hisa-aki shinkai- black objects and hoop conjecture in five-dimensional space-time

8/3/2019 Hisa-aki Shinkai- Black Objects and Hoop Conjecture in Five-dimensional Space-time

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http://www.is.oit.ac.jp/~shinkai/

GR19 @ Mexico City, July 2010

Hisa-aki Shinkai(Osaka Institute of Technology, Japan)

work with Yuta Yamada (OIT)

Initial DataYamada & HS, CQG 27 (2010) 045012

EvolutionYamada & HS, in preparation.


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1. Motivation and Goal

Higher-Dim Black Holes have Rich Structures

LHC experiments will (or will not) reveal Higher-DimBHs in near future

4-dim BH : horizon is S^2,stable solutions

Schwarzschild --- Birkoff theorem (M)Kerr --- uniqueness theorem (M, J)

Brane-World models give new viewpoints to gravity

and cosmology


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4-dim BHs Higher-dim BHs :

Schwarzschild Tangherlini

--- unique & stable

Kerr Myers-Perry--- maybe unstable in higher J

black ring (Emparan-Reall)black Saturndi-rings, orthogonal di-rings, ...

"Black Objects"




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4-dim BHs Higher-dim BHs :

Schwarzschild Tangherlini

--- unique & stable

Kerr Myers-Perry--- maybe unstable in higher J

black ring (Emparan-Reall)black Saturndi-rings, orthogonal di-rings, ...

"Black Objects"




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black holeblack stringblack ringblack Saturndi-rings, orthogonal di-rings ...

"Black Objects"

Uniqueness (only in spherical sym.)Stability?Formation Process?Dynamical Features? ...




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black holeblack stringblack ringblack Saturndi-rings, orthogonal di-rings ...

"Black Objects"

Uniqueness (only in spherical sym.)Stability?Formation Process?Dynamical Features? ...



No Hair Conjecture?Cosmic Censorship?Hoop Conjecture?


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2. Initial Data Construction

- time symmetric, asymptotically flat- conformal flat- non-rotating homogeneous dust- in spheroidal shape orring shape

- solve the Hamiltonian constraint eq. 512^2 grids- Apparent Horizon Searchboth forRing Horizon and Common Horizon

- Define Hoop and check the Hoop Conjecture


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2.A: Initial Data Constructionmetric & Hamiltonian constraint

Toroidal Cases

Spheroidal Cases


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Apparent Horizons Search

Toroidal Cases

Spheroidal Cases

Common Horizon

Ring Horizon

2.A: Initial Data Construction


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Area of Horizons

Toroidal Cases

Spheroidal Cases

Common Horizon

Ring Horizon

2.A: Initial Data Construction


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2.B: Initial Data Results

Spheroidal Cases

cf. (3-dim.) Nakamura-Shapiro-Teukolsky (1988)


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Spheroidal Cases

cf. (3-dim.) Nakamura-Shapiro-Teukolsky (1988)


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Toroidal Cases


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Toroidal Cases


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Hoop Conjecture

Hyper-Hoop conjecture ?

2.C. Initial Data Analysis

Hyper-Hoop Conjecture

Thorne (1972)

Penrose (1969)

Ida-Nakao (2002)


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Hoop Conjecture



Hyper-Hoop Conjecture

Thorne (1972)

Penrose (1969)

Ida-Nakao (2002)

In 5-D, if mass gets compacted

in some area, ....


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Spheroidal Cases



Define Hyper-Hoop as the surface


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Spheroidal Cases





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Spheroidal Cases





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Toroidal Cases



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Toroidal Cases



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Toroidal Cases


Hyper-Hoopdoes not work for

ring horizons.


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3. Evolution Code

- ADM full 4+1, ADM 2+1 Double Axisym Cartoon- 33^4 grids, 65^2 x 2^2 grids- Maximal slicing condition, zero shift vectors- asymptically flat

- Collisionless Particles (5000)- the same total mass

- no rotation

- Apparent Horizon Searchboth forRing Horizon and Common Horizon


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3. Evolution (case I)t=0 No Horizon


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t=0.2 Common Horizon


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t=0 No Horizon

3. Evolution (case II)


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t=0 No Horizon

t=0.9 Ring Horizon



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t=0 No Horizon

t=0.9 Ring Horizon



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t=0 No Horizon

t=0.9 Ring Horizon




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t=0 No Horizon

t=0.9 Ring Horizon




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t=0 No Horizon

t=0.9 Ring Horizon




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t=0 No Horizon

3. Evolution (case III)


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t=0 No Horizon

t=1.19 Ring Horizon



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t=0 No Horizon

t=1.19 Ring Horizon



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t=0 No Horizon

t=1.19 Ring Horizon



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t=0 No Horizon

t=1.19 Ring Horizon



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t=0 No Horizon

t=1.19 Ring Horizon



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t=0 No Horizon

t=1.19 Ring Horizon



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4. Summary and Future Plans

Initial Data:Topology of horizon changes with matter configurationsHyper-Hoop predictionworks well for formations of spheroidal black holes

but not for rings.

Evolution:no horizon common horizon

ring horizon

Future Plans:include rotation, change slicing conditionssearch event horizon,

Towards Dynamics of 5-dim Black Objects

hisa-aki shinkai- black objects and hoop conjecture in five-dimensional space-time

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