Bardeen solution with a cloud of strings Manuel E. Rodrigues12 Henrique A. Vieira1y 1Faculdade de F sica Programa de P os-Gradua c ao em F sica

2025-04-24 0 0 702.64KB 16 页 10玖币
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Bardeen solution with a cloud of strings
Manuel E. Rodrigues(1,2)∗, Henrique A. Vieira(1)†
(1)Faculdade de F´ısica, Programa de P´os-Gradua¸c˜ao em F´ısica,
Universidade Federal do Par´a,
66075-110, Bel´em, Par´a, Brazil
(2)Faculdade de Ciˆencias Exatas e Tecnologia,
Universidade Federal do Par´a Campus Universit´ario de Abaetetuba,
68440-000, Abaetetuba, Par´a, Brazil
(Dated: October 14, 2022)
In this paper we present a Bardeen solution surrounded by a cloud of strings fluid. We show
how this model has the same event horizon characteristic as the Bardeen solution, however, the
parameter of the strings make the solution singular at the origin. We also analyze the solution from
a thermodynamic point of view. We calculate the system’s state function, enthalpy; the temperature;
and the other potentials; as a function of entropy. By analyzing the thermodynamic coefficients,
we show that the solution presents three distinct phases, two of which are stable and one unstable.
These three phases could be best visualized using a plot of Gibbs free energy versus temperature.
In the end, we calculate the critical exponents and find that they are the same as those found in
Van der Walls theory.
PACS numbers: 04.70.BW, 04.70.-s
I. INTRODUCTION
With the development of general relativity at the beginning of the last century [1], we started to see the Universe
in a completely different way. Schwarzschild [2], solving Einstein’s equations, proposed the existence of a region of
space time with singular density from which not even electromagnetic radiation could escape. Later, this region
would become known as black holes, solutions to the Einstein equations that describe regions of space time covered
by an event horizon and that may have a singularity inside [3]. This field challenged the understanding of physicists
from that time and is still today subject to theoretical studies and motivation for the improvement of observational
astronomical apparatus. And it has recently gained strength with the detection of gravitational waves [4–10], and
with the release, by the Event Horizon Telescope team, of the first images of the shadow of a black hole [11–16].
Until the mid-1970s we thought it was impossible for any object to come out of a black hole, even electromagnetic
radiation. However, Hawking [17] showed that these bodies emit thermal radiation and this started a new area in
gravitation known today as black hole thermodynamics [18–27]. There are several reasons why this area is relevant
today. First, any physicist knows the difficulty of trying to describe, for example, one gas using the equations of motion
from mechanics. Now, imagine trying to evaluate the contribution of each star, or even each galaxy, separately, in
the formation and functioning of the Universe. This is why thermodynamics is an essential part of the analysis of
cosmology, which deals with distances of the order 1 mega parsec. Second, a purely classical black hole has no entropy
and therefore would violate the 2nd law of thermodynamics. Last but not least, it is possible to construct purely
classical laws for black holes [28] that can only be interpreted by adding quantum effects, and the reason for this is
not yet known since we do not have a quantum theory of gravity, neither one that can explain all forces of nature in
a simple and condensed way.
Recently, the string theory is the most referred possibility to be “the final theory”. On it, the Universe is thought of
as a collection of extended objects instead of pointlike particles. A promising candidate is a one-dimensional continua
string object. M. G¨urses and F. G¨ursey [29] first derived the string equation of motion in General Relativity, then
they showed [30] that a fluid governed by this equation could model the interior of a Kerr-Schild metric. Later,
J. Stachel [31] proposed an extension of the relativistic “dust cloud” model for a perfect fluid. With this in mind,
Letelier [32] obtained a solution of Einstein’s equations for clouds of strings and used it to construct a model of a star.
Subsequently, many other papers in the literature have considered clouds of strings a fluid serving as a background for
black holes, charged or not. In the Einstein-Gaus-Bonet theory: it was first considered on [33], where the authors also
compute Hawking temperature, entropy, heat capacity, and Helmholtz free energy; then [34] calculated quasinormal
∗E-mail address: esialg@gmail.com
†E-mail address: henriquefisica2017@gmail.com
arXiv:2210.06531v1 [gr-qc] 12 Oct 2022
2
modes for a scalar field in such space-time, highlighting the function of the parameter associated with the clouds of
strings; at [35] they included charge, which could be a monopole, and focused on thermodynamic analysis. There are
also solutions for higher orders considering Lovelock theory: Ghosh, Papnoi, and Maharaj [36] found first to third-
order solutions of D≥4 dimensions and they made thermodynamic stability considerations; following the previous
work, Ghosh and Maharaj [37] generalized the solution to Ndimensions, evaluated the energy conditions, and studied
the role played by clouds of strings in the structure of the event horizon. Another class of solutions involving clouds
of strings are those that are also considered the so-called quintessence. Today, we have experimental data pointing
to an accelerating expanding universe [38–40], and yet no theoretical possibility has been widely accepted as the
explanation for this phenomenon, with the ideas of matter and dark energy being the most widespread candidates
within the literature. Quintessence is currently seen as the candidate to explain dark energy [41] and thereby solve
the question of the accelerating expansion of the Universe. Toledo and Bezerra [42] find a Reissner–Nordstr¨om type
solution then studied its thermodynamics and calculated the quasinormal frequencies for a scalar field; the same
authors extended the previous work to higher dimensions using the Lovelock theory [43]; on [44] we have an extension
of Letellier space-time and is showed that this model corresponds to a black hole with global monopole surrounded by
quintessence; finally, on [45] they presented a charged AdS black hole and studied it’s thermodynamics aspects. Cai
and Miao [46] have further shown, using Rastall gravitation, that it is possible to obtain a black hole solution where
the clouds of strings become quintessence in a given limit. It is also worth mentioning that Richarte and Simeone [47]
built a traversable wormhole model using Letelier’s idea.
There are several types of black hole solutions today, however, the one proposed by James Bardeen[48] was revo-
lutionary because there is no curvature singularity. Initially, the model lacked a source, but a few years later Beato
and Garcia [49] proposed a self-gravitating magnetic charge, described by nonlinear electrodynamics [50], as a source
and this make it a exact solution of Einstein’s equations. It was realized that one can use nonlinear electrodynamics
to construct regular black hole [51], and now we have solutions that: possess electric charge as source [52, 53]; it’s
obtained using f(r) theory [54–56], and also using f(G) theory [56, 57]; possess angular momentum [58, 59]; and on
Rainbow Gravity [60]. There is also in the literature a Bardeen solution surrounded by the quintessence [61], a work
that shows how the regularity of the model is lost when a singular solution is attached to it. Therefore, we aim here
to build a solution that is Bardeen surrounded by clouds of strings.
The paper is organized as follows: in section II we will comment on the general aspects of how we did to obtain
our solution; in section III we will find the event horizons and make considerations about regularity; in section IV
we will work on the solution from a thermodynamic point of view; and finally, in section V we will make our final
considerations. We will consider throughout this work the metric signature (+,−,−,−). The Riemann tensor is
defined as Rα
βµν =∂µΓα
βν −∂νΓα
βµ + Γσ
βν Γα
σµ −Γσ
βµΓα
σν , where Γα
µν =1
2gαβ (∂µgνβ +∂νgµβ −∂βgµν ) is the
Levi-Civita connection. Also, we will use geometrodynamic units where G=~=c= 1.
II. SPACE TIME WITH CLOUDS OF STRINGS
We are looking for a Bardeen solution in a spacetime surrounded by clouds of strings, this type of solution can be
derived from general relativity minimally coupled to non linear electrodynamics (NED) and the cloud of strings by
the action
S=Zd4x√−g[R+ 2λ+L(F)] + SCS ,(1)
where gis the metric determinant, Ris the curvature scalar, λis the cosmological constant, L(F) is the nonlinear
general Lagrangian of electromagnetic theory, function of the scalar F=Fµν Fµν /4, and SCS is the Nambu–Goto
action to used to describe stringlike objects, given by [32]
SSC =Z√−γMdΛ0dΛ1.(2)
Where, γis the determinant of γAB, which is a induced metric on a submanifold given by
γAB =gµν
∂xµ
∂ΛA
∂xν
∂ΛB,(3)
Mis a dimensionless constant that characterize the string, and Λ0and Λ1are a timelike and spacelike parameter.
Note that here each string is associated with a world sheet and is described by xµ(ΛA). It’s possible to rewrite (2) as
3
SSC =ZM−1
2Σµν Σµν dΛ0dΛ1,(4)
where Σµν is a bi-vector written as
Σµν =AB ∂xµ
∂ΛA
∂xν
∂ΛB,(5)
note that AB is the Levi–Civita symbol, 01 =−10 = 1.
Varying the action (1) with respect to the metric we find
Rµν −1
2gµν R+gµν λ= 8πTµν + 8πT SC
µν ,(6)
where Tµν is the stress-energy tensor of the matter sector, defined for NED as
Tµν =gµν L(F)−dL
dF Fα
µFνα,(7)
and TSC
µν is the stress-energy tensor of the cloud of string, which has the form [32]
TSC
µν =ρΣα
µΣαν
8π√−γ,(8)
with ρbeing the proper density of the cloud. Requiring a zero divergence of the stress-energy tensor (8), we get the
equations
∇µ(ρΣµν ) = ∂µ√−gρΣµν = 0,(9)
and
Σµβ∇µΣν
β
(−γ)1/2= 0.(10)
For a spherically symmetric spacetime we have
ds2=f(r)dt2−1
f(r)dr2−r2dθ2−r2sin2θdφ2,(11)
we will have only one nonzero component of the Σµν bivector which is Σ01 and it depends exclusively on the radial
coordinate. Solving the two previous equations for Σ, we get
Σ01 =√−γ=a
ρr2,(12)
where ais an integration constant related to strings, being limited to the interval 0 < a < 1 [32].
Now, defining the Lagrangian for the Bardeen solution as
L(F) = 3
8πsq2 p2q2F
2 + p2q2F!5/2
,(13)
where s=|q|/(2m), we can solve the Einstein equations (6) getting only two nontrivial differential equations
−a
r2+−rf0(r)−f(r) + λr2+ 1
r2−6M
q3r2
q2+ 15/2= 0,(14)
4
−f00(r)
2−f0(r)
r+λ−3Mqq2
r2+ 1 −5q10 + 2q2r8+ 2r10
q3r2(q2+r2)4= 0.(15)
Solving the system of differential equations above we have
f(r)=1−a−2M1
r−2Mr2
(q2+r2)3/2−λr2
3,(16)
where M1is a integration constant. In order to get a Bardeen-AdS solution for when a= 0 we do from here M1= 0.
So, we have
f(r) = 1 −a−2M r2
(q2+r2)3/2−λr2
3.(17)
We see that if a= 0 on (17) the Bardeen’s solution [49] is recovered.
III. EVENT HORIZONS AND REGULARITY
To obtain the horizons, we need to solve f(r) = 0. Although we were not able to obtain an analytical expression,
we show the behavior of f(r) as a function of rin the figure 1. It is clear that the number of horizons depends on the
values of the parameters, we see that it can be up to two.
For any extreme solution we have a degeneracy in the event horizon, the condition to find the event horizon is, as
said before
f(r+) = 0,(18)
where r+represents the horizon radius. And furthermore, we must also impose
df(r+)
dr+
= 0.(19)
From this we can obtain both the value of the radius of the event horizon, and that of the critical charge qc. For those
values specified on the figure 1 we have qc= 0.9985.
To check for singularities we need to analyze the Kretschmann scalar [62], K=Rµναβ Rµναβ , which for this case is
K=4
3"3a2
r4+
2aλ+6M
(q2+r2)3/2
r2+6λMq24q2−r2
(q2+r2)7/2+2λ2+9M2−4q6r2+ 47q4r4−12q2r6+ 8q8+ 4r8
(q2+r2)7#.(20)
We can expand this function in Taylor series to r >> 1 and r << 1 around the value zero
K(r∼0) = 4a2
r4+83mq 4λq2−3a+ 36m2+λ2q6
3q6+
8aλ+6m
q3
3r2−30mr2q4λq2−a+ 24m
q8+O(r3),(21)
K(r∼ ∞) = 8λ2
3+8aλ
3r2+O 1
r3!.(22)
We see in (21) two terms proportional to a2/r4and a/r2that account for the divergence of the scalar when r→0. It is
clear that if a= 0 we return to the Bardeen case where spacetime is regular everywhere. In general, the Kretschmann
scalar has the limits
lim
r→0K=∞,
lim
r→∞ K=8λ2
3.
(23)
摘要：

BardeensolutionwithacloudofstringsManuelE.Rodrigues(1;2),HenriqueA.Vieira(1)y(1)FaculdadedeFsica,ProgramadePos-Graduac~aoemFsica,UniversidadeFederaldoPara,66075-110,Belem,Para,Brazil(2)FaculdadedeCi^enciasExataseTecnologia,UniversidadeFederaldoParaCampusUniversitariodeAbaetetuba,68440-00...

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