Charged string loops in Reissner–Nordström black hole background
Eur. Phys. J. C
Charged string loops in ReissnerNordström black hole background
Tursinbay Oteev 0
Martin Kološ 0
Zdeneˇk Stuchlík 0
0 Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava , Bezrucˇovo nám.13, 74601 Opava , Czech Republic
We study the motion of current carrying charged string loops in the ReissnerNordström black hole background combining the gravitational and electromagnetic field. Introducing new electromagnetic interaction between central charge and charged string loop makes the string loop equations of motion to be nonintegrable even in the flat spacetime limit, but it can be governed by an effective potential even in the black hole background. We classify different types of the string loop trajectories using effective potential approach, and we compare the innermost stable string loop positions with loci of the charged particle innermost stable orbits. We examine string loop small oscillations around minima of the string loop effective potential, and we plot radial profiles of the string loop oscillation frequencies for both the radial and vertical modes. We construct charged string loop quasiperiodic oscillations model and we compare it with observed data from microquasars GRO 165540, XTE 1550564, and GRS 1915+105. We also study the acceleration of current carrying string loops along the vertical axis and the string loop ejection from RN black hole neighbourhood, taking also into account the electromagnetic interaction.
1 Introduction
Detailed studies of relativistic currentcarrying string loops
moving axisymmetrically along the symmetry axis of Kerr or
Schwarzschild–de Sitter black holes appeared currently [
8–
10
]. Tension of such string loops prevents their expansion
beyond some radius, while their worldsheet current
introduces an angular momentum barrier preventing collapse
into the black hole. Such a configuration was also studied
in [
7,13,21
]. There is an important possible astrophysical
relevance of the currentcarrying string loops [
8
] as they could
in a simplified way represent plasma that exhibits associated
stringlike behavior via dynamics of the field lines in the
plasma [
4,19
] or due to thin isolated flux tubes of plasma that
could be described by an onedimensional string [
5,19,20
].
In the previously mentioned articles the string loop was
electromagnetically neutral and there was no external
electromagnetic field. Motion of electromagnetically charged string
loops in combined external gravitational and
electromagnetic fields has been recently studied [
30,31
]. Now we would
like to extend such research and we examine dynamic
properties of electromagnetically charged and current carrying
string loop also in combined electromagnetic and
gravitational fields of Reissner–Nordström background
representing a pointlike electric charge Q source. Our work
demonstrates the effect of the black hole charge Q on the string loop
dynamic in general; the discussion of the black hole charge
relevance is given in Appendix A. We discuss two
astrophysically crucial limiting cases of the dynamics of the charged
string loops related to phenomena observed in microquasars:
small oscillations around equilibrium radii that can be
relevant for the observed quasiperiodic highfrequency
oscillations, and strong acceleration of the string loops along the
symmetry axis of the black hole–string loop system that can
be relevant for creation of jets.
The general dynamics of motion for relativistic current
and charge carrying string loop with tension μ and scalar
field ϕ was introduced by [
13
] for the spherically
symmetric Schwarzschild BH spacetimes, for the Kerr spacetimes
it is discussed in [
8,10,11,14
]. General Hamiltonian form
for all axially symmetric spacetimes also with
electromagnetic field is introduced by [13]. To show properly how
the string loops interact electromagnetically, we will
compare charged particle motion with the charged string loop in
the same Reissner–Nordström black hole background, using
results already obtained in [
1,2,17,18
]. We show that there
are similarities in the dynamics of the charged string loops
and charged test particles, as the dynamics can be described
in both cases by the Hamiltonian formalism with a relatively
simple effective potential. There is a fundamental difference
in the RN backgrounds: while the test particle motion is
regular, the string loop motion has in general chaotic character
[
10, 13
], where “islands” of regularity occur only for small
oscillations near the string loop stable equilibrium points.
Throughout the present paper we use the spacelike
signature (−, +, +, +), and the system of geometric units in which
G = 1 = c. However, for expressions having an
astrophysical relevance we use the constants explicitly. Greek indices
are taken to run from 0 to 3.
2 Dynamics in spherically symmetric spacetimes
Gravitational interaction of the string loop with the central
electrically charged black hole occurs through the spherically
symmetric Reissner–Nordström (RN) metric given by the
line element expressed in geometric units
ds2 = − f (r )dt 2 + f −1(r )dr 2 + r 2(dθ 2 + sin2 θ dφ2), (1)
μ, momenta are
(6)
(7)
(2)
(3)
(5)
ton for example the gravitational interaction is physically
irrelevant. Therefore the central charge Q will significantly
electrically interact with the charged string loop even when
the black hole charge Q is too small to make relevant
contribution to the metric (1).
We thus examine different physically relevant situations
according to the gravitational/electric field strength ratio:
Flat There is no black hole and hence no gravitational
interaction. The electric field of the charge Q is so weak, that it
will not contribute to the metric. We will use the flat
metric (1), with M = 0, Q = 0, while the electromagnetic
interaction will be given by (5). Discussed in Sect. 3.1.
RN There will be black hole, with the gravitational field
influenced by the strong electromagnetic field. We will use full
RN metric (1) with electromagnetic interaction given by
(5). Discussed in Sect. 3.2.
The relevance of the individual three cases for realistic values
of RN black hole metric/string loop, all string loop quantities
and their dimensions in physical units, will be discussed in
detail in Appendix A.
2.1 Hamiltonian formalism for charged particle motion
The dynamics of axially symmetric charged current carrying
string loops can be enlightened by comparison with charged
test particle motion, as both these dynamics can be
formulated in the framework of Hamiltonian formalism. Recall that
evolution of axisymmetric string loops adjusted to
axisymmetric backgrounds can be represented by evolution of a
single point of the string [
10, 21
].
We can also compare electromagnetic forces acting on
charge test particles or string loops. Since the motion of a
charged test particle in the RN black hole background has
been intensively studied in literature [
1, 2, 17, 18
], we will
give just short summary.
Motion of a charged particle with mass m and charge q is
given by the Hamiltonian [
15
]
Hp = 21 gαβ ( α − q Aα )( β − q Aβ ) + 21 m2,
where mechanical, P μ, and canonical,
related as
P μ = mU μ = m dτ
dx μ
=
μ − q Aμ.
Due to the spherical symmetry of the RN background (1), the
charged particles move in central planes only. For a single
particle the central plane can be chosen as the equatorial
plane. Since the Hamiltonian (6) does not contain coordinate
φ (axial symmetry) and coordinate t explicitly, two constants
where the metric function reads
f (r ) = 1 −
2M
r
Q2
+ r 2 .
In the metric function f (r ), the parameter M stands for the
black hole mass, while Q stands for the black hole charge.
For 0 ≤ Q < M the metric (1) describes black hole with
two event horizons, located at
rh± = M ±
M 2 − Q2,
for Q = M there is just one degenerate event horizon
solution, for Q > M we have naked singularity without horizons.
Hereafter in this paper we will use for simplicity the system
of units in which the mass of the black hole M = 1, i.e., we
express the related quantities in units of the black hole mass.
In order to clearly show the trajectory of string loops, it is
useful to use the Cartesian coordinates x , y, z related to the
Schwarzschild coordinates
x = r sin(θ ) cos(φ),
y = r sin(θ ) sin(φ),
z = r cos(θ ).
(4)
The electromagnetic field related to the Reissner–Nordström
(RN) metric is given by the covariant electromagnetic
fourvector potential Aα [
15
] that takes the simple form
Q
Aα = r (−1, 0, 0, 0).
Recall that electromagnetic interaction is much more stronger
than the gravitational interaction – between electron and
pro√
−hhab(ϕa + Aa ) b = 0.

The assumption of axisymmetry implies ϕσ σ = 0 and Aφ =
Aσ = Aσ (φ), from (13) we have conserved quantities and
n, given by
= ϕτ + Aτ , n = ϕσ ,
Varying the action (10) with respect to the induced metric hab,
we obtain the worldsheet stress–energy tensor density (being
of density weight one with respect to worldsheet coordinate
transformations)
of motion exist – particle energy E = − t , and particle
axial angular momentum L = φ . Now one can write the
Hamiltonian of the charged particle equatorial motion in the
form
1 1 1
H = 2 f (r ) Pr2 − 2 f (r )
Using the H = 0 condition, we obtain immediately equation
of the radial motion in the form
,
corresponding to the motion in 1D effective potential
determining the turning points of the radial motion where dr/dτ =
0.
2.2 Hamiltonian formalism for relativistic string loop
Dynamics of relativistic, charged, current carrying string is
described by the action S with Lagrangian L [
12,13
]
S =
L dσ dτ,
L = −μ
√
1 √
−h − 2
−hhab(ϕa + Aa )(ϕb + Ab),
(10)
where Aa = Aγ X γa . The string worldsheet is described by
the spacetime coordinates X α(σ a ) with α = 0, 1, 2, 3 given
as functions of two worldsheet coordinates σ a with a = 0, 1.
This implies induced metric on the worldsheet in the form
hab = gαβ Xαa Xβb,
where a = ∂ /∂a. The string current localized on the
2D worldsheet is described by a scalar field ϕ(σ a ). The 2D
worldsheet with coordinates τ, σ is immersed into 4D metrics
with coordinates t, r, θ , φ using
X α(τ, σ ) = (t (τ ), r (τ ), θ (τ ), σ ).
The action (10) is inspired by an effective description
of superconducting strings representing topological defects
occurring in the theory with multiple scalar fields undergoing
spontaneous symmetry breaking [
32,33
] and can be used as
effective description of current created by bosons or fermions
on superconducting string. Contrary to the formalism used in
[
8,10
], we rescale scalar field ϕ → ϕ/2. First part of (10) is
classical Nambu–Goto string action for string with tension μ
only, second part describes interaction of scalar field ϕ with
fourpotential Aα of electromagnetic field.
In the conformal gauge, the equation of motion of the
scalar field, given by the variation of the action (10) against
field ϕ, reads
(11)
(12)
dτ =
τ τ dζ,
we can define for the string loop dynamics define the
Hamiltonian
The contribution from the string tension μ > 0 gives a
positive energy density and a negative pressure (tension). The
current contribution is traceless, due to the conformal
invariance of the action – it can be considered as a 1 + 1
dimensional massless radiation fluid with positive energy density
and equal pressure [
8
].
Electromagnetic properties of the charged circular string
loop are obtained by varying the action (10) with respect to
the fourpotential Aα:
J μ
δL μ
= δ Aμ = −ρ Xτ + j Xμσ ,
∂ρ ∂ j
∂τ = ∂σ
,
where the string loop electric current is j = n + Aφ and
string loop electric charge density is ρ = [
12
].
Varying the action (10) with respect to X μ implies
equations of motion in the form
D
dτ
(τ ) D
μ + dσ
(μσ ) = 0,
where the string loop momenta are defined by the relations
(σ )
μ
(τ ) ∂L
μ ≡ ∂ X˙ μ =
∂L
≡ ∂ X μ =
τ a gμλ Xλa +
Aμ,
σ a gμλ Xλa − (n + Aφ ) Aμ.
Defining affine parameter ζ , related to the worldsheet
coordinate τ by the transformation
(13)
(14)
(15)
(16)
(17)
(18)
(19)
From the first equation in (21) we obtain relation between
the canonical μ and mechanical momenta Pμ in the form
Pμ =
μ
−
Aμ.
2.3 Conserved quantities and effective potential for string
loop dynamics
Now we restrict our study to the string loop dynamics in the
RN black hole background using metric (1) and
electromagnetic fourpotential (5). The metric (1) does not depend on
coordinates t (static) and φ (axial symmetry) and only one
nonzero covariant component of the electromagnetic
fourvector potential is At (5). Such symmetries imply existence
of conserved quantities during string motion–string energy
E and string axial angular momentum L, determined by the
relations
− E =
L =
t = Pt +
φ = gφφ
τ σ
At ,
+
Aφ
= −
n = −2 J 2ω 1 − ω2.
The string loop does not rotate in Schwarzschild coordinates,
d X φ /dζ = 0 – see Eq. (12), but the string loop has a
nonzero angular momentum generated completely by the scalar
field living on the string loop. Instead of string loop
electric charge and current n (16), we can introduced new
conserved quantities – “angular parameter” J and “charge
parameter” ω, given by
J 2 ≡
2 + n2 ,
2
1
ω ≡ √2 J
The parameter J ≥ 0 has simple interpretation as combined
magnitude of charge and current on the string loop, and as we
will see later, J is a generator of the centrifugal force, acting
against contraction caused by the string loop tension μ. Note
that even though string loop is not rotating mechanically, J
parameter is acting as an angular momentum due to internal
properties of the string. For this reason we call it angular
momentum J parameter. Further, the new parameter ω is
string loop charge rescaled by parameter J , such that −1 ≤
ω ≤ 1. We can distinguish three limiting cases of parameter
ω:
H = 21 gαβ ( α −
Aα)( β −
Aβ )
1
+ 2 gφφ ( τ τ )2 − ( τ σ )2
and the related Hamilton equations
Pμ ≡ dζ
d X μ
∂ H
= ∂ μ
,
d μ
dζ
∂ H
= − ∂ X μ
ω = −1 There is no electric current on the string loop, n =
0, only negative electric charge < 0 uniformly
distributed along the loop. Since we consider the
central object charge Q > 0 to be positive, there
acts an electromagnetic attractive force between the
central object and the string loop.
ω = 0 There is no charge on the string loop, = 0, only
current n, and there is no electromagnetic
interaction between the string loop and the central object
electric charge Q. The black hole charge Q can
affect the string loop dynamic only through changes
in (1) metric. This case was already studied in [
22
]
for the so called “tidal charge” black hole scenario.
ω = 1 There is no electric current on the string loop,
n = 0, only positive electric charge > 0
uniformly distributed along the loop. there will be
electromagnetic repulsive force between the
central object and the string loop.
The electric force between the central object with charge
Q is attractive for −1 ≤ ω < 0, while it is repulsive for
0 < ω ≤ 1. We will focus on string loop dynamics for
ω ∈ {−1, 0, 1} limiting values, and we will assume the string
loop behaviour for another value of ω, will be combination of
the limiting values. It is interesting that for all three limiting
cases ω ∈ {−1, 0, 1}, the string loop angular momentum L
is zero (23).
The string dynamics depends on the J parameter (14)
through the worldsheet stress–energy tensor. Using the two
constants of motion (23), we can rewrite the Hamiltonian (20)
into the form related to the r and θ momentum components
H = 21 grr Pr2 + 21 gθθ Pθ2 + 21 gtt (E +
2
1
+ 2 gφφ
Assuming μ > 0, one can also express all quantities in the
terms of string loop tension μ, divide the whole Hamiltonian
(25) with μ, and hence get rid of extra parameter μ with
transformations like E → E /√μ and J → J /√μ. Hence,
in all following equations, we will take μ = 1 and we will
discuss the string loop quantities and their dimensions in
physical units in the Appendix A.
The equations of motion (21) following from the
Hamiltonian (25) are very complicated and can be solved only
numerically in general case, although there exist analytical
solutions for simple cases of the motion in the flat or de Sitter
spacetimes [
9
]. However, we can tell a lot about the string
loop dynamics even without solving the equation of motion
(21) by studying properties of the effective potential
governing the turning points of the string loop motion that is implied
by the Hamiltonian. It is useful to express the Hamiltonian
(25) in the form
(25)
(26)
H = HD + HP,
where we split H into “dynamical” HD and “potential” HP
parts. The “dynamical” part HD contains all terms with
momenta α or Pα, while the “potential” part HP depends
on the coordinates and conserved quantities only.
The positions where a string loop has zero velocity (HD =
0 and r˙ = 0, θ˙ = 0) forms a boundary for the string motion.
Since the total Hamiltonian is zero, H = 0, the potential part
of the Hamiltonian is also zero HD = 0 at the boundary points
with zero velocity. This will allow us to to express string loop
energy E at the turning (boundary) points in the form
E = Veff (r, θ ) ≡
−gtt gφφ
+ 1 −
At ,
(27)
where we define the effective potential function Veff (r, θ ).
The condition (27), for energy E , creates an unbreakable
boundary (curve) in the x –z plane, restricting the string
loop motion. In the previous works [
8,10
], the term “energy
boundary function” was used for the effective potential,
Eb(r, θ ) = Veff (r, θ ).
Stationary points of the effective potential function Veff
(x , z) are determined by two conditions
(Veff )x = 0
(Veff )z = 0,
where the prime ()m denotes derivation with respect to the
coordinate m. In order to determine character of the
stationary points at (xe, ze) given by the stationarity conditions (28),
i.e., whether we have a maximum (“hill”) or minimum
(“valley”) of the effective potential function Veff (x , z), we have
to examine additional conditions
[(Veff )zz (Veff )x x − (Veff )zx (Veff )xz ](xe, ze) > 0,
[(Veff )zz ](xe, ze) < 0 (max) > 0 (min).
The curve E = Veff (x , z), forming unbreakable energetic
boundary for the string loop motion, can be open in the x
direction in the equatorial plane (z = 0), allowing the string
loop to move towards horizon and be captured by the black
hole. The energetic boundary can be open in zdirection,
allowing the string loop to escape to infinity from the black
hole neighbourhood.
(28)
(29)
(30)
3 String loop in combined electric and gravitational
field
3.1 Charged string loop in flat spacetime
We discuss the flat spacetime case separately, as establishing
the flat space limit requires M = 0 and Q = 0
simultaneously, but this means vanishing of the electromagnetic field.
We can use cylindrical coordinates (t, x , z, φ) in flat
spacetime, and compare string loop Hamiltonian with
Hamiltonian for particle on circular geodesic – this will be very
helpful for exploring the situation and for identification of
acting forces. In the electrostatic field of point charge Q
(5) we have for the “potential”, HP, parts of the
Hamiltonian determining the charged particle and charged string loop
motion, the simple expressions
1
Hparticle = − 2
1
Hstring = − 2
While particle can move only in the central plane, taken to be
equatorial plane for simplicity, and particle motion remain
regular, the string loop can move also outside the
equatorial plane, and string loop dynamics is generally chaotic.
In Hamiltonian (31) we clearly see radial Coulombic force
∼ Q/r 2, acting on the element of string loop with
electric charge ; Coulombic force is attractive for Q < 0,
and repulsive for Q > 0. The radial force field breaks the
symmetry of the string loop translation along the z axis, and
the string loop dynamics can not be regular even in the flat
spacetime.
Using the condition Hstring = 0, we find the effective
potential (boundary function) Veff (x , z) for a string loop
electrically interacting with the point charge Q in flat spacetime
in the form (y coordinate can be suppressed by fixing at y = 0
then r = √x 2 + z2)
Veff (x , z) = μx + x +
√2ω J Q
r
.
(31)
(32)
The case ω = −1 corresponds to the configuration of
opposite charges Q and , the ω = 0 case to uncharged string
loop, and the case ω = 1 corresponds to configurations with
the same sign of the electric charges. We give examples of
the effective potential function Veff (x , z) in Figs. 1 and 2.
In Fig. 1 the left graph represents the string loop effective
potential function Veff (x , z = 0) as section at the
equatorial plane. In the x direction, we have one minimum of the
effective potential, depending on values of J an ω. In middle
graph is plotted the string loop effective potential function
Veff (x = xmin, z) as section at its equatorial minimum. The
stationary points of the Veff (x , z) function are located in the
equatorial plane, z = 0, only; in the zdirection, for ω = −1
we have minima, for ω = 0 we have constant behaviour
in z direction, and for ω = 1 we have saddle point. The
behaviour of the effective potential along the vertical z axis
Veff (x = x0, z), as section at x0 = 2 (right Fig.), is also
plotted for all three limiting values of ω parameter.
For visualizing the regions where the string loop motion
is possible, we demonstrate in Fig. 2. the E = const.
secω = −1. The x section of Veff is taken at z = 0, while z section is taken
at corresponding minima x = xmin (middle fig.) or at x = 2 (right fig.)
tions of the effective potential full 2D function Veff (x , z) for
both x and z coordinates. In the left, picture we give the
Veff profile for ω = −1, E = 3.5 case. The electrostatic
interaction between the black hole and string loop charges is
attractive and the string loop is trapped in closed area (light
grey) – string loop is located in effective potential “lake”. In
the middle figure, the case ω = 0, E = 4.5 is presented. In
this case, trapped motion in x axis is observed, and there is no
motion in vertical z axis, if we consider zero initial velocity
in z direction, the effective potential (32) does not depend
on z in absence of the electric interaction between the black
hole and string loop. In the right picture, we shown the case
ω = 1, E = 5. The string loop is allowed to oscillate in a
limited x interval, while, due to the electrostatic repulsion
between the black hole and string loop charges, the string
loop is escaping along the vertical z axis. Depending on the
initial position, the string loop can move in the upper or lower
half spaces and it can never cross the equatorial plane.
Coming from Fig. 2, we draw in Fig. 3 the trajectories of
the string loop within their energy boundaries for the same
values of the parameters J , Q and ω. We can conclude that in
the case of opposite (attractive) charges (ω = −1), electric
attraction resists the string loop to escape to infinity. In the
absence of electric interaction (ω = 0), there is no force
in vertical direction and the trajectory of the string loop is
always on the plane parallel to the z plane. In the repulsively
charged case (ω = 1), the electric repulsive potential barrier
pushes the string loop away from the center.
3.2 Charged string loops in Reissner–Nordström
background
For string loop motion in the Reissner–Nordström
background, the general form of the Hamiltonian (25) reduces
to
J 2
r sin θ + r sin θ
2
H = 21 f (r ) Pr2 + 21r2 Pθ2 + 21
1
− 2 f (r )
E −
Q 2
r
.
As the whole axisymmetric string loop can be represented
by a single point that can be characterized by a coordinate
y = 0 (see e.g. [
22
]), we can introduce the effective potential
for charged string loop in the form
Veff (x, z; Q, J, ω) =
2 Q2
1 − r + r 2
μx + x
J 2
+
where r is radial distance r 2 = x 2 + z2 and parameters
J, ω were already introduced and explained in Eq. (24). We
put for simplicity M = 1 (expressing r in units of mass
parameter). The effective potential Veff (x , z) is not defined
in the dynamical region, between the inner and outer RN
black hole horizons, where f (r ) < 0 [
22
]. For the black
hole spacetimes, Q ≤ 1, we will consider string loop motion
in the region above the outer horizon, r > r+, see Eq. (3). For
the RN naked singularity spacetimes, Q > 1, the dynamical
region ceases to exist, and Veff (x , z) is defined for any r > 0.
First we need to explore asymptotic behaviour of the
effective potential (34). Reissner–Nordström spacetime is
asymptotically flat, hence in the xdirection there is
Veff (x → ∞, Q, J, ω) → +∞,
and in the zdirection we obtain
J 2
Veff (z → ∞, Q, J, ω) → x + x
= Veff(flat),
(35)
(36)
for details see [
9
].
Here we consider all possible types of the charged string
loop motion around the RN black holes as well as the RN
naked singularities. The case of the charged string loop
motion in the field of RN black holes and naked singularities
is included in the related study of behavior of string loops in
the braneworld spherically symmetric black holes studied in
[
22
], that where it was demonstrated that the string loop can
oscillate in the closed area, fall down into the black hole, or
escape to infinity in the vertical direction, while oscillating
in the x direction. Exploring the effective potential the type
of string loop motion can be estimated.
Stationary points of the 2D effective potential function
Veff (x , z) are given by Eq. (28). The stationary points can be
found in the equatorial plane, z = 0, and their x coordinate
is given by the relation
H 2
x
J 2
μ − x 2
+ (x − Q2)
J 2
x 2 + μ −
√2 J Qω
x
(33)
From Eq. (37) one can easily find the corresponding condition
for string loop angular momentum parameter J
H = 0.
(37)
(38)
(39)
(40)
J = Jext,
where
Jext ≡
−Q H ωx ± x 2 P(x − 1)x + Q2 H 2ω2
√2 P
.
Here we have used the following notations:
P(x , Q) = 2Q2 − 3x + x 2,
H (x , Q) =
Q2 − 2x + x 2.
The Jext(r ; ω, Q) function determines both stable and
unstable stationary positions of the string loop. Radial profiles of
trapped motion boundaries (dashed). If the Jext’s profile is within the
shaded lines string loop’s motion is always in some toroidal space
otherwise it escapes to infinity
Jext(r ; ω, Q) function are plotted in Fig. 4. for various
combinations of ω and Q charges.
Depending on the black hole charge Q and string loop
charge parameter ω, there can exist one, two or three
stationary points of the effective potential in the equatorial plane.
The sign of d Jext defines type of the extrema, the positive
d x
derivation term, d Jext > 0, determines the effective
potend x
tial equatorial minima while negative derivative, ddJexxt < 0,
determines the maxima. The extremal point of the Jext
function, given by d Jext = 0, defines the innermost stable string
d x
loop position.
Exploration of the effective potential Veff allows us to
determine all possible types of the string loop trajectories in
the RN backgrounds. In addition to the effective potential
(41)
(42)
(43)
extrema, given by the Jext function, we need to explore when
the string loop can escape to infinity along the z axis. From
Eq. (36) we know that the energetic boundary will be open
in the z direction for the string loop motion, if string energy
E satisfies the condition
E > Veff(flat) = 2 J.
Therefore, condition Veff (x , z; Q, ω) < 2 J gives the
boundaries of string loop’s trapped motion. Solving the Eq. (41)
with respect to J , we find the regions where string loop’s
motion is trapped. We thus derive the loop trapping function
in the form
JL1,2 =
−Qx 3 H ± x (Q2ω2 H − 2(x 2 − x ) P)
√2 P
.
The trapped oscillatory string loop motion occurs under the
circumstance given by the relations
JL1(x ; Q, ω) > J > JL1(x ; Q, ω).
In Fig. 4 we demonstrate the behavior of the extrema Jext
function, along with the boundary functions JL1 and JL2
giving trapped motion boundaries; if the string loop angular
momentum is located within the dashed boundaries the string
loop can never escape to infinity in the vertical direction at
the corresponding fixed x .
In Fig. 5 we use the diagonal pictures of Fig. 4 and
construct the corresponding effective potential along x axis and
zaxis at xmin given for the chosen values of J , where xmin
is position of the effective potential minima. In Fig. 6 we
give some typical trajectories of the string loop motion. In
the top row of the Fig. 5, we consider Q = 0.3, ω = −1
case. First we take J = 10 as it is crossing the Jext profile at
two points (Fig 5). In this case, there is one stable and one
unstable equilibrium position.
In Fig. 6a we show trajectories of string loop’s motion
crosssection along with the boundary energy profiles – we
observe that the motion is trapped in the closed region. Then
we consider J = 7.44, as this value of J is touching the
extremal point of the function Jext (Fig 5), corresponding to
the innermost stable equilibrium position (ISEP) – any small
deviation from this position causes the string to collapse to
the black hole. String loop’s trajectory for this case is given
in Fig. 6b; we can conclude that the motion is finite in the
zdirection and the energy boundary profile is open to the
black hole, and the string finally falls down to the black hole.
And last case of Q = 0.3, ω = −1 configuration, we take
J = 4 value as it is not crossing the Jext profile at all. In this
case, there is no possible trapped motion and the string loop
has to escape to infinity in the vertical direction (Fig. 6c).
Another possible string loop trajectory around the black hole
is given for Q = 0.3, ω = 1, J = 10 case in Fig. 6d, with
string escaping to infinity.
Medium line elements of Fig. 5 represent the naked
singularity Q = 1.0677, ω = 0 case. The most distinctive
behavior of the effective potential is given by the presence of
two minima for J = 5.5. This indicates that the string loop
has in the x direction two stable positions around the naked
singularity, and escape along vertical direction is
impossible for sufficiently low string loop energy. The trajectory of
the string loop for this type of motion is given in Fig. 6e.
This type of energy boundary profile corresponds to trapped
motion – the trapped motion can take place in one of two
possible closed toroidal spaces around the RN naked
singularity. At the bottom line on Fig. 5 we consider Q = 1.414,
ω = 1 situation. There is one minimum of the potential well
corresponding to stable equilibrium position. String loop’s
motion in x direction is limited. There is also small potential
barrier resisting the string loop to cross the z = 0
equatorial plane. String loop escapes to infinity loosing oscillatory
energy in the x direction (Fig. 6f) [
8,9,13
].
Effective potential Veff (x , z = 0) has real extrema only
for real values of the extreme angular momentum function
Jext given by expression (39). We thus find the condition
relating the limiting values of RN charge parameter Q and
the string loop charge parameter ω in the form
ω2 = ωc2rit(x , Q) ≡ −
Q2(2Q2 − 3x + x 2)(x − 1)
Q2x 2(Q2 − 2x + x 2)
(44)
along with the condition Q2 − 2x + x 2 ≥ 0. This allow
us to distinguish regions with different string loop effective
potential behavior for any central charge Q, as demonstrated
in Fig. 7, where the region around a black hole or naked
singularity is separated into regions where effective potential
has minima (light grey region), or maxima (grey region), or
there are no extrema at all (white region). The line dividing
grey and light grey regions gives the location of innermost
stable equilibrium position (ISEP).
It is useful to compare the ISEP for charged string loops
with its particle equivalent – the charged particle innermost
stable circular orbit (ISCO), in the same RN black hole
background [
17,18
]. In Fig. 8, such comparison is given for all
three (positive, neutral, negative) variants of the charged test
object. As it can be seen, the charged string loop ISEP is
always located between the photon orbit and the charged
particle ISCO in the RN spacetime, revealing true about the
string loop real nature.
4 Quasiperiodic oscillations of string loops
The quasiperiodic oscillatory motion of the string loops
trapped in a toroidal space (or in “lake”) around the minima
of the effective potential Veff (x , z) function could be used
to interpret interesting astrophysical phenomenon –
highfrequency quasiperiodic oscillations (HF QPOs). Most of
compact Xray binaries that contain a black hole or a
neutron star demonstrate quasiperiodic variability of the Xray
flux in the kHz frequency range. Some of these HF QPOs
appear in pairs as upper and lower frequencies (νU, νL) and
in Fourier spectra are observed twin peaks. Since the peaks
of high frequencies are close to the orbital frequency of
the marginally stable circular orbit representing the inner
edge of Keplerian discs orbiting black holes (or neutron
stars), the strong gravity effects must be relevant to
interpret HF QPOs [23]. So far, many models have been
proposed to explain HF QPOs in black hole binaries: the
relativistic precession model, the warped disc model,
resonance model [
16, 24–26, 28
]. Usually, Keplerian orbital and
epicyclic (radial and latitudinal) frequencies of geodetical
circular motion are assumed in models explaining the HF
261
Fig. 7 Local extrema position x and type (maxima/minima) of the
effective potential Veff (r, θ = π/2) for different values of RN charge
parameter Q and for all three considered values of ω parameter. Thick
black line corresponds to event horizon and restrict the dynamical
region. Darker grey colour denotes region where maxima of the Veff
function can exist, while lighter grey colour denotes region where
minima can exist. Only in the lighter grey areas can exist stable string loop
position – the boundary between darker/light areas act as innermost
stable string loop position. In white areas above RN black hole horizon
there are no extrema point of Veff function
QPOs in both black hole and neutron star systems [
27
].
However, neither of these models is able to explain the HF QPOs in
all microquasars [
29
]. On the other hand, there is possibility
of the relevance of string loop’s oscillations, characterized by
their radial and vertical (latitudinal) frequencies that are
comparable to the epicyclic geodetical frequencies, but slightly
Fig. 8 ISEP for string loop and ISCO for charged particle with respect
to black hole charge Q. ISEP for string are given for attracting ω = −1,
not interacting ω = 0 and repulsing ω = 1 interaction types between
different, enabling thus some corrections to the predictions of
the models based on the geodetical epicyclic frequencies. Of
course, the frequencies of string loops oscillations in physical
units have to be related to distant observers.
Let have string loop located in the equatorial plane and
at a effective potential Veff (r, θ ) minimum (r = r0, θ =
θ0 = π/2). Slight displacement from minima position r =
r0 + δr, θ = θ0 + δθ , causes small string loop oscillations
around the stable equilibrium positions, determined by the
equations of harmonic oscillations
δ¨r + ωr 2δr = 0,
δ¨θ + ωθ 2δθ = 0,
where locally measured frequencies of the oscillatory motion
are given by
ωr2 =
∂ 2 Veff
∂r 2 ,
ωθ2 = r 2 f1(r ) ∂ ∂2 Vθ2eff .
Local observers, at the position of the string loop in the RN
spacetime measure angular frequencies
(45)
(46)
,
(47)
string and charged black hole. ISCO for charged particles are defined
for three different charge to mass ratio values
Table 1 Observed twin HF QPOs data for three microquasars, and the
restrictions on mass of black holes located in them, based on
independent measurements on the HF QPO measurements
Source
νU (Hz)
νL (Hz)
M/M
GRO 165540
XTE 1550564
GRS 1915+105
here E = E (r0, θ0) is the energy of the string loop on its
minima position and f (r ) is the characteristic lapse function
of the RN metric (1). The frequencies for observers at infinity
, have to be multiplied by the factor c3/G M to be expressed
in the standard physical units
1 c3
ν(r,θ ) = 2π G M
(r,θ ).
We focus our attention to resonance frequencies with ratio
3:2 observed in Xray data from GRO 165540, XTE
1550564, and GRS 1915+105 that require the string loop
frequencies corresponding to twin peaks appears in the νr :νθ = 3:2
or νθ : νr = 3:2 ratio – see Table 1. We explore the
displacement of resonance frequencies with respect to black hole’s
charge Q and string loop ω parameter. In Fig. 9 we
illustrate the radial coordinates of equilibrium positions where
νr :νθ = 3:2 or νθ :νr = 3:2 ratios appear in dependence on
parameters Q and ω. On the top row of Fig. 9 we present
Q = 0.5 case. As it is seen, with changing from strong
electric attraction, ω = −1, to absence of interaction, ω = 0,
and finally to strong electric repulsion, ω = 1, the position
(48)
(49)
√2 Jext Qr 2ω + Q2r 2
ωr2 =
ωθ2 =
2 Je2xt 6Q2 + (r − 6)r +
r 5
( Jext − r )( Jext + r )
r 3
.
where Jext(r ) gives J parameter for equatorial minima.
Frequencies measured by static observers at infinity are related
to the locally measured frequencies (47) by the gravitational
redshift transformation
261
of resonance frequencies tend to come closer to the black
hole. For the naked singularity case of Q = 1.0677 on the
middle row of Fig. 9, in the similar step of changes of ω,
we observe different scenario from Q = 0.5 situation. Here,
for ω = −1 case three equilibrium points satisfying the 3:2
ratio condition for νr and νθ occur. Further, in ω = 0 case,
the resonance frequencies occur at four positions. Finally,
when ω = 1, the resonance frequencies does not appear at
all. Bottom line elements on Fig. 9 represent the Q = 1.414
loops are considered around Reissner–Nordström black hole of 10 solar
masses. Radii of 3:2 and 2:3 resonances are given as dashed lines
naked singularity case. In this scenario, the resonance 3:2
frequencies appear at two locations for the ω = −1 case,
while for the ω = 0 case they occur only at one position, and
in the ω = 1 case vertical frequencies disappear, the string
loops are unstable relative to vertical perturbations.
For fixed black hole charge Q and fixed string loop charge
parameter ω, upper frequency of the twin HF QPOs can be
given as a function of black hole’s mass M . If the black hole
mass is restricted by separated observations, as is commonly
the case, we obtain some restrictions on the string loop
resonant oscillations model, as illustrated in Fig. 10. Here, the
situation is demonstrated for several values of black hole’s
charge Q and limits on the black hole mass as given in Table 1.
We can see that for the Schwarzschild black hole (Q = 0),
the string loop model can explain only the HF QPOs in GRS
1915+105. Introducing black hole charge Q and parameter ω,
the string loop resonant oscillations model widens the area of
its applicability. For ω = 1 and Q = 0.5, 0.8 case, the model
fully describes observed values from GRO 165540 source.
It contains the whole range of expected mass range from
Table 1. Nevertheless, the string loop resonant oscillation
model in Reissner–Nordström background can not explain
the observed values from XTE 1550564 source. For any
value of ω parameter and for any low values of black hole
charge Q, the string loop model can not fit observed mass
range for the XTE 1550564 source and an additional
influence of the black hole rotation has to be expected.
Moreover, in Fig. 10 we can clearly see that the predicted
value of the black hole mass is increasing with the black
hole charge Q increase. It will become harder and harder
to fit the observed HF QPOs as the Q parameter increases,
hence we can conclude that introducing new parameter Q
into the string loop HF QPOs model is not successfully
efficient in explaining the observed HF QPOs in microquasars,
and inclusion of the black hole spin that can be sufficiently
efficient as demonstrated in [
23
] is necessary.
5 String loop acceleration and asymptotical ejection
speed
From the astrophysical perspective, one of the most relevant
applications of the axisymmetric string loop motion is the
possibility of strong acceleration of the linear string motion
due to the transmutation process in the strong gravity field
of immensely compact objects that arises due to the chaotic
character of the string loop motion and could well
simulate acceleration of relativistic jets in Active Galactic Nuclei
(AGN) and microquasars [
8,10,21
]. Since the RN spacetime
is asymptotically flat, we have to examine the linear string
loop motion in the flat spacetime; as the Columbian
electric field disappears asymptotically at the RN spacetime, this
approximation is sufficient to understood the results of the
acceleration process. The energy of string loop (32) in the
Cartesian coordinates reads
E 2 = z˙2 + x˙2 +
J 2
x + x
2
= Ez2 + Ex2,
where dot denotes derivative with respect to the affine
parameter ζ . The energies related to the x − and z− directions are
given by the relations
Ez2 = z˙2,
Ex2 = x˙2 +
J 2
x + x
2
= (xi + xo)2 = E02,
where xi (xo) represents inner (outer) boundary of the
oscillatory motion. The energy E0 representing the internal energy
of the string loop is minimal when the inner and outer radii
coincide, leading to the relation
E0(min) = 2 J
that determines the minimal energy needed for escaping of
the string loop to the infinity in the spacetimes related to
black holes or naked singularities.
Clearly, Ex = E0 and Ez are constants of string motion in
the flat spacetime and transmutation between energy modes
(50)
(51)
(52)
of black hole charge Q = 1. The Lorenz factor γ is calculated for
string loops with angular momentum J = 1.1 and starting from initial
position x0 = 1.9, z0 = 0
Fig. 12 Escaping trajectories of string loop in RN(top line elements)
and flat spacetime(bottom line elements). Asymptotical speed of
transmitted string loops given for three values of ω parameter for charged
black hole Q and flat spacetime. The asymptotical ejection speed of
string loop Vz is calculated for the values of angular momentum J = 7,
energy E = 20 and initial position z0 = 7. x0 is found from the
expressions of energy in RN (34) and flat spacetime (32)
are not possible there. However, in the vicinity of black holes,
the kinetic energy of oscillating string can be transformed
into the kinetic energy of the translational linear motion.
The energy in the xdirection E0 can be interpreted as
an internal energy of the oscillating string, consisting from
the potential and kinetic parts; only in the limiting case of
xi = xo, the internal energy has zero kinetic component.
The string internal energy can quite reasonably represent the
rest energy of string moving in the zdirection in flat
spacetime [
21
]. The final Lorentz factor of the translational motion
of an accelerated string loop as observed in asymptotically
flat region of the Reissner–Nordström spacetimes is, from
(51) defined by the relation [
8,21
].
E E
γ = E0 = xi + xo
,
where E is the total energy of the string loop moving with the
internal energy E0 in the zdirection, with the velocity
corresponding to the Lorenz factor γ . Apparently, the maximal
Lorentz factor of the transitional motion reads [
21
]
E
γmax = 2 J .
From this equation we can see that for observing
ultrarelativistic acceleration of the string loop large ratio of the
string energy E versus its angular momentum parameter J
is needed. In Fig. 11 we illustrate asymptotic linear speed of
transmitted string loops. We demonstrate the influence of the
ω parameter on ejection speed of string loops for extremal
black holes with charge Q = 1, string angular momentum
parameter J = 1.1, starting from position x0 = 1.9, z0 = 0.
Ejection speeds are expressed by the Lorentz factor (γ =
2
1/ 1 − vejection). As presented in Fig. 11, for bigger values
of ω we observe greater values of ejection speed. This can
be explained due to repulsion from the center of the
acceleration of the string loop. Nevertheless, the observed ejection
speeds are not so highly relativistic as they are in the Kerr
naked singularity spacetimes [
10
].
In Fig. 12 we give escaping trajectories of the
transmitted string loops in the RN background and flat spacetime
for attracting, ω = −1, not interacting, ω = 0, and
repulsing, ω = 1, types of the string loop interaction with the
charged black hole and central point charge in flat spacetime.
It is expected to observe bigger ejection speed in repulsing
case(ω = 1) than attracting one(ω = −1). However,
surprisingly the string loop acceleration is higher in ω = −1 case
than ω = 1 case. This can be explained by studying their
trajectories within the energy boundaries. In the ω = −1
case, due to attraction by the black hole, the string loops
enters deeper in a black hole’s potential well and the
transition effect of oscillating energy to escaping translational
energy in the zdirection becomes more effective. Due to the
chaotic nature of string loop dynamics, we can expect
completely different set of velocities for different set of initial
conditions.
6 Conclusion
The astrophysically relevant problems of current carrying
string loops in spherically symmetric spacetimes have been
studied recently [
9,21,22
]. In the present paper we
investigate the relevant issues for Reissner–Nordström background,
giving the attention on the influence of black hole charge Q
and its electromagnetic interaction with string loop charge ω
created by scalar field ϕ living on the string loop.
Scalar field ϕ, living on the string loops and represented
by the angular momentum parameter J , is essential for
creating the centrifugal forces, and therefore for existence of
stable string loop positions. In RN background is the charged
string loop innermost stable equilibrium position (ISEP)
located between the photon circular orbit at rph and
innermost stable charged particle orbit (ISCO). The condition
rph < rISSP < rISCO, already proven for rotating Kerr black
hole background [
10
], is supporting consideration of string
loop model as a composition of charged particles and their
electromagnetic fields [
5
].
We have shown different types of string loop energy
boundaries and different string loop trajectories in RN
backgrounds. There are not any new types of the string loop
motion for RN black hole background [
8
], but in the field
of RN naked singularities two closed toroidal regions for
the string loop motion are possible (Q = 1.0677, ω = 1,
J = 10) .
String loop harmonic oscillations around stable equilibria,
defined by Eq. (47), could be one of the perspective
explanations of the HF QPOs observed in binary systems containing
black holes or neutron starts. In the present paper, we applied
the string loop resonant oscillations model to fit observed
data from GRO 165540, XTE 1550564, GRS 1915+105
microquasar sources. Our fittings are substantially
compatible with the observed data from the GRO 165540 source and
partially coincide with the GRS 1915+105 data. For the latter
source the values of ω parameter are significant. Observed
data from XTE 1550564 can be explained only for Q ∼ 0.9
values. We can conclude that the twin HF QPOs could be
efficiently explained by the string loop oscillatory model,
if we consider interaction of an electrically charged current
carrying axisymmetric string loop with the combined
gravitational and electromagnetic fields of Kerr–Newman black
hole where due to the combination of the black hole spin
even small electric charge of the black hole can cause
relevant modifications of the frequencies of the string loop
oscillations.
String loop acceleration to the relativistic escaping
velocities in the black hole neighbourhood, is one of the possible
explanations of relativistic jets coming from AGN. We have
studied the effect of black hole and string loop charges
interaction on to the acceleration process. Due to the chaotic
character of equations of motion, the positively charged string
loop ω > 0 can be ejected from equatorial plane even in
flat background. The RN black hole charge Q does not
contribute to the string loop acceleration speeds due to
electrostatic repulsion, it only modifies the effective potential Veff
and allows the string loop to came closer to the black hole,
where the transmutation is more effective. This implies a
surprising phenomena: the transmutation effect is more
efficient, and the string loop is more significantly accelerated for
the electric attraction of the string loop and the black hole, as
the transmission process can occur in deeper regions of the
gravitational potential well than in the case of electric
attraction. Note that contrary to the standard Blandford–Znajek
mechanism of jet acceleration to high velocities [
3
], where
fast rotating black hole must be assumed, in the stringloop
acceleration model rotation of the black hole is nor required.
The RN solution is simple and elegant solution of
combined Einstein and Maxwell equations and by studying
charged string loop dynamics in this solution, we would like
to just complete our previous string loop studies in order to
map potential role of the Coulombic electric interaction. We
explore theoretical properties of charged string loop motion
in RN background and we show that unrealistically high
values of RN charge are needed to explain real astrophysical
data. In some dynamic situations, as those corresponding to
unstable states of accretion disks wider ionization processes,
the electric charge could be momentarily larger as indicated
above for stationary situations.
Acknowledgements T.O. acknowledges the Silesian University in
Opava Grant no. SGS/14/2016, M.K. acknowledges the Czech Science
Foundation Grant no. 1603564Y and Z.S. acknowledges Albert
Einstein Centre for Gravitation and Astrophysics supported by the Czech
Science Foundation Grant no. 1437086G.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Dimensional analysis and estimates of string
loop parameters
In the geometrized units the gravitational constant G and the
speed of light c are taken to be dimensionless units. Their
values and values of the Coulomb or electrostatic constant
ke and the Sun mass M in the SI units are
G = 6.67 × 10−11 m3 · kg−1 · s−2,
c = 3.00 × 108 m · s−1,
ke = 8.99 × 108 kg · m3 · C−2 · s−2,
M
= 2.00 × 1030 kg.
(A.1)
For central Schwarzschild black hole with M = 10M , black
hole length scales can be calculated in SI units
rhor = 2M G/c2 = 3 × 104 m,
rloop = 2π · 6M G/c2 = 5.6 × 105 m,
where rhor Schwarzschild horizon radius (black hole size)
and rloop is length of loop located at ISCO (inner edge of
Keplerian accretion disc). For Schwarzschild or RN black
hole, the radial coordinate r is circumferential.
Appendix A.1: Astrophysical relevance of black hole charge
One can compare the characteristic length scale given by
the charge of the RN black hole QG with its gravitational
radius. This gives the charge, whose gravitational effect is
comparable with the spacetime curvature of a black hole.
For the black hole of mass M this condition implies that
the gravitational effect of the charge Q on the background
geometry can be neglected if
(A.2)
Q << QG = 2
G M ≈ 1020 M
ke M
C.
If Q << QG, the electric field cannot modify the background
geometry of the black hole, but still there can be electrostatic
interaction between black hole and particle/string charges.
Reissner–Nordström black hole charge Q is assumed to
be small or even negligible for realistic black holes. Since
gravitation interaction is quite weak compared to the
electromagnetic interaction, with ration e/√Gmp ∼ 1018, any
RN charged black hole will easily separate electrons and
protons from surrounding plasma and neutralizing the RN black
hole with charge Q > 10−18 QG quickly [
6
]. The amount of
material, necessary for neutralization of maximally charged
RN black hole Q = QG is small
(A.3)
(A.4)
Macretion ∼ 10−18 M ∼ 1012 M
M
kg.
Appendix A.2: String loop parameters
The string loop model enables to apply and compare the
derived solutions for different physical mechanisms to obtain
the estimates of the parameters characterizing the string loop
dynamics. In order to make the estimate of the tension μ
strength, one can use, e.g., the similarity between the role of
the parameter μ and the Lorentz force acting on a charged
particle in the action governing the string loop dynamics.
On the other hand, we can find estimates of the fundamental
string loop parameter values related to the so called cosmic
strings, giving the upper limit of the application of the string
loop model. Realistic estimations give in the SI units the
string loop tension μ in order [
31
]
μ(L) ≤ μ < μ(CS),
μ(L) = 10−14 kg m−1,
To prove that string loop is test object only, we give some
examples of string loop mass and charge. Total string loop
mass mloop will be related to the total string loop energy Eloop
by massenergy equivalence formula
Eloop = mloopc2.
mloop = 2π r μ/c2.
For the Nambu–Goto string loop with radius r the total string
energy is just string length 2π r times tension μ, giving for
string loop mass formula
For our choice (A.2) of black hole M = 10Msun, we
have extremely light string loop in Lorentz case mloop(L) ∼
10−10 kg, while “Earth mass” loop mloop(CS) ∼ 1024 kg in
cosmic string case.
We can give ratio between the total mass of the charged
string loop mloop and the mass of RN black hole M , and ratio
between string loop charge qloop and charge of RN black hole
Q in SI units by formulas
mloop
M
= 2π
μG
c4
E ,
2
qloop
Q2 =
4π 2 μG
a2
c4
2,
(A.9)
where E and are previously used dimensionless string
loop energy and charge density and a is ratio between charge
and mass of the RN black hole a = Q/M ∈ (0, 1). For
charged string loop ω = 1 at stable position r = 6 around
RN black hole with a = 0.5, the dimensionless parameters
. .
are E = 13, = 10. Since the term Gμ/c4 is very small,
10−31 for Lorentz and 10−7 for cosmic strings, the string
loop total mass mloop and total charge qloop are negligible in
comparison the the RN black hole mass M and charge Q.
Only if a → 0 then RN black hole charge Q will become
comparable to the charge of string loop qloop.
(A.5)
(A.6)
(A.7)
(A.8)
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