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22017/10/12 @ Einstein Toolkit, Mallorca, Spain

Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

http://www.oit.ac.jp/is/~shinkai/

Hisaaki Shinkai (Osaka Inst. Tech., Japan)

Who am I ? 1996-99 Postdoc at WashU.  Newman-Penrose scalar in 3+1 language  Cactus developer at the very initial phase Formulation Problem of Einstein equations Higher-dimensional numerical relativity Alternative Gravity Theories, Approaches Chair of the Board, KAGRA Scientific Congress

32017/10/12 @ Einstein Toolkit, Mallorca, Spain

Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

http://www.oit.ac.jp/is/~shinkai/

Hisaaki Shinkai (Osaka Inst. Tech., Japan)

4-dim, 5-dim, 6-dim,  … how dimensionality affects to dynamics? Gauss-Bonnet terms  … how higher-order curvature terms affects to dynamics? 2 models Colliding scalar pulses / Fate of wormholes Ref: HS & Torii , PRD 96(2017)044009 [arXiv:1706.02070] 

Spheroidal matter collapse 4D vs 5D

5D collapses -- proceed rapidly. -- towards spherically. -- AH forms in wider ranges.

I = RabcdRabcd

at I(tend)

Spheroidal matter collapse 4D vs 5D

Dynamics in Gauss-Bonnet gravity?

E>O PEB OFI HBOP HB>AFJD PB IO C I P FJD EBE>O PS O H PF J >J EBO 9 J J 9 E>O IFJFI I I>OO C OP>PF O EB F >H -3 O H PF J  

FO BT B PBA P E>RB OFJD H> FP >R FA>J B CB>P B  P E>O JBRB BBJ ABI JOP >PBA FJ C HH D >RFP

JBS P F FJ J IB F >H BH>PFRFP  A 7R RH B MTE 4 .+ (& ( & & +  

4ISSI 4 ., (& ( & &  6 9 EWTMI E 5 RHTMKWI (&- ,

I E >PPBJPF JO FJ 3 II JFP  8 EIHE R EYE 4 -. (&&. &( &&+  

E M 2 IMLEW : W &- (& (- &  E M 2 IMLEW : W 4 .+ (& ( & &&-  

　Introduction

B BRTMM 8 EIHE 4 - (&&+ ( &&(C 1L 2 7YEN 2 8 II C II 5WT L : 3-+ (& + )-(

Formulation for evolution [N+1] 　Introduction (2)

JFPF>H >H B . JOP PF J RF> . JC I>H > > E 2 EGN LR I M M ME HE E0 8 DR LM R 4 .) (& & & &

BP C 0M >PF JO TIEH FW GRPS MGE IH 

Formulation for evolution [dual null] 　Field Equations (1)

x

+x

�

⌃0

Evolution

Formulation for evolution [dual null] 　Field Equations (1)

matter variables 　Field Equations (2)

evolution equations (1) 　Field Equations (3)

evolution equations (2) 　Field Equations (4)

@+ @�

I(5) = RijklRijkl

GR 5d: small amplitude waves 　Colliding Scalar Waves

flat background, normal scalar field

x

+x

+x

�x

�

@+ @�

I(5) = RijklRijkl

GR 5d: large amplitude waves 　Colliding Scalar Waves

��

��

��

��

��

��

��

��

��

GR 5d GaussBonnet 5d

I(5) = RijklRijkl

I(5) at origin

x

�

I(5) = RijklRijkl

GaussBonnet 5d (negative α)

↵GB = +1

↵GB = �1

↵GB = 0

↵GB = �1

↵GB = +1

↵GB = 0

　Colliding Scalar Waves

max (RijklRijkl

)

＊4dim, 5dim, 6dim,… higher dim ＊Gauss-Bonnet coupling (α>0) ̶> less growth of curvature

↵GB = �1

↵GB = +1

↵GB = 0

��

��

��

��

��

��

��

��

��

I(5) at origin

↵GB = �1

↵GB = +1

↵GB = 0

maximum of Kretschmann invariant

BH & WH are interconvertible?

EB > B RB OFIFH> PE JP>FJ I> DFJ>HH P > BAO C> BO >JA >J B ABWJBA P > FJD E FV JO 3

7JH PEB > O>H J>P B C PEB 3O AF B O SEBPEB 3OBR HRB FJ H O IFJ O ABJOFP SEF E FO DFRBJ H >HH

, 3> S> A JP A 8E O / ) (

Black Hole WormholeLocally defined

by

Achronal (spatial/null) outer TH ⇨ 1-way traversable

Temporal (timelike) outer THs ⇨ 2-way traversable

Einstein eqs.

Positive energy density  normal matter (or vacuum)

Negative energy density  “exotic” matter

Appear-ance occur naturally

Unlikely to occur naturally. but constructible??

　Wormhole Evolution

Wormhole evolution

#+ = 0 #� = 0#+ = 0#� = 0

x

+x

�x

+x

�

Positive Energy Input= add normal scalar field= subtract ghost field

Negative Energy Input= add ghost scalar field

Black Hole InflationaryExpansion

4d GR ↵GB = 0 HS-Hayward PRD 66(2002) 044005

⌃(t)

#+ = 0 #� = 0

⌃(t)

#+ = 0#� = 0

N RY JEG

�E > 0

�E < 0

4d GR ↵GB = 0 HS-Hayward PRD 66(2002) 044005

#+ = 0 #� = 0#+ = 0#� = 0

x

+x

�x

+x

�

Positive Energy Input= add normal scalar field= subtract ghost field

Negative Energy Input= add ghost scalar field

Black Hole InflationaryExpansion

⌃(t)

#+ = 0 #� = 0

⌃(t)

#+ = 0#� = 0

Wormhole evolution N RY JEG

真貝著タイムマシンと時空の科学

4-dim.

5-dim.

6-dim.

x

+x

�

Sn�2

#+ = 0#� = 0

Positive Energy

Black Hole

In higher dim, large instability. （linear perturbation analysis）

Wormhole evolution in n-dim n-dim GR Torii-HS PRD 88 (2013) 064027↵GB = 0

N RY JEG

4-dim.

5-dim.

6-dim.

x

+x

�

Sn�2

#+ = 0#� = 0

Positive Energy

Black Hole

In higher dim, large instability. （linear perturbation analysis）

Wormhole evolution in n-dim n-dim GR Torii-HS PRD 88 (2013) 064027↵GB = 0

N RY JEG

Confirmed Numerically

↵GB = +0.001

Wormhole evolution in Gauss-Bonnet のおさらい 5d Gauss-Bonnet WH : positive energy injection

MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031

(a)

0.0

0.5

1.0

1.5

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Circumference radius of throat [5dim GB alpha=+0.001](with perturabation of positive-energy pulse)

alpha=0.001, c1=0.1, E=+0.11alpha=0.001, c1=0.2, E=+0.46alpha=0.001, c1=0.3, E=+1.04alpha=0.001, c1=0.5, E=+2.85alpha=0.001, c1=0.7, E=+5.49

circ

umfe

renc

e ra

dius

of t

he th

roat

proper time at the throat

(a)

ΔE > ΔE5 > 0 --> BH collapse ΔE < ΔE5 --> Inflationary expansion

�E < +0.5

�E > +0.5

↵GB = +0.001

Wormhole evolution in Gauss-Bonnet のおさらい 6d Gauss-Bonnet WH : positive energy injection

MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031

ΔE > ΔE6 > ΔE5 > 0 --> BH collapse ΔE < ΔE5 < ΔE6 --> Inflationary expansion

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Circumference radius of throat [6dim GB alpha=+0.001](with perturabation of positive-energy pulse)

alpha=0.001, c1=0.50, E=+1.5276alpha=0.001, c1=0.60, E=+2.1954alpha=0.001, c1=0.65, E=+2.5790alpha=0.001, c1=0.70, E=+2.9896

circ

umfe

renc

e ra

dius

of t

he th

roat

proper time at the throat

(b)(b)

�E > +2.2

�E < +2.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

horiz

on lo

catio

ns

(alph

a=+0

.01, w

ith p

ertur

batio

n of

posit

ive e

nerg

y)

theta_

+=0 :

alph

a=0.0

1, a=

0.0

theta_

-=0 : a

lpha=

0.01,

a=0.0

theta_

+=0 :

alph

a=0.0

1, a=

0.5

theta_

-=0 : a

lpha=

0.01,

a=0.5

theta_

+=0 :

alph

a=0.0

1, a=

0.6

theta_

-=0 : a

lpha=

0.01,

a=0.6

theta_

+=0 :

alph

a=0.0

1, a=

0.7

theta_

-=0 : a

lpha=

0.01,

a=0.7

theta_

+=0 :

alph

a=0.0

1, a=

0.8

theta_

-=0 : a

lpha=

0.01,

a=0.8

xminu

sxp

lus

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

horiz

on lo

catio

ns

(alph

a=+0

.01, w

ith p

ertur

batio

n of

posit

ive e

nerg

y)

theta_

+=0 :

alph

a=0.0

1, a=

0.77

theta_

-=0 : a

lpha=

0.01,

a=0.7

7

theta_

+=0 :

alph

a=0.0

1, a=

0.78

theta_

-=0 : a

lpha=

0.01,

a=0.7

8

theta_

+=0 :

alph

a=0.0

1, a=

0.79

theta_

-=0 : a

lpha=

0.01,

a=0.7

9

theta_

+=0 :

alph

a=0.0

1, a=

0.80

theta_

-=0 : a

lpha=

0.01,

a=0.8

0

xminu

sxp

lus

critical behavior

existence of trapped surface ̶> not necessary to form a BH

↵GB = 0.01

5d Gauss-Bonnet WH : trapped surface

　Colliding Scalar Waves

Summary

　Wormhole Evolution

max (RijklRijkl

)

1 PE I ABHO PEB J I>H B PF JO - S G C>R FAFJD PEB > B> >J B C OFJD H> FP >HPE DE FP FO FJBRFP> HB

CI JRW H LE M LI GTM MGE M WE MR JRTJRTPM K E 28 LI I M I GI RJ LI TESSIHTIKMR M LI 5M IM 72 KTE M HRI RHMTIG M HMGE I E JRTPE MR RJ E 28

ΔE > ΔE6 > ΔE5 > 0 --> BH collapse ΔE < ΔE5 < ΔE6 --> Inflationary expansion

Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity › is › shinkai › Viewgraphs ›...

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