Spinning Strings, Black Holes and Stable Closed Timelike Geodesics

0.12 MB·English

by  Valeria M. Rosa

Spinning Strings, Black Holes and Stable Closed Timelike Geodesics.
Val´eria M. Rosa∗
Departamento de Matem´atica, Universidade Federal de Vic¸osa, 36570-000 Vic¸osa, M.G., Brazil
Patricio S. Letelier†
Departamento de Matem´atica Aplicada-IMECC, Universidade Estadual de Campinas, 13081-970 Campinas, S.P., Brazil
The existence and stability under linear perturbation of closed timelike curves in the spacetime
associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the super-
position of the black hole, we find that the spinning string spacetime is deformed in such a way to
allow theexistence of closed timelike geodesics.
PACSnumbers: 04.20.Gz, 04.20.Dg,04.20.Jb
8 Keywords: ClosedTimelikeGeodesics,LinearStability,TimeMachines,BlackHoles,CosmicStrings,Torsion
0 Lines
0
2
The existance of closed timelike curves (CTCs) in the to angularmomentandmassequalto charge(Perjeons),

n G¨odel universe and other apacetimes is a worrying fact in particular, they present a explicit CTG. This partic-
a
since these curves show a clear violation of causality. ular CTG is not stable, but there exist many other that
J
In some cases these CTCs can be disregarded by en- are stable [11]. d) There are linearly stable CTGs [8] in
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ergyconsiderations. Theirexistancerequiresanexternal one of the G¨odel-type metrics with not flat background
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forceactingalongthewholeCTC,processthatmaycon- studied by Gu¨rses et al. [12][13]. For CTGs in a space-

] sume a great amount of energy. The energy needed to time associated to a cloud of strings with negative mass
c travel along a CTC in G¨odel’s universe is computed in density see [14]. These CTGs are not stable [8].
q
[1]. When the externalforce is null the energy needed to The existence of CTCs in a spacetime whose source is
-
r travelisalsonull. Therefore,inprinciple,theexistenceof a spinning string has been investigatedby many authors
g
closedtimelike geodesics(CTGs)presentsa biggerprob- (see for instance [15]-[19]). The interpretation of these
[
lem of breakdown of causality. strings as torsion line defects can be found in [20], [21],

2 The classical problem of the existence of closed see also [22][23]. These torsion line defects appear when
v geodesics in Riemannian geometry was solved by one tries to stabilize two rotating black holes kept apart
9
Hadamard[2] in two dimensions and by Cartan[3] in an by spin repulsion [24]. Also, the black hole thermody-
0
1 arbitrary number of dimensions. As a topological prob- namics associatedto astatic blackhole piercedby anon
1 lem, the existence of CTGs was proved by Tipler [4] in rotating string was studied some time ago by Aryal et
. a class of four-dimensional compact Lorentz manifolds al. [25].
4
0 with covering space containing a compact Cauchy sur- In the present workwe study the existence and stabil-
7 face. In a compact pseudo-Riemaniann manifold with ity of CTCs under linear perturbations in the spacetime
0 Lorentzian signature (Lorentzian manifold) Galloway [5] associated to Schwarzschild black hole (BH) pierced by
: found sufficient conditions to have CTGs, see also [6]. a spinning string. Even though this spacetime is more a
v
mathematical curiosity than an example of a real space-
i Tothebestofourknowledgetherearefoursolutionto
X time we believe that the study of stability of CTCs and
theEinsteinequationsgeneratedbymatterwithpositive
r CTGscanshedsomelightintotheexistenceofthisrather
mass density that contain CTGs: a) Soares [7] found
a
pathological curves. In particular, we study sufficient
a class of cosmological models, solutions of Einstein-
conditions to have linearly stable CTGs. We find that
Maxwell equations, with a subclass where the timelike
these conditions are not very restrictive and can be eas-
paths of matter are closed. For these models the exis-
ily satisfied. Furthermore, we compared them with the
tenceofCTGsisdemonstratedandexplicitexamplesare
same conditions studied by Galloway [5] for a compact
given. These CTGs are not linearly stable [8]. b) Stead-
Lorentzian manifold.
man [9] described the behavior of CTGs in a vacuum
Let us consider the spacetime with metric,
exterior for the van Stockum solution that represents an
infinite rotating dust cylinder. For this solution explicit 2m dr2
2 2 2 2 2 2 2
examplesofCTCsandCTGsareshown. Therearestable ds =(1 )(dt αdϕ) r (dθ +β sin θdϕ ),
− r − −1 2m−
CTGsinthisspacetime[8]. c)BonnorandSteadman[10] − r
(1)
studied the existence of CTGs in a spacetime with two
where α=4S and S is the string’s spin angular momen-
spinning particleseachone withmagnetic momentequal
tum per unit of length, β = 1 4λ and λ is the string’s
−
linearmassdensitythatisequaltoitstension(λ 1/4).
≤
Inthe particularcase,α=0 andβ =1,the metric (1)
∗Electronicaddress: e-mail: [email protected] reducestotheSchwarzschildsolution. Whenm=0, Eq.
†Electronicaddress: e-mail: [email protected] (1)representsaspinning string,withthe furtherspecial-
2
ization β =1 (not deficit angle)we have a pure massless Acurveγ parametrizedbythe propertime,s, istime-
torsionline defect [20][21]. Thereforethe metric (1) can likewhenx˙µx˙ =1. Forthecurveγ(s)wehavethatthis
µ
beconsideredasrepresentingthespacetimeassociatedto last condition gives us,
a Schwarzschild black hole pierced by a spinning string.
m
Letusdenotebyγ aclosedcurvegiveninitsparamet- ϕ˙2 = . (13)
ric form by, β2r02(r0 3m)
−
π The solution of (9)-(11) is
t=t0, r =r0, ϕ [0,2π], θ = , (2)
∈ 2
0
ξ = k1(c3sin(ωs+c4)/ω+λs)+c1s+c5,
where t0 and r0 are constants. When γ is parametrized ξ1 =c−3cos(ωs+c4)+λ,
with an arbitrary parameter σ, we have a timelike curve ξ2 = k4(c3sin(ωs+c4)/ω+λs)+c2s+c6, (14)
when ddxσµddxσµ > 0. This condition reduces to gϕϕ > 0, ξ3 =c−7cos(√k5s+c8),
i.e.,
where c , i=1,...,8 are integration constants,
i
2 2 2
(1 2m/r0)α r0β >0. (3)
− −
ω = k3 k1k2
p −
AgenericCTCγsatisfiesthesystemofequationsgiven = [β2(r0 6m)ϕ˙2/r0]1/2, (15)
by −
x¨µ+Γµ x˙αx˙β =Fµ(x), (4) and λ = −k2c1/ω2. Thus when r0 > 6m, the constant
αβ ω is real and the solution (14) shows the typical behav-
ior for stability, i.e., vibrational modes untangled with
where the overdot indicates derivation with respect to
translational ones that can be eliminated by a suitable
s, Γµ are the Christoffel symbols and Fµ is a specific
αβ choice of the initial conditions.
externalforce (aµ =Fµ).The nonzerocomponentofthe
When the black hole is removed, we are left with the
four-accelerationof γ is
spacetime of the spinning string whose line element is,
ar = r13(r0−2m)(α2m−r03β2)ϕ˙2. (5) ds2 =(dt−αdϕ)2−dr2−r2(dθ2+β2sin2θdϕ2). (16)
0
Theclosedcurve,γ,istimelikewhenα2 r2β2 >0. The
0
Our goal is to study the behavior of closed timelike ar-component of the four-acceleration i−s given by ar =
geodesics. Thereforetakingαasoneofthe twosolutions β2r0ϕ˙2. Thus for r < α/β we have closed timelike
of − | |
curves, which are not geodesics.
For the closed curve (2) the system (7) is written now
2 3 2
α m r β =0, (6)
− 0 as in (9)-(11) replacing equation (10) by
rw0e>ha3vmea,rth=at0.pUutndtheertChTisGcoonudtistiidoent(h3)eibslsaactkishfioelde.when ξ¨1+k2ξ˙2+k3ξ1 =∂r(Γ122ϕ˙2)ξ1, (17)
Let γ˜ be the curve obtained from γ after a small per- where now k2 =2Γ122ϕ˙ and ϕ˙2 =(α2 r02β2)−1. In this
turbation ξ, i.e., x˜µ = xµ+ξµ. From equations (4) one −
particularcasethesolutionof(7)hasthesameformthat
findsthatthesystemofdifferentialequationssatisfiedby (14)withω2 =2β2ϕ˙2(2+β2r2ϕ˙2). Therefore,theCTCs
0
the perturbation ξ is [26],
are stable.
In summary, there exist linearly stable CTCs in the
d2ξα dξβ
+2Γα uµ+Γα ξλuβuµ =Fαξλ. (7) spacetime related to a spinning string and these curves
ds2 βµ ds βµ,λ ,λ are restricted to a small region of the spacetime. Closed
timelike geodesics do not exist in this spacetime.
For the above mentioned closed timelike geodesic the
For the nonlinear superposition of a spinning string
system (7) reduces to
with a Schwarzschild black hole the new spacetime has
ξ¨0+k1ξ˙1 =0, (8) loifntehaerlyussutaalbcleirCcuTlaGrsg.eTodheesricesgiionnthofesStcahbwilaitryzsicshtihldebsalamcke
ξ¨1+k2ξ˙0+k3ξ1 =0, (9) hole alone. The presence of the spinning string does not
ξ¨2+k4ξ˙1 =0, (10) affect the stability of the orbits. It seems that torsion
lines defects superposedto matter (not strings,β =1)is
ξ¨3+k5ξ3 =0, (11) a main ingredient to have stable CTGs. Loosely speak-
ing,wehavethatatorsionline defectalonemakespossi-
where
bletheexistenceofCTCs. Whentheblackholeispresent
the spinning string spacetime is deformed in such a way
0 1 1 2
k1 =2Γ21ϕ˙, k2 =2Γ20ϕ˙, k3 =Γ22,1ϕ˙ , to allow the existence of a CTG. This fact is also con-
2 3 2
k4 =2Γ21ϕ˙, k5 =Γ22,3ϕ˙ . (12) firmed in the case of the two Perjeons solutions studied
3
m= 0.3; β= 0.9; α= 25; r = 6.1401 m= 1; β= 0.4; α= 25; r = 15.749
m m
200 200
150 150
s(r)0100 s(r)0100
50 50
0 0
0 5 10 15 20 25 30 35 0 10 20 30 40 50 60 70
r r
0 0
m= 1; β= 0.9; α= 25; r = 9.172 m= 1; β= 0.7; α= 25; r = 10.845
m m
200 200
150 150
s(r)0100 s(r)0100
50 50
00 5 10 15 20 25 30 35 00 10 20 30 40 50 60 70
r r
0 0
m= 4; β= 0.9; α= 25; r = 14.5597 m= 1; β= 0.9; α= 25; r = 9.172
m m
200 200
150 150
s(r)0100 s(r)0100
50 50
00 5 10 15 r0 20 25 30 35 00 10 20 30 r0 40 50 60 70
FIG. 1: The function s(r0) for a spinning string (solid line) FIG. 2: The function s(r0) for a spinning string (solid line)
and for a black hole pierced by the string (dashed line). We and for a black hole pierced by the string (dashed line). We
see howthepresenceof themass shift themaximum of s(r0) see how thesize of thedeficit angle parameter β changes the
forthestringthatislocatedatr0 =0toapositionoutsidethe region for CTCs and the valueof rm.
black hole horizon. The maximum, rm represent the radius
of the CTGs, thefirst two are stable and thesecond is not.
rameters α = 25 (spin parameter) and deficit angle pa-
rameter β = 0.9 and different values of the black hole
in[10]whereinthetorsionlinedefectisamainingredient
mass (m = 0.3,1,4). We see how the presence of the
to have CTCs and CTGs.
mass shift the maximum for the string located at r0 =0
It is instructive to look the previous results in a more
to a position r0 >3m. Also the points under the curves
direct and graphic way. The length of CTC in (2) only
represent the pairs [r0,s(r0)] for CTCs in each case. We
depends on the value of r =r0. We find,
note that the region for CTCs for the black hole pierced
by the string diminishes when the mass increases. The
s(r0) = 2πqgϕϕ(r0), maximumofthedashedlinerepresentstheCTG.Wesee,
= 2π[(1 2m/r0)α2 r02β2]1/2. (18) thatinthefirsttwocasestheCTGsarestable(rm >6m)
− − and in the last case the CTG is not stable (r <6m).
m
This function has a local maximum for In Fig. 2 we keep the value of the black hole mass
constant, m=1, as well as, the spin parameter, α=25,
r =(mα2/β2)1/3. (19)
m and change the deficit angle parameter β = 0.4,0.7,0.9.
We see that the larger the string density, λ=(1 β)/4,
Notethatthisequationisequivalentto(6),thecondition −
the larger the region for CTCs.
to have a geodesic.
The role of the black hole mass, in the appearance In Fig. 3 we keep the value of the black hole mass
of CTGs, is to produce a local maximum in the length constant, m = 1, as well as, the deficit angle parameter
function, s(r0). This maximum gives us the position of β = 0.9 and change the spin parameter α = 15,20,25.
the CTGthatinourcaseis locatedoutsideofthe source WeseethattheregionswheretheCTCsappeararelarger
of the spacetime, beyond the black hole horizon. for biggerspin parameter. This parameteris essentialto
In Fig. 1 we present, as a solid line the function s(r0) have CTCs and CTGs in this case.
foraspinningstring,andasadashedlinethesamefunc- AswesaidbeforetheexistenceofaCTGsdoesnotput
tion for the superpositionof the black hole with the pre- restrictionsontheenergytotravelalongthiscurve. Fur-
viously mentioned string for the same values of the pa- thermore,theforceneededtomovenearastablegeodesic
4
m= 1; β= 0.9; α= 15; r = 6.5248 A result from Galloway [5] states that in a compact
m Lorentzian manifold, each stable free t-homotopy class
100
contains a longest closed timelike curve, and this curve
80
isnecessarilyaclosedtimelikegeodesic. Theassumption
)0 60 thatM becompactcanbeweakened,itissufficienttoas-
s(r 40 sumethatthereexistsanopensetU inM withcompact
20 closure such that each curve γ (the free t-homotopy
∈C
0 class) is contained in U. In our case th G¨odel universe
0 5 10 15 20 25 30 35
r0 and other apacetimes e region containing the CTCs in
m= 1; β= 0.9; α= 20; r = 7.9042 is not compact. Therefore Galloway’s conditions do noCt
m
150 apply in this case, they too strong.
We want to point out that the stability of the circu-
100
)0 lar orbits does not depend on the fact of the orbit be a
s(r 50 CTG. We found the same regionof stability of the usual
circular geodesics. This result is not surprising since our
0
0 5 10 15 20 25 30 35 pierced black hole is locally identical to a usual black
r
0 hole. Moreover one can consider black holes surrounded
m= 1; β= 0.9; α= 25; r = 9.172
m bydifferentaxiallysymmetricdistributionsofmatter[27]
200
pierced by a spinning string. In this case, depending on
150 thedifferentparametersofthesolution,wecanalsohave
s(r)0100 CTGs and their stability will be the same as the usual
circular orbits considered in [27].
50
Furthermore, we analyze if the CTGs studied in the
0
0 5 10 15 20 25 30 35
r present work satisfy the sufficient conditions of Gal-
0
loway’s theorem for the existence of CTGs. We found
thatoursCTGsdonotsatisfytheseconditions. Thepos-
sibility of an example that satisfy exactly the conditions
FIG. 3: The function s(r0) for a spinning string (solid line) ofthis theoremisunderstudy. We wantto mentionthat
and for a black hole pierced by the string (dashed line). We
thesolutionofEinsteinequationsconsideredinthiswork
see how the size of the spin parameter α changes the region
is much simpler that the ones listed in the introduction.
for CTCs and the value of rm. The spin parameter, in this
case, is theessential ingredient to haveCTCs and CTGs. Finally,wenoticethatthe spacetimeassociatedto the
black hole pierced by a spinning string is not a counter
example to the Chronology Protection Conjecture [28]
issmall. Therefore,theenergyrequiredwillbealsosmall. that essentially says that the laws of the physics do not
In principle this small force can be provided by and en- allow the appearance of closed timelike curves. A valid
gine, say a rocket. Hence there will be not a severe en- dynamic to built this spacetime is not known.
ergyrestrictiontotravelneartoageodesic. Furthermore,
when moving along a stable CTG the control problem is V.M.R.thanksDepartamentodeMatema´tica-UFVfor
a trivial one. Small trajectory corrections require small giving the conditions to finish this work which was par-
energy, also we do not have the danger to enter into a tiallysupportedbyPICDT-UFV/CAPES.P.S.L.thanks
run away situation. the partial financial support of FAPESP and CNPq.
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