e-mail : [email protected] abstract Grav itational Co llapse in Fiv e-dimensional Sp Yuta Yamada (Osaka Insti f Technology, ( ) Hisa-aki Shinkai (Osaka Ins lo gy, Japa ( ) We numerically inv estigate black-hole (and black-ring) fo rmatio n in fiv e-dimensional spacetime. We mo del the ini tial matter distribution in no n-ro tati ng homo geneo us sphero idal an to ro idal co nfi guratio ns under the mo mentarily static assumptio n, and express the matter with co llisionless particles. We evo lv e the spacetime using the m ng condi tio n fo r the lapse functi on and the minimal strain conditio n fo r shift vecto r. We search apparent horizo ns bo th o f S 3 and S 1 S 2 to polo gy during time ev olutio ns. Fo r to ro idal matter cases, we o bserv e the to polo gy of apparent ho rizo ns depends on the ring radius, and also we find the to polo gy change of ho rizo n fro m ri ng to spherical shape evolutio n. For sphero idal co nfigurations, we can no t find apparent ho rizon for highly pro late matter, which indicates also the fo rmatio n o f naked singularity. We ho le o r black-ring is fo rmed and also repo rt the po ssibility of a naked singularity fo rmatio n. 1. Motiv atio n Formatio n o f naked singularities fo rmatio n o f naked singulatities in 4D Grav itational co llapse o f collisio nless particles with sphero idal configuration. (Shapi ro , Teuko lsky, PRL66, 994, 1991) possibility of naked singularities in 5D Suggest the fo rmatio n o f naked singularities by spindle co llapse. (Yo o, Ida, Nakao , PRD71, 104014, 2005) Dynamics of Black-Objects Stable? Unstable? 3. Initial data (Yamada, Shinkai, CQG27, 045012) Formatio n process ? Dynamical features ? 2. Our Numerical Appro ach 5. Time evo lutio n o f sphero idal co nfiguration Evolutio n o f two kinds o f no n-ro tati ng matter co nfigurations. Using the (4 + 1) ADM fo rmalism. Express the matter with co llisio nless particles. Search apparent ho rizo ns. 4. Ev olutio n Evolutio n equatio ns Maximal time slicing co ndi tio n fo r lapse functio n Minimal strain co ndi tio n fo r shift v ecto r We assume axi-symmetric space-time using th on metho d. We find three different cases for rizon formati n depending the ring radius R c at t = 0. Case1 commo n horizo n (small radius) Case2 ring horizo n co mmo n hori zo n Case3 ring horizo n (large radius) Lapse functio fo r large radi us case We co nstruct sequences of ini tial data with co nfo rmally flat, mo ment of time symmetry, asympto tically flat Confo rmal transfo rmatio n The Hamilto ni an co nstraint equation *boundary co ndi tio n Initial data sequence o f sphero idal co nfiguration Initial data sequence o f ring co nfiguration The ho rizon is no t formed when the matter is highly pro late shape. The asterisk indicates the lo catio n o f the maximum Kretschmann inv ari ant. Rc > 0.78rs o nly the ring horizo n Rc < 0.78rs o nly the co mmo n horizo n Both ho rizon's area are smo othly co nnected. scenario of gravitational co llapse collapse black ring ? collapse black ring black ho le ? Case1 ho rizo n fo rms 6. Time evo lutio n o f ring co nfi guratio n Case2 No ho rizon (Naked singularity?) Kretschmann Invariant Kretschmann Invariant COSMO / Cos