Hisaaki Shinkai　真貝寿明 (Osaka Inst. Tech., Japan)

12017/7/6 The 13th International Conference on Gravitation, Astrophysics, & Cosmology (ICGAC-XIII)

@ Seoul, Korea

Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

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

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 , arXiv:1706.02070 


Dynamics in Gauss-Bonnet gravity?

•  has  the  simplest  leading  terms  from  String  Theory      •  has  two  solution  branches  (GR/non-‐‑‒GR).    •  has  minimum  mass  for  static  spherical  BH  solution 


•  is  expected  to  have  singularity  avoidance  feature.   
(but  has  never  been  demonstrated  in  full  gravity.)      •new  topic  in  numerical  relativity. 
S  Golod  &  T  Piran,  PRD  85  (2012)  104015 
N  Deppe+,  PRD  86  (2012)  104011
F  Izaurieta  &  E  Rodriguez,  1207.1496  

•much  attentions  in  WH  community
H  Maeda  &  M  Nozawa,  PRD  78  (2008)  024005  
P  Kanti,  B  Kleihaus  &  J  Kunz,  PRL  107  (2011)  271101 
P  Kanti,  B  Kleihaus  &  J  Kunz,  PRD  85  (2012)  044007   


　Introduction

T  Torii  &  H  Maeda,  PRD  71  (2005)  124002  W-‐‑‒K  Ahn,  B  Gwak,  B-‐‑‒H  Lee,  W  Lee,  Eur.  Phys.  J.  C75  (2015)  372

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

@+ @�

I(5) = RijklRijkl

GR 5d: large amplitude waves 　Colliding Scalar Waves

��

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��

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��

�� �� �� �� �� �� ��

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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?

They  are  very  similar    -‐‑‒-‐‑‒  both  contain  (marginally)  trapped  surfaces  and  can  be  defined  by  trapping  horizons  (TH)  

Only  the  causal  nature  of  the  THs  differs,  whether  THs  evolve  in  plus  /  minus  density  which  is  given  locally.  

S.A.  Hayward,  Int.  J.  Mod.  Phys.  D  8  (1999)  373

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

BH & WH are interconvertible?

They  are  very  similar    -‐‑‒-‐‑‒  both  contain  (marginally)  trapped  surfaces  and  can  be  defined  by  trapping  horizons  (TH)  

Only  the  causal  nature  of  the  THs  differs,  whether  THs  evolve  in  plus  /  minus  density  which  is  given  locally.  

S.A.  Hayward,  Int.  J.  Mod.  Phys.  D  8  (1999)  373

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

 (known  fact)

�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  (known  fact)

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

 (known  fact)

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

 (known  fact)

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

)

For  both  models,  the  normal  corrections  (αGB  >  0)  work  for  avoiding  the  appearance  of  singularity,  although  it  is  inevitable.  

We  found  that  in  the  critical  situation  for  forming  a  BH,  the  existence  of  the  trapped  region  in  the  Einstein-‐‑‒GB  gravity  does  not  directly  indicate  a  formation  of  a  BH.  

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