Spinning Strings, Black Holes and Stable Closed Timelike Geodesics
Spinning Strings, Black Holes and Stable Closed Timelike Geodesics.
Valéria M. Rosa∗
Departamento de Matemática, Universidade Federal de Viçosa, 36570-000 Viçosa, M.G., Brazil
Patricio S. Letelier†
Departamento de Matemática 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 the existence of closed timelike geodesics.
PACS numbers: 04.20.Gz, 04.20.Dg, 04.20.Jb
Keywords: Closed Timelike Geodesics, Linear Stability, Time Machines, Black Holes, Cosmic Strings,Torsion
Lines
The existance of closed timelike curves (CTCs) in the
Gödel universe and other apacetimes is a worrying fact
since these curves show a clear violation of causality.
In some cases these CTCs can be disregarded by en-
ergy considerations. Their existance requires an external
force acting along the whole CTC, process that may con-
sume a great amount of energy. The energy needed to
travel along a CTC in Gödel’s universe is computed in
[1]. When the external force is null the energy needed to
travel is also null. Therefore, in principle, the existence of
closed timelike geodesics (CTGs) presents a bigger prob-
lem of breakdown of causality.
The classical problem of the existence of closed
geodesics in Riemannian geometry was solved by
Hadamard [2] in two dimensions and by Cartan [3] in an
arbitrary number of dimensions. As a topological prob-
lem, the existence of CTGs was proved by Tipler [4] in
a class of four-dimensional compact Lorentz manifolds
with covering space containing a compact Cauchy sur-
face. In a compact pseudo-Riemaniann manifold with
Lorentzian signature (Lorentzian manifold) Galloway [5]
found sufficient conditions to have CTGs, see also [6].
To the best of our knowledge there are four solution to
the Einstein equations generated by matter with positive
mass density that contain CTGs: a) Soares [7] found
a class of cosmological models, solutions of Einstein-
Maxwell equations, with a subclass where the timelike
paths of matter are closed. For these models the exis-
tence of CTGs is demonstrated and explicit examples are
given. These CTGs are not linearly stable [8]. b) Stead-
man [9] described the behavior of CTGs in a vacuum
exterior for the van Stockum solution that represents an
infinite rotating dust cylinder. For this solution explicit
examples of CTCs and CTGs are shown. There are stable
CTGs in this spacetime [8]. c) Bonnor and Steadman [10]
studied the existence of CTGs in a spacetime with two
spinning particles each one with magnetic moment equal
∗Electronic address: e-mail: vmrosa@ufv.br
†Electronic address: e-mail: letelier@ime.unicamp.br
to angular moment and mass equal to charge (Perjeons),
in particular, they present a explicit CTG. This partic-
ular CTG is not stable, but there exist many other that
are stable [11]. d) There are linearly stable CTGs [8] in
one of the Gödel-type metrics with not flat background
studied by Gürses et al. [12][13]. For CTGs in a space-
time associated to a cloud of strings with negative mass
density see [14]. These CTGs are not stable [8].
The existence of CTCs in a spacetime whose source is
a spinning string has been investigated by many authors
(see for instance [15]-[19]). The interpretation of these
strings as torsion line defects can be found in [20], [21],
see also [22][23]. These torsion line defects appear when
one tries to stabilize two rotating black holes kept apart
by spin repulsion [24]. Also, the black hole thermody-
namics associated to a static black hole pierced by a non
rotating string was studied some time ago by Aryal et
al. [25].
In the present work we study the existence and stabil-
ity of CTCs under linear perturbations in the spacetime
associated to Schwarzschild black hole (BH) pierced by
a spinning string. Even though this spacetime is more a
mathematical curiosity than an example of a real space-
time we believe that the study of stability of CTCs and
CTGs can shed some light into the existence of this rather
pathological curves. In particular, we study sufficient
conditions to have linearly stable CTGs. We find that
these conditions are not very restrictive and can be eas-
ily satisfied. Furthermore, we compared them with the
same conditions studied by Galloway [5] for a compact
Lorentzian manifold.
Let us consider the spacetime with metric,
ds2 = (1−
)(dt−αdϕ)2−
1− 2m
−r2(dθ2+β2 sin2 θdϕ2),
where α = 4S and S is the string’s spin angular momen-
tum per unit of length, β = 1 − 4λ and λ is the string’s
linear mass density that is equal to its tension (λ ≤ 1/4).
In the particular case, α = 0 and β = 1, the metric (1)
reduces to the Schwarzschild solution. When m = 0, Eq.
(1) represents a spinning string, with the further special-
http://arxiv.org/abs/0704.1109v2
mailto:e-mail: vmrosa@ufv.br
mailto:e-mail: letelier@ime.unicamp.br
ization β = 1 (not deficit angle) we have a pure massless
torsion line defect [20] [21]. Therefore the metric (1) can
be considered as representing the spacetime associated to
a Schwarzschild black hole pierced by a spinning string.
Let us denote by γ a closed curve given in its paramet-
ric form by,
t = t0, r = r0, ϕ ∈ [0, 2π], θ =
, (2)
where t0 and r0 are constants. When γ is parametrized
with an arbitrary parameter σ, we have a timelike curve
when dx
> 0. This condition reduces to gϕϕ > 0,
i.e.,
(1− 2m/r0)α2 − r20β2 > 0. (3)
A generic CTC γ satisfies the system of equations given
ẍµ + Γ
αβ ẋ
αẋβ = Fµ(x), (4)
where the overdot indicates derivation with respect to
αβ are the Christoffel symbols and F
µ is a specific
external force (aµ = Fµ). The nonzero component of the
four-acceleration of γ is
(r0 − 2m)(α2m− r30β2)ϕ̇2. (5)
Our goal is to study the behavior of closed timelike
geodesics. Therefore taking α as one of the two solutions
α2m− r3
β2 = 0, (6)
we have ar = 0. Under this condition (3) is satisfied when
r0 > 3m, that put the CTG outside the black hole.
Let γ̃ be the curve obtained from γ after a small per-
turbation ξ, i.e., x̃µ = xµ + ξµ. From equations (4) one
finds that the system of differential equations satisfied by
the perturbation ξ is [26],
+ 2Γαβµ
uµ + Γαβµ,λξ
λuβuµ = Fα,λξ
λ. (7)
For the above mentioned closed timelike geodesic the
system (7) reduces to
ξ̈0 + k1ξ̇
1 = 0, (8)
ξ̈1 + k2ξ̇
0 + k3ξ
1 = 0, (9)
ξ̈2 + k4ξ̇
1 = 0, (10)
ξ̈3 + k5ξ
3 = 0, (11)
where
k1 = 2Γ
ϕ̇, k2 = 2Γ
ϕ̇, k3 = Γ
22,1ϕ̇
k4 = 2Γ
ϕ̇, k5 = Γ
22,3ϕ̇
2. (12)
A curve γ parametrized by the proper time, s, is time-
like when ẋµẋµ = 1. For the curve γ(s) we have that this
last condition gives us,
ϕ̇2 =
(r0 − 3m)
. (13)
The solution of (9)-(11) is
ξ0 = −k1(c3 sin(ωs+ c4)/ω + λs) + c1 s+ c5,
ξ1 = c3 cos(ωs+ c4) + λ,
ξ2 = −k4(c3 sin(ωs+ c4)/ω + λs) + c2 s+ c6,
ξ3 = c7 cos(
k5s+ c8),
where ci, i = 1, . . . , 8 are integration constants,
k3 − k1k2
= [β2(r0 − 6m)ϕ̇2/r0]1/2, (15)
and λ = −k2c1/ω2. Thus when r0 > 6m, the constant
ω is real and the solution (14) shows the typical behav-
ior for stability, i.e., vibrational modes untangled with
translational ones that can be eliminated by a suitable
choice of the initial conditions.
When the black hole is removed, we are left with the
spacetime of the spinning string whose line element is,
ds2 = (dt− αdϕ)2 − dr2 − r2(dθ2 + β2 sin2 θdϕ2). (16)
The closed curve, γ, is timelike when α2− r2
β2 > 0. The
ar-component of the four-acceleration is given by ar =
−β2r0ϕ̇2. Thus for r < |α/β| we have closed timelike
curves, which are not geodesics.
For the closed curve (2) the system (7) is written now
as in (9)-(11) replacing equation (10) by
ξ̈1 + k2ξ̇
2 + k3ξ
1 = ∂r(Γ
ϕ̇2)ξ1, (17)
where now k2 = 2Γ
ϕ̇ and ϕ̇2 = (α2 − r2
β2)−1. In this
particular case the solution of (7) has the same form that
(14) with ω2 = 2β2ϕ̇2(2+β2r2
ϕ̇2). Therefore, the CTCs
are stable.
In summary, there exist linearly stable CTCs in the
spacetime related to a spinning string and these curves
are restricted to a small region of the spacetime. Closed
timelike geodesics do not exist in this spacetime.
For the nonlinear superposition of a spinning string
with a Schwarzschild black hole the new spacetime has
linearly stable CTGs. The region of stability is the same
of the usual circular geodesics in the Schwarzschild black
hole alone. The presence of the spinning string does not
affect the stability of the orbits. It seems that torsion
lines defects superposed to matter (not strings, β = 1) is
a main ingredient to have stable CTGs. Loosely speak-
ing, we have that a torsion line defect alone makes possi-
ble the existence of CTCs. When the black hole is present
the spinning string spacetime is deformed in such a way
to allow the existence of a CTG. This fact is also con-
firmed in the case of the two Perjeons solutions studied
0 5 10 15 20 25 30 35
m= 0.3;  β= 0.9;  α= 25;  r
= 6.1401
0 5 10 15 20 25 30 35
m= 1;  β= 0.9;  α= 25;  r
= 9.172
0 5 10 15 20 25 30 35
m= 4;  β= 0.9;  α= 25;  r
= 14.5597
FIG. 1: The function s(r0) for a spinning string (solid line)
and for a black hole pierced by the string (dashed line). We
see how the presence of the mass shift the maximum of s(r0)
for the string that is located at r0 = 0 to a position outside the
black hole horizon. The maximum, rm represent the radius
of the CTGs, the first two are stable and the second is not.
in [10] wherein the torsion line defect is a main ingredient
to have CTCs and CTGs.
It is instructive to look the previous results in a more
direct and graphic way. The length of CTC in (2) only
depends on the value of r = r0. We find,
s(r0) = 2π
gϕϕ(r0),
= 2π[(1− 2m/r0)α2 − r20β2]1/2. (18)
This function has a local maximum for
rm = (mα
2/β2)1/3. (19)
Note that this equation is equivalent to (6), the condition
to have a geodesic.
The role of the black hole mass, in the appearance
of CTGs, is to produce a local maximum in the length
function, s(r0). This maximum gives us the position of
the CTG that in our case is located outside of the source
of the spacetime, beyond the black hole horizon.
In Fig. 1 we present, as a solid line the function s(r0)
for a spinning string, and as a dashed line the same func-
tion for the superposition of the black hole with the pre-
viously mentioned string for the same values of the pa-
0 10 20 30 40 50 60 70
m= 1;  β= 0.4;  α= 25;  r
= 15.749
0 10 20 30 40 50 60 70
m= 1;  β= 0.7;  α= 25;  r
= 10.845
0 10 20 30 40 50 60 70
m= 1;  β= 0.9;  α= 25;  r
= 9.172
FIG. 2: The function s(r0) for a spinning string (solid line)
and for a black hole pierced by the string (dashed line). We
see how the size of the deficit angle parameter β changes the
region for CTCs and the value of rm.
rameters α = 25 (spin parameter) and deficit angle pa-
rameter β = 0.9 and different values of the black hole
mass (m = 0.3, 1, 4). We see how the presence of the
mass shift the maximum for the string located at r0 = 0
to a position r0 > 3m. Also the points under the curves
represent the pairs [r0, s(r0)] for CTCs in each case. We
note that the region for CTCs for the black hole pierced
by the string diminishes when the mass increases. The
maximum of the dashed line represents the CTG. We see,
that in the first two cases the CTGs are stable (rm > 6m)
and in the last case the CTG is not stable (rm < 6m).
In Fig. 2 we keep the value of the black hole mass
constant, m = 1, as well as, the spin parameter, α = 25,
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,
the larger the region for CTCs.
In Fig. 3 we keep the value of the black hole mass
constant, m = 1, as well as, the deficit angle parameter
β = 0.9 and change the spin parameter α = 15, 20, 25.
We see that the regions where the CTCs appear are larger
for bigger spin parameter. This parameter is essential to
have CTCs and CTGs in this case.
As we said before the existence of a CTGs does not put
restrictions on the energy to travel along this curve. Fur-
thermore, the force needed to move near a stable geodesic
0 5 10 15 20 25 30 35
m= 1;  β= 0.9;  α= 15;  r
= 6.5248
0 5 10 15 20 25 30 35
m= 1;  β= 0.9;  α= 20;  r
= 7.9042
0 5 10 15 20 25 30 35
m= 1;  β= 0.9;  α= 25;  r
= 9.172
FIG. 3: The function s(r0) for a spinning string (solid line)
and for a black hole pierced by the string (dashed line). We
see how the size of the spin parameter α changes the region
for CTCs and the value of rm. The spin parameter, in this
case, is the essential ingredient to have CTCs and CTGs.
is small. Therefore, the energy required will be also small.
In principle this small force can be provided by and en-
gine, say a rocket. Hence there will be not a severe en-
ergy restriction to travel near to a geodesic. Furthermore,
when moving along a stable CTG the control problem is
a trivial one. Small trajectory corrections require small
energy, also we do not have the danger to enter into a
run away situation.
A result from Galloway [5] states that in a compact
Lorentzian manifold, each stable free t-homotopy class
contains a longest closed timelike curve, and this curve
is necessarily a closed timelike geodesic. The assumption
thatM be compact can be weakened, it is sufficient to as-
sume that there exists an open set U in M with compact
closure such that each curve γ ∈ C (the free t-homotopy
class) is contained in U . In our case th Gödel universe
and other apacetimes e region containing the CTCs in C
is not compact. Therefore Galloway’s conditions do not
apply in this case, they too strong.
We want to point out that the stability of the circu-
lar orbits does not depend on the fact of the orbit be a
CTG. We found the same region of stability of the usual
circular geodesics. This result is not surprising since our
pierced black hole is locally identical to a usual black
hole. Moreover one can consider black holes surrounded
by different axially symmetric distributions of matter [27]
pierced by a spinning string. In this case, depending on
the different parameters of the solution, we can also have
CTGs and their stability will be the same as the usual
circular orbits considered in [27].
Furthermore, we analyze if the CTGs studied in the
present work satisfy the sufficient conditions of Gal-
loway’s theorem for the existence of CTGs. We found
that ours CTGs do not satisfy these conditions. The pos-
sibility of an example that satisfy exactly the conditions
of this theorem is under study. We want to mention that
the solution of Einstein equations considered in this work
is much simpler that the ones listed in the introduction.
Finally, we notice that the spacetime associated to the
black hole pierced by a spinning string is not a counter
example to the Chronology Protection Conjecture [28]
that essentially says that the laws of the physics do not
allow the appearance of closed timelike curves. A valid
dynamic to built this spacetime is not known.
V.M.R. thanks Departamento de Matemática-UFV for
giving the conditions to finish this work which was par-
tially supported by PICDT-UFV/CAPES. P.S.L. thanks
the partial financial support of FAPESP and CNPq.
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