Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

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

1

2017/7/6  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⁾ = R_ijklR^ijkl

GR 5d: small amplitude waves

  Colliding Scalar Waves

flat background, normal scalar field

@₊

@

I⁽⁵⁾ = R_ijklR^ijkl

GR 5d: large amplitude waves

  Colliding Scalar Waves

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GR 5d GaussBonnet 5d

I⁽⁵⁾ = R_ijklR^ijkl

I⁽⁵⁾ at origin

x I⁽⁵⁾ = R_ijklR^ijkl

GaussBonnet 5d (negative α)

↵_GB = +1

↵_GB = 1

↵_GB = 0

↵_GB = 1

↵_GB = +1

↵_GB = 0

  Colliding Scalar Waves

max (R_ijklR^ijkl)

＊4dim, 5dim, 6dim,…  higher dim 

＊Gauss-Bonnet coupling (α>0)     ̶> less growth of curvature 

↵_GB = 1

↵_GB = +1

↵_GB = 0

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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 Wormhole

Locally  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 Wormhole

Locally  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 Inflationary

Expansion

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 Inflationary

Expansion

⌃(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

S

^{n 2}

#

⁺

= 0

# = 0

Positive Energy Black Hole

In higher dim, large instability. 

（linear perturbation analysis）

Wormhole evolution in n-dim

n-dim GR ↵ _GB = 0 Torii-HS PRD 88 (2013) 064027

 (known  fact)

4-dim.

5-dim.

6-dim.

x

⁺

x

S

^{n 2}

#

⁺

= 0

# = 0

Positive Energy Black Hole

In higher dim, large instability. 

（linear perturbation analysis）

Wormhole evolution in n-dim

n-dim GR ↵ _GB = 0 Torii-HS PRD 88 (2013) 064027

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

circumference radius of the throat

proper time at the throat (a)

ΔE > ΔE

⁵  

> 0 --> BH collapse 

ΔE < ΔE

⁵

       --> 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 > ΔE

⁶  

> ΔE

⁵  

> 0 --> BH collapse 

ΔE < ΔE

⁵

 < ΔE

⁶

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

circumference radius of the throat

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

horizon

locations

(alpha=+0.01, with

perturbation

of positive energy)

theta_+=0 : alpha=0.01,

a=0.0

theta_-=0

: alpha=0.01, a=0.0

theta_+=0 : alpha=0.01,

a=0.5

theta_-=0 : alpha=0.01,

a=0.5

theta_+=0 : alpha=0.01,

a=0.6

theta_-=0

: alpha=0.01, a=0.6

theta_+=0 : alpha=0.01,

a=0.7

theta_-=0 : alpha=0.01,

a=0.7

theta_+=0 : alpha=0.01,

a=0.8

theta_-=0

: alpha=0.01, a=0.8

xminus xplus

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

horizon

locations

(alpha=+0.01, with

perturbation

of positive energy)

theta_+=0 : alpha=0.01, a=0.77 theta_-=0 : alpha=0.01, a=0.77

theta_+=0 : alpha=0.01, a=0.78 theta_-=0 : alpha=0.01, a=0.78

theta_+=0 : alpha=0.01, a=0.79 theta_-=0 : alpha=0.01, a=0.79

theta_+=0 : alpha=0.01, a=0.80 theta_-=0 : alpha=0.01, a=0.80

xminus xplus

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 (R_ijklR^ijkl)

For  both  models,  the  normal  corrections  (α

^GB

 >  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 > ΔE

6  

> ΔE

5  

> 0 --> BH collapse 

ΔE < ΔE

5

 < ΔE

6

       --> Inflationary expansion