$\displaystyle
T^{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\partial(\mathcal{L}\sqrt{-g})}{\partial
g_{\mu\nu}}=D^{\mu}\phi\overline{D}^{\nu}\overline{\phi}+\overline{D}^{\mu}\overline{\phi}D^{\nu}\phi-F^{\mu}{}_{\alpha}F^{\nu\alpha}-\frac{1}{2}\left(j^{\mu}A^{\nu}+j^{\nu}A^{\mu}\right)-g^{\mu\nu}\mathcal{L},$
(3.10)

where $\mathcal{L}$ is the Lagrangian density, that is, the sum of terms
inside the curly brackets in Eq. (3.1).

####  3.1.2 The Nielsen-Olesen solution ($j^{\mu}=0$)

In cylindrical polar coordinates $(t,r,\theta,z)$ and assuming a Minkowski
background

$\displaystyle{\rm d}s^{2}=\eta_{\mu\nu}dx^{\mu}{\rm d}x^{\nu}={\rm
d}t^{2}-{\rm d}r^{2}-r^{2}{\rm d}\theta^{2}-{\rm d}z^{2},$ (3.11)

the ansatz for the Nielsen-Olesen string is [1]

$\displaystyle\phi(r,\theta)=\eta f(r)e^{in\theta},\ \ \
A_{\theta}=\frac{n}{e}\alpha_{\theta}(r),\ \ \ A_{r}=A_{z}=A_{0}=0,$ (3.12)

with $j^{\mu}=0$, where $f(r)$ and $a_{\theta}(r)$ are dimensionless functions
which obey the boundary conditions

$f(r)=\left\\{\begin{array}[]{rl}0,&\ r=0\\\ 1,&\
r\rightarrow\infty,\end{array}\right.$ (3.13)
$a_{\theta}(r)=\left\\{\begin{array}[]{rl}0,&\ r=0\\\ 1,&\
r\rightarrow\infty.\end{array}\right.$ (3.14)

The EOM may be written as [2]

$\displaystyle\frac{{\rm d}^{2}f}{{\rm
d}R_{s|n|}^{2}}+\frac{1}{R_{s|n|}}\frac{{\rm d}f}{{\rm
d}r}-\frac{n^{2}f}{R_{s|n|}^{2}}(1-\alpha_{\theta})^{2}-\frac{1}{2}\left(\frac{r_{s|n|}}{r_{s}}\right)^{2}f(f^{2}-1)=0,$
(3.15) $\displaystyle\frac{{\rm d}^{2}a_{\theta}}{{\rm
d}R_{v|n|}^{2}}-\frac{1}{R_{v|n|}}\frac{{\rm d}a_{\theta}}{{\rm
d}R_{v|n|}}+2\left(\frac{r_{v|n|}}{r_{v}}\right)^{2}f^{2}(1-a_{\theta})-j^{\nu}=0,$
(3.16)

where we have defined the dimensionless coordinates

$\displaystyle R_{i|n|}=\frac{r}{r_{i|n|}},\ i\in\left\\{s,v\right\\},$ (3.17)

and where $r_{s|n|}$ and $r_{v|n|}$ denote the values of the scalar and vector
core radii for an $|n|$-vortex string, respectively. We also use the
simplified notation, $r_{s}$ and $r_{v}$, to refer to the scalar and vector
core radii of $|n|=1$ strings and define the parameters

$\displaystyle\beta_{|n|}=\frac{r_{v|n|}^{2}}{r_{s|n|}^{2}},\ \ \
\beta=\frac{r_{v}^{2}}{r_{s}^{2}},$ (3.18)

for later convenience. Equations (3.15)-(3.16) are solved, to leading order in
the uncoupled regime, by [64]

$f(r)\simeq\left\\{\begin{array}[]{rl}(r/r_{s|n|})^{|n|},&\ r\lesssim
r_{s|n|}\\\ 1,&\ r\gtrsim r_{s|n|},\end{array}\right.$ (3.19)
$a_{\theta}(r)\simeq\left\\{\begin{array}[]{rl}(r/r_{v|n|})^{2},&\ r\lesssim
r_{v|n|}\\\ 1,&\ r\gtrsim r_{v|n|}.\end{array}\right.$ (3.20)

Strictly, the power law solutions for $f(r)$ and $a_{\theta}(r)$ given in Eqs.
(3.19)-(3.20) are valid only in the ranges $r\ll r_{i|n|}$,
$i\in\left\\{s,v\right\\}$, while the asymptotic forms defined in Eqs.
(3.13)-(3.14) hold only for $r\gg r_{i|n|}$. However, by assuming that each
holds approximately, up to the limiting value $r=r_{i|n|}$, and ignoring the
discontinuity of the derivatives, we can easily obtain order of magnitude
estimates for the physical parameters of the string without the need for
detailed numerical calculations.
The topological winding number $n\in\mathbb{Z}$ is given by

$\displaystyle
n=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial\vartheta}{\partial\theta}\Bigg{|}_{z={\rm
const.}}{\rm d}\theta,$ (3.21)

where $\vartheta$ is the phase of the scalar field $\phi$. In general, the
core radii are of the order of the Compton wavelengths of the associated
bosons, $m_{s}^{-1}$ and $m_{v}^{-1}$, but may also depend on $|n|$. The
$|n|$-dependence is expected to take a simple form [64, 65, 66] so that here
we assume

$\displaystyle r_{s|n|}$ $\displaystyle\simeq$
$\displaystyle|n|^{\xi}m_{s}^{-1}=|n|^{\xi}(\sqrt{\lambda}\eta)^{-1},$
$\displaystyle r_{v|n|}$ $\displaystyle\simeq$
$\displaystyle|n|^{\epsilon}m_{v}^{-1}=|n|^{\epsilon}(e\eta)^{-1},$ (3.22)

where $\xi\geq 0$, $\epsilon\geq 0$ are constants.
Substituting the ansatz (3.12) together with the approximate solutions in Eqs.
(3.19)-(3.20) into Eq. (3.10), the only nonzero component of the energy-
momentum tensor is $T^{00}$ and the only constant of motion is the Hamiltonian
$E_{|n|}$, which depends on the absolute value of the topological winding
number. For $r_{s|n|}$ and $r_{v|n|}$ given by Eq. (3.1.2), it is straight
forward to show that, assuming $r_{v}\geq r_{s}$ ($\beta\geq 1$), which
corresponds to a Type II superconducting regime [2], the (constant) mass-
energy per unit length of the string, $\mu_{|n|}=\int T^{00}r{\rm d}r{\rm
d}\theta$, is:

$\displaystyle\mu_{|n|}=2\pi\eta^{2}\left[|n|+|n|^{2-2\epsilon}+|n|^{2\xi}+|n|^{2}\ln\left(\sqrt{\beta_{|n|}}\right)\right].$
(3.23)

However, Bogomol’nyi showed that for a Nielsen-Olesen string at critical
coupling $e=\sqrt{\lambda}$, for which

$\displaystyle r_{s|n|}=r_{s|n|}\equiv r_{c|n|},\quad(\beta_{|n|}=1),$ (3.24)

must satisfy the condition [65, 66]

$\displaystyle\mu_{|n|}\geq 2\pi\eta^{2}|n|,$ (3.25)

also known as the Bogomol’nyi-Prasad-Sommerfield (BPS) bound [67]. Since
stability implies saturation of the bound [2, 68], we require

$\displaystyle 0\leq\xi\leq 1/2,\ \ \ \epsilon\geq 1/2,$ (3.26)

for $|n|\sim\mathcal{O}(1)$. Alternatively, we may set

$\displaystyle\xi=\epsilon=1/2,$ (3.27)

for arbitrary $|n|$. It is common in the literature to take $\xi=0$ and
$\epsilon=1/2$, following Bogomol’nyi’s original assumptions [65, 66], but in
the present work we instead adopt the conditions given in Eq. (3.27) to allow
for large $|n|$. For either choice of parameters, the order of magnitude
estimate for the mass-energy per unit length of a Nielsen-Olesen string with
winding number $n$, at critical coupling, is

$\displaystyle\mu_{|n|}=2\pi\eta^{2}|n|.$ (3.28)

Though, in general, the effective tensions/pressures in the string
$\mathcal{T}^{r}{}_{r}=\int T^{r}{}_{r}r{\rm d}r{\rm d}\theta$,
$\mathcal{T}^{\theta}{}_{\theta}=\int T^{\theta}{}_{\theta}r{\rm d}r{\rm
d}\theta$ and $\mathcal{T}^{z}{}_{z}=\int T^{z}{}_{z}r{\rm d}r{\rm d}\theta$,
are not necessarily conserved, is straightforward to show that, for the
Nielsen-Olesen string

$\displaystyle\mu_{|n|}=-\mathcal{T}^{z}{}_{z},$ (3.29)

for any value of $\beta_{|n|}$, and

$\displaystyle\mathcal{T}^{r}{}_{r}=\mathcal{T}^{\theta}{}_{\theta}=-2\pi\eta^{2}\left[|n|^{2\xi}-|n|^{2-2\epsilon}-|n|^{2}\ln\left(\sqrt{\beta_{|n|}}\right)\right].$
(3.30)

Hence, for $\xi=\epsilon=1/2$, as in Eq. (3.27), and at critical coupling, the
only nonzero component of the tension is $\mathcal{T}^{z}{}_{z}$, which is
equal to minus the energy per unit length of the string. Substituting the
ansatz (3.12) with $j^{\mu}=0$ into Eq. (3.7), it is straightforward to show
that $J^{\mu}=0$ for all $\mu\in\left\\{0,1,2,3\right\\}$ so that the string
is uncharged.

###  3.2 String effective actions, with and without currents

An effective action for the Nielsen-Olesen string can be constructed by
switching to a set of dimensionless world-sheet coordinates $\zeta^{a}$,
$a\in\left\\{0,1\right\\}$, where $\zeta^{0}=\tau$ is time-like and
$\zeta^{1}=\sigma$ is space-like, as for $F$-strings. These parameterize the
two-dimensional sheet swept out by the line $\langle\phi\rangle=0$, which
represents the central axis of the string core. Erecting two normals to this
sheet, denoted $n_{\alpha}{}^{\mu}$, we may describe the small volume of
space-time swept out by the finite-width string using the coordinates
$Y^{\mu}=X^{\mu}(\tau,\sigma)+n_{\alpha}{}^{\mu}\rho^{\alpha}$, where
$X^{\mu}$ are the embedding coordinates of the core central axis and
$\rho^{\alpha}$ probe the core region, e.g.
$\rho^{\alpha}\in\left\\{r,\theta\right\\}$ where $r\in[0,r_{c|n|}]$ and
$\theta\in[0,2\pi)$ [49, 69].
If the curvature radius of the string is much greater than its thickness, we
may approximate the integral over this volume as ${\rm d}^{4}x\sqrt{-g}={\rm
d}^{2}\zeta{\rm d}^{2}\rho\sqrt{-\tilde{g}}$, where
$\tilde{g}_{\mu\nu}(\tilde{\zeta})$ is the induced metric with respect to the
coordinates $\tilde{\zeta}^{\mu}=\left\\{\zeta^{a},\rho^{\alpha}\right\\}$. To
zeroth order in $\rho^{\alpha}$, this metric is given by a block diagonal
matrix, such that $\tilde{g}_{\mu\nu}=$
diag$(\gamma_{ab},\delta_{\alpha\beta})+\mathcal{O}(\rho)$, where
$\delta_{\alpha\beta}$ is the usual Kronecker delta symbol and $\gamma_{ab}$
is the induced metric on the world-sheet defined by $\langle\phi\rangle=0$,
$\gamma_{ab}(\zeta)=g_{\mu\nu}(X)(\partial X^{\mu}/\partial\zeta^{a})(\partial
X^{\mu}/\partial\zeta^{b})$. Substituting from (3.12) and (3.19)-(3.20) and
performing the transverse integration over ${\rm d}^{2}\rho$, then gives [49,
69]

$\displaystyle S=-\mu\int{\rm d}^{2}\zeta\sqrt{-\gamma}+\ .\ .\ .$ (3.31)

where $\mu\simeq\mu_{|n|}$ is given by Eq. (3.28) and we have again assumed
critical coupling, $r_{s|n|}=r_{v|n|}=r_{c|n|}$ and $|n|\sim\mathcal{O}(1)$,
for the sake of simplicity. Hence, to leading order, the effective action for
a Nielsen-Olesen vacuum string, in the Abelian-Higgs model, is simply the
Nambu-Goto action for an $F$-string with no electric or magnetic world-sheet
fluxes [7, 8, 9].
The presence of additional world-sheet fluxes then generates an additional
contribution to the action (3.31), given by [70]

$\displaystyle\Delta S=\frac{1}{e^{2}\mu}\int{\rm
d}^{2}\zeta\sqrt{-\gamma}\gamma_{ab}J^{a}J^{b},$ (3.32)

where $J^{a}$ denotes the world-sheet components of the physical current. The
total effective action may then be written as

$\displaystyle S\simeq-\mu\int{\rm
d}^{2}\zeta\sqrt{-\gamma}(1-\gamma_{ab}j^{a}j^{b}),$ (3.33)

where $j^{a}=e^{-1}J^{a}$ is the dimensionless current. The world-sheet energy
momentum tensor for the superconducting string is defined as
$\theta^{ab}=2j^{a}j^{b}-\gamma^{ab}(\gamma_{cd}j^{c}j^{d})$, so that the
physical energy-momentum tensor of the effective model is given by

$\displaystyle T^{\mu\nu}=\frac{-2}{\sqrt{-g}}\int{\rm
d}^{2}\zeta\sqrt{-\gamma}(\gamma^{ab}+\theta^{ab})\partial_{a}X^{\mu}\partial_{b}X^{\nu}\delta^{4}(x-X),$

where $X^{\mu}$ denotes a string embedding coordinate, as before, and
$x^{\mu}$ denotes a space-time background coordinate [49, 69, 70].
In [44], Witten outlined, from a microphysical perspective, how strings can
carry fermionic currents, with fermionic charge carriers trapped as zero modes
along the string. However, due to Bose-Fermi equivalence in $(1+1)$
dimensions, the effective action (3.33) developed in [49, 69] is valid for
strings with both bosonic and fermionic currents. It is therefore valid in
general, as a first order approximation, for any species of superconducting
string.
Before concluding this section, we note that, for superconducting defect-
string loops, the persistent current is linked to the existence of a second
topological invariant $N\neq n$, given by

$\displaystyle
N=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial\vartheta}{\partial\sigma}\Bigg{|}_{\theta={\rm
const.}}{\rm d}\sigma,$ (3.34)

where $\sigma\in[0,2\pi)$ again denotes the space-like parameter that
parameterizes the string length. Thus, $|N|$ gives the number of twists in the
phase $\vartheta$ of $\phi$ within the loop and its conservation follows
directly from the imposition of periodic boundary conditions to ensure
continuity. For long strings, $N$ is no longer a topological invariant, but
the imposition of periodic boundary conditions over a finite section of string
still ensures the existence of a nonzero integer winding number.
In the zero-width approximation (3.33), the role of $N$ is obscured, but, in
Sec. 5, we will explore the relationship between the dynamics of these
‘internal’ field lines and the macroscopic dynamics of the string in detail.
Using Finsler and generalised Finsler constructions we derive the same
macroscopic EOM as the effective action approach, using a microphysical
ansatz, and without the need for dimensional reduction or (approximate)
effective actions.

##  4 Finsler and generalised Finsler geometries

In this section we give a brief review of Finsler geometry, which is used to
model circular superconducting string loops in Sec. 5.2, and the generalised
Finsler geometry used to model noncircular loops in Sec. 5.3. For both brevity
and clarity, we introduce a minimum of technical detail, concentrating only on
the specific properties of Finsler and generalised Finsler spaces that will be
utilised in our description of current-carrying defect strings, presented in
Sec. 6. The interested reader is referred to the many excellent technical
summaries of Finsler geometry, such as [71, 72, 73], for further details.

###  4.1 Finsler geometry

Finsler geometry is a natural generalisation of Riemannian geometry, which,
roughly speaking, is equivalent to “Riemannian geometry without the quadratic
restriction” [74]. Hence, while the action for a non-relativistic point-
particle in Riemannian space is 555Here, the multiplicative constant with
units of mass has been absorbed into the definition of the metric tensor
(4.2).

$\displaystyle S=\int{\rm d}s,$ (4.1)

where the line-element ${\rm d}s$ is quadratic in form,

$\displaystyle{\rm d}s^{2}=g_{ij}(x){\rm d}x^{i}{\rm d}x^{j},\quad
i,j\in\left\\{1,2,3,\dots\right\\},$ (4.2)

the corresponding action in Finsler space is given by the general functional,

$\displaystyle S=\int F(x,\partial_{\tau}x).$ (4.3)

The metric tensor $g_{ij}(x)$ in (4.2) is a function of the spatial
coordinates only whereas the Finlser space action (4.3) depends on both $x$
and $\partial_{\tau}x$, where $\tau$ is an ‘internal’ dynamical variable, and
the derivative terms belong to the tangent space at $x$, $\partial_{\tau}x\in
T_{x}M$. Thus, in Finsler geometry, the more familiar Riemannian space is
‘non-localised’ by allowing the line-element to depend a single vector, which
is defined on the tangent bundle [75, 76].
However, strictly speaking, standard Finsler geometries are defined as
generalised Riemannian spaces and the question of how to extend Finslerian
structures to manifolds with pseudo-Riemannian signatures is nontrivial and by
no means settled (see [78, 79, 80, 81, 82, 83, 84] for further discussion).
Nonetheless, it seems reasonable to generalise the standard space-time line-
element,

$\displaystyle{\rm d}s^{2}=g_{\mu\nu}(x){\rm d}x^{\mu}{\rm
d}x^{\nu},\quad\mu,\nu\in\left\\{0,1,2,3,\dots\right\\},$ (4.4)

to

$\displaystyle{\rm d}\tilde{s}^{2}=\tilde{g}_{\mu\nu}(x,\partial_{\tau}x){\rm
d}x^{\mu}{\rm d}x^{\nu}.$ (4.5)

This corresponds to a minimally modified Riemannian structure, in which the
space-time interval is ‘almost’ quadratic in the coordinate intervals, but is
not exactly so due to the presence of an effective Finsler metric,
$\tilde{g}_{\mu\nu}(x,\partial_{\tau}x)$. In this case, the internal time-like
variable $\tau$ may, or may not, be identified with the external space-time
coordinate, depending on the choice of embedding for the particle, string, or
other extended object (e.g., $D$-brane).
As we will show, explicitly, in Sec. 5.2, it is precisely this freedom that
allows us to describe the embedded motion of circular superconducting string
loops, in a physical pseudo-Riemannian space-time, in terms of an effective
Finsler metric (4.5). However, this is only possible because, in the case of
circular symmetry, the space-like world-sheet coordinate, $\sigma$, does not
appear explicitly in the embedding ansatz. Therefore, in order to treat non-
circular loops in a similar way, we must consider a form of ‘generalised
Finsler geometry’, with an effective metric that depends on both
$\partial_{\tau}x$ and $\partial_{\sigma}x$, as well as $x$.

###  4.2 Generalised Finsler geometry

Based on our preliminary discussions in Sec. 4.1, we introduce the generalised
Finsler geometry in which geodesic motion is characterised by the action
666Here, again, the multiplicative constant with units of mass has been
absorbed into the definition of the effective metric, $\tilde{g}_{\mu\nu}$.

$\displaystyle S=\int{\rm
d}\tilde{s}=\int\tilde{g}_{\mu\nu}(x,\partial_{\tau}x,\partial_{\sigma}x){\rm
d}x^{\mu}{\rm d}x^{\nu}.$ (4.6)

Here, the time- and space-like internal variables, $\tau$ and $\sigma$, may or
may not be identified with space-time background coordinates, according to the
chosen embedding. In Sec. 5.2, we will identify them with the string world-
sheet coordinates. This allows us to describe non-circular superconducting
string loops, in a physical pseudo-Riemannian space-time, in terms of an
effective generalised Finsler metric,
$\tilde{g}_{\mu\nu}(x,\partial_{\tau}x,\partial_{\sigma}x)$.
However, before continuing, we address a possible source of confusion that
arises through the inconsistent use of terminology in the mathematical
literature. What we have called ‘generalised Finsler geometry’, for want of a
better term, is sometimes referred to simply as Finsler geometry (see for
example [77]), though, strictly speaking, it is not. We stress that, in
Finsler geometry proper, the line-element functional depends on a single
vector, which is defined on the tangent bundle. By contrast, in further
extensions of the canonical Riemannanian structure, this can be generalised to
include multiple vectors, which may or may not be defined on the tangent
space, as well as spinors or other mathematical objects. For the rigorous
mathematical treatment of these Finsler-inspired spaces the interested reader
is referred to [77], and references therein, for further details.

##  5 Topological defect strings with internal currents, revisited

In this Section, we consider the effect of additional charged matter,
represented by the current $j^{\mu}\neq 0$, which becomes localized within the
Abelian-Higgs string core. This is not intended as a realistic model of a
superconducting string but, rather, an illustrative one, which captures
certain generic features of the string phenomenology. In particular, results
obtained by Nielsen [46, 48] suggest that, generically, the
$(3+1)$-dimensional dynamics of higher-dimensional wound-strings should be
equivalent to the dynamics of superconducting defect strings under dimensional
reduction.
With this in mind, we aim to go one step further than the dimensional
reduction program: we search for ansatzes for the Abelian-Higgs fields which
yield EOM for the macroscopic string dynamics that are _identical_ to those
obtained previously for both long (i.e., formally infinite) wound strings, and
wound-string loops [52, 47]. We show that an appropriate ansatz for the scalar
and vector fields (and the ‘external’ current $j^{\mu}$), describing long
superconducting defect strings, exists in the usual sense. That is, ignoring
gravitational effects, such strings live in a globally Minkowski space-time
and the field ansatz is independent of the background metric. However, in
order to obtain EOM for circular loops of superconducting defect string, which
are the same as those obtained previously for wound-string loops, we must
consider a generalization of the geometry to an effective Finsler metric. To
treat noncircular superconducting loops, an appropriately generalised Finsler
geometry is required.
Remarkably, by selecting an appropriate ansatz/metric combination, and
combining the Euler-Lagrange equations for the Abelian-Higgs fields with the
conservation of the energy-momentum tensor, we obtain EOM describing the
macroscopic (external) string dynamics which are identical to those for wound-
strings obtained in Sec. 3, without the need for dimensional reduction.
Furthermore, we also obtain EOM for the microscopic (internal) field evolution
within the string core, which are identical, under the exchange of field
theory and embedding coordinate variables, to those which determine the
evolution of the $F$-string in the compact extra dimensions of the wound-
string model.
We argue that the selection of our ansatz / Finsler or generalised Finsler
metric combination is valid, in that it reflects an alternative geometric
formulation of the Abelian-Higgs theory, in terms of an extended (Finsler or
generalised Finsler) space, which is physically _equivalent_ to standard
formulation in pseudo-Riemannian space. Although this is an unconventional
approach, we note that it does not require the formulation of an effective
action and that, therefore, if the method itself is valid, the resulting
string EOM are _exact_.

###  5.1 Long defect strings: Riemann geometry

Let us first consider a metric with line element

$\displaystyle{\rm d}s^{2}=P^{2}(\tau){\rm d}\tau^{2}-{\rm d}r^{2}-r^{2}{\rm
d}\theta^{2}-Q^{2}(\tau){\rm d}\sigma^{2}=P^{2}(t){\rm d}t^{2}-{\rm
d}r^{2}-r^{2}{\rm d}\theta^{2}-Q^{2}(t){\rm d}\sigma^{2},$ (5.1)

where we define the dimensionless world-sheet variables $\tau\in[0,\infty)$
and $\sigma\in[0,2\pi)$ and the second equality follows from the
identifications ${\rm d}t=\zeta{\rm d}\tau$, $P(t)=\zeta^{-1}P(\tau)$ and
$Q(t)=Q(\tau)$. These, in turn, follow from the adoption of the static gauge
for the string, $X^{0}=t=\zeta\tau$, and the final spatial coordinate is
directly proportional to the space-like world-sheet coordinate $\sigma$
(2.27). This metric is more general than the one we need to treat long
strings, either with or without currents, but it is useful to derive the more
general EOM first and then to impose more restrictive conditions later. The
corresponding Jacobian is $\sqrt{-g}=rP(t)Q(t)$.
Now consider an ansatz of the form

$\displaystyle\phi(r,\theta,z,t)=\eta f(r)e^{in\theta+in\Gamma(z,t)},\ \ \
A_{\theta}=\frac{n}{e}a_{\theta}(r),\ \ \ A_{r}=0,$ $\displaystyle
A_{z}(r,z,t)=\frac{n}{e}a_{\theta}(r)\Gamma^{\prime}(z,t),\ \ \
A_{0}(r,z,t)=\frac{n}{e}a(r)\dot{\Gamma}(z,t),$ (5.2)

where we have defined the coordinate $z(\sigma,t)=Q(t)\sigma$, for
convenience, and a dash and a dot represent differentiation with respect to
$z$ and $t$, respectively. (Clearly, this denotes the true Cartesian $z$-axis
only when $Q={\rm const.}$) The functions $f(r)$ and $a_{\theta}(r)$ obey the
boundary conditions in Eqs. (3.13)-(3.14), and the new radial function $a(r)$
obeys analogous conditions, i.e.

$a(r)=\left\\{\begin{array}[]{rl}0,&\ r=0\\\ 1,&\
r\rightarrow\infty.\end{array}\right.$ (5.3)

Equivalently, we may write the ansatz (5.1) as

$\displaystyle\phi(r,\theta,\sigma,\tau)=\eta
f(r)e^{in\theta+in\Gamma(\sigma,\tau)},\ \ \
A_{\theta}=\frac{n}{e}a_{\theta}(r),\ \ \ A_{r}=0,$ $\displaystyle
A_{\sigma}(r,\sigma,\tau)=\frac{n}{e}a_{\theta}(r)Q^{-1}(\tau)\partial_{\sigma}\Gamma(\sigma,\tau),\
\ \
A_{\tau}(r,\sigma,\tau)=\frac{n}{e}a(r)P^{-1}(\tau)\partial_{\tau}\Gamma(\sigma,\tau).$
(5.4)

The two notations are completely interchangeable and, in the analysis that
follows, we use whichever one is most convenient for our purpose.
Defining the world-sheet gauge covariant derivatives and electromagnetic field
tensor via

$\displaystyle D_{a}=\partial_{a}-ieA_{a},\ \ \
F_{a\nu}=\partial_{a}A_{\nu}-\partial_{\nu}A_{a},$ (5.5)

where $a\in\left\\{\tau,\sigma\right\\}$, the scalar EOM (3.4) may be written
as

$\displaystyle\frac{1}{r}\partial_{r}(r\partial_{r}\phi)+\frac{1}{r^{2}}D_{\theta}(D_{\theta}\phi)+\frac{1}{PQ}D_{\sigma}\left(\frac{P}{Q}D_{\sigma}\phi\right)-\frac{1}{PQ}D_{\tau}\left(\frac{Q}{P}D_{\tau}\phi\right)-\frac{\lambda}{2}\phi(|\phi|^{2}-\eta^{2})=0,$
(5.6)

and the vector EOM (3.5) becomes

$\displaystyle\frac{1}{r}\partial_{r}(rF_{r}{}^{\nu})+\frac{1}{r^{2}}\partial_{\theta}(F_{\theta}{}^{\nu})+\frac{1}{PQ}\partial_{\sigma}\left(\frac{P}{Q}F_{\sigma}{}^{\nu}\right)-\frac{1}{PQ}\partial_{\tau}\left(\frac{Q}{P}F_{\tau}{}^{\nu}\right)+ie(\overline{\phi}D^{\nu}\phi-\phi\overline{D}^{\nu}\phi)-j^{\nu}=0,$
(5.7)

where $j^{\mu}$ again denotes the current sourced by the externally coupled
charge, which is confined to the string core.

####  5.1.1 Long defect strings without internal currents

We may consider a finite section $\Delta={\rm const.}$ of a (formally)
infinitely long string by setting

$\displaystyle P(t)=1,\ \ \ Q(t)=(2\pi)^{-1}\Delta,$ (5.8)

so that

$\displaystyle{\rm d}t=\zeta{\rm d}\tau,\ \ \ {\rm d}z=(2\pi)^{-1}\Delta{\rm
d}\sigma.$ (5.9)

In this case, the scalar EOM separates into real and imaginary parts

$\displaystyle\frac{{\rm d}^{2}f}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}f}{{\rm
d}r}-\frac{n^{2}f}{r^{2}}(1-a_{\theta})^{2}-n^{2}f(1-a)^{2}[\Gamma^{\prime
2}-\dot{\Gamma}^{2}]-\frac{1}{2r_{s}^{2}}f(f^{2}-1)=0,$ (5.10) $\displaystyle
inf(1-a)[\Gamma^{\prime\prime}-\ddot{\Gamma}]=0.$ (5.11)

The $\theta-$, $r-$, $z-$ and $t-$components of the vector EOM are:

$\displaystyle\frac{{\rm d}^{2}a_{\theta}}{{\rm d}r^{2}}-\frac{1}{r}\frac{{\rm
d}a_{\theta}}{{\rm d}r}+\frac{2f^{2}}{r_{v}^{2}}(1-a_{\theta})+j_{\theta}=0$
(5.12) $\displaystyle\frac{n}{e}\frac{{\rm d}a}{{\rm
d}r}[\Gamma^{\prime\prime}-\ddot{\Gamma}]+j_{r}=0,$ (5.13)
$\displaystyle\frac{{\rm d}^{2}a}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}a}{{\rm
d}r}+\frac{2f^{2}}{r_{v}^{2}}(1-a)+\frac{e}{n}\Gamma^{\prime-1}j_{z}=0,$
(5.14) $\displaystyle\frac{{\rm d}^{2}a}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}a}{{\rm
d}r}+\frac{2f^{2}}{r_{v}^{2}}(1-a)+\frac{e}{n}\dot{\Gamma}^{-1}j_{0}=0,$
(5.15)

respectively. The consistency of the ansatz then requires $\Gamma^{\prime
2}-\dot{\Gamma}^{2}={\rm const}$. However, as we will show explicitly at the
end of the present section, the conservation of energy and momentum requires
us to set

$\displaystyle\dot{\Gamma}^{2}=\Gamma^{\prime 2},\ \ \
\dot{\Gamma}=\pm\Gamma^{\prime},$ (5.16)

This automatically implies

$\displaystyle\Gamma^{\prime\prime}-\ddot{\Gamma}=0,$ (5.17)

but Eq. (5.16) represents the stronger condition and Eqs. (5.16)-(5.17) may be
compared with the analogous conditions (2.53)-(2.54) for the ‘internal’
angular coordinate in the wound-string model.
The simplest way to impose (5.16) is to set $\Gamma(z,t)=0$, which corresponds
to the zero current scenario, $j^{\mu}=0$ [52]. The EOM (5.10)-(5.11) and
(5.12)-(5.15) then reduce to the standard EOM for the Nielsen-Olesen string
the energy and tension of the finite string section are,

$\displaystyle E=\mu_{|n|}\Delta,\quad\mathcal{T}^{z}=-\mu_{|n|}\Delta,$
(5.18)

and all other effective pressures and shears are zero, as expected. Clearly,
this is exactly analogous to the corresponding $F$-string case, with the
effective warped-geometry string tension $a\mathcal{T}$ replaced by
$\mu_{|n|}$.

####  5.1.2 Long defect strings with internal currents

To construct the long-string solution with internal current, we now note that
Eq. (5.16) is satisfied by any function of the form

$\displaystyle\Gamma(z,t)=\Gamma(k_{z}z+\omega_{z}t)\equiv\Gamma(\sigma,t)=\Gamma(m\sigma+\omega_{z}t),\
\ \ m\in\mathbb{Z}$ (5.19)

where

$\displaystyle\omega_{z}^{2}=k_{z}^{2},\ \ \ \omega_{z}=\pm k_{z},$ (5.20)

but that it does not admit superpositions of left and right movers,
$\Gamma(z,t)=\Gamma_{L}(k_{z}z+\omega_{z}t)+\Gamma_{R}(k_{z}z-\omega_{z}t)$,
which are solutions to the relativistic wave equation (5.17). The constant
$k_{z}$ is naturally interpreted as a wave number and the associated
wavelength is defined via

$\displaystyle k_{z}=2\pi/\lambda_{z}$ (5.21)

The imposition of periodic boundary conditions over the length $\Delta$ then
implies

$\displaystyle\Delta=m\lambda_{z}.$ (5.22)

The expression for $\Gamma(z,t)$ in Eq. (5.19) may be compared with (2.55) and
(5.20)-(5.22) are formally identical to (2.56), (2.57) and (2.59), since we
have used the same notation to refer to analogous, but physically different,
variables. The physical interpretation of the parameters $\omega_{z}$, $k_{z}$
and $\lambda_{z}$ in the defect-string model is considered later in this
section.
By analogy with Sec. 3, we note that different plane wave modes in the Fourier
expansion of $\Gamma(z,t)$ 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 $m_{j}\in\mathbb{Z}$. By
writing $\Gamma(z,t)$ in the form (5.17), which implies
$\partial\Gamma/\partial t=\omega_{z}{\rm d}\Gamma/{\rm d}u$,
$\partial\Gamma/\partial z=k_{z}{\rm d}\Gamma/{\rm d}u$,
$\partial\Gamma/\partial\sigma=m{\rm d}\Gamma/{\rm d}u$, 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. For functions with only nonlinear terms in
$u$, the natural choice is the mode with the highest amplitude, which then
gives the approximate wavelength of any fluctuation in current density. If
$\Gamma$ 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 in $\Gamma(z,t)$ correspond to a uniform current and
any additional nonlinear terms describe local fluctuations in current density
around the mean value.
The relationship between the model parameters, in particular $\lambda_{z}$ and
$m$, and the internal structure of the string (that is, the configuration of
the scalar and vector fields) will be discussed in detail later in this
section. We argue that the function $\Gamma(z,t)$ determines the space-time
embedding of lines of constant phase in the scalar field $\phi$ and that these
are analogous to the space-time embedding of the angular coordinate for wound
strings in higher-dimensional theories [52, 70]. We aim to show that
equivalent phase-line/angular coordinate embeddings give rise to identical
macroscopic EOM for different string species. This enables us to specify
precisely the conditions under which topological defect strings, originating
in field theory, and fundamental Nambu-Goto strings, originating in string
theory, are phenomenologically equivalent from an observational point of view.
An advantage of this method is that it does not require us to make any
approximations. In particular, we need not resort to an effective action to
determine the EOM for the superconducting defect string.
If we now set

$\displaystyle j_{\theta}=0,\ \ \ j_{r}=0,$ (5.23) $\displaystyle
j_{z}(r,z,t)=\frac{n}{e}\Gamma^{\prime}(t,z)j(r),\ \ \
j_{0}(r,z,t)=\frac{n}{e}\dot{\Gamma}(t,z)j(r),$ (5.24)

the EOM for the current-carrying string reduce to Eqs. (3.15)-(3.16), as in
the Nielsen-Olesen case, plus a new EOM in $a(r)$,

$\displaystyle\frac{{\rm d}^{2}a}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}a}{{\rm d}r}+\frac{2f^{2}}{r_{v}^{2}}(1-a)+j=0.$ (5.25)

It may be shown that a small distribution of ‘external’ charge, that is,
charge not generated specifically by the configuration of the Abelian-Higgs
fields, located near the centre of the string,

$j(r)=\left\\{\begin{array}[]{rl}k,&\ 0\leq r\leq\delta\\\ 0,&\
r>\delta.\end{array}\right.$ (5.26)

where $k<0$ is a constant and $\delta\ll r_{s}\leq r_{v}$, is sufficient to
prevent divergence of the electric and magnetic field densities (i.e. the
divergence of $a(r)$) as $r\rightarrow 0$ [52]. Alternatively, a
phenomenologically equivalent model may be obtained by taking the pure
Abelian-Higgs action, together with the ansatz Eq. (5.1), and imposing a cut-
off $r_{\rm min}=\delta\ll r_{s}\leq r_{v}$, assumed to be due to quantum
gravity effects at small distances [85]. This allows the pure Abelian-Higgs
string to support a zero-mode, so that no additional ‘external’ charge is
required.
In the additional current model, $f(r)$ and $a_{\theta}(r)$ are given by Eqs.
(3.19)-(3.20) and the approximate solution for $a(r)$ is [52]

$a(r)\simeq\left\\{\begin{array}[]{rl}-kr^{2},&\ r\lesssim\delta\\\
1-CK_{0}((r/r_{c|n|})^{1+|n|}),&\ \delta\lesssim r\lesssim r_{c|n|},\\\ 1,&\
r\gtrsim r_{c|n|},\end{array}\right.$ (5.27)

where $C$ is a constant of integration, which must be fixed via the matching
condition at $r=\delta$, and $K_{0}$ denotes the zeroth order modified Bessel
function of the second kind. (Here, again, we assume critical coupling
$r_{s|n|}=r_{v|n|}=r_{c|n|}$, for simplicity.) Typically, $j(r)$ gives
contributions to the relevant components of the energy-momentum tensor of
order $\pm r_{c|n|}^{2}aj\Gamma^{\prime 2}$. Integrating over $r{\rm d}r$ then
yields $\pm r_{c|n|}^{2}k^{2}\delta^{4}\Gamma^{\prime 2}$. For
$|k|\lesssim\delta^{-2}$, which is automatically implied by Eq. (5.27), these
are subdominant to all other contributions. In this case,
$J^{0}\simeq\mathcal{J}^{0}\equiv J^{z}\simeq\mathcal{J}^{z}$ since $j^{0}$,
$j^{z}$ give subdominant contributions to constants of motion and effective
pressures and shears inside the string, as well as to the total current
$J^{\mu}+\mathcal{J}^{\mu}+j^{\mu}$.
We now define the parameters $\omega^{2}(z,t)$ and $\Omega^{2}(t)$ via

$\displaystyle\omega^{-2}(z,t)=\frac{\Delta^{2}+(2\pi)^{2}r_{c|n|}^{2}(\partial_{\sigma}\Gamma)^{2}}{\Delta^{2}},\
\ \ \Omega^{-2}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\omega^{-2}(z,t){\rm
d}\sigma,$ (5.28)

by analogy with (2.60). For the non-superconducting Nielsen-Olesen string,
both the lines of constant phase in $\phi$ and the magnetic field lines are
parallel to the central axis of the string whereas, in the superconducting
case, both become twisted. In the constant current case, the magnetic field
lines form static helices and the phase lines are helices that rotate around
the central axis at a constant angular velocity. Thus, nonlinear terms in
$\Gamma(z,t)$ induce local fluctuations in angular velocity and, hence, local
compressions and rarefractions in the both the phase and magnetic field lines.
The integer $m$ which characterises the linear term in $\sigma$ in the
function $\Gamma$, if present, may then be interpreted as the number of
windings or ‘twists’ in the field lines over the length $\Delta$. Physically,
the parameter $\omega^{2}(z,t)$ represents the local fraction of the total
length of a line of constant phase that lies parallel to the string axis. It
is therefore analogous to the parameter, defined for wound strings in higher
dimensions [52, 53, 64, 86, 87], which gives the local fraction of the total
string length in the non-compact directions. The parameter $\Omega^{2}(t)$
represents the spatially-averaged (and, in principle, still time-dependent)
value of $\omega^{2}(z,t)$.
In order to make the analogy with the wound string case more explicit, it is
useful to define the physical observables $U^{\mu}{}_{\nu}$ as integrals (over
the space-like world-sheet parameter ${\rm d}\sigma$) of components of an
effective energy-momentum tensor for the string,
$\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}$. This must therefore be defined in terms
of the fundamental energy-momentum tensor for the fields
$T^{\mu}{}_{\nu}\sqrt{-g}$ and the relevant Killing vectors, so that

$\displaystyle U^{\mu}{}_{\nu}=\int\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}{\rm
d}\sigma=\int k_{(\nu)\alpha}T^{\mu\alpha}\sqrt{-g}{\rm d}r{\rm d}\theta{\rm
d}z,$ (5.29)

where $k_{(\nu)\alpha}$ denotes a set of vectors labelled by the index $\nu$,
not a two-index tensor. We then have, for example, $E_{|n|}\equiv
U^{0}{}_{0}=\int\mathcal{T}^{0}{}_{0}{\rm d}\sigma=\int
k_{(0)\alpha}T^{0\alpha}\sqrt{-g}{\rm d}r{\rm d}\theta{\rm d}z$, etc. In this
model, the relevant Killing vectors are

$\displaystyle k_{(0)\alpha}=[1,0,0,0]=[\eta_{00},0,0,0]=\eta_{(0)\alpha},$
$\displaystyle
k_{(\theta)\alpha}=[0,0,-r^{2},0]=[0,0,\eta_{\theta\theta},0]=\eta_{(\theta)\alpha},$
$\displaystyle k_{(z)\alpha}=[0,0,0,-1]=[0,0,0,\eta_{zz}]=\eta_{(z)\alpha},$
(5.30)

which may be written in a compact form as

$\displaystyle k_{(\nu)\alpha}=\eta_{(\nu)\alpha},$ (5.31)

where $\eta_{\mu\nu}$ is the Minkowski metric, here expressed in the
coordinates $\left\\{t,r,\theta,z\right\\}$.

In terms of $\omega^{2}(z,t)$, the nonzero components of the string energy-
momentum tensor may be written in a compact form as:

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=\eta^{2}|n|\Delta\omega^{-2},\ \ \ \ \ \ \ \ \ \ \ \ \
\mathcal{T}^{z}{}_{0}\sqrt{-g}=\mp\eta^{2}|n|\Delta\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.32a) $\displaystyle\mathcal{T}^{0}{}_{z}\sqrt{-g}$
$\displaystyle=\pm\eta^{2}|n|\Delta\left(\frac{1-\omega^{2}}{\omega^{2}}\right),\
\
\mathcal{T}^{z}{}_{z}\sqrt{-g}=\eta^{2}|n|\Delta\left(\frac{2\omega^{2}-1}{\omega^{2}}\right),$
(5.32b) $\displaystyle\mathcal{T}^{0}{}_{\theta}\sqrt{-g}$
$\displaystyle=\pm\eta^{2}|n|\Delta
r_{c|n|}\frac{\sqrt{1-\omega^{2}}}{\omega},\ \ \
\mathcal{T}^{z}{}_{\theta}\sqrt{-g}=\mp\eta^{2}|n|\Delta
r_{c|n|}\frac{\sqrt{1-\omega^{2}}}{\omega}.$ (5.32c)

Clearly, these expressions should be compared with Eqs. (2.61a)-(2.61c) in the
wound-string case. Strictly, they represent the order of magnitude values of
the components of the energy-momentum tensor but the approximate equalities
simply reflect the fact that we have used the approximate solutions, Eqs.
(3.19)-(3.20) and (5.27), when performing the integrals over ${\rm d}r$,
together with the condition $|k|\lesssim\delta^{-2}$. The expressions may be
made exact by determining the appropriate numerical solutions for scalar and
vector EOM functions, $f(r)$, $a_{\theta}(r)$ and $a(r)$, but this does not
affect the macroscopic dynamics of the string since each component
$\mathcal{T}^{I}{}_{J}\sqrt{-g}$ is scaled by the same multiplicative factor
of order unity.
Interestingly, the EOM (5.16) is precisely the condition required to ensure
that the Lagrangian $\mathcal{L}$ is _independent_ of $\dot{\Gamma}$ and
$\Gamma^{\prime}$. In fact, imposing (5.16) ensures that $\mathcal{L}$ reduces
_exactly_ to the Lagrangian of the Nielsen-Olesen string, at the level of the
energy-momentum tensor. This is analogous to the wound string case considered
in Sec. 2, where the EOM governing the motion of the windings was found to be
precisely the condition required to ensure that the string Lagrangian is
independent of the higher-dimensional embedding coordinate.
The conservation equation for the fundamental fields

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}\sqrt{-g}=\frac{\partial}{\partial
x^{\mu}}\left(T^{\mu}{}_{\nu}\sqrt{-g}\right)-\frac{1}{2}\frac{\partial
g_{\alpha\beta}(x)}{\partial x^{\nu}}T^{\alpha\beta}\sqrt{-g}=0,$ (5.33)

implies the conservation law for the string

$\displaystyle\nabla_{\mu}\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}=\frac{\partial}{\partial
X^{\mu}}\left(\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}\right)-\frac{1}{2}\frac{\partial
g_{\alpha\beta}(X)}{\partial X^{\nu}}\mathcal{T}^{\alpha\beta}\sqrt{-g}=0,$
(5.34)

where $T^{\mu}{}_{\nu}\sqrt{-g}$ and $\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}$ are
related via Eq. (5.29) and the $x^{\nu}$ denote the space-time background
coordinates whereas the $X^{\nu}$ denote embedding coordinates. For long
strings, this gives

$\displaystyle\partial_{0}(\mathcal{T}^{0}{}_{0}\sqrt{-g})+\partial_{z}(\mathcal{T}^{z}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (5.35a)
$\displaystyle\partial_{0}(\mathcal{T}^{0}{}_{z}\sqrt{-g})+\partial_{z}(\mathcal{T}^{z}{}_{z}\sqrt{-g})$
$\displaystyle=0,$ (5.35b)
$\displaystyle\partial_{0}(\mathcal{T}^{0}{}_{\theta}\sqrt{-g})+\partial_{z}(\mathcal{T}^{z}{}_{\theta}\sqrt{-g})$
$\displaystyle=0.$ (5.35c)

each of which, it may be verified directly, is equivalent to Eq. (5.16). (See
Eqs. (5.35a)-(5.35c), for comparison, in the wound-string case.) Equation
(5.35c) explicitly ensures the conservation of electric charge, since

$\displaystyle J^{0}\sqrt{-g}=\frac{e}{(2\pi)^{2}\Delta
r_{c|n|}^{2}\eta^{2}}\mathcal{T}^{0}{}_{\theta}\sqrt{-g}=-J^{z}\sqrt{-g}=\frac{e}{(2\pi)^{2}\Delta
r_{c|n|}^{2}\eta^{2}}\mathcal{T}^{z}{}_{\theta}\sqrt{-g}.$ (5.36)

Finally, for convenience, we split $\Gamma(\sigma,t)$ into linear and
nonlinear parts

$\displaystyle\Gamma(\sigma,t)=m\sigma+\omega_{z}t+\Gamma_{NL}(m\sigma+\omega_{z}t)$
(5.37)

and introduce the parameter $\alpha^{2}$, which quantifies the nonlinearity of
the windings in the field lines, via

$\displaystyle\alpha^{-2}=\frac{1}{2\pi
m}\int_{0}^{2\pi}(\partial_{\sigma}\Gamma)^{2}{\rm d}\sigma=\frac{1}{2\pi
m}\int_{0}^{2\pi}\left(1+\frac{\partial_{\sigma}\Gamma_{NL}}{m}\right)^{2}{\rm
d}\sigma,$ (5.38)

by analogy with (2.88). The number of complete windings over the length
$\Delta$ is given by

$\displaystyle n_{z}=\frac{1}{2\pi}\int_{0}^{2\pi}\partial_{\sigma}\Gamma{\rm
d}\sigma,$ (5.39)

so that the nonlinear terms in $\Gamma$ make no contribution and we see that

$\displaystyle n_{z}=m,$ (5.40)

as in the wound-string case. However, here, $n_{z}$ refers to the number of
complete twists in the lines on constant phase, over the interval
$z\in[0,\Delta]$ at $\theta={\rm const.}$, as opposed to the number of
windings in the compact internal space. The effective local and spatially
averaged values of the model parameters, $(Q^{\rm eff})^{2}(\sigma,t)$ and
$\langle Q^{2}\rangle(t)$, are related as in the wound-string case, by Eq.
(2.107), and it is possible to show that

$\displaystyle\langle\lambda_{z}^{2}\rangle=\alpha^{2}\lambda_{z}^{2},\ \ \
\langle n_{z}^{2}\rangle=n_{z}^{2}/\alpha^{2},$ (5.41)
$\displaystyle\langle\omega_{z}^{2}\rangle=\langle k_{z}^{2}\rangle,\ \ \
\langle k_{z}^{2}\rangle=\frac{(2\pi)^{2}}{\langle\lambda_{z}^{2}\rangle}.$
(5.42)

The constants of motion and integrated pressures and shears inside the string
may then be written in a compact form as:

$\displaystyle
E_{|n|}=2\pi\eta^{2}|n|\Delta\left(1+\frac{(2\pi)^{2}R^{2}}{\langle\lambda_{z}^{2}\rangle}\right)=2\pi\eta^{2}|n|\Delta\Omega^{-2},$
(5.43) $\displaystyle P^{z}=\pm 2\pi\eta^{2}|n|\langle
n_{z}\rangle\times\frac{(2\pi)^{2}r_{c|n|}^{2}}{\langle\lambda_{z}\rangle}=\pm
2\pi\eta^{2}|n|\Delta\left(\frac{1-\Omega^{2}}{\Omega^{2}}\right),$ (5.44)
$\displaystyle\Lambda^{\theta}=\pm(2\pi)^{2}\eta^{2}|n|r_{c|n|}n_{z}=\pm
2\pi\eta^{2}|n|\Delta\frac{\sqrt{1-\Omega^{2}}}{\Omega}\alpha$ (5.45)
$\displaystyle\Pi_{|n|}=2\pi\eta^{2}|n|\Delta\sqrt{1+\frac{(2\pi)^{2}r_{c|n|}^{2}}{\lambda_{z}^{2}}(2\alpha^{-2}-1)}=2\pi\eta^{2}|n|\Delta\Omega^{-1}\sqrt{2-\alpha^{2}-(1-\alpha^{2})\Omega^{2}},$
(5.46)
$\displaystyle\mathcal{T}^{z}=-2\pi\eta^{2}|n|\Delta\left(1-\frac{(2\pi)^{2}r_{c|n|}^{2}}{\langle\lambda_{z}^{2}\rangle}\right)=-2\pi\eta^{2}|n|\Delta\left(\frac{2\Omega^{2}-1}{\Omega^{2}}\right),$
(5.47)

which should be compared with (2.89)-(2.93), using (2.88) where necessary.
Here, $E_{|n|}$ denotes the total energy, $P^{z}$ the linear momentum of the
current, $\Lambda^{\theta}$ the total angular momentum in the
$\theta-$direction, and $\Pi_{|n|}$ the total $4-$momentum of the string in
the length $\Delta$, while $\mathcal{T}^{z}=U^{z}{}_{z}$ is the integrated
pressure in the $z-$direction. The integrated pressure in the
$\theta-$direction is $\mathcal{T}^{\theta}=U^{\theta}{}_{\theta}=0$, as
expected for a string with a stable fixed radius. Note that, technically,
$\Lambda^{\theta}$, as given in (2.91), is defined as the geometric mean
$\Lambda^{\theta}=\pm\sqrt{-U^{0}{}_{\theta}U^{0\theta}}$, for convenience.
However, all three quantities $\Lambda^{\theta}$, $U^{0}{}_{\theta}$ and
$U^{0\theta}$ are conserved, differing only by factors of $r_{c|n|}$.
Analogous expressions also hold for the energy and momentum densities and
local pressures and shears. These may be obtained by replacing $\Omega^{2}(t)$
with $\omega^{2}(z,t)$ or by replacing the quantities $\langle Q^{2}\rangle$
with $(Q^{\rm eff})^{2}$ and setting $\alpha^{2}=1$ in Eqs. (2.89)-(2.93). In
the simple linear case, corresponding to constant current, we have $\langle
Q^{2}\rangle=(Q^{\rm eff})^{2}=Q^{2}$. In particular, we note the existence of
a critical tensionless solution for

$\displaystyle\omega^{2}(t,z)=\Omega^{2}(t)=1/2,\ \forall z,t$ (5.48)

or, equivalently

$\displaystyle\lambda_{z}=\sqrt{\langle\lambda_{z}^{2}\rangle}=2\pi r_{c|n|},$
(5.49)

whereas zero _net_ tension requires only the weaker condition

$\displaystyle\sqrt{\langle\lambda_{z}^{2}\rangle}=2\pi
r_{c|n|}\leftrightarrow\Omega^{2}(t)=1/2,\forall t.$ (5.50)

The parallels with the wound-string case are obvious. For
$\omega^{2}(t,z)>1/2$ the local ‘tension’ of the string becomes a repulsive
pressure. However, though such a solution exists in the classical model,
quantum mechanical considerations suggest it can never be realised physically,
since superconducting strings with currents exceeding the threshold value

$\displaystyle J_{\rm max}\simeq\frac{e}{2\pi r_{c|n|},},$ (5.51)

where $e$ is the coupling (i.e. charge) and $r_{c|n|}$ is the string radius,
become unstable and decay [2]. In our formulation, this threshold is
equivalent to a minimum wavelength for the phase- and magnetic field-line
‘twists’,

$\displaystyle\lambda_{z}^{\rm min}=2\pi r_{c|n|},$ (5.52)

which is precisely the wavelength required to satisfy the tensionless
condition (5.48).
The fundamental equations of motion and the expressions for the physical
properties of the (long) current-carrying Abelian-Higgs string are completely
analogous to those obtained for wound $F$-strings in Sec. 2, under the
identifications

$\displaystyle\Gamma(z,t)\leftrightarrow\varphi(z,t),\ \ \
r_{c|n|}\leftrightarrow R,$ (5.53)

where $\varphi_{(}z,t)$ represents the angular embedding coordinate in the
higher-dimensional space and $R$ is compactification radius (and, where
necessary, the identifications $\Delta\leftrightarrow a\Delta$,
$\lambda_{z}\leftrightarrow a\lambda_{z}$ etc, where $a$ denotes the ‘warp
factor’ of the wound-string background geometry). In some sense, this is
unsurprising, since the motion of higher-dimensional windings is formally
equivalent to an effective world-sheet current in $(3+1)$ dimensions under
dimensional reduction [46, 47, 48], as stated previously. Nonetheless, it is
common in the literature to construct effective actions for either current-
carrying defect strings [49, 88, 89, 69] or dimensionally reduced strings [48]
in order to determine (and solve) their approximate equations of motion. The
approach outlined here demonstrates an exact equivalence between the two
(albeit in a specific case, that of long straight strings) without the need
for an effective action. In the following sections, we extend this result for
circular, then arbitrary planar loop configurations, though a complete
generalization to arbitrary configurations in $(3+1)$ dimensions is left to
future work. A similar approach may, in principle, be extended to any other
species of defect string in which the flow of current is associated with
‘twists’ in an order parameter, for example chiral strings or vorton models
[52, 53, 62, 63, 50, 92, 93, 90, 91, 94, 95, 96].

###  5.2 Circular defect-string loops: Finsler geometry

To treat circular superconducting loops, we again consider the line-element
(5.1) but, instead of imposing the condition (5.8), we instead allow $P(t)$
and $Q(t)$ to be genuine functions of time.

####  5.2.1 Circular defect-string loops without currents

As in the case of long strings, it is in many ways simpler to obtain the zero-
current solution as a particular limit of the more general case, in which a
nonzero superconducting current exists. This avoids the needless repetition of
similar calculations. We therefore refer the reader to the following
subsection, in which the EOM for the non-superconducting circular string loop
is recovered by substituting $\Gamma(\sigma,t)=0$ ($j^{\mu}=0$) into the more
general formulae that are valid for $\Gamma(\sigma,t)\neq 0$ ($j^{\mu}\neq
0$).

####  5.2.2 Circular defect-string loops with currents

We may consider a circular loop by interpreting $\sigma\in[0,2\pi)$ as an
angular coordinate and $Q(t)$ as a (time-dependent) radial factor in (5.1), so
that the background metric contains a genuinely time-dependent $g_{00}$
component. The real and imaginary parts of the scalar EOM then become

$\displaystyle\frac{{\rm d}^{2}f}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}f}{{\rm
d}r}+\frac{n^{2}f}{r^{2}}(1-a_{\theta})^{2}+n^{2}f(1-a)^{2}\left[\frac{1}{Q^{2}}(\partial_{\sigma}\Gamma)^{2}-\frac{1}{P^{2}}\dot{\Gamma}^{2}\right]+\frac{1}{2r_{s}^{2}}f(f^{2}-1)=0,$
(5.54) $\displaystyle
inf(1-a)\left[\frac{1}{Q^{2}}\partial_{\sigma}^{2}\Gamma+\frac{1}{P^{2}}\ddot{\Gamma}+\frac{1}{PQ}\frac{{\rm
d}}{{\rm d}t}\left(\frac{Q}{P}\right)\dot{\Gamma}\right]=0.$ (5.55)

Adopting the conditions in Eq. (5.23), the $r-$ and $\sigma-$components of the
vector EOM are:

$\displaystyle\frac{n}{e}\frac{{\rm d}a}{{\rm
d}r}\left[\frac{1}{Q^{2}}\partial_{\sigma}^{2}\Gamma+\frac{1}{P^{2}}\ddot{\Gamma}+\frac{1}{PQ}\frac{{\rm
d}}{{\rm d}t}\left(\frac{Q}{P}\right)\dot{\Gamma}\right]=0,$ (5.56)
$\displaystyle\frac{{\rm d}^{2}a}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}a}{{\rm
d}r}+\frac{2f^{2}}{r_{v}^{2}}(1-a)+\frac{e}{n}(\partial_{\sigma}\Gamma)^{-1}j_{\sigma}=0.$
(5.57)

The $\theta-$component is again as in the Nielsen-Olesen case and the
$t-$component is identical to Eq. (5.15). We then define

$\displaystyle
j_{\sigma}(r,\sigma,t)=\frac{n}{e}\partial_{\sigma}\Gamma(t,\sigma)j(r),\ \ \
j_{0}(r,\sigma,t)=\frac{n}{e}\dot{\Gamma}(t,\sigma)j(r),$ (5.58)

and impose the condition

$\displaystyle\dot{\Gamma}^{2}=\frac{P^{2}}{Q^{2}}(\partial_{\sigma}\Gamma)^{2},\
\ \ \dot{\Gamma}=\pm\frac{P}{Q}\partial_{\sigma}\Gamma,$ (5.59)

which automatically implies

$\displaystyle\partial_{\sigma}\Gamma^{2}+\frac{Q^{2}}{P^{2}}\ddot{\Gamma}+\frac{Q}{P}\frac{{\rm
d}}{{\rm d}t}\left(\frac{Q}{P}\right)\dot{\Gamma}=0.$ (5.60)

This may be shown explicitly by differentiating Eq. (5.59) with respect to
both $\sigma$ and $t$, then eliminating $\partial_{\sigma}\dot{\Gamma}$.
Equations (5.59)-(5.60) represent generalisations of Eqs. (5.16)-(5.17) and,
as we will soon show, can be cast in ana analogous form to (2.149) for an
appropriate choice of $P(t)$ and $Q(t)$.
We again note that (5.59) is precisely the condition required to ensure that
the Lagrangian $\mathcal{L}$ reduces to that of the Nielsen-Olesen string at
the level of the energy-momentum tensor, being independent of $\dot{\Gamma}$,
$\partial_{\sigma}\Gamma$ and $P^{2}$, $Q^{2}$. Thus, subject to Eq. (5.59),
the remaining EOM in the radial functions $f(r)$, $a_{\theta}(r)$ and $a(r)$
reduce to those obtained in Sec. 3.1. Adopting the same ansatz for $j(r)$, Eq.
(5.26), the solutions for all three functions, plus $j(r)$ itself, are those
obtained previously.
The parameters $\omega^{2}(\sigma,t)$ and $\Omega^{2}(t)$ are now defined as

$\displaystyle\omega^{-2}(\sigma,t)=\frac{Q^{2}+r_{c|n|}^{2}(\partial_{\sigma}\Gamma)^{2}}{Q^{2}},\
\ \ \Omega^{-2}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\omega^{-2}(\sigma,t){\rm
d}\sigma,$ (5.61)

and the physical properties and effective energy-momentum tensor for the
string may be defined using only a slight modification of Eq. (5.29):

$\displaystyle U^{\mu}{}_{\nu}=\int\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}{\rm
d}\sigma=\int\kappa_{(\nu)\alpha}T^{\mu\alpha}\sqrt{-g}{\rm d}r{\rm
d}\theta{\rm d}\sigma.$ (5.62)

Here we have replaced the integral over ${\rm d}z$ with an integral over ${\rm
d}\sigma$ and the set of vectors $\kappa_{(\nu)\alpha}$ are defined by analogy
with Eqs. (5.1.2)-(5.31), but with an important difference, related to the
$g_{\sigma\sigma}$ component of the new metric. Thus, we set

$\displaystyle\kappa_{(0)\alpha}=[1,0,0,0]=[\eta_{00},0,0,0]=\eta_{(0)\alpha},$
$\displaystyle\kappa_{(\theta)\alpha}=[0,0,-r^{2},0]=[0,0,\eta_{\theta\theta},0]=\eta_{(\theta)\alpha},$
$\displaystyle\kappa_{(\sigma)\alpha}=[0,0,0,-Q^{2}]=[0,0,0,g_{\sigma\sigma}]=g_{(\sigma)\alpha},$
(5.63)

or, equivalently

$\displaystyle\kappa_{(\nu)\alpha}=\eta_{(\nu)\alpha}+\delta^{(\sigma)}{}_{(\nu)}(g_{(\sigma)\alpha}-\eta_{(\sigma)\alpha}).$
(5.64)

We note that, for $Q^{2}(t)=1$, Eqs. (5.2.2) and (5.64) reduce to Eqs. (5.1.2)
and (5.31), respectively. Though the definition of $\kappa_{(\sigma)\alpha}$,
especially, in Eqs. (5.2.2)-(5.64) may appear arbitrary, we will show that it
is in fact _necessary_ , in order to yield a self-consistent set of
conservation equations.
However, as we shall see, the consistency of the Euler-Lagrange equations and
the conservation of the energy-momentum tensor also require us to impose a
condition relating the metric component $g_{00}=P^{2}(t)$ to the
‘$Q$-coordinate’ velocity, namely $P^{2}=1-\dot{Q}^{2}$. Imposing this
condition, _and_ interpreting $Q(t)$ as a genuine space-time coordinate, the
metric becomes Finsler and, strictly speaking, we cannot call the set
$\kappa_{(\nu)\alpha}$ ‘Killing vectors’, since the _original_ concept of a
Killing vector, defined by Killing’s equation (see, for example [97]), applies
only to Riemannian and pseudo-Riemannian geometries. However, the concept of a
Killing-like vector field, which characterizes an isometry, has been extended
to Finsler spaces in [98, 99].
Using the definitions in Eqs. (5.2.2)-(5.64), the components of
$\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}$ that are analogous to those in Eqs.
(5.32a)-(5.32c) are

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=2\pi\eta^{2}|n|\frac{Q}{P}\omega^{-2},\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \mathcal{T}^{\sigma}{}_{0}\sqrt{-g}=\pm
2\pi\eta^{2}|n|\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$ (5.65a)
$\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}$ $\displaystyle=\pm
2\pi\eta^{2}|n|Q^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),\ \ \
\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}=2\pi\eta^{2}|n|PQ\left(\frac{2\omega^{2}-1}{\omega^{2}}\right),$
(5.65b) $\displaystyle\mathcal{T}^{0}{}_{\theta}\sqrt{-g}$ $\displaystyle=\pm
2\pi\eta^{2}|n|Qr_{c|n|}\frac{\sqrt{1-\omega^{2}}}{\omega},\ \
\mathcal{T}^{\sigma}{}_{\theta}\sqrt{-g}=\mp
2\pi\eta^{2}|n|Pr_{c|n|}\frac{\sqrt{1-\omega^{2}}}{\omega},$ (5.65c)

and the relevant conservation equations are

$\displaystyle\partial_{0}(\mathcal{T}^{0}{}_{0}\sqrt{-g})+\partial_{\sigma}(\mathcal{T}^{\sigma}{}_{0}\sqrt{-g})$
$\displaystyle=0,$ (5.66a)
$\displaystyle\partial_{0}(\mathcal{T}^{0}{}_{\sigma}\sqrt{-g})+\partial_{\sigma}(\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g})$
$\displaystyle=0,$ (5.66b)
$\displaystyle\partial_{0}(\mathcal{T}^{0}{}_{\theta}\sqrt{-g})+\partial_{\sigma}(\mathcal{T}^{\sigma}{}_{\theta}\sqrt{-g})$
$\displaystyle=0.$ (5.66c)

Substituting from Eqs. (5.65a)-(5.65c) into Eqs. (5.66a)-(5.66c), it may be
verified that Eqs. (5.66b) and (5.66c) are each equivalent to

$\displaystyle\frac{Q}{\dot{Q}}\frac{\dot{\omega}}{\omega}=1-\omega^{2},$
(5.67)

and that Eq. (5.66a) gives

$\displaystyle
P^{2}\left(1-\omega^{2}-2\frac{Q}{\dot{Q}}\frac{\dot{\omega}}{\omega}\right)+P\dot{P}\frac{Q}{\dot{Q}}=0.$
(5.68)

In fact, Eq. (5.67) follows directly from the definition of $\omega^{2}$ under
the assumption that $\omega^{2}=\omega^{2}(t)$ or, equivalently,
$\partial_{\sigma}\omega=0$. For circular loops, this is required by symmetry
and $\Gamma(\sigma,t)$ contains no nonlinear terms, so that

$\displaystyle\Gamma(\sigma,t)=n_{\sigma}\sigma+\omega_{\sigma}t,\ \ \
(n_{\sigma}\in\mathbb{Z})$ (5.69)

Eqs. (5.67)-(5.68) together imply

$\displaystyle P(2\omega^{2}-1)+P\dot{P}\frac{Q}{\dot{Q}}=0.$ (5.70)

We now note that this is equivalent to the EOM for a circular wound string
loops obtained in Sec. 2 (see also [52, 86]), _if_ we impose the conditions

$\displaystyle P^{2}(t)=1-\dot{Q}^{2}(t),\ \ \
P(t)=\pm\sqrt{1-\dot{Q}^{2}(t)},$ (5.71)

which, together with Eq.(5.67), implies

$\displaystyle\frac{Q}{\dot{Q}}\frac{\dot{P}}{P}=\frac{Q\ddot{Q}}{P^{2}}=2\omega^{2}-1.$
(5.72)

The two unique EOM that satisfy both the fundamental Abelian-Higgs EOM and the
conservation equations (5.66a)-(5.80), and which are sufficient to determine
the two free functions $Q(t)$ and $\Gamma(\sigma,t)$, may then be written as

$\displaystyle(1-\dot{Q}^{2})(2\omega^{2}-1)+Q\ddot{Q}=0,$ (5.73a)
$\displaystyle\dot{\Gamma}^{2}=\frac{(1-\dot{Q}^{2})}{Q^{2}}(\partial_{\sigma}\Gamma)^{2}.$
(5.73b)

Equations (5.73a)-(5.73b) are, clearly, formally analogous the two EOM that
determine the dynamics of the circular wound-string loop, i.e., Eqs. (2.149)
under the identification $Q(t)\leftrightarrow\rho(t)$. Together with the
Finsler condition (5.71) this implies
$P(t)\leftrightarrow\sqrt{1-\dot{\rho}^{2}}$ and Eqs. (5.67), (5.68) and
(5.70) are formally equivalent to Eqs. (2.163), (2.165) and (2.166),
respectively.
The _normalized_ components of the current density are

$\displaystyle
J^{0}\sqrt{-g}=\frac{e}{2\pi}\frac{\sqrt{1-\dot{Q}(t_{i})}}{Q(t_{i})}\frac{Q}{\sqrt{1-\dot{Q}}}\dot{\Gamma},\
\ \
J^{\sigma}\sqrt{-g}=\frac{e}{2\pi}\frac{\sqrt{1-\dot{Q}(t_{i})}}{Q(t_{i})}\frac{\sqrt{1-\dot{Q}}}{Q}\partial_{\sigma}\Gamma,$
(5.74)

by analogy with (2.161), so that using the EOM in $\Gamma$ gives

$\displaystyle
J^{0}\sqrt{-g}=\pm\frac{e}{2\pi}\frac{\sqrt{1-\dot{Q}(t_{i})}}{Q(t_{i})}\partial_{\sigma}\Gamma=-J^{\sigma}\sqrt{-g}=\mp\frac{e}{2\pi}\frac{\sqrt{1-\dot{Q}(t_{i})}}{Q(t_{i})}\dot{\Gamma},$
(5.75)

by analogy with (2.162). The components of the $4$-current may be expressed in
terms of $\mathcal{T}^{0}{}_{\theta}\sqrt{-g}$ and
$\mathcal{T}^{\sigma}{}_{\theta}\sqrt{-g}$, given by Eq. (5.65c), and it is
straightforward to show that the current conservation equation is equivalent
to Eq. (5.66c). As for long strings, the EOM in $\Gamma(\sigma,t)$ may be
written as a dispersion relation

$\displaystyle\omega_{\sigma}^{2}=P^{2}k_{\sigma}^{2},\ \ \
\omega_{\sigma}=\pm Pk_{\sigma},$ (5.76)

where we have defined the (time-dependent) wavenumber and wavelength as

$\displaystyle k_{\sigma}(t)=\frac{2\pi}{\lambda_{\sigma}(t)},\ \ \
\lambda_{\sigma}(t)=\frac{2\pi Q(t)}{n_{\sigma}}.$ (5.77)

However, since we have decided to interpret $\sigma$ as an angular coordinate,
and hence to interpret $Q(t)$ as a radial coordinate, we are free to redefine
the metric such that

$\displaystyle{\rm d}s^{2}=P^{2}(t){\rm d}t^{2}-{\rm d}r^{2}-r^{2}{\rm
d}\theta^{2}-Q^{2}(t){\rm d}\sigma^{2}\rightarrow P^{2}(t){\rm d}t^{2}-{\rm
d}x^{2}-{\rm d}Q^{2}(t)-Q^{2}(t){\rm d}\sigma^{2},$ (5.78)

where $x$ is a standard (static) Cartesian coordinate. Note that, here, $Q$ is
really a function of the ‘internal’ world-sheet coordinate $\tau$, but that
this is identified directly with the physical time coordinate $t$ via the
choice of the static gauge for the string embedding, $t=\zeta\tau$. Hence,
both $\left\\{t,r,\theta,\sigma\right\\}$ and $\left\\{t,x,Q,\sigma\right\\}$
are legitimate coordinate systems for the string. The former contains the
coordinates that are most useful to describe the string core, $r$ and
$\theta$, but it is more convenient to adopt the latter to describe the
macroscopic string dynamics. Using the conditions (5.71), the effective
Finsler-type metric may then be written as:

$\displaystyle{\rm d}s^{2}=(1-\dot{Q}^{2}(t)){\rm d}t-dx^{2}-{\rm
d}Q^{2}(t)-Q^{2}(t){\rm d}\sigma^{2}.$ (5.79)

In the new coordinate system the components of the effective energy-momentum
tensor involving only $t$ and $\sigma$ are the same as before, but there also
exist additional components whose conservation equation,

$\displaystyle\partial_{0}(T^{0}{}_{Q}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{Q}\sqrt{-g})-\frac{1}{2}\partial_{Q}g_{\sigma\sigma}T^{\sigma\sigma}\sqrt{-g}=0,$
(5.80)

must be satisfied by the macroscopic dynamics of the string. Although
$\mathcal{T}^{\sigma\sigma}\sqrt{-g}$ is well defined (albeit implicitly) by
Eqs. (3.10) and (5.62), the additional components $T^{0}{}_{Q}\sqrt{-g}$ and
$T^{\sigma}{}_{Q}\sqrt{-g}$ cannot be specified using these. In order to apply
the same method as that used to determine
$\mathcal{T}^{\sigma\sigma}\sqrt{-g}$, etc, we would first need to write down
a new ansatz for the field variables $\phi$, $A_{\mu}$ (and $j^{\mu}$) in the
new coordinate system, after the transformation
$\left\\{t,r,\theta,\sigma\right\\}\rightarrow\left\\{t,x,Q,\sigma\right\\}$.
Nonetheless, we _do_ know that, whatever the correct definitions of these
quantities, the resulting conservation equation must be compatible with
(5.70). Based on the results derived in Sec. 2, we propose the set

$\displaystyle\mathcal{T}^{0}{}_{Q}\sqrt{-g}=-2\pi\eta^{2}|n|\frac{Q\dot{Q}}{P}\omega^{-2},\
\ \ \mathcal{T}^{\sigma}{}_{Q}\sqrt{-g}=\pm
2\pi\eta^{2}|n|\dot{Q}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
$\displaystyle\mathcal{T}^{\sigma\sigma}\sqrt{-g}=-2\pi\eta^{2}|n|\frac{P}{Q}\left(\frac{2\omega^{2}-1}{\omega^{2}}\right).$
(5.81)

by analogy with Eq. (2.159c). The conservation equation (5.80) then gives

$\displaystyle
P^{2}(2\omega^{2}-1)+Q\ddot{Q}+\dot{Q}^{2}\left(1-\frac{Q}{\dot{Q}}\frac{\dot{P}}{P}-2\frac{Q}{\dot{Q}}\frac{\dot{\omega}}{\omega}\right)=0.$
(5.82)

Applying the ‘Finsler conditions’ (5.71) / (5.79), Eq. (5.82) reduces to
(5.73a). Our definitions of $T^{0}{}_{Q}\sqrt{-g}$ and
$T^{\sigma}{}_{Q}\sqrt{-g}$ are therefore consistent with both the Euler-
Lagrange equations of the Abelian-Higgs model _and_ the conservation equations
for the energy-momentum tensor corresponding to the previous set of
coordinates $\left\\{t,r,\theta,\sigma\right\\}$, in which the microscopic
field ansatz was defined.
It is straightforward to verify that the general solution to Eq. (5.73b) is of
the form

$\displaystyle\Gamma(\sigma,t)=\Gamma\left(n_{\sigma}\sigma\pm
n_{\sigma}\int\frac{\sqrt{1-\dot{Q}^{2}}}{Q}{\rm d}t\right),$ (5.83)

whereas, at least for $\dot{Q}(t_{i})=0$, the solution of Eq. (5.73a) takes
the simple form

$\displaystyle
Q(t)=Q(t_{i})\sqrt{1-\left(\frac{1-2\omega^{2}(t_{i})}{\omega^{4}(t_{i})}\right)\sin^{2}\left(\frac{\omega^{2}(t_{i})}{Q(t_{i})}(t-t_{i})\right)}.$
(5.84)

These solutions have been studied in detail in [52, 64, 86] and Eqs. (5.83)
and (5.84) should be compared with Eqs. (2.154)-(2.155) and (2.192),
respectively.
After applying the Finsler conditions, both the EOM (5.73a)-(5.73b) and the
expressions for the components of the energy momentum tensor (5.66a)-(5.80)
are equivalent to those obtained in Sec. 2, for circular wound string loops
with time-dependent radius $\rho(t)$, under the identifications

$\displaystyle\Gamma(\sigma,t)\leftrightarrow\varphi(\sigma,t),\ \ \
Q(t)\leftrightarrow\rho(t),\ \ \ r_{c|n|}\leftrightarrow R,$ (5.85)

or, strictly, $Q(t)=a\rho(t)$ and $P(t)=a\sqrt{1-\dot{\rho}^{2}(t)}$ if $a<1$
in the wound string model.
This, in turn, justifies our ‘guess’ for the $T^{0}{}_{Q}\sqrt{-g}$ and
$T^{\sigma}{}_{Q}\sqrt{-g}$ components of the effective string energy-momentum
tensor. The definitions proposed in Eq. (5.2.2), together with the conditions
(5.71), ensure we obtain a self-consistent set of Euler-Lagrange and
conservation equations for the superconducting defect string, which turn out
to be formally equivalent to the EOM obtained in the wound-string case,
despite the fact that they are derived from radically different actions, i.e.,
the fundamental action for the Abelian-Higgs fields (3.1) and the Nambu-Goto
action for the $F$-string (2.2).
We note that this result was by no means _certain_. The formal correspondence
in [46] demonstrates that the motion of windings in the compact extra
dimensions can be interpreted as a charge density on the effective
$(3+1)$-dimensional world sheet of the string. It does _not_ show that the
resulting EOM will be identical to those obtained for any particular defect
string model. In principle, any differences in the macroscopic evolution
equations could give rise to specific higher-dimensional signatures in the
wound-string case. This could have occurred, for example, if it were not
possible to reconcile Eq. (5.80) with Eqs. (5.66a)-(5.66c) to form a self-
consistent set.
Using the procedure above, we obtain precisely the same EOM as in the wound-
string case, but derived from the fundamental Euler-Lagrange and conservation
equations for the Abelian-Higgs fields. In particular, our results suggest
that the EOM coupling the evolution of the windings to the $(3+1)$-dimensional
dynamics of the $F$-string are _precisely_ equivalent to the EOM coupling the
internal field evolution to the macroscopic dynamics of the superconducting
$U(1)$ string. This makes sense if we consider the configuration of the
$F$-string in the internal space as determining its ‘internal’ structure from
an effective $(3+1)$-dimensional perspective.
Thus, together with $f(r)$, $a_{\theta}(r)$, $a(r)$ and $j(r)$, the functions
$\Gamma(\sigma,t)$ and $Q(t)=\rho(t)$ determine both the microscopic string
structure and its macroscopic evolution. It follows that the expressions for
the constants of motion also take the same form as those given in Sec. 2. We
may study the behavior of $f(r)$, $a_{\theta}(r)$, $a(r)$, $j(r)$ and
$\Gamma(\sigma,t)$ in the coordinate system covering the internal structure of
the string core, $\left\\{t,r,\theta,\sigma\right\\}$, whereas $Q(t)$
determines the macroscopic motion in the $\left\\{t,x,Q,\sigma\right\\}$
coordinate system. Eqs. (5.73a)-(5.73b) therefore describe the _interplay_
between the large scale dynamics of the string, as a composite structure, and
the microscopic configuration of its constituent fields. In practice, we can
reconstruct the former from the latter. Since the EOM in $Q(t)$ depends only
on the constants $\partial_{\sigma}\Gamma=n_{\sigma}$ and $r_{c|n|}$, it may
be solved independently. We then substitute the solution into the second EOM
to determine $\Gamma(\sigma,t)$ and, together with the solutions (either
approximate or numerical) for $f(r)$, $a_{\theta}(r)$, $a(r)$ and $j(r)$, the
microscopic field evolution is precisely determined.
It is intriguing, and suggestive of a more fundamental link, that the same set
of equations describing current-carrying string loops can be obtained from
such different fundamental theories (without the need for approximations or
effective actions) when one theory, first defined in Minkowski space, then has
a ‘Finsler condition’ imposed upon it at the level of the EOM. As we shall see
in the next Section, the EOM governing arbitrary planar loops, in the wound
string case, may also be obtained from the fundamental Abelian-Higgs
equations. In this model, one first defines these in Minkowski space-time,
before imposing a ‘generalised Finsler condition’.

###  5.3 Noncircular defect-string loops: generalised Finsler geometry

To treat noncircular superconducting loops, we must consider a generalisation
of the line-element (5.1), in which the free functions depend also on an
‘internal’ space-like variable, i.e., such that $P(t)\rightarrow P(t,\sigma)$
and $Q(t)\rightarrow Q(t,\sigma)$.

####  5.3.1 Noncircular defect-string loops without currents

As for circular string loops, it is again simpler to obtain the zero-current
solution as a particular limit of a more general scenario, which in this case
corresponds to an arbitrary planar loop with nonzero current. We derive the
EOM for this configuration in the following subsection, and the corresponding
non-superconducting solution is again recovered by substituting
$\Gamma(\sigma,t)=0$ ($j^{\mu}=0$).

####  5.3.2 Noncircular defect-string loops with currents

Let us now consider a metric with line element of the form

$\displaystyle{\rm d}s^{2}=P^{2}(\sigma,t){\rm d}t^{2}-{\rm d}r^{2}-r^{2}{\rm
d}\theta^{2}-2U(t,\sigma){\rm d}t{\rm d}\sigma-Q^{2}(\sigma,t){\rm
d}\sigma^{2},$ (5.86)

giving $\sqrt{-g}=r\sqrt{P^{2}Q^{2}+U^{2}}$. Using the same field ansatz, Eq.
(5.1), but identifying $\Gamma(z,t)\equiv\Gamma(\sigma,t)$ and substituting
$A_{\sigma}(r,\sigma,t)=(n/e)a(r)\partial_{\sigma}\Gamma$ for
$A_{z}(r,z,t)=(n/e)a(r)\Gamma^{\prime}$, the scalar EOM again separates into
real and imaginary parts

$\displaystyle\frac{{\rm d}^{2}f}{{\rm d}r^{2}}+\frac{1}{r}\frac{{\rm
d}f}{{\rm
d}r}-\frac{n^{2}f}{r^{2}}(1-a_{\theta})^{2}-\frac{n^{2}f(1-a)^{2}}{(-X)}[P^{2}(\partial_{\sigma}\Gamma)^{2}-Q^{2}\dot{\Gamma}^{2}+2U\dot{\Gamma}\partial_{\sigma}\Gamma]-\frac{1}{2r_{s}^{2}}f(f^{2}-1)=0,$

and

$\displaystyle(-X)[Q^{2}\ddot{\Gamma}-P^{2}\partial_{\sigma}^{2}\Gamma-2U\partial_{\sigma}\dot{\Gamma}-\partial_{\sigma}U\dot{\Gamma}-\dot{U}\partial_{\sigma}\Gamma+2Q\dot{Q}\dot{\Gamma}-2P\partial_{\sigma}P\partial_{\sigma}\Gamma]$
$\displaystyle-\frac{1}{2}\frac{\partial(-X)}{\partial
t}[Q^{2}\dot{\Gamma}-U\partial_{\sigma}\Gamma]-\frac{1}{2}\frac{\partial(-X)}{\partial\sigma}[-P^{2}\partial_{\sigma}\Gamma-U\dot{\Gamma}]=0,$
(5.88)

respectively, where we have denoted $(-X)=P^{2}Q^{2}+U^{2}$.
Next, we again adopt the conditions (5.23) and define in the remaining nonzero
components of $j^{\mu}$ as in Eq. (5.58). Under these conditions, it may be
also shown that, _if_ Eq. (5.3.2) is satisfied, the $r-$ component of the
vector EOM vanishes identically, while the $\sigma-$, and $t-$components both
reduce to (5.25) and the $\theta-$component again reduces to that obtained for
the Nielsen-Olesen string.
For the new metric (5.86), we modify the definition of the effective energy-
momentum tensor for the string in Eq. (5.62), so that

$\displaystyle U^{\mu}{}_{\nu}=\int\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}{\rm
d}\sigma=\int\tilde{\kappa}_{(\nu)\alpha}T^{\mu\alpha}{}_{\nu}\sqrt{-g}{\rm
d}r{\rm d}\theta{\rm d}\sigma.$ (5.89)

The new set of vectors $\tilde{\kappa}_{(\nu)\alpha}$ are defined by analogy
with Eqs. (5.2.2)-(5.64), but with an important difference, related to the
$g_{\sigma\sigma}$ and $g_{0\sigma}$ components of the new metric. Thus, we
set

$\displaystyle\tilde{\kappa}_{(0)\alpha}=[1,0,0,-U(\sigma,t)]=[\eta_{00},0,0,0]=\eta_{(0)\alpha},$
$\displaystyle\tilde{\kappa}_{(\theta)\alpha}=[0,0,-r^{2},0]=[0,0,\eta_{\theta\theta},0]=\eta_{(\theta)\alpha},$
$\displaystyle\tilde{\kappa}_{(\sigma)\alpha}=[0,0,0,[-Q^{2}+(P^{2}Q^{2}+U^{2})]/(1-P^{2})]=[0,0,0,(g_{\sigma\sigma}+\eta_{\theta\theta}(-g))/(1-g_{00})],$
(5.90)

or, equivalently

$\displaystyle\tilde{\kappa}_{(\nu)\alpha}=\eta_{(\nu)\alpha}+\delta^{(\sigma)}{}_{(\nu)}\delta^{\sigma}{}_{\alpha}\left[(g_{(\sigma)\sigma}+\eta_{\theta\theta}(-g))/(1-g_{00})-\eta_{(\sigma)\sigma}\right],$
(5.91)

where $g_{\mu\nu}$ refers to the metric giving the line element (5.86). Note
that, when $U(\sigma,t)=0$ and $Q(\sigma,t)=Q(t)$, Eqs. (5.3.2) and (5.91)
reduce to Eqs. (5.2.2) and (5.64), respectively. Again, the definition of
$\tilde{\kappa}_{(\sigma)\alpha}$ may appear arbitrary at first, but we will
show that it is necessary in order to obtain a consistent set of conservation
equations.
If we now assume that

$\displaystyle
P^{2}(\partial_{\sigma}\Gamma)^{2}-Q^{2}\dot{\Gamma}^{2}+2U\dot{\Gamma}\partial_{\sigma}\Gamma=0,$
(5.92)

the scalar EOM (LABEL:Kawa-scalar-real) reduces to the usual one for the
Nielsen-Olesen string, as before. Equation (5.92) may be solved in the
variable ($\dot{\Gamma}/\partial_{\sigma}\Gamma$), so that

$\displaystyle\frac{\dot{\Gamma}}{\partial_{\sigma}\Gamma}=\frac{U\pm\sqrt{P^{2}Q^{2}+U^{2}}}{Q^{2}}=\frac{P^{2}}{-U\pm\sqrt{P^{2}Q^{2}+U^{2}}},$
(5.93)

which we note is formally analogous to (2.210) for appropriate choices of $P$,
$Q$ and $U$. (These will be considered, in detail, shortly.) Next, adopting
the definitions given in Eqs. (5.3.2) and (5.91), the components of
$\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}$ that are analogous to those in Eqs.
(5.32a)-(5.32c) and (5.65a)-(5.65c) can be written as

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}$
$\displaystyle=2\pi\eta^{2}|n|\frac{Q^{2}}{\sqrt{P^{2}Q^{2}+U^{2}}}\omega^{-2},$
$\displaystyle\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}$
$\displaystyle=-2\pi\eta^{2}|n|\left[\frac{U}{\sqrt{P^{2}Q^{2}+U^{2}}}\omega^{-2}\mp\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\right],$
(5.94a) $\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}$
$\displaystyle=2\pi\eta^{2}|n|\left[\frac{U}{\sqrt{P^{2}Q^{2}+U^{2}}}\omega^{-2}\pm\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\right]\left(\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{1-P^{2}}\right),$
$\displaystyle\mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}$
$\displaystyle=2\pi\eta^{2}|n|\frac{P^{2}}{\sqrt{P^{2}Q^{2}+U^{2}}}\left[1-\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(U\pm\sqrt{P^{2}Q^{2}+U^{2}})^{2}}{P^{2}Q^{2}}\right]\left(\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{1-P^{2}}\right),$
(5.94b) $\displaystyle\mathcal{T}^{0}{}_{\theta}\sqrt{-g}$ $\displaystyle=\mp
2\pi\eta^{2}|n|r_{c|n|}Q\frac{\sqrt{1-\omega^{2}}}{\omega},$
$\displaystyle\mathcal{T}^{\sigma}{}_{\theta}\sqrt{-g}$ $\displaystyle=\pm
2\pi\eta^{2}|n|r_{c|n|}\frac{U\pm\sqrt{P^{2}Q^{2}+U^{2}}}{Q}\frac{\sqrt{1-\omega^{2}}}{\omega},$
(5.94c)

where we have defined $\omega^{2}(\sigma,t)$ as in Eq. (5.61) but with
$Q^{2}=Q^{2}(\sigma,t)$. The relevant conservation equations are Eqs.
(5.66a)-(5.66c). Once more, we note that the condition which ensures the
consistency of the field ansatz, in this case Eq. (5.92), is precisely the
condition required to ensure that the Lagrangian reduces to that of the
Nielsen-Olesen string at the level of the energy-momentum tensor.
For noncircular loops, the normalised components of the current density may be
defined as

$\displaystyle\mathcal{J}^{0}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi}\frac{U(\sigma,t_{i})\pm\sqrt{P^{2}(\sigma,t_{i})Q^{2}(\sigma,t_{i})+U^{2}(\sigma,t_{i})}}{Q^{2}(\sigma,t_{i})}\frac{Q^{2}}{\sqrt{P^{2}Q^{2}+U^{2}}}\dot{\Gamma},$
$\displaystyle\mathcal{J}^{\sigma}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi}\frac{U(\sigma,t_{i})\pm\sqrt{P^{2}(\sigma,t_{i})Q^{2}(\sigma,t_{i})+U^{2}(\sigma,t_{i})}}{Q^{2}(\sigma,t_{i})}\frac{\sqrt{P^{2}Q^{2}+U^{2}}}{Q^{2}}\partial_{\sigma}\Gamma,$
(5.95)

by analogy with Eq. (2.4.2), so that using the EOM in $\phi$ gives

$\displaystyle\mathcal{J}^{0}\sqrt{-g}=\pm\frac{e}{2\pi}\frac{U(\sigma,t_{i})\pm\sqrt{P^{2}(\sigma,t_{i})Q^{2}(\sigma,t_{i})+U^{2}(\sigma,t_{i})}}{Q^{2}(\sigma,t_{i})}\partial_{\sigma}\Gamma$
$\displaystyle=$
$\displaystyle-\mathcal{J}^{\sigma}\sqrt{-g}=\mp\frac{e}{2\pi}\frac{U(\sigma,t_{i})\pm\sqrt{P^{2}(\sigma,t_{i})Q^{2}(\sigma,t_{i})+U^{2}(\sigma,t_{i})}}{Q^{2}(\sigma,t_{i})}\dot{\Gamma}.$
(5.96)

by analogy with (2.4.2). The components of the $4$-current may again be
expressed in terms of $\mathcal{T}^{0}{}_{\theta}\sqrt{-g}$ and
$\mathcal{T}^{\sigma}{}_{\theta}\sqrt{-g}$, now given by Eq. (5.94c), and it
is straightforward to show that the current conservation equation is then
equivalent to Eq. (5.66c).
Although the full system of evolution equations, that is, the remaining Euler-
Lagrange equations of the Abelian-Higgs model, (5.3.2) and (5.92), and the
conservation equations, (5.94a)-(5.94c), appear incredibly complicated, we
will show that careful treatment allows us to demonstrate their consistency.
Furthermore, this will allow us to write the general solution for
$\Gamma(t,\sigma)$ in terms of $Q(t,\sigma)$, and its first and second
derivatives in $t$ and $\sigma$, via the imposition of a suitable ‘generalised
Finsler condition’.
As a first step towards demonstrating the self-consistency of Eqs.
(5.66a)-(5.66c), using the components $\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}$
obtained in Eqs. (5.94a)-(5.94c), 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 $Q$ and $\omega$, in addition to the EOM in $\Gamma$. Even
in this more general case, we still have three EOM and two quantities
($\dot{\omega}$ and $\partial_{\sigma}\omega$) we wish to eliminate, so this
poses no problem.
We begin the elimination of $\dot{\omega}$ and $\partial_{\sigma}\omega$ by
first writing each of the components of $\mathcal{T}^{\mu}{}_{\nu}\sqrt{-g}$
in Eqs. (5.94a)-(5.94b) 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

$\displaystyle\mathcal{T}^{0}{}_{0}\sqrt{-g}=\beta+\beta(\omega^{-2}-1),\ \ \
\mathcal{T}^{\sigma}{}_{0}\sqrt{-g}=\gamma+\delta(\omega^{-2}-1),$ (5.97a)
$\displaystyle\mathcal{T}^{0}{}_{\sigma}\sqrt{-g}=\epsilon+\zeta(\omega^{-2}-1),\
\ \ \ \mathcal{T}^{\sigma}{}_{\sigma}\sqrt{-g}=\eta+\theta(\omega^{-2}-1),$
(5.97b)

where the values of $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$
and $\theta$ are obtained by comparison of Eqs. (5.97a)-(5.97b) with Eqs.
(5.94a)-(5.94b), giving

$\displaystyle\beta=\frac{Q^{2}}{\sqrt{P^{2}Q^{2}+U^{2}}},$ (5.98)
$\displaystyle\gamma=\delta\mp 1,\ \ \
\delta=-\frac{U}{\sqrt{P^{2}Q^{2}+U^{2}}}\pm 1.$ (5.99)
$\displaystyle\epsilon=-\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{1-P^{2}}(\delta\mp
1),\ \ \ \zeta=-\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{1-P^{2}}\delta,$ (5.100)
$\displaystyle\eta=\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{\sqrt{P^{2}Q^{2}+U^{2}}},\
\ \ \theta=-\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{1-P^{2}}\frac{\delta^{2}}{\beta}.$
(5.101)

Equations (5.97a)-(5.97b) are formally equivalent to (2.215a)-(2.215b) and the
definitions of $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$ and
$\theta$ (5.98)-(5.101) are analogous to those in Eqs. (2.216)-(2.219).
Substituting from Eqs. (5.97a)-(5.97b) into Eqs. (5.66a)-(5.66b), rearranging
to make $\partial_{t}(\omega^{-2}-1)$ the subject, and equating the results,
gives

$\displaystyle[\beta(\dot{\zeta}+\partial_{\sigma}\theta)-\zeta(\dot{\beta}+\partial_{\sigma}\delta)](1-\omega^{2})+[\beta(\dot{\epsilon}+\partial_{\sigma}\eta)-\zeta(\dot{\beta}+\partial_{\sigma}\gamma)]\omega^{2}=0.$
(5.102)

where we have used the fact that $\zeta\delta-\beta\theta=0$, which may be
explicitly verified by direct substitution. This equation involves only
$\omega$ and is formally equivalent to (2.222).
Next, we interpret the function $Q(\sigma,t)$ as an effective ‘local’ radial
coordinate (i.e. depending on both $t$ and $\sigma$), and make the coordinate
transformation
$\left\\{t,r,\theta,\sigma\right\\}\rightarrow\left\\{t,x,Q,\sigma\right\\}$.
This corresponds to the change of line element

$\displaystyle{\rm d}s^{2}$ $\displaystyle=P^{2}(\sigma,t){\rm d}t-{\rm
d}r^{2}-r^{2}{\rm d}\theta^{2}-2U(\sigma,t){\rm d}t{\rm
d}\sigma-Q^{2}(\sigma,t){\rm d}\sigma^{2}$ (5.103) $\displaystyle\rightarrow
P^{2}(\sigma,t){\rm d}t-{\rm d}x^{2}-{\rm d}Q^{2}(\sigma,t)-2U(\sigma,t){\rm
d}t{\rm d}\sigma-Q^{2}(\sigma,t){\rm d}\sigma^{2}.$

We then impose the ‘generalised Finsler condition’, which relates the metric
components $g_{00}=P^{2}(\sigma,t)$, $g_{0\sigma}=-U(\sigma,t)$ and
$g_{\sigma\sigma}=-Q^{2}(\sigma,t)$ to both the temporal and spatial derivates
of the $Q-$coordinate, via the identifications

$\displaystyle P^{2}=1-\dot{\rho}^{2},\ \ \
Q^{2}=\rho^{2}+(\partial_{\sigma}\rho)^{2},\ \ \
U=\dot{\rho}\partial_{\sigma}\rho,$ (5.104)

where $\rho(\sigma,t)$ may be interpreted as the genuine local radius of the
string. Together with the conditions (5.104), Eq. (5.103) implies that the
metric is a generalised Finsler metric. This may be expressed as:

$\displaystyle{\rm d}s^{2}=\left(1-\left[\frac{\partial}{\partial
t}\left(\frac{Q^{2}-(P^{2}Q^{2}+U^{2})}{1-P^{2}}\right)^{1/2}\right]^{2}\right){\rm
d}t-{\rm d}x^{2}-{\rm d}Q^{2}-Q^{2}{\rm d}\sigma^{2},$ (5.105)

where the behaviour of the functions $P(\sigma,t)$, $Q(\sigma,t)$ and
$U(\sigma,t)$ must be reconstructed by solving the EOM for $\rho(\sigma,t)$.
The method used from here on is precisely analogous to that used in Sec. 2 and
we begin by rewriting the line element (5.105) as:

$\displaystyle{\rm d}s^{2}=(1-\dot{\rho}^{2}){\rm d}t^{2}-{\rm d}x^{2}-{\rm
d}\rho^{2}-(\rho^{2}+(\partial_{\sigma}\rho)^{2}){\rm d}\sigma^{2}.$ (5.106)

Next, we note that substituting for $(\dot{\epsilon}+\partial_{\sigma}\eta)$,
$(\dot{\beta}+\partial_{\sigma}\gamma)$, $\beta$ and $\zeta$ from Eqs.
(5.98)-(5.101), the terms in the second set of square brackets in Eq. (5.102)
are proportional to the quantity

$\displaystyle\chi=(1-\dot{\rho}^{2})\left(1-\frac{\partial^{2}_{\sigma}\rho}{\rho}\right)+\left(1+\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}}\right)\rho\ddot{\rho}+2\left[\frac{(\partial_{\sigma}\rho)^{2}}{\rho^{2}}-\frac{\dot{\rho}}{\rho}\partial_{\sigma}\rho\partial_{\sigma}\dot{\rho}\right],$
(5.107)

which was defined previously in Eq. (2.224). It may be verified that $\chi=0$
is the unique EOM obtained from the conservation equations (5.66a)-(5.66c) in
the limit $\omega^{2}\rightarrow 1$. In other words, this corresponds to the
macroscopic evolution equation of either a Nambu-Goto or Nielsen-Olesen
string, at least in the classical theories, ignoring gravitational effects. We
then have

$\displaystyle\beta(\dot{\epsilon}+\partial_{\sigma}\eta)-\zeta(\dot{\beta}+\partial_{\sigma}\gamma)=\left[\beta\partial_{\sigma}\rho-\rho^{2}\dot{\rho}\delta\right]\times\frac{\rho^{3}}{[(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}]^{\frac{3}{2}}}\chi,$
(5.108)

The first set of square brackets in Eq. (5.102) may again be written purely in
terms of the quantities $\beta$ and $\delta$ (c.f. (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:

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

This is formally equivalent to (2.226). We see, therefore, that the term
proportional to $(1-\omega^{2})/\omega^{2}$ quantifies the contribution of the
superconducting current. Specifically, this is expressed via the embedding
function for the lines of constant phase, $\Gamma(\sigma,t)$, in terms of the
associated parameter $\omega^{2}(\sigma,t)$. This, in turn, may be thought of
as a kind of ‘current parameter’, whereby $\omega^{2}=1$ implies
$J^{0}=-J^{z}=0$, while $\omega^{2}\rightarrow 0$ gives
$J^{0}=-J^{z}\rightarrow\infty$. (As noted in Sec. 2.) It is straightforward
to show 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. (5.109).
Substituting the generalised Finsler conditions (5.104) into Eq. (5.92), we
obtain

$\displaystyle(1-\dot{\rho}^{2})(\partial_{\sigma}\Gamma)^{2}-(\rho^{2}+(\partial_{\sigma}\rho)^{2})^{2}\dot{\Gamma}^{2}+2\dot{\rho}\partial_{\sigma}\rho\dot{\Gamma}\partial_{\sigma}\Gamma=0,$
(5.110)

and it is straightforward, but tedious, to demonstrate that, under these
conditions, this is precisely equivalent to Eq. (5.3.2) (see analogous results
in Sec. 2). Thus, Eqs. (5.109) and (5.110) alone are sufficient to completely
specify the dynamics of the string.
We now solve for $\Gamma(\sigma,t)$ in terms of $\rho(\sigma,t)$ and its
derivatives. For the sake of notational simplicity, we first rewrite Eq.
(5.109) as

$\displaystyle-Z\omega^{2}+Y(1-\omega^{2})=0,$ (5.111)

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

$\displaystyle\left(\frac{Z}{Y}\right)=\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.112)

c.f. Eqs. (2.227)-(2.228). Using the definition of $\omega^{2}(\sigma,t)$
(5.61), but with $Q(\sigma,t)$ given by Eq. (5.104), Eq. (5.112) is equivalent
to

$\displaystyle(\partial_{\sigma}\Gamma)^{2}=\frac{1}{r_{c|n|}^{2}}(\rho^{2}+(\partial_{\sigma}\rho)^{2})\left(\frac{1-\omega^{2}}{\omega^{2}}\right).$
(5.113)

This expression has physical significance: as we we will see, it allows us to
interpret the EOM in $\Gamma(t,\sigma)$ as dispersion relation governing
longitudinal ‘waves’ (i.e. compressions and rarefractions) in the helical
phase lines within the string core, as in the case of long strings and
circular loops. We then have

$\displaystyle\dot{\Gamma}^{2}=\frac{1}{r_{c|n|}^{2}}\left(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}\right)^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.114)

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

Clearly, Eqs. (5.113)-(LABEL:phi_lastX) are precisely analogous to
(2.229)-(LABEL:phi_last*).
For circular strings ($\partial_{\sigma}\rho=0$), we recover

$\displaystyle
n_{\sigma}^{2}=\frac{\rho^{2}}{r_{c|n|}^{2}}\left(\frac{1-\dot{\rho}^{2}+\rho\ddot{\rho}}{1-\dot{\rho}^{2}-\rho\ddot{\rho}}\right)=\frac{\rho^{2}}{r_{c|n|}^{2}}\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.116)

which is equivalent to Eq. (5.70), using the expression for $\omega^{2}(t)$
from Eq. (5.61). This motivates the following definitions for noncircular
loops:

$\displaystyle(n_{\sigma}^{\rm eff})^{2}=\frac{(v_{\theta}^{\rm
eff})^{2}(l_{\sigma}^{\rm
eff})^{2}}{(v_{\sigma}^{eff})^{2}(2\pi)^{2}r_{c|n|}^{2}}=(\partial_{\sigma}\Gamma)^{2},$
(5.117)

where

$\displaystyle(l_{\sigma}^{\rm
eff})^{2}=(2\pi)^{2}(\rho^{2}+(\partial_{\sigma}\rho)^{2}),$ (5.118)

and

$\displaystyle\frac{(v_{\theta}^{\rm eff})^{2}}{(v_{\sigma}^{\rm
eff})^{2}}=\left(\frac{1-\omega^{2}}{\omega^{2}}\right).$ (5.119)

Here $n_{\sigma}^{\rm eff}$ and $l_{\sigma}^{\rm eff}$ denote the effective
local winding number and string radius, respectively, for given values of
$\sigma$ and $t$, while $v_{\theta}^{\rm eff}$ and $v_{\sigma}^{\rm eff}$
denote the local velocities of the phase lines in the $\theta-$direction
(perpendicular to the string core axis) and the $\sigma-$direction (parallel
to the string core axis).
Together with Eqs. (5.117)-(5.119), the following definitions then form a
self-consistent set:

$\displaystyle(l_{\sigma}^{\rm eff})^{2}=(n_{\sigma}^{\rm
eff})^{2}(\lambda_{\sigma}^{\rm eff})^{2},\ \ \ (v_{\theta}^{\rm
eff})^{2}(\lambda_{\sigma}^{\rm eff})^{2}=(v_{\sigma}^{\rm
eff})^{2}(2\pi)^{2}R^{2},$ (5.120) $\displaystyle(k_{\sigma}^{\rm
eff})^{2}=\frac{(2\pi)^{2}}{(\lambda_{\sigma}^{\rm eff})^{2}},$ (5.121)
$\displaystyle(\omega_{\sigma}^{\rm eff})^{2}=(v_{\sigma}^{\rm
eff})^{2}(k_{\sigma}^{\rm eff})^{2}\equiv(\omega_{\theta}^{\rm
eff})^{2}=\frac{(v_{\theta}^{\rm eff})^{2}}{r_{c|n|}^{2}}=\dot{\Gamma}^{2},$
(5.122)

which are compeletely analogous to their counterparts in the wound-string
model. Combining Eqs. (5.119) and (5.120), we then have

$\displaystyle(\lambda_{\sigma}^{\rm
eff})^{2}=(2\pi)^{2}r_{c|n|}^{2}\left(\frac{1-\omega^{2}}{\omega^{2}}\right).$
(5.123)

where $\lambda_{\sigma}^{\rm eff}(\sigma,t)$ denotes the effective local
wavelength of the phase line ‘twists’. Eq. (5.123) reproduces the tensionless
condition, $(\lambda_{\sigma}^{\rm
eff})^{2}=\lambda_{\sigma}=(2\pi)^{2}r_{c|n|}^{2}$, when
$\omega^{2}(\sigma,t)=1/2$ for all $\sigma,t$.
We would like to be able to verify that the expressions for $(l_{\sigma}^{\rm
eff})^{2}$, $(n_{\sigma}^{\rm eff})^{2}$, $(v_{\sigma}^{\rm eff})^{2}$,
$(v_{\theta}^{\rm eff})^{2}$, $(\lambda_{\sigma}^{\rm eff})^{2}$ and
$(\omega_{\sigma}^{\rm eff})^{2}\equiv(\omega_{\theta}^{\rm eff})^{2}$ reduce
to their equivalents for both circular loops and straight strings, in
appropriate limits. However, from Eqs. (5.117)-(5.123), we do not have enough
information to write down explicit expressions for $(\omega_{\sigma}^{\rm
eff})^{2}$, $(v_{\sigma}^{\rm eff})^{2}$, or $(v_{\theta}^{\rm eff})^{2}$. By
analogy with Sec. 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_lastX). Hence, we have that

$\displaystyle\Gamma(\sigma,t)=\int n_{\sigma}^{\rm eff}{\rm
d}\sigma+\int\omega_{\sigma}^{\rm eff}{\rm d}t,$ (5.124)

where

$\displaystyle(n_{\sigma}^{\rm
eff})^{2}=\frac{1}{r_{c|n|}^{2}}(\rho^{2}+(\partial_{\sigma}\rho)^{2})\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.125) $\displaystyle(\omega_{\sigma}^{\rm
eff})^{2}=\frac{1}{r_{c|n|}^{2}}\left(\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}})^{2}}{\rho^{2}+(\partial_{\sigma}\rho)^{2}}\right)\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.126)

and the sign of $\omega_{\sigma}^{\rm eff}$ relative to $n_{\sigma}^{\rm eff}$
depends on the sign of $\dot{\Gamma}$ relative to $\partial_{\sigma}\Gamma$.
These are completely equivalent to their wound-string counterparts
(2.240)-(2.242), under the exchange $\rho\leftrightarrow a\rho$ and
$r_{c|n|}\leftrightarrow R$.
It may then be shown explicitly that the expressions for $(l_{\sigma}^{\rm
eff})^{2}$, $(n_{\sigma}^{\rm eff})^{2}$, $(v_{\sigma}^{\rm eff})^{2}$,
$(v_{\phi}^{\rm eff})^{2}$, $(\lambda_{\sigma}^{\rm eff})^{2}$ and
$(\omega_{\sigma}^{\rm eff})^{2}\equiv(\omega_{\phi}^{\rm eff})^{2}$, defined
in Eqs. (5.117)-(5.118) and (5.120)-(5.122), reduce to their equivalents, for
circular loops and long strings, in the limits
$\partial_{\sigma}\rho\rightarrow 0$ and $(2\pi)^{2}\rho\rightarrow\Delta$,
respectively. As in previous Sections, 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 $\rho(\sigma,t)$.
We now note that, interpreting $\rho(t,\sigma)$ as the genuine radial
coordinate for the loop, the associated conservation equation takes the form

$\displaystyle\partial_{0}(T^{0}{}_{\rho}\sqrt{-g})+\partial_{\sigma}(T^{\sigma}{}_{\rho}\sqrt{-g})-\frac{1}{2}\partial_{\rho}g_{\sigma\sigma}T^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=0.$ (5.127)

This, in turn, requires

$\displaystyle\mathcal{T}^{0}{}_{\rho}\sqrt{-g}$
$\displaystyle=-2\pi\eta^{2}|n|\frac{\rho^{2}\dot{\rho}}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}}\omega^{-2}\mp
2\pi\eta^{2}|n|\partial_{\sigma}\rho\left(\frac{1-\omega^{2}}{\omega^{2}}\right),$
(5.128a) $\displaystyle\mathcal{T}^{\sigma}{}_{\rho}\sqrt{-g}$
$\displaystyle=2\pi\eta^{2}|n|\frac{\partial_{\sigma}\rho}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}}\left[1-\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}))^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})}\right]$
$\displaystyle\pm
2\pi\eta^{2}|n|\dot{\rho}\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}))^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})},$
(5.128b) $\displaystyle\mathcal{T}^{\sigma\sigma}\sqrt{-g}$
$\displaystyle=-2\pi\eta^{2}|n|\frac{(1-\dot{\rho}^{2})}{\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2})}}\left[1-\left(\frac{1-\omega^{2}}{\omega^{2}}\right)\frac{(\dot{\rho}\partial_{\sigma}\rho\pm\sqrt{(1-\dot{\rho}^{2})\rho^{2}+(\partial_{\sigma}\rho)^{2}}))^{2}}{(1-\dot{\rho}^{2})(\rho^{2}+(\partial_{\sigma}\rho)^{2})}\right].$
(5.128c)

The expression for $\mathcal{T}^{\sigma\sigma}\sqrt{-g}$ follows directly from
the definitions in Eqs. (5.89)-(5.91), whereas it may be shown that those for
$\mathcal{T}^{0}{}_{\rho}\sqrt{-g}$ and
$\mathcal{T}^{\sigma}{}_{\rho}\sqrt{-g}$ are necessary for consistency. Using
these definitions, Eq. (5.127) is equivalent to (5.109). Yet again, _all_
expressions obtained in this Section are equivalent to those obtained for
arbitrary planar loops of wound-string in Sec. 2, under the identifications
$\Gamma(\sigma,t)\leftrightarrow\varphi(\sigma,t)$ and
$r_{c|n|}\leftrightarrow R$, together with $\rho(\sigma,t)\leftrightarrow
a\rho(\sigma,t)$, etc, where necessary in order to account for a warp factor
of $a^{2}<1$ in the string theory model.
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 phase / magnetic field-line ‘twists’, 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. (5.117)-(5.118) and (5.120)-(5.122), namely
$\langle l_{\sigma}^{2}\rangle(t)$, $\langle n_{\sigma}^{2}\rangle(t)$,
$\langle v_{\sigma}^{2}\rangle(t)$, $\langle v_{\phi}^{2}\rangle(t)$,
$\langle\lambda_{\sigma}^{2}\rangle(t)$ and
$\langle\omega_{\sigma}^{2}\rangle(t)\equiv\langle\omega_{\phi}^{2}\rangle(t)$,
as well as a generalized expression for $\Omega^{2}(t)$, which is valid for
arbitrary current-carrying loop configurations. As before, based on our
previous findings, we expect the condition of zero net tension to correspond
to $\Omega^{2}=1/2$, for which
$a^{2}\langle\lambda_{\sigma}^{2}\rangle=(2\pi)^{2}r_{c|n|}^{2}$ and
$\langle\omega_{\sigma}^{2}\rangle=1/r_{c|n|}^{2}$ for all $t$. Such a
definition would allow us to write the constants of motion and _bulk_
properties of noncircular superconducting strings in terms of spatially-
averaged values of the parameters which determine the microscopic ‘internal’
structure of the string core. 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.

###  5.4 Summary of the defect-string model

We have shown that, by adopting an appropriate ansatz for the Abelian-Higgs
fields, and imposing an appropriate Finsler condition on the space-time metric
such that

$\displaystyle{\rm d}s^{2}={\rm d}t^{2}-{\rm d}r^{2}-r^{2}{\rm
d}\theta^{2}-\rho^{2}(t){\rm d}\sigma^{2}$ (5.129) $\displaystyle\rightarrow$
$\displaystyle{\rm d}\tilde{s}^{2}=(1-\dot{\rho}^{2}(t)){\rm d}t^{2}-{\rm
d}r^{2}-r^{2}{\rm d}\theta^{2}-\rho^{2}(t){\rm d}\sigma^{2},$

the evolution equations for a circular superconducting string loop can be
derived, exactly, from the standard Abelian-Higgs field equations. The
appropriate ansatz for scalar field describes rotating ‘twists’ in the lines
of constant phase, resulting in the existence of a second topological
invariant, $N=\int\partial\vartheta/\partial\sigma|_{\theta={\rm const.}}{\rm
d}\sigma$ ($N\in\mathbb{Z}$), analogous to that found in the case of chiral
strings [44, 50].
The time-dependent embedding of the lines of constant phase that are situated
at the edge of the string core, $r=r_{c|n|}$, is exactly analogous to the
embedding of the higher-dimensional windings of an $F$-string with identical
macroscopic motion in the large space-time dimensions: the only difference is
the replacement $R\leftrightarrow r_{c|n|}$, where $R$ is the radius of the
compact internal space, and $a\rho(t)\rightarrow\rho(t)$, where necessary, due
to the presence of warping caused by flux-compactifications. We conclude that
the similarity (dissimilarity) of the embedding of the lines of constant
phase, in the defect-string model, and the higher-dimensional windings, in the
$F$-string model, is the key determinant of the similarity (dissimilarity) of
the macroscopic string dynamics. Crucially, this similarity is quantitative:
when the embedding of the phase-lines is exactly equivalent to the embedding
of the windings, under the exchange of variables given above, then the
macroscopic dynamics and all observable string parameters, including the
$(3+1)$-dimensional energy and momentum densities, pressures, and shears, are
identical. In this case, there is no observable difference, from a
$(3+1)$-dimensional perspective, between the species of topological defect
string and its higher-dimensional $F$-string counterpart.
We conjecture that a similar equivalence holds in even more complicated
scenarios, for example, when the modified Abelian-Higgs strings considered
here are replaced by chiral strings, or vortons, etc. [92, 93, 94, 95, 96],
and when simple $F$-strings are replaced by general $(p,q)$-strings, i.e.,
bound states of $p$ $F$-strings and $q$ $D$-strings [30, 31, 32, 33]. For more
complex defect string species, one must consider the embedding(s) of ‘twisted
lines’ in which the relevant order parameter(s) of the model remain constant.
For more complex $F/D$-string bounds states, one must substitute the relevant
effective string tensions and, where necessary, consider multiple embeddings,
possibly corresponding to the presence of multiple order parameters in the
defect string counterpart.
In the case of noncircular loops, we imposed an appropriate generalised
Finsler condition,

$\displaystyle{\rm d}s^{2}={\rm d}t^{2}-{\rm d}r^{2}-r^{2}{\rm
d}\theta^{2}-\rho^{2}(t,\sigma){\rm d}\sigma^{2}$ $\displaystyle\rightarrow$
$\displaystyle{\rm d}\tilde{s}^{2}=(1-\dot{\rho}^{2}(t,\sigma)){\rm
d}t^{2}-{\rm d}r^{2}-r^{2}{\rm
d}\theta^{2}-\dot{\rho}(t,\sigma)\partial_{\sigma}\rho(t,\sigma){\rm d}t{\rm
d}\sigma-(\rho^{2}(t,\sigma)+(\partial_{\sigma}\rho)^{2}(t,\sigma)){\rm
d}\sigma^{2},$

and used this together with the previous ansatz for the microscopic structure
of the Abelian-Higgs fields in the superconducting string. Imposing this
condition, we again found that the fundamental Abelian-Higgs field equations
reduce to those for the standard Nielsen-Olsen string, plus the EOM describing
the macroscopic dynamics of an arbitrary planar wound-string loop. Yet again,
the correspondence is exact: the key determinants of equivalent dynamics are
the embeddings of the lines of constant phase (in the defect-string model) and
the higher-dimensional windings (in the $F$-string case). Equivalent
embeddings, under the replacements $R\leftrightarrow r_{c|n|}$ and
$a\rho(t)\rightarrow\rho(t)$, give identical macroscopic dynamics, observable
energy and momentum densities, pressure and shears.
Though tentative, these results suggest that the method we have outlined above
is valid much more generally. If so, it may be extended to both arbitrary
string configurations and arbitrary defect and $F/D$-string species in future
works. This would allow us to determine, exactly, which string species are
distinguishable, and which are not, via their observable cosmological and
astrophysical signatures.

##  6 Conclusions

We have outlined a detailed model of wound $F$-strings, in a physically
intuitive coordinate system based on cylindrical polars in the large space-
time dimensions, and have identified the key physical parameters which
determine their macroscopic dynamics and physical observables. These include
the effective $(3+1)$-dimensional energy and momentum densities, pressures,
and shears, which may be expressed compactly in terms of the parameter
$\omega^{2}(t,\sigma)\in[0,1]$. This represents the local fraction of the
string length lying parallel to the macroscopic directions, so that
$\omega^{2}=1$ represents an unwound string and the limit
$\omega^{2}\rightarrow 0$ represents a string contained entirely within the
compact space.
Equivalently, for fixed winding radius $R$, the local fraction
$\omega^{2}(t,\sigma)$ can be expressed in terms of an effective local winding
number, $n_{\sigma}^{\rm eff}(t,\sigma)$, and wavelength,
$\lambda_{\sigma}^{\rm eff}(t,\sigma)$, and the equation of motion for the
angular embedding coordinate in the compact internal space takes the form of a
canonical dispersion relation. This is, essentially, the dispersion relation
for waves ‘in’ the string, i.e., for compressions and rarefractions in the
rotating higher-dimensional windings which, from a $(3+1)$-dimensional
perspective, appear as superconducting currents. However, our treatment goes
beyond the usual paradigm of dimensional reduction and we consider the full
higher-dimensional string configuration, rather than integrating out the
higher-dimensional variables to construct an approximate effective action [49,
69].
This motivates us to consider a similarly detailed ansatz for superconducting
defect strings, i.e., one that is capable of describing both the macroscopic
‘external’ evolution of the string, as an extended object, and the evolution
of the ‘internal’ variables that determine the microscopic structure of the
string core. This is equivalent to describing the macroscopic string dynamics
purely in terms of the evolution of fundamental fields, again without the use
of an approximate effective action. The major advantage of this approach is
that it allows us, in conjunction with our previous results for wound-strings,
to determine exactly when two inequivalent string species are observationally
indistinguishable, from a $(3+1)$-dimensional perspective.
We found two key factors that determine whether a given wound-string
configuration is distinguishable from a superconducting defect string. These
are (a) the embedding of the higher-dimensional windings (for $F$-strings) and
(b) the embedding of the ‘twists’ in the lines of constant phase along the
string length (for defects). When the two embeddings are equivalent under the
exchange $r_{c|n|}\leftrightarrow R$, where $r_{c|n|}$ is the radius of the
defect-string core and $R$ is the radius of the windings in the compact space,
the dispersion relation for the superconducting current in the former exactly
matches the dispersion relation that governs the evolution of the windings in
the latter. In this scenario, it is physically reasonable to expect the
macroscopic dynamics of each string species to also be identical, thus
rendering them indistinguishable to observers in $(3+1)$ dimensions: but how
can such an exact equivalence be demonstrated? This is the question that was
asked and answered in the final section of our paper, and its solution is the
main achievement of this work.
We have shown that, when equivalent embeddings for the phase-lines and higher-
dimensional windings are imposed, the same equations of motion for the
macroscopic string variables, e.g., the local effective loop radius
$\rho(t,\sigma)$, can be obtained in each model. Remarkably, with the use of
an appropriate ansatz, the Euler-Lagrange equations for the fundamental
Abelian-Higgs fields reduce to the EOM of the standard Nielsen-Olesen string
plus the EOM for the macroscopic string radius, $\rho(t,\sigma)$, obtained
from the wound-string model.
However, there is a catch. In order to derive the correct EOM for
$\rho(t,\sigma)$ from the Abelian-Higgs field equations, we must impose an
appropriate Finsler condition on the space-time metric, for circular loops,
and an appropriate generalised Finsler condition for non-circular loops.
Mathematically, this trick is equivalent to modelling a time-evolving string,
embedded in a fixed background geometry, as a fixed (comoving) string embedded
in a time-evolving background. Interestingly, in the case of circular loops,
the effective space-time background required is of Finsler type, with an
effective Finsler metric of the form $\tilde{g}_{\mu\nu}(x,\partial_{\tau}x)$.
For non-circular loops, an effective generalised Finsler metric of the form
$\tilde{g}_{\mu\nu}(x,,\partial_{\tau}x,\partial_{\sigma}x)$ must be imposed.
These results demonstrate the potential importance of these novel geometric
structures in modelling complex physical systems in astrophysics and
cosmology, but also hint at a possible deeper connection between the
mathematics of string theory, including higher-dimensional embeddings [100,
101], and the mathematics of Finsler and generalised Finsler spaces [102,
103]. In future work we will attempt to define, in more abstract and general
terms, the conditions under which a given higher-dimensional configuration of
a string, field, or other extended object (e.g. a $D$-brane), embedded in a
physical pseudo-Riemannian space, can be described, from a $(3+1)$-dimensional
viewpoint, in terms of an effective Finsler or generalised Finsler geometry.

Finally, because the text of this paper is rather long and involved, we
present a bullet point summary of the our main conclusions, below, for added
clarity. The new and novel results presented in this work are:

  1. 1.

The demonstration that the macroscopic EOM for current-carrying strings, or,
equivalently, strings with higher-dimensional windings, can be obtained from a
microscopic field theory ansatz, without the the need for the standard
approximations used in constructing an effective action.

  2. 2.

This provides a unified framework for studying various string-theoretic and
field-theoretic string species in which both macroscopic ‘external’ and
microscopic ‘internal’ variables are treated at the same level, i.e, , without
integrating out the latter in order to obtain approximate dynamical equations
for the former.

  3. 3.

This, in turn, permits us to identify the precise conditions under which
string-theoretic and field-theoretic string species become indistinguishable,
observationally, to a macroscopic observer.

  4. 4.

For Nambu-Goto strings with higher-dimensional windings and superconducting
Abelian-Higgs strings this occurs when the helical embedding of the windings
exactly matches the embedding of the lines of constant phase in the vortex-
string core. However, a similar method of comparison can be used, in
principle, to compare any two string-theoretic and field-theoretic string
species.

  5. 5.

Remarkably, the derivation of the macroscopic EOM for the field-theoretic
string species requires us to model these in an effective Finsler or
generalised Finsler geometry, where former applies to circular string loops
and the latter to non-circular configurations.

  6. 6.

This demonstrates the utility of these ‘exotic’ geometries in modelling
complex physical systems in cosmology and astrophysics, but the exact
mechanism behind this correspondence, i.e., why it works, requires further
investigation.

  7. 7.

Nonetheless, the correspondence discovered here, in a few test cases, hints at
a deeper and as yet unexplored connection between Finsler-type geometries and
string theory, perhaps along the lines considered decades ago by the
mathematician Matsumoto [104]. Though more work is needed to confirm this, it
opens up an interesting new direction for research in both string theory and
the theory of topological defects.

Acknowledgments

My sincere thanks to Tiberiu Harko for help, advice, and interesting
discussions, throughout the course of this project. This work was supported by
the Natural Science Foundation of Guangdong Province, grant no. 008120251030.

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