hideaki kudoh- doubly spinning black rings and beyond
TRANSCRIPT
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
1/29
Doubly Spinning Black Rings
and Beyond
Doubly Spinning Black Rings
and Beyond
Hideaki Kudoh
(UC Santa Barbara / U. of Tokyo)
Hideaki Kudoh
(UC Santa Barbara / U. of Tokyo)
19 Feb. 2007 @Jerusalem19 Feb. 2007 @Jerusalem
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
2/29
Aspects of gravity in higher dimensions
Physics of event horizons in higher-dimensional gravityis far richer and complex, compared with those in 4D
Tip of the iceberg of a rich landscape of solutions
various types of black holes may be easily produced inhigher dimensions
black hole, black rings, black branes, ... w/ various fields
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
3/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
z
Brane/domain wall
Brane - an interesting object in string theory
BPS domain wallInteresting and realistic objects in higher dimensional early
universe and in strings theory
Collision of branes
one of fundamental process and topic
[reconnection, annihilation , tachyon condensation, etc]
Creation of big bang universe, inflation , .
Dynamics of brane and wall
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
4/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
-4 -2 2 4
-2
-1.5
-1
-0.5
Model of Colliding walls Collision of walls has been studied using toy models, focusing on
reheating process and/or without self-gravity [e.g. Takamizu-Maeda 04, 05]
Colliding walls including gravity
exact BPS domain-wall of 5D supergravity [Arai et. al.03] A non-trivial field is only a scalar field in hypermultiplets.
Integrating out trivial/irrelevant fields, the system can be reduced to a simpleEinstein-scalar system
Width of wall AdS Minkowski
AdS
Flat space glued
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
5/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Model of Colliding walls (2)
Initial data
By a boost, a wall get velocity
Superposing two walls, sufficiently smooth initial data can beobtained
- 4 - 2 2 4
-2
- 1 . 5
-1
- 0 . 5
Wall
AdS
-4-224
-2
-1.5
-1
-0.5
AdS
AdSflatAdS
5D-spacetimes
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
6/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Time evolution of colliding walls
symmetric collision with the same velocity,width & amplitude.
density
A sharp peak of density implies a curvature singularity
In AdS, a singularity would be easier to form, while a BH formation is not easier than inflat spacetime.
Cosmic censorship : break down of predictability (particularly cosmological context)
asymmetric collision with different width
HK, Takamizu, Maeda (gr-qc/0702***)
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
7/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Spacetime structure -black brane production-
Black brane production Analogy to gravitational collapse The similar results for other different
model For non-relativistic cases, walls just
pass through or multiple bounce takeplace without singularity.
Walls are trappedaround the horizon
Generic consequence of colliding walls
big bang universe after the collisionwill be largely affected by the blackhole (Ekpyrotic, cyclic universe, relaxingto 3-brane scenario)
The picture is quite different fromthe nave expectation [silent collision ]
They might fragment into black holes.We will get BHs stuck on a brane/wall
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
8/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Stationary black holesin higher dimensions
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
9/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
What do we know ?
4D
Uniqueness of Kerr (Newman) BH
Stability of BH Hawkings topology theorem : Horizon is S2
(only for connected horizon)
The Kerr BH will be a generic BH, formed after a gravitational collapse.
5D (or higher)So far, many examples and suggestions have been obtained.
Break down of uniqueness (BH & BR)
A topology theorem [e.g. Helfgott,Oz,Yanay06]
stability of Schwarzschild BH
In stationary cases, deformed horizon might be possible.
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
10/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Spinning Black Holes Stationary system in higher dimensions may have
many vacuum structure of spacetimes itself.
Ultra spinning limit
Onset of Instability, fragmentation into BHs (?)[even for 5D case which has Kerr bound like 4D]
Effective bound on J ? or wavy horizon ? [Reall]
Myers-Perry, Emparan-Myers
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
11/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Stationary axisymmetric vacuum solution in 5D
found mainly based on an educational guesswork(and knowledge of 4D instanton)
Extended to dipole charges, supersymmetry, etc.
Until recently, no systematic method finding it hadnot been known.
Emparan & ReallBlack Rings
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
12/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Myers-Perry BHs
Black Rings
Fat and thin BRs 3 black holes (MP + two BRs) for given J and S Perhaps, these are unstable ... [Gregory-Laflamme instability for thin black rings]
wavy horizon (?) or fragmentation into MP. charged black ring may be stable even under the broken symmetry.
e.g. magnetically charged non-uniform black strings become stable. [Miyamoto, Kudoh]
Dipole charged black rings [Emparan 04]
Stabilization issues (Elvang,Emparan and Virmani; Arcioni etal; Nozawa&Maeda; Hovdebo&Myers)
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
13/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Stationary black holes in higher dimensions
D=5, 3-Killing vectors
MP, BR.
Black di-rings [Iguchi-Mishima],
Black Saturn [Elvang-Figueras]
D>5
Myers-Perry BH Less symmetric states ?[Reall]
Presumably, more large class ofsolutions, with new type of topologies.
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
14/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Systematic methods Solution-generating techniques
(Belinsky&Zakharov 78, .., Pomeransky, Iguchi&Mishima, Tomizawa et.al. etc.)
are useful for higher dimensional Weyl spacetimes.
[D-dimensional spacetime with D-2 Killing vectors]
BR, MP etc. in 5D are Weyl spacetimes But in higher dimensions (D>5), such objects are not Weyl type.
It is important to provide a feasible method which will have widerapplications Numerical approaches have some advantages
(e.g. static cases are successful)
Can we re-formulate the problem in a suitable manner?
As is often the case in GR, coordinates are important.
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
15/29Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Purpose
Formulation of a feasible method by whichwe can explore the broad range of higher
dimensional gravity
10
Systematic method
Doubly spinning black rings
Extension to higher dimensions
Summary
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
16/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Ring vs Canonical coordinates
Z=0
Z>0
Z
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
17/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Canonical form of metric (I)
is defined by
r
z
Mink
Symbolically, we write these
conditions as
Emparan-Reall
R d
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
18/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
z-axis is divided into intervals.
The rod-structure is
and va is called direction of rod.
E.g.
Rod
More precise definition of rod is as follows
Minkowski black ring
r
z
Timelike rod on horizon
Spacelike rod on axis
Harmark, Emparan-Reall
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
19/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Canonical form of metric (II)
Einstein eqs
Rod-structure (boundary conditions) all known solutions can be classified by its rod-structure
Solution rod
Once a rod-structure is provided, a corresponding solution will exist.
rod solution
But, a general method to find an explicit solution has (had) not beenknown.
Minkowski black ring
Harmark, Emparan-Reall
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
20/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
To demonstrate how we can use these in numerics,the doubly spinning BR is the best example to study
S2-rotating black ring without S1-rotation[Iguchi-Mishima, Figueras]
Supersymmetric BRs has two spins, but they are not independent charges.
S2-rotation
S1-rotation
Doubly Spinning Black Rings
HK, gr-qc/0611136 [PRD2007]
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
21/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Doubly Spinning Black Rings
Taking the BR with single spin as a backgroundmetric.
The rods appropriate for DS BR are
easily guessed
A free parameter is the 2nd angularvelocity.
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
22/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Doubly Spinning Black Rings- slow rotation -
Perturbations will make the analysis and results clear.[Non-linear extension is straightforward]
At linear order,
Expand by angular velocity on S2
generates the second new spin
shift the mass / spin in the background.[ turn off at 1st order]
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
23/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
To see the consistency and effects of S2-rotation, theperturbations are solved upto 2nd order.
Reduced AreaVS. Reduced angular momentum
Backreaction
Ergosurface at a sectionof the ring (no CTC, no deficit)
horizon
Ergosurface
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
24/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Caveat Without an exact solution, we cannot fix the absolute value of periodicity of
angle
Amplitude (and periodicity) can be chosen arbitrary for each BR consistent with EOM, BCs, Smarr relation,
The absolute value affects all physical quantities. However, reduced area & angular momentum are invariant at 2nd order
The absolute value is fixed by taking flat (Minkowski) limit continuously
This is only possible for exact solutions [because we have discreet set of sol. ]
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
25/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Exact solution
The exact solutions of doubly spinning BR is now available.[Pomeransky & Senkov hep-th/0612005 ]
side-view top-view
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
26/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Extension of the method
Fully non-linear solutions
Possible other new solutions in 5D
black hole-black ring co-existence ( found)
Stationary KK bubbles (?) Black Rings in AdS
Wavy horizon . (at least 3-dimensional problem)
Generalization to higher dimensions Coordinates are important (as usual in GR)
No canonical form
even Mink. and Schwarz. BH cannot be written down in thecanonical form
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
27/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Extension
Conformal form of metric still holds
5D Minkowski is
Direct extension to coordinates suited for SnSm
topology.
Similar coordinates in AdS are also available.
6D MP
7D MP
8D MP
5D MP & BR
E t i (2)
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
28/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Trial of black ring with S1Sntopology[Minkowski]
Extension (2)
Rod-structure is no more available Still, physical requirements provide feasible boundary conditions.
Axes has curvature singularity if proper regularity (periodicity) is notimposed
There is no ambiguity for the constant mode
Single rotation seems to be much simpler than DSBR
A@3D=m33
0
0.2
0.4
0.60.8
1
x
12
3
0.92
0.94
0.96
0.981
A@4D=m12
0
0.2
0.4
0.60.8
1
x
12
3
-1
-0.5
0
0.5
1
-
-
A@0D=mcc
0
0.2
0.4
0.60.8
1
x
12
34 y
1
1.0005
1.001
A@1D=m11
0
0.2
0.4
0.60.8
1
x
12
34 y
0
0.5
1
1.5
2
A@2D=m22
0
0.2
0.4
0.60.8
1
x
12
34 y
0.85
0.9
0.95
-
8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond
29/29
Hideaki Kudoh - Doubl S innin Black Rin s and Be ond -
Summary
Stationary axisymmetric vacuum solutions
General frame work which is suited for numerics
Doubly Spinning Black Rings
Further applications in more than 5-dims are ongoing.