#### Self-force on an arbitrarily coupled scalar charge in cylindrical thin-shell spacetimes

Eur. Phys. J. C
Self-force on an arbitrarily coupled scalar charge in cylindrical thin-shell spacetimes
C. Tomasini 0
E. Rubín de Celis 0
C. Simeone 0
0 Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, CONICET , Ciudad Universitaria, Buenos Aires 1428 , Argentina
We consider the arbitrarily coupled field and selfforce of a static massless scalar charge in cylindrical spacetimes with one or two asymptotic regions, with the only matter content concentrated in a thin-shell characterized by the trace of the extrinsic curvature jump κ . The self-force is studied numerically and analytically in terms of the curvature coupling ξ . We found the critical values ξc(n) = n/ (ρ (rs ) κ ), with n ∈ N and ρ (rs ) the metric's profile function at the position of the shell, for which the scalar field is divergent in the background configuration. The pathological behavior is removed by restricting the coupling to a domain of stability. The coupling has a significant influence over the self-force at the vicinities of the shell, and we identified ξ = 1/4 as the value for which the scalar force changes sign at a neighborhood of rs ; if κ (1 − 4ξ ) > 0 the shell acts repulsively as an effective potential barrier, while if κ (1 − 4ξ ) < 0 it attracts the charge as a potential well. The sign of the asymptotic self-force only depends on whether there is an angle deficit or not on the external region where the charge is placed; conical asymptotics produce a leading attractive force, while Minkowski regions produce a repulsive asymptotic self-force.
1 Introduction
A field theory including a complex scalar field coupled to
a gauge field predicts that spontaneous symmetry breaking
can lead to cylindrical topological defects known as local or
gauge cosmic strings [1]. The gravitational effects of such
objects have been considered of particular physical
interest within the study of structure formation in the early
Universe, since they could have acted as possible “seeds” for
density fluctuations [2–4]; besides, strings could be, in
principle, observed by gravitational lensing effects. Though in
the present day theoretical framework cosmic strings are not
considered as the main source of the primordial
cosmological matter fluctuations, they are still taken as a possible
secondary source of fluctuations [5]; this motivates a recently
renewed interest in their study.
Local or gauge strings are characterized by having an
energy-momentum tensor whose only non null components
are T00 = Tzz . The metric around a gauge string was
first calculated by Vilenkin [6] in the linear
approximation of general relativity, using a Dirac delta to
approximate the radial distribution of the energy-momentum
tensor for a cosmic string along the z axis. Thus, it was
proposed T˜μν = δ(x )δ(y) diag(μ, 0, 0, μ), where μ is the linear
mass density, which is determined by the energy scale at
which the symmetry breaking process took place. Under this
assumption, and working up to first order in Gμ, Vilenkin
obtained a spacetime around the string which is flat but
presents a deficit angle ϕ = 8π Gμ. Some years later,
Hiscock [7], motivated by the possibility of theories which
may lead to large values of Gμ (i. e. Gμ ∼ 1),
considered a thick cylinder of radius a with uniform tension
and linear mass density, whose energy-momentum tensor is
Tμν (x , y) = diag(μ, 0, 0, μ)θ (r − a)/a2, and solved the full
Einstein equations in the interior and matched the resulting
static metric with the vacuum solution for the exterior; no
matter layer was assumed to exist at the radius a, so that the
adopted matching conditions imposed the continuity of both
the metric and the extrinsic curvature. His work showed that
Vilenkin’s results for the metric outside the string core were
actually valid to all orders in Gμ, that is, the conical
geometry was not an artifact introduced by the linearized
approximation, but an essential feature of gauge strings. Since the
corresponding metric has g00 = 1, i.e. the Newtonian
potential is null, non charged rest particles would not be affected
by the gauge string gravity. However, strings moving at
relativistic speeds would induce waves which could eventually
lead to observable matter density variations.
Things change considerably, however, when charged
particles are taken into account. The local flatness of a conical
background manifold does not imply zero force on a rest
charge. A vanishing force requires a field symmetric around
the charge, and this is not possible when a deficit angle exists.
While the field equations of a given point source are locally
those of Minkowski spacetime, the globally correct solution
in a conical geometry is not symmetric around the source,
and as a result a self-force appears on the charged particle
[8]. Moreover, as shown in [9–11] for electrically charged
point particles, the behavior of such force in relation with
the position of the charge allows to distinguish between two
locally identical geometries which differ in their global
topological aspects. This provides a sort of tool for detecting
wormholes of the thin-shell kind [12], whose geometries are
locally the same of those of the type which they would
connect. These wormholes are characterized by connecting two
exterior geometries of a given class by a throat, which is a
minimal area or circumference surface (see below), where
a matter layer of negligible thickness is placed [13,34,35].
Hence, for example, by studying the force on a static charged
particle an observer could determine if the background is that
of a gauge cosmic string or that of a thin-shell wormhole
connecting two conical spacetimes. Also, in the case of
topologically trivial spacetimes, an analogous analysis would allow
to distinguish different interior geometries by studying the
self-force on a charge placed at the exterior region beyond a
matter shell [11,14,15]. The problem of the self-force on a
scalar point charge appears as a natural but not at all trivial
extension of previous work.
Scalar fields are widely used in cosmological models
to explain the early Universe and the present accelerated
expansion; non-minimally coupled to gravity scalar fields
are included in inflationary models [16], while the unknown
nature of dark matter is sometimes described by classical
light or massless scalar fields [17,18]. Besides the physical
interest of scalar charges and the self-force problem [19–24],
a central aspect motivating a detailed analysis is the different
form in which the scalar field couples to gravity: while the
field of an electric charge is continuous across a non charged
matter shell separating two regions of a spacetime, this is not
the case for a scalar charge, and matter shells induce a sort
of new sources for the corresponding scalar field; this
necessarily translates to an, in principle, different behaviour of the
self-force. Some peculiarities due to non minimal coupling
of the scalar and the gravitational fields were pointed out
in [19,25–29]. In particular, some authors who studied the
scalar self-force and propagation of a scalar field in a fixed
background found that the stability regions of solutions for
the non minimally coupled wave equation are restricted by
the value of the coupling constant. An anomalous divergence
was first realized by Bezerra and Khusnutdinov [30] who
reported an infinite scalar self-force for some critical values
of the coupling constant in a class of spherically symmetric
wormhole spacetimes. For example, in a spherical
wormhole with an infinitely short throat of the thin-shell kind they
found critical values at ξ = n/4, for n ∈ N. Continuing to
previous work, Taylor [31] notice that solutions to the scalar
wave equation sourced by a point charge are unstable in the
Ellis wormhole for the discrete set of couplings ξ = n2/2.
The pathological behavior was removed by restricting the
coupling constant to the domain of stability where no poles
are encountered [32,33]. In the present work we find that a
similar feature occurs in cylindrical wormholes and also in
trivial spacetimes.
In this article we study the self-force acting on a static point
like scalar charge in a background constructed by cutting and
pasting two cylindrical manifolds with different deficit angle,
and we analyze the dependance on the non minimal coupling
to gravity. The considered geometries have all the matter
content isolated in a thin-shell (see for example [34,35]) and
in this way we could analyze the influence of the coupling on
the scalar field solutions and the consequences over the
selfforce. Additionally, the obtained results clarify some aspects
of the self-force in terms of the global properties of a given
background geometry. On the other hand, resonances of the
field appear at critical values ξc of the coupling which depend
on the shape of the profile function of the metric or, more
specifically, on the value of the deficit angles. We established
that the stable domain of the coupling is related to the kind of
matter to which it couples or, equivalently, to the sign of the
trace of the extrinsic curvature tensor jump over the shell.
We shall consider two models which are presented in Sect.
2; in the first one, an interior and an exterior submanifolds
are joined at an hypersurface where a thin-shell is placed and,
in the second one, two exterior submanifolds are joined. The
thin-shell is characterized by the value of κ, the trace of the
jump on the extrinsic curvature tensor. Section 3 is divided in
three sections; in Sects. 3.1 and 3.2 the field equation sourced
by the static scalar charge is solved in both types of manifolds
by the method of separation of variables in cylindrical
coordinates. The field is split in two terms, one homogeneous
and the other inhomogeneous at the position of the charge.
Resonant configurations for which the field diverges at
critical values ξc of the coupling are found in Sect. 3.3. These
are associated to instabilities of the coupled scalar field
equation in the corresponding background geometries. The scalar
field is regularized at the position of the particle in Sect. 4 by
subtracting the Detweiler–Whiting singular Green function
to the actual field. Finally the self-force over the scalar charge
is calculated evaluating the gradient of the regular field at the
position of the particle. The results are analyzed analytically
and numerically in terms of the coupling constant for
different configurations of the background spacetime in Sects. 4.1
and 4.2. Throughout the article the geometrized unit system
is used where c = G = 1.
2 Approach
The system is given by the action S = S + Sm0 + Sq in
a fixed background. The first term is the action for the free
massless scalar field
S = − 81π d4x √−g gαβ ∂α ∂β + ξ R 2 , (
1
)
where the integration is over all the spacetime with metric
gαβ , g is its determinant and ξ the arbitrary coupling of the
field to the curvature scalar R. The particle action Sm0 is
Sm0 = − m0
γ
dτ,
where γ is the scalar particle’s world line, m0 its bare mass
and dτ = −gαβ z˙α z˙β dτ is the proper time differential
along γ . The interaction term Sq coupling the field to the
particle’s charge q is written as
Sq =
dτ q
(z(τ ))
= q
= q
γ
γ
dτ
d4x √−g δ4(x , z(τ )) (x )
d4x √−g
γ
dτ δ4(x , z(τ )) (x ).
The wave equation for the coupled to gravity massless
scalar field is obtained demanding the total action to be
stationary under a variation δ (x ). We are interested in a scalar
charge q at rest; this yields the inhomogeneous equation
(
δ3(x − x )
− ξ R) (x ) = − 4π q z˙0 √−g
,
where is the d’Alambertian operator of the metric, x is
the spatial position of the charge q, and z˙0 = dt /dτ . On
the other hand, demanding the action to be stationary under
a variation δzα(τ ) of the world line yields the equations of
motion
m(τ )
Dz˙α
dτ
= q gαβ + z˙α z˙β
∇β (z)
for the scalar particle. The otherwise dynamical mass m(τ ) =
m0−q (z) is constant if the particle is at rest in a static
spacetime [36]. To hold the charge fixed, the total force exerted by
a mechanical strut is
Fsαtrut = m(τ ) 0α0z˙0z˙0 − q gαβ ∂β ,
the second term corresponds to minus the scalar force over the
static particle and will need to be regularized at the position
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
=
=
(
8
)
(
9
)
(
10
)
of the charge, while the term with the Christoffel symbol 0α0
α
is zero in locally flat geometries. Specifically, 00 = 0 is the
case of the cylindrical thin-shell spacetimes we will consider
which are described by line elements
In an infinitely thin straight cosmic string geometry (without
a shell), the profile function is simply ρ(r ) = r ω with the
radial coordinate 0 < r < ∞, 0 ≤ θ < 2π , and the
parameter 0 < ω ≤ 1 which gives the deficit angle 2π(1 − ω) of
the conical manifold. For thin-shell spacetimes we will have
two types of profile function. The first one is
Type I : ρ(r )
r ωi , if 0 < r ≤ ri (Mi )
(r − ri + re) ωe , if ri ≤ r < ∞(Me),
where the parameters ωi and ωe give the deficit angles in
the interior Mi and exterior Me regions, and the internal
and external radii are related by the first fundamental form
over the thin-shell hypersurface [34,35]; ρ(ri ) = ri ωi =
reωe. The second type of profile corresponds to a wormhole
spacetime with two asymptotic regions, this is
Type II : ρ(r )
(r− − r ) ω−, if − ∞ < r ≤ 0(M−)
(r + r+) ω+ , if 0 ≤ r < + ∞(M+),
where two exterior regions M− and M+, with ω− and
ω+ respectively, are joined over the hypersurface of a thin
throat by the condition ρ(0) = r−ω− = r+ω+. For
example, a cylindrical wormhole which is symmetric across the
throat has a radius r0 = r− = r+ and profile function
ρ(r ) = (r0 + |r |) ω. Either Type I or Type II geometries are
everywhere flat except at the shell hypersurface and, for Type
I spacetimes only, in the case of conical interiors (ωi = 1) the
central axes represents a conical singularity. For both types,
the Ricci scalar is R(r ) = −2 κ δ(r − rs ), where rs is the
corresponding shell’s radial position and κ is the trace of
the jump on the extrinsic curvature tensor over the thin-shell
[12,34]. In general: for Type I we have κ = (ωe − ωi )/ρ(rs )
with rs = ri , and for Type II κ = (ω+ + ω−)/ρ(rs ), with
rs = 0.
3 Scalar field in conical spacetimes
The solution to Eq. (
5
) in the conical manifold of an infinitely
thin straight cosmic string without a thin-shell, ρ(r ) = r ω,
is known from [37] and can be written in a close form as1
sinh(ζ /ω) ω−1 dζ
[cosh(ζ /ω) − cos (θ − θ )] (cosh ζ − cosh u)1/2
where: cosh u =
r 2 + r 2 + (z − z )2
2rr
, u
0.
(
12
)
For the static charge placed in Type I or Type II geometries,
the field Eq. (
5
) in cylindrical coordinates becomes
∂2 ρ (r ) ∂ 1 ∂2 ∂2
∂r 2 + ρ(r ) ∂r + ρ2(r ) ∂θ 2 + ∂ z2 − ξ R(r )
δ3(x − x )
= −4π q ρ(r ) .
We look for a solution of the form
= q
+∞
n=0 π 1 + δ0,n
4
+∞
0
Qn(θ )
dk Z (k, z)χn(k, r ),
where Fz = {Z (k, z) = cos[k(z − z )]} and Fθ = {Qn(θ ) =
cos[n(θ − θ )]} are a complete set of orthogonal functions of
the coordinates z and θ . Then, the radial functions χn(k, r )
are obtained from the radial equation
∂
∂r
ρ(r )
∂
∂r
−ρ(r )
= −δ(r − r ).
n
ρ(r )
2
+ k2 + ξ R(r )
χn(k, r )
Integrating over an infinitesimal radial interval around the
position of the shell r = rs on (
15
) we obtain the condition
∂
∂r
χn(k, r )
r=rs +
= − 2ξ κ χn(k, r ) r=rs
r=rs −
for the radial solutions, where we used the Ricci scalar
R(r ) = −2κδ(r − rs ) and the continuity of the profile
function ρ(r ) and of χn(k, r ) at r = rs . Analogously, from (
15
),
1 The axis at r = 0 of the conical geometry, where the infinitely thin
cosmic string would be placed, is intentionally excluded. The field in
the manifold which includes the axis couples to the Ricci scalar R(r ) =
− 2(ω − 1)δ(r )/(ωr ) at r = 0 and it can be shown to be = ω + ξ ,
with
q u lim (1 − ω)ξ
ξ = − π ω rr sinh u →0 ω + 2(1 − ω)ξ K0( )
where the term in brackets becomes identically null, i.e. the complete
solution which accounts for the coupling at the conical peak does not
affect the relevant part of the field [38].
(
17
)
∂
∂r
χn(k, r )
r=r +
1
r=r − = − ρ(r ) ,
assuming the continuity of χn(k, r ) at the radial position
r = r of the particle. The radial solutions in Type I and
Type II geometries will be obtained from (
15
) plus the
latter conditions and the requirements over the corresponding
asymptotic regions. In a conical manifold without shell, (
15
)
reduces to the nth order inhomogeneous Bessel equation
∂2 1 ∂
∂r 2 + r ∂r −
2 υ2
k + r 2
χ ω = −
δ(r − r )
,
ωr
(
18
)
where υ = n/ω, for each n and k. Requiring finiteness if
r → 0 and r → ∞, the continuity at r = r and its derivative
discontinuity (
17
), we obtain the scalar field in the infinitely
thin straight cosmic string spacetime given as a series
expansion with the radial solutions:
χ ω = ω1 Kυ (kr>)Iυ (kr<) , (
19
)
where Kυ (kr ) and Iυ (kr ) are the usual modified Bessel
functions of order υ = n/ω with n N0, two independent
solutions of the homogeneous version of Eq. (
18
), while
r< = min r ; r and r> = max r ; r . The field ω given
as the series expansion (
14
) with χ ω is equivalent to the
integral form (
12
).
3.1 Scalar field in Type I spacetimes
The radial solutions in Type I spacetimes will be obtained
defining an interior radial coordinate r1 = r for 0 < r < ri ,
in the interior region Mi of parameter ωi , and an exterior
radial coordinate r2 = r − ri + re for ri < r < ∞, in
the exterior region Me with parameter ωe. Using these new
coordinates we write the radial functions as
for each n and k. From (
15
), the respective equations become
=
=
with λ = n/ωi and ν = n/ωe. To solve them we impose: the
boundary conditions at the axis and infinity
rl1i→m0 χ i = ∞ ,
lim
r2→+∞
χ e = 0 ,
(
13
)
(
14
)
(
15
)
(
16
)
(
11
)
(
20
)
(
21
)
(
22
)
continuity of the field and its derivative discontinuity (
16
) at
the shell
χ i (r1 = ri ) = χ e(r2 = re) ,
∂ ∂
∂r2
χ e
r2=re − ∂r1
χ i r1=ri = −2κξ χ i (ri ),
plus the conditions for the inhomogeneity at the position of
the charge. Considering first the particle placed in Mi , the
inhomogeneity discontinuity (
17
) at r1 = r is
χ e
1
= χ ωe + ωe Kν (kr2)Bn(k), ν = n/ωe;
Iν (kr2< )Kν (kr2> ), rr22<> == mmianx{{rr22,,rr22}}
χ i
while χ i is required to be continuous at r1 = r1. From (
21
)
and (
22
), with the charge in Mi , the radial solutions are given
by
χ e = ωi
1
χ i = χ ωi + ω1i Iλ(kr1) An(k) , λ = n/ωi ;
Kν (kr2)Cn(k) , ν = n/ωe ;
Iλ(kr1< )Kλ(kr1> ) ,
r1< = min{r1, r1}
r1> = max{r1, r1}
account for the derivative’s jump (
25
) of χ i . The terms χ ωi
of the interior solution are intentionally left apart because,
inside the Fourier series, they sum up the integral expression
(
12
), as in (
19
). Finally, the coefficients An(k) and Cn(k) are
determined from conditions (
23
) and (
24
) at the shell;
An(k) = −Iλ(kr1)
Kλ(kri )Kν (kre) − Kν (kre)Kλ(kri ) + 2κξ Kν (kre)Kλ(kri ) ,
× Iλ(kri )Kν (kre) − Kν (kre)Iλ(kri ) + 2κξ Kν (kre)Iλ(kri )
Cn(k) =
An(k)Iλ(kri ) + Iλ(kr1)Kλ(kri ) ,
Kν (kre)
with the prime over Bessel functions implying derivative with
respect to the corresponding radial coordinate. Alternatively,
with the particle in Me the inhomogeneity is at r2 = r −
ri + re and now
while χ e is required to be continuous at r2 = r2. From (
21
)
and (
22
), with the charge in Me, the radial solutions are given
by
Either with the particle in the interior or exterior region,
we see from the two terms explicitly written in the respective
radial solutions, (
26
) or (
33
), that in the four-dimensional
region where the charge is placed we can separate the field
in two parts as
=
ω +
ξ if (x , x ) ∈ Mi or Me.
The field ω, corresponding to the series with radial
functions called χ ω, is inhomogeneous at x and locally
equivalent to the field (
12
) of a charge in a cosmic string manifold
with the corresponding parameter ω = ωi or ω = ωe. The
remaining part of the field, the series called ξ , is
homogeneous at x and accounts for the distortion produced in
the specific thin-shell geometry with information about the
coupling and boundary conditions.
3.2 Scalar field in Type II thin-shell wormholes
The radial solutions in Type II geometries, corresponding
to wormholes with two asymptotic regions, will be obtained
defining the radial coordinate r1(r ) = r− − r for r < 0, in
the region M− of parameter ω−, and the radial coordinate
r2(r ) = r + r+ for r > 0, in the region M+ of parameter
ω+. Using these coordinates we write the radial functions as
with λ = n/ω− and ν = n/ω+. The boundary conditions at
the infinities are
lim χ − = 0 , lim χ + = 0 , (
40
)
r1→+∞ r2→+∞
continuity of the field and its derivative discontinuity (
16
) at
the shell are
χ −(r1 = r−) = χ +(r2 = r+),
∂ ∂
χ +
r2=r+ + ∂r1
∂r2
respectively, and the inhomogeneity discontinuity (
17
) is
χ − r1=r− = − 2κξ χ −(r−) ,
while χ + is required to be continuous at r2 = r2. From (
39
)
and (
40
) the radial solutions are given by
Kλ(kr1)En(k), λ = n/ω−;
Kν (kr2)Wn(k), ν = n/ω+;
1
χ − = ω
+
1
χ + = χ ω+ + ω
+
where the terms
χ ω+ =
1
ω+
Wn(k) = −Kν (kr2)
Iν (kr2< )Kν (kr2> ),
r1< = min{r2, r2}
r1> = max{r2, r2}
account for the derivative’s jump (43) of χ +. The coefficients
En(k) and Wn(k) are determined from (
41
) and (42) at the
shell;
Iν (kr+)Kλ(kr−) + Kλ(kr−)Iν (kr+) + 2κξ Iν (kr+)Kλ(kr−) ,
× Kλ(kr−)Kν (kr+) + Kν (kr+)Kλ(kr−) + 2κξ Kν (kr+)Kλ(kr−)
(47)
En(k) =
Iν (kr+)Kν (kr2) + Wn(k)Kν (kr+) ,
Kλ(kr−)
with the prime over Bessel functions implying derivative with
respect to the corresponding radial coordinate. As we pointed
out earlier in Sect. 3.1 for the solutions in Type I geometries,
in the four-dimensional region where the charge is placed,
we can separate the field as
=
ω+ +
ξ if (x , x ) ∈ M+,
with ω+ constructed with the radial solutions χ ω+ ,
inhomogeneous at x and locally equivalent to the field (
12
) of a
charge in a cosmic string manifold, and the term ξ which
is homogeneous at x .
(
39
)
(
41
)
(42)
(43)
(44)
(45)
(46)
(48)
(49)
3.3 Resonant configurations The scalar field produced by a static scalar particle of charge q in Type I or Type II spacetimes is
=
π n=0
−z )],
4q +∞ cos[n(θ − θ )]
with the radial functions χn(k, r ) specified in the previous
subsections. The dependence of the field on the curvature
coupling is seen from the respective coefficients (
29
)–(
30
),
(
35
)–(
36
) or (47)–(48) in each of the radial solutions. From
a simple inspection on the results obtained in Type I
spacetimes, we see that the denominators of coefficients An(k)
and Bn(k), given in (
29
) and (
35
) respectively, do not
vanish if κ is negative and the coupling ξ is positive. The same
is guaranteed with a positive κ and a negative value for ξ ,
including denominators of coefficients Wn(k) obtained for
Type II, given in (47), where κ is always positive. In the
contrary, if the product κ ξ is positive the coefficients manifest
the possibility of encountering poles in the integrands of (50)
for some value k = k p. Nevertheless, if the integrand of a
given mode n presents a pole at k p > 0, the divergency can
be circumvented splitting the integral at k p ∓ , with → 0,
by canceling out the contributions of the lateral limits. As
an alternative to the last method, contour integrals for the
complex-valued integrand function can be used to obtain the
positive real half-line integral. But if the denominator in the
integrand coefficient is null for k = 0 we have, inevitably,
a divergent mode. We can examine directly the integrand
of each mode in the limit k → 0+ to identify these
divergencies. For example, in Type I geometries with the particle
placed at r1 = r1, we obtained the internal radial solutions
χni (k, r1) = χnωi (k, r1) + ω1i Iλ(kr1) An(k), and the external
ones χne(k, r2) = ω1i Kν (kr2)Cn(k), given in (
26
) and (
27
),
from which we can compute:
−
2 ξ −ξc(n) (ωe−ωi )
.
(52)
(53)
,
(54)
r2<
r2>
n/ωe
.
(55)
The critical value ξc(n) is given by
ξ (n)
c
n n
= ρ(ri ) κ = ωe − ωi
,
for n ∈ N, in Type I spacetimes. Equivalently, with the
particle placed at r2 = r2 we had the internal and external radial
solutions: χni (k, r1) = ω1e Iλ(kr1)Dn(k) and χne(k, r2) =
χnωe (k, r2) + ω1e Kν (kr2)Bn(k), given in (
32
) and (
33
), from
where we can see analogously
These limits show a divergence for a mode n 1 if the
curvature coupling takes the critical value ξc(n) = n/(ωe − ωi ),
which is identified with a resonant configuration for the nth
mode of the scalar field in Type I spacetimes. For n = 0 there
is no value of ξ associated to this kind of resonances (the
denominator of radial coefficients do not vanish for k = 0
in the n = 0 mode). In terms of the extrinsic curvature on
the thin-shell in Type I spacetimes; if κ < 0 (ordinary
matter thin-shell) the coupling can take values ξ > ξc(n=1) =
1/(κωi ri ) to avoid encountering a resonant mode which
makes the field divergent and, on the other hand, if κ > 0
(exotic matter thin-shell) couplings ξ < ξc(n=1) = 1/(κωi ri )
ensure that the configuration does not produce some
resonant mode. Repeating the analysis in Type II geometries,
with the radial solutions obtained with the particle placed
at r2 = r2: χn−(k, r1) = ω1+ Kλ(kr1)En(k) for region M−,
and χn+(k, r2) = χnω+ (k, r2) + ω1 Kν (kr2)Wn(k) for region
M+, given in (45) and (45), we ca+n see the same dependance
k→0+
χn−(k, r1) −−−→
−
−
k→0+
χn+(k, r2) −−−→
ξ
r−
r1
n/ω+
2n/ω+ ⎤
⎦
r2<
r2>
n/ω+
,
,
(56)
(57)
with
ξc(n) = ρ(0n) κ = ω+ +n ω− (58)
for n ∈ N, in the wormhole spacetime. These limits show a
resonant mode n 1 for the scalar field if the coupling takes
the critical value ξc(n) = n/(ω+ +ω−). To avoid encountering
a resonant mode, which makes the field divergent in Type II
spacetimes, the coupling must take values ξ < ξc(n=1).
4 Regular field and scalar self-force
The static scalar self-force over a charged particle in a curved
spacetime is obtained regularizing the actual field with a
singular field S as
fα = q xl→imx ∇α
−
S ,
where the coincidence limit takes the coordinate spatial
components of the point x to the charge’s position x along the
shortest geodesic connecting them [39]. The field S =
q 4π G DW (x ; x ) for a static particle in a static spacetime
is constructed with the Detweiler–Whiting singular Green
function in three dimensions over the normal convex
neighborhood of x [40]. This Green function has the same
singularity structure as the particle’s actual field, exerts no force
on the particle and, in the considered static problem, can be
calculated as [36]
(59)
where σ = σ (x, x ) is half the squared geodesic distance
between x and x as measured in the purely spatial sections
of the spacetime, the Van-Vleck Morette determinant has the
following expansion
1
G DW (x ; x ) = 4π
1/2
√2σ
τadv
1
+ 2 τret
V (x , x (τ ))dτ ,
(60)
1/2
1
= 1 + 12 Rαβ σ ;ασ ;β + O
while the tail part is given by
τadv
τret
V (x , x (τ ))dτ =
√
The terms of order O (σ ) /√2σ would be irrelevant for the
renormalization of the field since they vanish in the
coincidence limit. To regularize the field of a point scalar charge
in conical spacetimes we will need the regular part of the
inhomogeneous field in the locally identical geometry of a
infinitely thin cosmic string manifold. Metric (
8
) is locally
flat, except over the thin-shell, and an expression for the
singular field can be simply given by the Green function over
Minkowski spacetime, i.e. S = q/ √2σ . Then, this
singular field can be represented as
S
≡
Mink =
q
= π √2rr
ω=1
+∞
u
sinh(ζ )dζ
[cosh(ζ )− cos (θ −θ )] (cosh ζ − cosh u)1/2 ,
where for ω=1 we used the integral expression (
12
) with
ω = 1.
We first compute the self-force over a scalar charge in the
infinitely thin straight cosmic string background, which only
has a radial component due to cylindrical symmetry and is
equal to
(68)
(69)
(70)
dinates are assumed in advance. The renormalized field (66)
is restored from the last result putting r = r , while the
selfforce is fω = q ∂r ωR |r . We note that the self-energy of
a scalar particle in a conical manifold coincides with that of
an electric charge, known as Linet’s result on a cosmic string
geometry [8].
The singular structure in (63) is encountered in every
conical geometry and the regularization procedure will be
analogous in each of the conical thin-shell spacetimes because
they are locally indistinguishable. In virtue of the explicit
splitting of the field as = ω + ξ , done in Sect. 3.1 for
Type I (
37
), or (49) in 3.2 for Type II geometries, the singular
part at the position of the charge appears in the term ω and
the regularization procedure yields
R
=
R
ω +
ξ ,
with the regular field
ing parameter ω.
ωR given in (67), with the
correspond
4.1 Self-force in Type I geometries
The regularization procedure applied to the field obtained
in Type I thin-shell geometries, with radial functions (
20
)
given in terms of coordinates r1(r ) = r for Mi or r2(r ) =
r − ri + re for Me, yields the regular field at a general
radial position r of the particle:
R (r, r )
=
⎧ q Lωi 2 − ln rr
⎪⎪⎪⎪⎪⎪⎪ 4π4rq +∞ +∞
⎨⎪⎪⎪ + πwi n=0 0
q Lωe 2 − ln r2r(r)
4πr2 2
⎪⎪⎪⎪⎪⎪⎪ 4q +∞ +∞
⎩⎪⎪⎪ + πwe n=0 0
dk Iλ(kr ) 1A+nδ(nk,)0
if r < ri
dk Kν (kr2(r )) 1B+nδ(nk,)0 if r > ri
fω = − q
with
Lω =
+∞
0
2 Lω 1
4π r 2
sinh(ζ /ω)
ω [cosh(ζ /ω) − 1]
sinh ζ dζ
− cosh ζ − 1 sinh(ζ /2)
,
which was obtained evaluating θ = θ and z = z , in (
12
)
for the actual field and in (63) for the singular field, to take
the coincidence limit (59) over radial geodesics. The
selfenergy is defined as U = q2 rωen, with the renormalized
field obtained from
ren
ω
= xl→imx
ω −
S
q Lω
= 2π r .
This self-field can be represented as a regular field at the
position of the particle given by
ωR (r, r ) = 4qπ Lrω 2 − ln rr , (67)
which is an homogeneous solution of the wave equation
(
5
), where the coincidence limits over the irrelevant
coorfrom where the self-energy is obtained as Usel f = q2 R |r=r .
If the charge is placed in a region without deficit angle (ω =
1), then Lω = 0 and the first term is null. The radial scalar
self-force f = q ∂r R |r over the particle is
⎧⎪ − q42πLrω2i + π4qw2i
⎪⎪⎪
f = ⎪⎪⎪⎪⎪⎪⎪⎨ n+=∞0 q+02∞Lωe 4q2
⎪⎪⎪⎪⎪⎪⎪⎪ −+∞4π+[r∞2(r )]2 + πwe
⎪⎪
⎩⎪ n=0 0
dk Iλ(kr ) 1A+nδ(nk,)0 ,
if r < ri ,
dk Kν (kr2(r )) 1B+nδ(nk,)0 , if r > ri .
This is the only force acting over the particle in the locally flat
Type I spacetime. To see the results the self-force is plotted
(a) vicinities of the shell.
(b) far from the shell.
for different values of the coupling constant in two distinct
Type I geometries; one corresponding to a thin-shell of
ordinary matter, i.e. κ < 0, and the other with a shell of exotic
matter, κ > 0. In the first case, presented in Fig. 1, the
background is a Minkowski interior region (ωi = 1) and a conical
exterior with ωe = 1/2, while the latter, shown in Fig. 2, is a
conical interior with ωi = 1/2 joined to a Minkowski
exterior. The force over the particle is plotted as π4 rqi22 f against
the dimensionless position r/ri of the charge.
In Fig. 1 we show the results in a Type I geometry with an
ordinary matter thin-shell and coupling values in the stable
domain ξ > ξc(n=1) = −2. The self-force vanishes at the
central axis of the geometry because the first term in (70)
is null after regularization in a Minkowski interior. When
approaching the shell from either sides we observe a
divergent force, being attractive or repulsive depending on the
value of the coupling, Fig. 1a. Nevertheless, in the conical
exterior region the force is asymptotically attractive to the
center for every value of the coupling constant due to the
leading term ∼ −L ωe r −2 obtained after renormalization, Fig. 1b.
The solid line in Fig. 1 corresponds to minimal coupling;
in the interior region with r < ri the force increases from
cero at the center to infinity as r → ri−, while in the exterior
we see a negative force diverging at the vicinities of the shell
and vanishing at the asymptotic conical infinity. This shows
that for ξ = 0 the curvature jump at the ordinary matter
shell acts attracting the particle. We observed that negative
values of ξ present qualitatively the same results but with an
intensified force near the shell. A richer spectrum appears
for positive couplings. If 0 < ξ 1/4 the force changes
sign in either region but becomes attractive to the shell in its
vicinities. For ξ > 1/4, the force diverges if r → ri but with
the opposite sign i.e., the particle is repeled in the vicinities
of the ordinary shell. Despite this local difference, the leading
asymptotic term proportional to −L ωe=1/2 in (70) produces
an attraction sufficiently far from the shell in the deficit angle
exterior region.
In Fig. 2 we show the results in a Type I geometry with
a shell of exotic matter and couplings in the stable range
ξ < ξc(n=1) = 2. The self-force diverges at the center with
the particle been attracted to the conical singularity at the
interior, and approaching the shell from either sides it diverges
showing different behaviors depending on the value of the
coupling, Fig. 2a. In an exterior region without angle deficit
Lωe=1 = 0, and the force becomes repulsive from the
center sufficiently far from the shell for any value of ξ , Fig. 2b,
differing from the previous case shown in Fig. 1b of a deficit
angle exterior.
The solid plotted line corresponds to minimal coupling;
the force is negative in the interior, i.e. attractive to the conical
axis and repulsive from the shell, and in the exterior region
the force is positive. For ξ = 0 the exotic shell repels the
particle and the force diverges if r → ri . Negative values of
ξ present similar results but with a greater intensity of the
force. The richer spectrum appears, again, for ξ > 0. With
couplings in the range 0 < ξ 1/4, the force may vanish and
change sign in a same region, but sufficiently near to the shell
it becomes repulsive. Oppositely, for ξ > 1/4 we see that the
shell attracts the particle at its vicinities and sufficiently far
from it, in the exterior region, a leading asymptotic repulsion
from the center is observed due to the influence of the angle
deficit of the core geometry.
In either example we observed a smooth change in the
self-force plot as long as the coupling takes values in the
corresponding stable domain, with the exception of the change
at ξ = 1/4. If the coupling is ξc(n=1) = 1/(ωe − ωi ) we
obtain a divergent self-force due to the contribution of the
unstable n = 1 mode of the field. As expected, a divergency
is also encountered at ξc(n) = n/(ωe − ωi ) for n ∈ N in the
respective mode, corresponding to the critical configurations
for Type I geometries. For the spacetime with an ordinary
tmheatcterirtitchailnc-oshueplllin(gwsiathreωξic(n)= 1, ωe = 1/2 and κ < 0),
= −2n and observing plots for
couplings in the successive intervals − 2(n + 1) < ξ < − 2n
we found that the self-force is qualitatively the same as the
one produced with couplings in the interval − 2 < ξ < 0.
Analogously, in the spacetime with exotic matter (ωi = 1/2,
ωe = 1 and κ > 0), ξc(n) = 2n and varying the coupling
in the intervals 2n < ξ < 2(n + 1) we observed
qualitatively the same self-force found for couplings in the interval
1/4 < ξ < 2.
4.2 Self-force in Type II geometries
The regularization procedure over the field obtained in Type
II wormhole geometries of radial functions (
38
), for the
charged particle placed in M+ where r2(r ) = r + r+ and
r2 = r + r+, yields
R (r, r ) = q4πLrω2+ 2 − ln r2r(2r )
4q
+ π w
+∞ +∞
The self-energy is given by Usel f = q2 R |r=r , and the radial
scalar self-force f = q ∂r R |r is
To analyze the results it is convenient to focus on
symmetric wormholes across the infinitely short throat of radius r0 =
r+ = r− and, furthermore, we first consider the self-force in a
Type II geometry without deficit angle (ω+ = ω− = 1). The
force over the particle placed in M+ of a cylindrically
symmetric wormhole constructed with two Minkowski regions is
plotted in Fig. 3 as π4rq022 f against the dimensionless position
r/r0 of the charge; couplings ξ < 1/4 are shown in Fig. 3a,
and couplings ξ > 1/4 are in Fig. 3b. In this geometry the
first term of the self-force in (72) is null, and the common
feature is an asymptotically repulsive force from the throat of
exotic matter for every value of ξ . The critical configurations
for a cylindrically symmetric Minkowski wormhole
corre(a) Couplings ξ < 1/4.
(b) Couplings 1 /4 < ξ < 1/2 and 1/2 < ξ < 1.
(a) Couplings ξ
(b) Couplings in the range 0 < ξ < 1/4.
sponds to couplings ξc(n) = n/(ω+ + ω−) = n/2, n ∈ N.
An instability is manifested in the n = 1 mode at the value
ξc(n=1) = 1/2 for which the self-force diverges. While
varying the coupling in the safe domain ξ < 1/2, the self-force
varies smoothly with the exception of the qualitative change
produced at ξ = 1/4. Instabilities are repeated at ξc(n) = n/2,
and in the successive intervals n/2 < ξ < (n + 1)/2 we
observed qualitatively the same self-force of those cases with
couplings 1/4 < ξ < 1/2; to show this, in Fig. 3b, we present
results for couplings in the interval 1/4 < ξ < 1/2 and for
the next interval, 1/2 < ξ < 1, as well.
The solid line in Fig. 3a corresponds to minimal coupling;
the force is repulsive from the throat and diverges
approaching the shell in the limit r → 0. We see that negative values
of ξ produce the same behavior with an increasing repulsion
with decreasing value of the coupling. Couplings in the range
0 < ξ < 1/4 show an attractive force in some finite region
but produce a repulsion in the vicinities of the infinitely short
throat as well. Oppositely, the particle is attracted in the
vicinities of the shell for couplings ξ > 1/4, as shown in
Fig. 3b, diverging in the limit r → 0. Despite the different
local behavior near the throat for ξ > 1/4, we find a leading
asymptotic repulsion associated to the positive jump κ > 0
and the non trivial topology of this spacetime.
Finally we present the scalar self-force over a charge
placed in a symmetric wormhole constructed with conical
regions given by the parameter ω+ = ω− = 2/3. The force
over the particle in M+ is represented as π4rq022 f against
the dimensionless position r/r0 of the charge; for couplings
ξ < 1/4 in Fig. 4, and for couplings ξ > 1/4 in Fig. 5. In
comparison with the previous Minkowski symmetric
wormhole, the main global difference is the asymptotically
leading term in (72) given by f ∼ − L2/3 r −2 in the deficit
angle regions, which produces an attractive force far from
the throat for every value of ξ . The critical configurations
for this conical symmetric wormhole appear for couplings
ξc(n) = n/(ω+ + ω−) = 3n/4, n ∈ N. An instability is
manFig. 5 Dimensionless scalar self-force π4qr022 f as a function of r/r0 in a
conical wormhole with couplings in the range 1/4 < ξ < 3/4 = ξc(n=1),
and 3/4 < ξ < 3/2 = ξc(n=2). The force is asymptotically attractive
ifested in the n = 1 mode at the value ξc(n=1) = 3/4 for
which the self-force diverges. In the stable domain ξ < 3/4,
the self-force varies smoothly with the significant qualitative
change at ξ = 1/4, as in previous cases. In Fig. 5 we include
cases beyond the stable domain.
The solid line in Fig. 4a corresponds to minimal coupling;
the force is repulsive in the vicinities of the throat, diverges
in the limit r → 0 at the thin-shell, and sufficiently far it
becomes attractive. Negative values of ξ present the same
qualitative behavior with an increasing force with decreasing
value of the coupling. Figure 4b shows results for couplings
in the range 0 < ξ < 1/4; the force is also repulsive near
the infinitely thin throat but it may vanish in one or three
positions in an intermediate region becoming asymptotically
attractive as r −2. In Fig. 5 we show the self-force for
couplings ξ > 1/4; the force is attractive in the vicinities of
the shell for all these cases, we find an intermediate region
with a repulsive force and the leading asymptotic attraction
associated to the deficit angle term. We plotted cases with
couplings beyond the stable domain; this is 3/4 < ξ < 3/2,
between the critical values for the n = 1 and n = 2 modes,
which show qualitatively the same self-force of those with
couplings in the interval 1/4 < ξ < 3/4. This last feature is
repeated in the successive intervals 3n/4 < ξ < 3(n + 1)/4.
4.3 The ξ = 1/4 coupling from the radial equation
The analysis of the scalar self-force over a charged
particle with the field arbitrarily coupled to the Ricci curvature
scalar, in the cylindrically symmetric spacetimes with
matter content concentrated over a thin-shell, pointed out the
specific value ξ = 1/4 as the coupling for which the force
changes sign at the vicinities of the shell. The presence of
the thin-shell produces a deformation of the field lines and
the curvature coupling determines some details of this
distortion: in the previously studied examples we saw that the
effect over the particle near the shell changes from attraction
to repulsion at ξ = 1/4. To understand this result
analytically we can study the radial Eq. (
15
) by the substitution
χn(k, r ) = ψ (r )/√ρ(r ) for each n and k, to take it to the
form
d2
dr 2 − V (r, ξ ) ψ (r ) = −
δ(r − r )
√ρ(r )
.
The influence of the coupling can be interpreted from the
homogeneous version of (73); a one dimensional radial wave
equation for ψ propagating in the potential
V (r, ξ ) =
4 n2 − ρ 2(r )
4ρ2(r )
+ k2 + 21 ρρ((rr)) (1 − 4ξ ) ,
where we have used that the Ricci scalar is R(r ) =
−2 ρ (r )/ρ(r ) = − 2κ δ(r − rs ), to put it in terms of the
profile function. The last term in (74), which is proportional
to a delta function, changes sign at ξ = 1/4 in the
considered geometries. Note that in a Minkowski (ω = 1) or
conical spacetime where there is not a thin-shell, this potential is
simply
Vvac(r ) =
4 n2 − ω2
(2r ω)2
+ k2.
In comparison with what the radial wave equation would
be in a vacuum cylindrical spacetime without the shell, the
presence of the matter shell introduces a finite change in
V (r, ξ ) accounted by ρ (r ) in the first term of (74), and an
infinitely narrow potential barrier or well at r = rs given
by the delta function on ρ (r ). Nevertheless, if ξ = 1/4 the
point-like infinite term disappears from the potential; for this
specific value the contribution from the curvature coupling
at the thin-shell exactly cancels out the effect of the extrinsic
curvature discontinuity in the potential for ψ . Without track
of the delta function, the potential
(73)
(74)
(75)
has, at most, a finite jump given by ρ (r ) (as if it were an
interface change at the hypersurface r = rs ), and no
singularity from the extrinsic curvature discontinuity. In terms of
a boundary contribution for ψ (r ) at the shell, the jump of the
radial derivative of ψ at r = rs can be obtained from
(77)
rs +
While for any value of ξ = 1/4 the non null jump is positive
or negative, for ξ = 1/4 there is no jump on ψ (r ) across the
shell.2
Focusing in Type I thin-shell spacetimes where we used
the global radial coordinate 0 < r < +∞, the profile
function and derivatives are
ρ(r ) = ωi r (ri − r ) + ωe (r − ri + re) (r − ri ), (78)
ρ (r ) = ωi (ri − r ) + ωe (r − ri ),
ρ (r ) = (ωe − ωi ) δ(r − ri ) = κ ρ(ri ) δ(r − ri ).
(79)
(80)
We can see from the potential (74) that the effective infinite
term, localized at r = ri , has sign given by the product:
κ(1 − 4ξ ). For example, in Type I geometries with
ordinary matter thin-shells (κ < 0), there is an infinite barrier
if ξ > 1/4 and an infinite well if ξ < 1/4. In terms of the
force produced by the scalar field over a charge q, in Type
I geometries with κ < 0; a coupling ξ > 1/4 (barrier) repels
the particle from the shell in a neighborhood of r = ri , and
a coupling ξ < 1/4 (well) attracts the particle to the shell.
This conclusion is in accordance with the results found in
the plots of Fig. 1a of a shell with ordinary matter.
Similarly, the opposite was found in terms of the coupling in the
examples plotted on Fig. 2a; the shell of exotic matter and
κ > 0 produces an infinite potential with negative sign if
ξ > 1/4 (well) which attracts the particle, or with positive
sign if ξ < 1/4 (barrier) which repels it, in a neighborhood
of r = ri . In wormhole spacetimes, Type II, the analogous
analysis applies; the potential well or barrier with ξ = 1/4
is manifested in the sign of the self-force at the vicinities of
the throat of κ > 0, producing an attraction (ξ > 1/4) or
repulsion (ξ < 1/4) from the shell.
5 Summary and conclusions
The arbitrarily coupled massless scalar field produced by a
static point charge in cylindrically symmetric backgrounds
V (r, ξ = 1/4) =
(76)
2 The value ξ = 1/4 is known as the coupling that eliminates the Robin
boundary energy for a scalar field [41].
with a thin-shell of matter was found in spacetimes with one
(Type I) or two (Type II) asymptotic regions. The self-force
over the charged particle was calculated from the scalar field
and studied numerically and analytically in terms of the
curvature coupling ξ . The fixed background geometries used are
everywhere flat except over the thin-shell where the trace of
the extrinsic curvature jump is κ and, for Type I spacetimes
only, over the central axes of the geometry in case of
conical interiors. Type I geometries have deficit angle interior
and exterior regions, characterized by parameters ωi and ωe
respectively, with κ = (ωe − ωi )/ρ (rs ), where ρ (rs ) is the
metric’s profile function at the position of the shell. Type II
wormhole spacetimes, with exterior regions of parameters
ω− and ω+ respectively, have κ = (ω− + ω+)/ρ (rs ). We
found the critical values ξc(n) = n/ (ρ (rs ) κ ), with n ∈ N,
for which the coupled to curvature scalar field is unstable in
the background configuration. For a Type I geometry with
an ordinary matter thin-shell (κ < 0) the safety domain of
the coupling is ξ > ξc(n=1), while for exotic matter shells
(κ > 0), either in geometries of Type I or II, the field is
stable if the coupling takes values in the range ξ < ξc(n=1).
These results add to those in [30–33] (mentioned in the
introduction) relative to the stable domain of solutions for
a scalar field coupled to spherically symmetric wormhole
backgrounds but, in our case, studying spacetimes with
cylindrical symmetry and in trivial topologies as well.
The only force over the charged particle in these
geometries is the scalar self-force, which we obtained from the
regularization of its own scalar field. The sign of the asymptotic
force does not depend on the coupling nor on the topological
difference between Type I and Type II, it only depends on
whether there is an angle deficit or not on the external region
where the charge is placed; conical asymptotics produce a
leading attractive force, while Minkowski regions produce
a repulsive asymptotic self-force. This clarify some aspects
of the self-force in terms of global properties of the given
background geometry. In thin-shell spherical wormholes the
scalar self-force of a massless particle changes sign at the
value ξ = 1/8 (corresponding to the conformal flatness of
the 3D section of the constant time manifold, see [30–33]),
in our cylindrical geometries there is no value of ξ for which
this occurs globally and the asymptotic sign of the force only
depends on the conicity or not of the external region. On the
other hand, there is a relevant local influence of the coupling
over the self-force at the vicinities of the shell of matter. The
specific coupling ξ = 1/4 was identified as the value for
which the scalar force changes sign at a neighborhood of
the shell; if κ (1 − 4ξ ) > 0 the shell acts repulsively as an
effective potential barrier, while if κ (1 − 4ξ ) < 0 it attracts
the charge as a potential well. Finally we note that beyond
the stable domain of the curvature coupling, in the intervals
ξ ∈ (ξc(n); ξc(n+1)) between two successive critical values,
the self-force shows qualitatively the same behavior as the
one produced with couplings in some interval of the safety
range; this refers, precisely, to the interval ξc(n=1) < ξ < 0
in cases with κ < 0, or to the interval 1/4 < ξ < ξc(n=1) in
cases with κ > 0.
Acknowledgements This work was supported by the National
Scientific and Technical Research Council of Argentina.
Open Access This article is distributed under the terms of the Creative
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