Nonlinear dynamics
in the Einstein-Gauss-Bonnet gravity
Hisaaki Shinkai 真貝寿明
(Osaka Inst. Tech., Japan)
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
2017/7/6 The 13th International Conference on Gravitation, Astrophysics, & Cosmology (ICGAC-XIII)@ Seoul, Korea
1Introduction

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

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

Evolutionxx⌃0+Field Equations (1)Formulation for evolution [dual null]Field Equations (2)matter variables

Field Equations (3)evolution equations (1)Field Equations (4)evolution equations (2)Colliding Scalar Waves

GR 5d: small amplitude waves
flat background, normal scalar field@+@

I (5) = Rijkl Rijkl

Colliding Scalar Waves

GR 5d: large amplitude waves@+@

I (5) = Rijkl Rijkl

I (5) = Rijkl Rijkl↵GB = 0I (5) = Rijkl Rijkl

GR 5d↵GB = +1

GaussBonnet 5d��↵GB =I1(5)����������������������������������������
��������������������

at origin

����

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↵GB =1��

↵GB = 0����GaussBonnet 5d (negative α)��↵GB = +1
��

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x��

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

Colliding Scalar Waves

max (Rijkl Rijkl )maximum of Kretschmann invariant↵GB =1��↵GB = 0I(5)��������������������
��������������������
��������������������

at origin����

��

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↵GB = +1

↵GB =1��↵GB = 0������↵GB = +1
��

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Wormhole Evolution

BH & WH are interconvertible?
S.A.  Hayward,  Int.  J.  Mod.  Phys.  D  8  (1999)  373They  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.  

Black Hole

Wormhole

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

Temporal (timelike)outer THs
⇨ 2-way traversable

Einsteineqs.

Positive energy density
normal matter (orvacuum)

Negative energy density“exotic” matteroccur naturally

Unlikely to occur
naturally.
but constructible??

AppearanceWormhole Evolution

BH & WH are interconvertible?
S.A.  Hayward,  Int.  J.  Mod.  Phys.  D  8  (1999)  373They  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.  

Black Hole

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

Temporal (timelike)outer THs
⇨ 2-way traversable

Einsteineqs.

Positive energy density
normal matter (orvacuum)

Negative energy density“exotic” matteroccur naturally

Unlikely to occurnaturally.
but constructible??

Appearance

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

   ⌃(t)⌃(t)#+ = 0# =0#+ = 0InflationaryExpansion

Black Hole

# =0x

# =0

#+ = 0x+Positive Energy Input= add normal scalar field= subtract ghost fieldx#+ = 0# =0x+

Negative Energy Input= add ghost scalar field↵GB = 0
HS-Hayward PRD 66(2002) 044005
Wormhole evolution   (known  fact)

4d GR	    ⌃(t)⌃(t)#+ = 0# =0#+ = 0InflationaryExpansion

Black Hole

# =0x

# =0

#+ = 0x+

Positive Energy Input= add normal scalar field= subtract ghost fieldx#+ = 0# =0x+

Negative Energy Input= add ghost scalar field

E<0E>0真貝著
タイムマシンと時空の科学

Torii-HS PRD 88 (2013) 064027
n-dim GR ↵GB = 0
Wormhole evolution in n-dim   (known  fact)
4-dim.
Black Hole

5-dim.
6-dim.

x+x#+ = 0# =0Positive Energy

Sn 2

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

Torii-HS PRD 88 (2013) 064027
n-dim GR ↵GB = 0
Wormhole evolution in n-dim   (known  fact)
4-dim.
Black Hole

5-dim.
6-dim.

x+x#+ = 0# =0Positive Energy

Sn 2

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

Conﬁrmed
Numerically5d WormholeGauss-Bonnet
evolution
WH : positivein Gauss-Bonnet
energy injectionのおさらい↵GB = +0.001
MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031
Circumference radius of throat [5dim GB alpha=+0.001](with perturabation of positive-energy pulse)

(a)
2.0

circumference radius of the throat

(a)

1.5

E < +0.5

1.0

E > +0.5
0.5alpha=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

0.00.01.02.0

3.04.0proper time at the throat

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

5.06.06d Wormhole
Gauss-Bonnetevolution
WH : positivein Gauss-Bonnet
energy injectionのおさらい↵GB = +0.001
MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031
Circumference radius of throat [6dim GB alpha=+0.001](with perturabation of positive-energy pulse)

(b)
2.0

circumference radius of the throat

(b)

1.5

E < +2.2
1.0

E > +2.20.5
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
0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

proper time at the throat

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

3.5

4.0

00

0.

5

55

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lus

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xp

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3.

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a_+=

a_-=

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a_:a0,+=01:aw ho
ith ripe zonrt l
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3.

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a_(aet-=haa_=+,lp

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ith r
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lp
ha=0h0a
et: a lph = 0 .01
tha +et _ - =0.
lath
a_ = 0 : a p h a = 0 0 1 , , a =e: a lph = 0 .01 a = 0.0
th t a _ +=e0t h ta_ - = 0 : a l p h a = 0 . 0 1 , a = 0 . 0
et: a lph a = 0 .01 , a = 0.5
a _ +=, a 0.
lp
a= . 0-= 0 :
5
1=00a h: a lph a = 0 .01 , a = 0.6, a 0.
lp
a= . 061
ha=03.
= 0 .01 , a = 0.7
5.0, a 0.
71,=a = 0.8
0.
4.
8ththetet

et

0.

th

=+th

hath

lp

th(athth etath et _+th eta a_- =0th et _+ =0 :th eta a_- =0 : a alp
th et _+ =0 : lph hath eta a_- =0 : a alp a= =0.
et _ =0 : lp ha 0. 01
a_ += : al ha = 01 ,-= 0 alp ph =0 0.0 , a a=2.
0 : a h a= .0 1, = 0.
5
: a lp a= 0 1 a 0. 77
lp ha 0. .01 , a =0 77
ha = 01 , =0 .7
3.
=0 0.0 , a a= .7 8
0.0 1, =0 0.7 81, a .7 9
a= =0 93.
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↵GB = 0.01

0.

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usin
xm

3.

3.

4.

5d Gauss-Bonnet WH : trapped surface
critical behavior

existence of trapped surface
̶> not necessary to form a BH

SummaryColliding Scalar Waves
max (Rijkl Rijkl )Wormhole Evolution
ΔE > ΔE6 > ΔE5 > 0 --> BH collapse
ΔE < ΔE5 < ΔE6--> Inﬂationary expansion
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.  

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