-1}\frac{U_\alpha}{X_\alpha}dy_\alpha^2-\frac{X}{U}\left[W d\bar{t}-\sum_{i=1}^{n-1}\gamma_i d\bar{\phi}_i\right]^2$$ $$+\sum_{\alpha=1}^{n-1}\frac{X_\alpha}{U_\alpha}\left[\frac{(1+g^2r^2)W}{1-g^2y_\alpha^2}d\bar{t}-\sum_{i=1}^{n-1}\frac{(r^2+a_i^2)\gamma_i}{a_i^2-y_\alpha^2}d\bar{\phi}_i\right]^2 $$ }} In odd dimensions $$D=2n+1$$, the metric is #center(){{ $$ds^2=\frac{U}{X}dr^2+\sum_{\alpha=1}^{n-1}\frac{U_\alpha}{X_\alpha}dy_\alpha^2-\frac{X}{U}\left[W d\bar{t}-\sum_{i=1}^na_i^2\gamma_i d\bar{\phi}_i\right]^2$$ $$+\sum_{\alpha=1}^{n-1}\frac{X_\alpha}{U_\alpha}\left[\frac{(1+g^2r^2)W}{1-g^2y_\alpha^2}d\bar{t}-\sum_{i=1}^n\frac{a_i^2(r^2+a_i^2)\gamma_i}{a_i^2-y_\alpha^2}d\bar{\phi}_i\right]^2 $$ $$+\frac{\prod_{k=1}^na_k^2}{r^2\prod_{\alpha=1}^{n-1}y_\alpha^2}\left[(1+g^2r^2)Wd\bar{t}-\sum_{i=1}^n(r^2+a_i^2)\gamma_id\bar{\phi}_i\right]^2$$ }} where &aname(id_2){} **5d vacuum black hole solutions In five dimensions, following the topology theorems, the event horizon topology must be either a sphere $$S^3$$, a ring $$S^1\times S^2$$, a lens space $$S^3/\Gamma$$ or their connected sums. &aname(id_2_1){Black rings} |Dim|Metric|Authors (Age)|Symmetry|Parameters|Λ|Other features| |5|Singly spnning black ring|Emparan, Reall (2002)|stationary, axially symmetric|mass, 1 rotation|no|$$S^1$$-rotation| |5|Singly spnning black ring|Mishima, Iguchi (2006), Figueras (2005)|stationary, axially symmetric|mass, 1 rotation|no|$$S^2$$-rotation, conical singularity| |5|Doubly spnning black ring|Pomeransky, Sen'kov (2006) |stationary, axially symmetric|mass, 2 rotations|no|| -&bold(){Black rings with $$S^1$$-rotation} [R. Emparan, H. S. Reall, PRL 88 (2002) 101101] The metric is written in the C-metric coordinates in the form #center(){{ $$ds^2 = -\frac{F(y)}{F(x)}\left(dt-CR\frac{1+y}{F(y)}d\psi\right)^2+\frac{R^2F(x)}{(x-y)^2}\left[-\frac{G(y)}{F(y)}d\psi^2+\frac{G(x)}{G(y)}d\phi^2+\frac{dx^2}{G(x)}-\frac{dy^2}{G(y)}\right]$$ }} where #center{{{ $$F(\xi)=1+\lambda \xi \,, \qquad G(\xi)=(1-\xi^2)(1+\nu \xi) \,, \qquad C= \sqrt{\lambda(\lambda-\nu)\frac{1+\lambda}{1-\lambda}}$$ }}} The regularity requires a balance condition, that is, $$\lambda=2\nu/(1+\nu^2)$$. The mass and angular momentum are #center(){{{ $$M = \frac{3\pi R^2}{4G}\frac{\lambda}{1-\nu}\,, \qquad J_\psi = \frac{\pi R^3}{2G}\frac{\sqrt{\lambda(\lambda-\nu)(1+\lambda)}}{(1-\nu)^2}$$ }}} -&bold(){Black rings with $$S^2$$-rotation} [T. Mishima, H. Iguchi, Phys. Rev. D 73 (2006) 044030; P. Figueras, JHEP 07 (2005) 039] The metric is written in the C-metric coordinates as #center(){{ $$ds^2 = -\frac{H(y,x)}{H(x,y)}\left(dt-\frac{\lambda a y(1-x^2)}{H(y,x)}d\phi\right)^2+\frac{R^2H(x,y)}{(x-y)^2}\left[-\frac{(1-y^2)F(x)}{H(x,y)}d\psi^2+\frac{(1-x^2)F(y)}{H(y,x)}d\phi^2+\frac{dx^2}{(1-x^2)F(x)}-\frac{dy^2}{(1-y^2)F(y)}\right]$$ }} where #center{{{ $$H(\xi,\eta)=1+\lambda\xi+\frac{a^2\xi^2\eta^2}{R^2} \,, \qquad F(\xi)=1+\lambda \xi+\frac{a^2\xi^2}{R^2} \,.$$ }}} The conical singularity can not be avoided. The mass and angular momentum are #center(){{{ $$M = \frac{3\pi R^2}{4G}\frac{\lambda}{1-\lambda+a^2/R^2}\,, \qquad J_\phi = -\frac{\pi R^2}{G}\frac{\lambda a}{(1-\lambda+a^2/R^2)^{3/2}}$$ }}} &aname(id_2_2){Black lenses} &aname(id_3){} **5d Kaluza-Klein black holes |Dim|Metric|Authors (Age)|Symmetry|Parameters|Λ|Other features| 2013-08-01T11:10:35+09:00 1375323035 https://w.atwiki.jp/exactsolutions/pages/1.html **Welcome to the Spacetime Solutions at Glance Wiki This website aims to give a thorough map of the existing solutions at glance. Since Einstein established the General Relativity Theory in 1916, many different classes of solutions have been found for about a century. Nowadays, people expand their thinking with gravity to various gravitational theories without sticking to GR. This website collects solutions in Einstein's general relativity and alternative gravitational theories in four dimensions and higher. **How to edit Anyone can edit the contents of this site. -Edit from [編集] > [ページ編集] of the above menu. -Create a new page by clicking [@メニュー] > [新規ページ作成]. -Mathematical formulae can be used in the circumstance between $$ with use of the TeX commands. $$ds^2 = -dt^2 + dx^2+dy^2+dz^2$$ **References -Thomas Mueller and Frank Grave, Catalogue of Spacetimes, [[arXiv:0904.4184>http://arxiv.org/abs/0904.4184]] -Tomáš Málek, Exact Solutions of General Relativity and Quadratic Gravity in Arbitrary Dimension, [[arXiv:1204.0291>http://arxiv.org/abs/1204.0291]] -Shin'ya Tomizawa, Hideki Ishihara, Exact solutions of higher dimensional black holes, [[Prog. Theor. Phys. Suppl. 189 (2011) 7-51>http://ptp.ipap.jp/journal/PTPS-189.html]] ([[arXiv:1104.1468>http://arxiv.org/abs/1104.1468]]) **Books -[[Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt, Exact Solutions of Einstein's Field Equations, July 2009, Cambridge Univ Press>http://www.cambridgejapan.org/academicproduct.html?isbn=9780521467025]] -[[Jerry B. Griffiths and Jiří Podolský, Exact Space-Times in Einstein's General Relativity, August 2012, Cambridge Univ Press>http://www.cambridge.org/us/academic/subjects/physics/cosmology-relativity-and-gravitation/exact-space-times-einsteins-general-relativity]] Counter Total：&counter() Today：&counter(today) Yesterday：&counter(yesterday) 2013-08-01T11:04:20+09:00 1375322660 https://w.atwiki.jp/exactsolutions/pages/2.html **[[Top page]] ---- **Einstein gravity ***4 dim. #region([[Einstein metrics>Einstein metrics in four dimensions]]) Einstein metrics -&link_anchor(id_1,page=Einstein metrics in four dimensions){Black hole solutions} -bbb -ccc #endregion #region([[Einstein-Maxwell theory>Einstein-Maxwell theory in four dimensions]]) Einstein-Maxwell theory -&link_anchor(id_1,page=Einstein-Maxwell theory in four dimensions){Charged black hole solutions} -bbb -ccc #endregion #region(Einstein-Maxwell-Dilaton theory) Einstein-Maxwell-Dilaton theory -aaa -bbb -ccc #endregion ***D dim. #region([[Einstein metrics>Einstein metrics in higher dimensions]]) Einstein metrics -&link_anchor(id_1,page=Einstein metrics in higher dimensions){Black hole solutions} -&link_anchor(id_2,page=Einstein metrics in higher dimensions){Black ring solutions} -ccc #endregion #region(5d Einstein-Maxwell systems) 5d Einstein-Maxwell systems #region(5d pure Einstein-Maxwell theory) 5d pure Einstein-Maxwell theory -aaa -bbb -ccc #endregion #region([[5d Einstein-Maxwell theory with CS term>Five-dimensional Einstein-Maxwell theory with CS term]]) 5d Einstein-Maxwell theory with CS term -&link_anchor(id_1,page=Five-dimensional Einstein-Maxwell theory with CS term){Charged black hole solutions} -bbb -ccc #endregion #region(5d U1xU1xU1 supergravity) 5d U(1)xU(1)xU(1) supergravity -aaa -bbb -ccc #endregion #endregion #region(Einstein-Maxwell-Dilaton theory) Einstein-Maxwell-Dilaton theory -aaa -bbb -ccc #endregion ---- **Ohter gravitational theories ---- **LINK 2013-07-30T10:48:50+09:00 1375148930 https://w.atwiki.jp/exactsolutions/pages/15.html *Einstein metrics Einstein metrics are vacuum solutions of the Einstein equation with cosmological constant. In particular, if the cosmological constant vanishes, the solutions are Ricci-flat, $$R_{\mu\nu} = 0$$. &aname(id_1){} **Black hole solutions |Metric|Author (Age)|Symmetry|Parameters|Λ|Other features| |Schwarzschild|Schwarzschild (1916)|static, spherically symmetric|mass|no|| |Schwarzschild-(A)dS|Kottler (1918)|static, spherically symmetric|mass|yes|| |Taub-NUT|Taub (1951), Newman, Tamburino, Unti (1963)|stationary, axially symmetric|mass, NUT|no|| |Kerr|Kerr (1963)|stationary, axially symmetric|mass, rotation|no|| |Kerr-NUT|Demianski (1966), Kramer, Neugebauer (1968), Robinson, Robinson, Zund (1969)|stationary, axially symmetric|mass, rotation, NUT|no|| |Schwarzschild-NUT-(A)dS|Demianski (1972)|static, spheircally symmetric|mass, NUT|yes|| |Kerr-(A)dS|Demianski (1973)|stationary, axially symmetric|mass, rotation|yes|| |Kerr-NUT-(A)dS|Frolov (1973)|stationary, axially symmetric|mass, rotation, NUT|yes|| -&bold(){Schwarzschild metric} [Schwarzschild (1916), Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.) 1916, 189-196 (arXiv:physics/9905030)] #center(){{$$ds^2 = -\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)$$}} -&bold(){Kerr metric} [Kerr (1963), Physical Review Letters 11 (5) 237–238] #center(){{$$ds^2 = -\frac{\Delta}{\Sigma}(dt-a \sin^2\theta d\phi)^2+\frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2+\frac{\sin^2\theta}{\Sigma}(adt-(r^2+a^2)d\phi)^2$$}} where #center(){{$$\Delta=r^2+a^2-2Mr \,, \quad \Sigma=r^2+a^2\cos^2\theta$$}} 2013-07-26T17:43:24+09:00 1374828204 https://w.atwiki.jp/exactsolutions/pages/17.html *Einstein-Maxwell theory $$S = \int d^4x \sqrt{-g} (R + \Lambda - \frac{1}{4}F_{\mu\nu}F^{\mu\nu})$$ &aname(id_1){} **Charged black hole solutions |Metric|Author (Age)|Symmetry|Parameters*|Λ|Other features| |Reissner-Nordstrom|Reissner (1916), Nordstrom (1918)|static, spherically symmetric|mass, electric charge|no|| |Kerr-Newman|Newman, Couch, Chinnapared, Exton, Parakash, Torrence (1965)|stationary, axially symmetric|mass, rotation, electric charge|no|| |Carter|Carter (1968)|stationary, axially symmetric|mass, rotation, NUT, electric charge|yes|| |Kinnersley|Kinnersley (1969)|stationary, axially symmetric|mass, rotation, NUT, electric and magnetic charges, acceleration|no|| |Plebanski|Plebanski (1975)|stationary, axially symmetric|mass, rotation, NUT, electric and magnetic charges|yes|| |Plebanski-Demianski|Plebanski, Demianski (1976)|stationary, axially symmetric|mass, rotation, NUT, electric and magnetic charges, acceleration|yes|| 2013-07-26T17:07:13+09:00 1374826033 https://w.atwiki.jp/exactsolutions/pages/18.html *5D Eisntein-Maxwell with Chern-Simons term $$S_5=\int *(R+\Lambda)-\frac{1}{2} *F \wedge F +\frac{\lambda}{3\sqrt{3}}F\wedge F\wedge A $$ where $$\Lambda$$ is the cosmological constant and $$\lambda$$ is the Chern-Simons coupling. -$$\lambda=0$$; pure Einstein-Maxwell theory -$$\lambda=1$$; minimal supergravity &aname(id_1){} **Charged black hole solutions |Dim|Metric|Authors (Age)|Symmetry|Parameters|Λ|Other features| 2013-07-26T16:43:12+09:00 1374824592 https://w.atwiki.jp/exactsolutions/pages/3.html **Updates #recent(20) 2013-07-24T15:18:34+09:00 1374646714 https://w.atwiki.jp/exactsolutions/pages/7.html * アーカイブ @wikiのwikiモードでは #archive_log() と入力することで、特定のウェブページを保存しておくことができます。 詳しくはこちらをご覧ください。 ＝＞http://atwiki.jp/guide/25_171_ja.html ----- たとえば、#archive_log()と入力すると以下のように表示されます。 保存したいURLとサイト名を入力して"アーカイブログ"をクリックしてみよう #archive_log() 2013-07-24T14:43:34+09:00 1374644614 https://w.atwiki.jp/exactsolutions/pages/9.html * 動画(youtube) @wikiのwikiモードでは #video(動画のURL) と入力することで、動画を貼り付けることが出来ます。 詳しくはこちらをご覧ください。 ＝＞http://atwiki.jp/guide/17_209_ja.html また動画のURLはYoutubeのURLをご利用ください。 ＝＞http://www.youtube.com/ ----- たとえば、#video(http://youtube.com/watch?v=kTV1CcS53JQ)と入力すると以下のように表示されます。 #video(http://youtube.com/watch?v=kTV1CcS53JQ) 2013-07-24T14:43:34+09:00 1374644614 https://w.atwiki.jp/exactsolutions/pages/10.html @wikiにはいくつかの便利なプラグインがあります。 ----- #ls ----- これ以外のプラグインについては@wikiガイドをご覧ください =>http://atwiki.jp/guide/ 2013-07-24T14:43:34+09:00 1374644614