# Wormhole solution free of ghosts in Einstein’s gravity with two scalar
fields

Shin’ichi Nojiri *  —<EMAIL_ADDRESS>Department of
Physics, Nagoya University, Nagoya 464-8602, Japan
&
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan  G. G. L. Nashed *  —
<EMAIL_ADDRESS>Centre for Theoretical Physics, The British University in
Egypt, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt

(August 28, 2024)

###### Abstract

In this paper, we construct models that admit the traversable wormhole
geometries in the framework of Einstein’s gravity with two scalar fields. As
well known, the energy conditions are broken and we show that there appears a
ghost. The ghost can be, however, eliminated by imposing a constraint on the
ghost field, which is a scalar. The constraint is similar to the mimetic one
proposed by Chamseddine and Mukhanov to construct an alternative description
of cold dark matter. We explicitly show that there does not appear any
unstable mode although the energy conditions are broken. Therefore we obtain a
model that realizes the traversable and stable wormhole.

##  I Introduction

Recently, observations involving the relativistic collision of two compact
objects have resulted in the production of gravitational waves (GWs). These
waves serve as invaluable instruments for examining the properties of the
colliding entities. Moreover, the recent findings reported by LIGO Abbott et
al. (2016a, b, 2017a, 2017b, 2017c) have provided compelling evidence that the
field of gravitational wave astronomy will play a substantial role in
advancing our understanding of gravitational interactions and extreme-gravity
astrophysical phenomena. However, despite these notable advancements, recent
observations have not ventured into the intricate intricacies of spacetime
beyond the photon sphere. The expected forthcoming gravitational wave (GW)
observations are in a position to record the ringdown phase. This phase is
recognized by a series of damped oscillatory modes in the initial stages,
commonly known as quasinormal modes (QNMs) Vishveshwara (1970);
Kokkotas:1999bd; Berti et al. (2009); Mazur and Mottola (2023); Morris et al.
(1988); Damour and Solodukhin (2007); Konoplya and Zhidenko (2011). This stage
has the capacity to provide vital information about the composition of compact
objects Cardoso et al. (2016), particularly elucidating the physics in the
vicinity of black holes’ event horizons (BHs) and the potential presence of
unexpected structural characteristics. Future gravitational wave (GW)
observations hold the promise of providing us with a deeper understanding of
compact objects that differ from black holes (BHs). These distinct compact
entities, which lack event horizons, are commonly known as exotic compact
objects (ECOs) Mazur and Mottola (2023); Damour and Solodukhin (2007); Holdom
and Ren (2017).

Among the notable exotic compact object (ECO) solutions, wormholes (WHs) stand
out. They are solutions to the Einstein equations that enable connections
between different regions of the Universe or even between entirely separate
Universes Misner (1960); Wheeler (1957). Although they have distinct causal
structures when compared to black holes (BHs), wormholes (WHs) do possess
photon spheres. As a result, in gravitational wave (GW) data, the early stage
of the ringdown signal has the potential to mask the ability to distinguish
between wormholes (WHs) and black holes (BHs). Previous research has delved
into the examination of Lorentzian wormholes within the framework of General
Relativity (GR), as documented in prior studies Bronnikov (1973); Ellis
(1973); Morris et al. (1988); Sahni and Starobinsky (2000); Carroll (2001);
Peebles and Ratra (2003); Gonzalez-Diaz (2002); González et al. (2022);
Poisson and Visser (1995). In these investigations, the establishment of
conditions for traversable wormholes was accomplished by introducing a static
spherically symmetric metric. Significantly, these conditions necessitate the
inclusion of exotic or phantom matter, which violates the null energy
condition (NEC). Wormholes that incorporate ordinary matter adhering to the
null energy condition (NEC)Poisson and Visser (1995); Antoniou et al. (2020);
Mehdizadeh et al. (2015a) have been introduced and explored within the
framework of modified gravity theories. These theories encompass Brans-Dicke
theoryNandi et al. (1997); Lobo and Oliveira (2010); Sushkov and Kozyrev
(2011); Papantonopoulos and Vlachos (2020), $f(R)$ gravity Garcia and Lobo
(2010); Montelongo Garcia and Lobo (2011), Einstein-Gauss-Bonnet theory
Mehdizadeh et al. (2015b); Kord Zangeneh et al. (2015), Einstein-Cartan
theory, and general scalar-tensor theories Bronnikov and Galiakhmetov (2015,
2016); Mehdizadeh and Ziaie (2017); Yaqin and Gunara (2017). It’s worth
highlighting that wormhole (WH) solutions within the context of $f(R)$
theories were extensively investigated in the work presented in Karakasis et
al. (2022). Furthermore, wormholes (WHs) with a self-interacting scalar field
were the focus of scrutiny in the study presented in Anabalon and Cisterna
(2012).

Another pivotal element concerning a WH involves the breach of energy
conditions within the context of GR, at least in the vicinity of the WH’s
throat Morris and Thorne (1988); Lobo (2007). This implies that it is
necessary to possess a quantity of exotic matter (where the stress-energy
tensor (SET) of matter contravenes the null energy condition (NEC)) in order
to maintain the stability of the WH throat. As a result, in certain reference
frames, the energy density of matter can be perceived as negative. It is
evident that there has been significant interest in constructing WHs while
minimizing the requirement for exotic matter Visser et al. (2003). In
reference Visser et al. (2003), the authors have theoretically demonstrated
that it is possible to minimize the exotic matter requirement to an
infinitesimal level and confine it precisely at the throat of the WH by
carefully selecting the WH’s geometry. The mathematical procedure is referred
to as the “cut and paste technique”, and the resulting WH is termed a “thin-
shell WH”. Research on thin-shell WHs can be located in references Visser
(1989a, b); Lobo and Crawford (2004); Dias and Lemos (2010). Nandi et al.
Nandi et al. (2004) subsequently introduced an enhanced quantification method
to precisely determine the amount of exotic matter present in a specific
spacetime. Consequently, numerous arguments have been presented to
substantiate the breach of energy conditions. In this context, the use of the
phantom energy equation of state (EoS) has been employed to maintain
traversable WHs, as demonstrated in references Sushkov (2005); Lobo (2005a);
Gonzalez et al. (2009).

One intriguing aspect to contemplate is that within the phantom regime, the
energy density grows over time, offering a conceptual basis for the existence
of WHs. The stability analysis of a phantom WH geometry has been explored in
reference Lobo (2005b). While an array of WH solutions has been identified for
the generalized Chaplygin gas Lobo (2006); Kuhfittig (2009), changing
cosmological constants Rahaman et al. (2007), polytropic phantom energy Jamil
et al. (2010), and ghost scalar fields Carvente et al. (2019). Subsequently,
the phantom energy EoS was employed to formulate precise, evolving WH
structures in reference Cataldo et al. (2008). In this theoretical framework,
it was determined that phantom energy can facilitate the existence of evolving
WHs.

Nonetheless, physicists consistently strive to evade energy condition
violations or provide appropriate justifications for them. However, up to the
present time, constructing a static WH geometry that complies with the energy
conditions remains an unattained goal. Consequently, scientists are exploring
various strategies to address this challenge. This observation prompted
consideration of the potential existence of WH solutions within alternative
theories of gravity. Examples include higher-order gravity theories Hochberg
(1990); Ghoroku and Soma (1992), cosmological WHs in higher dimensions
Zangeneh et al. (2014), and the Einstein-Gauss-Bonnet theory Bhawal and Kar
(1992); Maeda and Nozawa (2008); Mehdizadeh et al. (2015b). When examining
$f(R)$ gravity, it is conceivable to theoretically create traversable WHs
without the necessity of exotic matter Pavlovic and Sossich (2015); Lobo and
Oliveira (2009) or by utilizing dark matter as a source Muniz and Maluf
(2022). Alternatively, researchers can explore the quest for WHs within the
framework of third-order Lovelock gravity Kord Zangeneh et al. (2015);
Mehdizadeh and Lobo (2016), hybrid metric-Palatini gravity Rosa (2021); Kord
Zangeneh and Lobo (2021), $f(Q)$ gravity Banerjee et al. (2021a); Parsaei et
al. (2022); Hassan et al. (2022), and extended theories of gravity De Falco et
al. (2021a, b). The investigation of traversable WHs within the realm of
$f(R,T)$ gravity was conducted in references Moraes and Sahoo (2017); Elizalde
and Khurshudyan (2018); Moraes and Sahoo (2019). Simultaneously, in references
Rosa and Kull (2022); Banerjee et al. (2021b), authors identified precise WH
solutions in $f(R,T)$ gravity without the need for exotic matter. It is the
aim of the present study to derive WH in the frame of Einstein’s gravity
coupled with two scalar fields.

In Nojiri et al. (2021), a general formulation to construct a model that
admits arbitrarily given spherically symmetric and time-dependent geometry as
a solution has been given in the framework of Einstein’s gravity coupled with
two scalar fields. We apply the formulation to the static wormhole
formulation. As expected, the model includes ghost mode. The ghost mode often
plays the role of the phantom and is consistent with the breakdown of the
energy conditions. The ghost mode has, however, negative kinetic energy
classically and generates negative norm states as a quantum theory. Therefore
the existence of the ghost mode tells that the model is physically
inconsistent. In this paper, we eliminate such ghosts using the mimetic
constraint and make the ghost mode non-dynamical. We show that there does not
appear unstable mode corresponding to the ghost mode although the energy
conditions are still broken. This may tell that the breakdown of the energy
conditions could not always imply physical inconsistency.

The organization of this paper is as follows: In the next section, based on
the formulation in Nojiri et al. (2021), we construct a model whose solutions
include a well-known wormhole geometry. We show that there appears a ghost in
the model. In Section III, we show that one of the two scalar fields can be
canonical and not a ghost. Because another one is a ghost in general, we
propose a model to make the scalar field non-dynamical by imposing the mimetic
constraint, and as a result, unstable modes corresponding to the existence of
the ghost disappear. In Section IV, we confirm the absence of the ghost mode
by using the perturbation from the wormhole geometry although the energy
conditions are broken. The breakdown of the energy conditions may not always
imply any physical inconsistency. The last section is devoted to the summary
and discussion.

##  II Wormhole based on Einstein’s theory with two scalar fields

Einstein’s GR with two scalar fields $\phi$ and $\chi$ is described by the
action as follows Nojiri et al. (2021)111Throughout this study, we will use
the relativistic units in which we will put $\kappa=1$.,

$\displaystyle S_{\mathrm{GR}\phi\chi}=\int d^{4}x\sqrt{-g}$
$\displaystyle\left[\frac{R}{2\kappa^{2}}-\frac{1}{2}\,A(\phi,\chi)\partial_{\mu}\phi\partial^{\mu}\phi-B(\phi,\chi)\,\partial_{\mu}\phi\partial^{\mu}\chi\right.$
$\displaystyle\left.\quad-\frac{1}{2}\,C(\phi,\chi)\partial_{\mu}\chi\partial^{\mu}\chi-V(\phi,\chi)\right]\,.$
(1)

In this context, $g$ represents the determinant of the metric tensor
$g_{\mu\nu}$, $R$ denotes the Ricci scalar, and $V(\phi,\chi)$ represents the
potential of the scalar doublet. The values of the coefficients $A$, $B$, and
$C$ are contingent upon the properties of the scalars. Upon varying the action
(II) with respect to the metric $g_{\mu\nu}$, we derive the ensuing Einstein
equation:

$\displaystyle\frac{1}{\kappa^{2}}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)=$
$\displaystyle\,A(\phi,\chi)\partial_{\mu}\phi\partial_{\nu}\phi+B(\phi,\chi)\left(\partial_{\mu}\phi\partial_{\nu}\chi+\partial_{\nu}\phi\partial_{\mu}\chi\right)+C(\phi,\chi)\partial_{\mu}\chi\partial_{\nu}\chi$
$\displaystyle\,-g_{\mu\nu}\left[\frac{1}{2}\,A(\phi,\chi)\partial_{\rho}\phi\partial^{\rho}\phi+B(\phi,\chi)\partial_{\rho}\phi\partial^{\rho}\chi+\frac{1}{2}\,C(\phi,\chi)\partial_{\rho}\chi\partial^{\rho}\chi+V(\phi,\chi)\right]\,,$
(2)

Through the variation of action (II) concerning the scalar fields $\phi$ and
$\chi$, we acquire the following expressions:

$\displaystyle 0=$
$\displaystyle\,\frac{A_{\phi}}{2}\,\partial_{\mu}\phi\partial^{\mu}\phi+A\nabla^{\mu}\partial_{\mu}\phi+A_{\chi}\partial_{\mu}\phi\partial^{\mu}\chi+\left(B_{\chi}-\frac{1}{2}\,C_{\phi}\right)\partial_{\mu}\chi\partial^{\mu}\chi+B\nabla^{\mu}\partial_{\mu}\chi-
V_{\phi}\,,$ (3) $\displaystyle 0=$
$\displaystyle\,\left(-\frac{1}{2}\,A_{\chi}+B_{\phi}\right)\partial_{\mu}\phi\partial^{\mu}\phi+B\nabla^{\mu}\partial_{\mu}\phi+\frac{1}{2}\,C_{\chi}\partial_{\mu}\chi\partial^{\mu}\chi+C\nabla^{\mu}\partial_{\mu}\chi+C_{\phi}\partial_{\mu}\phi\partial^{\mu}\chi-
V_{\chi}\,.$ (4)

Here, let us define $A_{\phi}$ as $\partial A(\phi,\chi)/\partial\phi$, and
similarly for other derivatives. It’s worth noting that Eqs.(3) and (4) can be
derived from the Bianchi identity in conjunction with Eq.(II).

In the following, we identify

$\displaystyle\phi=t\,,\quad\chi=r\,.$ (5)

As explained in the reference Nojiri et al. (2021), making the assumption (5)
doesn’t result in any loss of generality. In the case of a spacetime with a
general spherically symmetric yet time-dependent solution, the scalar fields
$\phi$ and $\chi$ exhibit dependencies on both the time coordinate, denoted as
$t$, and the radial coordinate, denoted as $r$. In the context of a given
solution, the specific dependencies of $\phi$ and $\chi$ on both the time
variable $t$ and the radial variable $r$ are determined as functions:
$\phi=\phi(t,r)$ and $\chi=\chi(t,r)$. We may redefine the scalar fields to
replace $t$ and $r$ with new scalar fields, $\tilde{\phi}$ and $\tilde{\chi}$,
$\phi\left(\tilde{\phi},\tilde{\chi}\right)\equiv\phi\left(t=\tilde{\phi},r=\tilde{\chi}\right)$
and
$\chi\left(\tilde{\phi},\tilde{\chi}\right)\equiv\chi\left(t=\tilde{\phi},r=\tilde{\chi}\right)$.
Subsequently, we can associate the new scalar fields with the time and radial
coordinates in (5). The transformation of variables from
$\left(\phi,\chi\right)$ to $\left(\tilde{\phi},\tilde{\chi}\right)$ can be
integrated into the redefinitions of $A$, $B$, $C$, and $V$ within the action
(II). This demonstrates that making the assumption (5) doesn’t lead to any
loss of generality. Furthermore, as we will observe later, even when $\phi$ is
identified with $t$ a static spacetime can still be achieved.

Now we consider the following spherically symmetric line-element

$\displaystyle
ds^{2}=-\mathrm{e}^{2\Phi(r)}dt^{2}+\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,.$ (6)

Here, $\Phi\left(r\right)$ and $b\left(r\right)$ represent arbitrary functions
of the radial coordinate, and they are referred to as the redshift and shape
functions, respectively. At the minimum wormhole throat, denoted as $r_{0}$
with the condition $b(r_{0})=r_{0}$, the wormhole serves as a connection
between two distinct Universes, and the radial coordinate range follows the
inequality $0<r_{0}\leq r\leq\infty$. To prevent the formation of an event
horizon or any singularities at the wormhole throat $r_{0}$, the redshift
function should remain well-defined across all points. To ensure the
traversability of the wormhole, two conditions must be met:
$\frac{b\left(r\right)-b^{\prime}\left(r\right)}{2b\left(r\right)^{2}}>0$, and
$1-\frac{b\left(r\right)}{r}>0$ (where ′ represents the derivative with
respect to $r$). Additionally, for asymptotically flat wormhole solutions, the
condition $\frac{b\left(r\right)}{r}\to 0$ as $r\to\infty$ is imposed, as
outlined in Shamir and Zia (2018).

Applying the field equation (II) to the line element (6), we obtain,

$\displaystyle\frac{b^{\prime}\left(r\right)}{r^{2}}=$
$\displaystyle\,\frac{\left(\left(\left(r-b\left(r\right)\right)C\left(r\right)+2V\left(r\right)r\right)\mathrm{e}^{2\Phi\left(r\right)}+A\left(r\right)r\right)\kappa^{2}}{2\mathrm{e}^{2\Phi\left(r\right)}r}\,,$
$\displaystyle B\left(r\right)=$ $\displaystyle\,0\,,$
$\displaystyle\frac{2r\left(r-b\left(r\right)\right)\Phi^{\prime}\left(r\right)-b\left(r\right)}{r^{3}}=$
$\displaystyle\,\frac{\left(\left(\left(r-b\left(r\right)\right)C\left(r\right)-2V\left(r\right)r\right)\mathrm{e}^{2\Phi\left(r\right)}+A\left(r\right)r\right)\kappa^{2}}{2\mathrm{e}^{2\Phi\left(r\right)}r}\,,$
$\displaystyle\frac{1}{r^{3}}\Bigl{\\{}2r^{2}\left(r-b\left(r\right)\right)\Phi^{\prime\prime}\left(r\right)$
$\displaystyle+2\Bigl{(}r\left(r-b\left(r\right)\right)\Phi^{\prime}\left(r\right)-\frac{b^{\prime}\left(r\right)r}{2}+\frac{b\left(r\right)}{2}\Bigr{)}\left(1+\Phi^{\prime}\left(r\right)r\right)\Bigr{\\}}$
$\displaystyle=$
$\displaystyle\,{\frac{\left(\left(\left(-r+b\left(r\right)\right)C\left(r\right)-2\,V\left(r\right)r\right)\mathrm{e}^{2\Phi\left(r\right)}+A\left(r\right)r\right)\kappa^{2}}{\mathrm{e}^{2\Phi\left(r\right)}r}}\,.$
(7)

Assuming the WH geometry to has the form Tangphati et al. (2020); Capozziello
et al. (2021)

$\displaystyle\Phi\left(r\right)=\frac{r_{0}}{2r}\,,\quad
b\left(r\right)=r\mathrm{e}^{-\gamma\left(r-r_{0}\right)}\,,$ (8)

with positive constants $r_{0}$ and $\gamma$ and substituting these
expressions into Eq. (II), we obtain

$\displaystyle A\left(r\right)=$
$\displaystyle\,\frac{\mathrm{e}^{\frac{r_{0}}{r}}}{4\kappa^{2}r^{4}\mathrm{e}^{\gamma
r}}\left(-2r_{0}r\mathrm{e}^{\gamma r_{0}}+4r^{2}\mathrm{e}^{\gamma
r_{0}}+2r_{0}r\mathrm{e}^{\gamma r}-2\gamma r^{3}\mathrm{e}^{\gamma
r_{0}}-{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}-\gamma
r_{0}r^{2}\mathrm{e}^{\gamma r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma
r}\right)\,,$ $\displaystyle B\left(r\right)=$ $\displaystyle\,0\,,$
$\displaystyle C\left(r\right)=$
$\displaystyle\,-\frac{4r^{2}\mathrm{e}^{\gamma
r_{0}}-6r_{0}r\mathrm{e}^{\gamma r_{0}}+6r_{0}r\mathrm{e}^{\gamma r}+2\gamma
r^{3}\mathrm{e}^{\gamma r_{0}}-{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}-\gamma
r_{0}r^{2}\mathrm{e}^{\gamma r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma
r}}{4\kappa^{2}r^{4}\left(\mathrm{e}^{\gamma r}-\mathrm{e}^{\gamma
r_{0}}\right)}\,,$ $\displaystyle V\left(r\right)=$
$\displaystyle\,\frac{-r_{0}\mathrm{e}^{\gamma r_{0}}+2r\mathrm{e}^{\gamma
r_{0}}+r_{0}\mathrm{e}^{\gamma r}-\gamma r^{2}\mathrm{e}^{\gamma
r_{0}}}{{2\kappa}^{2}r^{3}\mathrm{e}^{\gamma r}}\,.$ (9)

We are assuming $r\geq r_{0}$. Eq. (II) tells that if we consider the model,

$\displaystyle A\left(\chi\right)=$
$\displaystyle\,\frac{\mathrm{e}^{\frac{r_{0}}{\chi}}}{4\kappa^{2}\chi^{4}\mathrm{e}^{\gamma\chi}}\left(-2r_{0}\chi\mathrm{e}^{\gamma
r_{0}}+4\chi^{2}\mathrm{e}^{\gamma
r_{0}}+2r_{0}\chi\mathrm{e}^{\gamma\chi}-2\gamma\chi^{3}\mathrm{e}^{\gamma
r_{0}}-{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}-\gamma
r_{0}\chi^{2}\mathrm{e}^{\gamma
r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma\chi}\right)\,,$ $\displaystyle
B\left(\chi\right)=$ $\displaystyle\,0\,,$ $\displaystyle C\left(\chi\right)=$
$\displaystyle\,-\frac{4\chi^{2}\mathrm{e}^{\gamma
r_{0}}-6r_{0}\chi\mathrm{e}^{\gamma
r_{0}}+6r_{0}\chi\mathrm{e}^{\gamma\chi}+2\gamma\chi^{3}\mathrm{e}^{\gamma
r_{0}}-{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}-\gamma
r_{0}\chi^{2}\mathrm{e}^{\gamma
r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma\chi}}{4\kappa^{2}\chi^{4}\left(\mathrm{e}^{\gamma\chi}-\mathrm{e}^{\gamma
r_{0}}\right)}\,,$ $\displaystyle V\left(\chi\right)=$
$\displaystyle\,\frac{-r_{0}\mathrm{e}^{\gamma r_{0}}+2r\mathrm{e}^{\gamma
r_{0}}+r_{0}\mathrm{e}^{\gamma\chi}-\gamma\chi^{2}\mathrm{e}^{\gamma
r_{0}}}{{2\kappa}^{2}\chi^{3}\mathrm{e}^{\gamma\chi}}\,,$ (10)

the wormhole spacetime given by (6) with (8) and (5) is an exact solution of
the model.

We should note both $A\left(r\right)$ and $C\left(r\right)$ become negative in
some region of $r$, as shown in Fig. 1 0(a), for example, when $r\to r_{0}$,
we obtain,

$\displaystyle
A\left(r\right)\to\frac{\mathrm{e}^{\frac{r_{0}}{r}}}{4\kappa^{2}{r_{0}}^{2}}\left(4-3\gamma
r_{0}\right)\,,\quad C\left(r\right)\to-\frac{4+\gamma
r_{0}}{4\kappa^{2}\gamma{r_{0}}^{2}\left(r-r_{0}\right)}\,.$ (11)

Therefore $C\left(r\right)$ is negative and $A\left(r\right)$ becomes negative
if $4-3\gamma r_{0}<0$.

(a) A$\&$ C

(b) Density, radial and tangential pressures

(c) some components of the energy conditions

Figure 1: 0(a) The general behaviors of the two functions $A$ and $C$ as $r\to
r_{0}$; 0(b) is the behavior of the density, radial, and tangential pressure;
0(c) represents some components of the energy conditions.

When $A\left(r\right)$ or $C\left(r\right)$ is negative, the scalar field
$\phi$ or $\chi$ becomes a ghost, which generates the breakdown of the energy
conditions in general. Using Eq. (II), we obtain the form of the energy
density $\rho$ and the radial and tangential components of the pressure,
$p_{\mathrm{r}}$ and $p_{\mathrm{t}}$ as follows, 222 We may express Eq. (II)
as $G_{\mu\nu}=\kappa^{2}T^{\mathrm{sc}}_{\mu\nu}$ by using the Einstein
tensor $G_{\mu\nu}:={R}_{\mu\nu}-g_{\mu\nu}{R}/2$ and the energy-momentum
tensor of the scalar field $T^{\mathrm{sc}}_{\mu\nu}$. By writing the energy-
momentum tensor as $T^{\mathrm{sc}\nu}_{\ \
\mu}=\mathrm{diag}\left(-\rho,p_{\mathrm{r}},p_{\mathrm{t}},p_{\mathrm{t}}\right)$,
we can extend the energy conditions to Einstein’s gravity with scalar two
fields as follows,  1. ${\rho}+{p}_{r}>0$, ${\rho}+{p}_{t}>0$, and ${\rho}\geq
0$, which is the Weak Energy Condition (WEC). 2. ${\rho}+{p}_{t}\geq 0$, and
${\rho}+{p}_{r}\geq 0$, which is the Null Energy Condition (NEC). 3.
${\rho}+{p}_{r}\geq 0$, ${\rho}+{p}_{t}\geq 0$, and
${\rho}+{p}_{r}+2{p}_{t}\geq 0$, which is the Strong Energy Condition (SEC).
4. ${\rho}-{p}_{r}\geq 0$, ${\rho}-{p}_{t}\geq 0$,and ${\rho}\geq 0$, which is
the Dominant Energy Conditions (DEC).

$\displaystyle\rho=$ $\displaystyle\,\frac{-\mathrm{e}^{\gamma
r_{0}}+\mathrm{e}^{\gamma r_{0}}\gamma
r+\mathrm{e}^{-\gamma\left(r-2r_{0}\right)}-\mathrm{e}^{-\gamma\left(r-2r_{0}\right)}r\gamma}{r^{2}\left(\mathrm{e}^{\gamma
r}-\mathrm{e}^{\gamma r_{0}}\right)}\,,$ $\displaystyle p_{\mathrm{r}}=$
$\displaystyle\,-\frac{r\mathrm{e}^{\gamma r_{0}}+r_{0}\mathrm{e}^{\gamma
r}-2r_{0}\mathrm{e}^{\gamma
r_{0}}-\mathrm{e}^{-\gamma\left(r-2r_{0}\right)}r+\mathrm{e}^{-\gamma\left(r-2r_{0}\right)}r_{0}}{r^{3}\left(\mathrm{e}^{\gamma
r}-\mathrm{e}^{\gamma r_{0}}\right)}\,,$ $\displaystyle p_{\mathrm{t}}=$
$\displaystyle\,\frac{\left({r_{0}}^{2}+\left(\gamma
r^{2}+2r\right)r_{0}-2\gamma
r^{3}\right)\mathrm{e}^{-\gamma\left(r-2r_{0}\right)}+\left(2\gamma
r^{3}-2{r_{0}}^{2}-\left(\gamma r^{2}+4r\right)r_{0}\right)\mathrm{e}^{\gamma
r_{0}}+r_{0}\left(2r+r_{0}\right)\mathrm{e}^{\gamma
r}}{4r^{4}\left(\mathrm{e}^{\gamma r}-\mathrm{e}^{\gamma r_{0}}\right)}\,.$
(12)

Then we find the energy conditions are broken when $r\gtrsim r_{0}$ as shown
in Fig. 1 0(b) and Fig. 1 0(c).

##  III Eliminating ghosts

We consider the conditions that $A\left(r\right)$ becomes non-negative. We
define a function $N\left(r\right)$ by
$A\left(r\right)=\frac{\mathrm{e}^{\frac{r_{0}}{r}}N\left(r\right)}{4\kappa^{2}r^{4}\mathrm{e}^{\gamma
r}}$, that is

$\displaystyle N\left(r\right)\equiv$
$\displaystyle\,-2r_{0}r\mathrm{e}^{\gamma r_{0}}+4r^{2}\mathrm{e}^{\gamma
r_{0}}+2r_{0}r\mathrm{e}^{\gamma r}-2\gamma r^{3}\mathrm{e}^{\gamma
r_{0}}-{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}-\gamma
r_{0}r^{2}\mathrm{e}^{\gamma r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma r}\,.$ (13)

Then we find

$\displaystyle N^{\prime\prime\prime\prime}\left(r\right)=$
$\displaystyle\,8\gamma^{3}r_{0}\mathrm{e}^{\gamma
r}+2\gamma^{4}r_{0}r\mathrm{e}^{\gamma
r}+\gamma^{4}{r_{0}}^{2}\mathrm{e}^{\gamma r}>0\,,$ (14)

and

$\displaystyle N(r_{0})=$ $\displaystyle\,\left(4-3\gamma
r_{0}\right){r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}\,,\quad
N^{\prime}(r_{0})=\left(8-5\gamma r_{0}\right)r_{0}\mathrm{e}^{\gamma
r_{0}}\,,\quad N^{\prime\prime}(r_{0})=\left(8-10\gamma
r_{0}+3\gamma^{2}{r_{0}}^{2}\right)\mathrm{e}^{\gamma r_{0}}\,,$
$\displaystyle N^{\prime\prime\prime}(r_{0})=$
$\displaystyle\,\gamma\left(-12+6\gamma
r_{0}+3\gamma^{2}{r_{0}}^{2}\right)\mathrm{e}^{\gamma r_{0}}\,,$ (15)

The conditions $N(r_{0})\geq 0$, $N^{\prime}(r_{0})\geq 0$,
$N^{\prime\prime}(r_{0})\geq 0$, and $N^{\prime\prime\prime}(r_{0})\geq 0$
give $\gamma r_{0}\leq\frac{4}{3}\approx 1.33$, $\gamma
r_{0}\leq\frac{8}{5}=1.6$, $\gamma r_{0}\leq\frac{4}{3}$ or $\gamma r_{0}\geq
2$, and $\gamma r_{0}\leq-1-\sqrt{5}$ or $\gamma r_{0}\geq-1+\sqrt{5}\approx
1.236$, respectively. Therefore if

$\displaystyle\frac{4}{3}\geq\gamma r_{0}\geq-1+\sqrt{5}\,,$ (16)

$N\left(r\right)$ and therefore $A\left(r\right)$ are not negative, which
tells $\phi$ is not ghost but canonical scalar field. For the WH to be
traversable, $r_{0}$ should be large enough. Such $r_{0}$ can be realized by
choosing $\gamma$ small enough to satisfy the condition (16).

On the other hand, $\chi$ is a ghost in general. In order to avoid the ghost,
we propose a model where the scalar field $\chi$ becomes non-dynamical by
imposing the mimetic constraint on $\chi$ as follows333An alternative
conceptualization to the concept of cold dark matter emerges through the
mimetic modification of General Relativity (GR), as originally introduced by
Chamseddine and Mukhanov Chamseddine and Mukhanov (2013). Subsequent
investigations into this theoretical framework have been conducted in a series
of works Chamseddine et al. (2014); Nojiri and Nashed (2022); Nashed and
Nojiri (2022); Nashed (2023a, b); Nashed and Nojiri (2021); Nashed (2021). In
their paper Chamseddine and Mukhanov (2013), Chamseddine and Mukhanov isolated
the conformal degree of freedom of Einstein-Hilbert gravity in a covariant
way, and in the resulting theory, the physical metric is defined with the
account of an auxiliary scalar field, which appears through its first
derivatives. In their publication Chamseddine and Mukhanov (2013), Chamseddine
and Mukhanov adeptly extracted the conformal degree of freedom inherent in
Einstein-Hilbert gravity in a covariant manner. Within this developed
framework, the physical metric is defined while considering the presence of an
auxiliary scalar field, which emerges through its primary derivatives.,

$\displaystyle\left(1-\frac{b(\chi)}{\chi}\right)g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi=1\,,$
(17)

whose solution is consistently $\chi=r$. The constraint can be realized by
introducing a multiplier field $\lambda$ and add the following term
$S_{\mathrm{mim}}$ to the action (II), $S_{\mathrm{GR}\phi\chi}\to
S_{\mathrm{GR}\phi\chi}+S_{\mathrm{mim}}$,

$\displaystyle S_{\mathrm{mim}}\equiv\int
d^{4}x\sqrt{-g}\lambda\left\\{\left(1-\frac{b(\chi)}{\chi}\right)\partial_{\rho}\chi\partial^{\rho}\chi-1\right\\}\,.$
(18)

By the term (18), Eq. (II) is modified as

$\displaystyle\frac{1}{\kappa^{2}}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)=g_{\mu\nu}$
$\displaystyle\,\left[-\frac{1}{2}\,A(\phi,\chi)\partial_{\rho}\phi\partial^{\rho}\phi-B(\phi,\chi)\partial_{\rho}\phi\partial^{\rho}\chi-\frac{1}{2}\,C(\phi,\chi)\partial_{\rho}\chi\partial^{\rho}\chi-V(\phi,\chi)\right]$
$\displaystyle\,+A(\phi,\chi)\partial_{\mu}\phi\partial_{\nu}\phi+B(\phi,\chi)\left(\partial_{\mu}\phi\partial_{\nu}\chi+\partial_{\nu}\phi\partial_{\mu}\chi\right)+C(\phi,\chi)\partial_{\mu}\chi\partial_{\nu}\chi$
$\displaystyle\,+\frac{1}{2}g_{\mu\nu}\lambda\left\\{\left(1-\frac{b(\chi)}{\chi}\right)\partial_{\rho}\chi\partial^{\rho}\chi-1\right\\}-\lambda\left(1-\frac{b(\chi)}{\chi}\right)\partial_{\mu}\chi\partial_{\nu}\chi\,,$
(19)

but we can always consider the solution with $\lambda=0$ and therefore the
spacetime with the wormhole (6) becomes an exact solution even if we add the
term in (18).

We may consider more general models where $\lambda$ does not vanish. Applying
the field equations (III) to the line element (6), we obtain,

$\displaystyle\frac{b^{\prime}\left(r\right)}{r^{2}}=$
$\displaystyle\,\frac{\left(\left(r\left(r-b\left(r\right)\right)C\left(r\right)+2V\left(r\right)r^{2}+b\left(r\right)\lambda\left(r\right)\left(2r-b\left(r\right)\right)\right)\mathrm{e}^{2\Phi\left(r\right)}+A\left(r\right)r^{2}\right)\kappa^{2}}{2\mathrm{e}^{2\Phi\left(r\right)}r^{2}}\,,$
$\displaystyle B\left(r\right)=$ $\displaystyle\,0\,,$
$\displaystyle\frac{2r\left(r-b\left(r\right)\right)\Phi^{\prime}\left(r\right)-b\left(r\right)}{r^{3}}=$
$\displaystyle\,\frac{\left(\left(r\left(r-b\left(r\right)\right)C\left(r\right)-2V\left(r\right)r^{2}-\lambda\left(r\right)\left(2r^{2}-2rb\left(r\right)+b^{2}\right)\right)\mathrm{e}^{2\Phi\left(r\right)}+A\left(r\right)r^{2}\right)\kappa^{2}}{2\mathrm{e}^{2\Phi\left(r\right)}r}\,,$
$\displaystyle\frac{1}{r^{3}}$
$\displaystyle\left\\{2r^{2}\left(r-b\left(r\right)\right)\Phi^{\prime\prime}\left(r\right)+2\left(r\left(r-b\left(r\right)\right)\Phi^{\prime}\left(r\right)-\frac{b^{\prime}\left(r\right)r}{2}+\frac{b\left(r\right)}{2}\right)\left(1+\Phi^{\prime}\left(r\right)r\right)\right\\}$
$\displaystyle=$
$\displaystyle\,\frac{\left(\left(r\left(b\left(r\right)-r\right)C\left(r\right)-2V\left(r\right)r^{2}+b\left(r\right)\lambda\left(r\right)(2r-b\left(r\right))\right)\mathrm{e}^{2\Phi\left(r\right)}+A\left(r\right)r^{2}\right)\kappa^{2}}{\mathrm{e}^{2\Phi\left(r\right)}r}\,.$
(20)

The solution of the above system takes the form:

$\displaystyle A\left(r\right)=$
$\displaystyle\,-\frac{\mathrm{e}^{\frac{r_{0}}{r}}}{4\mathrm{e}^{\gamma
r}\kappa^{2}r^{4}}\left(2\mathrm{e}^{\gamma r_{0}}\gamma
r^{3}+2r_{0}\mathrm{e}^{\gamma r_{0}}r-4r^{2}\mathrm{e}^{\gamma
r_{0}}-2r_{0}r\mathrm{e}^{\gamma r}+r_{0}\,\mathrm{e}^{\gamma r_{0}}\gamma
r^{2}+{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}-{r_{0}}^{2}\mathrm{e}^{\gamma
r}\right)\,,$ $\displaystyle B\left(r\right)=$ $\displaystyle\,0\,,$
$\displaystyle C\left(r\right)=$
$\displaystyle\,\frac{\left(\mathrm{e}^{\gamma r}-\mathrm{e}^{\gamma
r_{0}}\right)\lambda\left(r\right)}{\mathrm{e}^{\gamma
r}}-\frac{2\mathrm{e}^{\gamma r_{0}}\gamma r^{3}-6r_{0}\mathrm{e}^{\gamma
r_{0}}r+4r^{2}\mathrm{e}^{\gamma r_{0}}+6r_{0}r\mathrm{e}^{\gamma
r}-r_{0}\mathrm{e}^{\gamma r_{0}}\gamma r^{2}-{r_{0}}^{2}\mathrm{e}^{\gamma
r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma
r}}{4{\kappa}^{2}r^{4}\left(\mathrm{e}^{\gamma r}-\mathrm{e}^{\gamma
r_{0}}\right)}\,,$ $\displaystyle V\left(r\right)=$
$\displaystyle\,-\frac{\lambda\left(r\right)}{2}-\frac{r_{0}\mathrm{e}^{\gamma
r_{0}}-2\mathrm{e}^{\gamma r_{0}}r-r_{0}\mathrm{e}^{\gamma
r}+\mathrm{e}^{\gamma r_{0}}{r}^{2}\gamma}{2\mathrm{e}^{\gamma
r}{\kappa}^{2}{r}^{3}}\,.$ (21)

where $\lambda$ can take any value. Of course, the solution (III) gives the
energy density and pressure components with those given by Eq. (II) because
the geometry is not changed. By putting $r=\chi$ in (III), we obtain a class
of models where the wormhole spacetime given by (6) with (8) and (5) is an
exact solution of the models. Because $\lambda(\chi)=\lambda(r=\chi)$ is an
arbitrary function, we may choose $\lambda(r)$ so that $C(\chi)=0$, that is

$\displaystyle\lambda\left(r\right)=\frac{2\mathrm{e}^{\gamma r_{0}}\gamma
r^{3}-6r_{0}\mathrm{e}^{\gamma r_{0}}r+4r^{2}\mathrm{e}^{\gamma
r_{0}}+6r_{0}r\mathrm{e}^{\gamma r}-r_{0}\mathrm{e}^{\gamma r_{0}}\gamma
r^{2}-{r_{0}}^{2}\mathrm{e}^{\gamma r_{0}}+{r_{0}}^{2}\mathrm{e}^{\gamma
r}}{4{\kappa}^{2}r^{4}\left(\mathrm{e}^{\gamma r}-\mathrm{e}^{\gamma
r_{0}}\right)^{2}\mathrm{e}^{-\gamma r}}\,,$ (22)

or so that $V(\chi)=0$,

$\displaystyle\lambda\left(r\right)=\frac{r_{0}\mathrm{e}^{\gamma
r_{0}}-2\mathrm{e}^{\gamma r_{0}}r-r_{0}\mathrm{e}^{\gamma
r}+\mathrm{e}^{\gamma r_{0}}{r}^{2}\gamma}{\mathrm{e}^{\gamma
r}{\kappa}^{2}{r}^{3}}\,.$ (23)

In the next section, we confirm that the ghost mode does not appear and
therefore the wormhole geometry in the model could be stable.

##  IV Absence of ghost

In order to investigate the (in)stability of the wormhole geometry, we focus
on the model where $\lambda=0$ is a solution. We now consider the perturbation
around the solution (6) with (8) and (5) as follows,

$\displaystyle g_{\mu\nu}\to
g_{\mu\nu}+h_{\mu\nu}\,,\quad\phi\to\phi+\tau\,,\quad\chi\to\chi+\xi\,,\quad\lambda\to\lambda+\zeta\,.$
(24)

Then Eq. (III) in the background where $A=A(\chi)$, $B=0$, $C=C(\chi)$,
$V=V(\chi)$, and $\lambda=0$ gives,

$\displaystyle\frac{1}{\kappa^{2}}$
$\displaystyle\,\left[\frac{1}{2}\left\\{\nabla_{\mu}\nabla^{\rho}h_{\nu\rho}+\nabla_{\nu}\nabla^{\rho}h_{\mu\rho}-\nabla^{2}h_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\left(g^{\rho\sigma}h_{\rho\sigma}\right)-2R^{\sigma\
\rho}_{\ \nu\ \mu}h_{\sigma\rho}+R^{\rho}_{\ \mu}h_{\rho\nu}+R^{\rho}_{\
\nu}h_{\rho\mu}\right\\}\right.$
$\displaystyle\,\left.-\frac{1}{2}h_{\mu\nu}R-\frac{1}{2}g_{\mu\nu}\left\\{-h_{\rho\sigma}R^{\rho\sigma}+\nabla^{\rho}\nabla^{\sigma}h_{\rho\sigma}-\nabla^{2}\left(g^{\rho\sigma}h_{\rho\sigma}\right)\right\\}\right]$
$\displaystyle=$
$\displaystyle\,h_{\mu\nu}\left[-\frac{1}{2}A(\chi)\partial_{\rho}\phi\partial^{\rho}\phi-\frac{1}{2}C(\chi)\partial_{\rho}\chi\partial^{\rho}\chi-V(\chi)\right]-g_{\mu\nu}\left[-\frac{1}{2}A(\chi)\partial^{\rho}\phi\partial^{\sigma}\phi-\frac{1}{2}C(\chi)\partial^{\rho}\chi\partial^{\sigma}\chi\right]h_{\rho\sigma}$
$\displaystyle\,-g_{\mu\nu}A(\chi)\partial_{\rho}\phi\partial^{\rho}\tau+A(\chi)\left(\partial_{\mu}\tau\partial_{\nu}\phi+\partial_{\mu}\phi\partial_{\nu}\tau\right)-g_{\mu\nu}C(\chi)\partial_{\rho}\chi\partial^{\rho}\xi+C(\chi)\left(\partial_{\mu}\xi\partial_{\nu}\chi+\partial_{\mu}\chi\partial_{\nu}\xi\right)$
$\displaystyle\,+\left[g_{\mu\nu}\left\\{-\frac{1}{2}A^{\prime}(\chi)\partial_{\rho}\phi\partial^{\rho}\phi-\frac{1}{2}C^{\prime}(\chi)\partial_{\rho}\chi\partial^{\rho}\chi-V^{\prime}(\chi)\right\\}+A^{\prime}(\chi)\partial_{\mu}\phi\partial_{\nu}\phi+C^{\prime}(\chi)\partial_{\mu}\chi\partial_{\nu}\chi\right]\xi$
$\displaystyle\,-\zeta\left(1-\frac{b(\chi)}{\chi}\right)\partial_{\mu}\chi\partial_{\nu}\chi\,.$
(25)

Here we also used the constraint (17). Under the perturbation (24), the
constraint (17) has the following form,

$\displaystyle
0=\left(1-\frac{b(\chi)}{\chi}\right)g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\xi\,.$
(26)

By using the background solution (6) with (8) and (5), the constraint (26)
gives,

$\displaystyle 0=\partial_{r}\xi\,,$ (27)

whose solution is $\xi=\xi(t,\theta,\phi)$ and $\xi$ does not depend on $r$.
Therefore if we put the boundary condition that $\xi\to 0$ when $r\to\infty$,
we find $\xi$ identically vanishes,

$\displaystyle\xi=0\,.$ (28)

This is because $\chi$ is not dynamical due to the mimetic constraint (17). We
now choose a condition to fix the gauge as follows,

$\displaystyle 0=\nabla^{\mu}h_{\mu\nu}\,.$ (29)

Then Eq. (IV) with (28) has the following form,

$\displaystyle\frac{1}{\kappa^{2}}$
$\displaystyle\,\left[\frac{1}{2}\left\\{-\nabla^{2}h_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\left(g^{\rho\sigma}h_{\rho\sigma}\right)-2R^{\sigma\
\rho}_{\ \nu\ \mu}h_{\sigma\rho}+R^{\rho}_{\ \mu}h_{\rho\nu}+R^{\rho}_{\
\nu}h_{\rho\mu}\right\\}-\frac{1}{2}h_{\mu\nu}R-\frac{1}{2}g_{\mu\nu}\left\\{-h_{\rho\sigma}R^{\rho\sigma}-\nabla^{2}\left(g^{\rho\sigma}h_{\rho\sigma}\right)\right\\}\right]$
$\displaystyle=$
$\displaystyle\,h_{\mu\nu}\left[-\frac{1}{2}A(\chi)\partial_{\rho}\phi\partial^{\rho}\phi-\frac{1}{2}C(\chi)\partial_{\rho}\chi\partial^{\rho}\chi-V(\chi)\right]-g_{\mu\nu}\left[-\frac{1}{2}A(\chi)\partial^{\rho}\phi\partial^{\sigma}\phi-\frac{1}{2}C(\chi)\partial^{\rho}\chi\partial^{\sigma}\chi\right]h_{\rho\sigma}$
$\displaystyle\,-g_{\mu\nu}A(\chi)\partial_{\rho}\phi\partial^{\rho}\tau+A(\chi)\left(\partial_{\mu}\tau\partial_{\nu}\phi+\partial_{\mu}\phi\partial_{\nu}\tau\right)-\zeta\left(1-\frac{b(\chi)}{\chi}\right)\partial_{\mu}\chi\partial_{\nu}\chi\,.$
(30)

By multiplying Eq. (IV) with $g^{\mu\nu}$ and using the mimetic constraint
(17), we obtain

$\displaystyle\zeta=$
$\displaystyle\,-\frac{1}{\kappa^{2}}\left[\nabla^{2}\left(g^{\mu\nu}h_{\mu\nu}\right)-\frac{1}{2}\left(g^{\mu\nu}h_{\mu\nu}\right)R+2h_{\mu\nu}R^{\mu\nu}\right]$
$\displaystyle\,+\left(g^{\mu\nu}h_{\mu\nu}\right)\left[-\frac{1}{2}A(\chi)\partial_{\rho}\phi\partial^{\rho}\phi-\frac{1}{2}C(\chi)\partial_{\rho}\chi\partial^{\rho}\chi-V(\chi)\right]-4\left[-\frac{1}{2}A(\chi)\partial^{\rho}\phi\partial^{\sigma}\phi-\frac{1}{2}C(\chi)\partial^{\rho}\chi\partial^{\sigma}\chi\right]h_{\rho\sigma}$
$\displaystyle\,-2A(\chi)\partial_{\rho}\phi\partial^{\rho}\tau\,.$ (31)

This tells that $\zeta$ is not an independently propagating mode.

We may consider the perturbation of (3)

$\displaystyle
0=A(\chi)\nabla^{2}\tau-A(\chi)h_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi-\frac{1}{2}A(\chi)\nabla^{\rho}\left(g^{\mu\nu}h_{\mu\nu}\right)\partial_{\rho}\phi+A_{\chi}\partial_{\mu}\tau\partial^{\mu}\chi\,.$
(32)

Therefore $\tau$ behaves as a massless mode. And as long as the condition (16)
is satisfied, which tells $A$ is positive, the scalar mode $\tau$ is not a
ghost but a canonical scalar.

The remaining mode could be only massless spin-two mode corresponding to the
standard graviton. Therefore there is no mode generating ghost instability.

In order to investigate the causality, we often use the speed of sound in a
fluid. The radial and tangential speeds of sound could be defined by

$\displaystyle
v_{r}^{2}=\frac{dp_{r}}{d\rho}=\frac{p^{\prime}_{r}}{\rho^{\prime}}\,,\quad
v_{t}^{2}=\frac{dp_{t}}{d\rho}=\frac{p^{\prime}_{t}}{\rho^{\prime}}\,.$ (33)

As we have seen, however, nothing corresponds to the sound wave. The existing
waves are gravitational waves and massless scalar waves, whose propagating
speeds are different from the above sound speeds. Therefore the arguments of
the sound speed are not applicable here. In other words, the dynamics of the
fields cannot be approximated by the dynamics of the fluid, in general. Even
if we can approximate the field(s) by any fluid the EoS could not be so
simple. In the simple case, the pressure $p$ only depends on the energy
density $\rho$ but in general, $p$ depends on other parameters, and therefore
the expressions in (33) are not approved for the general cases. Even in our
model, because $p_{\mathrm{r}}\neq p_{\mathrm{t}}$, the pressure depends on
the direction and therefore the field cannot be expressed by the perfect
fluid, which has no direction dependence.

##  V Summary and discussions

Studying stable wormholes in the context of Einstein’s theory of GR with two
scalar fields is a complex and challenging endeavor. Such research would
likely involve a deep dive into theoretical physics, differential geometry,
and advanced mathematical techniques.

Stable wormholes have been a popular subject in science fiction, often
depicted as portals to other parts of the Universe or alternate dimensions.
While these portrayals are speculative, the idea of traversable wormholes has
captured the imagination of both scientists and the general public.
Investigating the theoretical possibility of stable wormholes can be seen as a
step toward understanding the Universe’s potential intricacies.

In this study, we constructed the model where the standard and traversable
wormhole geometries are included in the exact solutions. We have used the
formulation in Nojiri et al. (2021), where it has been shown how the model
reproducing general spherically symmetric and even time-dependent solutions
can be constructed. The model based on the original formulation in Nojiri et
al. (2021), however, includes ghosts.

In order to eliminate the ghosts, we have imposed the mimetic constraint on
the scalar field so that the ghost fields become non-dynamical. As a result,
although the energy conditions are broken, we have obtained models without the
instability due to the ghosts. This could tell that there are stable models
even if the energy conditions are broken.

In this study, we succeeded in constructing realistic a stable wormhole using
Einstein GR with two scaler fields. Can this procedure be applied in the frame
of $f(R)$ gravitational theory with two scalar fields or in the frame of
Gauss-Bonnet theory with two scalar fields? All these questions may be
answered elsewhere.

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# On the Provenance of Linked Data Statistics

William Waites
<EMAIL_ADDRESS>
School of Informatics, University of Edinburgh
Open Knowledge Foundation

(October 31st 2010)

###### Abstract

As the amount of linked data published on the web grows, attempts are being
made to describe and measure it. However even basic statistics about a graph,
such as its size, are difficult to express in a uniform and predictable way.
In order to be able to sensibly interpret a statistic it is necessary to know
how it was calculate. In this paper we survey the nature of the problem and
outline a strategy for addressing it.

##  1 Background and Motivation

For the past several years datasets of Linked Open Data on the web have been
catalogued and made into a diagram [6] to illustrate their proliferation and
interconnectedness. More recently some statistics about these datasets have
been calculated [3]. Amongst the published statistics are, for example the
number of triples in various graphs or unions of graphs across a particular
domain of interest. Some more sophisticated statistics are also given in
absolute terms, e.g. the absolute number of links outgoing from a particular
dataset.

In conjunction with this work, a vocabulary [1] has been developed for
describing RDF datasets. This vocabulary contains predicates for describing
common statistics, for example the number of triples or number of distinct
subjects, as well as some more generic facilities for annotating a dataset
description with other types of statistical information.

Inasmuch as these statistics help to understand some of the properties of
these data at a coarse grained level and get a rough idea of their dimensions
they are quite useful and indeed valuable contributions. However as always we
must ask what they mean. As, for example, void:triples denotes the size of the
dataset, intuitively we might think that this gives some idea of the amount of
information contained in it. But, most datasets carry a greater or lesser
amount of redundant information. It might be included to make querying easier
or to make extracts more easily readable by a human. In some sense it could be
argued that when counting triples that this redundant information should be
left out. It is also easily demonstrated that an unlimited amount of redundant
triples can easily be added to any dataset without really changing the
information content. Clearly the meaning of void:triples is somewhat of a
moving target.

The problem is exacerbated when more sophisticated statistics are calculated.
In the example of the outgoing links from one dataset to another, one might
want to normalise their count by dividing by the size of the dataset – to
arrive at a measure that might be called “out-link density”. Perhaps such a
measure would tell us something about the character of the dataset itself,
independent of its size. But because this measure is built on the basic notion
of the size of the dataset we need to do some work before arriving at a
meaningful value for it.

> Notation Conventions: Throughout the following, where RDF data is explicitly
> represented, the Notation 3 [2] syntax is used and declarations for common
> namespaces such as rdf, rdfs, owl, foaf, dct are ommitted. In addition RDF
> terms and statements are represented in a fixed-with font.

##  2 Redundancy in Graphs

A well known trivial example of adding redundancy to graphs uses blank nodes.
Blank nodes are to be read as existential variables [7].

######  Example 1.

Production Rules

If we start with a graph containing one statement,

bob a foaf:Person.

we can then add a statement with a blank node,

_:b1 a foaf:Person.

which simply says “there exists someone that is a foaf:Person”. This
information was already contained in the original graph and really adds
nothing new. And we can add as many more blank nodes, _:b2, _:b3, ..., _:bn.
What does this do to a count of the size of the graph?

Such a production rule that generated these redundant triples would called
unsafe [4] but it is quite possible to add a finite amount redundant
information with safe rules as well. The question of how much redundant
information has been added remains.

######  Example 2.

Graph Reduction

As more realistic example, elaborated with the opposite strategy of removing
redundancy, consider the following graph, and description logic fragment:

$s_{1}$ | bob a foaf:Person.
---|---
$s_{2}$ | bob foaf:knows alice.
$s_{3}$ | alice a foaf:Person.
$s_{4}$ | alice foaf:knows bob.
$s_{5}$ | foaf:knows rdfs:domain foaf:Person.
$s_{6}$ | foaf:knows rdfs:range foaf:Person.

If this is accompanied by the RDF semantic rules [7] (Section 4) concerning
domains and ranges, namely,

$r_{1}$ | {?s ?p ?o. ?p rdfs:domain ?A}
---|---
|  => {?s a ?A }.
$r_{2}$ | {?s ?p ?o. ?p rdfs:range ?B}
|  => {?o a ?B }.

we can immediately see that, under the given rules, statements $s_{1}$ and
$s_{3}$ are redundant as they can be derived from the statements involving the
predicate foaf:knows and knowledge about its domain and range.

We need to make some distinction here between the graph under consideration,
${s_{1},...,s_{4}}$ and the extra information we have drawn upon,
${s_{5},s_{6}}$. In general the latter will come from a source external to the
former, being referenced as a vocabulary. It can also easily be seen that the
description logic fragment could just as well be expressed as a rule itself,

$r_{3}$ | {?a foaf:knows ?b}
---|---
|  => {?a a foaf:Person.
|  ?b a foaf:Person.}

We can further see that if we have $r_{3}$, we don’t need $r_{1}$ or $r_{2}$.

So far we have shown that for,

$G$ | $=\\{s_{1},...,s_{4}\\}\cup\\{s_{5},s_{6}\\}$
---|---
${\mathcal{R}}$ | $=\\{r_{1},r_{2}\\}$
$G^{\prime}$ | $=\\{s_{2},s_{4}\\}$
${\mathcal{R}}^{\prime}$ | $=\\{r_{3}\\}$

the $G$ and $G^{\prime}$ under $\mathcal{R}$ and $\mathcal{R}^{\prime}$
respectively are in some sense equivalent, though it remains to state
explicitly what is meant by that. By inspection we can see that $G^{\prime}$
is half the size of $G$, in terms of number of statements, so we might say
that $50\%$ of $G$ was redundant. It is not immediately clear if $\mathcal{R}$
and ${\mathcal{R}}^{\prime}$ are the same or different sizes.

##  3 Rules and Redundancy

The examples above rely on rules or what are often called entailment regimes.
Some common ones are defined in a placeholder vocabulary by the W3C [8]. Rules
may be applied in the usual way, as in the first example, to produce
statements and this is known as calculating the closure of the graph.

######  Definition 1.

The closure of a graph, $G$ with respect to a set of rules, ${\mathcal{R}}$ is
the set of all statements produced by applying ${\mathcal{R}}$ to $G$ until
exhaustion, in other words adding all statements that it is possible to infer
given the data and the rules. We will denote this operation as
$G^{{\mathcal{R}}+}$.

Calculating $G^{{\mathcal{R}}+}$ is computationally expensive but tractable
with safe rules. Note that the closure with respect to an empty ruleset is
just the graph itself, or $G=G^{\emptyset+}$.

######  Definition 2.

The cardinality of a graph, $G$ is simply the number of triples it contains
and is denoted $|G|$.

This is enough to provide a stable notion of the size of the graph by adding
in a predictable proportion of redundancy by specifying the entailment regime.
Comparing the cardinality of two graphs side-by side can thus be done in a
meaningful way.

##  4 The Minimisation Problem

What might be better than considering graphs which have had redundancy added,
however predictable it might be, is to consider graphs with all possible
redundancy eliminated. This question was first considered in [10] where Meier
proposed applying rules negatively.

######  Algorithm 1.

Meier’s Algorithm111 Meier didn’t explicitly state the use of backwards
chaining but simply said to check if a given triple was possible to infer from
the remaining triples under the given rules. The formulation given here comes
from an actual implementation by the author of the present paper [13].

    def reduce(graph, rules):
        for triple in graph:
            graph.remove(triple)
            if not backchain(graph, rules,
                             goal=triple):
                graph.add(triple)
        return graph

The problem was further considered by Polleres et al. in [11] and given the
name MINI-RDF. The formulation builds on Meier’s work and asks how difficult
it is to find an irreducible graph given a set of rules and a set of
constraints222 They also considered the problem of rule reduction in addition
to graph reduction, the general case of the simplifications to the rules given
in the second example above. . Constraints are simply rules that specify that
a certain amount of redundancy must be left in the graph when a reduction has
been completed. It turns out that the problem is not tractable in general.

Whilst Meier’s algorithm can be completed in polynomial time, it will only
find an irreducible graph that entails the same closure of the original. It
was proven in [11] that finding the smallest irreducible graph is in general
also intractable.

######  Definition 3.

A minimisation of a graph, $G$ with respect to a set of rules, $\mathcal{R}$
is the smallest possible graph that has the same closure with respect to
$\mathcal{R}$ as $G$ and is written, $G^{\mathcal{R}-}$. $G^{\mathcal{R}-}$ is
a solution to $\mathrm{MINI\mathchar 45\relax RDF}(G,\mathcal{R},\emptyset)$.

######  Example 3.

It can be easily seen that $G^{R-}$ is not, in general, unique. For example,
consider the following graph, $G=\\{s_{1},s_{2}\\}$ and rule,
${\mathcal{R}}=\\{r_{1}\\}$,

$s_{1}$ | a links_to b.
---|---
$s_{2}$ | b linked_from a.
$r_{1}$ | {?x links_to ?y}
|  <=> {?y linked_from ?x}

either of $s_{1}$ or $s_{2}$ could be deleted from $G$ to obtain
$G^{\mathcal{R}-}$.

Since finding $G^{\mathcal{R}}$ is in general intractable, a potentially
fruitful avenue of future research is to consider the circumstances under
which it can be solved in polynomial time. Obviously the trivial case,
$G^{\emptyset-}$ is solvable. For some kinds of rules, such as those in the
second example above, the minimisation will be unique. For a broader set of
rules, such as those with loops as in the third example above,
$|G^{\mathcal{R}-}|$ will be unique. This last, the set of cardinality-
preserving rules for which calculating the minimisation of a graph is
tractable, is the broadest set of interest for the present purposes – rules in
this class are practical to apply to remove as much redundancy as possible
from the graph.

##  5 Redundancy Revisited

We are now in a position to make some formal definitions of what we mean by
redundancy in graphs.

######  Definition 4.

The redundancy contained in a graph, $G$ with respect to a set of rules,
${\mathcal{R}}$ is given by $1-{|G^{{\mathcal{R}}-}}|/{|G|}$.

The foregoing considerations give rise to four fundamental statistics about a
graph, given a set of rules,

  * •

The cardinality of the graph as published.

  * •

The cardinality of the closure of the graph under a given set of rules.

  * •

The cardinality of a minimal graph under a given set of rules.

  * •

The redundancy of the graph under a given set of rules.

From these it is possible to build up more elaborate statistics in a
predictable way. To take the “out-link density” example from the introduction,
this might be expressed as,

$D^{+}_{out}(G)=\frac{|G^{{\mathcal{R}}+}_{out}|}{|G^{{\mathcal{R}}+}|}$

or alternatively,

$D^{-}_{out}(G)=\frac{|G^{{\mathcal{R}}-}_{out}|}{|G^{{\mathcal{R}}-}|}$

Where the numerator has been constructed by selecting triples whose objects
are resources in a different graph from the minimised graph. If the rules are
as strong as possible, such a statistic might tell us something charactistic
of the graph, if it is closer to $0$ the graph contains mostly internal
information as might be the case with large datasets such as DBpedia or
OpenCyc. A void:Linkset on the other hand might have a characteristic out-link
density closer to $1$ as most of its statements express the relationships
between other datasets.

##  6 Vocabulary Considerations

The fact that the three of the fundamental statistics depend on the rules used
means that in order to express them unambiguously we need also to mention the
rules or entailment regimes. This is not a large burden but does mean that we
need a vocabulary for it. Such a vocabulary would need to have predicates for
including both Horn rules and description logics. As support for the Rule
Interchange Format [5] becomes more common it will be necessary to include
rules expressed in this language as well.

The placeholder vocabulary for entailment regimes [8] is a good starting
point. The URIs defined there are useful as recognisable unique identifiers
but as yet have no formal descriptions beyond pointers to the human readable
documentation – there is no automated way to discover which rules each regime
implies.

The [9] vocabulary could be adapted for this but it relies heavily on
modelling SPARQL CONSTRUCT queries. While it has been shown [12] that there is
a mapping from these types of queries to FOPL, most rulesets aren’t written
this way and it doesn’t make so much sense to map rules from their native
representation to this vocabulary simply in order to indicate their use.

We therefore propose a lightweight vocabulary [14] for Graph Normalisation,
gn, as exemplified below333 Recent work on the voiD vocabulary tends to
deprecate the void:statItem predicate and the use of the SCOVO vocabulary for
expressing statistics. In our view this mechanism should be retained or
replaced with something similar to support the expression of statistical
provenance. ,

######  Example 4.

Graph Normalisation Vocabulary

    eg:dataset a void:Dataset ;
      void:statItem [
        scovo:dimension eg:redundancy ;
          rdf:value 0.3 ;
          gn:normalisation [
            a gn:MiniRDF ;
              gn:rules [
                a gn:RuleSet ;
                gn:n3 <.../rdfs-rules.n3> ;
                gn:dlogic owl:, foaf: ;
                gn:rif <.../rif-rules.rif>
              ]
          ]
      ].

Additionally, gn:constraints is defined for completeness to support
specification of the MINI-RDF problem though in practice this would probably
never be used.

In this way, to check or recreate this statistic one might procede from such a
description as follows,

######  Example 5.

Redundancy Calculation

  1. 1.

Fetch the dataset in question, $G$.

  2. 2.

Fetch Horn (N3, RIF) rules.

  3. 3.

Fetch description logics, making sure to follow owl:imports, collectively $D$.

  4. 4.

Transform the description logics into their equivalent Horn rules.

  5. 5.

Construct $\mathcal{R}$ as the set all the rules fetched.

  6. 6.

Run Meier’s algorithm using $G\cup D$ as the graph for the backchaining step.

  7. 7.

Compute $1-|G^{\mathcal{R}-}|/|G|$.

This example of course assumes that the all rules used are tractable for
$G^{\mathcal{R}-}$ and cardinality preserving.

It should be noted that the current practice with respect to void:triples is
simply the above with an empty ruleset. In this way some amount of backwards
compatibility with current practice is maintained.

##  7 Optimising with Graph Diffs

If finding a $G^{{\mathcal{R}}-}$ is tractable, it is still a computationally
expensive operation. If provenance information is kept for datasets such that
given the previous version and the provenance metadata it is possible to
reconstruct the current version there is a significant optimisation to be had,
especially for large datasets that experience incremental change.

Starting with a minimisation of the previous version,
$G^{{\mathcal{R}}-}_{i}$, and a pair of graphs $I$ and $D$ representing
triples to be inserted or deleted444 Assuming blank nodes are handled via some
sort of skolemisation mechanism.  such that $G_{i+1}=G_{i}-D+I$, we can
construct $G^{{\mathcal{R}}-}_{i+1}$ by first calculating,
$G^{{\mathcal{R}}-}_{i}-D$ since if they aren’t in $G_{i+1}$ they won’t be in
its minimisation. This intermediate graph is a possibly non-minimal subgraph
of $G_{i+1}$. Because the order of the triples in minimise() doesn’t matter we
can now run it only testing the triples in $I$.

##  8 Conclusion

We have reviewed the truism that in order to be able to sensibly interpret
statistics one must know how they are calculated. Looking at how this applies
to descriptions of RDF graphs we have seen that apparently simple statistics
can be calculated in a number of ways. Thus the importance of provenance of
the statistics has been highlighted. A proposal for how this provenance
information might be expressed was put forward and some interesting areas for
further theoretical research were noted.

## References

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