Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

 1

2018/7/2 @ MG15, Rome

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] 


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 [N+1]

  Introduction (2)

•Initial Value Construction via Conformal approach



Black hole initial data: H Yoshino , PRD 83 (2011) 104010

•Set of Equations 


ready, but complicated


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

_ijkl

R

^ijkl

GR 5d: small amplitude waves

  Colliding Scalar Waves

flat background, normal scalar field

x ⁺ x ⁺

x x

@

₊

@

I

⁽⁵⁾

= R

_ijkl

R

^ijkl

GR 5d: large amplitude waves

  Colliding Scalar Waves

��

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GR 5d GaussBonnet 5d

I

⁽⁵⁾

= R

_ijkl

R

^ijkl

I

⁽⁵⁾

at origin

x I

⁽⁵⁾

= R

_ijkl

R

^ijkl

GaussBonnet 5d (negative α)

↵

_GB

= +1

↵

_GB

= 1

↵

_GB

= 0

↵

_GB

= 1

↵

_GB

= +1

↵

_GB

= 0

  Colliding Scalar Waves

max (R

_ijkl

R

^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

_ijkl

R

^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