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"
1. Motivation and Goal
Higher-Dim Black Holes have Rich Structures
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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"
1. Motivation and Goal
Higher-Dim Black Holes have Rich Structures
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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? ...
1. Motivation and Goal
Higher-Dim Black Holes have Rich Structures
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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? ...
1. Motivation and Goal
Higher-Dim Black Holes have Rich Structures
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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2.B: Initial Data Results
Spheroidal Cases
cf. (3-dim.) Nakamura-Shapiro-Teukolsky (1988)
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2.B: Initial Data Results
Toroidal Cases
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2.B: Initial Data Results
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 ?
2.C. Initial Data Analysis
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
Hyper-Hoop conjecture ?
2.C. Initial Data Analysis
Define Hyper-Hoop as the surface
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Spheroidal Cases
Hyper-Hoop conjecture ?
2.C. Initial Data Analysis
Define Hyper-Hoop as the surface
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Spheroidal Cases
Hyper-Hoop conjecture ?
2.C. Initial Data Analysis
Define Hyper-Hoop as the surface
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Hyper-Hoop conjecture ?
Toroidal Cases
2.C. Initial Data Analysis
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Hyper-Hoop conjecture ?
Toroidal Cases
2.C. Initial Data Analysis
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Hyper-Hoop conjecture ?
Toroidal Cases
2.C. Initial Data Analysis
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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3. Evolution (case I)t=0 No Horizon
t=0.2 Common Horizon
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3. Evolution (case I)t=0 No Horizon
t=0.2 Common Horizon
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3. Evolution (case I)t=0 No Horizon
t=0.2 Common Horizon
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3. Evolution (case I)t=0 No Horizon
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
3. Evolution (case II)
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t=0 No Horizon
t=0.9 Ring Horizon
3. Evolution (case II)
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t=0 No Horizon
t=0.9 Ring Horizon
t=1.1 Common Horizon
3. Evolution (case II)
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t=0 No Horizon
t=0.9 Ring Horizon
t=1.1 Common Horizon
3. Evolution (case II)
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t=0 No Horizon
t=0.9 Ring Horizon
t=1.1 Common Horizon
3. Evolution (case II)
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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
3. Evolution (case III)
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t=0 No Horizon
t=1.19 Ring Horizon
3. Evolution (case III)
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t=0 No Horizon
t=1.19 Ring Horizon
3. Evolution (case III)
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t=0 No Horizon
t=1.19 Ring Horizon
3. Evolution (case III)
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t=0 No Horizon
t=1.19 Ring Horizon
3. Evolution (case III)
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t=0 No Horizon
t=1.19 Ring Horizon
3. Evolution (case III)
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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
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