Dynamics in n-dimensional Gauss-Bonnet gravityshinkai/Viewgraphs/201503_JPS.pdfDynamics in...

Dynamics in n-dimensional Gauss-Bonnet gravity

真貝寿明　（大阪工大情報科学部）鳥居隆　　（大阪工大工学部）

Part I Field Equations (dual-null formulation) Part II Colliding Scalar Waves Part III Wormhole dynamics

2015年3月24日　物理学会@早稲田http://www.oit.ac.jp/is/~shinkai/

論点＊4dim, 5dim, 6dim, … ダイナミクスはどう変化するか＊Gauss-Bonnet項は，ダイナミクスにどう影響するか

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

Dynamics in Gauss-Bonnet gravity?

S A J . II E IDJ I D I E 7A IPS A J J C E I E A J . E E . S J OG A M J E C I P M E I

A J E M I E D EJ I

SE G E E D I C I C M P 4MJMD H L 1 & & ) & 1ENNE 1 + & & ) &&

3 6 R HE 2 MD HFRE & , &).+

SD A E EJ E 9/ DD E P5 : ED : M S 1 , ) 8 L H 0 8JEHG RP 7 8RL 9 & , && ,&& & 8 L H 0 8JEHG RP 7 8RL 1 & )) ,

　Introduction (1)

Formulation for evolution [N+1] 　Introduction (2)

S E C 8 C EJ I E M E ID C GGI A0J I GMJE HLH H J D 5 MPGHLM 1 ( && & ) &

S -‐‑‒H EJE DT BR MKNJH ED

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

efef

large amplitude wavesGR 5d GaussBonnet 5d

↵GB = +1↵GB = 0

GR 5d GaussBonnet 5d

I(5) = RijklRijkl

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I(5) at origin I(5) at origin

x

�x

�

I(5) = RijklRijkl

↵GB = +1↵GB = 0

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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

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��I(5) at origin

x

�

↵GB = �1

↵GB = +1

↵GB = 0

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↵GB = �1

↵GB = +1

↵GB = 0

I(6) at origin

x

�

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↵GB = �1

↵GB = +1↵GB = 0

x

�

I(4) at origin

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x

�

I(4), I(5), I(6), I(7) at origin

GR4dGR5dGR6dGR7d

GR & GB 5d GR & GB 6d

GR & GB 4d GR 4d̶7d

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I(5) = RijklRijkl

at origin I(5) = RijklRijkl

at origin

↵GB = 2

↵GB = 1

↵GB = 0.5

↵GB = 0.25

x

�/

p↵̃

x

�

BH & WH are interconvertible?

7A P I M IP J D C I A E E D I E CCP I GGJ I J E E RE P I GG E A I EJ 7/

3ECP A J C E I A 7/J T IJ A A I 7/JM CM E GC J D E J EJ P A A J M E C CCP

/ P I E 1 APJ , ( )))

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

initial data: wormhole cases 　Wormhole Evolution

Ghost pulse input -- Bifurcation of the horizons（4d)

more negative field -> throat expansion

less negative field -> throat shrink

wormhole configurations (4dim. GR)

Bifurcation of the horizons -- go to a Black Hole or Inflationary expansion

Normal pulse (a traveller) input -- Forming a Black Hole

0

1

2

3

4

5

6

0

1

2

3

4

5

6

- ghost

pulse (c

_1=-0.

1)

4dim

, c1=

-0.1,

theta+ =0

4dim

, c1=

-0.1,

theta- =0

5dim

, c1=

-0.1,

theta+ =0

5dim

, c1=

-0.1,

theta- =0

6dim

, c1=

-0.1,

theta+ =0

6dim

, c1=

-0.1,

theta- =0

x minu

s

x plus

0

1

2

3

4

5

6

0

1

2

3

4

5

6

- ghost

pulse (c

_1=-0.

01)

4dim

, c1=

-0.01

, theta

+ =0

4dim

, c1=

-0.01

, theta

- =0

5dim

, c1=

-0.01

, theta

+ =0

5dim

, c1=

-0.01

, theta

- =0

6dim

, c1=

-0.01

, theta

+ =0

6dim

, c1=

-0.01

, theta

- =0

x minu

s

x plus

ghost pulse (negative amp.) input

positive energy input --> BH formation

4d 5d 6d GR

4-d

6-d

4-d

6-d

double trapping horizon

ghost pulse (positive amp.) input

negative energy input --> throat inflates

0

1

2

3

4

5

6

0

1

2

3

4

5

6

+ ghost

pulse (c

_1=0.1

)

4dim

, c1=

+0.1, th

eta+ =0

4dim

, c1=

+0.1, th

eta- =0

5dim

, c1=

+0.1, th

eta+ =0

5dim

, c1=

+0.1, th

eta- =0

6dim

, c1=

+0.1, th

eta+ =0

6dim

, c1=

+0.1, th

eta- =0

x minu

s

x plus

0

1

2

3

4

5

6

0

1

2

3

4

5

6

+ ghost

pulse (c

_1=0.0

1)

4dim

, c1=

+0.01,

theta+ =0

4dim

, c1=

+0.01,

theta- =0

5dim

, c1=

+0.01,

theta+ =0

5dim

, c1=

+0.01,

theta- =0

6dim

, c1=

+0.01,

theta+ =0

6dim

, c1=

+0.01,

theta- =0

x minu

sx p

lus

4d 5d 6d GR

double trapping horizon

4-d

6-d

4-d

6-d

0.0

1.0

2.0

3.0

4.0

0.0 2.0 4.0 6.0 8.0 10.0

initial data (5-dim, Gauss-Bonnet alpha)

theta_+ alpha=+1.00theta_+ alpha=+0.50theta_+ alpha=+0.10theta_+ alpha=+0.05theta_+ GR5dtheta_+ alpha=-0.05theta_+ alpha=-0.10

thet

a_+

(exp

ansi

on)

x-plus

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10.0

initial data (5-dim, Gauss-Bonnet alpha)

Omega alpha=+1.00Omega alpha=+0.50Omega alpha=+0.10Omega alpha=+0.05Omega GR5dOmega alpha=-0.05Omega alpha=-0.10

omega

x-plus

-0.5

0.0

0.5

1.0

1.5

2.0initial data (5-dim, Gauss-Bonnet alpha)

phi alpha=+1.00phi alpha=+0.50phi alpha=+0.10phi alpha=+0.05phi GR5dphi alpha=-0.05phi alpha=-0.10

0.0 2.0 4.0 6.0 8.0 10.0

phi (

scal

ar fi

eld)

x-plus

conformal factor scalar field

Initial Data Part 3. Wormhole in Gauss-Bonnet gravity

throat inflates

5d GR vs Gauss-Bonnet instability appears

BH formation

�GB > 0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

horiz

on loca

tions

(Gau

ss-B

onnet

n=5, a

lpha >

0)

theta+=0, GR5

theta-=0, GR5

theta+=0, GB5, alpha=+0.01

theta-=0, GB5, alpha=+0.01





x-minu

s / th

roat ra

dius

x-plus

/ thro

at rad

ius

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

horiz

on loca

tions

(Gau

ss-B

onnet

n=5, a

lpha <

0)

theta+=0, GR5

theta-=0, GR5

theta+=0, GB5, alpha=-0.01

theta-=0, GB5, alpha=-0.01





x-minu

s / th

roat ra

dius

x-plus

/ thro

at rad

ius

wormhole configurations (5dim. GaussBonnet)�GB > 0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

horiz

on loca

tions

(Gau

ss-B

onnet

n=5, a

lpha >

0)

theta+=0, GR5

theta-=0, GR5







x-minu

s / th

roat ra

dius

x-plus

/ thro

at rad

ius

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

horiz

on loca

tions

(Gau

ss-B

onnet

n=5, a

lpha <

0)

theta+=0, GR5

theta-=0, GR5







x-minu

s / th

roat ra

dius

x-plus

/ thro

at rad

ius

6d 7d Gauss-Bonnet instability appears easily

�GB > 0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

horiz

on loca

tions

(Gau

ss-B

onnet

n=6, a

lpha >

0)

theta+=0, GR6

theta-=0, GR6







x-minu

s / th

roat ra

dius

x-plus

/ thro

at rad

ius

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

horiz

on loca

tions

(Gau

ss-B

onnet

n=7, a

lpha >

0)

theta+=0, GR7

theta-=0, GR7





x-minu

s / th

roat ra

dius

x-plus

/ thro

at rad

ius

�GB > 0

6 dim. 7 dim.

　Colliding Scalar Waves

Summary

　Wormhole Evolution

max (RijklRijkl

)

＊4dim, 5dim, 6dim,… 高次元化＊Gauss-Bonnet項（正αの項）は，どちらも特異点形成条件を緩くさせる

5,6,7次元 Gauss-Bonnet負αの GB coupling --> BH collapse 正αの GB coupling --> Inflationary expansion

massless scalar waveの衝突による特異点形成

wormhole解(ghost scalar)に摂動を加える

5,6,7次元 Gauss-Bonnet

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