# Twin stau as a self-interacting dark matter

Michał Łukawski
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,
ul. Pasteura 5, PL-02-093 Warsaw, Poland
<EMAIL_ADDRESS>
###### Abstract

Supersymmetric Twin Higgs models allow for reducing the fine-tuning with
respect to Minimal Supersymmetric Standard Model by protecting the mass of the
Higgs boson by additional, accidental global symmetry. This class of models
introduce numerous new states, some of which might be candidates for dark
matter. Since those reside in twin sector, they are not charged under Standard
Model gauge group. We proposed twin stau as a candidate for dark matter. Even
though twin stau is charged under twin electromagnetism, since supersymmetric
partners obtain large masses from supersymmetry breaking they can easily
escape bounds for self-interacting dark matter. The mass of twin stau which
reproduces correct relic abundance is usually between 300 and 500 GeV. This
scenario can be probed by future direct detection experiments such as Lux-
Zepelin.

##  1 Introduction

The mass of the Higgs particle is not protected from the large corrections
from higher energy scales such as Planck mass. One of the most successful
mechanisms protecting the mass of the Higgs is supersymmetry, which allows for
cancellation of the these corrections above the scale of supersymmetry
breaking, $m_{\mathrm{SUSY}}$, at which new particles enter [1]. Naturalness
requires that $m_{\mathrm{SUSY}}$ is not much above the EW scale, usually at
below 1 TeV. Hence, one of the main features of supersymmetric models are
relatively light coloured particles which should be produced at LHC in hadron
collisions. Since no such signal has been found [2], the minimal
implementations of supersymmetry such as Minimal Supersymmetric Standard Model
(MSSM) requires large fine-tuning of the parameters to reproduce the correct
electroweak scale.
One of the mechanisms which can relax the fine-tuning of supersymmetric models
is Twin Higgs (TH), first introduced in non-supersymmetric context [3]. TH
mechanism extends the particle content with partners of all MSSM states and
imposes $Z_{2}$ symmetry between sectors, which we refer to as visible (even
though supersymmetric particles have not been observed yet) and twin. The
Higgs particle in this class of models is then the pseudo-Nambu-Goldstone
boson (pNGB) of an accidental, global symmetry of the potential. Its mass is
generated by explicit breaking of the global symmetry. Since the mass of the
Higgs is proportional to small breaking of the symmetry, it is protected from
large quantum corrections.
Twin Higgs models predict existence of numerous new states, some of which
could be dark matter (DM) candidates, including dark matter in twin
supersymmetric sector[4]. However, in cases of charged DM, one has to either
break or eliminate twin electromagnetism gauge symmetry to escape bounds on
self-interacting dark matter. In the following, I will show that in
Supersymmetric Twin Higgs (SUSY TH) models twin stau is a viable DM candidate
with interesting properties such as long range self-interactions mediated by
the twin photon.
More detailed discussion of the results presented here can be found in [5].

##  2 Supersymmetric Twin Higgs

First, let’s take a look at the scalar potential of a TH model without
referring to supersymmetry for simplicity. As mentioned before, the particle
content of the Standard Model is doubled by adding a second, twin sector. In
particular, the scalar sector is extended by twin Higgs doublet $H^{\prime}$.
Additionally, $Z_{2}$ symmetry interchanging particles between sectors is
imposed. The potential is given by [6]

$V(H,H^{\prime})=\lambda\big{(}H^{2}+H^{\prime
2}\big{)}^{2}-m_{\mathcal{H}}^{2}\big{(}H^{2}+H^{\prime
2})+\Delta\lambda\big{(}H^{4}+H^{\prime 4}\big{)}+\Delta m^{2}H^{2}$ (1)

Note that these doublets form $SU(4)$ fundamental representation
$\mathcal{H}_{\mathrm{SU(4)}}=(H,H^{\prime})$. Having that in mind, we clearly
see that first and second terms are $SU(4)$ and $Z_{2}$ invariant. These terms
form Mexican hat potential, which leads to the spontaneous symmetry breaking.
The third term breaks $SU(4)$ while preserving $Z_{2}$ and is responsible for
generation of the mass of the Higgs particle. The last term breaks $Z_{2}$,
generating misalignment of the vacuum expectation values (vevs) between
sectors parameterized by ratio $v^{\prime}/v$. It is necessary since otherwise
exact $Z_{2}$ symmetry would imply equal decay branching ratio of the Higgs
into visible and twin sectors, which is in contradiction with LHC data.
Current constraint on invisible Higgs decays imply that $v^{\prime}/v\gtrsim
3$, [6]. Misalignment of vevs requires fine-tuning of parameters needed to
reproduce the mass of the visible Higgs, which can be parameterized by
$\Delta_{v^{\prime}/v}=(v^{\prime 2}/v^{2}-2)/2$ which leads to
$\mathcal{O}(30)$% fine-tuning for $v^{\prime}/v=3$ which will be our
benchmark for discussion. Fine-tuning at that level means that model is
essentially fully natural. It should be mentioned that ratios
$v^{\prime}/v\geq 7$ are disfavored since they require fine-tuning worse than
5%.
As mentioned before, the minimum of the potential breaks symmetry $SU(4)$ to
$SU(3)$ generating 7 pNGBs. Six of them give masses to $SU_{L}(2)$ and
$SU_{L}^{\prime}(2)$ gauge bosons, while the remaining one is identified with
the SM Higgs.
In supersymmetric models, the scalar potential is fully fixed by the gauge
interactions (D-terms) and the particle content (F-terms). Hence, the scalar
potential of the form (1) cannot be simply added to the lagrangian and should
come from either D-term [8, 7, 9] or F-term [10, 11]. In both cases, the tree-
level mass of the Higgs is proportional to $\cos^{2}(2\beta)$, which
approaches 1 in the limit of large $\tan\beta=v_{u}/v_{d}$. The key difference
between those cases is captured by $\lambda$ dependence on $\tan\beta$. In
general, the fine-tuning in SUSY TH is inversely proportional to the quartic
coupling $\lambda$ of the $SU(4)$ invariant term of the potential.
In F-term SUSY TH, quartic term is proportional to $\sin^{2}(2\beta)$, which
is maximized for $\tan\beta=1$ for which tree-level mass of the Higgs
vanishes. It is then necessary to use moderate values of $\tan\beta$, which
inevitably lead to fine-tuning which requires further model building to
alleviate.
In contrast, in D-term SUSY TH, the quartic term is proportional to
$\cos^{2}(2\beta)$ and is maximized for large values of $\tan\beta$. As a
result, fine-tuning in these models can be as low as 20% with heavy stops
masses of 2 TeV. We will focus on this case since it seems far more
preferable, however most of the analysis presented here is not UV dependent.

##  3 Twin stau

Twin stau is $Z_{2}$ partner of the supersymmetric scalar partner of tau
lepton. Hence, it is charged under twin electromagnetism and twin weak
interactions $U_{Y}^{\prime}(1)\times SU_{L}^{\prime}(2)$ and its mass gets
large contributions from the supersymmetry breaking. We will consider $Z_{2}$
symmetric SUSY breaking, thus soft masses twin stau obtains are equal in
visible and twin sectors. To be more specific, the mass matrix of twin stau is
given by [13]

$m_{\tilde{\tau^{\prime}}}^{2}=\left(\begin{matrix}m_{L_{3}}^{2}+\Delta_{\tilde{\tau}_{L}}+m_{\tau^{\prime}}^{2}&-\mu
v^{\prime}y_{\tau}\sin(\beta)\\\ -\mu
v^{\prime}y_{\tau}\sin(\beta))&m_{\bar{{e}}_{3}}^{2}+\Delta_{\tilde{\tau}_{R}}+m_{\tau^{\prime}}^{2}\end{matrix}\right)$
(2)

where $m_{L_{3}}$ and $m_{R_{3}}$ are soft SUSY breaking masses, $\mu$ is
Higgs mass term, $v^{\prime}$ is vacuum expectation value of twin Higgs and
$m_{\tau^{\prime}}$ is mass of the twin tau. Note, that it is assumed that
there is no $Z_{2}$ breaking in Yukawa sector $y_{\tau^{\prime}}=y_{\tau}$ and
$\tan\beta^{\prime}=\tan\beta$. The D-term contributions to the mass of the
twin stau are given by
$\Delta_{\tilde{\tau}_{L}^{\prime}}=(-1/2+\sin^{2}\theta_{W})\cos 2\beta
m_{Z}^{\prime 2}$ and
$\Delta_{\tilde{\tau}_{R}^{\prime}}=-\sin^{2}\theta_{W}\cos 2\beta
m_{Z}^{\prime 2}$, where $\theta_{W}$ is Weinberg angle. Mass matrix of stau
is obtained by removing all primes.
Note that for $v^{\prime}/v>1$, the off-diagonal terms are larger in twin
sector than in visible one. It leads to twin stau lighter than stau in the
parameter space with large mixing. By making a common assumption that lightest
supersymmetric particle is stable, the twin stau can be candidate for a dark
matter.
Note that while twin stau DM prefers large off-diagonal mass matrix, $\mu$
cannot be arbitrarily large if the model is to remain natural. Assumption of
equal Yukawa couplings in both sectors is not necessary and might even be
preferable (see [14] for $\Delta N_{\mathrm{eff}}$ problem in TH models and
how it could be solved by breaking $Z_{2}$ in Yukawas[14, 15]), however I will
not cover this case.
As mentioned in the introduction, twin stau is charged under twin
electromagnetism, which implies that there are long-range self-interactions
mediated by massless twin photons. As discussed in [16] the bound on mass of
self-interacting DM have been overestimated and currently the strongest
constraint comes from measurements of non-zero ellipticity of gravitational
potential of NGC720. For equal couplings of electromagnetism and twin
electromagnetism, the lower bound on mass of twin stau is approximately 210
GeV. That is the reason why breaking of twin electromagnetism is necessary in
case of charged, non-supersymmetric candidates for dark matter, such as twin
taus or mesons.

###  3.1 Direct detection

Since twin stau belongs to the twin sector, its only interactions with the
visible sector must be mediated by the Higgs portal, Fig. 1. Thus, the
interactions of twin stau with atomic nuclei are suppressed by mixing between
Higgs and twin Higgs, which is roughly given by $v/v^{\prime}$. In particular,
the effective interaction between the visible SM Higgs and twin stau is given
by

$\lambda_{h\tilde{\tau}^{\prime}\tilde{\tau}^{\prime}}=\frac{g}{m_{W^{\prime}}}\left[\left(\frac{1}{2}c_{\theta_{\tilde{\tau}^{\prime}}}^{2}-s_{W}^{2}c_{2\theta_{\tilde{\tau}^{\prime}}}\right)m_{Z^{\prime}}^{2}c_{2\beta}-m_{\tau^{\prime}}^{2}+\frac{m_{\tau^{\prime}}}{2}\mu\tan\beta
s_{2\theta_{\tilde{\tau}^{\prime}}}\right]\frac{v}{v^{\prime}}$ (3)

where $c_{\alpha}$ and $s_{\alpha}$ are shortcuts for $\cos\alpha$ and
$\sin\alpha$, respectively, and $m_{W^{\prime}}$ is twin W boson mass. Note
that this coupling is maximized for large mixing angle
$\theta_{\tilde{\tau}^{\prime}}$, so we expect DD bounds to be strongest in
that region.
In our results, we will assume that in the region with $\Omega h^{2}>0.12$,
there exists a mechanism which dilutes DM to observed relic abundance. For
region with $\Omega h^{2}<0.12$, we rescale the DM cross section with factor
$\Omega h^{2}/0.12$. It is justified by the fact that it is possible that twin
stau constitutes a fraction of DM, but overproduced DM is not physical. A more
detailed discussion of the issue can be found in [5].

Figure 1: Diagram of twin stau interactions with visible sector quarks. The
interaction is mediated by the Higgs portal

##  4 Results

For the calculation of the relic density, we have modified Micromegas [17, 18,
19]. Micromegas takes into account twin stau coannihilations within a twin
sector, but since in a large portion of the parameter space stau is almost
degenerate with twin stau it is necessary to adjust the relic abundance. Upon
justified assumption that the annihilation cross section for stau and twin
stau is the same, the effective cross section for twin stau is given by

$\sigma_{\mathrm{eff}}=\sigma\frac{1+(1+\Delta)^{3}e^{-2x_{f}\Delta}}{[1+(1+\Delta)^{3/2}e^{-x_{f}\Delta}]^{2}},$
(4)

where $x_{f}=m_{{\tilde{\tau}}^{\prime}}/T_{f}\sim 25$, with $T_{f}$ the
freeze-out temperature. The relic density can then be approximated using
$\Omega_{\mathrm{coann}}h^{2}=\Omega_{0}h^{2}\sigma/\sigma_{\mathrm{eff}}$.

Figure 2: Contour of relic abundance $\Omega h^{2}=0.12$ (blue line) in plane
of soft twin stau masses $m_{R3}$ and $m_{L3}$. In purple region twin stau is
tachyonic, in red twin stau is not LSP. Mass contours of twin stau are black.
Direct detection bounds from Xenon1T and LZ are coloured orange and green,
respectively. Dashed green contour corresponds to new, first results from LZ
[21]

First, we will consider the decoupling case where all SUSY breaking masses
except the stau are set to 10 TeV, left panel of Fig. 2. We consider $\mu=1.5$
TeV which is unnatural but is a good starting point for the analysis. As
mentioned before, large $\tan\beta$ is preferred both due to naturalness and
large twin stau mixing necessary for twin stau LSP. Some of the parameter
space is excluded due to mass spectrum. For mostly left-handed twin stau twin
sneutrino is always LSP while for mostly right-handed
${\tilde{\tau}}^{\prime}$ stau is the LSP and is excluded. For very small soft
masses, twin stau is tachyonic due to off-diagonal mass terms proportional to
the large $\mu$. In this case, the correct relic abundance is obtained for
$m_{{\tilde{\tau}}^{\prime}}$ between 260 and 400 GeV, depending on the
mixing. Some of the parameter space is excluded due to the direct detection
bounds coming from Xenon1T [20] and primary results from Lux-Zepelin (LZ)
[21]. Predicted sensitivity of LZ will allow for probing whole parameters
space shown on this plot. The Big Bang Nucleosynthesis (BBN) bound comes from
the fact that in this region of parameter space, the difference between masses
of stau and twin stau becomes too small to allow
${\tilde{\tau}}\rightarrow{\tilde{\tau}}^{\prime\dagger}\tau\tau^{\prime}$
decay. As a result, stau becomes long-lived, and its late decay would change
the nuclei composition of the universe. However, in the decoupled case, the
lifetime of stau is generically too long. Charged long-lived particles (with
decay length above $\mathcal{O}(1)$m) could have been seen at LHC as charged
disappearing tracks if their mass is lower than approximately 430 GeV [22].
Since stau decay is mediated by twin bino and higgsino big, $\mu$ and very
large $M_{1}$ lead to large decay length across whole plot.

A more realistic scenario with $M_{1}=\mu=700$ GeV is shown on right panel of
Fig. 2. A large portion of that parameter space has decay length of stau below
1 m. Note that in this plot only small part is not excluded by either new
results from LZ nor by decay length of stau. However, the reason for that is
that we keep the most natural, non-excluded value of vevs ratio
$v^{\prime}/v=3$. One can trade off naturalness for the opening up of the
parameter space. I decided to keep the ratio $v^{\prime}/v=3$ as originally
presented at 7th Young Researchers’ Workshop, as well as in [5]. However, one
must keep in mind that new LZ results indicate that the scenario with minimal
tuning is now strongly constrained. The final results from Lux-Zepelin will
probe the whole parameters space.

## Acknowledgements

This work was partially supported by the National Science Centre, Poland,
under research grant no. 2020/38/E/ST2/00243.

## References

  * [1] R. K. Kaul, Phys. Lett. B 109 (1982), 19-24 doi:10.1016/0370-2693(82)90453-1
  * [2] G. Aad et al. [ATLAS], Eur. Phys. J. C 80 (2020) no.8, 737 doi:10.1140/epjc/s10052-020-8102-8
  * [3] Z. Chacko, H. S. Goh and R. Harnik, Phys. Rev. Lett. 96 (2006), 231802 doi:10.1103/PhysRevLett.96.231802
  * [4] M. Badziak, G. Grilli Di Cortona and K. Harigaya, Phys. Rev. Lett. 124 (2020) no.12, 121803 doi:10.1103/PhysRevLett.124.121803
  * [5] M. Badziak, G. Grilli di Cortona, K. Harigaya and M. Łukawski, JHEP 10 (2022), 057 doi:10.1007/JHEP10(2022)057
  * [6] N. Craig, A. Katz, M. Strassler and R. Sundrum, JHEP 07 (2015), 105 doi:10.1007/JHEP07(2015)105
  * [7] M. Badziak and K. Harigaya, JHEP 06 (2017), 065 doi:10.1007/JHEP06(2017)065
  * [8] M. Badziak and K. Harigaya, JHEP 10 (2017), 109 doi:10.1007/JHEP10(2017)109
  * [9] M. Badziak and K. Harigaya, Phys. Rev. Lett. 120 (2018) no.21, 211803 doi:10.1103/PhysRevLett.120.211803
  * [10] A. Falkowski, S. Pokorski and M. Schmaltz Phys. Rev. D 74 (2006)
  * [11] S. Chang, L. J. Hall and N. Weiner, Phys. Rev. D 75 (2007), 035009 doi:10.1103/PhysRevD.75.035009
  * [12] R. Barbieri, L. J. Hall and K. Harigaya, JHEP 11 (2016), 172 doi:10.1007/JHEP11(2016)172
  * [13] S. P. Martin, Adv. Ser. Direct. High Energy Phys. 18 (1998), 1-98 doi:10.1142/9789812839657_0001
  * [14] Z. Chacko, N. Craig, P. J. Fox and R. Harnik, JHEP 07 (2017), 023 doi:10.1007/JHEP07(2017)023
  * [15] R. Barbieri, L. J. Hall and K. Harigaya, JHEP 10 (2017), 015 doi:10.1007/JHEP10(2017)015
  * [16] P. Agrawal, F. Y. Cyr-Racine, L. Randall and J. Scholtz, JCAP 05 (2017), 022 doi:10.1088/1475-7516/2017/05/022
  * [17] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 149 (2002), 103-120 doi:10.1016/S0010-4655(02)00596-9
  * [18] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 174 (2006), 577-604 doi:10.1016/j.cpc.2005.12.005
  * [19] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 176 (2007), 367-382 doi:10.1016/j.cpc.2006.11.008
  * [20] E. Aprile et al. [XENON], Phys. Rev. Lett. 121 (2018) no.11, 111302
  * [21] J. Aalbers et al. [LZ], [arXiv:2207.03764 [hep-ex]].
  * [22] M. Aaboud et al. [ATLAS], Phys. Lett. B 788 (2019), 96-116


# Linearization stability of reflection-asymmetric thin-shell wormholes with
double shadows

Naoki Tsukamoto1<EMAIL_ADDRESS>1Department of General Science and
Education, National Institute of Technology, Hachinohe College, Aomori
039-1192, Japan

###### Abstract

Wormholes are hypothetical objects which can be black hole mimickers with
strong gravitational fields. Recently, Wielgus et al. have constructed
reflection-asymmetric thin-shell wormholes which are composed of the parts of
a Schwarzschild spacetime and a Reissner-Nordstrom spacetime with two photon
spheres with different sizes each other in [M. Wielgus, J. Horak, F. Vincent,
and M. Abramowicz, Phys. Rev. D 102, 084044 (2020)]. They have discussed
observational property of the shadows with two photon rings in different sizes
as seen from an observer and they have named their shadows double shadows. In
this paper, we study the linearization stability of the reflection-asymmetric
thin-shell wormholes with the double shadows.

##  I Introduction

Recently, LIGO and VIRGO Collaborations have detected gravitational waves from
black hole binaries Abbott:2016blz  and Event Horizon Telescope Collaboration
has reported the shadow image of a black hole candidate at the center of a
giant elliptical galaxy M87 Akiyama:2019cqa . The theoretical and
observational aspects of compact objects in general relativity will be more
important than before.

Wormholes are hypothetical objects with a non-trivial topology in general
relativity Visser_1995 ; Morris:1988cz . Morris and Thorne have discussed the
passability of wormholes and they have also shown that energy conditions
violate at least at the throat of static and spherically symmetric wormholes
if we assume general relativity without a cosmological constant Morris:1988cz
. The wormholes can be black hole mimickers because they can have strong
gravitational fields. For example, spherically symmetric wormholes with strong
gravitational fields can have unstable (stable) circular light orbits named
photon spheres (antiphoton spheres) Perlick:2003vg ; Nandi:2006ds ;
Tsukamoto:2012xs ; Tsukamoto:2016qro ; Nandi:2018mzm ; Shaikh:2018oul ;
Shaikh:2019jfr ; Tsukamoto:2020uay ; Tsukamoto:2020bjm .

A thin-shell wormhole whose energy conditions are broken only at a throat
which is supported by a thin shell Lanczos:1922 ; Lanczos:1924 ; Israel:1966rt
; Poisson:2004  was considered by Visser Visser:1989kg  with Darmois-Israel
matching Darmois:1927 ; Israel:1966rt ; Poisson:2004 . The linearization
stability of the thin shell of a Schwarzschild wormhole was investigated by
Poisson and Visser Poisson:1995sv  and then the stability of thin-shell
wormholes such as Reissner-Nordstrom wormholes Eiroa:2003wp ; Eiroa:2007qz ;
Eiroa:2008ky ; Eiroa:2009hm ; Kuhfittig:2010pb ; Sharif:2013lna ;
Sharif:2013tva ; Eid:2016axb ; HabibMazharimousavi:2017qfb ; Wang:2017gdp ;
Forghani:2019wgt , the other static and spherically symmetric wormholes
Kim:1992sh ; Barcelo:2000ta ; Ishak:2001az ; Lobo:2003xd ; Lobo:2005zu ;
Eiroa:2005pc ; Lobo:2005yv ; Rahaman:2006xb ; Rahaman:2007bf ; Eiroa:2007qz ;
Eiroa:2008ky ; Eiroa:2009hm ; Garcia:2011aa ; Eiroa:2015hrt , plane symmetric
wormholes Lemos:2008aj , cylindrical symmetric wormholes Eiroa:2004at ;
Bejarano:2006uj , higher-dimensional wormholes Thibeault:2005ha ;
Rahaman:2006vg ; Mazharimousavi:2010bm ; Kokubu:2014vwa ; Mehdizadeh:2015dta ;
Kokubu:2015spa , lower-dimensional wormholes Perry:1991qq ; Kim:1993bz ;
Rahaman:2011yh ; Eiroa:2013xj ; Tsukamoto:2018lsg , and wormholes in an
expanding spacetime LaCamera:2011qi  have been discussed. The passability
Nakao:2013hba  and nonlinear stability Akai:2017aro  of thin-shell wormholes
have been studied.

Usually two copied manifolds are used to construct the thin-shell wormholes.
It would be difficult to distinguish a reflection-symmetric wormhole spacetime
constructed by the parts of the copied black hole spacetimes from the original
black hole spacetime by astronomical observations if the wormhole has photon
spheres and if there are few light sources in the other side of the throat.
111In principle, we would distinguish wormholes from black holes. For an
example, if there are stars bounded by a wormhole in both sides of the
wormhole, the orbit of the star in one side can be affected by the gravity of
the star of the other side. Simonetti:2020vhw ; Dai:2019mse

There were few researches on reflection-asymmetric thin-shell wormholes such
as Garcia:2011aa ; Eid:2017xcg ; Forghani:2018gza . Recently, the shadow of a
reflection-asymmetric thin-shell wormhole which is composed of the parts of
two Schwarzschild spacetime with different masses were discussed by Wang et
al. Wang:2020emr . Wielgus et al. have constructed a reflection-asymmetric
thin-shell wormhole which is composed of the parts of the Schwarzschild
spacetime with a photon sphere and a Reissner-Nordstrom spacetime with another
photon sphere and they have discussed the observational property of the shadow
Wielgus:2020uqz . They have concluded that the shadow of the asymmetric
wormhole has two photon rings in different sizes as seen from an observer and
they have named the shadow double shadow. The asymmetric thin-shell wormhole
with the photon spheres can be distinguished from the black holes by the
observations since light rays can be reflected by a potential wall near the
throat Wielgus:2020uqz .

In this paper, we investigate the linearization stability of the reflection-
asymmetric thin-shell wormhole which is composed of the parts of the
Schwarzschild spacetime and the Reissner-Nordstrom spacetime with the double
shadow constructed in Wielgus:2020uqz .

This paper is organized as follows. We review the Reissner-Nordstrom spacetime
very briefly in Sec. II and we construct the reflection-asymmetric thin-shell
wormhole with the double shadow in Sec. III. We investigate the linearization
stability of the reflection-asymmetric thin-shell wormhole in Sec. IV and we
conclude our results in Sec. V. In this paper we use the units in which a
light speed and Newton’s constant are unity.

##  II Reissner-Nordstrom spacetime

A Reissner-Nordstrom spacetime has a line element

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)

where $f(r)$ is given by

$\displaystyle f(r)\equiv 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$ (2)

and where $M>0$ and $Q$ are a mass and a charge, respectively. It is a black
hole spacetime with an event horizon at $r=r_{\mathrm{EH}}\equiv
M+\sqrt{M^{2}-Q^{2}}$ for $Q^{2}\leq M^{2}$ while it has naked singularity for
$M^{2}<Q^{2}$. There is a photon sphere at

$\displaystyle r=r_{\mathrm{PS}}\equiv\frac{3M+\sqrt{9M^{2}-8Q^{2}}}{2}$ (3)

for $Q^{2}\leq 9M^{2}/8$ and there is an antiphoton sphere at

$\displaystyle r=r_{\mathrm{APS}}\equiv\frac{3M-\sqrt{9M^{2}-8Q^{2}}}{2}$ (4)

for $M^{2}<Q^{2}\leq 9M^{2}/8$.

##  III Reflection-Asymmetric thin-shell wormhole

In this section, we construct a wormhole spacetime without reflection symmetry
or Z2 symmetry by the Darmois-Israel matching Darmois:1927 ; Israel:1966rt ;
Poisson:2004 . We make two manifolds
$\mathcal{M}_{\pm}\equiv\left\\{r>a\right\\}$, where $a$ is a constant
satisfying $a>r_{\mathrm{EH}\pm}$ by removing $\Omega_{\pm}\equiv\left\\{r\leq
a\right\\}$ from the Reissner-Nordstrom spacetimes. The boundaries of the
manifolds $\mathcal{M}_{\pm}$ are timelike hypersurfaces
$\Sigma_{\pm}\equiv\left\\{r=a\right\\}$ and we identify the hypersurfaces
$\Sigma\equiv\Sigma_{+}=\Sigma_{-}$. As a result, we obtain a manifold
$\mathcal{M}$ by gluing the manifolds $\mathcal{M}_{\pm}$ at a throat located
at $\Sigma$. The hypersurface $\Sigma$ is filled with a Dirac distribution
matter and it is called thin shell. We permit $a=a(\tau)$, where $\tau$ is the
proper time of the thin shell since we are interested in the stability of the
thin shell $\Sigma$.

The line elements in the domains $\mathcal{M}_{\pm}$ are given by

$ds_{\pm}^{2}=-f_{\pm}(r)dt^{2}_{\pm}+\frac{dr^{2}}{f_{\pm}(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (5)

where $f_{\pm}(r)$ are given by

$\displaystyle f_{\pm}(r)\equiv
1-\frac{2M_{\pm}}{r}+\frac{Q_{\pm}^{2}}{r^{2}}.$ (6)

Note the time coordinates $t_{\pm}$ are discontinuous on the hypersurface
$\Sigma$ while coordinates $r$, $\theta$, and $\phi$ are continuous across the
hypersurface $\Sigma$.

We assume that we can set coordinates $y^{i}=(\tau,\theta,\phi)$ on the both
sides of $\Sigma$. Let the thin shell be at $t_{\pm}=T_{\pm}(\tau)$ and
$r=a(\tau)$. We set the unit normal vectors of the thin shells as

$\displaystyle
n_{\mu\pm}dx^{\mu}_{\pm}=\pm\left(-\dot{a}dt_{\pm}+\dot{T}_{\pm}dr\right),$
(7)

where the overdot is a differentiation with respect to $\tau$. The four
velocity of the thin shell is given by
$u^{\mu}_{\pm}\partial_{\mu\pm}=\dot{T}_{\pm}\partial_{t\pm}+\dot{a}\partial_{r}$.
From the normalization of the four velocity $u^{\mu}_{\pm}u_{\mu\pm}=-1$, we
obtain

$f_{\pm}(a)\dot{T}^{2}_{\pm}-\frac{\dot{a}^{2}}{f_{\pm}(a)}=1,$ (8)

where $\dot{T}_{\pm}$ should be

$\displaystyle\dot{T}_{\pm}=\frac{\sqrt{f_{\pm}+\dot{a}^{2}}}{f_{\pm}}.$ (9)

By using Eq. (8) and the basis vectors $e^{\mu}_{i\pm}\equiv\partial
x^{\mu}_{\pm}/\partial y^{i}$ given by

$\displaystyle
e^{\mu}_{\tau\pm}\partial_{\mu\pm}=\dot{T}_{\pm}\partial_{t\pm}+\dot{a}\partial_{r},$
(10) $\displaystyle e^{\mu}_{\theta\pm}\partial_{\mu\pm}=\partial_{\theta},$
(11) $\displaystyle e^{\mu}_{\phi\pm}\partial_{\mu\pm}=\partial_{\phi},$ (12)

the induced metric $h_{ij\pm}\equiv g_{\mu\nu\pm}e^{\mu}_{i\pm}e^{\nu}_{j\pm}$
on the hypersurface $\Sigma$ in $\mathcal{M}_{\pm}$ is given by

$\displaystyle ds_{\Sigma}^{2}$ $\displaystyle=$
$\displaystyle\left.ds_{\pm}^{2}\right|_{\Sigma}$ (13) $\displaystyle=$
$\displaystyle h_{ij\pm}dy^{i}dy^{j}$ $\displaystyle=$
$\displaystyle-d\tau^{2}+a^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$

It guarantees that the metric on the hypersurface $\Sigma$ is the same as
viewed from both sides.

The thin shell satisfies Einstein equations

$S^{i}_{j}=-\frac{1}{8\pi}\left(\left[K^{i}_{j}\right]-\left[K\right]\delta^{i}_{j}\right),$
(14)

where the bracket $\left[\textrm{\boldmath$F$}\right]$ denotes the jump of any
function $F$ across $\Sigma$,

$\left[\textrm{\boldmath$F$}\right]\equiv\left.\textrm{\boldmath$F$}_{+}\right|_{\Sigma}-\left.\textrm{\boldmath$F$}_{-}\right|_{\Sigma},$
(15)

where $\textrm{\boldmath$F$}_{+}$ and $\textrm{\boldmath$F$}_{-}$ are $F$ in
$\mathcal{M}_{+}$ and $\mathcal{M}_{-}$, respectively, and $S^{i}_{j}$ is the
surface stress-energy tensor of the thin shell given by

$S^{i}_{j}=(\sigma+p)U^{i}U_{j}+p\delta^{i}_{j},$ (16)

where $U_{i}$ is given by $U_{i}dy^{i}\equiv
u_{\mu\pm}e^{\mu}_{i\pm}dy^{i}=-d\tau$, and where $\sigma=-S^{\tau}_{\tau}$
and $p=S^{\theta}_{\theta}=S^{\phi}_{\phi}$ are the surface energy density and
the surface pressure of the thin shell, respectively. Here, $K_{ij}$ is the
extrinsic curvature given by

$K_{ij}\equiv n_{\mu;\nu}e^{\mu}_{i}e^{\nu}_{j},$ (17)

where ; is the covariant derivative. By using the normal vectors (7), the
extrinsic curvatures of the hypersurfaces in $\mathcal{M}_{\pm}$ are given by

$\displaystyle K^{\tau}_{\tau\pm}$ $\displaystyle=$ $\displaystyle\frac{\pm
1}{\sqrt{\dot{a}^{2}+f_{\pm}}}\left(\ddot{a}+\frac{f^{\prime}_{\pm}}{2}\right),$
(18) $\displaystyle K^{\theta}_{\theta\pm}$ $\displaystyle=$ $\displaystyle
K^{\phi}_{\phi\pm}=\frac{\pm\sqrt{\dot{a}^{2}+f_{\pm}}}{a},$ (19)

and the traces are obtained as

$K_{\pm}=\frac{\pm
1}{\sqrt{\dot{a}^{2}+f_{\pm}}}\left(\ddot{a}+\frac{f^{\prime}_{\pm}}{2}\right)\pm\frac{2}{a}\sqrt{\dot{a}^{2}+f_{\pm}}.$
(20)

From $(\tau,\tau)$ and $(\theta,\theta)$ components of the Einstein equations
(14), we obtain

$\sigma=-\frac{\sqrt{\dot{a}^{2}+f_{+}}}{4\pi
a}-\frac{\sqrt{\dot{a}^{2}+f_{-}}}{4\pi a}$ (21)

and

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\sqrt{\dot{a}^{2}+f_{+}}}\left(\ddot{a}+\frac{\dot{a}^{2}+f_{+}}{a}+\frac{f^{\prime}_{+}}{2}\right)$
(22)
$\displaystyle+\frac{1}{8\pi\sqrt{\dot{a}^{2}+f_{-}}}\left(\ddot{a}+\frac{\dot{a}^{2}+f_{-}}{a}+\frac{f^{\prime}_{-}}{2}\right)$

and then we get, from Eqs. (21) and (22),

$\frac{d(\sigma\mathcal{A})}{d\tau}+p\frac{d\mathcal{A}}{d\tau}=0,$ (23)

where $\mathcal{A}\equiv 4\pi a^{2}$ is the area of the throat. Equation (23)
can be expressed by

$a\sigma^{\prime}+2(\sigma+p)=0,$ (24)

where the prime denotes the differentiation with respect to $a$ and
$\sigma^{\prime}=\dot{\sigma}/\dot{a}$. We assume that the thin shell is
filled with a barotropic fluid with $p=p(\sigma)$. From Eq. (24), we notice
that the surface density of the barotropic fluid is expressed as a function of
$a$ or $\sigma=\sigma(a)$. The equation of motion of the thin shell is given
by, from Eq. (21),

$\dot{a}^{2}+V(a)=0,$ (25)

where $V(a)$ is an effective potential defined by

$V(a)\equiv\bar{f}-\left(\frac{\Delta}{4\pi a\sigma}\right)^{2}-\left(2\pi
a\sigma\right)^{2},$ (26)

where $\bar{f}$ and $\Delta$ are given by

$\displaystyle\bar{f}\equiv\frac{f_{-}+f_{+}}{2}$ (27)

and

$\displaystyle\Delta\equiv\frac{f_{+}-f_{-}}{2},$ (28)

respectively. The derivative of $V$ with respect to $a$ is obtained as

$V^{\prime}=\bar{f}^{\prime}-\frac{\Delta\left[\Delta^{\prime}a\sigma-\Delta(\sigma+a\sigma^{\prime})\right]}{8\pi^{2}a^{3}\sigma^{3}}-8\pi^{2}a\sigma(\sigma+a\sigma^{\prime})$
(29)

and, from Eq. (24), it can be rewritten as

$V^{\prime}=\bar{f}^{\prime}-\frac{\Delta\left[\Delta^{\prime}a\sigma+\Delta(\sigma+2p)\right]}{8\pi^{2}a^{3}\sigma^{3}}+8\pi^{2}a\sigma(\sigma+2p).$
(30)

By using Eq. (24) again, the second derivative of $V$ is obtained as

$\displaystyle V^{\prime\prime}=$
$\displaystyle\bar{f}^{\prime\prime}-\frac{\Delta^{\prime
2}}{8\pi^{2}a^{2}\sigma^{2}}-\frac{\Delta}{8\pi^{2}a^{4}\sigma^{4}}\left[4\Delta^{\prime}(\sigma+2p)a\sigma\right.$
(31)
$\displaystyle\left.+\Delta^{\prime\prime}a^{2}\sigma^{2}-2\Delta\sigma(\sigma+p)(1+2\beta^{2})+3\Delta(\sigma+2p)^{2}\right]$
$\displaystyle-8\pi^{2}\left[(\sigma+2p)^{2}+2\sigma(\sigma+p)(1+2\beta^{2})\right],$

where $\beta^{2}\equiv dp/d\sigma=p^{\prime}/\sigma^{\prime}$.

Here and hereafter, we impose a constraint

$\displaystyle f_{+}(a)=f_{-}(a)$ (32)

and we concentrate on the case that the manifold $\mathcal{M}_{-}$ is the part
of the Schwarzschild black hole spacetime, i.e., $Q_{-}=0$, as well as Ref.
Wielgus:2020uqz . The constraint is expressed as

$\displaystyle Q_{+}^{2}=2aM_{-}(\xi-1),$ (33)

where $\xi$ is an asymmetry parameter defined by $\xi\equiv M_{+}/M_{-}$. The
reflection-asymmetric thin-shell wormhole with a double shadow must have the
throat in domains $r_{\mathrm{EH}-}<a<r_{\mathrm{PS}-}$ and
$r_{\mathrm{EH}+}<a<r_{\mathrm{PS}+}$. Permitted parameters $(\xi,a/M_{-})$
for the reflectional-asymmetry thin-shell wormhole with the double shadow are
shown in Fig. 1.

Figure 1: Permitted parameters $(\xi,a/M_{-})$ of the asymmetric wormhole with
the double shadow are in shaded zones. $\mathcal{M}_{+}$ is the part of the
Reissner-Nordstrom black hole (BH) spacetime in a deep blue shaded zone while
it is the part of the Reissner-Nordstrom naked singularity (NS) spacetime in
light blue shaded zones. $\mathcal{M}_{-}$ is the subset of the Schwarzschild
black hole spacetime. A blue dotted line denotes $a=r_{\mathrm{PS}+}$. A red
dashed line is explained in Sec. IV.

##  IV Stability of thin-shell wormhole

We consider the linearization stability of a static wormhole with a thin shell
at $a=a_{0}$ under the constraint (32) and $Q_{-}=0$. The surface energy
density $\sigma_{0}$ and pressure $p_{0}$ of the thin shell are given by

$\sigma_{0}=-\frac{\sqrt{f_{0}}}{2\pi a_{0}}$ (34)

and

$\displaystyle
p_{0}=\frac{1}{8\pi\sqrt{f_{0}}}\left(\frac{2f_{0}}{a_{0}}+\bar{f}^{\prime}_{0}\right),$
(35)

respectively. Here and hereafter a function with subscript $0$ means the
function at $a=a_{0}$. Since $V_{0}=V^{\prime}_{0}=0$ is satisfied, the
effective potential can be expanded around $a=a_{0}$ as

$V(a)=\frac{V_{0}^{\prime\prime}}{2}(a-a_{0})^{2}+O\left(\left(a-a_{0}\right)^{3}\right),$
(36)

where $V_{0}^{\prime\prime}$ is given by

$V_{0}^{\prime\prime}=A_{0}-B_{0}\left(1+2\beta^{2}_{0}\right),$ (37)

where $A_{0}$ and $B_{0}$ are defined by

$A_{0}\equiv\bar{f}^{\prime\prime}_{0}-\frac{\Delta^{\prime
2}_{0}}{2f_{0}}-\frac{\bar{f}^{\prime 2}_{0}}{2f_{0}}$ (38)

and

$B_{0}\equiv\frac{2f_{0}}{a_{0}^{2}}-\frac{\bar{f}^{\prime}_{0}}{a_{0}},$ (39)

respectively. The thin shell is stable (unstable) when
$V_{0}^{\prime\prime}>0$ $(V_{0}^{\prime\prime}<0)$. Thus, the thin shell is
stable when

$\beta^{2}_{0}<\frac{1}{2}\left(\frac{A_{0}}{B_{0}}-1\right)$ (40)

and $B_{0}>0$ holds or

$\beta^{2}_{0}>\frac{1}{2}\left(\frac{A_{0}}{B_{0}}-1\right)$ (41)

and $B_{0}<0$ holds. Figure 2 shows the parameter zone of $(\xi,a_{0}/M_{-})$
for $B_{0}>0$, the one for $B_{0}<0$, and their boundary $B_{0}=0$.

Figure 2: $B_{0}<0$ holds in a deep blue shaded zone and $B_{0}>0$ holds in a
light blue shaded zone. The red dashed line $a_{0}/M_{-}=(7-\xi)/2$ for
$(32+6\sqrt{2})/17\leq\xi\leq 2+\sqrt{2}/2$ denotes the boundary $B_{0}=0$.

The boundary $B_{0}=0$ is given by

$a_{0}/M_{-}=(7-\xi)/2\quad\mathrm{for}\quad\frac{32+6\sqrt{2}}{17}\leq\xi\leq
2+\frac{\sqrt{2}}{2}.$ (42)

On the boundary, the thin shell is unstable for any $\beta_{0}^{2}$. The
parameters $(a_{0}/M_{-},\beta^{2}_{0})$ for the stable thin shell are shown
in Fig. 3.

Figure 3: The static thin shell is stable if it is in blue shaded zones of
$(a_{0}/M_{-},\beta^{2}_{0})$. Left top, right top, left middle, right middle,
left bottom, and right bottom panels show the cases of $\xi=1.0$ for
$2<a_{0}/M_{-}<3$, $\xi=1.2$ for $2<a_{0}/M_{-}<2.8$, $\xi=1.6$ for
$2<a_{0}/M_{-}<2.4$, $\xi=1.8$ for $2<a_{0}/M_{-}<729/320$, $\xi=2.5$ for
$25/12<a_{0}/M_{-}<75/32$, and $\xi=3.5$ for $49/20<a_{0}/M_{-}<441/160$,
respectively.

##  V Conclusion and Discussion

We have investigated the linearization stability of the reflection-asymmetric
thin-shell wormhole which is composed of the parts of the Schwarzschild and
Reissner-Nordstrom manifolds with the double shadow. We have imposed the
constraint (32) as well as Ref. Wielgus:2020uqz . The linearization stability
given by Fig. 3 is characterized by the boundary $B_{0}=0$ or
$a_{0}/M_{-}=(7-\xi)/2$ for $(32+6\sqrt{2})/17\leq\xi\leq 2+\sqrt{2}/2$ shown
as the red line in Fig. 2. In the boundary case $B_{0}=0$, the throat with
$a_{0}/M_{-}=2.25$ is unstable for any $\beta^{2}_{0}$ as shown the left
bottom panel with $\xi=2.5$ in Fig. 3. We also note the throat at
$a_{0}/M_{-}=r_{\mathrm{EH}-}/M_{-}=2$ for $1\leq\xi\leq 2$ is unstable for
any $\beta^{2}_{0}$ as shown the panels in Fig. 3.

We comment on the Schwarzschild thin-shell wormhole with reflection symmetry
or $\xi=1.0$. In Fig. 2, $B_{0}$ vanishes at a point
$(\xi,a_{0}/M_{-})=(1.0,3.0)$. Thus, the throat of the Schwarzschild
reflection-symmetric wormhole with $\xi=1.0$ at $a_{0}/M_{-}=3$ is unstable
for any $\beta_{0}^{2}$ as shown Fig. 3.

We also give comments on wormholes without a thin shell while we have
concentrated on the thin-shell wormhole on this paper. The earliest
traversable wormhole filled with a phantom scalar field Martinez:2020hjm  was
investigated by Ellis Ellis:1973yv  and Bronnikov Bronnikov:1973fh . The
Ellis-Bronnikov wormhole is a reflection-asymmetric wormhole when it has a
positive Arnowitt-Deser-Misner (ADM) mass in a side and a negative ADM mass in
another side and it is a reflection-symmetric wormhole when it has vanishing
ADM masses in the both sides. See Refs. Ellis:1973yv ; Nandi:2016uzg  for
gravitational lensing by the Ellis-Bronnikov wormhole with the positive ADM
mass, see Refs. Cramer:1994qj ; Bronnikov:2018nub  for the negative ADM mass,
and see Ref. Tsukamoto:2016qro  and references therein for the vanishing ADM
mass. The instability of the Ellis-Bronnikov wormhole was reported
Shinkai_Hayward_2002 ; Gonzalez_Guzman_Sarbach_2008_I ;
Gonzalez_Guzman_Sarbach_2008_II . Reliable stable wormholes without thin
shells have not reported yet in general relativity but we may construct stable
wormhole solution by choosing exotic matters other than the phantom scalar
field Bronnikov:2013coa . Recently, Shaikh et al. have suggested gravitational
lensing by reflection-symmetric wormholes with multiple photon spheres and the
shadow images of the multiple photon spheres Shaikh:2018oul  and Bronnikov and
Baleevskikh have considered that all asymptotically-flat, static, spherically
symmetric wormholes with reflection symmetry have a photon sphere on a throat
Bronnikov:2018nub . However, we notice that they overlook the possibility that
circular photon orbits on the throat can form an antiphoton sphere. The
gravitational lensing by wormholes with the antiphoton sphere on the throat
has been suggested by Shaikh et al. in Ref. Shaikh:2019jfr . Tsukamoto has
shown that the circular photon orbits on the throats of a Damour-Solodukhin
wormhole Damour:2007ap  and a Simpson-Visser wormhole Simpson:2018tsi  can be
the antiphoton spheres Tsukamoto:2020uay ; Tsukamoto:2020bjm . In Ref.
Bronnikov:2018nub , Bronnikov and Baleevskikh have discussed the deflection
angle of a light ray scattered by reflection-asymmetric wormholes with the
photon sphere which is slightly off the throat. Gravitational lensing by
reflection-asymmetric wormholes with the antiphoton sphere which is slightly
off the throat is left as a future work.

Note added. Recentely, related papers have appeared on arXiv. Guerrero et al.
have constructed reflection-asymmetric wormholes supported by a positive
energy thin shell with a double shadow in Palatini $f(R)$ gravity
Guerrero:2021pxt . Peng et al. have studied the observational appearance of an
accretion disk around a reflection-asymmetric thin-shell wormhole Peng:2021osd
.

## References

  * (1) B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, 061102 (2016).
  * (2) K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. 875, L1 (2019).
  * (3) M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
  * (4) M. Visser, Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, Woodbury, NY, 1995).
  * (5) V. Perlick, Phys. Rev. D 69, 064017 (2004).
  * (6) K. K. Nandi, Y. Z. Zhang, and A. V. Zakharov, Phys. Rev. D 74, 024020 (2006).
  * (7) N. Tsukamoto, T. Harada, and K. Yajima, Phys. Rev. D 86, 104062 (2012).
  * (8) N. Tsukamoto, Phys. Rev. D 94, 124001 (2016).
  * (9) K. K. Nandi, R. N. Izmailov, E. R. Zhdanov, and A. Bhattacharya, JCAP 07, 027 (2018).
  * (10) R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, Phys. Lett. B 789, 270 (2019). [erratum: Phys. Lett. B 791, 422 (2019)].
  * (11) R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, JCAP 07, 028 (2019).
  * (12) N. Tsukamoto, Phys. Rev. D 101, 104021 (2020).
  * (13) N. Tsukamoto, Phys. Rev. D 103, 024033 (2021).
  * (14) C. Lanczos, Phys. Zeits. 23, 539 (1922).
  * (15) C. Lanczos, Ann. Phys. 74, 518 (1924).
  * (16) W. Israel, Nuovo Cim. B 44, 1 (1966), Erratum: [Nuovo Cim. B 48, 463 (1967)].
  * (17) E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, Cambridge, 2004).
  * (18) M. Visser, Nucl. Phys. B 328, 203 (1989).
  * (19) G. Darmois, Les équations de la gravitation Einsteinienne, vol. XXV of Mémorial des sciences mathématiques (Gauthier-Villars, Paris, 1927).
  * (20) E. Poisson and M. Visser, Phys. Rev. D 52, 7318 (1995).
  * (21) E. F. Eiroa, Phys. Rev. D 78, 024018 (2008).
  * (22) E. F. Eiroa and C. Simeone, Phys. Rev. D 76, 024021 (2007).
  * (23) E. F. Eiroa, Phys. Rev. D 80, 044033 (2009).
  * (24) E. F. Eiroa and G. E. Romero, Gen. Rel. Grav. 36, 651 (2004).
  * (25) M. Sharif and M. Azam, Eur. Phys. J. C 73, 2554 (2013).
  * (26) A. Eid, Eur. Phys. J. Plus 131, 23 (2016).
  * (27) S. Habib Mazharimousavi and M. Halilsoy, Int. J. Mod. Phys. D 27, 1850028 (2018).
  * (28) M. Sharif and M. Azam, JCAP 05, 025 (2013).
  * (29) D. Wang and X. Meng, Phys. Dark Univ. 17, 46 (2017).
  * (30) S. D. Forghani, S. H. Mazharimousavi, and M. Halilsoy, Eur. Phys. J. Plus 134, 342 (2019).
  * (31) P. K. F. Kuhfittig, Acta Phys. Polon. B 41, 2017 (2010).
  * (32) N. M. Garcia, F. S. Lobo, and M. Visser, Phys. Rev. D 86, 044026 (2012).
  * (33) E. F. Eiroa and C. Simeone, Phys. Rev. D 71, 127501 (2005).
  * (34) F. S. Lobo, Phys. Rev. D 71, 124022 (2005).
  * (35) F. Rahaman, M. Kalam, and S. Chakraborti, Int. J. Mod. Phys. D 16, 1669 (2007).
  * (36) F. Rahaman, M. Kalam, K. Rahman, and S. Chakraborti, Gen. Rel. Grav. 39, 945 (2007).
  * (37) E. F. Eiroa and G. Figueroa Aguirre, Eur. Phys. J. C 76, 132 (2016).
  * (38) M. Ishak and K. Lake, Phys. Rev. D 65, 044011 (2002).
  * (39) F. S. N. Lobo and P. Crawford, Class. Quant. Grav. 22, 4869 (2005).
  * (40) S. Kim, Phys. Lett. A 166, 13 (1992).
  * (41) F. S. N. Lobo and P. Crawford, Class. Quant. Grav. 21, 391 (2004).
  * (42) C. Barcelo and M. Visser, Nucl. Phys. B 584, 415 (2000).
  * (43) J. P. S. Lemos and F. S. N. Lobo, Phys. Rev. D 78, 044030 (2008).
  * (44) E. F. Eiroa and C. Simeone, Phys. Rev. D 70, 044008 (2004).
  * (45) C. Bejarano, E. F. Eiroa, and C. Simeone, Phys. Rev. D 75, 027501 (2007).
  * (46) M. Thibeault, C. Simeone, and E. F. Eiroa, Gen. Rel. Grav. 38, 1593 (2006).
  * (47) F. Rahaman, M. Kalam, and S. Chakraborty, Gen. Rel. Grav. 38, 1687 (2006).
  * (48) S. H. Mazharimousavi, M. Halilsoy, and Z. Amirabi, Phys. Rev. D 81, 104002 (2010).
  * (49) T. Kokubu and T. Harada, Class. Quant. Grav. 32, 205001 (2015).
  * (50) M. R. Mehdizadeh, M. Kord Zangeneh, and F. S. N. Lobo, Phys. Rev. D 92, 044022 (2015).
  * (51) T. Kokubu, H. Maeda, and T. Harada, Class. Quant. Grav. 32, 235021 (2015).
  * (52) G. Perry and R. B. Mann, Gen. Rel. Grav. 24, 305 (1992).
  * (53) S. Kim, H. Lee, S. K. Kim, and J. Yang, Phys. Lett. A 183, 359 (1993).
  * (54) F. Rahaman, A. Banerjee, and I. Radinschi, Int. J. Theor. Phys. 51, 1680 (2012).
  * (55) E. F. Eiroa and C. Simeone, Phys. Rev. D 87, 064041 (2013).
  * (56) N. Tsukamoto and T. Kokubu, Phys. Rev. D 98, 044026 (2018).
  * (57) M. La Camera, Mod. Phys. Lett. A 26, 857 (2011).
  * (58) K.-i. Nakao, T. Uno, and S. Kinoshita, Phys. Rev. D 88, 044036 (2013).
  * (59) Y. Akai and K.-i. Nakao, Phys. Rev. D 96, 024033 (2017).
  * (60) D. C. Dai and D. Stojkovic, Phys. Rev. D 100, 083513 (2019).
  * (61) J. H. Simonetti, M. J. Kavic, D. Minic, D. Stojkovic, and D. C. Dai, [arXiv:2007.12184 [gr-qc]].
  * (62) A. Eid, New Astron. 53, 6 (2017).
  * (63) S. D. Forghani, S. Habib Mazharimousavi, and M. Halilsoy, Eur. Phys. J. C 78, 469 (2018).
  * (64) X. Wang, P. C. Li, C. Y. Zhang, and M. Guo, Phys. Lett. B 811, 135930 (2020).
  * (65) M. Wielgus, J. Horak, F. Vincent, and M. Abramowicz, Phys. Rev. D 102, 084044 (2020).
  * (66) C. Martinez and M. Nozawa, Phys. Rev. D 103, 024003 (2021).
  * (67) H. G. Ellis, J. Math. Phys. 14, 104 (1973).
  * (68) K. A. Bronnikov, Acta Phys. Polon. B 4, 251 (1973).
  * (69) K. K. Nandi, R. N. Izmailov, A. A. Yanbekov, and A. A. Shayakhmetov, Phys. Rev. D 95, 104011 (2017).
  * (70) J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, Phys. Rev. D 51, 3117 (1995).
  * (71) K. A. Bronnikov and K. A. Baleevskikh, Grav. Cosmol. 25, 44 (2019).
  * (72) H. Shinkai and S. A. Hayward, Phys. Rev. D 66, 044005 (2002);
  * (73) J. A. González, F. S. Guzmán, and O. Sarbach Class. Quant. Grav. 26, 015010 (2009).
  * (74) J. A. González, F. S. Guzmán, and O. Sarbach Class. Quant. Grav. 26, 015011 (2009).
  * (75) K. A. Bronnikov, L. N. Lipatova, I. D. Novikov, and A. A. Shatskiy, Grav. Cosmol. 19, 269 (2013).
  * (76) T. Damour and S. N. Solodukhin, Phys. Rev. D 76, 024016 (2007).
  * (77) A. Simpson and M. Visser, JCAP 02, 042 (2019).
  * (78) M. Guerrero, G. J. Olmo, and D. Rubiera-Garcia, [arXiv:2102.00840 [gr-qc]].
  * (79) J. Peng, M. Guo, and X. H. Feng, [arXiv:2102.05488 [gr-qc]].