Hideaki Kudoh- Doubly Spinning Black Rings and Beyond

8/3/2019 Hideaki Kudoh- Doubly Spinning Black Rings and Beyond

1/29

Doubly Spinning Black Rings

and Beyond


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


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


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


4/29


-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


5/29


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


6/29


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***)


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



Stationary black holesin higher dimensions



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.



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



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



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)



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.



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.



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


16/29


Ring vs Canonical coordinates

Z=0

Z>0

Z


17/29


Canonical form of metric (I)

is defined by

r

z

Mink

Symbolically, we write these

conditions as

Emparan-Reall

R d


18/29


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


19/29


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


20/29


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


HK, gr-qc/0611136 [PRD2007]


21/29



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.


22/29


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]


23/29


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


24/29


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. ]


25/29


Exact solution

The exact solutions of doubly spinning BR is now available.[Pomeransky & Senkov hep-th/0612005 ]

side-view top-view


26/29


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


27/29


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)


28/29


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


29/29


Summary

Stationary axisymmetric vacuum solutions

General frame work which is suited for numerics


Further applications in more than 5-dims are ongoing.

Hideaki Kudoh- Doubly Spinning Black Rings and Beyond

Documents

Hideaki Kudoh- Doubly Spinning Black Rings and Beyond