Fate

ofth

eTra

versib

leW

orm

hole

s—

–B

lack

-Hole

Colla

pse

or

Inflatio

nary

Expansio

n—

–

真貝寿明

Hisa

-akiShin

kai　

理化学研究所　基礎科学特別研究員　（計算科学技術推進室）

Com

putationalSci.

Div.,

RIK

EN

(The

Instituteof

Physical

andChem

icalResearch),

Japan

Sean

A.H

ayw

ard

Dept.

ofScience

Education,

Ew

haW

omans

Univ.,

Seoul,

Korea

OU

TLIN

E

•Traversible

worm

hole(M

orris-Thorne

worm

hole,1988)

•Black

Hole

-W

ormhole

synthesis(H

ayward,

1999)

•“D

ynamical

Worm

hole”

•A

numerical

approach,dual-null

formulation

•A

newtyp

eof

criticalbehaviour??

HS

andS.A

.H

ayward,

Phys.

Rev.

D.66

(2002)044005

Morris-T

horne’s“T

raversable”worm

hole

M.S

.M

orrisand

K.S

.T

horne,Am

.J.

Phys.

56(1988)

395

M.S

.M

orris,K

.S.T

horne,and

U.Yurtsever,

PRL

61(1988)

3182

H.G

.Ellis,

J.M

ath.Phys.

14(1973)

104

(G.Clem

ent,Am

.J.

Phys.

57(1989)

967)

Desired

properties

oftraversable

WH

s

1.Spherically

symm

etricand

Static⇒

M.Visser,

PRD

39(89)3182

&N

PB

328(89)

203

2.Einstein

gravity

3.Asym

ptoticallyflat

4.N

ohorizon

fortravel

through

5.T

idalgravitational

forcesshould

be

small

fortraveler

6.Traveler

shouldcross

itin

afinite

andreasonably

small

proper

time

7.M

usthave

aphysically

reasonablestress-energy

tensor

⇒W

eakEnergy

Condition

isviolated

atthe

WH

throat.

⇒(N

ullEC

isalso

violatedin

generalcases.)

8.Should

be

perturbatively

stable

9.Should

be

possible

toassem

ble

1W

hy

Worm

hole

?

•T

heym

akegreat

sciencefiction

–short

cutsbetw

eenotherw

isedistant

regions.M

orris&

Thorne

1988,Sagan

“Contact”

etc

•T

heyincrease

ourunderstanding

ofgravity

when

theusual

energyconditions

arenot

satisfied,due

toquantum

effects

(Casim

ireff

ect,H

awking

radiation)or

alternativegravity

theories,brane-w

orldm

odelsetc.

•T

heyare

verysim

ilarto

blackholes

–both

contain(m

arginally)trapp

edsur-

facesand

canbe

definedby

trappinghorizons

(TH

).

Worm

hole≡H

ypersurface

foliatedby

marginally

trapped

surfaces

•BH

andW

Hare

interconvertible?N

ewduality?

BH

and

WH

arein

terconvertib

le?

(New

Duality?)

S.A

.H

ayward,

Int.J.

Mod.

Phys.

D8

(1999)373

•T

heyare

verysim

ilar–

both

contain(m

arginally)trapp

ed

surfacesand

canbe

defined

bytrapping

horizons(T

H)

•O

nlythe

causalnature

ofthe

TH

sdiff

ers,w

hetherT

Hs

evolvein

plus/

minus

density.

Black

Hole

Worm

hole

Locally

defined

by

Achronal(spatial/null)

outerT

H

Tem

poral

(timelike)

outerT

Hs

⇒1-w

aytraversable

⇒2-w

aytraversable

Einstein

eqs.Positive

energydensity

Negative

energydensity

normal

matter

(orvacuum

)“exotic”

matter

App

earanceoccur

naturallyU

nlikelyto

occurnaturally.

butconstructible

???

2Fate

ofM

orris-T

horn

e(E

llis)w

orm

hole

?

•“D

ynam

icalw

orm

hole

”defined

by

loca

ltra

ppin

ghorizo

n

•sp

herica

llysy

mm

etric,

both

norm

al/

ghost

KG

field

•apply

dual-n

ull

form

ula

tion

inord

er

tose

ek

horizo

ns

•N

um

erica

lsim

ula

tion

2.1

ghost/

norm

alK

lein

-Gord

on

field

s

Lagrangian:

L=√−

g

R16π−

14π

12(∇

ψ)2+

V1 (ψ

)

︸︷︷

︸norm

al

+14π

12(∇

φ)2+

V2 (φ

)

︸︷︷

︸ghost

The

fieldequations

Gµν

=2

ψ,µ ψ

,ν −g

µν

12(∇

ψ)2+

V1 (ψ

)

−2

φ,µ φ

,ν −g

µν

12(∇

φ)2+

V2 (φ

)

ψ=

dV

1 (ψ)

dψ

,φ

=dV

2 (φ)

dφ

.(H

ereafter,we

setV

1 (ψ)

=0,V

2 (φ)

=0)

2.2

dual-n

ull

form

ula

tion,sp

herica

llysy

mm

etric

space

time

SA

Hayw

ard,CQ

G10

(1993)779,

PRD

53(1996)

1938,CQ

G15

(1998)3147

•T

hespherically

symm

etricline-elem

ent:

ds

2=

r2d

S2−

2e −fd

x+dx−,

where

r=

r(x+,x

−),f

=f(x

+,x

−),···

•T

heEinstein

equations:

∂±∂±

r+

(∂±f)(∂±

r)=

−r(∂±

ψ) 2

+r(∂±

φ) 2,

r∂+∂−

r+

(∂+r)(∂−

r)+

e −f/2

=0,

r2∂

+∂−

f+

2(∂+r)(∂−

r)+

e −f

=+

2r2(∂

+ψ

)(∂−ψ

)−2r

2(∂+φ)(∂−

φ),

r∂+∂−

φ+

(∂+r)(∂−

φ)+

(∂−r)(∂

+φ)

=0,

r∂+∂−

ψ+

(∂+r)(∂−

ψ)+

(∂−r)(∂

+ψ

)=

0.

•To

obtaina

systemaccurate

near±,we

introducethe

conformal

factorΩ

=1/r

.W

ealso

definefirst-order

variables,the

conformally

rescaledm

omenta

expansionsϑ±

=2∂±

r=−

2Ω−

2∂±Ω

(θ±=

2r −1∂±

r)(1)

inaffinities

ν±=

∂±f

(2)

mom

entaof

φ℘±

=r∂±

φ=

Ω−

1∂±φ

(3)

mom

entaof

ψπ±

=r∂±

ψ=

Ω−

1∂±ψ

(4)

The

setof

equations(cont.):

∂±ϑ±

=−

ν±ϑ±−

2Ωπ

2±+

2Ω℘

2±,

(5)

∂±ϑ∓

=−

Ω(ϑ

+ϑ−/2

+e −

f),(6)

∂±ν∓

=−

Ω2(ϑ

+ϑ−/2

+e −

f−2π

+π−

+2℘

+℘−),

(7)

∂±℘∓

=−

Ωϑ∓℘±/2,

(8)

∂±π∓

=−

Ωϑ∓π±/2.

(9)

andrem

ember

theidentity:

∂+∂−

=∂−

∂+:

2.3

Initia

ldata

on

x+

=0,

x−

=0

slices

and

on

S

Generally,

we

haveto

set:(Ω

,f,ϑ

±,φ

,ψ)

onS

:x

+=

x−

=0

(ν±,℘

±,π

±)

onΣ±:

x∓

=0,

x±≥

0

Grid

Structure

forN

umerical

Evolution

xplusxm

inus

worm

hole throat

S

2.4

Morris-T

horn

e(E

llis)w

orm

hole

as

the

initia

ldata

onΣ

+(x

−=

0surface)

onΣ−

(x+

=0

surface)

Ω1/ √

a2+

z2

1/ √a

2+

z2

f0

0

ϑ±

±√

2z/ √a

2+

z2

∓√

2z/ √a

2+

z2

ν+

0

ν−0

φtan −

1(z/a)

−tan −

1(z/a)

℘+

+a/ √

2 √a

2+

z2

℘−

−a/ √

2 √a

2+

z2

ψ0

0

π+

0

π−

0

where

z=

(x+−

x−)/ √

2.

We

putthe

perturbation

in℘

+:

δ℘+

=ca

exp(−

cb (z−

cc )

2)

where

ca ,c

b ,cc

areparam

eters.

2.5

Gra

vita

tionalm

ass-e

nerg

y

•Localizing,

thelocal

gravitationalm

ass-energyis

givenby

theM

isner-Sharp

energyE

,

E=

(1/2)r[1−g −

1(dr,d

r)]=

(1/2)r+

efr

(∂+r)(∂−

r)=

12Ω[1

+12efϑ

+ϑ−]

while

the(localized

Bondi)

conformal

fluxvector

components

ϕ±

ϕ±

=r

2T±±∂±

r=

r2e

2fT∓∓∂±

r=

e2f(π

2∓−

℘2∓)ϑ

±/8π

.

•T

heyare

relatedby

theenergy

propagationequations

orunified

firstlaw

.∂±

E=

4πϕ±,

E(x

+,x

−)

=a2

+4π

∫(x

+,x −

)

(0,0)(ϕ

+dx

++

ϕ−dx−),

where

theintegral

isindep

endentof

path,by

conservationof

energy.

–lim

x+→

∞E

isthe

Bondi

energy

–lim

x+→

∞ϕ−

theBondi

fluxfor

theright-hand

universe.

–For

thestatic

worm

hole,the

energyE

=a

2/2 √a

2+

z2

iseveryw

herepositive,

maxim

al

atthe

throatand

zeroat

infinity,z→

±∞

,i.e.

theBondi

energyis

zero.

–G

enerally,the

Bondi

energy-lossprop

erty,that

itshould

be

non-increasingfor

matter

satisfyingthe

nullenergy

condition,is

reversedfor

theghost

field.

Num

erica

lG

rid/

Converg

ence

test

x plusx minus

wormhole throatS

Σ+

Σ−

0 5 10

05

1015

20

grid=801

grid=1601

grid=3201

grid=6401

grid=9601

x plus

x minus Figu

re1:

Num

ericalgrid

structu

re.In

itialdata

aregiven

onnull

hypersu

rfacesΣ

±(x

∓=

0,x±

>0)

and

their

intersection

S.

Figu

re2:

Con

vergence

beh

aviou

rof

the

code

forex

actstatic

worm

hole

initial

data.

The

location

ofth

etrap

pin

ghorizon

ϑ−

=0

isplotted

forseveral

resolution

slab

elledby

the

num

ber

ofgrid

poin

tsfor

x+

=[0,20].

We

seeth

atnum

ericaltru

ncation

erroreven

tually

destroy

sth

estatic

configu

ration.

Sta

tionary

Configura

tions

24

68

51

01

5

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

x plusx m

inus

expansion plus

24

68

51

01

5

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

x plusx m

inus

Energy

Figu

re2:

Static

worm

hole

configu

rationob

tained

with

the

high

estresolu

tioncalcu

lation:

(a)ex

pan

sionϑ

+an

d(b

)lo

calgrav

itational

mass-en

ergyE

areplotted

asfu

nction

sof

(x+,x

−).

Note

that

the

energy

ispositive

and

tends

tozero

atin

finity.

Ghost

pulse

input

–B

ifurca

tion

ofth

ehorizo

ns

0 2 4 6 8 10

02

46

810

x minus

x plus

(a) pu

lse inp

ut w

ith n

egative en

ergy

ghost scalar pulse

θ+ =

0

θ- =

0

Inflationary

expansion

θ+ <

0

θ- >

0

θ+ >

0

θ- <

00 2 4 6 8 10

02

46

810

x minus

x plus

(b1) p

ulse in

pu

t with

po

sitive energ

y

ghost scalar pulse

θ+ =

0

θ- =

0

Black

Hole

x- H

= 4.46

θ+ >

0

θ- <

0

θ+ <

0

θ- >

0

Figu

re3:

Horizon

location

s,ϑ±

=0,

forpertu

rbed

worm

hole.

Fig.(a)

isth

ecase

we

supplem

ent

the

ghost

field

,ca

=0.1,

and

(b1)

and

(b2)

arew

here

we

reduce

the

field

,ca

=−

0.1an

d−

0.01.D

ashed

lines

and

solidlin

esare

ϑ+

=0

and

ϑ−

=0

respectively.

Inall

cases,th

epulse

hits

the

worm

hole

throat

at(x

+,x

−)

=(3,3).

A45

counterclo

ckw

iserotation

ofth

efigu

recorresp

onds

toa

partial

Pen

rosediagram

.

Bifu

rcatio

nofth

ehorizo

ns

–go

toa

Bla

ckH

ole

or

Inflatio

nary

expansio

n

x plusx minus

Black HoleBlack Hole or orInflationaryInflationaryexpansionexpansionpulse

input

throat "throat"

0 1 2 3 4 5 6

02

46

810

12

amplitude =

+0.10

amplitude =

+0.01

no perturbationam

plitude = -0.01

amplitude =

-0.10

proper time on the "throat"

Areal Radius at the "throat"Figu

re4:

Partial

Pen

rosediagram

ofth

eevolved

space-tim

e.Figu

re6:

Areal

radiu

sr

ofth

e“th

roat”x

+=

x−,

plotted

asa

function

ofprop

ertim

e.A

ddition

alnegative

energy

causes

inflation

aryex

pan

sion,w

hile

reduced

negative

energy

causes

collapse

toa

black

hole

and

central

singu

larity.

Loca

lEnerg

yM

easu

re–

Dete

rmin

atio

nofth

eB

lack

Hole

Mass

0.0010

0.010

0.10

1.0 10

102

01

23

45

67

8

x plus = 12

x plus = 16

x plus = 20

Energy ( x+, x- )

x minus

(a) pu

lse inp

ut w

ith n

egative en

ergy

0.010

0.10

1.0

01

23

45

6

x plus = 12

x plus = 16

x plus = 20

Energy ( x+, x- )x m

inus

horizon θ+ =

0

formed at x

-=4

.46

(b1) p

ulse in

pu

t with

po

sitive energ

y

Black hole m

assM

= 0.42

Figu

re7:

Energy

E(x

+,x

−)as

afu

nction

ofx−,for

x+

=12,16,20.

Here

ca

is(a)

0.05,(b

1)−0.1

and

(b2)−

0.01.T

he

energy

fordiff

erentx

+coin

cides

atth

efinal

horizon

location

x−H

,in

dicatin

gth

atth

ehorizon

quick

lyattain

scon

stant

mass

M=

E(∞

,x−H

).T

his

isth

efinal

mass

ofth

eblack

hole

orcosm

ologicalhorizon

.

Isth

ere

aM

inim

um

Bla

ckH

ole

Mass

tobe

form

ed?

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.010-5

10-4

10-3

10-2

10-1

xH

- - 3

E0

(ca , c

b ) = (10

-4, 9)

(10-4, 6)

(10-4, 3)

(10-1, 3)

(10-1, 6)

(10-1, 9)

(a)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.5010-5

10-4

10-3

10-2

10-1

M

E0

(b)

(10-1, 3)(10

-1, 6)

(10-1, 9)

(ca , c

b ) = (10

-4, 9)

(10-4, 6)

(10-4, 3)

Figu

re8:

Relation

betw

eenth

ein

itialpertu

rbation

and

the

final

mass

ofth

eblack

hole.

(a)T

he

trappin

ghorizon

(ϑ+

=0)

coord

inate,

x−H−

3(sin

cew

efixed

cc

=3),

versus

initial

energy

ofth

epertu

rbation

,E

0 .W

eplotted

the

results

ofth

eru

ns

ofca

=10 −

1,···,10 −4

with

cb

=3,6,

and

9.T

hey

lieclose

toon

elin

e.(b

)T

he

final

black

hole

mass

Mfor

the

same

exam

ples.

We

seeth

atM

appears

toreach

anon

-zerom

inim

um

forsm

allpertu

rbation

s.

Norm

alP

ulse

(atra

velle

r)In

put

–Form

ing

aB

lack

Hole

0 2 4 6 8

02

46

8

x minus

x plus

normal scalar

pulse

θ+ =

0

θ- =

0

Black

Hole

Figu

re9:

Evolu

tionof

aw

ormhole

pertu

rbed

by

anorm

alscalar

field

.H

orizonlo

cations:

dash

edlin

esan

dsolid

lines

areϑ

+=

0an

dϑ−

=0

respectively.

Critica

lM

inim

um

Bla

ckH

ole

Mass

again

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.010-5

10-4

10-3

10-2

10-1

100

xH

- - 3

E0

(a)

(ca , c

b ) = (10

- 2, 9)

(10- 2, 6)

(10-2, 3)

~ ~

(0.5, 3)

(0.5, 3)

(0.5, 6)

0.20

0.25

0.30

0.35

0.40

0.45

0.5010-5

10-4

10-3

10-2

10-1

100

M

E0

(b)

(0.5, 3)

(0.5, 6)

(0.5, 9)

(ca , c

b ) = (10

- 2, 3)

(10- 2, 9) (10

- 2, 6)

~ ~

Figu

re10:

The

same

plots

with

Fig.?

?for

the

small

conven

tional

field

pulses.

(a)T

he

trappin

ghorizon

(ϑ+

=0)

coord

inate,

x−H −

3(sin

cew

efixed

cc=

3),versu

sin

itialen

ergyof

the

pertu

rbation

,E

0 .W

eplotted

the

results

ofth

eru

nsof

ca

=0.5,···,10 −

2

with

cb=

3,6,an

d9.

They

lieclose

toon

elin

e.(b

)T

he

final

black

hole

mass

Mfor

the

same

exam

ples.

Tra

velth

rough

aW

orm

hole

–w

ithM

ain

tenace

Opera

tions!

0 2 4 6 8 10

02

46

810

x plus case A (no m

aintenance)

case C

case B

x minus

ghost scalar pulsefor m

aintenance

normal scalar pulse

(travellers)

Figu

re11:

Atrial

ofw

ormhole

main

tenan

ce.A

ftera

norm

alscalar

pulse,

we

signalled

agh

ostscalar

pulse

toex

tend

the

lifeof

worm

hole

throat.

The

travellerspulse

arecom

mon

lyex

pressed

with

anorm

alscalar

field

pulse,

(ca ,c

b ,cc )

=(+

0.1,6.0,2.0).H

orizonlo

cations

ϑ+

=0

areplotted

forth

reecases:

(A)

no

main

tenan

cecase

(results

ina

black

hole),

(B)

with

main

tenan

cepulse

of(c

a ,cb ,c

c )=

(0.02390,6.0,3.0)(resu

ltsin

anin

flation

aryex

pan

sion),

(C)

with

main

tenan

cepulse

of(c

a ,cb ,c

c )=

(0.02385,6.0,3.0)(keep

stationary

structu

reupto

the

end

ofth

isran

ge).

Discu

ssion

Dynam

icsofth

eEllis-M

orris-T

horn

etra

versib

lew

orm

hole

⇒W

His

Unsta

ble

(A)w

ithpositiv

eenerg

ypulse

⇒B

lack

Hole

(B)w

ithnegativ

eenerg

ypulse

⇒In

flatio

nary

expansio

n

⇒(A

)co

nfirm

sduality

conje

cture

betw

een

BH

and

WH

.

⇒(B

)pro

vid

es

am

ech

anism

for

enla

rgin

ga

quantu

mw

orm

hole

tom

acro

scopic

size.

•W

eansw

ere

dto

the

questio

nof:

what

happens

ifour

hero

(or

hero

ine)

atte

mpts

totra

verse

the

worm

hole

.

•N

ew

disco

verie

softh

ecritica

lbehavio

ur.

“Scie

nce

can

be

stranger

than

science

fictio

n.”