Doubly Spinning Black Rings and Beyondold.phys.huji.ac.il/~barak_kol/HDGR/proceedings/Kudoh.pdf · 2007. 2. 18. · Hideaki Kudoh, - Doubly Spinning Black Rings and Beyond - Time

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

Aspects of gravity in higher dimensions

• Physics of event horizons in higher-dimensional gravity is 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 in higher dimensions

• black hole, black rings, black branes, ... w/ various fields

Hideaki Kudoh, - Doubly Spinning Black Rings and Beyond -z

Brane/domain wallBrane - 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

Hideaki Kudoh, - Doubly Spinning Black Rings and Beyond -

-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 simple

Einstein-scalar system

Width of wall AdS Minkowski

←AdS

Flat space → glued


Model of Colliding walls (2) • Initial data

– By a boost, a wall get velocity – Superposing two walls, sufficiently smooth initial data can be

obtained

-4 -2 2 4

-2

-1.5

-1

-0.5

Wall

←AdS

-4-224

-2

-1.5-1

-0.5

AdS→

AdSflatAdS

5D-spacetimes


Time evolution of colliding walls

symmetric collision with the same velocity, width & amplitude.

dens

ity

• 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 in flat spacetime.

• Cosmic censorship : break down of predictability (particularly cosmological context)

asymmetric collision with different width

HK, Takamizu, Maeda (gr-qc/0702***)


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 take place without singularity.

– Walls are trapped around the horizon

Generic consequence of colliding walls– “big bang universe” after the collision

will be largely affected by the black hole… (Ekpyrotic, cyclic universe, “relaxing to 3-brane” scenario)

→ The picture is quite different from the naïve expectation [“silent” collision ]

– They might fragment into black holes.We will get BHs stuck on a brane/wall


Stationary black holes in higher dimensions


What do we know ?• 4D

– Uniqueness of Kerr (–Newman) BH– Stability of BH– Hawking’s 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,Yanay’06]– 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 had not 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 of solutions, 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 wider applications– 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 which we can explore the broad range of higher

dimensional gravity

10

• Systematic method

• Doubly spinning black rings

• Extension to higher dimensions

• Summary


Ring vs Canonical coordinates

•

Z=0

Z>0

Z<0

x=cosθ

ρ

Ring coordinates {x,y}• natural adopted coordinates

• simple form of exact solutions

• asymptotic infinity (x=y=-1) is degenerated at a single point. (not rectangular)

Canonical coordinates {r,z}• rectangular including asymptotic infinity

• very complicated solutions

& metric are apparently singular

• systematic understanding of solutions

“Hybrid” of these two


Canonical form of metric (I) is defined by

r

z

Mink

Symbolically, we write these conditions as →

Emparan-Reall


• 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


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 been known.

Minkowski black ring

Harmark, Emparan-Reall


• 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 RingsHK, gr-qc/0611136 [PRD2007]


Doubly Spinning Black Rings

• Taking the BR with single spin as a background metric.

• The rods appropriate for DS BR are easily guessed

• A free parameter is the 2nd angular velocity.


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]


• To see the consistency and effects of S2-rotation, the perturbations are solved upto 2nd order.

Reduced Area

VS. Reduced angular momentum

Backreaction

• Ergosurface at a sectionof the ring (no CTC, no deficit)

horizon

Ergosurface


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. ]


Exact solutionThe exact solutions of doubly spinning BR is now available.

[Pomeransky & Sen’kov hep-th/0612005 ]

“side-view” “top-view”


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 the canonical form


Extension• Conformal form of metric still holds

5D Minkowski is

– Direct extension to coordinates suited for Sn×Sm

topology.

– Similar coordinates in AdS are also available.6D MP

7D MP8D MP

5D MP & BR


Trial of black ring with S1×Sn topology[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 not

imposed ⇒ There is no ambiguity for the “constant mode”

– Single rotation seems to be much simpler than DSBR

A@3D=m33

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A@4D=m12

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A@0D=mcc

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A@1D=m11

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A@2D=m22

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0


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 on going.

Doubly Spinning Black Rings and Beyondold.phys.huji.ac.il/~barak_kol/HDGR/proceedings/Kudoh.pdf · 2007. 2. 18. · Hideaki Kudoh, - Doubly Spinning Black Rings and Beyond - Time

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