F ate of the T r a v ersible W ormholes
—– Blac k-Hole Collapse or Inflationary Expansion —–
真貝寿明
Hisa-aki Shink ai
理化学研究所 基礎科学特別研究員 （計算科学技術推進室）Computational Sci. Div., RIKEN (The Institute of Physical and Chemical Resea rch), Japan
Sean A. Ha yw ard
Dept. of Science Education, Ewha W omans Univ., Seoul, Ko rea
OUTLINE • T raversible w ormhole (Mo rris-Tho rne w ormhole, 1988) • Black Hole - W ormhole synthesis (Ha yw ard, 1999) • “Dynamical W ormhole” • A numerical app roach, dual-null fo rmulation • A new ty p e of critical b ehaviour??
HS and S.A. Ha yw ard, Phys. Rev. D. 66 (2002) 044005
Mo rris-Tho rne’s “T raversable” w ormhole
M.S. Mo rris and K.S. Tho rne, Am. J. Phys. 56 (1988) 395 M.S. Mo rris, K.S. Tho rne, and U. Y urtsever, PRL 61 (1988) 3182 H.G. Ellis, J. Math. Phys. 14 (1973) 104 (G. Cl ´ement, Am. J. Phys. 57 (1989) 967)
Desired p rop erties of traversable WHs
1. Spherically symmetric and Static ⇒ M. Visser, PRD 39(89) 3182 & NPB 328 (89) 203
2. Einstein gravit y
3. Asymptotically flat
4. No ho rizon fo r travel through
5. Tidal gravitational fo rces should b e small fo r traveler
6. T raveler should cross it in a finite and reasonably small p rop er time
7. Must have a physically reasonable stress-energy tenso r ⇒ W eak Energy Condition is violated at the WH throat. ⇒ (Null EC is also violated in general cases.)
8. Should b e p erturbatively stable
9. Should b e p ossible to assemble
1 Wh y W ormhole?
• They mak e great science fiction – sho rt cuts b et w een otherwise distant regions. Mo rris & Tho rne 1988, Sagan “Contact” etc • They increase our understanding of gravit y when the usual energy conditions ar e not satisfied, due to quantum effects (Casimir effect, Ha wking radiation) or alternative gravit y theo ries, b rane-w orld mo dels etc. • They ar e very simila r to black holes –b oth contain (ma rginally) trapp ed sur- faces and can b e defined b y trapping ho rizons (TH).
Wo rmhole ≡ Hyp ersurface foliated b y ma rginally trapp ed surfaces • BH and WH ar e interconvertible? New dualit y?
BH and WH are interconvertible ? (New Dualit y?)
S.A. Ha yw ard, Int. J. Mo d. Phys. D 8 (1999) 373
• They are very simila r – b oth contain (ma rginally) trapp ed surfaces and can b e defined by trapping ho rizons (TH)
• Only the causal nature of the THs differs, whether THs evolve in plus / minus densit y.
Black Hole W ormhole Lo cally defined by Achronal(spatial/null) outer TH T emp oral (timelik e) outer THs
⇒ 1-w ay traversable ⇒ 2-w ay traversable
Einstein eqs. P ositive energy densit y Negative energy densit y no rmal matter (o r vacuum) “exotic” matter
App ea rance occur naturally Unlik ely to occur naturally . but constructible ???
2 F ate of Morris-Thorne (Ellis) w ormhole?
• “Dynamical w ormhole” defined b y lo cal trapping horizon • spherically symmetric, b oth normal/ghost K G field • apply dual-n ull form ulation in order to seek horizons • Numerical sim ulation
2.1 ghost/normal Klein-Gordon fields
Lagrangian:
L = √ − g
R
16 π − 1
4 π
1
2 ( ∇ ψ )
2+ V
1( ψ )
normal
+ 1 4 π
1 2 ( ∇ φ )
2+ V
2( φ )
ghost
The field equations
G
µν=2
ψ
,µψ
,ν− g
µν 1
2 ( ∇ ψ )
2+ V
1( ψ )
− 2
φ
,µφ
,ν− g
µν 1
2 ( ∇ φ )
2+ V
2( φ )
ψ = dV
1( ψ )
dψ , φ = dV
2( φ )
dφ . (Hereafter, w e set V
1( ψ )=0 ,V
2( φ )=0 )
2.2 dual-n ull form ulation, spherically symmetric spacetime
SA Ha yw ard, CQG 10 (1993) 779, PRD 53 (1996) 1938, CQG 15 (1998) 3147 • The spherically symmetric line-element:
ds
2= r
2dS
2− 2 e
−fdx
+dx
−, where r = r ( x
+,x
−) ,f = f ( x
+,x
−) , ··· • The Einstein equations:
∂
±∂
±r +( ∂
±f )( ∂
±r )= − r ( ∂
±ψ )
2+ r ( ∂
±φ )
2, r∂
+∂
−r +( ∂
+r )( ∂
−r )+ e
−f/ 2=0 , r
2∂
+∂
−f +2 ( ∂
+r )( ∂
−r )+ e
−f=+ 2 r
2( ∂
+ψ )( ∂
−ψ ) − 2 r
2( ∂
+φ )( ∂
−φ ) , r∂
+∂
−φ +( ∂
+r )( ∂
−φ )+( ∂
−r )( ∂
+φ )=0 , r∂
+∂
−ψ +( ∂
+r )( ∂
−ψ )+( ∂
−r )( ∂
+ψ )=0 . • To obtain a system accurate nea r
±,w e intro duce the confo rmal facto r Ω=1 /r .W e also define first-o rder va riables, the confo rmally rescaled momenta
expansions ϑ
±=2 ∂
±r = − 2Ω
−2∂
±Ω( θ
±=2 r
−1∂
±r ) (1)
inaffinities ν
±= ∂
±f (2)
momenta of φ℘
±= r∂
±φ =Ω
−1∂
±φ (3)
momenta of ψπ
±= r∂
±ψ =Ω
−1∂
±ψ (4)
The set of equations (cont.):
∂
±ϑ
±= − ν
±ϑ
±− 2Ω π
2±+2 Ω ℘
2±, (5)
∂
±ϑ
∓= − Ω( ϑ
+ϑ
−/ 2+ e
−f) , (6)
∂
±ν
∓= − Ω
2( ϑ
+ϑ
−/ 2+ e
−f− 2 π
+π
−+2 ℘
+℘
−) , (7)
∂
±℘
∓= − Ω ϑ
∓℘
±/ 2 , (8)
∂
±π
∓= − Ω ϑ
∓π
±/ 2 . (9) and rememb er the identit y: ∂
+∂
−= ∂
−∂
+:
2.3 Initial data on x
+=0 , x
−=0 slices and on S
Generally , w e have to set :
(Ω ,f ,ϑ
±,φ ,ψ ) on S : x
+= x
−=0
( ν
±,℘
±,π
±) on Σ
±: x
∓=0 , x
±≥ 0
Grid Structure fo r Numerical Evolution
xplus xminus
wormhole throat S
2.4 Morris-Thorne (Ellis) w ormhole as the initial data on Σ
+( x
−=0 surface) on Σ
−( x
+=0 surface) Ω 1 / √ a
2+ z
21 / √ a
2+ z
2f 00 ϑ
±± √ 2 z/ √ a
2+ z
2∓ √ 2 z/ √ a
2+ z
2ν
+0 ν
−0 φ tan
−1( z/ a ) − tan
−1( z/ a ) ℘
++ a/ √ 2 √ a
2+ z
2℘
−− a/ √ 2 √ a
2+ z
2ψ 00 π
+0 π
−0 where z =( x
+− x
−) / √ 2 .
We put the p erturbation in ℘
+: δ℘
+= c
aexp( − c
b( z − c
c)
2) where c
a,c
b,c
car ep arameters.
2.5 Gra vitational mass-energy • Lo calizing, the lo cal gravitational mass-energy is given b y the Misner-Sha rp energy E ,
E =( 1 / 2) r [1 − g
−1( dr ,d r )]=( 1 / 2) r + e
fr ( ∂
+r )( ∂
−r )= 1
2Ω [1 + 1
2 e
fϑ
+ϑ
−]
while the (lo calized Bondi) confo rmal flux vecto r comp onents ϕ
±ϕ
±= r
2T
±±∂
±r = r
2e
2fT
∓∓∂
±r = e
2f( π
2∓− ℘
2∓) ϑ
±/ 8 π.
• They ar e related b y the energy p ropagation equations or unified first la w. ∂
±E =4 πϕ
±,
E ( x
+,x
−)= a
2 +4 π
(x +,x −)(0,0)
( ϕ
+dx
++ ϕ
−dx
−) ,
where the integral is indep endent of path, b y conservation of energy .
– lim
x +→∞E is the Bondi energy – lim
x +→∞ϕ
−the Bondi flux fo r the right-hand universe.
– Fo r the static w ormhole, the energy E = a
2/ 2 √ a
2+ z
2is everywhere p ositive, maximal at the throat and zero at infinit y, z →± ∞ , i.e. the Bondi energy is zero.
– Generally , the Bondi energy-loss p rop ert y, that it should b e non-increasing fo r matter satisfying the null energy condition, is reversed fo r the ghost field.
Numerical Grid / Con v ergence test
x plusx minus
wormhole throat S Σ+Σ−
0 5 10
05101520 grid=801grid=1601grid=3201grid=6401grid=9601
x plus
x minus
Figure1:Numericalgridstructure.InitialdataaregivenonnullhypersurfacesΣ±(x ∓=0,x ±>0)andtheirintersectionS.Figure2:Convergencebehaviourofthecodeforexactstaticwormholeinitialdata.Thelocationofthetrappinghorizonϑ−=0isplottedforseveralresolutionslabelledbythenumberofgridpointsforx +=[0,20].Weseethatnumericaltruncationerroreventuallydestroysthestaticconfiguration.
Stationary Configurations
2 4 6 8
5 10 15 -1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
x plus x minus expansion plus
2 4 6 8
5 10 15 0.1 0.2 0.3 0.4
0.1 0.2 0.3 0.4
x p lus x m in u s En e rg y
Figure2:Staticwormholeconfigurationobtainedwiththehighestresolutioncalculation:(a)expansionϑ+and(b)localgravitationalmass-energyEareplottedasfunctionsof(x +,x −).Notethattheenergyispositiveandtendstozeroatinfinity.
Ghost pulse input – Bifurcation of the horizons
0 2 4 6 8 10
0246810
x minus
x plus (a) pulse input with negative energy
ghost scalar pulse θ+ = 0
θ- = 0 Inflationary expansionθ+ < 0θ- > 0
θ+ > 0 θ- < 0
0 2 4 6 8 10
0246810
x minus
x plus (b1) pulse input with positive energy
ghost scalar pulse θ+ = 0 θ- = 0 BlackHole
x- H = 4.46 θ+ > 0
θ- < 0 θ+ < 0θ- > 0
Figure3:Horizonlocations,ϑ±=0,forperturbedwormhole.Fig.(a)isthecasewesupplementtheghostfield,ca=0.1,and(b1)and(b2)arewherewereducethefield,ca=−0.1and−0.01.Dashedlinesandsolidlinesareϑ+=0andϑ−=0respectively.Inallcases,thepulsehitsthewormholethroatat(x +,x −)=(3,3).A45 ◦counterclockwiserotationofthefigurecorrespondstoapartialPenrosediagram.
Bifurcation of the horizons – g o to a Blac k Hole or Inflationary expansion
x plusx minus Black HoleBlack Hole or orInflationaryInflationaryexpansionexpansionpulse input
throat "throat"
0 1 2 3 4 5 6
024681012 amplitude = +0.10amplitude = +0.01no perturbationamplitude = -0.01 amplitude = -0.10
proper time on the "throat"
Areal Radius at the "throat"
Figure4:PartialPenrosediagramoftheevolvedspace-time.Figure6:Arealradiusrofthe“throat”x +=x −,plottedasafunctionofpropertime.Additionalnegativeenergycausesinflationaryexpansion,whilereducednegativeenergycausescollapsetoablackholeandcentralsingularity.
Lo cal Energy Measure – Determination of the Blac k Hole Mass
0.0010 0.010 0.10 1.0 10 102
012345678 x plus = 12x plus = 16x plus = 20
Energy ( x+, x- )
x minus (a) pulse input with negative energy
0.010 0.10 1.0
0123456 x plus = 12x plus = 16x plus = 20
Energy ( x+, x- )
x minus horizon θ+ =0
formed at x -=4.46 (b1) pulse input with positive energy
Black hole massM = 0.42
Figure7:EnergyE(x +,x −)asafunctionofx −,forx +=12,16,20.Herecais(a)0.05,(b1)−0.1and(b2)−0.01.Theenergyfordifferentx +coincidesatthefinalhorizonlocationx −H,indicatingthatthehorizonquicklyattainsconstantmassM=E(∞,x −H).Thisisthefinalmassoftheblackholeorcosmologicalhorizon.
Is there a Minim um Blac k Hole Mass to b e formed?
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
10-510-410-310-210-1
xH- - 3
E0 (ca , cb ) = (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 0.50
10 -510 -410 -310 -210 -1
M
E0 (b)
(10 -1, 3) (10 -1, 6) (10 -1, 9) (ca , cb ) = (10 -4, 9) (10 -4, 6) (10 -4, 3)
Figure8:Relationbetweentheinitialperturbationandthefinalmassoftheblackhole.(a)Thetrappinghorizon(ϑ+=0)coordinate,x −H−3(sincewefixedcc=3),versusinitialenergyoftheperturbation,E0.Weplottedtheresultsoftherunsofca=10 −1,···,10 −4withcb=3,6,and9.Theylieclosetooneline.(b)ThefinalblackholemassMforthesameexamples.WeseethatMappearstoreachanon-zerominimumforsmallperturbations.
Normal Pulse (a tra v eller) Input – F orming a Blac k Hole
0 2 4 6 8
02468
x minus
x plus normal scalar pulse θ + = 0 θ - = 0 BlackHole
Figure9:Evolutionofawormholeperturbedbyanormalscalarfield.Horizonlocations:dashedlinesandsolidlinesareϑ+=0andϑ−=0respectively.
Critical Minim um Blac k Hole Mass again
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
10-510-410-310-210-1100
xH - - 3
E0 (a)
(ca , cb ) = (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 0.50
10 -510 -410 -310 -210 -110 0
M
E0 (b)
(0.5, 3)
(0.5, 6)
(0.5, 9) (ca , cb ) = (10 - 2, 3)
(10 - 2, 9) (10 - 2, 6) ~ ~
Figure10:ThesameplotswithFig.??forthesmallconventionalfieldpulses.(a)Thetrappinghorizon(ϑ+=0)coordinate,x −H−3(sincewefixed˜cc=3),versusinitialenergyoftheperturbation,E0.Weplottedtheresultsoftherunsof˜ca=0.5,···,10 −2
with˜cb=3,6,and9.Theylieclosetooneline.(b)ThefinalblackholemassMforthesameexamples.
Tr ave l through a W ormhole – with Main tenace Op erations!
0 2 4 6 8 10
0246810
x plus case A (no maintenance) case C
case B
x minus
ghost scalar pulsefor maintenance normal scalar pulse(travellers)
Figure11:Atrialofwormholemaintenance.Afteranormalscalarpulse,wesignalledaghostscalarpulsetoextendthelifeofwormholethroat.Thetravellerspulsearecommonlyexpressedwithanormalscalarfieldpulse,(˜ca,˜cb,˜cc)=(+0.1,6.0,2.0).Horizonlocationsϑ+=0areplottedforthreecases:(A)nomaintenancecase(resultsinablackhole),(B)withmaintenancepulseof(ca,cb,cc)=(0.02390,6.0,3.0)(resultsinaninflationaryexpansion),(C)withmaintenancepulseof(ca,cb,cc)=(0.02385,6.0,3.0)(keepstationarystructureuptotheendofthisrange).
