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See GWIC_Intro_Cover2.pdf

WHITE TEXT

Gravitational Wave International Committee

This document was produced by the GWIC 3G Subcommittee

Final release, April 2021

Cover: Robert Hurt/IPAC/Caltech

###### Contents

  1. 1 Introduction
    1. 1.1 The GWIC 3G Subcommittee Reports
    2. 1.2 Intended Audiences and Impacts
    3. 1.3 GWIC 3G Subcommittee Membership

##  Chapter 1 Introduction

The first direct detection of gravitational waves (GW) emitted from a pair of
merging black holes located 1.3 billion light years from Earth in September
2015 has been heralded as one of most significant scientific breakthroughs in
physics and astronomy of the 21st century. Coming nearly 100 years after
Albert Einstein predicted their existence, the first observations of directly
detectable GW signal are ushering in a completely new way to observe and study
the most violent and energetic astrophysical events in the Universe.

The opening of the GW window on the Universe has highlighted in dramatic
fashion the tremendous scientific potential of this new field. The discoveries
by LIGO and Virgo since the first detection have revealed stunning new
insights into the nature of black holes and neutron stars.

Motivated by these initial breakthroughs and recognizing that to fully exploit
the new field will require new observatories that may take 15 – 20 years from
conception until operations,the Gravitational Wave International Committee
(GWIC)111The Gravitational Wave International Committee was formed in 1997 to
facilitate international collaboration and cooperation in the construction,
operation and use of the major gravitational wave detection facilities world-
wide. It is associated with the International Union of Pure and Applied
Physics as its Working Group WG.11. Through this association, GWIC is
connected with the International Society on General Relativity and Gravitation
(IUPAP’s Affiliated Commission AC.2), its Commission C19 (Astrophysics), and
another Working Group, the AstroParticle Physics International Committee
(APPIC). convened a subcommittee222The members of the GWIC 3G Subcommittee are
provided in Section 1.3. in 2016 to examine the path to build and operate a
network of future ground-based observatories, capable of extending the
observational GW horizon well beyond that currently attainable with the
current generation of detectors.

The primary aims of the Third Generation (3G) Subcommittee were to identify
and frame the major issues and challenges in developing a coherent vision for
future worldwide network of GW observatories and, from that, produce a
resource for the ground-based GW community that can assist in planning and
executing a strategy to realize the vision. Over the period from November 2016
through March 2019, members of the subcommittee drew in researchers from the
GW and broader research communities to collect input and produce reports in
five major areas as described below. Each report was subsequently reviewed
independently and anonymously by a panel comprised of funding agency officers,
discipline experts and generalist scientists appointed by the Gravitational
Wave Agency Correspondents (GWAC).333The Gravitational Wave Agencies
Correspondents was formed in 2015 to provide a direct channel of communication
between funding agencies to coordinate the use of existing and explore new
funding opportunities for the gravitational wave science community. The final
reports were approved for release by the GWIC membership in September 2020.

###  1.1 The GWIC 3G Subcommittee Reports

The five reports commissioned by GWIC lay out an ambitious set of science
targets achievable by future ground-based detectors, identify the requisite
research end development in detector development and large-scale computing
needed to reach those targets, examine the synergies and complementarity with
other scientific disciplines, and assess possible governance models to manage
and operate a unified global network of future observatories:

  1. 1.

The Next Generation Global Gravitational Wave Observatory Science Book (59
pages) presents a comprehensive survey of scientific frontiers accessible and
the fundamental questions addressable by the future observatories, exploring
the fields of high energy astrophysics, cosmology, gravitational physics,
nuclear physics, and high energy physics. The report highlights a large number
of key science targets available to future observatories possessing ten times
the sensitivities attainable with today’s LIGO, Virgo, and KAGRA
observatories.

  2. 2.

Research and Development for the Next Generation of Ground-Based
Gravitational-Wave Detectors (77 pages) examines in detail the wide range of
nearer- and longer-term detector R&D programs needed for next generation GW
detectors commensurate with the key science targets presented in the Science
Book, including considerations of site selection and large-scale vacuum
infrastructure. The report makes a series of detailed recommendations on the
needed advances in detector technology and the timescales needed to achieve
those advances. It also identifies areas where larger-scale globally
coordinated R&D efforts will be critical to ensuring success while minimizing
costs.

  3. 3.

Future Ground-based Gravitational-Wave Observatories: Synergies with Other
Scientific Communities (11 pages) identifies a broad set of scientific
constituencies beyond the ground-based GW community having common scientific
interests where GW data have significant impact and makes a series of
recommendations for facilitating strong linkages with those relevant
scientific communities. It also presents communication and outreach plans to
help engage with those communities and motivate them in support of 3G GW
science.

  4. 4.

Gravitational-Wave Data Analysis: Computing Challenges in the 3G Era (23
pages) lays out the many computational challenges – required
computing/storage/networking resources, anticipated algorithm and software
needs, and commensurate person power – imposed by the orders of magnitude
greater GW event rates enabled by these new GW observatories. Key to meeting
these challenges will be establishing synergistic collaborations with the high
energy physics and astronomy communities as well as with industrial partners
to develop shared solutions.

  5. 5.

An Exploration of Possible Governance Models for the Future Global
Gravitational-Wave Observatory Network (13 pages) evaluates a variety of
possible governance models by carrying out a comprehensive survey of existing
organizational structures of large-scale scientific laboratories and
collaborations to propose a sustainable, centralized governance model for the
overall management of the construction and operations of the planned 3G
observatories.

###  1.2 Intended Audiences and Impacts

These reports, somewhat technical in nature, are primarily intended for
members of the GW community to serve as a basis for planning and executing
successful proposals to construct and operate future 3G observatories. These
reports are of particular relevance to two future ground-based observatory
efforts – the Einstein Telescope (ET) in Europe and the Cosmic Explorer (CE)
in the US. ET and CE are in different stages of planning toward project
approval at present, however both projects have had close interactions with
the GWIC 3G Subcommittee during the course of the development of these reports
and acknowledge benefiting from the material presented in them.

The reports may also serve to familiarize and educate other individuals and
groups, including researchers from other disciplines as well as funding
agencies, on specific themes and topics of overlapping interest with the GW
community.

Each report was written, to the extent possible, to be self-contained and
stand-alone, however when read collectively they will provide a broader and
more complete perspective on the scientific opportunities afforded by 3G
observatories and the steps needed to realize their operation in the 2030s.

###  1.3 GWIC 3G Subcommittee Membership

Steering Committee
Michele Punturo, INFN Perugia, Italy (Co-chair)
David Reitze, Caltech, USA (Co-chair)
Peter Couvares, Caltech, USA
Stavros Katsanevas, European Gravitational Observatory
Takaaki Kajita, University of Tokyo, Japan
Vicky Kalogera, Northwestern University, USA
Harald Lueck, AEI, Hannover, Germany
David McClelland, Australian National University, Australia
Sheila Rowan, University of Glasgow, UK
Gary Sanders, Caltech, USA
B.S. Sathyaprakash, Penn State University, USA and Cardiff University, UK
David Shoemaker, MIT, USA (Secretary)
Jo van den Brand, Nikhef, Netherlands

Science Book Subcommittee
Vicky Kalogera, Northwestern University, USA (Co-chair)
B.S. Sathyaprakash, Penn State USA and Cardiff University, UK (Co-chair)
Matthew Bailes, Swinburne, Australia
Marie Anne Bizouard, CNRS, France
Alessandra Buonanno, AEI, Potsdam, Germany, and University of Maryland, USA
Adam Burrows, Princeton, USA
Monica Colpi, INFN, Italy
Matt Evans, MIT, USA
Stephen Fairhurst, Cardiff University, UK
Stefan Hild, Maastricht University, Netherlands
Mansi M. Kasliwal, Caltech, USA
Luis Lehner, Perimeter Institute, Canada
Ilya Mandel, University of Birmingham, UK
Vuk Mandic, University of Minnesota, USA
Samaya Nissanke, University of Amsterdam, Netherlands
Marialessandra Papa, AEI, Hannover, Germany
Sanjay Reddy, University of Washington, USA
Stephan Rosswog, Oskar Klein Centre, Sweden
Chris Van Den Broeck, Nikhef, Netherlands

Detector Research and Development Subcommittee
David McClelland, Australian National University, Australia (Co-chair)
Harald Lueck, AEI, Hannover, Germany (Co-chair)
Rana Adhikari, Caltech, USA
Masaki Ando, University of Tokyo, Japan
GariLynn Billingsley, Caltech, USA
Geppo Cagnoli, ILM, Lyon, France
Matt Evans, MIT, USA
Martin Fejer, Stanford University, USA
Andreas Freise, University of Birmingham, UK
Paul Fulda, University of Florida, USA
Eric Genin, Virgo, Italy
Gabriela González, Louisiana State University, USA
Jan Harms, Università degli Studi di Urbino, Italy
Stefan Hild, University of Glasgow, UK
Giovanni Losurdo, INFN Pisa, Italy
Ian Martin, University of Glasgow, UK
Anil Prabhakar, IIT Madras, India
Stuart Reid, University of Strathclyde, UK
Fulvio Ricci, Università La Sapienza, and INFN Roma, Italy
Norna Robertson, Caltech, USA
Jo van den Brand, Nikhef, Netherlands
Benno Willke, AEI, Hannover, Germany
Mike Zucker, MIT, USA

Synergies with Scientific Communities Subcommittee
Michele Punturo, INFN Perugia, Italy (Co-chair)
David Reitze, Caltech, USA (Co-chair)
David Shoemaker, MIT, USA

Governance Models Subcommittee
Stavros Katsanevas, Astro Particle and Cosmology Laboratory, France (Co-chair)
Gary Sanders, Caltech, USA (Co-chair)
Beverly Berger, Stanford University, USA
Gabriela González, Louisiana State University, USA
James Hough, University of Glasgow, UK
Ajit K. Kembhavi, Inter-University Centre for Astronomy and Astrophysics,
India
Frank Linde, Nikhef, Netherlands
David McClelland, Australian National University, Australia
Masatake Ohashi, Institute of Cosmic Ray Research, Japan
Fulvio Ricci, Università La Sapienza, and INFN Roma, Italy
Stan Whitcomb, Caltech, USA

Data Analysis Computing Challenges Subcommittee
Peter Couvares, Caltech, USA (Co-chair)
Ian Bird, CERN, Switzerland (Co-chair)
Ed Porter, Université Paris Diderot, France (Co-chair)
Stefano Bagnasco, INFN, Italy
Geoffrey Lovelace, California State Fullerton, USA
Josh Willis, Caltech, USA


# High-Temperature Modifications of Charged Casimir Wormholes

Phongpichit Channuie<EMAIL_ADDRESS>School of Science, Walailak
University, Nakhon Si Thammarat, 80160, Thailand  College of Graduate Studies,
Walailak University, Nakhon Si Thammarat, 80160, Thailand

###### Abstract

In this letter, we extend the investigation of the consequences of thermal
fluctuations on the Casimir effect within the context of a traversable
wormhole, recently proposed by Garattini & Faizal, arXiv:2403.15174 [gr-qc],
subject to charge contributions. Specifically, our focus is on scenarios where
the plates exhibit radial variation. In our analysis, we initially concentrate
on the high temperature approximation, considering solely the influence of
charge on the thermal Casimir wormholes. Additionally, upon incorporating
Generalized Uncertainty Principle (GUP) corrections to the Casimir energy, we
obtain a new class of wormhole solutions. Notably, we establish that the
flare-out condition remains consistently satisfied. Intriguingly, our findings
reveal that both the charge and GUP contributions serve to further enlarge the
throat’s size.

##  I Introduction

The Casimir effect is based on the assumption of perfectly conducting
surfaces. However, real-world plates are never perfectly conducting; they
exhibit a complex permittivity, varying with frequency depending on the
material. This consideration alters the original Casimir force at zero
temperature, transforming it into a finite-temperature thermal effect. This
thermal effect incorporates both thermal and quantum fluctuations, as
described by the Lifshitz theory, which accounts for correlated fluctuating
charges and currents in the plates Sabisky1973 . This adaptation of the
conventional Casimir effect to finite temperature is termed the thermal
Casimir effect Canaguier-Durand:2010hwu ; Marino:2014rfa . It has been shown
that at separations beyond a critical value, the force generated by thermal
fluctuations surpasses that of zero-temperature quantum fluctuation
Sushkov:2010cv ; Bostrom:2000zza . This suggests the potential significance of
thermal effects in scaling up the size of traversable wormholes (TWs). Very
recently, the consequences of thermal fluctuations to the Casimir effect on a
traversable wormhole have been investigated Garattini:2024jkr . The authors
illustrated how thermal fluctuations alter the throat of the wormhole in both
high and low temperature regimes. Furthermore, they elucidated the impact of
finite temperature on the size of a wormhole. In this work, we investigate new
exact and analytic solutions of the Einstein–Maxwell field equations
describing Casimir wormholes including the high-temperature contribution to
the Casimir energy source for a traversable wormhole with the plates radially
varying and discuss the effect of the Generalized Uncertainty Principle (GUP).
Notice that the electromagnetic field combined with the Casimir source has
been used to examine the impacts of such an electrovacuum on a traversable
wormhole Samart:2021tvl ; Garattini:2023qyo .

The main purpose of this work is to extend the analysis present in
Ref.Garattini:2024jkr  for the thermal Casimir wormholes by incorporating the
electric charge underlying the GUP. This work is organized as follows: In
Section (II), we formulate the Einstein–Maxwell field equations describing
Casimir wormholes with charge. We next consider the thermal Casimir wormholes
employing energy densities featuring the thermal modifications to the chared
Casimir wormholes and solve to obtian the wormholes solutions in Section
(III). The results also include the GUP effects. We conclude of findings in
the last section.

##  II Setup

Our aim is to model a wormhole solution and typically we consider an
anisotropic fluid stress-energy tensor to describe the matter distribution,
which is described by the following form

$T^{\mu}_{\nu}=(\rho+p_{t})u^{\mu}u_{\nu}+p_{t}\delta^{\mu}_{\nu}-(p_{t}-p_{r})X^{\mu}X_{\nu},$
(1)

where $u^{\mu}$ is the fluid 4-velocity, $X_{\mu}$ is the unit space like
vector in the direction of radial vector and $g_{\mu\nu}$ is the metric
tensor. Furthermore, $\rho=\rho(r)$ is the energy density, $p_{r}=p_{r}(r)$
and $p_{t}=p_{t}(r)$ are the radial and tangential pressures, respectively.
The energy-momentum tensor of electromagnetic field defined by

${{T^{\rm
em}}^{\mu}}_{\nu}=\frac{1}{4}\big{(}-F^{\mu\gamma}F_{\nu\gamma}+\frac{1}{4}\delta^{\mu}_{\nu}F^{\beta\zeta}F_{\beta\zeta}\big{)}\,.$
(2)

We define an appropriate frame of the fluid velocity vectors Cadoni:2020izk

$\displaystyle u^{a}=e^{-\Phi}\delta^{a}_{0}\,,\quad
X^{a}=\sqrt{1-\frac{b(r)}{r}}\delta^{a}_{1}\,,$ (3)

so that $u_{a}u^{a}=-1$ and $X_{a}X^{a}=1$ as required. Now, rewriting the Eq.
(LABEL:eq8) in its covariant form we construct the geometrical Einstein tensor
on the left-hand side with an effective energy-momentum tensor on the right-
hand side, as

$R^{\mu}_{\nu}-\frac{1}{2}\delta^{\mu}_{\nu}R=8\pi\Big{(}T^{\mu}_{\nu}+{{T^{\rm
em}}^{\mu}}_{\nu}\Big{)},$ (4)

where

$\displaystyle{{T^{\rm em}}^{\mu}}_{\nu}$ $\displaystyle=$
$\displaystyle(\rho^{\rm em},p^{\rm em}_{r},p^{\rm em}_{t},p^{\rm em}_{t})$
(5) $\displaystyle=$ $\displaystyle\frac{q^{2}}{8\pi r^{4}}{\rm
diag.}(-1,-1,1,1)\,.$

Taking into account the Eqs. (1) and (5), the nonzero components of the field
equations (4) take the form:

$\displaystyle\frac{b^{\prime}}{r^{2}}=8\pi\Big{(}\rho+\frac{q^{2}}{8\pi
r^{4}}\Big{)},$ (6) $\displaystyle
2\left(1-\frac{b}{r}\right)\frac{\Phi^{\prime}}{r}-\frac{b}{r^{3}}=8\pi\Big{(}p_{r}-\frac{q^{2}}{8\pi
r^{4}}\Big{)},$ (7)
$\displaystyle\left(1-\frac{b}{r}\right)\left[\Phi^{\prime\prime}+\Phi^{\prime
2}-\frac{b^{\prime}r-b}{2r(r-b)}\Phi^{\prime}-\frac{b^{\prime}r-b}{2r^{2}(r-b)}+\frac{\Phi^{\prime}}{r}\right]=8\pi\Big{(}p_{t}+\frac{q^{2}}{8\pi
r^{4}}\Big{)},$ (8)

where $\rho$ represents the effective energy density, $p_{r}$ is the effective
radial pressure and $p_{t}$ is the effective tangential pressure respectively.

##  III Wormhole Solutions

In the same manner of previous work on zero temperature Casimir wormholes
Garattini:2019ivd , the plates separation distance $d$ can be promoted to a
radial variable $r$. We can compute the form of the energy density. For the
high temperature case, the thermal energy density is given by
Garattini:2024jkr

$\displaystyle\rho_{TC}=-\frac{k_{B}T}{8\pi r^{3}}\zeta(3)\,,$ (9)

where for in this case, we also have that the EoS leads to $\omega_{TC}=2$.
Following Ref.Garattini:2024jkr  for high temperature regimes, we come up with
the following shape function:

$\displaystyle b(r)=-\zeta(3)k_{B}T\log(r)-\frac{q^{2}}{r}+c_{1}\,,$ (10)

where $c_{1}$ is a integration constant. Taking $b(r_{0})=r_{0}$, it can be
determined to yield

$\displaystyle
c_{1}=\frac{\zeta(3)k_{B}Tr_{0}\log(r_{0})+q^{2}+r_{0}^{2}}{r_{0}}\,.$ (11)

Therefore, the first EFE leads to the following shape function

$\displaystyle b(r)$ $\displaystyle=$ $\displaystyle
r_{0}+q^{2}\left(\frac{1}{r_{0}}-\frac{1}{r}\right)$ (12)
$\displaystyle-\zeta(3)k_{B}T\log\left(\frac{r}{r_{0}}\right)\,,$

with $\zeta(x)$ being the Riemann zeta function. Having considered the charge
contribution, the above result is a generalization of what is found in
Ref.Garattini:2024jkr . We also find that $b(r)/r\rightarrow 0$ when
$r\rightarrow\infty$. From the shape function (12), one finds that the flare-
out condition is always satisfied since

$\displaystyle
b^{\prime}(r_{0})=\frac{q^{2}}{r_{0}^{2}}-\frac{\zeta(3)k_{B}T}{r_{0}}<1\,,$
(13)

implying that

$\displaystyle q^{2}-r^{2}_{0}<\zeta(3)k_{B}Tr_{0}\,.$ (14)

Moreover from the condition $1-b(r)/r>0$, it is possible to clarify that there
exists $r=r^{*}$ such that $b(r^{*})=0$, where

$\displaystyle
r^{*}=-\frac{q^{2}}{\zeta(3)k_{B}T\,W\left(-\frac{q^{2}e^{-\frac{q^{2}+r_{0}^{2}}{\zeta
r_{0}Tk_{B}}}}{\zeta r_{0}Tk_{B}}\right)}\,,$ (15)

with $W(x)$ is a Lambert $W$ function or product logarithm. Clearly, when
$q\rightarrow 0$, the above result reduces to that obtained in
Ref.Garattini:2024jkr , $r^{*}\rightarrow
r_{0}e^{\frac{r_{0}}{\zeta(3)k_{B}T}}$. To obtain a TW we need to compute the
redshift function. We have to impose the EoS $p_{r}(r)=\omega\rho(r)$. From
Eq.(7), one finds

$\displaystyle\Phi^{\prime}(r)=\frac{-b(r)r+\zeta(3)k_{B}Tr\omega+q^{2}}{2r^{2}(b(r)-r)}\,.$
(16)

Substituting $b(r)$ from Eq.(12), we come up with

$\displaystyle\Phi^{\prime}(r)=\frac{rr_{0}\left(r_{0}-\zeta(3)T\omega
k_{B}\right)-\zeta
rr_{0}Tk_{B}\log\left(\frac{r}{r_{0}}\right)+q^{2}\left(r-2r_{0}\right)}{2r\left(\zeta
rr_{0}Tk_{B}\log\left(\frac{r}{r_{0}}\right)+\left(r-r_{0}\right)\left(rr_{0}-q^{2}\right)\right)}\,.$
(17)

Close to the throat, the r.h.s. of Eq.(17) can be approximated by

$\displaystyle\Phi^{\prime}(r)\simeq\frac{r^{2}_{0}\left(r_{0}-\zeta(3)T\omega
k_{B}\right)-q^{2}r_{0}}{2r_{0}\left(r-r_{0}\right)\left(r^{2}_{0}-q^{2}\right)}\,.$
(18)

Following Ref.Garattini:2019ivd  we can choose $\omega$ such that
$\Phi^{\prime}(r)=0$ to avoid the appearance of a horizon. This can be seen if

$\displaystyle\omega=\frac{r_{0}\left(r^{2}_{0}-q^{2}\right)}{\zeta(3)r^{2}_{0}Tk_{B}}\,.$
(19)

Note that when $q=0$, the above result reduces to that of
Ref.Garattini:2024jkr , i.e., $\omega=r_{0}/(\zeta(3)r^{2}_{0}Tk_{B})$.
Interestingly, this allow us to compute the throat of the wormhole: $r_{0}$.
Here using Eq.(19) and incorporating with $\omega_{TC}=2$, we simply find a
positive solution:

$\displaystyle
r_{0}=\zeta(3)Tk_{B}\Big{(}1+\sqrt{1+\frac{q^{2}}{\zeta(3)^{2}T^{2}k_{B}^{2}}}\Big{)}\,.$
(20)

Having compared with Ref.Garattini:2024jkr , in our convention, we take
$l_{P}=1$. Notice from the above result that the size of the throat is
enlarged further by the effect of the charge. Moreover, it can be more
generalized by following the work present in Ref.Jusufi:2020rpw  for
elaborating in more details about the GUP corrected energy density. This allow
to construct a new class of Casimir wormholes. Following Samart:2021tvl , it
is straightforward to calculate the shape function of the traverasable
wormholes with the electric charge underlying a so-called Generalized
Uncertainty Principle (GUP). We finally find

$\displaystyle b(r)$ $\displaystyle=$ $\displaystyle\frac{\pi^{3}\beta
D_{i}}{270r^{3}}-\zeta(3)k_{B}T\log(r)$ (21)
$\displaystyle-\frac{q^{2}}{r}+c_{1}\,,$

where $c_{1}$ is a integration constant, $D_{i}$s are the constants, see
Ref.Jusufi:2020rpw . Taking $b(r_{0})=r_{0}$, it can be determined to yield

$\displaystyle c_{1}=\zeta Tk_{B}\log\left(r_{0}\right)-\frac{\pi^{3}\beta
D_{i}}{270r_{0}^{3}}+\frac{q^{2}}{r_{0}}+r_{0}\,.$ (22)

Therefore, the first Einstein Field Equation leads to the following shape
function

$\displaystyle b(r)$ $\displaystyle=$ $\displaystyle
r_{0}+q^{2}\left(\frac{1}{r_{0}}-\frac{1}{r}\right)$ (23)
$\displaystyle+\frac{\pi^{3}\beta
D_{i}}{270}\left(\frac{1}{r^{3}}-\frac{1}{r_{0}^{3}}\right)$
$\displaystyle-\zeta(3)k_{B}T\log\left(\frac{r}{r_{0}}\right)\,.$

Notice that the 3rd term is due to the GUP effect, while the last term is the
consequences of thermal fluctuations. It is also straightforward to show that
$b(r)/r\rightarrow 0$ when $r\rightarrow\infty$. From the shape function (23),
one finds that the flare-out condition is always satisfied since

$\displaystyle b^{\prime}(r_{0})=-\frac{\pi^{3}\beta
D_{i}}{90r_{0}^{4}}-\frac{\zeta(3)k_{B}T}{r_{0}}+\frac{q^{2}}{r_{0}^{2}}<1\,,$
(24)

implying that

$\displaystyle q^{2}-r^{2}_{0}<\frac{\pi^{3}\beta
D_{i}}{90r^{2}}+\zeta(3)k_{B}Tr_{0}\,.$ (25)

To obtain a TW, we are going to compute the redshift function. We have to
impose the EoS $p_{r}(r)=\omega\rho(r)$. From Eq.(7), one finds

$\displaystyle\Phi^{\prime}(r)=\frac{-b(r)r+\zeta(3)k_{B}Tr\omega+q^{2}}{2r^{2}(b(r)-r)}\,.$
(26)

Substituting $b(r)$ from Eq.(12), we come up with

$\displaystyle\Phi^{\prime}(r)=\frac{270r^{2}r_{0}^{2}\left(rr_{0}\left(\zeta
T\omega k_{B}-r_{0}\right)+{\cal
R}-q^{2}\left(r-2r_{0}\right)\right)+\pi^{3}\beta
D_{i}\left(r^{3}+r_{0}^{3}(3\omega-1)\right)}{2\pi^{3}\beta
r\left(r_{0}^{3}-r^{3}\right)D_{i}-540r^{3}r_{0}^{2}\left({\cal
R}+\left(r-r_{0}\right)\left(rr_{0}-q^{2}\right)\right)}\,,$ (27)

where we have defined

$\displaystyle{\cal R}=\zeta rr_{0}Tk_{B}\log\left(\frac{r}{r_{0}}\right)\,,$
(28)

Close to the throat, the r.h.s. of Eq.(27) can be approximated by

$\displaystyle\Phi^{\prime}(r)\simeq-\frac{3r_{0}^{3}\left(90\left(\zeta
r_{0}^{3}T\omega k_{B}+q^{2}r^{2}_{0}-r_{0}^{4}\right)+\pi^{3}\beta\omega
D_{i}\right)}{2r\left(r-r_{0}\right)\left(\pi^{3}\beta\left(r^{2}+r_{0}\left(r+r_{0}\right)\right)D_{i}+270r^{2}r_{0}^{2}\left(rr_{0}-q^{2}\right)\right)}\,.$
(29)

Following Ref.Garattini:2019ivd , we can choose $\omega$ such that
$\Phi^{\prime}(r)=0$ to avoid the appearance of a horizon. This can be done if

$\displaystyle\omega=\frac{90r_{0}^{2}\left(r_{0}^{2}-q^{2}\right)}{90\zeta
r_{0}^{3}Tk_{B}+\pi^{3}\beta D_{i}}\,.$ (30)

The result is more general compared to the previous case. Note that when
$q=0,\,\beta=0$, the above result reduces to that of Ref.Garattini:2024jkr ,
i.e., $\omega=r_{0}/(\zeta(3)r^{2}_{0}Tk_{B})$. Following the preceding
result, we can also compute the throat of the wormhole.

##  IV Concluding Remarks

In this work, following Ref.Garattini:2024jkr , we have investigated the
consequences of thermal fluctuations to the Casimir effect on a traversable
wormhole subject to the charge contribution. To do calculations, we focused
only the situation in which the plates have been taken radially varying. For
the high temperature approximation, we first considered only charge
contribution to the thermal Casimir wormholes. We investigated the effect of
such finite temperature effects incorporating with a charge on the size of a
wormhole. Having compared with Ref.Garattini:2024jkr , the size of the throat
is enlarged further by the effect of the charge contribution given by

$\displaystyle
r_{0}=\zeta(3)Tk_{B}\Big{(}1+\sqrt{1+\frac{q^{2}}{\zeta(3)^{2}T^{2}k_{B}^{2}}}\Big{)}\,.$
(31)

We have verified that the flare-out condition is always satisfied. Indeed, we
can restore constants, e.g, $G,\,c,\,\hbar$, so that our results reflect
physical meanings. This is done by introducing the Casimir thermal wave length
Ref.Garattini:2024jkr . The second scenario deals with the GUP corrections to
the thermal Casimir energy. In this case, we have also found a new class of WH
solutions. We have found that the flare-out condition is always satisfied in
this case. Interestingly, the wormhole solutions in the present work can be
further tested by the wormhole observations, see, e.g., Abe:2010ap ;
Toki:2011zu ; Godani:2021aub ; Ohgami:2015nra ; Dai:2019mse .

###### Acknowledgements.

This work is financially supported by Thailand NSRF via PMU-B under grant
number PCB37G6600138.

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