Modelling Cosmic Springs with Finsler and Generalised Finsler Geometries

Matthew J. Lake<EMAIL_ADDRESS>
_a National Astronomical Research Institute of Thailand,
260 Moo 4, T. Donkaew, A. Maerim, Chiang Mai 50180, Thailand
_ _ b Department of Physics, Faculty of Science, Chiang Mai University,
239 Huaykaew Road, T. Suthep, A. Muang, Chiang Mai 50200, Thailand
_ _ c School of Physics, Sun Yat-Sen University,
Guangzhou 510275, People’s Republic of China
_ _ d Department of Physics, Babeş-Bolyai University,
Mihail Kogălniceanu Street 1, 400084 Cluj-Napoca, Romania
_

###### Abstract

We show that the equations of motion governing the dynamics of strings in a
compact internal space can be written as dispersion relations, with a local
speed that depends on the velocity and curvature of the string in the large
dimensions. From a $(3+1)$-dimensional perspective these can be viewed as
dispersion relations for waves propagating in the string interior and are
analogous to those for current-carrying topological defects. This allows us to
construct a unified framework with which to study and interpret the internal
structure of various field-theoretic and fundamental string species, in a
simple physically intuitive coordinate system, without the need for
dimensional reduction or approximate effective actions. This, in turn, allows
us to identify the precise conditions under which higher-dimensional strings
and current-carrying defects are observationally indistinguishable, for
macroscopic observers. Our approach naturally incorporates the description of
so-called ‘cosmic springs’, whose dynamics are expressed in terms of an
effective Finsler geometry, for circular loops, or generalised Finsler
geometry, for non-circular configurations. This demonstrates the importance of
these novel geometric structures and their utility in modelling complex
physical phenomena in cosmology and astrophysics.

Keywords: superconducting cosmic string; cosmic spring; Finsler geometry;
higher-dimensional windings; mathematical modelling

Symmetry, MDPI special issue, Mathematical Modelling of Physical Systems,
P. Nicolini, ed., 2022.

###### Contents

  1. 1 Introduction
  2. 2 Fundamental strings with higher-dimensional windings
    1. 2.1 The background geometry, equations of motion, and physical observables for wound-strings
      1. 2.1.1 The background geometry, string action and the Euler-Lagrange equations for wound-strings
      2. 2.1.2 The space-time energy-momentum tensor, constants of motion, and effective pressures and shears for wound-strings
      3. 2.1.3 The one-to-one equivalence of the EOM and conservation equations in a physically intuitive coordinate system
      4. 2.1.4 Definition of the higher-dimensional wound-string model parameters
    2. 2.2 Long $F$-strings
      1. 2.2.1 Long $F$-strings in $(3+1)$ dimensions
      2. 2.2.2 Long $F$-strings with higher-dimensional windings
    3. 2.3 Circular $F$-string loops
      1. 2.3.1 Circular $F$-string loops in $(3+1)$ dimensions
      2. 2.3.2 Circular $F$-string loops with higher-dimensional windings
    4. 2.4 Noncircular $F$-string loops
      1. 2.4.1 Noncircular $F$-string loops in $(3+1)$ dimensions
      2. 2.4.2 Noncircular $F$-string loops with higher-dimensional windings
    5. 2.5 Summary of the wound-string model
  3. 3 Topological defect strings with internal currents
    1. 3.1 The Abelian-Higgs model and the Nielsen-Olesen string
      1. 3.1.1 The Abelian-Higgs model with externally coupled charge ($j^{\mu}\neq 0$)
      2. 3.1.2 The Nielsen-Olesen solution ($j^{\mu}=0$)
    2. 3.2 String effective actions, with and without currents
  4. 4 Finsler and generalised Finsler geometries
    1. 4.1 Finsler geometry
    2. 4.2 Generalised Finsler geometry
  5. 5 Topological defect strings with internal currents, revisited
    1. 5.1 Long defect strings: Riemann geometry
      1. 5.1.1 Long defect strings without internal currents
      2. 5.1.2 Long defect strings with internal currents
    2. 5.2 Circular defect-string loops: Finsler geometry
      1. 5.2.1 Circular defect-string loops without currents
      2. 5.2.2 Circular defect-string loops with currents
    3. 5.3 Noncircular defect-string loops: generalised Finsler geometry
      1. 5.3.1 Noncircular defect-string loops without currents
      2. 5.3.2 Noncircular defect-string loops with currents
    4. 5.4 Summary of the defect-string model
  6. 6 Conclusions

##  1 Introduction

Cosmic strings are line-like concentrations of mass-energy, thought to have
formed during symmetry-breaking phase transitions in the early Universe [1, 2,
3, 4, 5, 6]. Though the string width is determined by the inverse of the
symmetry-breaking energy scale, this is small compared to cosmological
distances, and they may be approximated as one-dimensional objects for many
purposes [7, 8]. Such strings may have been produced at the epoch of electro-
weak symmetry-breaking [9, 10, 11], or the GUT scale, and their formation is
also a generic feature of the phase transitions predicted by many extensions
of the Standard Model [12, 13, 14].
In field theory they are a type of topological defect, analogous to the
magnetic flux tubes and other types of vortex configuration created in
condensed matter systems [15, 16, 17, 18, 19, 20, 21, 22, 23]. They are
produced via the Kibble-Zurek mechanism [24] when the vacuum manifold of the
fields ($\mathcal{M}$) possesses a nontrivial first homotopy group, e.g.,
$\pi_{1}(\mathcal{M})=\mathbb{Z}$, in which each nonzero integer corresponds
to a possible winding number of the vortex-string cross-section. In recent
years, the formation of ‘cosmic’, i.e. horizon-sized, fundamental strings
($F$-strings) and one-dimensional $D$-branes ($D$-strings) has also been
extensively studied in string theory [25, 26, 27, 28, 29, 30, 31, 32, 33, 34],
particularly in the context of brane inflation [35, 36, 37, 38, 39, 40], in
which such defects can be copiously produced [41, 42, 43].
However, regardless of the precise details of individual models, the main
phenomenological and observationally relevant parameters that characterise the
string are its linear mass density, or energy per unit length, often denoted
as $\mu$ or $T$, and tension, which is often denoted, somewhat confusingly, by
the letters $U$ or $\mathcal{T}$. 222Different conventions are prevalent in
different areas of research, e.g., $\mu$ is the preferred notation for the
linear mass density of defect strings, whereas $\mathcal{T}$ is more commonly
used for $F$-strings in string theory. In this work, we employ various
notations, in keeping with the conventions of the relevant research fields,
but clearly define our notation at the beginning of the relevant section(s) of
the paper. For vacuum strings in field theory and the fundamental strings of
string theory, $\mu=-U$, but for current-carrying strings the mass density and
tension may differ so that $\mu\neq-U$ [44, 45]. The relationship between the
macroscopic string dynamics and the current at a given point on the string, at
a given time, is complicated, since the local tension is determined by the
local flux density. This can vary as a function of both position and time,
giving rise to so-called ‘cosmic springs’, i.e., strings with variable
effective local tension.
In this paper, we study the complex interplay between the spring-like
properties of current-carrying strings and their macroscopic evolution. We
begin by demonstrating the equivalence, under dimensional reduction, between
current-carrying defect strings and wound-strings in geometries with compact
extra dimensions, providing a unified framework for modelling a variety of
cosmic string species. However, the demonstration of equivalence between
wound-strings and superconducting defect strings, under dimensional reduction,
is not new and was first shown by Nielsen [46]. We then show how, by rewriting
the equations of motion (EOM) for the current as dispersion relations, the
correct EOM for the macroscopic evolution of the string can be obtained
without dimensional reduction and without the need to construct an approximate
effective action.
When the EOM for the fluxes on the effective string world sheet in $(3+1)$
dimensions are written as dispersion relations, the local speed of propagation
depends on the velocity and curvature of the string in the noncompact space.
For circular string loops, this implies that the macroscopic dynamics can be
modelled in terms of an effective Finsler metric, but for noncircular
configurations a generalised Finsler geometry is required. This method
provides a new set of tools, and a new geometric language, with which to
describe the dynamics of cosmic springs, i.e., strings with variable effective
tension due to the presence of fluxes within the string core.
The structure of this paper is as follows. In Sec. 2, we review the necessary
background for our model of $F$-strings with higher-dimensional windings,
including the space-time embedding, EOM, and physical observables of the
string. Sec. 2.1.1 covers the background geometry, embedding, string action
and covariant EOM, while in Sec. 2.1.2 we determine general expressions for
the constants of motion and the effective pressures and shears, from a
$(3+1)$-dimensional perspective. In Sec. 2.1.3, we demonstrate the one-to-one
equivalence of the Euler-Lagrange equations and the conservation equations for
the space-time energy-momentum tensor in a simple, physically intuitive
coordinate system. We adopt these coordinates, and the associated gauge
conditions for the string, throughout the rest of the paper. The wound-string
model parameters, which determine the physically observable properties such as
the effective linear mass density and tension, are defined in Sec. 2.1.4.
Sections 2.2$-$2.4 then deal with long strings, circular loops and arbitrary
planar loops, respectively. Each subsection is further divided into two. The
part first deals with unwound strings whereas the second considers the effects
of higher-dimensional windings on the chosen ansatz for the configuration of
the string in the macroscopic dimensions. Hence, in each case, we first review
the string dynamics in the absence of high-dimensional effects, before
considering the generalisation to strings with nontrivial embeddings in the
internal space. We explicitly show that the conservation of momentum in the
extra dimensions is equivalent to the conservation of current from a
$(3+1)$-dimensional perspective and that the string EOM take the form of
dispersion relations for waves ‘inside’ the string. A brief summary of these
results is given in Sec. 2.5.
In Sec. 3, we review the necessary background for our model of current-
carrying topological defect strings. Though many superconducting string models
exist in the literature, their basic phenomenology is similar [2], and can be
well approximated by considering the simplest field theory in which vortex-
line solutions exist, namely, the Abelian-Higgs model. In Sec. 3.1, we review
the dynamics of the Abelian-Higgs fields and couple them to an ‘external’
source of charge which is capable of generating current when confined to the
string core. The simplest vortex-line solution known as the Nielsen-Olesen
string, which does not contain a superconducting current, is also reviewed.
Section 3.2 considers the standard approach to modelling both non-current-
carrying and current-carrying strings. In this approach, the effective action
for a zero-width and non-superconducting string is first constructed, by
integrating over the ‘internal’ degrees of freedom that describe the
microscopic structure of the string core in the Nielsen-Olesen solution. It is
straightforward to show that this leads directly to the Nambu-Goto action for
the $F$-string. Current-carrying strings are then modelled by adding world-
sheet fluxes to the effective action.
To the best of our knowledge no detailed study has been made, in the existing
literature, of the evolution of the internal structure of a finite-width
string, as it undergoes dynamical evolution as a solitonic field state in the
background space-time. The first goal of this work is to construct such a
model, via an appropriate choice of ansatz for the fundamental field
variables. This allows us to explicitly demonstrate the interplay between the
dynamical evolution of the ‘microscopic’ degrees of freedom (i.e., field
states), that characterise the string interior, and the macroscopic variables
that describe the dynamics of the string as an extended solitonic object. Our
second goal is to show that the evolution of these internal degrees of freedom
is completely analogous to the evolution of $F$-string windings in the compact
‘internal’ space predicted by string theory. We stress that this program goes
beyond the simple process of dimensional reduction. We do not dimensionally
reduce either the $F$-string windings or the internal degrees of freedom of
the finite-width core of the defect string. Instead, both microscopic and
macroscopic, ‘internal’ and ‘external’ evolution are considered together.
In the case of long, straight strings, we find that the two levels of
description can easily be reconciled within the usual pseudo-Riemannian
geometry of special and general relativity. (For simplicity, we consider a
$(3+1)$-dimensional Minkowski or warped Minkowski background as the geometry
of the large dimensions, throughout this work.) However, for circular defect-
string loops, the evolution of the fundamental field variables takes place in
an effective Finsler geometry, in which the metric depends both on the space-
time coordinates, $x$, and on their first derivatives with respect to a time-
like internal parameter, $\partial_{\tau}x$. In the absence of circular
symmetry, the effective metric also depends on the first derivatives of the
coordinates with respect to a space-like internal variable,
$\partial_{\sigma}x$, corresponding to a form of generalised Finsler geometry.
The connection to string theory is then made by associating $\tau$ and
$\sigma$ with the time-like and space-like world-sheet parameters. Hence, long
strings are modelled in Minkowski space, circular loops are modelled using an
effective Finsler geometry, and arbitrary planar loops are modelled using an
effective generalised Finsler geometry. These cases are dealt with in Secs.
5.1-5.3, respectively, and a short summary of the defect string model is given
in Sec. 5.4. Our Conclusions, and a brief discussion of prospects for future
work, are given in Sec. 6.

##  2 Fundamental strings with higher-dimensional windings

In [46] it was shown that the higher-dimensional dynamics of a fundamental
string embedded in $M^{4}\times S^{1}$ can be reinterpreted from a four-
dimensional perspective as an effective world-sheet current. There follows a
formal equivalence between superconducting strings (of zero width) in $M^{4}$
and Nambu-Goto strings [7, 9] in $M^{4}\times S^{1}$ under dimensional
reduction. In [47], it was shown that the equations of motion (EOM) for
$F$-strings in $M^{4}\times S^{1}$ admit critical solutions in which the
effective four-dimensional tension vanishes locally everywhere on the string.
For the critical case, the string may adopt arbitrary static configurations in
the large dimensions. Phenomenologically, this is consistent with similar
results for current-carrying strings [48, 49, 50] via the general
correspondence demonstrated in [46].
In [52], specific higher-dimensional $F$-string configurations were
investigated together with corresponding configurations of superconducting
defect strings with finite width. For long strings and circular loops, the
embedding of the $F$-string in the higher-dimensional space was found to be
formally analogous to the phase structure of superconducting Nielsen-Olesen
strings [1] at critical coupling. A key feature of this correspondence is that
the radius of the compact space $R$ and the string width in the field-
theoretic model $r_{c}$ play equivalent roles in determining the constants of
motion. This suggests that the radius of the windings in the higher-
dimensional space provides an effective thickness for the Nambu-Goto string in
four dimensions.
Based on these suggestive results, we conjecture that $F$-strings, which (by
definition) have no internal structure, can mimic the internal structure of
current carrying defect strings via their embedding in the ‘internal’ higher-
dimensional space required by string theory, in which three spatial dimensions
are extended and $n$ are compactified. As a qualitative statement, it is
unsurprising that this may _sometimes_ be true and, since the number of string
models available both in string theory and in various field theories is huge,
it is too vague to be of much use in distinguishing string species (or in
determining when different species may be indistinguishable) via current or
future observations. However, it is clear that it would be useful to be able
to state under precisely which circumstances a given defect string will be
successfully mimicked by the embedding of a higher-dimensional $F$-string.
Since cosmic superstring phenomenology offers a probe of high energy physics
in the early universe in the presence of extra dimensions, the detection of
higher-dimensional signatures from strings would be a key piece of evidence in
favour of string theory. It is therefore vital for us to understand what
distinguishable higher-dimensional signatures may actually exist. In this
section of the paper, we extend the results for $F$-strings presented in [52]
to the case of arbitrary planar loops. Our purpose is to work towards
developing a general formalism through which the EOM governing higher-
dimensional $F$-string dynamics may be interpreted as dispersion relations for
waves ‘in’ the effective four-dimensional width of the string. These can then
be compared with dispersion relations which govern the flow of current (and
the evolution of genuine internal structure) in field-theoretic strings, and
the conditions under which two string species give rise to identical
phenomenology may be determined.
Any equivalence should also include the string constants of motion, which
should be formally analogous under a one-to-one correspondence between string
theory and field-theoretic parameters, as in [52]. The finite-width field-
theoretic counterparts to the higher-dimensional $F$-string configurations
considered here are presented in Sec. 3. Though, for the sake of brevity, we
restrict our present analysis to planar loop configurations, we expect no
significant obstacles in extending this treatment to arbitrary
$(3+1)$-dimensional configurations. Finally, we note that the results
presented here also hold for strings wrapping $S^{1}$ sub-cycles of constant
radius in _any_ compactified geometry. The only caveat required is that the
solutions must be dynamically stabilised, where necessary, if topological
stability is not guaranteed (as it is for $M^{4}\times S^{1}$
compactification). This is the case with the solutions obtained in [51, 52,
53], which are valid for strings wrapping great circles in the $S^{3}$
manifold which regularises the conifold tip of the Klebanov-Strassler geometry
[54].

###  2.1 The background geometry, equations of motion, and physical
observables for wound-strings

In this section we review the background geometry, EOM, and physical
observables of the wound $F$-string model.

####  2.1.1 The background geometry, string action and the Euler-Lagrange
equations for wound-strings

We use the metric signature $(+---)$ and consider a background space-time with
the line element

$\displaystyle{\rm d}s^{2}=g_{IJ}{\rm d}x^{I}{\rm
d}x^{J}=a^{2}\eta_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}-R^{2}{\rm
d}\varphi^{2},$ (2.1)

where $\mu,\nu\in\left\\{0,1,2,3\right\\}$, $\eta_{\mu\nu}$ is the Minkowski
metric, $\varphi\in[0,2\pi)$ is an angular coordinate in the internal space
and $R$ is a (constant) radius. 333The assumption of constant radius is made
for simplicity and this condition should be loosened in a more thorough
treatment. Interestingly, this could lead to the description of so-called
‘lumpy’ cosmic strings, from a $(3+1)$-dimensional perspective [55], but we
leave this study to a future work. The phenomenological ‘warp factor’,
$a\in(0,1]$, accounts for the fact that the internal dimensions may be flux-
compactified, as expected in string theory. In this case, $a^{2}<1$ represents
the back reaction of the fluxes on the large dimensions [56].
Note that we need not literally assume an $M^{4}\times S^{1}$ compactification
and, in principle, the internal space may be far more complicated. Equation
(2.1) represents the effective metric ‘seen’ by any string wrapping an $S^{1}$
sub-cycle of constant radius in an (almost) arbitrary Calabi-Yau manifold. The
internal space is almost arbitrary, since the only condition we impose is that
an $S^{1}$ sub-cycle of constant radius exists. As the string is one-
dimensional, it necessarily wraps (topologically) $S^{1}$ sub-cycles in the
any compact space, though these need not be of uniform shape or length. For
simplicity, we assume windings of constant radius, though it would be
interesting to consider the more general case in which the effective
compactification scale $R^{{\rm eff}}(\tau,\sigma)$ is a function of the
world-sheet coordinates (c.f. [57]). For a constant compactification radius,
the part of the background metric seen by the string is of the form given in
Eq. (2.1), regardless of its macroscopic structure [47]. 444For example, in
[52, 53], strings wrapping $S^{1}$ sub-cycles of constant radius at the tip of
the Klebanov-Strassler geometry [54] were considered. Here the target space
manifold is $M^{4}\times S^{3}$ and the line element is given by ${\rm
d}s^{2}=a^{2}\eta_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}-R^{2}{\rm
d}\Omega_{3}^{2}$, where ${\rm d}\Omega_{3}^{2}$ is the line-element on the
unit three-sphere. In Hopf coordinates [58], this is given by ${\rm
d}\Omega_{3}^{2}={\rm d}\psi^{2}+\sin^{2}(\psi){\rm
d}\chi^{2}+\cos^{2}(\psi){\rm d}\varphi^{2}$ where $\psi\in[0,\pi)$ is the
polar angle and $\chi,\varphi\in[0,2\pi)$ are the two azimuthal angles. Taking
$\psi(\tau,\sigma)$, $\chi(\tau,\sigma)$ and $\varphi(\tau,\sigma)$ as
embedding coordinates for the string, it is clear that the value of $\psi$
controls the effective radius of the windings, which may vary as a function of
both $\tau$ and $\sigma$. For $\psi={\rm const.}$, the winding radius is also
constant and for $\psi=0$ it takes the maximum value $R$, the radius of the
$S^{3}$. Similar arguments hold true for more complicated manifolds, as long
as they contain at least one $S^{1}$ submanifold of constant radius. The
advantage of using the effective metric (2.1) is that we do not need to make
any further assumptions regarding the general structure of the internal space.
Specifically, we do not assume that the string windings will (or will not) be
topologically stabilised.
In the absence of additional fluxes, the basic string action is the Nambu-Goto
action [7, 9]

$\displaystyle S=-\mathcal{T}\int{\rm d}^{2}\zeta\sqrt{-\gamma},$ (2.2)

where $\gamma$ is the the determinant of the induced metric on the world-sheet

$\displaystyle\gamma_{ab}(X)=g_{IJ}\left(X\right)\partial_{a}X^{I}\partial_{b}X^{J},$
(2.3)

with $I,J\in\left\\{0,1,2,3,\dots,d\right\\}$, $a,b\in\left\\{0,1\right\\}$,
$\partial_{a}X^{I}=\partial X^{I}/\partial\zeta^{a}$, and
$\zeta^{0}=\tau,\zeta^{1}=\sigma$, in an arbitrary number of space-time
dimensions, $D=1+d$. The intrinsic string tension is given by

$\displaystyle\mathcal{T}=\frac{1}{2\pi\alpha^{\prime}},$ (2.4)

where $\alpha^{\prime}$ is the Regge slope parameter, which is related to the
(fundamental) string scale via

$\displaystyle l_{\rm st}=\sqrt{\alpha^{\prime}}.$ (2.5)

In the formulae above, and throughout the rest of this paper, we work in
natural units, $\hbar=1$, $c=1$. Variation of the action with respect to the
embedding coordinates gives the canonical Euler-Lagrange equations, plus a
boundary term

$\displaystyle\left[\mathcal{P}^{\sigma}{}_{I}(\tau,\sigma)\delta
X^{I}\right]_{0}^{\sigma_{f}}=0,$ (2.6)

where

$\displaystyle\mathcal{P}^{a}{}_{I}(\tau,\sigma)=\frac{\partial(\mathcal{L}\sqrt{-\gamma})}{\partial(\partial_{a}X^{I})}$
(2.7)

is the canonical momentum of $X^{I}$ with respect to $\zeta^{a}$. To satisfy
the boundary term, we may impose Dirichlet, Neumann or periodic boundary
conditions

$\displaystyle X^{I}(\tau,0)={\rm const.},\ \ \ X^{I}(\tau,\sigma_{f})={\rm
const.},$ (2.8)
$\displaystyle\mathcal{P}^{\sigma}{}_{I}(\tau,0)=\mathcal{P}^{\sigma}{}_{I}(\tau,\sigma_{f})=0,$
(2.9) $\displaystyle X^{I}(\tau,\sigma)=X^{I}(\tau,\sigma+m\sigma_{f}),\ \ \
m\in\mathbb{Z},$ (2.10)

where the imposition of Neumann boundary conditions implies that the string
end points move at the speed of light [59]. Using the identity
$\delta(-\gamma)=(-\gamma)\gamma^{ab}\delta\gamma_{ab}$, the canonical EOM may
also be rewritten in the form [8],

$\displaystyle\frac{\partial}{\partial\zeta^{a}}\left(\sqrt{-\gamma}\gamma^{ab}g_{IJ}(X)\partial_{b}X^{J}\right)-\frac{1}{2}\sqrt{-\gamma}\gamma^{cd}\frac{\partial
g_{KL}(X)}{\partial X^{I}}\partial_{c}X^{K}\partial_{d}X^{L}=0.$ (2.11)

####  2.1.2 The space-time energy-momentum tensor, constants of motion, and
effective pressures and shears for wound-strings

The space-time energy-momentum tensor is defined by varying the action with
respect to the metric $g_{IJ}(x)$

$\displaystyle\delta S=\int T^{IJ}\delta g_{IJ}\sqrt{-g}{\rm d}^{d}x,$ (2.12)

where from here on we use $x^{I}$ to refer to space-time coordinates and
$X^{I}$ to refer to the embedding coordinates of the string. For the Nambu-
Goto action, this gives [8, 2]

$\displaystyle
T^{IJ}=\frac{1}{\sqrt{-g}}\mathcal{T}\int\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{I}\partial_{b}X^{J}\delta^{D}(x-X){\rm
d}\tau{\rm d}\sigma.$ (2.13)

In the static gauge, $X^{0}=t=\zeta\tau$, where $\zeta$ is constant with
dimensions of length, this gives

$\displaystyle
T^{IJ}=\frac{1}{\sqrt{-g}}\zeta^{-1}\mathcal{T}\int\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{I}\partial_{b}X^{J}\delta^{d}(x-X){\rm
d}\sigma.$ (2.14)

The conserved currents associated with the space-time symmetries are

$\displaystyle J^{(I)A}=K^{(I)}{}_{B}T^{AB}$ (2.15)

where $K^{(I)}{}_{B}$ are the Killing vectors, so that the constants of motion
are given by

$\displaystyle\Pi^{I}=\int K^{(I)}{}_{B}T^{0B}\sqrt{-g}{\rm
d}^{d}x=\mathcal{T}\int\sqrt{-\gamma}\gamma^{\tau
b}K^{(I)}{}_{B}\partial_{b}X^{B}{\rm d}\sigma.$ (2.16)

We also define the covariant components

$\displaystyle\Pi_{I}=\int K_{(I)B}T^{0B}\sqrt{-g}{\rm
d}^{d}x=\mathcal{T}\int\sqrt{-\gamma}\gamma^{\tau
b}g_{IJ}(X)K^{(J)}{}_{B}\partial_{b}X^{B}{\rm d}\sigma,$ (2.17)

though these are not conserved if the index $I$ refers to a non-Cartesian
coordinate. In general, indices in the space-time energy-momentum tensor are
raised and lowered using the background metric

$\displaystyle T^{I}{}_{J}=g_{JK}(x)T^{IK},$ (2.18)

but the integral over ${\rm d}^{d}x$ together with the factor of
$\delta^{d}(x-X)$ in the expressions above, fixes $x=X$.
Using the letters $E$, $P$ and $l$ to denote energy, linear momentum and
angular momentum, respectively, we then define the warp factor-averaged values
of the constants of motion via

$\displaystyle E=\sqrt{P^{0}P_{0}},\ \ \
\mathcal{P}^{I}=\pm\sqrt{-P^{I}P_{(I)}},\ \ \
\Lambda^{J}=\pm\sqrt{-l^{J}l_{(J)}(t_{i})},$ (2.19)

where the space-time index $I$ labels a Cartesian coordinate and $J$ an
angular coordinate in the background space. The presence of of brackets around
a repeated index implies that no summation is implied. For unwarped background
geometries, these recover the standard expressions when $a^{2}=1$. By
convention, we take the signs of $\mathcal{P}^{I}$ and $\Lambda^{J}$ to match
those of $P^{I}$ and $l^{J}$, and $t_{i}$ denotes the initial time at which we
consider the state of the system. Such definitions are problematic unless the
metric $g_{IJ}(X)$ is independent of $\sigma$. However, this can be achieved
by adopting a gauge in which $\sigma$ is identified directly with a background
space-time coordinate, such that $X^{I}\propto\sigma$ for some $I$, and the
implications of this for the string EOM and conservation equations are
considered in Sec. 2.1.3. It is then straightforward to define the warp
factor-averaged total linear and angular momentum

$\displaystyle\mathcal{P}=\pm\sqrt{-P^{I}P_{I}},\ \ \
\Lambda=\pm\sqrt{-l^{J}l_{J}(t_{i})},$ (2.20)

and the total $4$-momentum

$\displaystyle\Pi=+\sqrt{\Pi^{I}\Pi_{I}}=+\sqrt{E^{2}-\mathcal{P}^{2}-\Lambda^{2}}.$
(2.21)

Finally, we define the contravariant and covariant integrated pressures and
shears via

$\displaystyle T^{K}=\int T^{KK}\sqrt{-g}{\rm d}^{d}x,\ \ \ T_{K}=\int
T^{(K)}{}_{K}(t_{i})\sqrt{-g}{\rm d}^{d}x,$ (2.22)

for $K\neq 0$, and

$\displaystyle S^{KL}=\int T^{KL}\sqrt{-g}{\rm d}^{d}x,\ \ \ S^{K}{}_{L}=\int
T^{K}{}_{L}(t_{i})\sqrt{-g}{\rm d}^{d}x,$ (2.23)

for $K,L\neq 0$, $K\neq L$, so that the warp-averaged values are

$\displaystyle\mathcal{T}^{K}=\pm\sqrt{-T^{K}T_{(K)}(t_{i})},\ \ \
\Sigma^{KL}=\pm\sqrt{S^{K}{}_{(L)}S_{(K)}{}^{L}(t_{i})}.$ (2.24)

The magnitudes of the total integrated pressures and shears are given by
$\mathcal{T}=+\sqrt{-T^{K}T_{K}}$ and $\Sigma=+\sqrt{S^{K}{}_{L}S_{K}{}^{L}}$,
where summation is now permitted over repeated indices. In addition to giving
warp factor averaged values for the conserved quantities in warped geometries,
the definitions above are also convenient in the sense that the
$\mathcal{T}^{K}$ always have canonical units of tension/pressure
$[E][l]^{-1}=[l]^{-2}$, whereas neither $T^{K}$ nor $T_{K}$ do, individually,
unless $X^{K}$ represents a Cartesian coordinate.

####  2.1.3 The one-to-one equivalence of the EOM and conservation equations
in a physically intuitive coordinate system

The conservation of the space-time energy-momentum tensor may be expressed as
[60]

$\displaystyle\nabla_{J}T^{J}{}_{I}\sqrt{-g}=\frac{\partial}{\partial
x^{J}}\left(T^{J}{}_{I}\sqrt{-g}\right)-\frac{1}{2}\frac{\partial
g_{AB}(x)}{\partial x^{I}}T^{AB}\sqrt{-g}=0.$ (2.25)

Substituting the energy-momentum tensor for the Nambu-Goto string gives

$\displaystyle\frac{\partial}{\partial x^{J}}\left[\int{\rm d}\tau{\rm
d}\sigma\delta^{D}(x-X)\sqrt{-\gamma}\gamma^{ab}g_{IK}(x)\partial_{a}X^{J}\partial_{b}X^{K}\right]$
(2.26) $\displaystyle-$ $\displaystyle\frac{1}{2}\frac{\partial
g_{AB}(x)}{\partial x^{I}}\int{\rm d}\tau{\rm
d}\sigma\delta^{D}(x-X)\sqrt{-\gamma}\gamma^{cd}\partial_{c}X^{A}\partial_{d}X^{B}=0.$

Restricting ourselves to embeddings in which the world-sheet coordinates can
be directly identified with two background space-time coordinates

$\displaystyle X^{0}=x^{0}=\zeta\tau,\ \ \ X^{1}=x^{0}=\chi\sigma,$ (2.27)

where $\zeta,\chi={\rm const.}$, so that

$\displaystyle{\rm d}\tau{\rm d}\sigma=\zeta^{-1}\chi^{-1}{\rm d}X^{0}{\rm
d}X^{1},$ (2.28)

and integrating over ${\rm d}^{d}x$, gives

$\displaystyle\frac{\partial}{\partial
x^{0}}\left[g_{IK}(X)\sqrt{-\gamma}\gamma^{ab}\partial_{a}x^{0}\partial_{b}X^{K}\right]+\frac{\partial}{\partial
x^{1}}\left[g_{IK}(X)\sqrt{-\gamma}\gamma^{ab}\partial_{a}x^{1}\partial_{b}X^{K}\right]$
(2.29) $\displaystyle+$ $\displaystyle\int{\rm
d}^{D-2}x\left[\frac{\partial\delta^{D-2}(x-X)}{\partial
x^{M}}g_{IK}(x)+\frac{\partial g_{IK}(x)}{\partial
x^{M}}\delta^{D-2}(x-X)\right]\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{M}\partial_{b}X^{K}$
$\displaystyle-$ $\displaystyle\frac{1}{2}\frac{\partial g_{AB}(X)}{\partial
X^{I}}\sqrt{-\gamma}\gamma^{cd}\partial_{c}X^{A}\partial_{d}X^{B}=0,$

for $M\notin\left\\{0,1\right\\}$. Using
$\partial_{a}x^{0}\propto\delta^{0}{}_{a}$ and
$\partial_{a}x^{1}\propto\delta^{1}{}_{a}$, together with the identity [61],

$\displaystyle\int{\rm d}xf(x)\frac{{\rm d}\delta(x)}{{\rm d}x}=-\int{\rm
d}x\delta(x)\frac{{\rm d}f(x)}{{\rm d}x},$ (2.30)

this becomes

$\displaystyle\partial_{a}x^{0}\frac{\partial}{\partial
x^{0}}\left[g_{IK}(X)\sqrt{-\gamma}\gamma^{ab}\partial_{b}X^{K}\right]+\partial_{a}x^{1}\frac{\partial}{\partial
x^{1}}\left[g_{IK}(X)\sqrt{-\gamma}\gamma^{ab}\partial_{b}X^{K}\right]$ (2.31)
$\displaystyle+$ $\displaystyle\left\\{-\left[\frac{\partial
g_{IK}(X)}{\partial X^{M}}\right]_{\partial X^{M}}+\left[\frac{\partial
g_{IK}(X)}{\partial X^{M}}\right]_{\partial
X^{M}}\right\\}\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{M}\partial_{b}X^{K}$
$\displaystyle-$ $\displaystyle\frac{1}{2}\frac{\partial g_{AB}(X)}{\partial
X^{I}}\sqrt{-\gamma}\gamma^{cd}\partial_{c}X^{A}\partial_{d}X^{B}=0.$

Canceling the boundary terms and rewriting the summation over world-sheet
indices in a compact form yields the Euler-Lagrange equations obtained using
the metric variation identity, Eq. (2.11). Hence, in this simple gauge, there
is one-to-one correspondence between the string EOM and the conservation
equations for the space-time energy-momentum tensor. For a given index $I$,
the corresponding equations are identical.

####  2.1.4 Definition of the higher-dimensional wound-string model parameters

Let $l(\sigma)$ denote the length of string in the interval $[0,\sigma]$ at
time $t$. In a $D=1+d$ dimensional space-time

$\displaystyle{\rm
d}l^{2}=-g_{IJ}(X)\partial_{\sigma}X^{I}\partial_{\sigma}X^{J}{\rm
d}\sigma^{2},$ (2.32)

where $I,J\in\left\\{1,2,3...d\right\\}$ denote only spatial coordinates. This
may be split into the sum of a three-dimensional and an extra-dimensional part

$\displaystyle{\rm d}l^{2}={\rm d}l_{3}^{2}+{\rm d}l_{d-3}^{2},$ (2.33)

where

$\displaystyle{\rm
d}l_{3}^{2}=-g_{kl}(X)\partial_{\sigma}X^{k}\partial_{\sigma}X^{l}{\rm
d}\sigma^{2}$ (2.34)

for $k,l\in\left\\{1,2,3\right\\}$, and

$\displaystyle{\rm
d}l_{d-3}^{2}=-g_{mn}(X)\partial_{\sigma}X^{m}\partial_{\sigma}X^{n}{\rm
d}\sigma^{2}$ (2.35)

for $m,n\in\left\\{4,5,6,\dots,d\right\\}$. The (physical) perpendicular
velocity of the string is defined in the static gauge in terms of $l(\sigma)$
[59]

$\displaystyle v_{\perp}^{2}=g_{IJ}(X)\frac{\partial X^{I}}{\partial
t}\frac{\partial X^{J}}{\partial t}-\left(g_{IJ}(X)\frac{\partial
X^{I}}{\partial l}\frac{\partial X^{J}}{\partial l}\right)^{2},\
I,J\in\left\\{1,2,3...d\right\\},$ (2.36)

so that in warped backgrounds the action may be written as

$\displaystyle S=-\mathcal{T}a\int{\rm d}t{\rm d}\sigma\left(\frac{{\rm
d}l}{{\rm d}\sigma}\right)\sqrt{1-\frac{v_{\perp}^{2}}{a^{2}}}.$ (2.37)

This reduces to the standard form given in [59] for $a=1$. Thus, in warped
geometries, the velocity of open string end points obeying Neumann boundary
conditions is given by $v_{\perp}^{2}(t,0)=v_{\perp}^{2}(t,2\pi)=a^{2}$.
We note that it is not permissible to substitute the ansatz (or partial
ansatz) at the level of the action before varying to obtain the EOM. The EOM
used throughout the rest of this paper are those obtained from the covariant
Euler-Lagrange equations for the string, Eq. (2.11). However, this
representation of the action is often useful, as it provides physical insight
into the motion of the string.
In the study of higher-dimensional strings, it is also useful to define the
parameter $\Omega^{2}(t)\in[0,1]$, which represents the (time-dependent)
fraction of the string lying in the large spatial dimensions, via

$\displaystyle\Omega^{-2}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\omega^{-2}(t,\sigma){\rm
d}\sigma,$ (2.38)

where $\omega^{2}(t,\sigma)\in[0,1]$ represents the instantaneous local value,
at a given point $\sigma$ and at time $t$. In terms of our previously defined
notation, this is given as

$\displaystyle\omega^{-2}(t,\sigma)=\left(\frac{{\rm d}l}{{\rm
d}\sigma}\right)^{2}\left(\frac{{\rm d}l_{3}}{{\rm
d}\sigma}\right)^{-2}=\frac{g_{kl}(X)\partial_{\sigma}X^{k}\partial_{\sigma}X^{l}+g_{mn}(X)\partial_{\sigma}X^{m}\partial_{\sigma}X^{n}}{g_{kl}\partial_{\sigma}X^{k}\partial_{\sigma}x^{l}}.$
(2.39)

###  2.2 Long $F$-strings

In this section, we consider long straight strings, both with and without
higher-dimensional windings. Long strings with higher-dimensional windings are
considered in detail in Sec. 2.2.2, so that it is instructive to first review
the treatment of long straight strings without windings, a brief analysis of
which is presented in Sec. 2.2.1.

####  2.2.1 Long $F$-strings in $(3+1)$ dimensions

Let us adopt Cartesian coordinates for the background metric

$\displaystyle{\rm d}s^{2}=a^{2}({\rm d}t^{2}-{\rm d}x^{2}-{\rm d}y^{2}-{\rm
d}z^{2})-R^{2}{\rm d}\varphi^{2},$ (2.40)

and take the embedding

$\displaystyle X^{I}(\tau,\sigma)\equiv X^{I}(t,\sigma)\equiv
X^{I}(t,z)=\left(t(\tau)=\zeta\tau,x=0,y=0,z(\sigma)=(2\pi)^{-1}\Delta\sigma,\varphi=0\right).$
(2.41)

Physically, this represents either a string of length $\Delta$, lying along
the $z$-axis, whose end points are fixed by Dirichlet boundary conditions, or
a finite section of (formally infinite) string, obeying periodic boundary
conditions. The action may be written as

$\displaystyle S=-\mathcal{T}a^{2}\Delta\int{\rm d}t,$ (2.42)

which is equivalent to the form given in Eq. (2.37), with

$\displaystyle\frac{dl}{d\sigma}=(2\pi)^{-1}a\Delta,\ \ \
\frac{v_{\perp}^{2}}{a^{2}}=0.$ (2.43)

The EOM are

$\displaystyle\partial_{0}(T^{0}{}_{0}\sqrt{-g})+\partial_{z}(T^{z}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (2.44a)
$\displaystyle\partial_{0}(T^{0}{}_{z}\sqrt{-g})+\partial_{z}(T^{z}{}_{z}\sqrt{-g})$
$\displaystyle=0.$ (2.44b)

From here on, we use the following notation:

$\displaystyle
T^{I}{}_{J}\sqrt{-g}=\int\mathcal{T}^{I}{}_{J}\sqrt{-g}\delta^{d}(x-X){\rm
d}\sigma$ (2.45)

where $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ denotes the energy-momentum tensor
_density_. Thus, $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ has the same dimensions as
$T^{I}{}_{J}\sqrt{-g}$, but must be multiplied by an appropriate delta
function and integrated over ${\rm d}\sigma$ to give the true energy-momentum
tensor of the string. For the sake of notational simplicity, we quote only the
relevant components of $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ for the various
systems considered in the remainder of this work. For the long straight
string, the components of $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ appearing in the
EOM are given by

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}=\mathcal{T}a^{2}(2\pi)^{-1}\Delta,\
\ \ \mathcal{T}^{z}{}_{0}\sqrt{-g}=0,$ (2.46a)
$\displaystyle\mathcal{T}^{0}{}_{z}\sqrt{-g}=0,\ \ \
\mathcal{T}^{z}{}_{z}\sqrt{-g}=\mathcal{T}a^{2}(2\pi)^{-1}\Delta.$ (2.46b)

Clearly, the EOM are trivially satisfied. The only nonzero constants of motion
are the energy and the integrated pressure in the $z$-direction,

$\displaystyle E=\mathcal{T}a\Delta,\ \ \
\mathcal{T}^{z}=-\mathcal{T}a\Delta.$ (2.47)

These are equal in magnitude but opposite in sign, as for vacuum strings in
field-theoretic models [2].

####  2.2.2 Long $F$-strings with higher-dimensional windings

Let us again adopt Cartesian coordinates for the background space and consider
the new wound-string embedding

$\displaystyle X^{I}(\tau,\sigma)\equiv X^{I}(t,\sigma)\equiv
X^{I}(t,z)=\left(t(\tau)=\zeta\tau,x=0,y=0,z(\sigma)=(2\pi)^{-1}\Delta\sigma,\varphi(t,\sigma)\equiv\varphi(t,z)\right).$

Physically, this represents a finite section of an infinite string, so that we
must impose periodic boundary conditions on $\varphi(t,\sigma)$. To consider a
section of open string, obeying Neumann boundary conditions, we would need to
adopt an embedding for the $z$-coordinate of the form
$z(t,\sigma)=(2\pi)^{-1}\Delta\sigma+g(t)$. This corresponds to a comoving
coordinate system which ‘tracks’ the propagation of the higher-dimensional
windings from a four-dimensional perspective (i.e. along the $z$-axis). In
this case, the analysis is a straightforward generalization of that presented
below but, from an astrophysical/cosmological perspective, long strings are
greater interest than open ones. We therefore restrict our current analysis to
finite sections of long string.
For the embedding (LABEL:Emb_3.2), the action may be written as

$\displaystyle S=-\mathcal{T}\int{\rm d}t{\rm
d}\sigma\sqrt{a^{2}((2\pi)^{-2}a^{2}\Delta^{2}+R^{2}(\partial_{\sigma}\varphi)^{2})-a^{2}R^{2}(2\pi)^{-2}\Delta^{2}\dot{\varphi}^{2}},$
(2.49)

which is equivalent to the static gauge form (2.37) with

$\displaystyle\frac{dl}{d\sigma}=\sqrt{(2\pi)^{-2}a^{2}\Delta^{2}+R^{2}(\partial_{\sigma}\varphi^{2})},\
\ \
\frac{v_{\perp}^{2}}{a^{2}}=\frac{\Delta^{2}R^{2}\dot{\varphi}^{2}}{a^{2}\Delta^{2}+(2\pi)^{2}R^{2}(\partial_{\sigma}\varphi^{2})}.$
(2.50)

The EOM are

$\displaystyle\partial_{0}(T^{0}{}_{0}\sqrt{-g})+\partial_{z}(T^{z}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (2.51a)
$\displaystyle\partial_{0}(T^{0}{}_{z}\sqrt{-g})+\partial_{z}(T^{z}{}_{z}\sqrt{-g})$
$\displaystyle=0,$ (2.51b)
$\displaystyle\partial_{0}(T^{0}{}_{\varphi}\sqrt{-g})+\partial_{z}(T^{z}{}_{\varphi}\sqrt{-g})$
$\displaystyle=0.$ (2.51c) The EOM in $\phi$ may be written explicitly as
$\displaystyle(a^{2}+R^{2}(\varphi^{\prime
2}-\dot{\varphi}^{2}))(\phi^{\prime\prime}-\ddot{\varphi})+a^{2}R^{2}(2\dot{\varphi}\varphi^{\prime}\dot{\varphi}^{\prime}-\varphi^{\prime
2}\varphi^{\prime\prime}-\dot{\varphi}^{2}\ddot{\varphi})=0,$ (2.52)

where a dash represents differentiation with respect to $z$. This is satisfied
for any function obeying

$\displaystyle\dot{\varphi}^{2}-\varphi^{\prime 2}=0,\ \ \
\dot{\varphi}=\pm\varphi^{\prime}$ (2.53)

which automatically implies

$\displaystyle\ddot{\varphi}-\varphi^{\prime\prime}=0.$ (2.54)

However, Eq. (2.53) is a stronger condition than Eq. (2.54), since the latter
admits superpositions of both left and right-movers, whereas the former does
not. Equation (2.53) is satisfied for any function of the form

$\displaystyle\varphi(t,z)=\varphi(k_{z}z+\omega_{z}t)\equiv\varphi(t,\sigma)=\varphi(m\sigma+\omega_{z}t),$
(2.55)

where

$\displaystyle\omega_{z}^{2}=k_{z}^{2},\ \ \omega_{z}=\pm k_{z}$ (2.56)

and $m\in\mathbb{Z}$. Defining

$\displaystyle k_{z}=2\pi/\lambda_{z},$ (2.57)

and imposing periodic boundary conditions on $\varphi(t,z)$,

$\displaystyle\varphi(k_{z}z+\omega_{z}t)=\varphi(k_{z}(z+\Delta)+\omega_{z}t)\equiv\varphi(m\sigma+\omega_{z}t)=\varphi(m(\sigma+2\pi)+\omega_{z}t),$
(2.58)

it follows that

$\displaystyle\Delta=m\lambda_{z}.$ (2.59)

Different plane wave modes in the Fourier expansion of $\varphi(z\pm t)$
correspond to different frequencies $\omega_{j}$ and wavelengths
$\lambda_{j}$, such that $\omega_{j}=\pm k_{j}=\pm 2\pi/\lambda_{j}=\pm 2\pi
m_{j}/\Delta$, for some $m_{j}\in\mathbb{N}$. But, by writing $\varphi(z\pm
t)$ in the form (2.55), which implies $\partial\varphi/\partial
t=\omega_{z}{\rm d}\varphi/{\rm d}u$, $\partial\varphi/\partial z=k_{z}{\rm
d}\varphi/{\rm d}u$, $\partial\varphi/\partial\sigma=m{\rm d}\varphi/{\rm
d}u$, where $u=k_{z}z+\omega_{z}t=m\sigma+\omega_{z}t$, we select a single
mode as being characteristic of the wave form. The natural choice is the mode
with the highest amplitude which, for nonlinear functions of $u$, gives the
approximate wavelength of any fluctuation in current density, as seen form a
four-dimensional perspective. If $\varphi$ contains linear terms in $z$ and
$t$, it is most natural to use the associated $k_{z}$ and $\omega_{z}$ to give
the characteristic wavelength and frequency. Linear terms give rise to a
uniform current and any additional nonlinear terms describe local fluctuations
in current density around the mean value. In this case, the integer $m$ may be
interpreted as the number of windings present in a four-dimensional string
section of length $\Delta$.
Though we may express any functions of $t$ and $z$ as functions of $t$ and
$\sigma$, or vice-versa, from here on we choose the latter, as this makes the
connection between the long string EOM and their generalisation to loops in
Sec. 2.3 more explicit. The functions $\omega^{2}(t,\sigma)$ and
$\Omega^{2}(t)$ are given as

$\displaystyle\omega^{-2}(t,\sigma)=1+\frac{(2\pi)^{2}R^{2}}{a^{2}\Delta^{2}}(\partial_{\sigma}\varphi)^{2},\
\ \ \Omega^{-2}(t)=1+\frac{2\pi
R^{2}}{a^{2}\Delta^{2}}\int_{0}^{2\pi}(\partial_{\sigma}\varphi)^{2}{\rm
d}\sigma,$ (2.60)

and, after substituting for $\dot{\varphi}$, the components of
$\mathcal{T}^{I}{}_{J}\sqrt{-g}$ appearing in the EOM may be written as

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=\mathcal{T}a^{2}(2\pi)^{-1}\Delta\omega^{-2},\ \ \ \ \ \ \ \ \
\ \ \ \
\mathcal{T}^{z}{}_{0}\sqrt{-g}=\mp\mathcal{T}a^{2}(2\pi)^{-1}\Delta\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.61a) $\displaystyle\mathcal{T}^{0}{}_{z}\sqrt{-g}$
$\displaystyle=\pm\mathcal{T}a^{2}(2\pi)^{-1}\Delta\left(\frac{1-\omega^{2}}{\omega^{2}}\right),\
\mathcal{T}^{z}{}_{z}\sqrt{-g}=\mathcal{T}a^{2}(2\pi)^{-1}\Delta\left(\frac{2\omega^{2}-1}{\omega^{2}}\right),$
(2.61b) $\displaystyle\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}$
$\displaystyle=\pm\mathcal{T}a(2\pi)^{-1}\Delta
R\frac{\sqrt{1-\omega^{2}}}{\omega},\ \ \
\mathcal{T}^{z}{}_{\varphi}\sqrt{-g}=\mp\mathcal{T}a(2\pi)^{-1}\Delta
R\frac{\sqrt{1-\omega^{2}}}{\omega}.$ (2.61c)

Following [46], we define the components of the current density as

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\frac{e}{2\pi}\dot{\varphi},\ \ \
\mathcal{J}^{z}\sqrt{-g}=\frac{e}{2\pi}\varphi^{\prime}.$ (2.62)

By analogy with Eq. (2.45) the components of the effective world-sheet
current, from a $(3+1)$-dimensional perspective, are given by

$\displaystyle
J^{\mu}\sqrt{-g}=\int\mathcal{J}^{\mu}\sqrt{-g}\delta^{d}(x-X){\rm d}\sigma$
(2.63)

but, for the sake of notational simplicity, we quote only the relevant values
of $\mathcal{J}^{\mu}$ from here on. Using the EOM in $\varphi$, we have

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\pm\frac{e}{2\pi}\varphi^{\prime}=-\mathcal{J}^{z}\sqrt{-g}=\mp\frac{e}{2\pi}\dot{\varphi},$
(2.64)

which may be written in terms of the higher-dimensional string variables as

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\frac{e}{2\pi\Delta
R^{2}\mathcal{T}}\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}=-\mathcal{J}^{z}\sqrt{-g}=\frac{e}{2\pi\Delta
R^{2}\mathcal{T}}\mathcal{T}^{z}{}_{\varphi}\sqrt{-g}.$ (2.65)

Equation (2.65) makes the link between the components of the effective
$(3+1)$-dimensional current and the higher-dimensional components of the
energy-momentum tensor explicit. Specifically, we see that the
$(3+1)$-dimensional current density, at a given point on the string, is
proportional to the momentum density in the compact direction at that point.
Hence, the conserved current is proportional to the conserved momentum in the
internal space. From Eqs. (2.63)-(2.64), it follows that
$\partial_{\mu}(J^{\mu}\sqrt{-g})=0$ and the current conservation equation is
equivalent to Eq. (2.61c).
The EOM form a self-consistent set, since each equation is equivalent to

$\displaystyle\dot{\omega}^{2}=\frac{(2\pi)^{2}}{\Delta^{2}}(\partial_{\sigma}\omega)^{2},\
\ \ \dot{\omega}=\pm\frac{2\pi}{\Delta}\partial_{\sigma}\omega^{2},$ (2.66)

which is in turn equivalent to Eq. (2.53). Substituting for $\dot{\varphi}$ in
Eq. (2.50), the perpendicular velocity and string length may be written in
terms of the higher-dimensional string variables as

$\displaystyle\frac{{\rm d}l}{{\rm d}\sigma}=(2\pi)^{-1}a\Delta\omega^{-1},\ \
\ \frac{v_{\perp}^{2}}{a^{2}}=(1-\omega^{2}).$ (2.67)

The net number of windings in the compact space, which are distributed along
the $z$-direction, is given by

$\displaystyle
n_{z}=\frac{1}{2\pi}\int_{0}^{2\pi}\partial_{\sigma}\varphi\,{\rm d}\sigma,$
(2.68)

and for convenience we may split $\varphi$ into linear and nonlinear parts

$\displaystyle\varphi(t,\sigma)=m\sigma+\omega_{z}t+\varphi_{\rm
NL}(m\sigma+\omega_{z}t).$ (2.69)

However, by the periodic boundary conditions, $\varphi_{\rm NL}$ makes no
contribution to the net winding number and

$\displaystyle n_{z}=m.$ (2.70)

In terms of integrals over $\varphi_{\rm NL}$, the constants of motion are

$\displaystyle E=\mathcal{T}a\Delta\left[1+\frac{2\pi
R^{2}}{a^{2}\lambda_{z}^{2}}\int_{0}^{2\pi}\left(1+\frac{\partial_{\sigma}\varphi_{\rm
NL}}{n_{z}}\right)^{2}{\rm d}\sigma\right],$ (2.71)
$\displaystyle\mathcal{P}^{z}=\pm
2\pi\mathcal{T}\times\frac{n_{z}R^{2}}{a\lambda_{z}}\int_{0}^{2\pi}\left(1+\frac{\partial_{\sigma}\varphi_{\rm
NL}}{n_{z}}\right)^{2}{\rm d}\sigma,$ (2.72)
$\displaystyle\Lambda^{\varphi}=\pm 2\pi\mathcal{T}aRn_{z},$ (2.73)
$\displaystyle\Pi=\mathcal{T}a\Delta\sqrt{1+\frac{4\pi
n_{z}^{2}R^{2}}{a^{2}\Delta^{2}}\int_{0}^{2\pi}\left(1+\frac{\partial_{\sigma}\varphi_{\rm
NL}}{n_{z}}\right)^{2}{\rm
d}\sigma-\frac{(2\pi)^{2}R^{2}n_{z}^{2}}{a^{2}\Delta^{2}}}.$ (2.74)

The integrated tensions are

$\displaystyle\mathcal{T}^{z}=-\mathcal{T}a\Delta\left[1-\frac{2\pi
R^{2}}{a^{2}\lambda_{z}^{2}}\int_{0}^{2\pi}\left(1+\frac{\partial_{\sigma}\varphi_{\rm
NL}}{n_{z}}\right)^{2}{\rm d}\sigma\right],$ (2.75)
$\displaystyle\mathcal{T}^{\varphi}=0,$ (2.76)

since $\mathcal{T}^{\varphi\varphi}\sqrt{-g}=0$ by the EOM for $\varphi$, and
the only nonzero component of the shear, which is between the $z$\- and
$\varphi$-directions, is

$\displaystyle\Sigma^{z\varphi}=-\frac{a}{R}\Lambda^{\varphi}.$ (2.77)

In the linear case ($\varphi_{\rm NL}=0$), these may be written in a compact
form as

$\displaystyle
E=\mathcal{T}a\Delta\left(1+\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{z}^{2}}\right)=\mathcal{T}a\Delta\Omega^{-2},$
(2.78)
$\displaystyle\mathcal{P}^{z}=\pm\mathcal{T}n_{z}\times\frac{(2\pi)^{2}R^{2}}{a\lambda_{z}}=\pm\mathcal{T}a\Delta\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right),$
(2.79) $\displaystyle\Lambda^{\varphi}=\pm
2\pi\mathcal{T}aRn_{z}=\pm\mathcal{T}a\Delta\frac{\sqrt{1-\Omega^{2}}}{\Omega},$
(2.80)
$\displaystyle\Pi=\mathcal{T}a\Delta\sqrt{1+\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{z}^{2}}}=\mathcal{T}a\Delta\Omega^{-1},$
(2.81)
$\displaystyle\mathcal{T}^{z}=-\mathcal{T}a\Delta\left(1-\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{z}^{2}}\right)=-\mathcal{T}a\Delta\left(\frac{2\Omega^{2}-1}{\Omega^{2}}\right).$
(2.82)

Clearly, it is possible for the string to be effectively tensionless
(everywhere) when the distance between windings is equal to the circumference
of extra dimension

$\displaystyle a^{2}\lambda_{z}^{2}=(2\pi)^{2}R^{2},$ (2.83)

or equivalently when

$\displaystyle\Omega^{2}=\omega^{2}=1/2,\ \forall t,\sigma.$ (2.84)

Under these conditions we also have that

$\displaystyle\omega_{z}^{2}\equiv\omega_{\varphi}^{2}=\frac{v_{\varphi}^{2}}{R^{2}}=\frac{a^{2}}{R^{2}},$
(2.85)

where $\omega_{\varphi}$ and $v_{\varphi}$ denote the angular and linear
velocity of the string in the compact space.
In the classical theory, the four-dimensional string tension reverses sign and
becomes an effective pressure for $\omega^{2}<1/2$. This leads to qualitative
differences in the macroscopic (i.e. $(3+1)$-dimensional) string dynamics. For
example, circular string loops with initial winding configurations such that
$a\lambda_{z}(t_{i})<2\pi R$ oscillate between two critical radii,
$\rho_{c1}=\rho(t_{i})$ and $\rho_{c2}>\rho(t_{i})$, implying that loops
initially expand, rather than contract, after chopping off from the long
string network [52, 53]. This is also shown explicitly, in the classical
theory of circular loops, in Sec. 2.3. However, due to quantum effects,
superconducting defect strings carrying either bosons or fermions are expected
have critical currents of order

$\displaystyle J_{\rm max}\simeq\frac{e}{2\pi R},$ (2.86)

where $e$ is the coupling (i.e. charge) [2]. In field theory, an important
generalisation of the original (vacuum) cosmic string models was given by
Witten [44], who proposed that ordinary cosmic strings may carry electric
currents, thus behaving like superconducting wires. The charge carriers may
either be bosons, in which case a charged Higgs field with a nonzero vacuum
expectation value in the core of the string is required, or fermions, which
are trapped as zero modes along the string. For bosonic currents, exceeding
the critical threshold implies electric field strengths large enough to induce
pair production via the Schwinger process whereas, for trapped fermions, this
marks the critical point at which it becomes energetically favourable for
particles to leave the string [2].
Although Nielsen [46] established a formal correspondence between
superconducting strings and strings compactified on higher-dimensional cycles
at the classical level, it is reasonable to consider the quantum bound (2.86)
in relation to this model. For the long straight string, this may be rewritten
in terms of the higher-dimensional variable $\lambda_{z}$ as

$\displaystyle a\lambda_{z}^{\rm min}=2\pi R.$ (2.87)

In the defect string picture, this is equivalent to requiring the distance
between neighbouring twists in the magnetic field lines to be greater than the
circumference of the string core [52]. In the wound string picture, it implies
that neighbouring windings must be separated by a distance larger than the
compactification scale. Though a full quantum mechanical treatment of the
wound-string model is needed to confirm this result, we note that it is
physically intuitive and is analogous to a standard result for nonrelativistic
strings of finite width, i.e., that one cannot put ‘twists’ in a length of
rope over distances significantly shorter than its width. It is also
interesting to note that the maximum current allowed by quantum mechanics is
exactly that which leads to the tensionless condition, so that strings with
positive effective pressure, or ‘repulsive rods’, are unable to form.
For the treatment of nonlinear windings, it is useful to introduce the
parameter $\alpha\in[0,1]$, defined via

$\displaystyle\alpha^{-2}=\frac{1}{2\pi
n_{z}^{2}}\int_{0}^{2\pi}(\partial_{\sigma}\varphi)^{2}{\rm
d}\sigma=\frac{1}{2\pi}\int_{0}^{2\pi}\left(1+\frac{\partial_{\sigma}\varphi_{\rm
NL}}{n_{z}}\right)^{2}{\rm d}\sigma.$ (2.88)

We define $\alpha$ as the inverse of the positive square root of (2.88) so
that $\alpha=1$ when $\varphi_{\rm NL}=0$ and $\alpha<1$ when $\varphi_{\rm
NL}\neq 0$. It is then straightforward to show that in general

$\displaystyle
E=\mathcal{T}a\Delta\left(1+\frac{(2\pi)^{2}R^{2}}{a^{2}\alpha^{2}\lambda_{z}^{2}}\right)=\mathcal{T}a\Delta\Omega^{-2},$
(2.89)
$\displaystyle\mathcal{P}^{z}=\pm\mathcal{T}\frac{n_{z}}{\alpha}\times\frac{(2\pi)^{2}R^{2}}{a\alpha\lambda_{z}}=\pm\mathcal{T}a\Delta\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right),$
(2.90) $\displaystyle\Lambda^{\varphi}=\pm
2\pi\mathcal{T}aRn_{z}=\pm\mathcal{T}a\Delta\frac{\sqrt{1-\Omega^{2}}}{\Omega}\alpha$
(2.91)
$\displaystyle\Pi=\mathcal{T}a\Delta\sqrt{1+\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{z}^{2}}(2\alpha^{-2}-1)}=\mathcal{T}a\Delta\Omega^{-1}\sqrt{2-\alpha^{2}-(1-\alpha^{2})\Omega^{2}},$
(2.92)
$\displaystyle\mathcal{T}^{z}=-\mathcal{T}a\Delta\left(1-\frac{(2\pi)^{2}R^{2}}{a^{2}\alpha^{2}\lambda_{z}^{2}}\right)=-\mathcal{T}a\Delta\left(\frac{2\Omega^{2}-1}{\Omega^{2}}\right).$
(2.93)

Thus, the definition (2.88) allows us to express the physical parameters of
the string in a compact form in terms of $\Omega(t)$ and $\alpha$, even for
strings with complicated nonlinear windings.
In the nonlinear case - that is, for a string with linear extension in the
$z$-direction of the Minkowski space-time but with nonlinear windings
propagating along its $(3+1)$-dimensional length - it is not possible for the
string to be everywhere tensionless at any moment in time. However, it is
still possible for the average net tension in the $z$-direction to be zero and
this occurs when

$\displaystyle a^{2}\alpha^{2}\lambda_{z}^{2}=(2\pi)^{2}R^{2},$ (2.94)

or equivalently for

$\displaystyle\Omega^{2}=1/2,\ \forall t,\ (\omega^{2}\neq 1/2,\ \forall
t,\sigma).$ (2.95)

Note that although $\Omega^{2}$, as we have defined it for a general string
configuration, is a function of time, it is clear that for the embedding
considered here it is a constant, even in the nonlinear case:

$\displaystyle\Omega^{2}=\left(1+\frac{(2\pi)^{2}R^{2}}{a^{2}\alpha^{2}\lambda_{z}^{2}}\right)^{-1}={\rm
const.}$ (2.96)

The linear case corresponds to $\alpha=1$ and $\alpha<1$ quantifies the degree
of nonlinearity in the windings. Specifically, we may interpret
$a\alpha\lambda_{z}$ as the mean value of the local wavelength of the windings
(in the warped geometry), averaged across the entire length of the string and
$n_{z}/\alpha$ as the average value of the effective ‘local’ winding number at
a given point $z$. To see this, we now define the local values of the various
model parameters - that is, their effective values at a given point on the
string, labelled by $\sigma$ (or $z$), at a given time $t$. Thus, the local
string length in $(3+1)$-dimensions and the local winding number are defined
as

$\displaystyle(l_{z}^{\rm
eff})^{2}=(2\pi)^{2}(\partial_{\sigma}z)^{2}=\Delta^{2},\ \ \ (n_{z}^{\rm
eff})^{2}=\frac{\Delta^{2}}{(2\pi)^{2}}\varphi^{\prime
2}=(\partial_{\sigma}\varphi)^{2}.$ (2.97)

We define the effective wavelength, wavenumber, angular frequency and velocity
of the string in the $z$-direction via

$\displaystyle(\omega_{z}^{\rm eff})^{2}=(v_{z}^{\rm eff})^{2}(k_{z}^{\rm
eff})^{2},\ \ \ (k_{z}^{\rm eff})^{2}=\frac{(2\pi)^{2}}{(\lambda_{z}^{\rm
eff})^{2}}=\frac{(2\pi)^{2}}{\Delta}(\partial_{\sigma}\varphi)^{2}.$ (2.98)

The instantaneous angular and linear velocities in the $\varphi$-direction are
then related via

$\displaystyle(\omega_{z}^{\rm eff})^{2}=(\omega_{\varphi}^{\rm
eff})^{2}=\frac{(v_{\varphi}^{\rm eff})^{2}}{R^{2}}=\dot{\varphi}^{2},$ (2.99)

since the angular frequency associated with the movement of the windings in
the large dimensions must be equal to the angular velocity of the string in
the compact space.
In our new notation, the EOM for $\varphi$ is equivalent to the dispersion
relation

$\displaystyle(\omega_{z}^{\rm eff})^{2}=(k_{z}^{\rm eff})^{2},\ \ \
\omega_{z}^{\rm eff}=\pm k_{z}^{\rm eff},$ (2.100)

and

$\displaystyle(v_{z}^{\rm eff})^{2}=1,\ \forall t,\sigma.$ (2.101)

By combining the various relations above, it follows that

$\displaystyle(l_{z}^{\rm eff})^{2}=(n_{z}^{\rm eff})^{2}(\lambda_{z}^{\rm
eff})^{2},\ (v_{\varphi}^{\rm eff})^{2}(\lambda_{z}^{\rm
eff})^{2}=(2\pi)^{2}R^{2},\ (n_{z}^{\rm eff})^{2}=\frac{(v_{\varphi}^{\rm
eff})^{2}(l_{z}^{\rm eff})^{2}}{(2\pi)^{2}R^{2}}=\frac{(v_{\varphi}^{\rm
eff})^{2}\Delta^{2}}{(2\pi)^{2}R^{2}},$ (2.102)

which form a self-consistent set. The expressions for the various components
of $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ may now be written entirely in terms of
the new parameters since

$\displaystyle\omega^{-2}=\frac{a^{2}(l_{z}^{\rm
eff})^{2}+(2\pi)^{2}R^{2}(n_{z}^{\rm eff})^{2}}{a^{2}(l_{z}^{\rm
eff})^{2}}=1+\frac{(2\pi)^{2}R^{2}}{a^{2}(\lambda_{z}^{\rm eff})^{2}}.$
(2.103)

In general, we have that

$\displaystyle a^{2}(\lambda_{z}^{\rm
eff})^{2}=(2\pi)^{2}R^{2}\left(\frac{\omega^{2}}{1-\omega^{2}}\right),$
(2.104)

so that the tensionless condition $\omega^{2}=1/2$ is equivalent to
$a^{2}(\lambda_{z}^{\rm eff})^{2}=(2\pi)^{2}R^{2}$ which, for a string which
is ‘straight’ with respect to the Minkowski directions, is clearly only
possible when the windings are linear, $\lambda_{z}^{\rm eff}=\lambda_{z}$,
thus yielding the condition obtained previously, (2.83). It is straightforward
to show that the consistency of the relations above also requires

$\displaystyle(v_{\varphi}^{\rm
eff})^{2}=a^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\equiv\frac{v_{\perp}^{2}}{a^{2}}.$
(2.105)

This can be interpreted physically since, for windings formed dynamically by
the motion of the string in the compact space, rather than as an initial
condition at the time of string formation, we expect

$\displaystyle\frac{(v_{\varphi}^{\rm eff})^{2}}{a^{2}(v_{z}^{\rm
eff})^{2}}=\frac{1-\omega^{2}}{\omega^{2}}.$ (2.106)

This is equivalent to the previous relation (2.105) since $(v_{z}^{\rm
eff})^{2}=1$.
Next, we define the integrated (i.e. spatially-averaged) values of the
parameters on the left-hand sides of Eqs. (2.97)-(2.100) according to the
averaging condition

$\displaystyle\langle Q^{2}\rangle(t)=\frac{1}{2\pi}\int_{0}^{2\pi}(Q^{\rm
eff})^{2}{\rm d}\sigma.$ (2.107)

The integrated values of any parameters appearing on the right-hand-sides are
defined implicitly in terms of those on the left. Hence, we have

$\displaystyle\langle l_{z}^{2}\rangle=\Delta^{2},\ \ \ \langle
n_{z}^{2}\rangle=n_{z}^{2}/\alpha^{2},$ (2.108)

and

$\displaystyle\langle\omega_{z}^{2}\rangle=\langle k_{z}^{2}\rangle,\ \ \
\langle
k_{z}^{2}\rangle=\frac{(2\pi)^{2}}{\langle\lambda_{z}^{2}\rangle}=\frac{(2\pi)^{2}}{\Delta^{2}}\langle(\partial_{\sigma}\varphi)^{2}\rangle,$
(2.109)

where

$\displaystyle\langle\lambda_{z}^{2}\rangle=\alpha^{2}\lambda_{z}^{2},$
(2.110)

together with

$\displaystyle\langle\omega_{z}^{2}\rangle=\langle\omega_{\varphi}^{2}\rangle=\frac{\langle
v_{\varphi}^{2}\rangle}{R^{2}}=\langle\dot{\varphi}^{2}\rangle.$ (2.111)

Combining these expression as before gives

$\displaystyle\langle l_{z}^{2}\rangle=\langle
n_{z}^{2}\rangle\langle\lambda_{z}^{2}\rangle,\ \ \ \langle
v_{\varphi}^{2}\rangle\langle\lambda_{z}^{2}\rangle=(2\pi)^{2}R^{2},\ \ \
\langle n_{z}^{2}\rangle=\frac{\langle v_{\varphi}^{2}\rangle\langle
l_{z}^{2}\rangle}{(2\pi)^{2}R^{2}}=\frac{\langle
v_{\varphi}^{2}\rangle\Delta^{2}}{(2\pi)^{2}R^{2}},$ (2.112)

and

$\displaystyle\langle v_{\varphi}^{2}\rangle=v_{\varphi}^{2}/\alpha^{2}.$
(2.113)

In general, we have

$\displaystyle
a^{2}\langle\lambda_{z}^{2}\rangle=(2\pi)^{2}R^{2}\left(\frac{\Omega^{2}}{1-\Omega^{2}}\right).$
(2.114)

The case of zero net tension occurs when the average value of the wavelength,
along the entire string, is equal to the circumference of the compact space

$\displaystyle a^{2}\langle\lambda_{z}^{2}\rangle=(2\pi)^{2}R^{2}.$ (2.115)

As before, the consistency of the above relations also requires

$\displaystyle\langle
v_{\varphi}^{2}\rangle=a^{2}\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right)\equiv\frac{\langle
v_{\perp}^{2}\rangle}{a^{2}}=\frac{\Omega^{2}}{a^{2}}\langle
v_{\varphi}^{2}\rangle.$ (2.116)

This follows from the expression for the ratio of the instantaneous velocities
for dynamically formed windings. By integrating Eq. (2.105) over ${\rm
d}\sigma$, we obtain

$\displaystyle\frac{\langle v_{\varphi}^{2}\rangle}{a^{2}\langle
v_{z}^{2}\rangle}=\frac{1-\Omega^{2}}{\Omega^{2}},$ (2.117)

which is equivalent to Eq. (2.116), since $\langle v_{z}^{2}\rangle=1$.
The physical parameters $E$, $\mathcal{P}^{z}$, $\Pi$ and $\mathcal{T}^{z}$
can now be written in terms of the spatially-averaged variables as

$\displaystyle
E=\mathcal{T}a\Delta\left(1+\frac{(2\pi)^{2}R^{2}}{a^{2}\langle\lambda_{z}^{2}\rangle}\right),$
(2.118)
$\displaystyle\mathcal{P}^{z}=\pm\mathcal{T}a\Delta\times\frac{(2\pi)^{2}R^{2}}{a^{2}\langle\lambda_{z}^{2}\rangle}=\pm\mathcal{T}\sqrt{\langle
n_{z}^{2}\rangle}\times\frac{(2\pi)^{2}R^{2}}{a\sqrt{\langle\lambda_{z}^{2}\rangle}},$
(2.119)
$\displaystyle\Pi=\mathcal{T}a\Delta\sqrt{1+\frac{(2\pi)^{2}R^{2}}{a^{2}\langle\lambda_{z}^{2}\rangle}\left(\frac{2\langle
n_{z}^{2}\rangle-n_{z}^{2}}{\langle n_{z}^{2}\rangle}\right)},$ (2.120)
$\displaystyle\mathcal{T}^{z}=-\mathcal{T}a\Delta\left(1-\frac{(2\pi)^{2}R^{2}}{a^{2}\langle\lambda_{z}^{2}\rangle}\right).$
(2.121)

These expressions remain consistent with our earlier definitions, since

$\displaystyle\Omega^{-2}=\frac{a^{2}\langle
l_{z}^{2}\rangle+(2\pi)^{2}R^{2}\langle n_{z}^{2}\rangle}{a^{2}\langle
l_{z}^{2}\rangle}=1+\frac{(2\pi)^{2}R^{2}}{a^{2}\langle\lambda_{z}^{2}\rangle}.$
(2.122)

In the linear case, we have

$\displaystyle\langle Q^{2}\rangle=(Q^{\rm eff})^{2}=Q^{2}={\rm const.},$
(2.123) $\displaystyle\omega^{2}=\Omega^{2}={\rm const.},\ \ \ \forall
t,\sigma$ (2.124)

and $\alpha=1$, so that the expressions above reduce to

$\displaystyle\Delta^{2}=n_{z}^{2}\lambda_{z}^{2},\ \ \
v_{\varphi}^{2}\lambda_{z}^{2}=(2\pi)^{2}R^{2},\ \ \
n_{z}^{2}=\frac{v_{\varphi}^{2}\Delta^{2}}{(2\pi)^{2}R^{2}}.$ (2.125)
$\displaystyle\omega_{z}^{2}=k_{z}^{2}=\frac{(2\pi)^{2}}{\lambda_{z}^{2}}\equiv\omega_{\varphi}^{2}=\frac{v_{\varphi}^{2}}{R^{2}},$
(2.126)

and $v_{z}^{2}=1$. These last relations also hold true, for the parameters
contained within the argument of the function
$\varphi(t,z)\equiv\varphi(t,\sigma)$, even in the nonlinear case, for which
$(Q^{\rm eff})^{2}\neq\langle Q^{2}\rangle\neq Q^{2}$.

###  2.3 Circular $F$-string loops

In this section, we consider circular loops, both with and without higher-
dimensional windings. Loops with higher-dimensional windings are considered in
detail in Sec. 2.3.2 and the general solution to the string EOM for arbitrary
initial conditions is given. As it is instructive to first review the
treatment of circular loops without windings, a brief analysis of this is
presented in Sect. 2.3.1.

####  2.3.1 Circular $F$-string loops in $(3+1)$ dimensions

To treat string loops we switch from Cartesian to cylindrical polar
coordinates, so that the $(y,z)$-plane becomes the $(\rho,\sigma)$-plane

$\displaystyle{\rm d}s^{2}=a^{2}({\rm d}t^{2}-{\rm d}x^{2}-{\rm d}y^{2}-{\rm
d}z^{2})-R^{2}{\rm d}\varphi^{2}=a^{2}({\rm d}t^{2}-{\rm d}x^{2}-{\rm
d}\rho^{2}-\rho^{2}{\rm d}\sigma^{2})-R^{2}{\rm d}\varphi^{2}.$ (2.127)

For circular loops with no higher-dimensional windings, the embedding is

$\displaystyle X^{I}(\tau,\sigma)\equiv
X^{I}(t,\sigma)=\left(t(\tau)=\zeta\tau,x=0,\rho(t),\sigma,\varphi=0\right),$
(2.128)

and the action becomes

$\displaystyle S=-\mathcal{T}a^{2}\int{\rm d}t{\rm
d}\sigma\rho\sqrt{1-\dot{\rho}^{2}}.$ (2.129)

This can be written in the standard form (2.37) by setting

$\displaystyle\frac{{\rm d}l}{{\rm d}\sigma}=a\rho,\ \ \
\frac{v_{\perp}^{2}}{a^{2}}=\dot{\rho}^{2}.$ (2.130)

The EOM are

$\displaystyle\partial_{0}(T^{0}{}_{0}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (2.131a)
$\displaystyle\partial_{0}(T^{0}{}_{\sigma}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\sigma}\sqrt{-g})$
$\displaystyle=0,$ (2.131b)
$\displaystyle\partial_{0}(T^{0}{}_{\rho}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\rho}\sqrt{-g})-\frac{1}{2}\partial_{\rho}g_{\sigma\sigma}T^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=0,$ (2.131c)

where the relevant components of $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ are given by

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=\mathcal{T}\frac{a^{2}\rho}{\sqrt{1-\dot{\rho}^{2}}},\ \ \ \ \
\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}=0,$ (2.132a)
$\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}$ $\displaystyle=0,\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \
\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}=\mathcal{T}a^{2}\rho\sqrt{1-\dot{\rho}^{2}},$
(2.132b) $\displaystyle\mathcal{T}^{0}{}_{\rho}\sqrt{-g}$
$\displaystyle=-\mathcal{T}\frac{a^{2}\rho\dot{\rho}}{\sqrt{1-\dot{\rho}^{2}}},\
\ \mathcal{T}^{\sigma}{}_{\rho}\sqrt{-g}=0,\ \
\mathcal{T}^{\sigma\sigma}\sqrt{-g}=-\mathcal{T}\frac{\sqrt{1-\dot{\rho}^{2}}}{\rho}.$
(2.132c)

It is straightforward to show that each of these is equivalent to

$\displaystyle 1-\dot{\rho}^{2}+\rho\ddot{\rho}=0,$ (2.133)

which represents the conservation of energy, since the only nonzero constant
of motion is the Hamiltonian

$\displaystyle E=2\pi\mathcal{T}\frac{a\rho}{\sqrt{1-\dot{\rho}^{2}}}.$
(2.134)

The integrated tensions are

$\displaystyle\mathcal{T}^{\rho}=-2\pi\mathcal{T}\frac{a\rho\dot{\rho}}{\sqrt{1-\dot{\rho}^{2}}},\
\ \ \mathcal{T}^{\sigma}=-2\pi\mathcal{T}a\rho\sqrt{1-\dot{\rho}^{2}}.$
(2.135)

and the only possible shear, between the $\sigma$\- and $\rho$-directions, is
zero

$\displaystyle\Sigma^{\rho\sigma}=0,$ (2.136)

since $\mathcal{T}^{\rho\sigma}\sqrt{-g}=0$. The expressions for the
Hamiltonian and integrated tensions, and, in fact, for all the components of
the energy momentum tensor $\mathcal{T}^{I}{}_{J}\sqrt{-g}$, become equivalent
to those for the long straight string (with no higher-dimensional windings)
for

$\displaystyle\Delta=2\pi\rho(t_{i}),\ \ \ \dot{\rho}(t)=0,\ \ \
\ddot{\rho}(t)=0,\ \forall t,$ (2.137)

where $t_{i}$ denotes the initial time, which is assumed to be the time of
loop formation.
Due to the circular symmetry of the system, the Hamiltonian density is
independent of $\sigma$ and the EOM for $\rho(t)$ is directly integrable. We
may solve it, subject to the generic boundary conditions

$\displaystyle\rho(t)|_{t=t_{i}}=\rho(t_{i}),\ \ \
\dot{\rho}(t)|_{t=t_{i}}=\dot{\rho}(t_{i}),$ (2.138)

to give

$\displaystyle\rho(t)=\frac{\rho(t_{i})}{\sqrt{1-\dot{\rho}^{2}(t_{i})}}\left|\cos\left(\frac{\sqrt{1-\dot{\rho}^{2}(t_{i})}}{\rho(t_{i})}(t-t_{i})-\sin^{-1}(\dot{\rho}(t_{i}))\right)\right|.$
(2.139)

Note that we must make use of the identity

$\displaystyle\cos(\sin^{-1}(x))=\sqrt{1-x^{2}}$ (2.140)

to recover the correct initial radius at $t=t_{i}$. The loop oscillates
between two critical radii

$\displaystyle\rho_{c1}=\rho(t_{i}),\ \ \ \rho_{c2}=0,$ (2.141)

performing one full oscillation with a time period

$\displaystyle T_{c}=\frac{\rho(t_{i})}{\sqrt{1-\dot{\rho}^{2}(t_{i})}}\pi.$
(2.142)

####  2.3.2 Circular $F$-string loops with higher-dimensional windings

Using cylindrical polar coordinates, the general embedding for a circular loop
with higher-dimensional windings is

$\displaystyle X^{I}(\tau,\sigma)\equiv
X^{I}(t,\sigma)=\left(t(\tau)=\zeta\tau,x=0,\rho(t),\sigma,\varphi(\tau,\sigma)\equiv\varphi(t,\sigma)\right),$
(2.143)

so that the action is

$\displaystyle S=-\mathcal{T}\int{\rm d}t{\rm
d}\sigma\sqrt{a^{2}(1-\dot{\rho}^{2})(a^{2}\rho^{2}+R^{2}(\partial_{\sigma}\varphi)^{2})-a^{2}\rho^{2}R^{2}\dot{\varphi}^{2}},$
(2.144)

which can be written in the standard form (2.37) with

$\displaystyle\frac{{\rm d}l}{{\rm
d}\sigma}=\sqrt{a^{2}\rho^{2}+R^{2}(\partial_{\sigma}\varphi)^{2}},\ \ \
\frac{v_{\perp}^{2}}{a^{2}}=\dot{\rho}^{2}+\frac{\rho^{2}R^{2}\dot{\varphi}^{2}}{a^{2}\rho^{2}+R^{2}(\partial_{\sigma}\varphi)^{2}}.$
(2.145)

The EOM are

$\displaystyle\partial_{0}(T^{0}{}_{0}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (2.146a)
$\displaystyle\partial_{0}(T^{0}{}_{\sigma}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\sigma}\sqrt{-g})$
$\displaystyle=0,$ (2.146b)
$\displaystyle\partial_{0}(T^{0}{}_{\rho}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\rho}\sqrt{-g})-\frac{1}{2}\partial_{\rho}g_{\sigma\sigma}T^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=0,$ (2.146c)
$\displaystyle\partial_{0}(T^{0}{}_{\varphi}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\varphi}\sqrt{-g})$
$\displaystyle=0.$ (2.146d)

Writing out the EOM in $\varphi(t,\sigma)$ explicitly in terms of $(-\gamma)$
and its derivatives, we have

$\displaystyle(-\gamma)[(1-\dot{\rho}^{2})\partial_{\sigma}^{2}\varphi-\rho^{2}\ddot{\varphi}-2\rho\dot{\rho}\dot{\varphi}]+\frac{1}{2}\rho^{2}\dot{\varphi}\frac{\partial(-\gamma)}{\partial
t}-\frac{1}{2}(1-\dot{\rho}^{2})\partial_{\sigma}\varphi\frac{\partial(-\gamma)}{\partial\sigma}=0,$
(2.147)

which may be rearranged to give

$\displaystyle
a^{2}R^{2}[(1-\dot{\rho}^{2})\partial_{\sigma}^{2}\varphi-\rho^{2}\ddot{\varphi}-2\rho\dot{\rho}\dot{\varphi}][(1-\dot{\rho}^{2})(\partial_{\sigma}\varphi)^{2}-\rho^{2}\dot{\varphi}^{2}]$
(2.148) $\displaystyle+$ $\displaystyle
a^{2}R^{2}\rho^{2}\dot{\varphi}[(1-\dot{\rho}^{2})\partial_{\sigma}\varphi\partial_{\sigma}\dot{\varphi}-\rho\ddot{\rho}(\partial_{\sigma}\varphi)^{2}-\rho\dot{\rho}\dot{\varphi}^{2}-\rho^{2}\dot{\varphi}\ddot{\varphi}]$
$\displaystyle-$ $\displaystyle
a^{2}R^{2}(1-\dot{\rho}^{2})\partial_{\sigma}\varphi[(1-\dot{\rho}^{2})\partial_{\sigma}\varphi\partial_{\sigma}^{2}\varphi-\rho^{2}\dot{\varphi}\partial_{\sigma}\dot{\varphi}]$
$\displaystyle+$ $\displaystyle
a^{4}\rho^{2}\left\\{(1-\dot{\rho}^{2})[(1-\dot{\rho}^{2})\partial_{\sigma}^{2}\varphi-\rho^{2}\dot{\varphi}\partial_{\sigma}\dot{\varphi}]-\rho^{2}\dot{\rho}\ddot{\rho}\dot{\varphi}\right\\}.$

This is very complicated, but we may try to guess an equivalent compact form
by noting that Eq. (2.148) reduces to the EOM for $\varphi(t,z)$ for the long
straight string, when the conditions in Eq. (2.137) are satisfied. Thus, let
us assume that

$\displaystyle\dot{\varphi}^{2}=\frac{(1-\dot{\rho}^{2})}{\rho^{2}}(\partial_{\sigma}\varphi)^{2}.$
(2.149)

Differentiating with respect to $t$ and $\sigma$, respectively, then gives

$\displaystyle(1-\dot{\rho}^{2})\partial_{\sigma}\varphi\partial_{\sigma}\dot{\varphi}-\dot{\rho}\ddot{\rho}(\partial_{\sigma}\varphi)^{2}-\rho\dot{\rho}\dot{\varphi}^{2}-\rho^{2}\dot{\varphi}\ddot{\varphi}=0,$
(2.150)
$\displaystyle(1-\dot{\rho}^{2})\partial_{\sigma}\varphi\partial_{\sigma}^{2}\varphi-\rho^{2}\dot{\varphi}\partial_{\sigma}\dot{\varphi}=0,$
(2.151)

and the complicated EOM (2.148) simplifies to

$\displaystyle(1-\dot{\rho}^{2})^{2}\partial_{\sigma}^{2}\varphi-\rho^{2}\dot{\rho}\ddot{\rho}\dot{\varphi}-(1-\dot{\rho}^{2})[\rho\dot{\rho}\dot{\varphi}+\rho^{2}\ddot{\varphi}]=0.$
(2.152)

Rearranging Eqs. (2.150)-(2.151) to make $\partial_{\sigma}\dot{\varphi}$ the
subject and equating the results, gives

$\displaystyle(\partial_{\sigma}\varphi)^{2}[(1-\dot{\rho}^{2})^{2}\partial_{\sigma}^{2}\varphi-\rho^{2}\dot{\rho}\ddot{\rho}\dot{\varphi}]-\rho^{2}\dot{\varphi}^{2}[\rho\dot{\rho}\dot{\varphi}+\rho^{2}\ddot{\varphi}]=0.$
(2.153)

Substituting again from the compact expression Eq. (2.149), we recover the
simplified EOM Eq. (2.152). Hence, our guess (2.149) is genuinely equivalent
to the original complicated EOM for $\varphi(t,\sigma)$, Eq. (2.148). Eq.
(2.149) is solved by any function of the form

$\displaystyle\varphi(t,\sigma)=\varphi\left(n_{\sigma}\sigma+\int\omega_{\sigma}(t){\rm
d}t\right),$ (2.154)

where

$\displaystyle\omega_{\sigma}^{2}(t)=n_{\sigma}^{2}\left(\frac{1-\dot{\rho}^{2}}{\rho^{2}}\right),\
\ \ \omega_{\sigma}(t)=\pm n_{\sigma}\frac{\sqrt{1-\dot{\rho}^{2}}}{\rho},$
(2.155)

which for convenience may again be split into linear and nonlinear parts

$\displaystyle\varphi(t,\sigma)=n_{\sigma}\sigma+\int\omega_{\sigma}(t){\rm
d}t+\varphi_{\rm NL}\left(n_{\sigma}\sigma+\int\omega_{\sigma}(t){\rm
d}t\right).$ (2.156)

However, since we have imposed circular symmetry on the $(3+1)$-dimensional
part of the ansatz, we must also impose this with respect to distribution of
the the windings, from a $(3+1)$-dimensional perspective. Therefore, we may
set $\varphi_{\rm NL}=0$, so that

$\displaystyle\varphi(t,\sigma)=n_{\sigma}\sigma+\int\omega_{\sigma}(t){\rm
d}t,$ (2.157)

which is equivalent to the condition

$\displaystyle\partial_{\sigma}\omega=0.$ (2.158)

Since we have imposed three constraints on a system of four PDEs, i.e. fixing
the forms of $X^{0}=\zeta\tau$ and $X^{1}=\sigma$, together with the condition
(2.158), we should now be left with a single remaining (independent) EOM in
the Euler-Lagrange equations. In other words, if the requirement of circular
symmetry for the windings is physically necessary, as symmetry suggests, it
should, together with the constraints imposed so far, imply that any
combination of Eqs. (2.146a)-(2.146d) yields the same independent EOM.
After substituting for $\dot{\varphi}^{2}$ from Eq. (2.149), the components of
$\mathcal{T}^{I}{}_{J}\sqrt{-g}$ which occur in the EOM may be written as,

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=\mathcal{T}\frac{a^{2}\rho}{\sqrt{1-\dot{\rho}^{2}}}\omega^{-2},\
\ \ \ \ \ \ \ \
\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}=\mp\mathcal{T}a^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.159a) $\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}$
$\displaystyle=\pm\mathcal{T}a^{2}\rho^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),\
\ \
\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}=\mathcal{T}a^{2}\rho\sqrt{1-\dot{\rho}^{2}}\left(\frac{2\omega^{2}-1}{\omega^{2}}\right),$
(2.159b) $\displaystyle\mathcal{T}^{0}{}_{\rho}\sqrt{-g}$
$\displaystyle=-\mathcal{T}\frac{a^{2}\rho\dot{\rho}}{\sqrt{1-\dot{\rho}^{2}}}\omega^{-2},\
\ \ \ \ \ \
\mathcal{T}^{\sigma}{}_{\rho}\sqrt{-g}=\pm\mathcal{T}a^{2}\dot{\rho}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
$\displaystyle\mathcal{T}^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=-\mathcal{T}\frac{\sqrt{1-\dot{\rho}^{2}}}{\rho}\left(\frac{2\omega^{2}-1}{\omega^{2}}\right),$
(2.159c) $\displaystyle\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}$
$\displaystyle=\pm\mathcal{T}a\rho R\frac{\sqrt{1-\omega^{2}}}{\omega},\ \ \ \
\ \
\mathcal{T}^{\sigma}{}_{\varphi}\sqrt{-g}=\mp\mathcal{T}a\sqrt{1-\dot{\rho}^{2}}R\frac{\sqrt{1-\omega^{2}}}{\omega},$
(2.159d)

where

$\displaystyle\omega^{-2}(t)=\frac{a^{2}\rho^{2}+R^{2}n_{\sigma}^{2}}{a^{2}\rho^{2}}.$
(2.160)

We now define the _normalised_ components of the current density as

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\frac{e}{2\pi}\frac{\sqrt{1-\dot{\rho}(t_{i})}}{\rho(t_{i})}\frac{\rho}{\sqrt{1-\dot{\rho}}}\dot{\varphi},\
\ \
\mathcal{J}^{\sigma}\sqrt{-g}=\frac{e}{2\pi}\frac{\sqrt{1-\dot{\rho}(t_{i})}}{\rho(t_{i})}\frac{\sqrt{1-\dot{\rho}}}{\rho}\partial_{\sigma}\varphi,$
(2.161)

so that using the EOM in $\varphi$ gives

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\pm\frac{e}{2\pi}\frac{\sqrt{1-\dot{\rho}(t_{i})}}{\rho(t_{i})}\partial_{\sigma}\varphi=-\mathcal{J}^{\sigma}\sqrt{-g}=\mp\frac{e}{2\pi}\frac{\sqrt{1-\dot{\rho}(t_{i})}}{\rho(t_{i})}\dot{\varphi}.$
(2.162)

The components of the $4$-current may again be expressed in terms of
$\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}$ and
$\mathcal{T}^{\sigma}{}_{\varphi}\sqrt{-g}$, now given by Eq. (2.159d), and it
is straightforward to show that the current conservation equation is
equivalent to Eq. (2.146d).
By direct substitution of Eqs. (2.159a)-(2.159c), together with Eq. (2.158),
it may be verified that Eqs. (2.146b) and (2.146d) yield

$\displaystyle\frac{\rho}{\dot{\rho}}\frac{\dot{\omega}}{\omega}=1-\omega^{2},$
(2.163)

and that Eqs. (2.146a) and (2.146c) give, respectively,

$\displaystyle(1-\dot{\rho}^{2})(2\omega^{2}-1)+\rho\ddot{\rho}+2\dot{\rho}^{2}(1-\dot{\rho}^{2})\left(1-\omega^{2}-\frac{\rho}{\dot{\rho}}\frac{\dot{\omega}}{\omega}\right)=0,$
(2.164)

and

$\displaystyle(1-\dot{\rho}^{2})\left(1-2\frac{\rho}{\dot{\rho}}\frac{\dot{\omega}}{\omega}\right)+\rho\ddot{\rho}=0.$
(2.165)

In fact, Eq. (2.163) follows directly from the definition of $\omega^{2}$
under the assumption that $\omega^{2}=\omega^{2}(t)$ (Eq. (2.160)) and is
therefore equivalent to the condition $\partial_{\sigma}\omega=0$ (2.158).
Thus, the sole remaining independent EOM is

$\displaystyle(1-\dot{\rho}^{2})(2\omega^{2}-1)+\rho\ddot{\rho}=0.$ (2.166)

In this system, the EOM in $\rho(t)$ is redundant and is satisfied identically
according to the conservation of energy and angular momentum represented by
the EOM in $X^{0}=t$, $\sigma$ and $\varphi$. It is also possible, though time
consuming, to show that for $\partial_{\sigma}\omega\neq 0$ ($\varphi_{\rm
NL}\neq 0$), Eqs. (2.146a)-(2.146d) are _not_ self-consistent. Physically,
this is because nonlinear fluctuations in the winding density would induce
additional $\sigma$-dependence in the effective tension of the string, causing
it to contract or expand at different rates at different points on the
circumference of the loop. Therefore, even if the string began in an initially
circular configuration in Minkowski space, this symmetry would immediately be
broken if not reflected in the initial winding distribution.
Again utilising the EOM in $\varphi$, we may relate the parameter $\omega^{2}$
to the perpendicular velocity of the string

$\displaystyle\frac{{\rm d}l}{{\rm d}\sigma}=a\rho\omega^{-1},\ \ \
\frac{v_{\perp}^{2}}{a^{2}}=\dot{\rho}^{2}+(1-\dot{\rho}^{2})(1-\omega^{2}).$
(2.167)

The $\sigma$-independence of the winding distribution also implies that

$\displaystyle\omega^{-2}(t)=\Omega^{-2}(t)=\frac{a^{2}\rho^{2}+R^{2}n_{\sigma}^{2}}{a^{2}\rho^{2}}=1+\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{\sigma}^{2}},$
(2.168)

where

$\displaystyle\lambda_{\sigma}(t)=\frac{2\pi\rho(t)}{n_{\sigma}},$ (2.169)

which is analogous to the first relation in Eq. (2.125) for straight strings,
and we define the (time-dependent) wavenumber via

$\displaystyle k_{\sigma}(t)=\frac{2\pi}{\lambda_{\sigma}(t)}.$ (2.170)

As for long straight strings with higher-dimensional windings, the EOM in
$\varphi(t,\sigma)$ may be written as a dispersion relation

$\displaystyle\omega_{\sigma}^{2}=(1-\dot{\rho}^{2})k_{\sigma}^{2},\ \ \
\omega_{\sigma}=\pm\sqrt{1-\dot{\rho}^{2}}k_{\sigma},$ (2.171)

but now the instantaneous linear velocity of the string is given by

$\displaystyle v_{\sigma}^{2}=1-\dot{\rho}^{2},\ \ \
v_{\sigma}=\pm\sqrt{1-\dot{\rho}^{2}},$ (2.172)

i.e., it depends on velocity of the string in the macroscopic dimensions and,
also, implicitly on its curvature, since $\dot{\rho}$ is related to $\rho$.
Further expressions, analogous to those in Eqs. (2.125)-(2.126) which are
valid for the straight string in the case of linear windings, also hold:

$\displaystyle\omega_{\sigma}^{2}=\omega_{\varphi}^{2}=\frac{v_{\varphi}^{2}}{R^{2}},\
\ \ v_{\varphi}^{2}\lambda_{\sigma}^{2}=(1-\dot{\rho}^{2})(2\pi)^{2}R^{2},\ \
\
n_{\sigma}^{2}=\frac{v_{\varphi}^{2}}{(1-\dot{\rho}^{2})}\frac{\rho^{2}}{R^{2}}.$
(2.173)

In general, we have

$\displaystyle
a^{2}\lambda_{\sigma}^{2}=(2\pi)^{2}R^{2}\left(\frac{\Omega^{2}}{1-\Omega^{2}}\right),$
(2.174)

so that consistency requires

$\displaystyle
v_{\varphi}^{2}=a^{2}(1-\dot{\rho}^{2})\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right)\equiv\frac{v_{\perp}^{2}}{a^{2}}.$
(2.175)

This is automatically satisfied for string loops with dynamically formed
windings, for which the following relation holds

$\displaystyle\frac{v_{\varphi}^{2}}{a^{2}v_{\sigma}^{2}}=\frac{1-\Omega^{2}}{\Omega^{2}},$
(2.176)

and which is equivalent to Eq. (2.175), since
$v_{\sigma}^{2}=1-\dot{\rho}^{2}$. Equations (2.174)-(2.176) are the circular
loop equivalents of Eqs. (2.104)-(2.106), respectively. The string is
everywhere effectively tensionless when

$\displaystyle\frac{v_{\perp}^{2}}{a^{2}}=\omega^{2}=\Omega^{2}=1/2,\ \forall
t,$ (2.177)

or equivalently when

$\displaystyle a^{2}\lambda_{\sigma}^{2}=(2\pi)^{2}R^{2},$ (2.178)

and, from Eq. (2.166), it is immediately clear that under these conditions
$\dot{\rho}=0$, $\ddot{\rho}=0\ \forall t$, as expected.
In terms of the higher-dimensional string parameters, the constants of motion
may be written as

$\displaystyle E=\mathcal{T}\frac{2\pi
a\rho}{\sqrt{1-\dot{\rho}^{2}}}\left(1+\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{\sigma}^{2}}\right)=\mathcal{T}\frac{2\pi
a\rho}{\sqrt{1-\dot{\rho}^{2}}}\Omega^{-2},$ (2.179)
$\displaystyle\Lambda^{\sigma}=\pm
2\pi\mathcal{T}\frac{\rho}{\rho(t_{i})}n_{\sigma}\times\frac{(2\pi)^{2}R^{2}}{a\lambda_{\sigma}}=\pm
2\pi\mathcal{T}a\frac{\rho^{2}}{\rho(t_{i})}\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right),$
(2.180) $\displaystyle\Lambda^{\varphi}=\pm 2\pi\mathcal{T}n_{\sigma}=\pm
2\pi\mathcal{T}\frac{a\rho}{R}\frac{\sqrt{1-\Omega^{2}}}{\Omega},$ (2.181)
$\displaystyle\Pi=\mathcal{T}\frac{2\pi
a\rho}{\sqrt{1-\dot{\rho}^{2}}}\Omega^{-1}\sqrt{\frac{1}{\Omega^{2}}-(1-\dot{\rho}^{2})\left\\{\frac{\rho^{2}}{\rho^{2}(t_{i})}\frac{(1-\Omega^{2})^{2}}{\Omega^{2}}+(1-\Omega^{2})\right\\}},$
(2.182)

and the nonzero effective tensions are given by

$\displaystyle\mathcal{T}^{\sigma}=-2\pi\mathcal{T}a\rho\sqrt{1-\dot{\rho}^{2}}\left(1-\frac{(2\pi)^{2}R^{2}}{a^{2}\lambda_{\sigma}^{2}}\right)=-2\pi\mathcal{T}a\rho\sqrt{1-\dot{\rho}^{2}}\left(\frac{2\Omega^{2}-1}{\Omega^{2}}\right),$
(2.183)
$\displaystyle\mathcal{T}^{\rho}=-\mathcal{T}\frac{a\rho\dot{\rho}^{2}}{\sqrt{1-\dot{\rho}^{2}}}\Omega^{-2}=-\dot{\rho}^{2}E.$
(2.184)

Again, $\mathcal{T}^{\varphi}=0$ since
$\mathcal{T}^{\varphi\varphi}\sqrt{-g}=0$ by the EOM for $\varphi(t,\sigma)$.
The nonzero shears between the $\sigma$\- and $\rho$-, $\sigma$\- and
$\varphi$-, and $\rho$\- and $\phi$-directions are

$\displaystyle\Sigma^{\rho\sigma}=\mp
2\pi\mathcal{T}a^{2}\rho\dot{\rho}\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right)=-a\rho(t_{i})\frac{\dot{\rho}}{\rho}\Lambda^{\sigma},$
(2.185) $\displaystyle\Sigma^{\sigma\varphi}=\mp
2\pi\mathcal{T}a^{2}\rho\sqrt{1-\dot{\rho}^{2}}\frac{\sqrt{1-\Omega^{2}}}{\Omega^{2}}=-\frac{a}{R}\sqrt{1-\dot{\rho}^{2}}\Lambda^{\varphi},$
(2.186) $\displaystyle\Sigma^{\rho\varphi}=\mp
2\pi\mathcal{T}a^{2}\rho\dot{\rho}\frac{\sqrt{1-\Omega^{2}}}{\Omega^{2}}=-a\rho(t_{i})\frac{\dot{\rho}}{\rho}\Lambda^{\varphi}.$
(2.187)

As expected, _all_ effective pressures and shears vanish locally when
$\omega^{2}=\Omega^{2}=1/2$ and under these conditions we again have
$\omega_{\sigma}^{2}\equiv\omega_{\varphi}^{2}=a^{2}/R^{2}$, as in the
straight string case, (2.85).
We now note that, imposing the boundary conditions given in Eq. (2.137) and
identifying $\lambda_{z}\leftrightarrow\lambda_{\sigma}(t_{i})$, we may also
identify

$\displaystyle P^{z}\leftrightarrow\frac{l^{\sigma}}{2\pi\rho(t_{i})},$
(2.188)

which ensures that

$\displaystyle P^{z}P_{z}=l^{\sigma}l_{\sigma}(t_{i}),$ (2.189)

and

$\displaystyle\mathcal{P}^{z}\leftrightarrow\Lambda^{\sigma}.$ (2.190)

In this case, the total $4$-momentum, given by

$\displaystyle\Pi=2\pi\mathcal{T}a\rho(t_{i})\sqrt{1+\frac{(2\pi)^{2}R^{2}n_{\sigma}^{2}}{a^{2}\lambda_{\sigma}^{2}(t_{i})}}=2\pi\mathcal{T}a\rho(t_{i})\Omega^{-1}(t_{i})$
(2.191)

for $\dot{\rho}(t_{i})=0$, is conserved when a long-string section chops off
from the network to form a loop, despite the fact that linear momentum in the
$z$-direction is ‘converted’ into angular momentum in the $\sigma$-direction.
Finally, we consider the general solution of Eq. (2.166). This was solved in
[52, 53], for the specific initial velocity $\dot{\rho}(t_{i})=0$, by
performing an Eulerian substitution of the second kind, giving

$\displaystyle\rho(t)=\rho(t_{i})\sqrt{1-\left(\frac{2\Omega^{2}(t_{i})-1}{\Omega^{4}(t_{i})}\right)\sin^{2}\left(\frac{\Omega^{2}(t_{i})}{\rho(t_{i})}(t-t_{i})\right)}.$
(2.192)

In this case the loop oscillates between two critical radii

$\displaystyle\rho_{c1}=\rho(t_{i}),\ \ \
\rho_{c2}=\left(\frac{1-\Omega^{2}(t_{i})}{\Omega^{2}(t_{i})}\right)\rho(t_{i}),$
(2.193)

performing one full oscillation with time period

$\displaystyle T_{c}=\rho(t_{i})\Omega^{-2}(t_{i})\pi.$ (2.194)

The same technique can be used to obtain the general solution for arbitrary
initial velocities: $\rho(t)$ has the same functional form as in Eq. (2.192),

$\displaystyle\rho(t)=A\sqrt{1-B\sin(C(t-t_{i})+D)},$ (2.195)

but the constants $A-D$ are given by more complicated expressions involving
$\dot{\rho}(t_{i})$,

$\displaystyle A$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\rho(t_{i})}{\sqrt{1-\dot{\rho}^{2}(t_{i})}}\Omega^{-2}(t_{i})$
$\displaystyle\times\left[1-2(1-\dot{\rho}^{2}(t_{i}))\Omega^{2}(t_{i})(1-\Omega^{2}(t_{i}))+\sqrt{1-4(1-\dot{\rho}^{2}(t_{i}))\Omega^{2}(t_{i})(1-\Omega^{2}(t_{i}))}\right]^{\frac{1}{2}},$
(2.196a) $\displaystyle B$
$\displaystyle=\frac{2\sqrt{1-4(1-\dot{\rho}^{2}(t_{i}))\Omega^{2}(t_{i})(1-\Omega^{2}(t_{i}))}}{1-2(1-\dot{\rho}^{2}(t_{i}))\Omega^{2}(t_{i})(1-\Omega^{2}(t_{i}))+\sqrt{1-4(1-\dot{\rho}^{2}(t_{i}))\Omega^{2}(t_{i})(1-\Omega^{2}(t_{i}))}},$
(2.196b) $\displaystyle C$
$\displaystyle=\frac{\sqrt{1-\dot{\rho}^{2}(t_{i}))}}{\rho(t_{i})\Omega^{-2}(t_{i})},$
(2.196c) $\displaystyle D$
$\displaystyle=-\sin^{-1}(B^{-1/2}\sqrt{1-\rho^{2}(t_{i})/A^{2}}),$ (2.196d)

In this case, the string oscillates with time period

$\displaystyle
T_{c}=\frac{\rho(t_{i})}{\sqrt{1-\dot{\rho}^{2}(t_{i})}}\Omega^{-2}(t_{i})\pi.$
(2.197)

The second critical radius $\rho_{c2}$ may be deduced from Eq. (2.195), but,
since it is also a complicated function of the initial conditions we do not
quote it here, explicitly, for the sake of brevity. In the limit
$\Omega^{2}(t_{i})\rightarrow 1$ ($\dot{\rho}^{2}(t_{i})\geq 0$), we recover
the previous solution for unwound circular loops, Eq. (2.139) and for
$\dot{\rho}^{2}(t_{i})\rightarrow 0$ ($\Omega^{2}(t_{i})<1$), we recover Eq.
(2.192).

###  2.4 Noncircular $F$-string loops

In this section, we consider arbitrary planar loops, both with and without
higher-dimensional windings. Since the EOM for this (most general) case are
rather complicated in our chosen gauge, it is instructive to first review the
analysis of noncircular loops without windings. This is done in Sect. 2.4.1.
Arbitrary planar loops with higher-dimensional windings are considered in
detail in Sec. 2.4.2 and the general solution to the higher-dimensional part
of the string EOM is given in terms of $(3+1)$-dimensional observables.

####  2.4.1 Noncircular $F$-string loops in $(3+1)$ dimensions

The embedding for an arbitrary planar loop with no higher-dimensional
windings, in cylindrical polar coordinates, is

$\displaystyle X^{I}(\tau,\sigma)\equiv
X^{I}(t,\sigma)=\left(t(\tau)=\zeta\tau,x=0,\rho(t,\sigma),\sigma,\varphi=0\right).$
(2.198)

The action is

$\displaystyle S=-\mathcal{T}a^{2}\int{\rm d}t{\rm
d}\sigma\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}},$
(2.199)

which may be written in the standard form using

$\displaystyle\frac{{\rm d}l}{{\rm
d}\sigma}=a\sqrt{\rho^{2}+(\partial_{\sigma}\rho)^{2}},\ \ \
\frac{v_{\perp}^{2}}{a^{2}}=\dot{\rho}^{2}\left(1-\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\right).$
(2.200)

Written terms of the energy-momentum tensor, the EOM are the same as for
circular loops without windings, Eqs. (2.131a)-(2.131c), but the components of
$\mathcal{T}^{I}{}_{J}\sqrt{-g}$ are different:

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}=\mathcal{T}\frac{\rho^{2}+(\partial_{\sigma}\rho)^{2}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}},\
\ \
\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}=-\mathcal{T}\frac{\dot{\rho}\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}},$
(2.201a)
$\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}=\mathcal{T}a^{2}\frac{\rho^{2}\dot{\rho}\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}},\
\ \
\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}=\mathcal{T}a^{2}\frac{(1-\dot{\rho}^{2})\rho^{2}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}},$
(2.201b)
$\displaystyle\mathcal{T}^{0}{}_{\rho}\sqrt{-g}=-\mathcal{T}a^{2}\frac{\rho^{2}\dot{\rho}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}},\
\ \
\mathcal{T}^{\sigma}{}_{\rho}\sqrt{-g}=\mathcal{T}a^{2}\frac{\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}},$
$\displaystyle\mathcal{T}^{\sigma\sigma}\sqrt{-g}=-\mathcal{T}a^{2}\frac{(1-\dot{\rho}^{2})}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}}.$
(2.201c)

By direct substitution, it is straightforward to show that each EOM is
equivalent to

$\displaystyle(1-\dot{\rho}^{2})\left(1-\frac{\partial^{2}_{\sigma}\rho}{\rho}\right)+\left(1+\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}}\right)\rho\ddot{\rho}+2\left(\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}}-\frac{\dot{\rho}}{\rho}\partial_{\sigma}\rho\partial_{\sigma}\dot{\rho}\right)=0,$
(2.202)

which again expresses the conservation of energy, since the only nonzero
constant of motion is the Hamiltonian. Formally, we may write down expressions
for the constants of motion, integrated pressures and integrated shears, but
these cannot be evaluated without adopting a more specific ansatz for
$\rho(t,\sigma)$. It is sufficient for our purposes to note that in general
$E\neq-\mathcal{T}^{\sigma}$, and that there is no way to obtain a static
tensionless solution for an unwound string.

####  2.4.2 Noncircular $F$-string loops with higher-dimensional windings

In cylindrical polars, the general embedding for arbitrary planar loops with
higher-dimensional windings is

$\displaystyle X^{I}(\tau,\sigma)\equiv
X^{I}(t,\sigma)=\left(t(\tau)=\zeta\tau,x=0,\rho(t,\sigma),\sigma,\varphi(\tau,\sigma)\equiv\varphi(t,\sigma)\right).$
(2.203)

The action is

$\displaystyle S=-\mathcal{T}\int{\rm d}t{\rm
d}\sigma\bigl{[}a^{2}(1-\dot{\rho}^{2})(a^{2}\rho^{2}+R^{2}(\partial_{\sigma}\varphi)^{2})+a^{4}(\partial_{\sigma}\rho)^{2}$
$\displaystyle-(a^{2}\rho^{2}+a^{2}(\partial_{\sigma}\rho)^{2})R^{2}\dot{\varphi}^{2}+2a^{2}\dot{\rho}\partial_{\sigma}\rho
R^{2}\dot{\varphi}\partial_{\sigma}\varphi\bigr{]},$ (2.204)

which may be written in the standard form by setting

$\displaystyle\frac{{\rm d}l}{{\rm d}\sigma}$ $\displaystyle=$
$\displaystyle\sqrt{a^{2}\rho^{2}+a^{2}(\partial_{\sigma}\rho)^{2}+R^{2}(\partial_{\sigma}\varphi)^{2}},$
$\displaystyle\frac{v_{\perp}^{2}}{a^{2}}$ $\displaystyle=$
$\displaystyle\dot{\rho}^{2}+\frac{(\rho^{2}+(\partial_{\sigma}\rho)^{2})R^{2}\dot{\varphi}^{2}-a^{2}\dot{\rho}^{2}(\partial_{\sigma}\rho)^{2}-2\dot{\rho}\partial_{\sigma}\rho
R^{2}\dot{\varphi}\partial_{\sigma}\varphi}{a^{2}\rho^{2}+a^{2}(\partial_{\sigma}\rho)^{2}+R^{2}(\partial_{\sigma}\varphi)^{2}}.$
(2.205)

The EOM are

$\displaystyle\partial_{0}(T^{0}{}_{0}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (2.206a)
$\displaystyle\partial_{0}(T^{0}{}_{\sigma}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\sigma}\sqrt{-g})$
$\displaystyle=0,$ (2.206b)
$\displaystyle\partial_{0}(T^{0}{}_{\rho}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\rho}\sqrt{-g})-\frac{1}{2}\partial_{\rho}g_{\sigma\sigma}T^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=0,$ (2.206c)
$\displaystyle\partial_{0}(T^{0}{}_{\varphi}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\varphi}\sqrt{-g})$
$\displaystyle=0,$ (2.206d)

and Eq. (2.206d) may be written in terms of $(-\gamma)$ and its derivatives as

$\displaystyle(-\gamma)\biggl{[}(1-\dot{\rho}^{2})\partial_{\sigma}^{2}\varphi-(\rho^{2}+(\partial_{\sigma}\rho)^{2})\ddot{\varphi}-2(\rho\dot{\rho}+\partial_{\sigma}\rho\partial_{\sigma}\dot{\rho})\dot{\varphi}$
$\displaystyle(\ddot{\rho}\partial_{\sigma}\rho+\dot{\rho}\partial_{\sigma}\dot{\rho})\partial_{\sigma}\varphi+(\dot{\rho}\partial^{2}_{\sigma}\rho+\partial_{\sigma}\rho\partial_{\sigma}\dot{\rho})\dot{\varphi}-2\dot{\rho}\partial_{\sigma}\dot{\rho}\partial_{\sigma}\varphi+2\dot{\rho}\partial_{\sigma}\rho\partial_{\sigma}\dot{\varphi}\biggr{]}$
$\displaystyle-\frac{1}{2}\frac{\partial(-\gamma)}{\partial
t}[\dot{\rho}\partial_{\sigma}\rho\partial_{\sigma}\varphi-(\rho^{2}+(\partial_{\sigma}\rho)^{2})\dot{\varphi}]-\frac{1}{2}\frac{\partial(-\gamma)}{\partial\sigma}[(1-\dot{\rho}^{2})\partial_{\sigma}\varphi+\dot{\rho}\partial_{\sigma}\rho\dot{\varphi}]=0.$
(2.207)

Writing out $(-\gamma)$ and its derivatives explicitly, this expression is
even more complicated than Eq. (2.148). However, bearing in mind our previous
results, we may guess that Eq. (2.4.2) can be written in a compact form as

$\displaystyle(1-\dot{\rho}^{2})(\partial_{\sigma}\varphi)^{2}-(\rho^{2}+(\partial_{\sigma}\rho)^{2})\dot{\varphi}^{2}+2\dot{\rho}\partial_{\sigma}\rho\dot{\varphi}\partial_{\sigma}\varphi=0,$
(2.208)

and that imposing this condition on the remaining EOM, Eqs. (2.206a)-(2.206c),
will lead to a self-consistent set of equations. In fact, it is far simpler to
work directly with Eq. (2.206d) as the EOM in $\phi(t,\sigma)$ and to impose
the constraint in Eq. (2.208) _before_ multiplying out the derivatives. It is
then straightforward to show that, subject to Eq. (2.208),

$\displaystyle\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\mp\mathcal{T}R^{2}\partial_{\sigma}\varphi,$
$\displaystyle\mathcal{T}^{\sigma}{}_{\varphi}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\pm\mathcal{T}R^{2}\dot{\varphi},$ (2.209)

so that Eq. (2.206d) reduces to an identity. Thus, Eq. (2.4.2), which is
equivalent to Eq. (2.206d), is automatically satisfied by the imposition of
the constraint, Eq. (2.208).
As in the previous cases considered, for long straight strings and circular
loops with higher-dimensional windings, imposing the constraint which
satisfies the EOM in $\varphi(t,\sigma)$ implies that all higher-dimensional
terms (i.e. terms dependent on $\varphi(t,\sigma)$ or its derivatives) cancel
in $(-\gamma)$ at the level of the EOM. This condition also ensures
$\mathcal{T}^{\varphi\varphi}\sqrt{-g}=0$, so that there is no effective
pressure in either the $\varphi$-direction or the ‘$R$-direction’ (i.e. ${\rm
d}R=0$). Physically, this means that there is no effective pressure which can
act to change the radius of the windings in the compact space.
For strings compactified on an genuine $M^{4}\times S^{1}$ manifold, where the
$S^{1}$ is of constant radius $R$, it is physically meaningless to talk about
the string moving in the ‘$R$-direction’. However, the metric (2.1) is also
valid as an effective metric for embeddings in more complex internal spaces,
where the string wraps cycles of constant radius in the compact dimensions.
For example, in [52, 53] and [51], strings wrapping great circles of the
$S^{3}$ internal manifold which regularises the conifold at the tip of the
Klebanov-Strassler geometry [54] were considered. In such a geometry, the
effective winding radius of the string may vary as a function of both $t$ and
$\sigma$ and, since the windings are not topologically stabilised, they must
be stabilised (if at all), dynamically.
As we have seen, for circular strings with higher-dimensional windings,
nonzero effective pressure in the $\rho$-direction is associated with nonzero
$\dot{\rho}$, but both these quantities become zero when there is no effective
pressure in the $\sigma$-direction, $\mathcal{T}^{\sigma\sigma}\sqrt{-g}=0$.
Though we did not explicitly include a term proportional to ${\rm d}R^{2}$ in
the metric considered in this paper, or allow $R$ to be a function of the
world-sheet coordinates in the embedding, we made such a restriction purely
for simplicity and, in principle, the radius of the windings can evolve, in an
arbitrary internal manifold, as a function of both space and time. Thus,
nonzero $\dot{R}$ would be associated with nonzero
$\mathcal{T}^{\varphi\varphi}\sqrt{-g}$, just as nonzero $\dot{\rho}$ is
associated with nonzero $\mathcal{T}^{\sigma\sigma}\sqrt{-g}$. Viewed in this
way, the condition (2.208) and the EOM for $\varphi(t,\sigma)$ and
$\varphi(t,z)$ in Secs. 4.2 and 3.2, respectively, result from, and are
necessary conditions for, our initial assumption of a constant winding radius.
Interestingly, in each higher-dimensional case considered so far, the
constraint equation which ensures $\mathcal{T}^{\varphi\varphi}\sqrt{-g}=0$
takes the form of a quadratic equation in
$\dot{\varphi}/\partial_{\sigma}\varphi$ (or equivalently
$\partial_{\sigma}\varphi/\dot{\varphi}$), whose descriminant is directly
proportional to the value of $(-\gamma)$ _after_ the constraint has been
applied, i.e.
$(-\gamma)\propto(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}$, or
the equivalent thereof. Thus, from Eq. (2.208), we have that

$\displaystyle\frac{\dot{\varphi}}{\partial_{\sigma}\varphi}=\frac{\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}=\frac{(1-\dot{\rho}^{2})}{-\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}$
(2.210)

which reduces to the previous EOM in $\phi(t,\sigma)$ for circular loops in
the limit $(\partial_{\sigma}\rho)^{2}\rightarrow 0$ and to the EOM in
$\varphi(t,z)$ for long straight strings when, in addition,
$\dot{\rho}\rightarrow 0$ and $2\pi\rho=\Delta$.
We must now test the consistency of the Euler-Lagrange equations subject to
this constraint. After applying Eqs. (2.208)/(2.210) to the components of the
energy-momentum tensor, we have:

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=\mathcal{T}a^{2}\frac{\rho^{2}+(\partial_{\sigma}\rho)^{2}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\omega^{-2},$
$\displaystyle\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}$
$\displaystyle=-\mathcal{T}a^{2}\frac{\dot{\rho}\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\omega^{-2}\mp\mathcal{T}a^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.211a) $\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}$
$\displaystyle=\mathcal{T}a^{2}\frac{\rho^{2}\dot{\rho}\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\omega^{-2}\pm\mathcal{T}a^{2}\rho^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
$\displaystyle\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}$
$\displaystyle=\mathcal{T}a^{2}\frac{(1-\dot{\rho}^{2})\rho^{2}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\left[1-\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})}\right],$
(2.211b) $\displaystyle\mathcal{T}^{0}{}_{\rho}\sqrt{-g}$
$\displaystyle=-\mathcal{T}a^{2}\frac{\rho^{2}\dot{\rho}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\omega^{-2}\mp\mathcal{T}a^{2}\partial_{\sigma}\rho\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
$\displaystyle\mathcal{T}^{\sigma}{}_{\rho}\sqrt{-g}$
$\displaystyle=\mathcal{T}a^{2}\frac{\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\left[1-\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})}\right]$
$\displaystyle\pm\mathcal{T}a^{2}\dot{\rho}\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}))^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})},$
$\displaystyle\mathcal{T}^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=-\mathcal{T}\frac{(1-\dot{\rho}^{2})}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\left[1-\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})}\right]$
(2.211c) $\displaystyle\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}$
$\displaystyle=\mp\mathcal{T}aR\sqrt{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\frac{\sqrt{1-\omega^{2}}}{\omega},$
$\displaystyle\mathcal{T}^{\sigma}{}_{\varphi}\sqrt{-g}$
$\displaystyle=\pm\mathcal{T}aR\left(\frac{\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}{\sqrt{\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\right)\frac{\sqrt{1-\omega^{2}}}{\omega},$
(2.211d)

where

$\displaystyle\omega^{-2}(t,\sigma)=\frac{a^{2}(\rho^{2}+(\partial_{\sigma}\rho)^{2})+R^{2}(\partial_{\sigma}\varphi)^{2}}{a^{2}(\rho^{2}+(\partial_{\sigma}\rho)^{2})}.$
(2.212)

In appropriate limits, these reduce to the equivalent expressions given in
Secs. 2.2-2.3. For noncircular loops, the normalized components of the current
density are defined as

$\displaystyle\mathcal{J}^{0}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi}\frac{\dot{\rho}(t_{i},\sigma)\partial_{\sigma}\rho(t_{i},\sigma)\pm\sqrt{(1-\dot{\rho}^{2}(t_{i},\sigma))\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}}{\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}$
$\displaystyle\times$
$\displaystyle\frac{\rho^{2}+(\partial_{\sigma}\rho)^{2}}{\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\dot{\varphi},$
$\displaystyle\mathcal{J}^{\sigma}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi}\frac{\dot{\rho}(t_{i},\sigma)\partial_{\sigma}\rho(t_{i},\sigma)\pm\sqrt{(1-\dot{\rho}^{2}(t_{i},\sigma))\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}}{\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}$
(2.213) $\displaystyle\times$
$\displaystyle\frac{\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\partial_{\sigma}\varphi,$

so that using the EOM in $\varphi$ gives

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\pm\frac{e}{2\pi}\frac{\dot{\rho}(t_{i},\sigma)\partial_{\sigma}\rho(t_{i},\sigma)\pm\sqrt{(1-\dot{\rho}^{2}(t_{i},\sigma))\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}}{\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}\partial_{\sigma}\varphi$
$\displaystyle=$
$\displaystyle-\mathcal{J}^{\sigma}\sqrt{-g}=\mp\frac{e}{2\pi}\frac{\dot{\rho}(t_{i},\sigma)\partial_{\sigma}\rho(t_{i},\sigma)\pm\sqrt{(1-\dot{\rho}^{2}(t_{i},\sigma))\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}}{\rho^{2}(t_{i},\sigma)+(\partial_{\sigma}\rho(t_{i},\sigma))^{2}}\dot{\varphi}.$
(2.214)

The components of the $4$-current may again be expressed in terms of
$\mathcal{T}^{0}{}_{\varphi}\sqrt{-g}$ and
$\mathcal{T}^{\sigma}{}_{\varphi}\sqrt{-g}$, now given by Eq. (2.211d), and it
is straightforward to show that the current conservation equation is
equivalent to Eq. (2.206d).
It is tempting to think, based on the complexity of the expressions in Eqs.
(2.211a)-(2.211c), that Eqs. (2.206a)-(2.206c) are intractably complex, at
least analytically. However, as we will show in the remainder of this section,
careful treatment of the remaining EOM will allow us to write the general
solution of $\varphi(t,\sigma)$ in terms of $\rho(t,\sigma)$ and its first and
second derivatives in $t$ and $\sigma$. As a first step towards demonstrating
the self-consistency of Eqs. (2.206a)-(2.206c), we note that each contains
terms in $\omega$, $\dot{\omega}$ and $\partial_{\sigma}\omega$. In the case
of circular loops, we had $\partial_{\sigma}\omega=0$, so that the EOM
contained terms in only $\omega$ and $\dot{\omega}$. However, we were able to
eliminate the latter via appropriate manipulation, leaving us with a single
self-consistent EOM in $\rho$ and $\omega$, in addition to the constraint on
$\varphi$ (which also allowed $\varphi$ to be solved directly in terms of
$\rho$).
Even in the more general case, we still have three EOM and two quantities
(namely $\dot{\omega}$ and $\partial_{\sigma}\omega$) we wish to eliminate, so
this poses no problem. One interesting aspect of the circular string case was
that the EOM in $\rho$ was in fact redundant, being simply a ‘composite’ of
the expressions for the conservation of energy and momentum, which are
equivalent to the EOM in $t$, $\sigma$ and $\varphi$ in our chosen gauge. As
we shall see, the same is true even for noncircular loops, so that one of the
(extremely complex) EOM above may be immediately ignored.
We begin the elimination of $\dot{\omega}$ and $\partial_{\sigma}\omega$ by
first writing each of the components of $\mathcal{T}^{I}{}_{J}\sqrt{-g}$ in
Eqs. (2.211a)-(2.211b) as the sum of two terms, one independent of the factor
$(\omega^{-2}-1)$ and one directly proportional to $(\omega^{-2}-1)$. Thus, we
set

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}=\beta+\beta(\omega^{-2}-1),\ \ \
\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}=\gamma+\delta(\omega^{-2}-1),$ (2.215a)
$\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}=\epsilon+\zeta(\omega^{-2}-1),\
\ \ \ \mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}=\eta+\theta(\omega^{-2}-1),$
(2.215b)

where the values of $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$
and $\theta$ are obtained by comparison of Eqs. (2.215a)-(2.215b) with Eqs.
(2.211a)-(2.211b), i.e.

$\displaystyle\beta=\frac{\rho^{2}+(\partial_{\sigma}\rho)^{2}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}},$
(2.216) $\displaystyle\gamma=\delta\mp 1,\ \ \
\delta=-\frac{\dot{\rho}\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\pm
1.$ (2.217) $\displaystyle\epsilon=-\rho^{2}(\delta\mp 1),\ \ \
\zeta=-\rho^{2}\delta,$ (2.218)
$\displaystyle\eta=\frac{\rho^{2}(1-\dot{\rho})^{2}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}},\
\ \ \theta=-\frac{\rho^{2}\delta^{2}}{\beta}.$ (2.219)

Substituting from Eqs. (2.215a)-(2.215b) into Eqs. (2.206a)-(2.206b),
rearranging to make $\partial_{t}(\omega^{-2}-1)$ the subject and equating the
results, gives

$\displaystyle(\zeta\delta-\beta\theta)\partial_{\sigma}(\omega^{-2}-1)+[\beta(\dot{\zeta}+\partial_{\sigma}\theta)-\zeta(\dot{\beta}+\partial_{\sigma}\delta)](\omega^{-2}-1)+\beta(\dot{\epsilon}+\partial_{\sigma}\eta)-\zeta(\dot{\beta}+\partial_{\sigma}\gamma)=0.$

Then, substituting in for $\zeta$, $\delta$, $\beta$ and $\theta$ from Eqs.
(2.216)-(2.219), it may be shown that

$\displaystyle\zeta\delta-\beta\theta=0,$ (2.221)

and an EOM involving only $\omega$ can be obtained:

$\displaystyle[\beta(\dot{\zeta}+\partial_{\sigma}\theta)-\zeta(\dot{\beta}+\partial_{\sigma}\delta)](1-\omega^{2})+[\beta(\dot{\epsilon}+\partial_{\sigma}\eta)-\zeta(\dot{\beta}+\partial_{\sigma}\gamma)]\omega^{2}=0.$
(2.222)

Next, we note that substituting for $(\dot{\epsilon}+\partial_{\sigma}\eta)$,
$(\dot{\beta}+\partial_{\sigma}\gamma)$, $\beta$ and $\zeta$, the terms in the
second set of square brackets in Eq. (2.222) are proportional to the
expression on the left-hand-side of the EOM for noncircular strings _without_
higher-dimensional windings, Eq. (2.202). This is to be expected since, in the
limit $\omega^{2}\rightarrow 1$, consistency requires that we must recover the
EOM for unwound strings. Specifically, we have that

$\displaystyle\beta(\dot{\epsilon}+\partial_{\sigma}\eta)-\zeta(\dot{\beta}+\partial_{\sigma}\gamma)=\left[\beta\partial_{\sigma}\rho-\rho^{2}\dot{\rho}\delta\right]\times\frac{\rho^{3}}{[(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}]^{\frac{3}{2}}}\chi,$
(2.223)

where

$\displaystyle\chi=(1-\dot{\rho}^{2})\left(1-\frac{\partial^{2}_{\sigma}\rho}{\rho}\right)+\left(1+\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}}\right)\rho\ddot{\rho}+2\left(\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}}-\frac{\dot{\rho}}{\rho}\partial_{\sigma}\rho\partial_{\sigma}\dot{\rho}\right).$
(2.224)

The first set of square brackets in Eq. (2.222) may also be written purely in
terms of the quantities $\beta$ and $\delta$, i.e.

$\displaystyle\beta(\dot{\zeta}+\partial_{\sigma}\theta)-\zeta(\dot{\beta}+\partial_{\sigma}\delta)$
$\displaystyle=$
$\displaystyle\rho^{2}(\delta\dot{\beta}-\beta\dot{\delta})-2\delta(\rho\dot{\rho}\beta+\rho\partial_{\sigma}\rho\delta)+\frac{\rho^{2}\delta}{\beta}(\delta\partial_{\sigma}\beta-\beta\partial_{\sigma}\delta),$
(2.225)

so that the final remaining independent EOM (involving only derivatives of
$\rho$ and powers of $\omega$, but not $\dot{\omega}$ or
$\partial_{\sigma}\omega$), may be written in a _relatively_ compact form as

$\displaystyle\chi$ $\displaystyle+$
$\displaystyle\frac{\rho^{3}(\beta\partial_{\sigma}\rho-\rho^{2}\dot{\rho}\delta)}{[(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}]^{\frac{3}{2}}}\left[\rho^{2}(\delta\dot{\beta}-\beta\dot{\delta})-2\delta(\rho\dot{\rho}\beta+\rho\partial_{\sigma}\rho\delta)+\frac{\rho^{2}\delta}{\beta}(\delta\partial_{\sigma}\beta-\beta\partial_{\sigma}\delta)\right]\left(\frac{1-\omega^{2}}{\omega^{2}}\right)$
(2.226) $\displaystyle=$ $\displaystyle 0,$

where we make use of the definitions for $\chi$, $\beta$ and $\delta$, given
in Eqs. (2.224) and (2.216)-(2.217). Hence, the term proportional to
$(1-\omega^{2})/\omega^{2}$ quantifies the contribution of the higher-
dimensional motion to the macroscopic dynamics of the string. In the limit
$\partial_{\sigma}\rho\rightarrow 0$, Eq. (2.226) reduces to Eq. (2.166), as
required. It may be shown that $\omega^{2}=1/2$ corresponds to the tensionless
case, for which $\dot{\rho}=0$, $\partial_{\sigma}\dot{\rho}=0$ and
$\ddot{\rho}=0$ for all $t$, by directly substituting each of these conditions
into Eq. (2.226). The EOM then reduces to $\omega^{2}=1-\omega^{2}$, which is
satisfied if and only if $\omega^{2}=1/2$.
Thus, Eqs. (2.226) and (2.208) alone are sufficient to completely specify the
dynamics of the string. The latter arises directly from the conservation of
momentum in the compact space (the $\varphi$-direction) and the former
combines the expressions for energy and momentum conservation in the whole
space-time. As in the circular case, the EOM in $\rho$ is again redundant and
was not used in the derivation of Eq. (2.226).
Finally, we may solve for $\varphi(t,\sigma)$ in terms of $\rho(t,\sigma)$ and
its derivatives. For the sake of notational simplicity, we first rewrite Eq.
(2.226) as

$\displaystyle-X\omega^{2}+Y(1-\omega^{2})=0,$ (2.227)

where the factors $X$ and $Y$ are independent of $\partial_{\sigma}\varphi$
or, equivalently

$\displaystyle\left(\frac{X}{Y}\right)=\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.228)

where the factor $(1-\omega^{2})/\omega^{2}$ is also independent of
$\partial_{\sigma}\varphi$. Using the definition of $\omega^{2}(t,\sigma)$
(2.212), we may also rewrite $(\partial_{\sigma}\varphi)^{2}$ as

$\displaystyle(\partial_{\sigma}\varphi)^{2}=\frac{a^{2}}{R^{2}}(\rho^{2}+(\partial_{\sigma}\rho)^{2})\left(\frac{1-\omega^{2}}{\omega^{2}}\right).$
(2.229)

This expression has physical significance: as we we will see, it allows us to
interpret the EOM in $\varphi(t,\sigma)$ as dispersion relation, as in Secs.
2.1 and 2.2. From Eq. (2.210) we then have

$\displaystyle\dot{\varphi}^{2}=\frac{a^{2}}{R^{2}}\left(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}\right)^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.230)

which allows us to reconstruct the required form of $\varphi(t,\sigma)$:

For circular strings ($\partial_{\sigma}\rho=0$), Eqs. (2.228) and (2.227)
give

$\displaystyle
n_{\sigma}^{2}=\frac{a^{2}}{R^{2}}\rho^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right)=\frac{a^{2}}{R^{2}}\rho^{2}\left(\frac{1-\dot{\rho}^{2}+\rho\ddot{\rho}}{1-\dot{\rho}^{2}-\rho\ddot{\rho}}\right),$
(2.232)

which is equivalent to Eq. (2.166), using the expression for $\omega^{2}(t)$
in Eq. (2.168). This motivates the following definition for noncircular loops

$\displaystyle(n_{\sigma}^{\rm eff})^{2}=\frac{(v_{\varphi}^{\rm
eff})^{2}(l_{\sigma}^{\rm eff})^{2}}{(v_{\sigma}^{\rm
eff})^{2}(2\pi)^{2}R^{2}}=(\partial_{\sigma}\varphi)^{2},$ (2.233)

where

$\displaystyle(l_{\sigma}^{\rm
eff})^{2}=(2\pi)^{2}(\rho^{2}+(\partial_{\sigma}\rho)^{2}),$ (2.234)

and

$\displaystyle\frac{(v_{\varphi}^{\rm eff})^{2}}{a^{2}(v_{\sigma}^{\rm
eff})^{2}}=\left(\frac{1-\omega^{2}}{\omega^{2}}\right).$ (2.235)

Together with Eqs. (2.233)-(2.235), the following definitions then form a
self-consistent set:

$\displaystyle(l_{\sigma}^{\rm eff})^{2}=(n_{\sigma}^{\rm
eff})^{2}(\lambda_{\sigma}^{\rm eff})^{2},\ \ \ (v_{\varphi}^{\rm
eff})^{2}(\lambda_{\sigma}^{\rm eff})^{2}=(v_{\sigma}^{\rm
eff})^{2}(2\pi)^{2}R^{2},$ (2.236) $\displaystyle(k_{\sigma}^{\rm
eff})^{2}=\frac{(2\pi)^{2}}{(\lambda_{\sigma}^{\rm eff})^{2}},$ (2.237)
$\displaystyle(\omega_{\sigma}^{\rm eff})^{2}=(v_{\sigma}^{\rm
eff})^{2}(k_{\sigma}^{\rm eff})^{2}\equiv(\omega_{\varphi}^{\rm
eff})^{2}=\frac{(v_{\varphi}^{eff})^{2}}{R^{2}}=\dot{\varphi}^{2}.$ (2.238)

Combining Eqs. (2.235) and (2.236), we then have

$\displaystyle a^{2}(\lambda_{\sigma}^{\rm
eff})^{2}=(2\pi)^{2}R^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right).$
(2.239)

We would like to be able to verify that the expressions for $(l_{\sigma}^{\rm
eff})^{2}$, $(n_{\sigma}^{\rm eff})^{2}$, $(v_{\sigma}^{\rm eff})^{2}$,
$(v_{\varphi}^{\rm eff})^{2}$, $(\lambda_{\sigma}^{\rm eff})^{2}$ and
$(\omega_{\sigma}^{\rm eff})^{2}\equiv(\omega_{\varphi}^{\rm eff})^{2}$ above,
reduce to those for $l_{\sigma}^{2}$, $n_{\sigma}^{2}$, $v_{\sigma}^{2}$,
$v_{\varphi}^{2}$ and $\omega_{\sigma}^{2}\equiv\omega_{\varphi}^{2}$ for
circular strings, and to those for $(l_{z}^{\rm eff})^{2}$, $(n_{z}^{\rm
eff})^{2}$, $(v_{z}^{\rm eff})^{2}$, $(v_{\varphi}^{\rm eff})^{2}$,
$(\lambda_{z}^{\rm eff})^{2}$ and $(\omega_{z}^{\rm
eff})^{2}\equiv(\omega_{\varphi}^{\rm eff})^{2}$ for straight strings, in
appropriate limits. However, from Eqs. (2.233)-(2.239) alone, we do not have
enough information to write down explicit expressions for
$(\omega_{\sigma}^{\rm eff})^{2}$, $(v_{\sigma}^{\rm eff})^{2}$, or
$(v_{\varphi}^{\rm eff})^{2}$. The missing piece comes from identifying
$(\omega_{\sigma}^{\rm eff})^{2}$ with the integrand of the integral over
${\rm d}t$ in Eq. (LABEL:phi_last*). Hence, we have that

$\displaystyle\varphi(t,\sigma)=\int n_{\sigma}^{\rm eff}{\rm
d}\sigma+\int\omega_{\sigma}^{\rm eff}{\rm d}t,$ (2.240)

where

$\displaystyle(n_{\sigma}^{\rm
eff})^{2}=\frac{a^{2}}{R^{2}}(\rho^{2}+(\partial_{\sigma}\rho)^{2})\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.241) $\displaystyle(\omega_{\sigma}^{\rm
eff})^{2}=\frac{a^{2}}{R^{2}}\left(\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})^{2}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\right)\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(2.242)

and the sign of $\omega_{\sigma}^{\rm eff}$ relative to $n_{\sigma}^{\rm eff}$
depends on the sign of $\dot{\varphi}$ relative to $\partial_{\sigma}\varphi$.
It may then be shown explicitly that the expressions for $(l_{\sigma}^{\rm
eff})^{2}$, $(n_{\sigma}^{\rm eff})^{2}$, $(v_{\sigma}^{\rm eff})^{2}$,
$(v_{\varphi}^{\rm eff})^{2}$, $(\lambda_{\sigma}^{\rm eff})^{2}$ and
$(\omega_{\sigma}^{\rm\rm eff})^{2}\equiv(\omega_{\varphi}^{\rm eff})^{2}$,
defined in Eqs. (2.233)-(2.234) and (2.236)-(2.238), do in fact reduce to
their counterparts for circular strings and long straight strings in the
limits $\partial_{\sigma}\rho\rightarrow 0$ and
$(2\pi)^{2}\rho\rightarrow\Delta$, respectively. Yet again, we see that the
tensionless condition, $\omega^{2}=1/2$, corresponds to uniform rotation of
the windings, with angular frequency $\omega_{\sigma}^{2}=a^{2}/R^{2}$, and
the existence of a constant local wavelength, $a^{2}(\lambda_{\sigma}^{\rm
eff})^{2}=a^{2}\lambda_{\sigma}=(2\pi)^{2}R^{2}$. Using Eqs. (2.208) and
(2.227), we may also express the perpendicular velocity of the string as

$\displaystyle\frac{v_{\perp}^{2}}{a^{2}}=(1-\dot{\rho}^{2})(1-\omega^{2})+\dot{\rho}^{2}\left(1-\frac{(\partial_{\sigma}\rho)^{2}\omega^{2}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\right),$
(2.243)

which may be seen as a straightforward generalisation of the cases considered
previously. As in Sec. 2.4.1, formally we may write down exact expressions for
the constants of motion and integrated pressures and shears, but we are unable
to evaluate them explicitly without further specifying the ansatz for
$\rho(t,\sigma)$. It is therefore sufficient for our purposes to write down
the remaining nonzero components of the energy-momentum tensor, from which, in
conjunction with Eqs. (2.211a)-(2.211d), all physical properties of the string
can be calculated:

$\displaystyle\mathcal{T}^{\rho\rho}\sqrt{-g}$
$\displaystyle=\frac{-\mathcal{T}[(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})-\rho^{2}]}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}$
$\displaystyle+\frac{\mathcal{T}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\biggl{[}\dot{\rho}^{2}(\rho^{2}+(\partial_{\sigma}\rho)^{2})-2\dot{\rho}\partial_{\sigma}\rho(\dot{\rho}\partial_{\sigma}\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})$
$\displaystyle+\frac{(\partial_{\sigma}\rho)^{2}(\dot{\rho}\partial_{\sigma}\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})^{2}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\biggr{]}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
$\displaystyle\mathcal{T}^{\rho\varphi}\sqrt{-g}=\pm\mathcal{T}\frac{a}{R}\frac{\rho^{2}\dot{\rho}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\left(\frac{\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}}{\sqrt{\rho^{2}+(\partial_{\sigma}\rho)^{2}}}\right)\frac{\sqrt{1-\omega^{2}}}{\omega}.$
(2.244)

From Eqs. (2.211a)-(2.211d) and (2.244), we see explicitly that _all_
effective pressures and shears vanish locally for $\omega^{2}=1/2$, as
expected.
Finally we note that, though it is beyond the scope of the current work, it
would be useful to define a parameter $\alpha(t)$, the generalization of the
constant $\alpha$ defined for long straight strings, which quantifies the
degree of nonlinearity in the windings, at a given moment in time, for
noncircular loops. By analogy with our previous results, we could then use
this to determine the spatially-averaged values of the parameters defined in
Eqs. (2.233)-(2.234) and (2.236)-(2.238), namely $\langle
l_{\sigma}^{2}\rangle(t)$, $\langle n_{\sigma}^{2}\rangle(t)$, $\langle
v_{\sigma}^{2}\rangle(t)$, $\langle v_{\varphi}^{2}\rangle(t)$,
$\langle\lambda_{\sigma}^{2}\rangle(t)$ and
$\langle\omega_{\sigma}^{2}\rangle(t)\equiv\langle\omega_{\varphi}^{2}\rangle(t)$,
as well as a generalized expression for $\Omega^{2}(t)$, which is valid for
arbitrary loop configurations. Based on our previous findings, we expect the
condition of zero average tension to correspond to $\Omega^{2}=1/2$, for which
$a^{2}\langle\lambda_{\sigma}^{2}\rangle=(2\pi)^{2}R^{2}$ and
$\langle\omega_{\sigma}^{2}\rangle=a^{2}/R^{2}$ for all $t$. Such a definition
would allow us to write the constants of motion and _bulk_ properties of
noncircular strings in terms of spatially-averaged parameters. This could be
useful for describing the bulk properties of string networks and help yield
further insights into the relation between their microscopic and macroscopic
dynamics. However, for the sake of brevity, we do not quote the explicit
expressions here.

###  2.5 Summary of the wound-string model

We have shown that the EOM corresponding to the higher-dimensional embedding
of strings, with windings of constant radius in an internal manifold, take the
form of canonical dispersion relations. The local propagation velocity of
these extra-dimensional ‘waves’, as seen from a $(3+1)$-dimensional
perspective, is a function of the perpendicular velocity and curvature of the
string in the large dimensions. Via the formal correspondence between higher-
dimensional dynamics and an effective $(3+1)$-dimensional world-sheet current
[46], these dispersion relations may be regarded as governing the local
current flow, at a given point on the string, at a given time $t$. For
arbitrary planar loops, the EOM for the embedding coordinates in the large
dimensions and the compact dimensions are separable, in an intuitive gauge in
which the world-sheet parameters are directly identified with the
cylindrically polar space-time coordinates. The macroscopic string dynamics
can therefore be determined without reference to the motion of the string in
the compact space, and the latter can be determined (in principle) from
$(3+1)$-dimensional observables.
The constants of motion of the string and all observable $(3+1)$-dimensional
quantities, such as pressures, shears, etc, may be written in a compact form
in terms of ‘microscopic’ variables related to the higher-dimensional
embedding, together with the fundamental model parameters such as the
intrinsic tension $\mathcal{T}$ and compactification radius $R$. Particularly
useful is the parameter $\omega^{2}(t,\sigma)$, which represents the local
fraction of the string length lying in the large dimensions. The results
obtained for the wound-string model may be compared with those obtained for
current carrying defect strings in the Abelian-Higgs model, presented in Sec.
5. Such a comparison suggests that $R$ plays the role of an effective four-
dimensional thickness and that the higher-dimensional embedding parameter
$\omega^{2}(t,\sigma)$ corresponds to the embedding of the lines of constant
phase, or, equivalently, magnetic field lines, within the defect-string core.
In principle, a similar analysis to that conducted in Sec. 5 for Abelian-Higgs
strings may be conducted for any defect string species, including chiral
strings [62, 63], electroweak strings [10, 11] and GUT scale strings [12, 13,
14]. The analysis presented here may also be extended to include more complex
warped geometries or internal manifolds and varieties of $F$\- and $D$-string
bound states [25, 27]. Thus, by writing the EOM for higher-dimensional
dynamics in the form of canonical dispersion relations for waves ‘in’ strings,
and comparing these with genuine dispersion relations governing the flow of
current in topological defects, we are able to predict the conditions under
which a given $F/D$-string species, embedded in a higher-dimensional space,
can be successfully mimicked by a field-theoretic string living in
$(3+1)$-dimensions. In Sec. 5 we will see that, mathematically, the exact EOM
governing the evolution of the of the microscopic ‘internal’ variables for
current-carrying defect stings are most naturally expressed in terms of
effective Finsler and generalised Finsler metrics, for the large space-time
dimensions.

##  3 Topological defect strings with internal currents

In this section we review the simplest field theory theory that gives rise to
vortex-strings, i.e., the Abelian-Higgs model. The action, EOM and constants
of motion for the Abelian-Higgs fields, coupled to an external source of
charge, are reviewed in Sec. 3.1.1 and the uncharged vortex solution, known as
the Nielsen-Olesen string [1], is reviewed in Sec. 3.1.2. In Sec. 3.2 we
briefly review the standard derivation of the effective actions for both non-
superconducting and superconducting strings, in the zero-width limit.

###  3.1 The Abelian-Higgs model and the Nielsen-Olesen string

In this Section, we give a brief overview of the Abelian-Higgs action, with an
additional ‘external’ current coupled to the electromagnetic sector. The EOM
and energy-momentum tensor in the general case, $j^{\mu}\neq 0$, are obtained,
and a summary of the Nielsen-Olesen vortex solution, for $j^{\mu}=0$, is
presented.

####  3.1.1 The Abelian-Higgs model with externally coupled charge
($j^{\mu}\neq 0$)

In natural units, using the metric convention $(+---)$, the Abelian-Higgs
action with an additional current $j^{\mu}$ coupled to the electromagnetic
sector is

$\displaystyle S=\int{\rm
d}^{4}x\sqrt{-g}\left\\{D_{\mu}\phi\overline{D}^{\mu}\overline{\phi}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-V(|\phi|)-j^{\mu}A_{\mu}\right\\},$
(3.1)

where $\mu,\nu\in\left\\{0,1,2,3\right\\}$ and $V(|\phi|)$ is the potential
term, which is determined by the symmetry breaking energy scale, $\eta$, and
the scalar coupling, $\lambda$:

$\displaystyle V(|\phi|)=\frac{\lambda}{4}(|\phi|^{2}-\eta^{2})^{2}.$ (3.2)

The gauge covariant derivative $D_{\mu}$ and the electromagnetic field tensor
$F_{\mu\nu}$ are defined according to

$\displaystyle D_{\mu}=\partial_{\mu}-ieA_{\mu},\ \ \
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$ (3.3)

where $e$ is the vector coupling, and the covariant EOM are

$\displaystyle\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}D^{\mu}\phi\right)+\lambda\phi\left(|\phi|^{2}-\eta^{2}\right)=0,$
(3.4)
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\nu}\left(\sqrt{-g}F^{\mu\nu}\right)+ie\left(\overline{\phi}D^{\mu}\phi-\phi\overline{D}^{\mu}\overline{\phi}\right)-j^{\nu}=0.$
(3.5)

The Abelian-Higgs part of the action (3.1) is invariant under local $U(1)$
transformations of the form

$\displaystyle\phi\rightarrow\phi^{\prime}=\phi e^{i\Lambda(x)},\ \ \
A_{\mu}\rightarrow
A_{\mu}^{\prime}=A_{\mu}+\frac{1}{e}\partial_{\mu}\Lambda(x),$ (3.6)

where $\Lambda(x)$ is any single-valued real function, giving rise to the
conserved current
$\mathcal{J}^{\mu}=-ie\left(\overline{\phi}D^{\mu}\phi-\phi\overline{D}^{\mu}\overline{\phi}\right)$.
The total conserved current, including the externally coupled charge is then
given by

$\displaystyle
J^{\mu}=\mathcal{J}^{\mu}+j^{\mu}=-ie\left(\overline{\phi}D^{\mu}\phi-\phi\overline{D}^{\mu}\overline{\phi}\right)+j^{\mu}.$
(3.7)

The conservation of $J^{\mu}$ is expressed directly via the vector EOM, Eq.
(3.5), and the corresponding conserved charge is

$\displaystyle Q=\int J^{0}\sqrt{-g}{\rm d}^{3}x.$ (3.8)

The energy-momentum tensor is defined implicitly by varying the action with
respect to the metric,

$\displaystyle\delta S=\int T^{\mu\nu}\delta g_{\mu\nu}\sqrt{-g}{\rm d}^{4}x,$
(3.9)

so that