Phantom collapse of electrically charged scalar field in dilaton gravity

3.3 MB·English

by  Anna Nakonieczna

Phantom collapse of electrically charged scalar field
in dilaton gravity
Anna Nakonieczna1and Marek Rogatko2
InstituteofPhysics
FacultyofMathematics,PhysicsandComputerScience
MariaCurie-SkłodowskaUniversity
20-031Lublin,pl.MariiCurie-Sklodowskiej1,Poland
3
1 Abstract. Ourresearchfocusongravitationalcollapseofelectricallychargedscalarfieldindilatongravityandinthepresence
0 of phantom coupling. We examine dynamical behaviour of the scalar field coupled to Maxwell field when gravitational
2 interactionshaveformconsistentwiththelow-energylimitofthestringtheory.Moreover,weallowtheevolvingfieldstohave
negativesigninfrontoftherespectivekinetictermoftheLagrangian.Themainaimofourstudiesistoinvestigateinwhat
n
mannerdoesthephantomnatureofeitherMaxwellordilatonfields(orbothofthem)affecttheoutcomesofthecollapse.
a
It turns out that the influence is crucial to the obtained spacetime structures. Negative kinetic energy of one (or both)
J
ofthefieldsdelays,changesthecourseorevenpreventsthecollapse.
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Keywords: dynamicalcollapse,dilatongravity,phantomcoupling
PACS: 04.25.dg,04.40.-b
]
h
t
- INTRODUCTION framework of the general theory of relativity the struc-
p
e ture of spacetime containing such a realistic black hole
h Dark energy is a highly predominating constituent of is described by the Kerr metric [7, 8]. Unfortunately,
[ theUniverse.Hence,aninterestingissuearises:inwhat investigating gravitational collapse which leads to the

manner does its presence affect the processes, which formation of the Kerr black hole is extremely difficult
1
v are driven by gravitational force? In order to answer duetoanalyticalandnumericalobstacles.Forthisreason
1 this question the properties of dark energy ought to be some simplifications ought to be introduced. Basing on
6 unraveled. Although it seems to be so widespread, the striking similarities in causal structures of spacetimes
5 problem of its exact nature is still unresolved. There containing Kerr and Reissner-Nordström black holes
1
are numerous diverse theoretical models which attempt thelatterspacetimemayberegardedasatoymodelfor
.
1 to describe and explain the properties of the considered the former one [7, 9]. Since it is spherically symmetric
0 unknownconstituentoftheUniverse. the calculations simplify considerably. The dynamical
3
Dark energy is described by the equation of state Reissner-Nordström spacetime stems from the collapse
1
w=Pρ−1, where P and ρ correspond to its pressure ofelectricallychargedscalarfield[10].Hencespacetime
:
v anddensity,respectively[1].Althoughtheexactvalueof structure emerging from the evolution of such a field
Xi thebarotropicindexwisstillunknown,observationsdo actually imitates a structure resulting from the realistic
notexcludethepossibilityofw<−1[2].Inthisparticu- gravitationalcollapse.
r
a larcase,theso-calledphantomfieldsprovideatheoreti- We simulate the dynamical behaviour of complex
calmodelfordarkenergy[3,4,5].Ourresearchconcern scalar field coupled to Maxwell field when gravita-
theinfluenceofphantomnatureofevolvingfieldsondy- tionalinteractionstakeformofdilatongravityandphan-
namicalgravitationalcollapse.Weareparticularlyinter- tom coupling of Maxwell and dilaton fields is possible.
ested in simulating the collapse of electrically charged Weareprimarilyinterestedinsingularspacetimessince
scalar field, which allows us to draw conclusions about examiningthedynamicalevolutionoffieldsunderthein-
outcomesoftherealisticprocesstakingplaceinnature. fluenceofgravityallowsustodescribeinternalstructures
Astrophysical black holes are electrically neutral oftheobjectscontainedwithinthemproperly.
and rotating [6]. The non-zero angular momentum
results in axial symmetry of these objects. Within the
1 [email protected]
2 [email protected],[email protected]
EINSTEIN-MAXWELL-DILATON The equationsof motionof the complexscalar fieldare
THEORYWITHPHANTOMCOUPLING asfollows:
rψ +r ψ +r ψ +ierA ψ +ier A ψ+
The action, which allows us to investigate gravitational ,uv ,u ,v ,v ,u u ,v v u
collapse in the considered theory, represents complex Qa2
+ieψ =0, (7)
scalar field coupled to Maxwell field in dilaton grav- 4r
ity. Moreover, it takes phantom coupling of Maxwell rψ∗ +r ψ∗+r ψ∗ −ierA ψ∗−ier A ψ+
,uv ,u ,v ,v ,u u ,v v u
anddilatonfieldsintoaccount.Inthestringframeithas
Qa2
thefollowingform: −ieψ∗ =0, (8)
4r
S(SF) =(cid:90) d4x(cid:112)−g(SF)e−2φ(cid:104)R(SF)−2ξ1(cid:0)∇(SF)φ(cid:1)2+ whiletheequationofmotionofthedilatonfieldhasthe
(cid:105) followingform:
+e2αφL(SF) , (1)
ξ Q2a2
whereL(SF) standsforLagrangiandensity rφ,uv+r,vφ,u+r,uφ,v−α ξ2e2α φ 4r3 +
1
L(SF) =−12DµψDµψ∗−ξ2FµνFµν, (2) −α8ξ+11re2φ(α +1)(cid:104)ψ,uψ,∗v+ψ,vψ,∗u+
while φ is the dilaton field, ψ – complex scalar field, +ieA (cid:0)ψψ∗−ψ ψ∗(cid:1)(cid:105)=0. (9)
u ,v ,v
F ≡ A − A stands for the electromagnetic
µν ν,µ µ,ν
field tensor, where Aµ is the Maxwell field. The co- Onaccountofgaugefreedomofelectromagneticpoten-
variant derivative of the scalar field is defined as tialA →A +∇ Λ,whereΛisanarbitraryscalarfunc-
µ µ µ
Dµ ≡∇(µSF)+ieAµ. Coupling constants e and α charac- tion,Maxwellequationsmaybewrittenas
terize couplings between the scalar field and Maxwell
and dilaton fields, respectively. Phantom constants ξ 2r2
1 Q≡ A , (10)
and ξ2 refer to dilaton and Maxwell fields, respec- a2 u,v
tively, and are equal to +1 or −1. In the latter case the Q +2α φ Q+ 1 e2φier2(cid:0)ψ ψ∗−ψψ∗(cid:1)=0. (11)
respectivefieldisphantom. ,v ,v 4ξ ,v ,v
2
The equations of motion for the considered fields are
obtainedviavariationalprinciple.Duringtheirderivation The set of equations (3)–(11) describes the gravita-
thestringframeisexchangedfortheEinsteinframeac- tional collapse in Einstein-Maxwell-dilaton theory with
cording to conformal transformation g(EF) =e−2φg(SF). phantom coupling. Because of its complicated structure
µν µν
Due to the fact that the evolving fields are mass- it has to be solved numerically. The suitable algorithm
less, the collapse is simulated in double null coor- wasdescribedin[11].
dinates and the assumed line element has the form
ds2=−a(u,v)2dudv+r(u,v)2dΩ2,wherea(u,v)isan
arbitraryfunction,r(u,v)istheradialfunctionanddΩ2 SPACETIMESTRUCTURES
standsforthelineelementontheunitsphere.
Einstein equations obtained in accordance with the The initial conditions for analysed evolutions consist of
aboveassumptionsandconditionsaregivenby profiles for complex scalar field and dilaton field. They
were chosen as trigonometric type with an amplitude
2a r 1 (cid:104)
,au ,u −r,uu=ξ1rφ,2u+4re2φ(α +1) ψ,uψ,∗u+ tp˜ivψelaynd[12g]a.uSssiinacne ttyhpeedwynitahmaicnalamgrpavliittuadtieonp˜aφl,croelslappesce-
+ieA (cid:0)ψψ∗ −ψ ψ∗(cid:1)+e2A2ψψ∗(cid:105), (3) is universal, its outcomes do not depend on the specific
u ,u ,u u
formofinitialprofiles.Theyarealsoindependentofthe
2a r 1
,v ,v −r =ξ rφ2+ re2φ(α +1)ψ ψ∗, (4) valuesofanelectriccouplingconstanteandaparameter
a ,vv 1 ,v 4 ,v ,v δ providedthatthesearenotequaltozero.Forthisrea-
a2 r r Q2a2 son,thesetwoquantitieswerefixedinalltheconducted
r + + ,u ,v =ξ e2α φ , (5)
,uv 4r r 2 4r3 evolutions.
a a a r Q2a2 Ontheotherhand,itturnsoutthatstructuresofspace-
,au2,v − a,uv − ,ruv =ξ1φ,uφ,v+ξ2e2α φ 4r4 + times emerging from the collapse depend on initial am-
plitudesofthecollapsingfields p˜ and p˜ aswellason
1 (cid:104) ψ φ
+8e2φ(α +1) ψ,uψ,∗v+ψ,vψ,∗u+ the value of dilatonic coupling constant α. We consid-
+ieAu(cid:0)ψψ,∗v−ψ,vψ∗(cid:1)(cid:105). (6) eforermdetwr roefvearlsuetos tohfeitl,owna-meneelrygyαs=trin−g1thaenodryα, w=h0il.eTthhee
TABLE1. Characteristicsofthecon-
latter causes the evolution to run in the presence of un-
sideredevolutions.Overliningindicates
coupleddilatonfield.
phantomcouplingoftheparticularfield.
In order to examine the roles of evolving fields
and couplings among them in the considered gravita- Type ξ1 ξ2 p˜φ p˜ψ
tional collapse it is sufficient to describe its outcomes
EM − −1 − =(cid:54) 0
wheninitialamplitudeofcomplexscalarfieldisconstant
EMD +1 −1 (cid:54)=0 (cid:54)=0
and the amplitude of dilaton field varies. An amplitude
ED −1 − (cid:54)=0 −
p˜ψ wassetasequalto0.6inalltheevolutions. EMD −1 +1 (cid:54)=0 (cid:54)=0
The types of all phantom evolutions, which will be EMD −1 −1 (cid:54)=0 (cid:54)=0
analysed,arecharacterizedinTable1.
rary horizon in spacetime. The joint collapse of phan-
PhantomMaxwellfield
tom Maxwell field coupled to the complex scalar one
and non-phantom dilaton field leads to the formation of
Figure 1 presents the structure of singular space-
asimplestspacetimestructuredescribingablackhole.
time emerging from the gravitational collapse of com-
Since phantom Maxwell field favours the emergence
plexscalarfieldcoupledtoMaxwellfieldwithphantom
ofsuchastructure,thisresultindicatesthatnon-phantom
coupling to gravity, i.e. the result of an EM-evolution.
dilatonfieldeitheralsosupportstheprocessorisunable
The singular spacetimes emerge in this case for large
tocounteracteffectively.
enoughvaluesofinitialcomplexscalarfieldamplitudes.
It turns out that the obtained structure corresponds to
the dynamical Schwarzschild-type spacetime. An ap-
parenthorizonsurroundingcentralspacelikesingularity
settles along u = const. at v → ∞ indicating the loca-
tion of an event horizon. On the contrary, the outcome
of the collapse of complex scalar field coupled to the
non-phantom Maxwell field is the dynamical Reissner-
-Nordström spacetime [10]. Apart from the central
spacelikesingularitysurroundedbyanapparenthorizon
coinciding with an event horizon at v→∞ it possesses
aCauchyhorizonatv=∞. FIGURE 2. Dynamical singular spacetimes emerging from
Taking the above into account, it may be stated that theEMD-collapseforα=−1andα=0when p˜φ isequalto
(a)0.01and(b)0.09.
phantomnatureofMaxwellfieldpreventstheformation
of a Cauchy horizon and supports the emergence of the
simplestspacetimestructurecontainingablackhole.
Phantomdilatonfield
The ED-collapse represents the evolution of phan-
tom scalar field under the influence of Einstein gravity.
Itturnsoutthatregardlessofaninitialvalueofthefield’s
amplitudetheresultingspacetimeisnon-singular.Onac-
count of the fact that dynamical Schwarzschild space-
time emerges from the collapse of non-phantom scalar
field[13],theobtainedresultmeansthatphantomnature
ofthescalarfieldpreventssingularityformation.
FIGURE 1. Dynamical singular spacetime emerging from A collection of singular spacetime structures, which
theEM-collapse. are obtained during the dynamical gravitational col-
lapseinEMD-theoryforα =−1,ispresentedinFig.3.
TheresultsofanEMD-collapseareidenticalforboth
For small values of dilaton field amplitude the initially
considered values of α. They are shown in Fig.2. Re-
formed central singularity, which is surrounded by an
gardless of the initial value of dilaton field amplitude
apparent horizon coinciding at v → ∞ with the event
allemergingspacetimesaresingularandSchwarzschild-
horizon situated along u=const., bifurcates and forms
-type. For small values of the amplitude the spacetime
twowormholethroatsduringthecourseofthecollapse.
structureistypical.Foritsbiggervaluesthecollapseruns
Finally, there exists a dynamical wormhole within the
intwostages,whatresultsinanappearanceofatempo-
emergingspacetime[12].Forbiggerinitialvaluesofthe
amplitude of dilaton field the spacetime is singular, but
the singularity is not surrounded by the event horizon,
i.e. a naked singularity forms. For even larger values
of the dilaton field amplitude the resulting spacetime is
non-singular.
FIGURE 4. Dynamical singular spacetimes emerging from
the EMD-collapse for (a) α =−1 when p˜ is equal to 0.1
φ
and(b)α=0whenp˜ isequalto0.2.
φ
• phantomdilatonfieldpreventssingularityformation
duetoitsstronglyrepulsivenature,
FIGURE 3. Dynamical singular spacetimes emerging from
• non-phantom Maxwell field favours Reissner-
the EMD-collapse for α =−1 when p˜ is equal to (a) 0.15
φ
-Nordström-typestructureformation,
and(b)0.2.
• phantomMaxwellfieldsupportsthecreationofthe
In contrast, the collapse of an electrically charged
Schwarzschild-typesingularspacetimestructures,
scalarfieldinthepresenceofphantomdilatonfieldwhen
α =0doesnotleadtosingularspacetimes. • dilatoniccouplingconstantα =−1diminishesthe
influenceofMaxwellfieldonthecollapseandthus
Summarizingtheabovefindingsitmaybestatedthat
enhancestheroleofdilatonfieldinitscourse,
the tendency of phantom dilaton field to prevent the
singularity formation is stronger in the uncoupled case, • dilatonic coupling constant α =0 reduces the role
i.e.forα =0,thanforα =−1. ofdilatonfieldandindirectlyenlargesthemeaning
of Maxwell field by acting on electrically charged
scalarfield.
PhantomMaxwellanddilatonfields
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