#  Cosmic adventures in stringland:
Bosonic and fermionic scattering in gravitating cosmic string spacetimes
Departamento de Física
Universidade Federal de Pernambuco

Marcos Vinicius Santos Silva

(08/11/2021)

Cosmic adventures in stringland

Scattering of bosonic and fermionic fields in gravitating cosmic string
spacetimes

Marcos Vinicius Santos Silva

A thesis presented for the degree of
Master in Physics

Departamento de Física
Universidade Federal de Pernambuco
Brazil
08/11/2021

This thesis was supervised by

Azadeh Mohammadi,

and was examined internally by

Fernando Roberto de Luna Parisio Filho

and externally by

Ilya Shapiro

on the
$8^{\text{th}}$ of December, 2021.

Agradecimentos

À minha mãe, Eliana, por ter me apoiado em todos os caminhos tortos que decidi
seguir na vida. Por ser minha maior inspiração em tudo. E por me permitir
tentar. Enfim, por existir.
À minha irmã Elisangela, por ter me incentivado a estudar desde cedo e por ter
sido minha primeira referência.
À Maria Clara por ter me feito ver as coisas com mais cores e detalhes. Por
ter me tornado uma pessoa mais aberta, mais alegre e cheia de vida.
À Dorinha e Vladimir, que tanto me ajudaram na odisséia que foi morar em
Recife. Obrigado pelas ajudas, conversas e comidas.
À Azadeh por ter confiado em mim, pela paciência e compreensão na pesquisa e
na vida.
À Joan e Saullo, que mesmo após tantas idas e vindas da vida ainda me trazem
bons momentos. Obrigado pelas conversas, ajudas, sessões de cinema no fim da
noite, pelos lanches da madrugada, e por permanecerem comigo por tanto tempo.
À Marta Menezes, a matemática do bem, pelas conversas e apoios que têm durado
tanto tempo apesar da distância.
Aos amigos que Recife me deu: Kivia e Henrique, que sempre estiveram por perto
pra tudo.
À Julia Medeiros por ter me abrigado num dos momentos mais difíceis desse
mestrado insano.
Ao grupinho dos Coaches Quanticos: Matheus Valença, Lucas Gabriel, Ricardo
Silveira e Hugo Estevão, pelo apoio quase incondicional, pelas reflexões,
indicações de músicas, origamis, resoluções de listas, bebedeiras, piadas e
trocadilhos. Sou eternamente grato por tudo isso.
À Nicolas Pessoa e João Guilherme pela imensa ajuda computacional e por tornar
minha caminhada um pouco mais confortável.
Aos muitos amigos que o Departamento de Física me trouxe, Ruben, Gustavo,
Naudson, Tertius, Pollyanna, Matheus Martins, João Rebouças, e tantos outros.
Obrigado pelas conversas, palavras de apoio, brincadeiras, saídas pro bar, e
principalmente por tornar o DF-UFPE mais humano.

A todos, citados e não-citados, aquele abraço!

Cosmic adventures in stringland

Scattering of bosonic and fermionic fields in gravitating cosmic string
spacetimes

Marcos Vinicius Santos Silva

Abstract

Cosmic strings are line-like topological defects expected to have formed
during cosmological phase transitions in the early universe when the Higgs
field acquired a non-vanishing expectation value. Because of being thin, on a
cosmological scale, their gravitational effects are usually studied using the
wire approximation, in which the string has a negligible width. Although the
wire approximation seems reasonable in the cosmological context, local
gravitational interaction with matter fields can be significant when
considering a non-vanishing radius. Generally, the spacetime of a gravitating
vortex is flat at the center, has a localized curvature at some finite
distance from the core, and is flat and conical far from the center. The
conical structure is characterized by an angular variable ranging from 0 to
less than $2\pi$, i.e., the space has an angular deficit. This work shows that
when considering a scalar field scattering in a gravitating cosmic string
spacetime, the standard partial-wave approach’s scattering amplitude is
singular. In order to avoid the divergence caused by the spacetime
asymptotically conical structure, we propose a modification of the asymptotic
ansatz in the partial-wave formalism and find the corrections in the phase-
shift and total scattering cross-section. We also developed a toy model for
the spacetime metric of a cosmic string and showed how local interaction with
the vortex gauge field affects the scalar field total cross-section. Then we
apply this formalism to a Dirac field and show the explicit formula for the
fermionic total cross-section. Finally, we study the scattering of bosonic and
fermionic fields in the spacetime of an abelian and a nonabelian gravitating
cosmic strings and show that the cross-sections have damped oscillations. In
order to understand the origin of this behavior, we used the aforementioned
toy model to show that the spacetime particular asymptotical structure causes
the observed oscillations.

Keywords: boson; fermion; cosmic string; gravitation; topological defect;
phase transition.

Aventuras cósmicas no país das cordas

Espalhamento de campos bosônicos e fermiônicos em espaços-tempo de cordas
cósmicas

Marcos Vinicius Santos Silva

Resumo

Cordas cósmicas são defeitos topológicos cilindricamente simétricos que
espera-se que tenham se formado durante algumas transições de fase no universo
primordial quando o campo de Higgs adquiriu um valor não-nulo. Por serem
estreitas, numa escala cosmológica, elas são comumente estudadas utilizando a
aproximação de fio na qual considera-se que o raio da corda é nulo. Embora
essa aproximação faça sentido num contexto cosmológico, a interação
gravitacional local desses objetos com campos de matéria pode ser muito
relevante quando consideramos um raio não-nulo. De forma geral, o espaço-tempo
ao redor de um vórtice é plano no centro, possui curvatura localizada a uma
distância finita da origem, e é plano e cônico longe do centro. A estrutura
cônica é caracterizada por uma coordenada angular variando de 0 a menos que
$2\pi$, isto é, o espaço-tempo possui um déficit angular. Nosso trabalho
mostra que quando consideramos o espalhamento de um campo escalar no espaço-
tempo de uma corda cósmica, a amplitude de espalhamento encontrada a partir da
abordagem usual de ondas parciais é divergente. Para evitar a singularidade
causada pela estrutura assintoticamente cônica do espaço-tempo, nós propomos
uma modificação do ansatz assintótico no formalismo de ondas parciais e
encontramos correções no desvio de fase e na seção de choque total. Também
desenvolvemos um modelo simplificado para a métrica de uma corda cósmica e
mostramos como a interação com o campo de calibre do vórtice afeta a seção de
choque do campo escalar. Após isso aplicamos o formalismo para um campo de
Dirac e explicitamente mostramos a fórmula da seção de choque para o caso
fermiônico. Por fim, estudamos o espalhamento de campos bosônicos e
fermiônicos no espaço-tempo de uma corda cósmica abeliana e não-abeliana e
vimos que a seção de choque apresenta oscilações amortecidas. Para entender a
origem desse comportamento usamos nosso modelo simplificado para mostrar que a
estrutura assintótica do espaço-tempo causa essas oscilações.

Palavras-chave: boson; fermion; corda cósmica; gravitação; defeito topologico;
transições de fase;

###### Contents

  1. 1 Introduction
  2. 2 Down the cosmic hole: Topological defects and string formation
    1. 2.1 Topological defects
      1. 2.1.1 The simplest model: $\phi^{4}$
      2. 2.1.2 Derrick sends in a little bill: Derrick’s theorem
    2. 2.2 Spontaneous Symmetry Breaking
      1. 2.2.1 A simple model: Goldstone
      2. 2.2.2 A hard one: abelian-Higgs
      3. 2.2.3 Global strings
      4. 2.2.4 Local strings
    3. 2.3 Nielsen-Olesen vortex
      1. 2.3.1 Properties of the Nielsen-Olesen vortex
    4. 2.4 Kibble and the cosmic tale
      1. 2.4.1 The role of temperature
      2. 2.4.2 Kibble mechanism
  3. 3 Gravitating vortices
    1. 3.1 The ideal string
      1. 3.1.1 Wire approximation
      2. 3.1.2 Gravitating ideal string
      3. 3.1.3 Topology induces physics
    2. 3.2 Extended gravitating vortices
  4. 4 Scattering cross-section in gravitating cosmic string spacetimes
    1. 4.1 Bosonic cross-section
      1. 4.1.1 Scalar field scattering
      2. 4.1.2 Toy model
    2. 4.2 Fermionic cross-section
  5. 5 Bosonic and fermionic scattering in a gravitating cosmic string spacetime
    1. 5.1 Bosonic scattering
    2. 5.2 Fermionic scattering
  6. 6 Conclusion

## 1\. Introduction

In 1976, Thomas Kibble showed that cosmological phase transitions during the
early universe should lead to the formation of topological defects, namely
cosmic strings, domain walls and monopoles [1]. Among these, cosmic strings
have been showed to be the most promising ones [2, 3]. Along the years many
attempts have been made in order to develop reasonable observational
constraints on the parameters of real cosmic strings [4, 5]. Some works
suggested cosmic strings as possible sources for gravitational lensing
phenomena [6]; and although some of these claims have been debunked [7],
recent data from 12 years of observation of the NANOGrav collaboration [8]
suggests that cosmic strings gravitational interaction might be affecting the
detection of pulsar blinking [9, 10]. It appears the subject of cosmic strings
is about to be revived. However, the attention toward cosmic strings is not
exclusively by cosmological reasons. These objects have strong condensed
matter counterparts. The U(1) string solution, for instance, is also present
in the theory of superconductors [11, 12, 13]. Furthermore, even the
gravitational interaction of cosmic strings, in the wire approximation, have
condensed matter analogues called wedge disclinations and edge dislocations.
Therefore the interest in cosmic strings/vortex solutions, are _not_ exclusive
for the high-energy physicists.

Many cosmic string solutions have been found in many diverse models, specially
during the 20th century, such as the Nielsen-Olesen vortex in 1973 [14],
Semilocal strings in 1991 [15] and Eletroweak strings in 1993 [16].
Nonetheless their gravitational interaction with nearby matter have been most
studied using the wire approximation. This is possibly because it is too hard
to study the gravity-coupled system, since one usually has to solve 5 (or
more) coupled partial differential equations. Hence one has to resort to
numerical methods to have even a basic understanding of the field and metric
solutions. Even the flat-spacetime cosmic string solutions have not been
analytically solved completely. With a symmetry-breaking model in hands one
frequently resorts to proving the existence of such a line-like object without
actually finding the solution analytically. Only in some restricted situations
the equations of motion are analytically solvable, such that it is of no
surprise that the gravity-coupled system is only possible to be studied using
numerical methods.

Although Garfinkle [17] made significant advances in understanding the
asymptotical conical limit of an abelian string spacetime, there was still
plenty of aspects to be understood. For instance, how do the vortex internal
parameters affect the asymptotical metric limit or the curvature profile? Is
the conical limit a general property of any cosmic string solution? Do the
gravitational description changes significantly when we consider the vortex
size to be non-negligible? These are all legitimate questions, but one needs
to build a comprehensive understanding of gravitating cosmic string solutions
to answer them. The works of [18] and [19] on the abelian-Higgs model appeared
as a light in this long cosmic journey. [18] established the default notation
in studying gravitationally extended vortex solutions of the abelian-Higgs
model and following them [19] showed that the metric of the U(1) vortex is
asymptotically conical and how the conical parameters depend on the internal
ones. Subsequently many gravitating vortex solutions have been found to also
be asymptotically conical. This feature, which seems to be general to all
gravitating cosmic string solutions, is a key point to the work presented in
chapters 4 and 5 of this thesis.

This thesis is organized as follows. In Chapter 2 we present the essential
theory of cosmic strings without taking into account their gravitational
interaction. We study the Nielsen-Olesen vortex with a reasonable amount of
details and present the numerical solution to the scalar and gauge fields as
well as the energy density with two different winding numbers. In Chapter 3,
we study the gravitational aspects of cosmic strings. We present the wire
approximation and two of the local physical effects of a conical spacetime. In
the second part of Chapter 3 we dive into the world of gravitationally
extended vortices. We present the results from [18] and [19] on the
gravitating U(1) string, and the results of [20] on a non-abelian-Higgs model
developed recently. We see that both of these models present conical structure
far from the core. In Chapter 4 we study the scattering of scalar and
fermionic fields in the spacetime of a gravitating cosmic string. We show that
the partial-wave formalism presents inconsistencies when applied to the scalar
field scattering in this class of spacetimes and propose a modification to
solve them. We explicitly show the corrections in the phase-shift, scattering
amplitude and total cross-section. In order to apply the formalism we develop
a toy-model for the spacetime of a cosmic string and study the scattering of a
scalar field in this toy-model. We also take into account the interaction with
the gauge field that generates the vortex and compare the cross-section with
and without the local gauge-field interaction. In the last part of 4 we apply
the same formalism to the fermionic field and find the expression of the total
scattering cross-section.

In Chapter 5 we apply the formalism to the spacetime found by [20]. In the
first part we study the scattering of the scalar field and show how the mass
of the field affects the total cross-section. We then turn our attention to
analyze the reason behind the observed oscillations in the total cross-
section. In order to do that we use the aforementioned toy-model to conclude
that the oscillations are caused by the spacetime asymptotical conical
structure. In the second part of Chapter 5 we study the fermionic scattering
in the same spacetime. We show the dependence of the cross-section with the
fermion mass and also observe oscillations in the total cross-section. Finally
we present a crude estimation of why the cross-section is larger when the
deficit angle of the asymptotical spacetime is bigger.

Throughout the text we use natural units where $\hbar=c=1$. Hence mass is
measured in Planck mass, $m_{p}$, and length in Planck length, $l_{p}$. For
future reference, these values are approximately $m_{p}=2.1\times 10^{-8}$ kg
and $l_{p}=1.6\times 10^{-35}$ m.

## 2\. Down the cosmic hole: Topological defects and string formation

###  2.1 Topological defects

> Start with something that everyone knows and understands, people like to
> hear what they know. Then say something that only the experts understand,
> lest you be accused of talking trivia. Conclude with something no one, not
> even you, can understand, just to keep the proper respect for physics.
>
> Vicki Weisskopf [21]

In field theory, a topological defect is a field configuration that only
depends on the value of the field at the boundaries. Since, for a localized
configuration, the boundary field values are also the vacuum ones we can also
define topological defects by the non-trivial structure of the vacuum of the
theory, which is commonly expressed as a disconnectedness in the vacuum
manifold. This motivates a more general definition of topological defects
which is also valid outside the field theory context: a topological defect is
a discontinuity (defect) in the order parameter111In field theory the order
parameter is the value of field at the vacuum. that cannot be removed
(topological).

Furthermore, if the field configuration cannot be continuously transformed to
the vacuum keeping the energy finite then it is considered topologically
stable since, as we shall see, it is possible to define a conserved charge
that depends only on the field values at the boundaries.

The non-trivial topology of the vacuum is commonly connected to a form of
symmetry breaking, which can be related to phase transitions in a variety of
systems, ranging from condensed matter ones, as in liquid crystals, liquid
helium, and superconductors, to the formation of structures in the early
universe, such as domain walls and cosmic strings.
In this chapter, we will see the basic topological defects from a field-
theoretical point of view. In the last section, however, we encounter some
interesting connections between condensed matter systems and the universe’s
structure.

####  2.1.1 The simplest model: $\phi^{4}$

When confronted with a new subject, it is often illuminating to start with a
straightforward problem. A simple model that admits a topological defect
solution is the 1+1-dimensional $\phi^{4}$ model, i.e. we are searching for
static solutions in 1 spatial dimension.

$\begin{gathered}\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi-V(\phi),\quad\quad\text{with
}\quad
V(\phi)=\frac{\lambda}{4}\left(\phi^{2}-\eta^{2}\right)^{2}.\end{gathered}$
(2.1)

The $\phi^{4}$ Lagrangian describes a real scalar field with a quartic self-
interaction potential. The quadratic term in the field,
$\lambda\eta^{2}\phi^{2}/2$, means the mass of the field is
$\sqrt{\lambda}\eta$. Also, the minimum of the potential, $V(\phi)=0$, is
achieved when $\phi=\pm\eta$, hence we say $\phi=\pm\eta$ are the vacuum
states.
The Euler-Lagrange equation yields the solutions

$\phi(x)=\pm\eta\tanh\left(\sqrt{\frac{\lambda}{2}}\,\eta x\right),$ (2.2)

which are called _kink_ (positive sign) and _anti-kink_ (negative sign). The
kink is a one-dimensional, localized, time-independent, stable solution. We
shall see it is also topologically stable, i.e. the boundary conditions
enforce stability under field perturbations.

Figure 2.1: Profiles of the field, potential and energy density of the kink
solution. Source: The author (2021).

In Figure 2.1 we see that the kink interpolates between both vacuum states,
$\pm\eta$, which are at $\pm\infty$. Also, the order parameter is the vacuum
expectation value (VEV), $\langle\phi\rangle$, the field’s value at the
minimum of the potential as it labels the different equilibrium field
configurations.
The $\phi^{4}$ model has many applications. For example, it can represent
domains walls [22], 2-dimensional structures expected to have formed during
the early universe, but also non-linear excitations in polyacetylene [23]. For
a historical overview of the $\phi^{4}$ model we suggest [24].

Now, if we look at the vacuum manifold (the space of vacuum states), we see
that it is disconnected. It is impossible to go from one vacuum to the other
passing only through vacuum states. This means that the order parameter,
$\langle\phi\rangle$, has a discontinuity between the vacua; thus, the vacuum
manifold is disconnected.

It was mentioned earlier that topological defects are usually connected with
symmetry breaking, but which symmetry is broken by the quartic potential? The
answer is $Z_{2}$ parity, which is equivalent to $\phi\to-\phi$. If we make
$\eta\to 0$ we see that both vacua coalesce to the same value, namely
$\langle\phi\rangle=0$, and the configuration becomes $Z_{2}$ invariant. The
degenerate minima and the disconnected vacuum manifold only appear because
$\eta\neq 0$.

Figure 2.2: Representation of vacuum states interpolated by the kink solution.
Source: The author (2021).

Topological defects are topologically stable. In order to destroy the kink,
one would have to take every point of the solution and move to zero, which
would require an infinite amount of energy since the solution extends to
infinity. This guarantees the stability of the kink. This argument can be
tracked down to the disconnectedness of the vacuum, where the vacuum structure
ensures stability.

We can see topological stability more clearly defining the current

$j^{\mu}=\epsilon^{\mu\nu}\,\partial_{\nu}\phi$ (2.3)

where $\epsilon^{\mu\nu}$ is the two-dimensional Levi-Civita antisymmetric
tensor. One can easily see that the current (2.3) is conserved,
$\partial_{\mu}j^{\mu}=0$, which gives rise to a conserved charge

$\begin{gathered}Q=\int_{-\infty}^{\infty}{dx\,j^{0}}=\phi(\infty)-\phi(-\infty)=2\eta.\end{gathered}$
(2.4)

The conserved charge derived from current (2.3) only depends on the VEV.
Naturally (2.3) is called _topological current_ , and (2.4) the _topological
charge_. One can see that (2.3) is _not_ derived from Noether’s theorem, so
this is a purely topological conservation law. Once again, the boundary
conditions (topology) guarantees stability!

####  2.1.2 Derrick sends in a little bill: Derrick’s theorem

Naturally, one might want to explore solutions of the $\phi^{4}$ model in more
spatial dimensions. However, it is not possible to find such stable static
solutions. Derrick’s theorem [25] states that

> there are no non-linear scalar field models with stable, time-independent,
> and localized solutions in more than one dimension

which imposes a big restriction on higher-dimensional solutions to (2.1).
Essentially, if one has a non-linear scalar model in more than one spatial
dimension it is guaranteed no static and localized222”Localized” means the
energy density vanishes at infinity. solution is stable under field
perturbations.
Derrick’s argument is based on the fact that any D-dimensional pure scalar
field theory, with static solution is $\phi(x)$, has total energy given by

$E(\phi(x))=E=\int{d^{D}x\left[\left|\nabla_{x}\phi(x)\right|^{2}+V(\phi(x))\right]}.$
(2.5)

where $\nabla_{x}$ is the gradient with respect to the $x$-coordinates. For
further use we define the two quantities

$\displaystyle\begin{aligned}
I_{1}&=\int{d^{D}x\left|\nabla_{x}\phi(x)\right|^{2}}>0\,,\\\
I_{2}&=\int{d^{D}xV(\phi(x))}>0\,.\end{aligned}$ (2.6)

We can make the rescaling $x\to y=\alpha x$, $\alpha\neq 0$, and define the
deformed field $\phi_{\alpha}=\phi(\alpha x)$. The energy of the new
configuration is given by

$\displaystyle E(\phi_{\alpha})=E_{\alpha}$
$\displaystyle=\int{d^{D}x\left[\left|\nabla_{x}\phi(\alpha
x)\right|^{2}+V(\phi(\alpha x))\right]}$ (2.7)
$\displaystyle=\alpha^{-D}\int{d^{D}y\left[\alpha^{2}\left|\nabla_{y}\phi(y)\right|^{2}+V(\phi(y))\right]}$
(2.8) $\displaystyle=\alpha^{2-D}I_{1}+\alpha^{-D}I_{2}.$ (2.9)

Now we can analyze how the energy behaves with respect to the rescaling
parameter $\alpha$ and the spatial dimensions. The main point here is that if
the solution $\phi(x)$ is stable then $\alpha=1$ is a minimum of the energy.

Extremizing the energy we find the condition for the solution to be at the
equilibrium point:

$\left.\frac{dE}{d\alpha}\right|_{\alpha=1}=0\Rightarrow DI_{2}=(2-D)I_{1}$
(2.10)

Now we need to know if this point is a minimum, $\delta^{2}E>0$, or a maximum,
$\delta^{2}E<0$.

$\left.\frac{d^{2}E}{d\alpha^{2}}\right|_{\alpha=1}=2(2-D)I_{1}<0\text{for
}D<2,$ (2.11)

which is greater than zero only when $D<2$. This means that for $D\geq 2$ a
deformation of $\phi(x)$ shall decrease the energy of the configuration, hence
the solution is not stable.

The alternatives to construct stable non-linear scalar field solutions in
higher dimensions are split into two categories: 1) extending the model by
including more fields, like a gauge or spinor field, and b) relaxing the time-
independence or localized condition. Higher-dimensional topological defects
can be obtained following any of these two paths. Before diving into any of
these routes, we need to understand a straightforward generalization of (2.1).

###  2.2 Spontaneous Symmetry Breaking

> Come forth into the light of things, let nature be your teacher.
>
> William Wordsworth

Knowing the symmetries of the Lagrangian is often regarded as the first step
in getting a comprehensive description of a system. If a symmetry is
continuous, Noether’s theorem assures a related conserved quantity, making
some calculations a lot easier. Nevertheless, sometimes a symmetry of the
Lagrangian might not be the symmetry of the vacuum. In such cases, we say the
symmetry is spontaneously broken.

####  2.2.1 A simple model: Goldstone

We can illustrate the main features of spontaneous symmetry breaking using the
Goldstone model

$\begin{gathered}\mathcal{L}=\partial_{\mu}\phi^{*}\,\partial^{\mu}\phi-V(\phi),\quad\quad\text{with
}\quad
V(\phi)=\frac{\lambda}{4}\left(|\phi|^{2}-\eta^{2}\right)^{2},\end{gathered}$
(2.12)

which looks similar to the $\phi^{4}$ one, but now $\phi$ is a complex scalar
field instead of a real one. Although, according to Derrick’s theorem, model
(2.12) in more than one dimension is unstable, it is also pedagogically useful
to illustrate spontaneous symmetry breaking. We see that (2.12) is invariant
under the action of the global $U(1)$ group, i.e., (2.12) does not change if
we multiply $\phi$ by a constant phase $\alpha$, $\phi\rightarrow
e^{i\alpha}\phi$. The vacuum state, $\phi=\eta e^{i\theta}$, however, is _not_
invariant under $U(1)$. The global $U(1)$ symmetry is spontaneously broken.

Figure 2.3: Potential of the Goldstone model, also called mexican hat or Higgs
potential. Source: the author (2021)

Since the coupling constant $\lambda$ is considered to be small, we can use
the term _vacuum expectation value_ referring to the ground state, since when
$\lambda\rightarrow 0$, the classical and quantum values become the same. For
small $\eta$, the potential $V(\phi)$ can be approximated by

$V(\phi)\approx|\phi|^{4}-2\lambda\eta^{2}|\phi|^{2},$ (2.13)

which has a maximum $V_{max}=\lambda\eta^{2}$ at $\phi=0$ and a minimum
$V_{min}=0$ at $|\phi|=\eta$. In order to keep track of field excitations, we
expand $\phi$ around the specific ground state $\phi=\eta$,

$\phi=\eta+\frac{1}{\sqrt{2}}(\phi_{1}+i\phi_{2}),$ (2.14)

where $\phi_{1}$ and $\phi_{2}$ are treated as two independent real scalar
fields. Each term of (2.12) becomes

$\displaystyle\begin{aligned}
\partial_{\mu}\phi^{*}\partial^{\mu}\phi&=\frac{1}{2}\left[\partial_{\mu}\phi_{1}\partial^{\mu}\phi_{1}+\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}\right],\\\
V(\phi)&=\frac{\lambda}{4}\left[\frac{1}{2}(\phi_{1}^{2}-\phi_{2}^{2})+\sqrt{2}\eta\phi_{1}\right]^{2}\\\
&=\frac{\lambda}{4}\left[\frac{1}{4}(\phi_{1}^{4}+\phi_{2}^{4}-2\phi_{1}^{2}\phi_{2}^{2})+2\eta^{2}\phi_{1}^{2}+\sqrt{2}\eta\phi_{1}(\phi_{1}^{2}-\phi_{2}^{2})\right].\\\
\end{aligned}$ (2.15)

Now plugging (2.15) into (2.12) yields

$\mathcal{L}_{exc}=\underbrace{\frac{1}{2}(\partial_{\mu}\phi_{1})^{2}-\frac{1}{2}\eta^{2}\lambda\phi_{1}^{2}+\frac{1}{2}(\partial_{\mu}\phi_{2})^{2}}_{\text{free
part}}+\mathcal{L}_{int},$ (2.16)

where $\mathcal{L}_{exc}$ refers to the excitation of the scalar field and
$\mathcal{L}_{int}$ includes all the interaction terms. What is important to
notice here is that the free part describes a massive scalar field $\phi_{1}$,
with mass $\eta\sqrt{\lambda/2}$, and a massless scalar field $\phi_{2}$. This
suggests the symmetry breaking, which can be tracked down to the inclusion of
a complex scalar field, generates a massless excitation of the Higgs field,
$\phi_{2}$, called _Goldstone boson_. In 1961 Jeffrey Goldstone conjectured
this to be a general feature of models with symmetry breaking. The Goldstone
conjecture [26] states that “every spontaneous breaking of a continuous
symmetry generates a massless scalar particle”. It was demonstrated, becoming
a theorem, in 1962 by Goldstone, Salam and Weinberg [27].

We can give meaning to the Goldstone boson by looking at Figure 2.3. In order
to use the expansion (2.14) suppose that the system is at the vacuum with
$\theta=0$, i.e. at Im$(\phi)=0$ and Re$(\phi)=\eta$. In this situation, a
change in $\phi_{1}$ (the real part of the field) is energetically
disfavoured, since the state would have to climb the potential hill, while a
change in $\phi_{2}$ takes the field to an energetically equivalent state,
only changing the representative vacuum (value of $\theta$). The Goldstone
boson encodes the energetic symmetry of the vacua. Naturally, any
interpretation to $\phi_{1}$ or $\phi_{2}$, as defined in (2.14), _have_ to
consider them as perturbations around $\phi=\eta$. There is, however, a
loophole to the Goldstone theorem. If we consider a broken _local_ symmetry,
we can make the Goldstone boson disappear. One might take a moment and try to
figure out why.

####  2.2.2 A hard one: abelian-Higgs

Now we are going to see what happens when we impose a local symmetry in the
Goldstone model. First of all, local symmetry plus Lorentz-invariance forces
the existence of a gauge field communicating the symmetry. Consider the
abelian-Higgs model with the following Lagrangian

$\begin{gathered}\mathcal{L}_{AH}=(D_{\mu}\phi)^{*}(D^{\mu}\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-V(\phi),\quad\quad\text{with
}\quad V(\phi)=\frac{\lambda}{4}(|\phi|^{2}-\eta^{2})^{2},\end{gathered}$
(2.17)

where $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ is the covariant derivative in field
space, $e$ is the coupling to the gauge field (usually referred to as charge),
$A_{\mu}$ is the gauge field, and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the Faraday
tensor. Because we are dealing with a gauge field, there is a subtlety. The
Lagrangian (2.17) have $U(1)$ _local_ symmetry, which means invariance under
_gauge transformations_

$\displaystyle\left\\{\begin{aligned}
\phi&\rightarrow\phi^{\prime}=e^{i\alpha(x)}\phi\\\
\phi^{*}&\rightarrow{\phi^{*}}^{\prime}=e^{-i\alpha(x)}\phi^{*}\\\
A_{\mu}&\rightarrow
A_{\mu}^{\prime}=A_{\mu}+\frac{1}{e}\partial_{\mu}\alpha(x),\\\
\end{aligned}\right.$ (2.18)

for any arbitrary coordinate dependent function $\alpha(x)$. Transformations
(2.18) tell us that for every choice of the four-vector $A_{\mu}$, the vector
$A_{\mu}^{\prime}$ is equivalent. This is called _gauge freedom_. It means
that we have the freedom of choosing a function $\alpha$, the _gauge_ , that
better suits our intentions. In this case we can choose a gauge that
annihilates the imaginary part of $\phi$, the Goldstone boson. Expanding the
field around vacuum

$\phi=\eta+\frac{1}{\sqrt{2}}\phi_{1},$ (2.19)

results in the Lagrangian

$\mathcal{L}_{exc}=\frac{1}{2}(\partial_{\mu}\phi_{1})^{2}+\frac{\lambda}{2}\eta^{2}\phi_{1}^{2}+e^{2}\eta^{2}A_{\mu}A^{\mu}+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathcal{L}_{int}.$
(2.20)

One might notice that now we do not have any massless scalar boson in (2.20),
the gauge boson "ate" the Goldstone boson. In addition, the field $A_{\mu}$
has a self-interaction term, $e^{2}\eta^{2}A_{\mu}A^{\mu}$, meaning that the
gauge boson, initially massless, acquires a mass $m_{A}=\sqrt{2}e\eta$. This
result can be generalized to any number of non-abelian gauge fields. It can be
shown that the corresponding gauge boson acquires a mass proportional to the
vacuum energy for every broken symmetry.

####  2.2.3 Global strings

Now consider the Goldstone potential

$V(\phi)=\frac{\lambda}{4}(|\phi|^{2}-\eta^{2})^{2},$ (2.21)

and assume the field $\phi$ is near the minima, $\phi=\phi_{0}e^{i\theta}$,
where $\theta$ is the phase in the field space. We can map circles, in field
space, near the minima of the potential. which we denote by the letter $L$, to
loops in physical space, which we denote by $S$, since both are periodic
closed curves. Once we travel around a loop $S$, in physical space, we know
the angular coordinate, $\varphi$, must change by $2\pi$ while the phase of
$L$, the image of $S$ in field space, $\theta$, must change by an integer
multiple of $2\pi$, $\Delta\theta=2\pi n$. Now imagine one continuously deform
the loop $S$, in physical space, to a point $p$. Since the field has to be
uniquely determined at $p$ there are two options: $\phi_{0}(p)=0$ or
$\phi_{0}(p)\neq 0$. If $\phi_{0}(p)=0$ then we know there is an energy
excitation inside $S$.

If $\phi_{0}(p)\neq 0$, because $\phi$ is single-valued at $p$ then
$\Delta\theta=0$ at some intermediate loop $L^{\prime}$, which is mapped to
$S^{\prime}$ between $S$ and $p$ in physical space, but since $\Delta\theta$
has to be an integer multiple of $2\pi$, $\Delta\theta$ has to jump from
$2\pi$ to $0$ at $L^{\prime}$, which contradicts the hypothesis that $\phi$ is
a smooth function of the coordinates. Hence $\phi_{0}=0$ at least at one point
of a loop $L^{\prime\prime}$ located between $L$ and $L^{\prime}$. The
excitation confined by the loop $S^{\prime\prime}$, the image of
$L^{\prime\prime}$ in physical space, is called a _cosmic string_. This
reasoning can be better visualized in Figure 2.4 and works for any model that
possesses a symmetry breaking potential of the form ((2.21)).

Figure 2.4: $|\phi|$ has to vanish at some point when contracted from a
minimum of the potential $V(\phi)$ to a single point $p$.

(a) Loops in physical space.

(b) Loops in physical space mapped onto field space.

Source: The author (2021).

We can show the phase difference around the loop $L^{\prime\prime}$, or any
other above it, in Figure 2.4(b) is indeed zero. The phase $\theta$ of any
point in $L^{\prime\prime}$ is

$\theta=\tan^{-1}\left(\frac{\text{Im}(\phi)}{\text{Re}(\phi)}\right).$ (2.22)

So, starting at point $(\text{Re}(\phi),\text{Im}(\phi))=(0,0)$ and going
anticlockwise on the loop $L^{\prime\prime}$ we see that $\theta$ increases in
the first quarter of the circle, and so it does on the second quarter.
However, on the third quarter, when $\text{Re}(\phi)<0$, the phase difference
is negative, and so it is on the fourth quarter, yielding $\Delta\theta=0$.

The deformation argument outlined above assures that there must be an
excitation of the Higgs field at some point of the surface bounded by $L$, and
it is clear that this excitation should have cylindrical symmetry since $L$ is
on a bounded 2-surface. Moreover, the fact that $\Delta\theta$ should be an
integer multiple of $2\pi$ means that we can make the correspondence
$\theta=n\varphi$, where $n$ is called the _winding number_. Finally, the
vacuum manifold is a circle, which is _not_ simply connected because a loop on
the circle cannot be continuously shrunk to a point on the circle [28].

Figure 2.5: The vacuum manifold of the Goldstone model is a circle. Source:
the author (2020)

Now we turn our attention to the simplest vortex solution, the global string.
This solution arises from the Goldstone model (2.12), which breaks global
$U(1)$ symmetry. Consider a localized configuration,
$V(\phi)\xrightarrow{r\to\infty}0$, which means that far from the core of the
string the scalar field is approximately $\phi=\eta e^{in\varphi}$. The energy
density is given by333$V(\phi)$ being localized implies that the vacuum value
of $|\phi|$ is a constant, which in turn means that the energy density
$H(\phi)$ is localized.

$H=|\nabla\phi|^{2}+V(\phi),\,H(r\rightarrow\infty)\approx|\nabla\phi|^{2},$
(2.23)

which gives the total energy (excluding the contribution from the core)

$\displaystyle\begin{aligned}
E&=\int{|\nabla\phi|^{2}d^{2}x}=\int{\eta^{2}n^{2}\frac{1}{r^{2}}rdrd\varphi}=2\pi
n^{2}\eta^{2}\int_{\delta}^{R}{\frac{1}{r}dr}\\\ &=2\pi
n^{2}\eta^{2}\,\ln\left(\frac{R}{\delta}\right).\end{aligned}$ (2.24)

In the above expression, $\delta$ is the width of the string444We define the
width of the string as the point at which the contribution from the potential
energy is irrelevant., $R$ is the cutoff radius, and the winding number $n$
labels the kinetic energy of the string. We see that (2.24) diverges as
$R\to\infty$, suggesting the isolated global string is unstable, which is
expected from Derrick’s theorem since this a static solution of a non-linear
pure scalar model. We could anticipate the instability based on Derrick’s
theorem.

####  2.2.4 Local strings

A local string is a vortex solution with local gauge symmetry breaking. Here
we consider the abelian-Higgs model (2.17), which is the Goldstone model
coupled with the $U(1)$ gauge field. Notice that, because we are coupling the
scalar field with a gauge field, Derrick’s theorem no longer predicts
instability. The actual vacuum is the same as in the pure Goldstone case, so
the scalar field cannot be affected by the gauge field far from the core,
i.e., $|\phi|\xrightarrow{r\to\infty}\eta$. The effect of the gauge field
should be concentrated on the string core since we are searching for a
localized solution.

The Euler-Lagrange equations for the abelian-Higgs model are

$\displaystyle\partial_{\mu}\partial^{\mu}\phi-2ieA^{\mu}\partial_{\mu}\phi+[\lambda\,(|\phi|^{2}-\eta^{2})-e^{2}A_{\mu}A^{\mu}]\phi=0,$
(2.25a)
$\displaystyle-\frac{1}{2}ie\left(\phi\partial^{\mu}\phi^{*}-\phi^{*}\partial^{\mu}\phi\right)+\frac{1}{2}e^{2}A^{\mu}\phi^{*}\phi\propto\partial_{\nu}F^{\nu\mu}.$
(2.25b)

Now plugging the field far from the core $\phi=\eta e^{in\varphi}$ in (2.25b),
we get

$A_{\varphi}=-\frac{n}{e},$ (2.26)

which indicates a magnetic flux, $\phi_{B}$, on the string

$\phi_{B}=\int_{S}{\vec{B}.\vec{dS}}=\int_{S}{(\nabla\times\vec{A})\,.\vec{dS}}=\oint_{\partial
S}{\vec{A}\cdot\vec{dl}}=\int_{\partial
S}{A^{\varphi}rd\varphi}=\int_{0}^{2\pi}{\frac{n}{er}\,rd\varphi}=\frac{2\pi}{e}n.$
(2.27)

The additional minus sign in $A^{\varphi}$ appears because the metric has
signature -2. We have seen that the winding number $n$ labels the kinetic
energy of the string, and now we see that it also labels the magnetic flux,
which is quantized!

Denoting the field covariant derivative as $D_{\mu}=\partial_{\mu}-ieA_{\mu}$,
we know that far from the core

$\displaystyle\begin{aligned} D_{\mu}\phi(r\rightarrow\infty)&\approx 0,\\\
F_{\mu\nu}(r\rightarrow\infty)&\approx 0,\end{aligned}$ (2.28)

which makes the energy depend purely on $V(\phi)$, $H\approx V(\phi)$, which
has finite integral. It can be shown that the mass per unit length $\mu$ of
the local $U(1)$ string is given by [29]

$\mu\approx 2\pi\eta^{2}\ln\left(\frac{m_{\phi}}{m_{A}}\right),$ (2.29)

where $m_{\phi}=\sqrt{\lambda}\eta$ and $m_{A}=\sqrt{2}e\eta$. We have dodged
Derrick’s theorem by employing a gauge-field coupling.

In 1973 Nielsen and Olesen [14] found explicitly the existence of a static
cylindrically symmetric solution to the field equations (2.25). For this
reason, the $U(1)$ local string, or abelian-Higgs string, is usually called
the Nielsen-Olesen vortex. In the next section, we show the derivation of the
Nielsen-Olesen solution and discuss some of its properties. Also, from now on,
we use the words _string_ and _vortex_ interchangeably.

###  2.3 Nielsen-Olesen vortex

> The scientist does not study nature because it is useful to do so. He
> studies it because he takes pleasure in it, and he takes pleasure in it
> because it is beautiful. If nature were not beautiful it would not be worth
> knowing, and life would not be worth living. I am not speaking, of course,
> of the beauty which strikes the senses, of the beauty of qualities and
> appearances. I am far from despising this, but it has nothing to do with
> science. What I mean is that more intimate beauty which comes from the
> harmonious order of its parts, and which a pure intelligence can grasp.
>
> Henri Poincaré

The equations of motion of the abelian-Higgs model can be expressed in the
form

$\displaystyle(\partial_{\mu}+ieA_{\mu})(\partial^{\mu}+ieA^{\mu})\phi+\frac{\lambda}{2}(|\phi|^{2}-\eta^{2})\phi=0,$
(2.30a) $\displaystyle\partial_{\mu}F^{\mu\nu}=j^{\nu},$ (2.30b)
$\displaystyle j^{\nu}=2e\,Im[\phi^{*}(\partial^{\nu}-ieA^{\nu})\phi].$
(2.30c)

For convenience we rescale the quantities,
$(\phi,A_{\mu},\partial_{\mu})\rightarrow\frac{1}{\eta}\,(\phi,A_{\mu},\partial_{\mu})$,
and choose the Lorentz gauge $\partial_{\mu}A^{\mu}=0$. Besides that, we take
the following ansatz

$\displaystyle\begin{aligned} \phi(\vec{r})&=e^{in\varphi}f(r),\\\
A^{\varphi}(\vec{r})&=-\frac{n}{er}\alpha(r),\end{aligned}$ (2.31)

where $f(r)$ and $\alpha(r)$ are unknown functions that satisfy the boundary
conditions

$\displaystyle\begin{aligned} r&\to\infty:f(r)=\alpha(r)=1,\\\ r&\to
0:f(0)=\alpha(0)=0.\end{aligned}$ (2.32)

Notice that far from the string we recover $A^{\varphi}=\frac{n}{er}$,
$\phi=\eta e^{in\varphi}$. Also, the magnetic flux is in fact given by $2\pi
n/e$ since $\alpha(r\to\infty)=1$. Now we can determine $f(r)$ and $\alpha(r)$
by plugging the ansatz (2.31) in the equations of motion (2.30). One of the
terms is

$\displaystyle\begin{aligned}
(\partial_{\mu}+ieA_{\mu})(\partial^{\mu}+ieA^{\mu})\phi&=\partial_{\mu}\partial^{\mu}\phi-
ieA_{\mu}\partial^{\mu}\phi-
ie\partial_{\mu}(A^{\mu}\phi)-e^{2}A_{\mu}A^{\mu}\phi\\\
&=-\left(\nabla^{2}\phi-
ie\vec{A}\cdot\vec{\nabla}\phi-\nabla\cdot(\vec{A}\phi)-e^{2}|A|^{2}\phi\right)\\\
&=-\left(\nabla^{2}\phi-2ie\vec{A}\cdot\vec{\nabla}\phi-\phi\underbrace{\nabla\cdot\vec{A}}_{=\,0}-e^{2}|A|^{2}\phi\right)\\\
&=-e^{in\varphi}\Bigg{[}\frac{1}{r}\frac{df}{dr}+\frac{d^{2}r}{dr^{2}}+\underbrace{\frac{2n^{2}}{r^{2}}f\alpha-\frac{n^{2}}{r^{2}}\alpha^{2}-\frac{n^{2}}{r^{2}}f}_{\frac{n^{2}f}{r^{2}}(\alpha-1)^{2}}\Bigg{]},\end{aligned}$
(2.33)

giving rise to

$f^{\prime\prime}+\frac{1}{r}f^{\prime}-\frac{n^{2}}{r^{2}}(\alpha-1)^{2}f-\frac{\lambda}{2}(f^{2}-1)f=0$
(2.34)

for (2.30a). Moreover, we obtain (2.30c)

$\displaystyle\begin{aligned}
\partial_{\mu}F^{\mu\nu}&=\partial_{\mu}(\partial^{\mu}A^{\nu})-\partial_{\mu}(\partial^{\nu}A^{\mu}),\\\
&=-\nabla^{2}\vec{A}=-\left[\nabla^{2}(A^{\varphi})-\frac{A^{\varphi}}{r^{2}}\right]\hat{\varphi},\\\
&=-\left[\frac{1}{r}\frac{\partial
A^{\theta}}{\partial}+\frac{\partial^{2}A^{\theta}}{\partial
r^{2}}-\frac{A^{\varphi}}{r^{2}}\right]\hat{\varphi},\\\
&=\frac{n}{er}\left(\alpha^{\prime\prime}-\frac{1}{r}\alpha^{\prime}\right)\hat{\varphi},\\\
\end{aligned}$ (2.35)

and

$\displaystyle\begin{aligned}
\phi^{*}(\partial^{\nu}+ieA^{\nu})\phi&=-e^{-in\varphi}f\left[\frac{in}{r}fe^{in\varphi}-\frac{in}{r}\alpha
fe^{in\varphi}\right],\\\ &=-i\frac{n}{r}f^{2}(1-\alpha)\hat{\varphi}.\\\
\end{aligned}$ (2.36)

Finally, the equations of motion take the form

$\displaystyle
f^{\prime\prime}+\frac{1}{r}f^{\prime}-\frac{n^{2}}{r^{2}}f(\alpha-1)^{2}-\frac{\lambda}{2}f(f^{2}-1)=0,$
(2.37a)
$\displaystyle\alpha^{\prime\prime}-\frac{1}{r}\alpha^{\prime}-2e^{2}f^{2}(\alpha-1)=0.$
(2.37b)

An analitical solution to (2.37) can only be obtained in some restricted
situations.
In the regime $r\to\infty$ we have $f(r\to\infty)=1$, which simplifies
(2.37b). By rescaling the radius, $\tilde{r}=\sqrt{2}er$, and changing the
variable, $\alpha(r)-1=\tilde{r}\xi(r)$, where $\xi(r)$ is to be determined,
we arrive at a new form of (2.37b) in the asymptotic regime,

$\tilde{r}^{2}\xi^{\prime\prime}+\tilde{r}\xi^{\prime}-(\tilde{r}^{2}+1)\xi=0,$
(2.38)

where prime denotes differentiation with respect to $\tilde{r}$. Equation
(2.38) is the modified Bessel equation, and since we want the solution to be
finite at large $r$, it shall be proportional to the modified Bessel function
of the second kind of order 1, $K_{1}(\tilde{r})$. The asymptotic behavior of
$\alpha(r)$ is given by

$\xi(\tilde{r})=\pm
K_{1}(\tilde{r})\to\alpha(r)=1\pm\sqrt{2}erK_{1}\left(\sqrt{2}er\right),$
(2.39)

where the integration constant is set to $\pm 1$.

If we rescale the coordinates $x\to x/e$ in (2.30) we can see that the
solution to the field equations are only dependent on the parameter
$\beta=(m_{\phi}/m_{A})^{2}$, which measures the relative strength between
scalar and gauge fields. In Figure 2.6 we show the numerical solution of
$f(r)$, $\alpha(r)$ and their dependence on $\beta$. We shall see that the
parameter $\beta$ measures the ratio between disruptive and confining forces.

Figure 2.6: Dependence of the scalar and gauge fields on the parameter
$\beta$. Here we used $n=2$. Source: The author (2021).

In Figure 2.7 we see how the vortex energy density is affected by the winding
number. In general, the string has zero energy density at the axis and is
surrounded by a cylindrical energy density that goes to zero at infinity. This
feature will be important when coupling strings with gravity.

Moreover, we notice that the width of the vortex is proportional to the
winding number $n$. In fact, if we consider the magnetic flux uniformly
distributed over the vortex, the characteristic width is proportional to
$\sqrt{n}$. The reason for this is that the vortex width should increase with
$n$ to accommodate the increasing magnetic flux. Recently Alexander Penin and
Quinten Weller, in [30] and [31], used an asymptotic expansion of the field
functions in inverse powers of $n$ to find a piece-wise perturbative
analytical solution of (2.37) in the limit $n\to\infty$.

Figure 2.7: The width of the Nielsen-Olesen vortex is proportional to the
winding number $n$. Source: The author (2021).

####  2.3.1 Properties of the Nielsen-Olesen vortex

Equipped with the solution of the Nielsen-Olesen vortex, we can start
analyzing some of its properties.

##### Energy-momentum tensor

Given an action $S$ defined in a spacetime with metric $g_{\mu\nu}$ with
signature -2, the energy-momentum tensor
$\mathchoice{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}$
is given by

$\mathchoice{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}=\frac{2}{\sqrt{-g}}\frac{\delta
S}{\delta g^{\mu\nu}},$ (2.40)

where $g$ is the determinant of the metric. For a Lagrangian $\mathcal{L}$,
(2.40) yields

$\mathchoice{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}=2\frac{\partial\mathcal{L}}{\partial
g^{\mu\nu}}-g_{\mu\nu}\mathcal{L}.$ (2.41)

Now if we apply (2.41) to the Lagrangian (2.17) we get

$\mathchoice{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}{T^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\nu}}}=\partial_{\mu}\phi\partial_{\nu}\phi^{*}+ie(A_{\mu}\phi\partial_{\nu}\phi^{*}-A_{\nu}\phi^{*}\partial_{\mu}\phi)+e^{2}A_{\mu}A_{\nu}|\phi|^{2}+\mathchoice{F^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\sigma}}}{F^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\sigma}}}{F^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\sigma}}}{F^{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}}_{\leavevmode{\mu}\leavevmode{\sigma}}}\mathchoice{F^{\leavevmode{\sigma}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{F^{\leavevmode{\sigma}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{F^{\leavevmode{\sigma}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{F^{\leavevmode{\sigma}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.9999pt][c]{$\displaystyle$}}{\makebox[3.9999pt][c]{$\textstyle$}}{\makebox[2.79993pt][c]{$\scriptstyle$}}{\makebox[1.99994pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}-g_{\mu\nu}\mathcal{L},$
(2.42)

which for the ansatz (2.31) results in

$\displaystyle T_{rr}$
$\displaystyle=\frac{n^{2}}{2e^{2}r^{2}}\left(\frac{d\alpha}{dr}\right)^{2}+\frac{1}{2}\left(\frac{df}{dr}\right)^{2}-\frac{f^{2}}{2r^{2}}n^{2}(1+\alpha)^{2}+\frac{\lambda\eta^{2}}{2}-\frac{\lambda}{4}f^{4}-\frac{\lambda\eta^{4}}{4},$
(2.43a) $\displaystyle T_{\varphi\varphi}$
$\displaystyle=\frac{n^{2}}{2e^{2}}\left(\frac{d\alpha}{dr}\right)^{2}+\frac{1}{2}f^{2}n^{2}(1+\alpha)^{2}+r^{2}\left[\frac{\lambda\eta^{2}}{2}f^{2}-\frac{\lambda}{4}f^{4}-\frac{1}{2}\left(\frac{df}{dr}\right)^{2}-\frac{\lambda\eta^{4}}{4}\right],$
(2.43b) $\displaystyle T_{tt}$
$\displaystyle=-T_{zz}=\frac{n^{2}}{2e^{2}r^{2}}\left(\frac{d\alpha}{dr}\right)^{2}+\frac{1}{2}\left[\left(\frac{df}{df}\right)^{2}+\frac{f^{2}}{r^{2}}n^{2}(1+\alpha)^{2}\right]-\frac{\lambda\eta^{2}}{2}+\frac{\lambda}{4}f^{4}+\frac{\lambda\eta^{4}}{4}.$
(2.43c)

Equations (2.43) tell us that besides the energy density,
$\varepsilon=\mathchoice{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}$,
and the longitudinal tension,
$p^{z}=\mathchoice{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}$,
there is also radial flux of momentum,
$p^{r}=\mathchoice{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}$,
and azimutal momentum
$p^{\varphi}=\mathchoice{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}$.
Although in some restricted situations we can neglect $p^{r}$ and
$p^{\varphi}$, they are essential to analyze the vortex internal dynamics
which are non-vanishing in general.

##### Stability

One interesting feature of the Nielsen-Olesen solution is that if the winding
number $n$ is larger than 1, the vortex can _unwind_ , i.e: the string decays
into $n$ strings each with magnetic flux equal to $2\pi/e$. In 1976 Eugene
Bogomol’nyi [32] showed that Nielsen-Olesen strings are unstable to such
unwinding if $\beta=\left(\frac{m_{\phi}}{m_{A}}\right)^{2}>1$ and stable if
$\beta<1$. This can be explained by the following reasoning. The magnetic
lines repel each other while the Higgs field act to confine the vortex. Since
the field strength is inversely proportional to the mass, when
$m_{A}>m_{\phi}$ ($\beta<1$) the Higgs field is stronger than the repulsion of
the magnetic lines, while when $m_{A}<m_{\phi}$ ($\beta>1$) the repulsion
overcomes the confinement and the string unwind. Hence we can see $\beta$ as
the ratio between disruptive and confining forces. At this point, we can see
the connection between cosmic strings and vortex lines in superconductors.
Type-II superconductors ($\beta<1$) at certain values of the applied magnetic
field present stable vortex lines with quantized magnetic flux, while in
type-I superconductors ($\beta>1$), the quantized vortex lines are unstable
and immediately unwind due to Meissner effect. In 1976, Vega and Schaposnik
[33] showed that in the critical coupling, $\beta=1$, the equations of motion
decouple, and it is possible to construct an analytical power-series solution
to the field equations. In this situation $T_{rr}=0$555It implies
$T_{\varphi\varphi}=0$, via energy-momentum conservation
$\partial_{\mu}\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=0$.
suggesting an equilibrium in the radial direction, i.e: there is no radial net
flux.

Here we finish our discussion on the Nielsen-Olesen solution. In the next
section, we will see a simple mechanism for the cosmological creation of
cosmic strings.

###  2.4 Kibble and the cosmic tale

> …so many out-of-way things had happened lately, that Alice begun to think
> very few things indeed were really impossible.
>
> From "Alice’s adventures in Wonderland", by Lewis Carrol.

The discussion of the Goldstone potential up to now has been simplified for
pedagogical purposes. When the temperature is above 0K, the Goldstone
potential has a different form caused by the interaction of the matter with
the environment energy. Furthermore, according to the standard model of
cosmology, the universe, with all its fields, started in a highly dense, hot
state and cooled through a series of epochs, during which matter interaction
drastically changed. In what follows, we will see how the temperature affects
the Higgs field and how it is related to the formation of topological defects
in the early universe.

####  2.4.1 The role of temperature

Before getting to the Kibble mechanism in the next section, we sketch the
essential temperature-dependence of the Goldstone potential. The discussion of
this section is summarized and for further details and calculations we refer
the reader to [29] The Goldstone potential at non-zero temperature can be
approximated by [29]

$V(\phi,T)=m^{2}(T)|\phi|^{2}+\frac{\lambda}{4}|\phi|^{4}-\frac{2\pi^{2}}{45}T^{4},$
(2.44)

via perturbative corrections, where $m(T)$ is the effective temperature-
dependent mass of the field

$m^{2}(T)=\frac{\lambda}{12}\left(T^{2}-6\eta^{2}\right).\\\ $ (2.45)

For $T>T_{c}=\sqrt{6}\eta$, $m^{2}(T)$ is positive and the VEV is
$\langle\phi\rangle=0$. As the universe evolved, the temperature dropped and,
for $T<T_{c}$, $m^{2}(T)<0$. When $m^{2}(T)<0$ the state
$\langle\phi\rangle=0$ is not stable anymore and the scalar field acquires a
non-zero VEV, leading to $U(1)$ symmetry breaking. Minimizing $V(\phi,T)$
results in

$|\phi|=\frac{1}{\sqrt{6}}\sqrt{T_{c}^{2}-T^{2}},$ (2.46)

which reduces to $\eta$ by setting $T=0$. In summary, decreasing temperature
drives symmetry breaking.

Finally, when $T\approx T_{c}$ the approximation is not valid anymore because
this regime represents a second-order phase transition and we need to include
further corrections to (2.44).

####  2.4.2 Kibble mechanism

Looking at a piece of iron at high temperatures, we see that the magnetization
direction varies from point to point; magnetization does not have a preferred
direction. This is the most symmetric state of the system since it possesses
rotational symmetry, i.e., we can rotate the iron, and magnetization will look
the same. Now suppose we cool it; the molecules have less kinetic energy and,
if the cooling is uniform throughout the material, the magnetization in each
region ends up in a common direction: the iron becomes a magnet! The
magnetized metal is the less symmetric state of the system since we no longer
have rotational symmetry. But if things happen this way, why do not all pieces
of iron we encounter are magnets?

The loophole of the reasoning is the assumption of uniform cooling. In nature,
such uniform cooling never happens. We certainly can achieve uniformity in a
laboratory with ever-increasing levels of precision, but this is not favored
spontaneously. In nature or most simple industrial processes, the non-uniform
cooling leads to different regions choosing different magnetization
directions, leading to a non-magnetic material. In the early universe, things
were quite similar. As the universe cooled down, the Higgs field went from a
symmetric state, $\langle\phi\rangle=0$, to an asymmetric one
$\langle\phi\rangle\neq 0$. Nevertheless, the phase of the field, i.e., the
specific representative vacuum it will be in, is not _a priori_ defined.
Random fluctuations of $\phi$ will determine it. As one can anticipate, this
settling process is not uniform through the universe. Causally disconnected
regions may choose different vacuum states, and the way these regions organize
themselves must give rise to topological defects.

Consider the $\phi^{4}$ model. Two nearby regions in space might end up in
opposite minima of the potential. Between them, there can emerge a kink
interpolating both vacua. The emerging 2-dimensional (though the dynamics is
restricted to 1 dimension) kink-like structure is called _domain wall_.

Now consider the Higgs potential. We can imagine a region of space where the
field is in the symmetric state $\langle\phi\rangle=0$ surrounded by patches
where the field is in the asymmetric state $|\langle\phi\rangle|\neq\eta$.
This gives birth to 1-dimensional defects, called cosmic strings. This
mechanism for the cosmological formation of topological defects is called
_Kibble mechanism_ , after Thomas Kibble, who proposed it in 1976 [1].

Figure 2.8: Casually disconnected regions in space can end up in different
vacua, creating topological defects.

(a)

(b)

Source: The author (2021).

## 3\. Gravitating vortices

###  3.1 The ideal string

> When it comes to the world around us, is there any choice but to explore?
>
> Lisa Randall, in "Warped Passages".

In this section, we will see how a zero-thickness cosmic string affects the
geometry of spacetime. To this end, we use the wire approximation, which
treats the vortex as a thin pipe with negligible radius and constant energy
density. This is a good starting point since some physical properties can be
worked out analytically, and it gives us some valuable physical intuition when
studying more complicated, non-analytical scenarios. We start with the
consequences of the wire approximation on the energy-momentum tensor and then
proceed to its gravitational effects.

####  3.1.1 Wire approximation

By treating the vortex as a thin wire we can average the energy-momentum (EM)
tensor
$\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}$
over the string cross-section and consider all the energy to be localized at
one point in the x-y plane,

$\mathchoice{\tilde{T}^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{\tilde{T}^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{\tilde{T}^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{\tilde{T}^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=\delta(x)\delta(y)\int{\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}\,dxdy},$
(3.1)

This is called the _wire approximation_ which is specially important when
considering gravitational effects of cosmic strings. Because the string is
supposed to be invariant under boosts along the z-axis we must have
$\mathchoice{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}=\mathchoice{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}$,
such that when we apply a Lorentz transformation to (3.1) it does not mix
these components. In addition, we assume the string does not have any internal
shear forces,
$\mathchoice{T^{\leavevmode{3}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}{T^{\leavevmode{3}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}{T^{\leavevmode{3}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}{T^{\leavevmode{3}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}=0$
for i = 1,2 or energy flux in any direction
$\mathchoice{T^{\leavevmode{0}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}{T^{\leavevmode{0}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}{T^{\leavevmode{0}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}{T^{\leavevmode{0}\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{i}}}=0$
for i = 1, 2, 3. These assumptions guarantee the string does not have any
considerable internal dynamics. In addition, because the divergence of the EM
tensor vanishes,
$\partial_{\mu}\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=0$,
we have

$0=\int{x\left(\partial_{x}\mathchoice{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}+\partial_{y}\mathchoice{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}\right)\,dxdy}=\int{(x\mathchoice{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}})\Big{|}_{x=-\infty}^{x=\infty}dy}+\int{(x\mathchoice{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}})\Big{|}_{y=-\infty}^{y=\infty}dx}-\int{\mathchoice{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}\,dxdy},$
(3.2)

where we used integration by parts. Assuming that the components of the EM
tensor fall exponentially when $r\rightarrow\infty$, one can say
$\mathchoice{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{x}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.00069pt][c]{$\displaystyle$}}{\makebox[4.00069pt][c]{$\textstyle$}}{\makebox[2.80048pt][c]{$\scriptstyle$}}{\makebox[2.00034pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=0$,
and, by a similar reasoning, also
$\mathchoice{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{y}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.6831pt][c]{$\displaystyle$}}{\makebox[3.6831pt][c]{$\textstyle$}}{\makebox[2.57817pt][c]{$\scriptstyle$}}{\makebox[1.84154pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=0$
for $\nu=1,2$. Hence
$\mathchoice{T^{\leavevmode{i}\mathchoice{\makebox[3.28334pt][c]{$\displaystyle$}}{\makebox[3.28334pt][c]{$\textstyle$}}{\makebox[2.29834pt][c]{$\scriptstyle$}}{\makebox[1.64166pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}\leavevmode{j}}}{T^{\leavevmode{i}\mathchoice{\makebox[3.28334pt][c]{$\displaystyle$}}{\makebox[3.28334pt][c]{$\textstyle$}}{\makebox[2.29834pt][c]{$\scriptstyle$}}{\makebox[1.64166pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}\leavevmode{j}}}{T^{\leavevmode{i}\mathchoice{\makebox[3.28334pt][c]{$\displaystyle$}}{\makebox[3.28334pt][c]{$\textstyle$}}{\makebox[2.29834pt][c]{$\scriptstyle$}}{\makebox[1.64166pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}\leavevmode{j}}}{T^{\leavevmode{i}\mathchoice{\makebox[3.28334pt][c]{$\displaystyle$}}{\makebox[3.28334pt][c]{$\textstyle$}}{\makebox[2.29834pt][c]{$\scriptstyle$}}{\makebox[1.64166pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.41159pt][c]{$\displaystyle$}}{\makebox[2.41159pt][c]{$\textstyle$}}{\makebox[1.68811pt][c]{$\scriptstyle$}}{\makebox[1.2058pt][c]{$\scriptscriptstyle$}}\leavevmode{j}}}=0$
for i, j = 1, 2. The only non-vanishing components of
$\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}$
are
$\mathchoice{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}$
and
$\mathchoice{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}{T^{\leavevmode{3}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{3}}}$.
Denoting the mass per unit length by $\mu$ and knowing
$\mathchoice{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}{T^{\leavevmode{0}\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5pt][c]{$\displaystyle$}}{\makebox[3.5pt][c]{$\textstyle$}}{\makebox[2.45pt][c]{$\scriptstyle$}}{\makebox[1.75pt][c]{$\scriptscriptstyle$}}\leavevmode{0}}}=\varepsilon(r)$
is the energy density, we end up with

$\begin{gathered}\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=\mu\delta(x)\delta(y)\,diag(1,0,0,1),\\\
\mu=2\pi\int_{0}^{\infty}{\varepsilon(r)rdr}.\end{gathered}$ (3.3)

Notice that the only conditions we used were that the EM should fall
exponentially at $r\rightarrow\infty$ and that it is invariant under boosts
along the $z$-axis, so these results are valid for every localized narrow
111The vortex width should be small compared to the other parameters of the
system. string configuration. The string solution with these simplifications
is an _ideal string_. Alternatively, one could use the lorentz-invariance,
localizability of the energy, the no-shear and no disruptive force conditions
to show that any static infinite straight string should have an EM tensor
proportional to diag(1, 0, 0, 1).

We have seen that the Nielsen-Olesen vortex does have momentum in $r$ and
$\varphi$ directions, so the wire approximation could only represent this
solution when $\beta=1$, which is also called supersymmetric limit. However,
it could be useful when considering large-scale gravitational effects, when
the internal dynamic of the string is not important.

####  3.1.2 Gravitating ideal string

In this section, we shall see how an infinite straight wire affects spacetime
around itself. We will follow the approach independently done by Richard Gott
[5] and William Hiscock [34]. The general procedure is straightforward. We
consider a cylindrically symmetric energy density $\varepsilon(r)$ with finite
radius $r_{0}$, find the metric in $r<r_{0}$, and match it with the exterior,
$r>r_{0}$, vacuum solution of the Einstein field equations (EFEs). In the end,
we take the limit $r_{0}\to 0$ in the exterior metric, which yields the metric
of a zero-thickness infinite straight string.

Figure 3.1: The figure approximates a cosmic string as an energy density
confined in a tube of radius $r_{0}$. Source: The author (2021).

We need to solve EFEs

$\mathchoice{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}=\mathchoice{R^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{R^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{R^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{R^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}-\frac{1}{2}R\delta^{\mu}_{\nu}=\mathchoice{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{T^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}},$
(3.4)

with an energy-momentum tensor inside the tube given by

$\mathchoice{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{T^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}=\mathchoice{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{T^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}=\varepsilon(r)\quad\text{
and
}\quad\mathchoice{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{T^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}=\mathchoice{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{T^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}=0,$
(3.5)

where $\varepsilon$ is considered to be any $r$-dependent function. Though
Gott and Hiscock used constant energy density we will stick to a general
approach and consider a constant energy density
$\varepsilon(r)=\varepsilon_{0}$, only when needed. The ansatz for the line-
element proposed by [5, 34] is

$ds^{2}=e^{2\chi}dt^{2}-e^{2\psi}\left(dr^{2}+dz^{2}\right)-e^{2\omega}d\phi^{2},$
(3.6)

where $\chi(r),\psi(r),\omega(r)$ are functions of $r$ only. We use
exponentials just to ensure the metric components are positive. Equation (3.6)
yields the following non-zero components of the Einstein tensor
$\mathchoice{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}{G^{\leavevmode{\mu}\mathchoice{\makebox[3.45787pt][c]{$\displaystyle$}}{\makebox[3.45787pt][c]{$\textstyle$}}{\makebox[2.42052pt][c]{$\scriptstyle$}}{\makebox[1.72893pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.21783pt][c]{$\displaystyle$}}{\makebox[4.21783pt][c]{$\textstyle$}}{\makebox[2.95248pt][c]{$\scriptstyle$}}{\makebox[2.10892pt][c]{$\scriptscriptstyle$}}\leavevmode{\nu}}}$

$\displaystyle\mathchoice{G^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{G^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{G^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}{G^{\leavevmode{t}\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[2.52779pt][c]{$\displaystyle$}}{\makebox[2.52779pt][c]{$\textstyle$}}{\makebox[1.76944pt][c]{$\scriptstyle$}}{\makebox[1.26389pt][c]{$\scriptscriptstyle$}}\leavevmode{t}}}$
$\displaystyle=-\left[\left(\omega^{\prime}\right)^{2}+\omega^{\prime\prime}+\psi^{\prime\prime}\right]e^{-2\psi}$
(3.7a)
$\displaystyle\mathchoice{G^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{G^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{G^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}{G^{\leavevmode{r}\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.35255pt][c]{$\displaystyle$}}{\makebox[3.35255pt][c]{$\textstyle$}}{\makebox[2.34679pt][c]{$\scriptstyle$}}{\makebox[1.67627pt][c]{$\scriptscriptstyle$}}\leavevmode{r}}}$
$\displaystyle=-\left[\chi^{\prime}\omega^{\prime}+\chi^{\prime}\psi^{\prime}+\omega^{\prime}\psi^{\prime}\right]e^{-2\psi}$
(3.7b)
$\displaystyle\mathchoice{G^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{G^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{G^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}{G^{\leavevmode{\varphi}\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[4.57916pt][c]{$\displaystyle$}}{\makebox[4.57916pt][c]{$\textstyle$}}{\makebox[3.20541pt][c]{$\scriptstyle$}}{\makebox[2.28958pt][c]{$\scriptscriptstyle$}}\leavevmode{\varphi}}}$
$\displaystyle=-\left[\left(\chi^{\prime}\right)^{2}+\chi^{\prime\prime}+\psi^{\prime\prime}\right]e^{-2\psi}$
(3.7c)
$\displaystyle\mathchoice{G^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{G^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{G^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}{G^{\leavevmode{z}\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}}_{\mathchoice{\makebox[3.5632pt][c]{$\displaystyle$}}{\makebox[3.5632pt][c]{$\textstyle$}}{\makebox[2.49423pt][c]{$\scriptstyle$}}{\makebox[1.7816pt][c]{$\scriptscriptstyle$}}\leavevmode{z}}}$