Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

2017/10/12 @ Einstein Toolkit, Mallorca, Spain 2

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

2017/10/12 @ Einstein Toolkit, Mallorca, Spain 3

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 = R

_abcd

R

^abcd

at I (t end)

Spheroidal matter collapse

4D vs 5D

Dynamics in Gauss-Bonnet gravity?

E>O PEB OFI HBOP HB>AFJD PB IO C I P FJD EB E>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⁾

= 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

��

����

��

����

��

����

��

�� �� �� �� �� �� ��

��������������������

��������������������

��������������������

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

��

����

��

����

��

����

��

�� �� �� �� �� �� ��

��������������������

��������������������

��������������������

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 > BA O 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 3O BR HRB FJ H O IFJ O ABJOFP SEF E FO DFRBJ H >HH

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

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?

EB > B RB OFIFH> PE JP>FJ I> DFJ>HH P > BA O 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 3O BR HRB FJ H O IFJ O ABJOFP SEF E FO DFRBJ H >HH

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

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

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 Inflationary

Expansion

⌃(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

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

N RY JEG

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

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=+0.11 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

)

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 JRT JRTPM K E 28 LI I M I GI RJ LI TESSIH TIKMR M LI 5M IM 72 KTE M HRI R HMTIG M HMGE I E JRTPE MR RJ E 28

ΔE > ΔE

⁶  

> ΔE

⁵  

> 0 --> BH collapse 

ΔE < ΔE

⁵

 < ΔE

⁶

       --> Inflationary expansion