Dynamics of domain walls intersecting black holes

0.18 MB·English

by  Antonino Flachi

YITP-05-71
KUNS-2008
Dynamics of domain walls intersecting black holes
Antonino Flachi1∗, Oriol Pujol`as1,2†, Misao Sasaki1‡ and Takahiro Tanaka3§
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8503, Japan
2Center for Cosmology and Particle Physics, Department of Physics,
New York University, 4 Washington Place, New York, NY 10003 US
3Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Previousstudiesconcerningtheinteractionofbranesandblackholessuggestedthatasmallblack
holeintersectingabranemayescapeviaamechanismofreconnection. Hereweconsiderthisproblem
by studyingthe interaction of a small black hole and a domain wall composed of a scalar field and
simulate the evolution of this system when the black hole acquires an initial recoil velocity. We
test and confirm previousresults, however, unlikethecases previously studied, in themoregeneral
set-upconsideredhere,weareabletofollow theevolutionofthesystemalsoduringtheseparation,
6 and completely illustrate how theescape of the black hole takes place.
0
0
2
Introduction. It is well known that primordial black morelikely to haveoccurredinthe earlyuniverse,where

n holes and domain walls may have formed in the early the adiabatic approximationmay not be adequate.
a universe: on the one hand, cosmological density per- Aside from the motivations mentioned above, the ex-
J
turbations may have collapsed and formed small black citing possibility of observing mini black holes at forth-
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holes [1, 2], on the other hand, during the cooling phase coming collider experiments, as predicted by low scale
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after the big bang, the series of phase transitions that gravity theories [10, 11, 12, 13, 14], has rejuvenated

1 have occurred may have produced extended topological interest in the subject. In such models our universe
v structures like domain walls [3]. In view of the stringent has a domain wall structure, the brane, that is embed-
4 constraintsthatboth primordialblackholes anddomain ded in a higher dimensional space; the standard model
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walls provide, understanding how these objects interact particles are confined on the brane and gravity propa-
1
1 could help in clarifying many issues regarding the early gatesthroughoutthehigherdimensionalbulkspacetime.
0 universeand,possibly,beofphenomenologicalrelevance. Many realizations of this scenario result in lowering the
6 Planck scale to a few TeV and share the common pre-
In the past few years, the interaction between black
0
diction that a small black hole forms when two particles
/ holes and domain walls has been object of some study.
h collide at sufficiently high energy with small impact pa-
Of particular significance, in relation to our work, are
t rameter. Once the black hole is produced, it will emit
- the results of Refs. [4, 5, 6, 7, 8]. Ref. [4] considers the
p Hawkingradiation,andthis canin principlebe observed
Dirac-Nambu-Goto approximation for the domain wall
e
atthe CERN LHC orin cosmicrayfacilities. The radia-
h and proves the existence of a family of static wall solu-
tionoftheblackholewillpartlygointolowerdimensional
: tions intersecting the black hole event horizon. Subse-
v
fields, localised on the brane, and partly into higher di-
i quent work extended those results to the case of thick
X mensionalmodes,whichwillcausetheblackholetorecoil
walls, Refs. [5, 6, 7, 8].
r into the extra dimensions.
a Animportantproblemwherethepreviousstudiesmay The recoil of the black hole in the context described
findapplicationisthescatteringofablackholeandado- above was first studied in Ref. [15] within a toy model
main wall [9]. When the domain wall moves towardsthe consisting of two scalar fields, one describing the black
black hole, it is expected to be captured by the black holeandtheotherapossiblequantaemittedbytheblack
hole and the static configurations mentioned above may hole in the process of evaporation; the coupling between
describethisprocessintheadiabaticlimit, namelywhen them is fixed as to reproduce the probability of emis-
the relative motion is very slow. It is likely that the sion of a scalar quanta from the black hole as prescribed
membrane will experience large deformations and topol- by the Hawking formula, and a delta function potential
ogychangingprocesseswilltake place,but the adiabatic is used to mimic the interaction between the black hole
limit cannot easily resolve this issue. A full dynamical and the brane. In the approximation that the interac-
simulation is necessary, and it would also help to clar- tion with the brane is negligibly small, it is shown that,
ify what happens in more extreme situations, which are assoonasaquantaisemittedintheextradimension,the
black hole will slide off the brane. Although this has the
important observable effect of a sudden interruption of
the radiation on the brane (Ref. [16] explores some pos-
∗fl[email protected]
siblephenomenologicalconsequences),itisnotclearhow
†[email protected]
‡[email protected] the separation process occurs. To clarify this issue and
§[email protected] to have further evidence for the conclusions of Ref. [15],
2
twoofthepresentauthorshaveconsideredtheinteraction coupling constant λ is of order Λ4−d, with Λ . M be-
between ‘mini’ black holes and branes from the different ing the cutoff of the effective field theory. We can ne-
perspective of studying the dynamics of branes in black glect the quantum corrections to the domain wall-type
hole spacetimes, Ref.[17]. Specifically, by treating the configurations at weak coupling Λ(d−2)/2 η. This
branein the Dirac-Nambu-Gotoapproximation,we have implies σ t1−d, and hence we have σ≫rd−2/M
shown that, once the black hole acquires an initial re- (r /t)d−1(≫M /M)(d−2)/(d−3), where we introHduced t≫he
H d
coil velocity perpendicularly to the brane, an instability d-dimensionalPlanckmassM =G1/(d−2). Requirement
d
develops and the brane tends to envelop the black hole. ofnegligibleself-gravitythereforeimpliesthatthewallis
This suggests a mechanism of escape for the black hole not very thin and/or the mass of the black hole is suffi-
due to the reconnection of the brane. In some approxi- ciently large.
mation the time of escape can be estimated and it was Theevolutionequationsdescribingthedynamicsofthe
foundto be shorterthanthe evaporationtime. Thisfact domain wall intersecting a black hole, are given by
may have important phenomenologicalconsequences.
1
The previous claim certainly deserves reconsideration 0 = f−1(r)∂2Φ+ ∂ rd−2f(r)∂ Φ
and one of our motivations is to test the robustness of − t rd−2 r r
the conclusions of Ref. [17] by modeling the brane as a + 1 ∂ (sinθ)d−(cid:0)3∂ Φ V′(Φ(cid:1)) , (3)
domain wall composed of a scalar field. Also, unlike the r2(sinθ)d−3 θ θ −
Dirac-Nambu-Gotocasepreviouslystudied,modelingthe (cid:2) (cid:3)
wherewe assumed O(d 2)-rotationalsymmetryaround
brane as a domain wall allows one to describe reconnec- −
theaxisperpendiculartotheequatorialplaneoftheblack
tion phenomena. In our case, this means that we can
hole.
follow the evolution of the system while the separation
The initial conditions are specified so as to mimic the
of the black hole occurs, and completely illustrate how
initial recoil of the black hole. For this purpose, we im-
the escape takes place.
pose that Φ and ∂Φ at t = 0 are given by a static kink
Domainwalldynamics. Thesystemweintendtostudy
profile in the flat spacetime boosted in the direction of
consists of a domain wall intersecting a black hole. As
the symmetry axis
customary, we consider black holes whose mass m ex-
ceeds the fundamental Planck scale M, in orderto make
λ rcosθ vt
quantum gravity corrections negligible. Also, the size Φ (t,r,θ)=ηtanh η − , (4)
v
of the mini black hole is assumed to be smaller than r2 √1 v2 !
−
the characteristic length of the extra dimensions. We
shall restrict our analysis to the case when the wall ten- with v constant.
sion is small enough, as this allows us to ignore the self- Boundary conditions have to be specified on the outer
gravity of the wall. In this regime, the mini black hole boundary, on the symmetry axis and on the horizon. At
is completely immersed in the higher dimensional space- ‘infinity’, since the gravity of the black hole is expected
time and can be adequately described by asymptotically to be negligible, we assume that Φ reduces to the flat
flat solutions [18, 19] boosted form (4). The regularity conditions at the sym-
metry axis and on the horizon do require
ds2 = f(r)dt2+f−1(r)dr2+r2dΩ2 , (1)
− d−2 ∂ Φ(t,r,θ) = 0 , at θ =0, π ,
θ
wheredΩd−2 isthelineelementofa(d 2) dimensional ∂tΦ(t,r,θ) = 0 , at r =rH . (5)
− −
unit sphere,
In order to solve Eq. (3) numerically, we use a mixed
r d−3 spectral and finite difference method. The first step is
H
f(r)=1
− r todecomposethegeneralsolutionintermsofacomplete
(cid:16) (cid:17) basis of smooth global functions [20]. Here, for conve-
and r is the horizon radius. We use units in which
H nience, we adopt the Chebyshev polynomials of second
c=~=1.
kind, and write the general solution as
Since the character of vacuum structures is generally N
insensitive to the details ofthe model, we limit ourselves Φ(t,r,θ)= ϕn(t,r)Un(cosθ) , (6)
to a scalar effective field theory with quartic potential n=0
X
λ whereN isthenumberofharmonicsused. Theboundary
V(Φ)= (Φ2 η2)2 . (2) condition along the symmetry axis is trivially satisfied,
4 −
as the Chebyshev polynomials U (cosθ) are regular at
n
The domain wall solutions of this model in flat space θ = 0,π. The decomposition (6) is used as ansa¨tze in
have tension σ √λη3 and thickness t 1/√λη. The equation(3), andsometrivialalgebraallowsustorecast
≃ ≃
3
Eq. (3) in matrix form Aslaststep,V′(Φ)isconvolutedwiththebasisfunctions
according to formula (9).
d 2
ϕ¨ = f2(r)ϕ′′+f(r) f′(r)+ − f(r) ϕ′ The details of the simulations will be reported else-
r where, here we briefly illustrate the main result. Fig. 1
(cid:20) (cid:21)
f(r) shows six snapshots describing the evolution of the en-
+ L ϕ+F , (7)
r2 · ergy density of the domain wall,
where we have defined
1 f(r) 1
ρ(t,r,θ)= Φ2 + Φ2 + Φ2 +V(Φ) . (12)
2 1 ∂x (1 x2)d−22∂xUn Um 2f(r) ,t 2 ,r 2r2 ,θ
L = − dx, (8)
nm π Z−1 h (1−x2)d−21−2 i The plot presents the result of a simulation obtained by
2f(r) 1 using N = 30 harmonics and 100 grid points in the ra-
F = U(x)V′(Φ) 1 x2dx , (9) dialdirection. Standardnumericalcheckshavebeenper-
− π −
Z−1 formedandthe evolutionhasbeen carriedoutinvarious
p
x=cosθ, and the vectors ϕ and U are given by caseswiththe thicknessofthe domainwallrangingfrom
0.01to1timesthesizeofthehorizonradiusandforvari-
ϕ = (ϕ (t,r),ϕ (t,r), ,ϕ (t,r)) , ousinitialvelocities,vrangingfrom0.01to1. Theresults
0 1 N
···
U = (U (x),U (x), ,U (x)) . obtainedinthecaseofadomainwallconfirmthesugges-
0 1 N
··· tion of Ref. [15, 17]: the domain wall envelops the black
Equation(7)isasystemofcoupledtwo-dimensionalnon- hole, that completely separates from the mother brane;
linear partial differential equations, which can be solved after the pinching occurs, the mother brane relaxes into
by using finite difference methods. a stable configuration.
We define ϕ−, ϕ and ϕ+, which represent the con- Modeling the brane as a domain wall, we have given
figurations at three subsequent time steps, t = t δ , furtherevidenceinsupportoftheconclusionsofRefs.[15,
− t
−
t and t = t+δ , respectively. We now define a one- 16, 17] that the black hole intersecting a brane may es-
+ t
dimensionalgridalongtheradialdirection,andlabelthe cape once it acquires an initial recoil velocity. As in the
points as r . Then the values of a function g(t,r) on a previous work, we have neglected the self-gravity of the
i
grid point r at t = t ,t and t will be denoted by g−, wall, but the present treatment allows us to describe re-
i − + i
g and g+. In terms of the coefficients of the expansion, connection processes of the domain wall. By following
i i
the initial conditions are written as theevolutionofthesystemduringtheseparation,wesuc-
ceeded in illustrating completely how this process takes
2 1
ϕ− = Φ (0,r ,x)U(x) 1 x2dx , (10) place.
i π v i −
Z−1 There is one important question which has not been
p
considered here: what happens when the tension is non-
and
negligible? Intuitionsuggeststhatthebackreactionmay
2 1 preventtheseparationatleastforsmallinitialvelocities,
ϕ = Φ (δ ,r ,x)U(x) 1 x2dx , (11)
i π v t i − and an energy barrier has to be overcome to induce in-
Z−1
p stability leading to pinching. In other words, when the
where Φv(t,r,x) is the boosted domain wall solution, tension of the brane is switched on, it might be possi-
Eq. (4). As for the boundary conditions, we have ble to find a sequence of static configurations before we
encounter a critical unstable configuration above which
2 1
ϕ(t,r r )= Φ (t,r r ,x)U(x) 1 x2dx . the instability sets in. However this problem is rather
∞ v ∞
≥ π Z−1 ≥ − complicated and beyondthe scope of the present article.
p
Studytounderstandthisissueisinprogressandwehope
at the outer boundary r , and
∞ to report on this soon.
ϕ˙(t,r =r )=0 , Acknowledgements. We thank J.J. Blanco-Pillado, H.
H
Kudoh, T. Shiromizu and H. Yoshino for useful discus-
on the horizon. sions and for raising interesting questions. OP acknowl-
To solve the system of Eqs. (7) we use a central dif- edges the kind hospitality of the CCPP at NYU. This
ference discretization scheme. In this way the R.H.S. of work is supported in part by Grant-in-Aid for Scientific
(7) can solely be expressed in terms of ϕ, whereas the Research, Nos. 1604724 and 16740165 and the Japan-
L.H.S. will depend on ϕ− and ϕ+. Therefore, once the U.K. Research Cooperative Program both from Japan
boundaryandinitial conditionsarespecified, the system Society for Promotion of Science. This work is also sup-
of equations (7) is completely determined. The poten- ported by the 21st Century COE “Center for Diversity
tial term F is evaluated by first reconstructing the field and Universality in Physics” at Kyoto university, from
profile Φ and then use it to evaluate the function V′(Φ). the Ministry of Education, Culture, Sports, Science and
4
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