Can we see super-Planckian domains?

Takehara Workshop June 　 6-8, 2011

Ken-ichi Nakao(Osaka City Unviersity)

In collaboration with Tomohiro Harada and Umpei Miyamoto (Rikkyo University)

Hirotada Okawa and Masaru Shibata (YITP)

If GR correctly describes gravitational phenomena…

Concentration of mass Collapse due to self-gravity

Large spacetime curvature, large energy density, large stress

All know theories of physics including GR are not available

→ New Physics (Superstring? Brane World? …..)

Generation of spacetime singularities

Near spacetime singularities

§Introduction

by Penrose(1965), Hawking & Penrose (1970)…..

Are spacetime singularities observable?

Cosmic censorship hypothesis by Penrose (1969)

Weak version: spacetime singularities generated by gravitational collapses are hiden behind event horizons

Strong version ： spacetime is globally hyperbolic

？

×

in 4-dim spaecetime.

§Cosmic　censorship　hypothesis

““Domain of super-Planckian (SP) scale Domain of super-Planckian (SP) scale = Border of spacetime”= Border of spacetime”

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Domain A　is　called　the　border　or　SP　domain,　if

EP = fundamental Planck scale = a positive consitant of

Practical　spacetime　singularity　=　Domain　in　which　GR　is　not　available

Harada and Nakao, PRD70, 041501 (2004)

where

Visible super-Planckian domain ≈ Naked singularity

Large extra-dimension scenario Large extra-dimension scenario

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Arkani-Hammed, Dimopoulos and Dvali 1998

Relation between D(>4)-dimensional Planck energy and 4-dimensional one:

(R: length scale of extra-dimansions)

Collisions of high energy particles Collisions of high energy particles in large extra-dimension scenario in large extra-dimension scenario

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Giddings & Thomas (2001), Dimopoulos and Landsberg (2001)

Gravitational radius of the center of mass energy E

Production rate much larger than 4-dimansional theory

Cross section of black hole production

Visible SP Domains generated by Visible SP Domains generated by collisions of high energy particlescollisions of high energy particles

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particle

particle

Domain into which two particles can enter at once

If　≤a　black　hole　forms.If　no　black　hole　forms.　

Nakao, Harada and Miyamoto (2010)

We call this the “collision domain”.

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Volume of the collision domain in (D-1)-dimensional space

Area of the base of the cylinder Height

of the cylinder

Volume in extra-dimensions

Average energy density in the collision domain

(VN : Volume of an N-dim sphere)

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Condition of visible SP domain formation

from 00-component of Einstein’s equations

If max >1, then is　allowed.SP domain with no black hole can

form !

『 Gravitational radius=Compton length, if M = EP. 』

: max2

where SN = Area of N dim sphere.

The　value　of　The　value　of　maxmax

Generation rate of visible SP domainsGeneration rate of visible SP domains

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Generation　rate　of　visible　SP　domain　>>　Generation　rate　of　BH

Generation rate of black Generation rate of black holesholes

Practically,　the　weak　version　of　cosmic　censorship　is　not　satisfied.

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High speed scattering of two black holes by higher dimensional numerical relativity

Scattering or merger of high speed BH’s（ e.g., classical counterpart of high energy scattering of elementary particles ）

Numerical relativity

Method to study various phenomena with strong gravity by numeically integrating Einsteinn’s equations

Evolution of binary composed of compact objects (NS, BH)

Main target of GW physics

＝＝

New target of numerical relativityNew techniques are necessary

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Scatter　or　merger　of　high　speed　BH’s

BH

BH

b

Rg

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For large impact parameter b

After scattering, two BH’s go apart to infinity

Scatter　or　merger　of　high　speed　BH’s

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BH horizon

For small impact parameter b

Formation of one larger BH

Scatter　or　merger　of　high　speed　BH’s

Initial　data

If　the　distance　between　two　BH’s　is　large　enough,　each　BH　and　its　neighborhood　is　very　similar　to　

Schwarzschild-Tangherlini　spacetime

Rg = EP (M/EP)1/2 ：　 gravitational radius

Scattering　of　high　speed　BH’s　in　5　dimensions　(equal　mass　M ）

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by Shibata, Okawa, Yamamoto (2008)

Boost transformation

translation

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Extrinsic curvature of t = const. hypersurface

Other components vanish.17

Initial　data　for　two　BH’s　approaching　to　each　other　with　velocity　v

andKij are　determined　so　that　constraint　equations　are　satidfied.

4-dim. spatial metric

Extrinsic curvature

However　here,　we　have　set　Kij .This　is　a　good　approximation　for　the　case　of　large　enough　distance　between　the　two　BH’s.

ここで、

In　this　initial　data,　there　is　almost　no　junk　radiation.　18

on the horizon of a spherically symmetric BH with M= EP in 5-dim spacetime.

SP domain:

is shown in the unit of (EP/M)1/2.

by　Okawa,　Shibata　&　KN　(2011)

BH’s

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Example: b=3.38Rg, v=0.7

Scattering　of　high　speed　BH’s　in　5　dimensions　(equal　mass　M ）

Visible　SP　domain

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After　scattering,　two　BH’s　will　go　apart　to　infinity

K2

[EP

/M]2

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Kv

[EP

/M]

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If　EP < M < 19EP , visible　SP　domain　forms　by　the　scattering　of　classical　BH’s　!

The　largest　value　of　K in　our　simulations　is　

If　finer　simulations　becomes　possible,　we　may　find　larger K.

v

Scatter

Simulations break down.

(Naked singularity?)Merger

b [

Rg]

bC

bB

Summary and discussionSummary and discussion We can see super-Planckian physics if We can see super-Planckian physics if

the spacetime dimension is larger than the spacetime dimension is larger than four (Naked singularity might form).four (Naked singularity might form).

It is unclear why visible super-It is unclear why visible super-Planckian domain forms by the Planckian domain forms by the scattering of 5-dim BH’s.scattering of 5-dim BH’s.

What is observed ?What is observed ?

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潮汐力による加速

L = 時空の曲率半径

mm

X = 粒子間の距離

大雑把に

重力源 9

Geodesic deviation equations

mm

粒子間の距離がコンプトン長でも、プランク時間内でプランクエネルギーまで加速される

時空の曲率半径　 L が プランク長さ lpl より短いとき

量子論的粒子生成で生まれた粒子は、高エネルギー衝突によりブラックホールを形成？

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