# Wormholes inside stars and black holes

Shin’ichi Nojiri1,2<EMAIL_ADDRESS>S.D. Odintsov3,4
<EMAIL_ADDRESS>Vladimir Folomeev5,6<EMAIL_ADDRESS>1) Department of
Physics, Nagoya University, Nagoya 464-8602, Japan
2) Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan
3) Institute of Space Sciences (ICE, CSIC) C. Can Magrans s/n, 08193
Barcelona, Spain
4) ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain
5) Institute of Nuclear Physics, Almaty 050032, Kazakhstan
6) Academician J. Jeenbaev Institute of Physics of the NAS of the Kyrgyz
Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan

###### Abstract

We construct models of two exotic objects: (i) a wormhole whose throat is
hidden by a stellar object like a neutron star; and (ii) a wormhole inside a
black hole. We work within Einstein’s gravity coupled to two scalar fields
with a specific choice of the scalar field Lagrangian. In general, the model
contains ghosts, but they are eliminated using the constraints given by the
Lagrange multiplier fields. The constraints are a generalization of the
mimetic constraint, where non-dynamical dark matter effectively appears. As a
result, in our model, instead of the non-dynamical dark matter, non-dynamical
exotic matter like a phantom effectively arises. For the mixed wormhole-plus-
star system, we find the corresponding mass-radius relations and show that it
is possible to get characteristics comparable to those of ordinary neutron
stars. For the wormhole inside the black hole, we find an extremal limit where
the radius of the throat coincides with the radius of the event horizon and
demonstrate that the Hawking temperature vanishes in this limit.

##  I Introduction

Wormholes are exotic compact objects possessing nontrivial spacetime topology
and connecting either different regions of the Universe or separate universes
[1, 2] (see also Refs. [3, 4, 5, 6, 7, 8, 9] and Ref. [10] for a general
overview of Lorentzian wormholes). In the framework of Einstein’s general
relativity, the existence of a static traversable wormhole requires exotic
matter which violates the null energy condition (at least in the neighborhood
of the wormhole throat [11]) and therefore all the energy conditions [12, 13,
14]. There are several attempts to construct the traversable wormhole by
modifying Einstein’s gravity, as in the Brans-Dicke theory [15, 16, 17] and
other theories [18].

In this paper, we consider more exotic objects, that is, a wormhole whose
throat is hidden by a stellar object like a neutron star and a wormhole inside
a black hole. The wormhole inside a stellar object has been studied in Refs.
[19, 20, 21, 22, 23, 24, 25, 26] numerically within Einstein’s gravity. Such
mixed neutron-star-plus-wormhole configurations containing a ghost scalar
field (which ensures the presence of a nontrivial spacetime topology) and
ordinary neutron star matter possess properties both of wormholes and of usual
stars: their masses and sizes are comparable to those of typical ordinary
neutron stars. However, the presence of a ghost field results in distressing
consequences: both the mixed configurations and pure wormholes are dynamically
unstable objects [27, 28, 29, 30, 21]. Such instability is caused essentially
by the presence in the system of ghost fields. This motivates one to study the
possibilities of providing a nontrivial spacetime topology without using ghost
fields.

The aforementioned mixed neutron-star-plus-wormhole configurations have been
constructed by solving self-consistently the Einstein equations and equations
for a scalar field and neutron matter. In doing so, as in the case of ordinary
neutron stars, some characteristic mass-radius relations are obtained, whose
form strongly depends on a specific choice of an equation of state of neutron
matter and the type of the scalar field. In the present paper, we will use
another approach when the gravitational field of a system is specified by
giving some ad hoc distribution of an energy density of neutron matter
threading the wormhole, as it is also done, for example, in Refs. [31, 32,
33]. In this case, the mass-radius relations will already differ substantially
from the dependencies typical of ordinary neutron stars or the mixed neutron-
star-plus-wormhole configurations mentioned above. In particular, for the
neutron-matter energy-density profile used in the present paper, the mass of
pure stars (i.e., configurations without a wormhole) turns out to be
proportional to the third power of the radius, whereas the ADM mass of the
wormhole whose throat is hidden by a stellar object under consideration is
proportional to the radius of the stellar object, as in the case of the
Schwarzschild black hole. In turn, the properties of the wormhole inside the
black hole are very similar to those of the Reissner-Nordström black hole,
although there is only one event horizon. For example, there appears an
extremal limit where the radius of the throat coincides with the radius of the
event horizon. In this limit, we find that the Hawking temperature vanishes.

The key idea for the construction of the exotic objects in the present paper
is that we employ the formulation by using two scalar fields suggested in Ref.
[34], where it has been shown that an arbitrarily given spherically symmetric
and static/time-dependent geometry can be realized in the framework of
Einstein’s gravity coupled with two scalar fields. However, in the model of
Ref. [34], there appear ghosts, and this means that the model is inconsistent.
In classical theories, the kinetic energy of the ghost is unbounded below. In
the framework of quantum theories, on the other hand, the ghosts generate
negative norm states, which break the Copenhagen interpretation of the quantum
theories [35]. The ghosts can be, however, eliminated by using constraints
given by the Lagrange multiplier fields, as shown in Refs. [36, 37, 38, 39].
These constraints can be regarded as a generalization of the mimetic
constraint of Ref. [40], where non-dynamical dark matter effectively appears.
In our model, non-dynamical exotic matter like a phantom effectively appears
as well.

The paper is organized as follows. In Sec. II, we review the formulation of
Einstein’s gravity coupled with two scalar fields and give the explicit forms
of the Einstein equations and the field equations for the two scalar fields.
Furthermore, we explain how the ghosts can be eliminated and how arbitrarily
given spherically symmetric and static/time-dependent geometries are realized.
In Sec. III, by using the formulation of Sec. II, we construct a model
realizing a wormhole where a stellar object like a neutron star could hide the
throat. Also, here we discuss the energy conditions and compare the mixed
object obtained with ordinary neutron stars. In Sec. IV, we consider a
wormhole inside the event horizon of a black hole. The wormhole could connect
two universes by the throat, but if the black hole in our universe is really a
black hole, the object in another universe connected by the throat can be
regarded as a white hole. Finally, Sec. V is devoted to summary and
discussion.

##  II Einstein’s gravity coupled to two scalars

The arbitrarily given spherically symmetric spacetimes can be realized in the
framework of Einstein’s gravity coupled with two scalar fields even if the
spacetime is dynamical or time-dependent. In the model of Ref. [34], there
appear ghosts, which make the model inconsistent because the kinetic energy of
the ghost is unbounded below in the framework of classical mechanics and the
ghosts generate negative norm states, which break the Copenhagen
interpretation of quantum theories [35]. In the subsequent works [36, 37, 38,
39], it has been shown that the ghosts can be eliminated by constraints given
by the Lagrange multiplier fields. These constraints are similar to the
mimetic constraint of Ref. [40], where non-dynamical dark matter effectively
appears.

The action in the model of Ref. [34] is given by that of four-dimensional
Einstein’s gravity coupled with two scalar fields $\phi$ and $\chi$,

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

Here $A(\phi,\chi)$, $B(\phi,\chi)$, and $C(\phi,\chi)$ are arbitrary
functions, $V(\phi,\chi)$ is the scalar-field potential, and
$\mathcal{L}_{\mathrm{matter}}$ is the matter Lagrangian density. The
gravitational coupling constant $\kappa$ is defined by using Newton’s
gravitational constant $G$, $\kappa^{2}=8\pi G$, although we mainly use
geometrized units $c=G=1$ throughout the paper.

By the variation of the action (II) with respect to the metric $g_{\mu\nu}$,
we obtain the Einstein equations,

$\displaystyle 0=$
$\displaystyle\,\frac{1}{2\kappa^{2}}\left(-R_{\mu\nu}+\frac{1}{2}g_{\mu\nu}R\right)$
$\displaystyle\,+\frac{1}{2}g_{\mu\nu}\left[-\frac{1}{2}A(\phi,\chi)\partial_{\rho}\phi\partial^{\rho}\phi-B(\phi,\chi)\partial_{\rho}\phi\partial^{\rho}\chi-\frac{1}{2}C(\phi,\chi)\partial_{\rho}\chi\partial^{\rho}\chi-V(\phi,\chi)\right]$
$\displaystyle\,+\frac{1}{2}\left[A(\phi,\chi)\partial_{\mu}\phi\partial_{\nu}\phi+B(\phi,\chi)\left(\partial_{\mu}\phi\partial_{\nu}\chi+\partial_{\nu}\phi\partial_{\mu}\chi\right)+C(\phi,\chi)\partial_{\mu}\chi\partial_{\nu}\chi\right]+\frac{1}{2}T_{\mathrm{matter}\,\mu\nu}\,,$
(2)

where the Greek indices run over $\mu,\nu,\ldots=0,1,2,3$ and
$T_{\mathrm{matter}\,\mu\nu}$ is the energy-momentum tensor of matter. On the
other hand, the variations of the action (II) with respect to the fields
$\phi$ and $\chi$ give the following equations:

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

where $A_{\phi}=\partial A(\phi,\chi)/\partial\phi$, etc. These field
equations are nothing but the Bianchi identities. In Appendix A, we show that
the field equations (II) can be surely obtained from the Einstein equations
(II).

###  II.1 Elimination of the ghosts

The metric of a general spherically symmetric and time-dependent spacetime is
given by

$\displaystyle
ds^{2}=-\mathrm{e}^{2\nu(t,r)}dt^{2}+\mathrm{e}^{2\lambda(t,r)}dr^{2}+r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\,.$ (4)

The argument that the metric in (4) is a general one is given in Ref. [34]
(see also the textbook [41]). We may also assume the ansatz

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

which does not give any loss of generality. The detailed arguments have been
given in the previous papers [34, 36, 37, 38, 39].

In Sec. III, we discuss how we can construct models realizing an arbitrarily
given spherically symmetric geometry expressed by the metric (4). In the
realizations, however, the functions $A$ and/or $C$ are often negative, and
therefore $\phi$ and/or $\chi$ become ghosts. In order to eliminate the
ghosts, we impose constraints by introducing the Lagrange multiplier fields
$\lambda_{\phi}$ and $\lambda_{\chi}$ and modifying the action (II)
$S_{\mathrm{GR}\phi\chi}\to S_{\mathrm{GR}\phi\chi}+S_{\lambda}$, where the
additional term $S_{\lambda}$ is given by

$\displaystyle S_{\lambda}=\int
d^{4}x\sqrt{-g}\left[\lambda_{\phi}\left(\mathrm{e}^{-2\nu(t=\phi,r=\chi)}\partial_{\mu}\phi\partial^{\mu}\phi+1\right)+\lambda_{\chi}\left(\mathrm{e}^{-2\lambda(t=\phi,r=\chi)}\partial_{\mu}\chi\partial^{\mu}\chi-1\right)\right]\,.$

By varying $S_{\lambda}$ with respect to $\lambda_{\phi}$ and
$\lambda_{\chi}$, we obtain the following constraints:

$\displaystyle
0=\mathrm{e}^{-2\nu(t=\phi,r=\chi)}\partial_{\mu}\phi\partial^{\mu}\phi+1\,,\quad
0=\mathrm{e}^{-2\lambda(t=\phi,r=\chi)}\partial_{\mu}\chi\partial^{\mu}\chi-1\,,$
(6)

which is consistent with the assumption (5). The constraints (6) are similar
to the mimetic constraint of Ref. [40], where non-dynamical dark matter
appears.

The constraints from Eq. (6) make the scalar fields $\phi$ and $\chi$ non-
dynamical, and the fluctuations of $\phi$ and $\chi$ around the background (5)
do not propagate. We now write the fluctuations as follows:

$\displaystyle\phi=t+\delta\phi\,,\quad\chi=r+\delta\chi\,,$

and, by using Eq. (6), we find

$\displaystyle\partial_{t}\left(\mathrm{e}^{-2\nu(t,r)}\delta\phi\right)=\partial_{r}\left(\mathrm{e}^{-2\lambda(t,r)}\delta\chi\right)=0\,.$

This equation tells us that, by imposing the initial condition $\delta\phi=0$
and by imposing the boundary condition $\delta\chi\to 0$ when $r\to\infty$, we
can find that both of $\delta\phi$ and $\delta\chi$ vanish in the whole
spacetime, $\delta\phi=0$ and $\delta\chi=0$. This tells us that both $\phi$
and $\chi$ are non-dynamical or frozen degrees of freedom.

As partially or completely shown in Refs. [36, 37, 38, 39], even in the model
given by the modified action $S_{\mathrm{GR}\phi\chi}+S_{\lambda}$,
$\lambda_{\phi}=\lambda_{\chi}=0$ consistently appear as a solution and
therefore any solution of Eqs. (II) and (II) which are based on the original
action (II) is a solution even for the modified model with the action
$S_{\mathrm{GR}\phi\chi}+S_{\lambda}$.

###  II.2 Reconstruction of models which realize any given spherically
symmetric and static/time-dependent spacetime

We now try to construct a model which has a solution realizing the metric
functions $\mathrm{e}^{2\nu(t,r)}$ and $\mathrm{e}^{2\lambda(t,r)}$ given in
Eq. (4).

The $(t,t)$, $(r,r)$, $(\vartheta,\vartheta)$, and $(t,r)$ components of Eqs.
(II) have the following form:

$\displaystyle\frac{\mathrm{e}^{-2\lambda+2\nu}}{\kappa^{2}}\left(\frac{2\lambda^{\prime}}{r}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)=$
$\displaystyle\,-\mathrm{e}^{2\nu}\left(-\frac{A}{2}\mathrm{e}^{-2\nu}-\frac{C}{2}\mathrm{e}^{-2\lambda}-V\right)+\mathrm{e}^{2\nu}\rho\,,$
$\displaystyle\frac{1}{\kappa^{2}}\left(\frac{2\nu^{\prime}}{r}-\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)=$
$\displaystyle\,\mathrm{e}^{2\lambda}\left(\frac{A}{2}\mathrm{e}^{-2\nu}+\frac{C}{2}\mathrm{e}^{-2\lambda}-V\right)+\mathrm{e}^{2\lambda}p\,,$
$\displaystyle\frac{1}{\kappa^{2}}\left\\{-r^{2}\mathrm{e}^{-2\nu}\left[\ddot{\lambda}\right.\right.$
$\displaystyle+\left.\left.\left(\dot{\lambda}-\dot{\nu}\right)\dot{\lambda}\right]+\mathrm{e}^{-2\lambda}\left(r\left(\nu^{\prime}-\lambda^{\prime}\right)+r^{2}\nu^{\prime\prime}+r^{2}\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}\right)\right\\}$
$\displaystyle=$
$\displaystyle\,r^{2}\left(\frac{A}{2}\mathrm{e}^{-2\nu}-\frac{C}{2}\mathrm{e}^{-2\lambda}-V\right)+r^{2}p\,,$
$\displaystyle\frac{2\dot{\lambda}}{\kappa^{2}r}=$ $\displaystyle\,B\,,$ (7)

where the dot and prime denote differentiation with respect to the time
coordinate $t$ and radial coordinate $r$, respectively. In this paper, we
assume that the matter is a perfect fluid with $\rho$ and $p$ being the energy
density and the pressure of matter, defined by

$\displaystyle T_{\mathrm{matter}\,tt}=-g_{tt}\rho\ ,\quad
T_{\mathrm{matter}\,ij}=p\,g_{ij}\,,$

where $i,j=r,\vartheta,\varphi$. The equations (II.2) can be algebraically
solved with respect to $A$, $B$, $C$, and $V$ as follows:

$\displaystyle A=$
$\displaystyle\frac{\mathrm{e}^{2\nu}}{\kappa^{2}}\left\\{-\mathrm{e}^{-2\nu}\left[\ddot{\lambda}+\left(\dot{\lambda}-\dot{\nu}\right)\dot{\lambda}\right]+\mathrm{e}^{-2\lambda}\left[\frac{\nu^{\prime}+\lambda^{\prime}}{r}+\nu^{\prime\prime}+\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right]\right\\}-\mathrm{e}^{2\nu}\left(\rho+p\right)\,,$
$\displaystyle B=$ $\displaystyle\,\frac{2\dot{\lambda}}{\kappa^{2}r}\,,$
$\displaystyle C=$
$\displaystyle\,\frac{\mathrm{e}^{2\lambda}}{\kappa^{2}}\left\\{\mathrm{e}^{-2\nu}\left[\ddot{\lambda}+\left(\dot{\lambda}-\dot{\nu}\right)\dot{\lambda}\right]-\mathrm{e}^{-2\lambda}\left[-\frac{\nu^{\prime}+\lambda^{\prime}}{r}+\nu^{\prime\prime}+\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right]\right\\}\,,$
$\displaystyle V=$
$\displaystyle\frac{\mathrm{e}^{-2\lambda}}{\kappa^{2}}\left(\frac{\lambda^{\prime}-\nu^{\prime}}{r}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)-\frac{1}{2}\left(\rho-p\right)\,.$
(8)

Then we can obtain a model that realizes the spacetime defined by the metric
(4) by finding $(t,r)$-dependence of $\rho$ and $p$ and by replacing $(t,r)$
in Eq. (II.2) with $(\phi,\chi)$.

##  III Wormhole inside a stellar object

We may consider a wormhole whose throat is hidden by a stellar object like a
neutron star. This can be done by choosing the radius of the neutron star to
be larger than the radius of the throat. Similar objects have been numerically
studied in Refs. [19, 20, 21, 22, 23, 24, 25, 26], where ghost scalar fields
are used to realize the objects. In the present paper, we construct the object
analytically, and the ghost is eliminated by using the constraint of Sec.
II.1.

To this end, consider a general spherically symmetric and static spacetime
whose metric can be obtained from Eq. (4) in the form

$\displaystyle
ds^{2}=-\mathrm{e}^{2\nu(r)}dt^{2}+\mathrm{e}^{2\lambda(r)}dr^{2}+r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\,.$ (9)

In the case of a wormhole whose throat has a radius $r_{0}$, near the throat
$r\sim r_{0}$, the metric functions $\mathrm{e}^{2\nu}$ and
$\mathrm{e}^{2\lambda}$ behave as

$\displaystyle\mathrm{e}^{2\nu}\sim\mathrm{e}^{2\nu_{0}}+\nu_{1}\left(r-r_{0}\right)\,,\quad\mathrm{e}^{2\lambda}\sim\frac{r_{0}}{r-r_{0}}\mathrm{e}^{2\lambda_{0}}\,,$
(10)

with constants $\nu_{0}$, $\nu_{1}$, and $\lambda_{0}$. If we redefine the
radial coordinate $r$ by using

$\displaystyle l=2\sqrt{r_{0}\left(r-r_{0}\right)}\,,$ (11)

near the throat, the metric (9) takes the form

$\displaystyle
ds^{2}\sim-\left(\mathrm{e}^{2\nu_{0}}+\frac{\nu_{1}l^{2}}{4r_{0}}\right)dt^{2}+\mathrm{e}^{2\lambda_{0}}dl^{2}+{r_{0}}^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\,.$ (12)

Notice that in Eq. (11) the radial coordinate $l$ is defined to be positive,
but in the expression (12), we may analytically continue $l$ into the region
where $l$ is negative. Let us assume that the region where $l$ is positive
corresponds to our universe, and then the region where $l$ is negative
corresponds to another universe connected with our universe by the wormhole.

We now consider a stellar-like object similar to a neutron star supported by a
perfect fluid. The conservation law of the perfect fluid with the energy
density $\rho$ and the pressure $p$ is given by

$0=\nabla^{\mu}T_{\mu r}=\nu^{\prime}\left(\rho+p\right)+p^{\prime}\,,$ (13)

while other components of the conservation law are trivially satisfied. If the
equation of state (EoS) $\rho=\rho(p)$ is given, Eq. (13) can be integrated as

$\nu=-\int^{p(r)}\frac{dp}{\rho(p)+p}\,.$ (14)

The EoS of compact stars, like neutron stars, can be approximated by a
polytropic EoS that gives a more or less realistic description of neutron
matter at high densities. Here we consider two simple EoSs:

  1. 1.

The rest-mass-polytrope,

$p_{1}=K_{1}\rho_{\mathrm{b}}^{1+\frac{1}{n_{1}}}\,.$ (15)

Here $\rho_{\mathrm{b}}=n_{\mathrm{b}}m_{\mathrm{b}}$ is the rest-mass density
of the neutron fluid, where $n_{\mathrm{b}}$ is the baryon number density and
$m_{\mathrm{b}}$ is the baryon mass. The values of the constants $K_{1}$ and
$n_{1}$ depend on the properties of the fluid under consideration. Such an EOS
was used, for instance, in Refs. [42, 43, 44, 45] in modeling general
relativistic isotropic and anisotropic fluid spheres. In what follows, we
refer to this choice as EoS1. It is known that for the neutron stars, $n_{1}$
can take values in the range $0.5\leq n_{1}\leq 3$ [45].

  2. 2.

The parametric relation between the pressure and energy density,

$\rho=\rho_{\mathrm{b}}+n_{2}p_{2}\,,\quad
p_{2}=K_{2}\rho_{\mathrm{b}}^{1+\frac{1}{n_{2}}}\,,$ (16)

with the constant
$K_{2}=kc^{2}(n_{\mathrm{b}}^{(\mathrm{ch})}m_{\mathrm{b}})^{-1/n_{2}}$, where
$c$ is the velocity of light, $n_{\mathrm{b}}^{(\mathrm{ch})}$ is a
characteristic value of $n_{\mathrm{b}}$, and $k$ and $n_{2}$ are parameters
whose values depend on the properties of the neutron matter (see Sec. III.2).
In what follows we refer to this choice as EoS2.

Upon substituting the EoSs (15) and (16) into the conservation law (14), one
can find the following expressions for the metric function $\nu$:

$\displaystyle\text{for the
EoS1}:\quad\nu=\nu_{c}-(n_{1}+1)\ln{\left(1+K_{1}\rho_{\mathrm{b}}^{1/n_{1}}\right)};$
(17) $\displaystyle\text{for the
EoS2}:\quad\nu=\nu_{c}-\ln{\left[1+K_{2}(n_{2}+1)\rho_{\mathrm{b}}^{1/n_{2}}\right]},$
(18)

where $\nu_{c}$ is an integration constant.

Given one of the aforementioned EoSs, let us consider an example case where we
assume the following profiles for $\rho=\rho(r)$ and $\lambda=\lambda(r)$:

$\displaystyle\rho=\left\\{\begin{array}[]{cc}\rho_{c}\left[1-\frac{\left(r-r_{0}\right)^{2}}{{\left(R_{s}-r_{0}\right)}^{2}}\right]&\
\mbox{when}\ r_{0}\leq r\leq R_{s}\\\ 0&\ \mbox{when}\
r>R_{s}\end{array}\right.\,,\quad\mathrm{e}^{-2\lambda}=1-\frac{r_{0}}{r}\,,$
(21)

where $r_{0}$ is a constant corresponding to the radius of the throat,
$\rho_{c}$ is the value of the energy density at the throat $r=r_{0}$, and
$R_{s}$ is the radius of the surface of the neutron fluid.

For both polytropes (15) and (16), consider the case $n_{1}=n_{2}=1$. Then, by
substituting the expression (21) into (17) and (18), we find in the region
$r_{0}\leq r\leq R_{s}$:

$\displaystyle\text{for the
EoS1}:\quad\mathrm{e}^{2\nu}=\frac{\mathrm{e}^{2\nu_{c}}}{\left[1+K_{1}\rho_{c}\left(1-\frac{\left(r-r_{0}\right)^{2}}{{\left(R_{s}-r_{0}\right)}^{2}}\right)\right]^{4}}\,,\quad\left(\mathrm{e}^{2\nu}\right)^{\prime}=\frac{8K_{1}\rho_{c}\mathrm{e}^{2\nu_{c}}\left(r-r_{0}\right)\left(R_{s}-r_{0}\right)^{-2}}{\left[1+K_{1}\rho_{c}\left(1-\frac{\left(r-r_{0}\right)^{2}}{\left(R_{s}-r_{0}\right)^{2}}\right)\right]^{5}}\,;$
(22) $\displaystyle\text{for the
EoS2}:\quad\mathrm{e}^{2\nu}=\frac{\mathrm{e}^{2\nu_{c}}}{\left[1+2K_{2}\rho_{c}\left(1-\frac{\left(r-r_{0}\right)^{2}}{{\left(R_{s}-r_{0}\right)}^{2}}\right)\right]^{2}},\quad\left(\mathrm{e}^{2\nu}\right)^{\prime}=\frac{8K_{2}\rho_{c}\mathrm{e}^{2\nu_{c}}\left(r-r_{0}\right)\left(R_{s}-r_{0}\right)^{-2}}{\left[1+2K_{2}\rho_{c}\left(1-\frac{\left(r-r_{0}\right)^{2}}{\left(R_{s}-r_{0}\right)^{2}}\right)\right]^{3}}\,.$
(23)

Then at the surface $r=R_{s}$, we find for both EoSs

$\displaystyle\mathrm{e}^{2\nu\left(r=R_{s}\right)}=\mathrm{e}^{2\nu_{c}}\,,\quad\left.\left(\mathrm{e}^{2\nu}\right)^{\prime}\right|_{r=R_{s}}=\frac{8K_{1,2}\rho_{c}}{R_{s}-r_{0}}\mathrm{e}^{2\nu_{c}}\,.$

Next, as an example, we may assume that outside the fluid, i.e., when
$r>R_{s}$, the metric function

$\mathrm{e}^{2\nu}=\mathrm{e}^{-2\lambda}=1-\frac{r_{0}}{r}.$ (24)

This implies that we deal with asymptotically flat spacetime. In turn, the
continuities of $\mathrm{e}^{2\nu}$ and
$\left(\mathrm{e}^{2\nu}\right)^{\prime}$ at the surface of the fluid give

$\displaystyle\mathrm{e}^{2\nu_{c}}=1-\frac{r_{0}}{R_{s}}\,,\quad
8K_{1,2}\rho_{c}R_{s}=r_{0}\,.$ (25)

These expressions determine the size of the throat $r_{0}$ and the value of
the integration constant $\nu_{c}$ in terms of $\rho_{c}$ and $R_{s}$. Note
here that in the case of $n_{1,2}\neq 1$ the ansatz (24) results in $r_{0}=0$.

The ansatz (24) for the metric functions outside the fluid implies that the
ADM mass $M$ is given by

$\displaystyle M=\frac{r_{0}}{2}=4K_{1,2}\rho_{c}R_{s}\,.$ (26)

This yields the mass-radius relation of the configuration under consideration.
In Sec. III.2, we will compare this mass-radius relation with that of ordinary
neutron stars described by the same EoSs.

But before note that, in ordinary stars modeled by the toy profiles of the
type (21), the mass could be proportional to the third power of the radius
$M\propto{R_{s}}^{3}$ if the matter is not so compressed because the mass $M$
could be proportional to the volume $\sim\frac{4\pi}{3}{R_{s}}^{3}$. For
example, instead of (21), we may consider the following matter profile in a
compact star without a wormhole $\left(r_{0}\to 0\right)$:

$\displaystyle\rho=\left\\{\begin{array}[]{cc}\rho_{c}\left(1-\frac{r^{2}}{R_{s}^{2}}\right)&\
\mbox{when}\ r\leq R_{s}\\\ 0&\ \ \mbox{when}\ r>R_{s}\end{array}\right.\,.$
(29)

Then the mass $M_{\mathrm{\star}}$ corresponding to ordinary matter of the
compact star, which is generally different from the ADM mass, is given by

$\displaystyle
M_{\mathrm{\star}}=4\pi\int_{0}^{R_{s}}dr\,r^{2}\rho(r)=4\pi\rho_{c}\int_{0}^{R_{s}}dr\,r^{2}\left(1-\frac{r^{2}}{R_{s}^{2}}\right)=\frac{8\pi\rho_{c}R_{s}^{3}}{15}\,,$
(30)

and we really find that $M_{\mathrm{\star}}\propto{R_{s}}^{3}$. Note that this
mass-radius relation differs strongly from that typical of realistic neutron
stars (see, e.g., Refs. [43, 44, 45, 46]). The reason is that we employ the
toy profile (29) which enables us to find the analytical expression (30), in
contrast to the realistic neutron stars whose models are constructed only
numerically.

In the case of the Schwarzschild black hole, the ADM mass $M$ is proportional
to the Schwarzschild radius. If the energy density $\rho_{c}$ at the throat
has a maximum, as it takes place for ordinary neutron stars, Eq. (26) tells us
that the mass $M$ is proportional to the radius $R_{s}$ of the stellar-like
object, as in the case of the Schwarzschild black hole, although $R_{s}$ must
be larger than the Schwarzschild radius $r_{0}$.

###  III.1 Energy conditions

Consider now the null, weak, strong, and dominant energy conditions, which
state that the energy-momentum tensor $T_{\mu\nu}$ satisfies, respectively,
the inequalities

$\displaystyle T_{\mu\nu}k^{\mu}k^{\nu}\geq 0\,,\quad
T_{\mu\nu}V^{\mu}V^{\nu}\geq
0\,,\quad\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)V^{\mu}V^{\nu}\geq
0\,,$ $\displaystyle T_{\mu\nu}V^{\mu}V^{\nu}\geq 0\,,\quad\text{and}\quad
T_{\mu\nu}V^{\nu}\quad\text{is not spacelike}\,,$

for any null vector $k^{\mu}$, $g_{\mu\nu}k^{\mu}k^{\nu}=0$, and for any
timelike vector $V^{\mu}$, $g_{\mu\nu}V^{\mu}V^{\nu}<0$ (see, e.g., Ref.
[10]). For the system under consideration, we have the energy-momentum tensor
$T^{\mu}_{\
\nu}=\text{diag}\left(-\rho_{\mathrm{tot}},p_{\mathrm{tot}}^{r},p_{\mathrm{tot}}^{\vartheta},p_{\mathrm{tot}}^{\vartheta}\right)$,
where the subscript “tot” denotes the total (the scalar fields plus the fluid)
energy density and pressure. In terms of these quantities, the above
conditions yield

$\displaystyle\text{NEC}:$
$\displaystyle\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{r}\geq
0\,,\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{\vartheta}\geq 0\,,$
$\displaystyle\text{WEC}:$
$\displaystyle\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{r}\geq
0\,,\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{\vartheta}\geq
0\,,\quad\rho_{\mathrm{tot}}\geq 0\,,$ $\displaystyle\text{SEC}:$
$\displaystyle\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{r}\geq
0\,,\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{\vartheta}\geq
0\,,\quad\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{r}+2p_{\mathrm{tot}}^{\vartheta}\geq
0\,,$ $\displaystyle\text{DEC}:$ $\displaystyle\quad\rho_{\mathrm{tot}}\geq
0\,,\quad\rho_{\mathrm{tot}}\geq|p_{\mathrm{tot}}^{r}|\,,\quad\rho_{\mathrm{tot}}\geq|p_{\mathrm{tot}}^{\vartheta}|\,.$

With the ansatz for $\lambda$ in the form (21) and (24), we have the following
expressions for the principal pressures valid over the whole space (inside and
outside the fluid):

$\displaystyle\rho_{\mathrm{tot}}=$ $\displaystyle\,0,\quad
p_{\mathrm{tot}}^{r}=\,\frac{2r(r-r_{0})\nu^{\prime}-r_{0}}{\kappa^{2}r^{3}},\quad
p_{\mathrm{tot}}^{\vartheta}=\,\frac{2r^{2}(r-r_{0})\left(\nu^{\prime\prime}+\nu^{\prime
2}\right)+r(2r-r_{0})\nu^{\prime}+r_{0}}{2\kappa^{2}r^{3}}\,.$ (31)

Using these expressions, one can find in the limit $r\to r_{0}$

$\left(\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{r}\right)\to-\frac{1}{\kappa^{2}r_{0}^{2}}\,,\quad\left(\rho_{\mathrm{tot}}+p_{\mathrm{tot}}^{\vartheta}\right)\to\frac{1+r_{0}\nu^{\prime}(r_{0})}{2\kappa^{2}r_{0}^{2}}\,.$

For the choice of the metric function $\nu$ in the form (22) and (23),
$\nu^{\prime}(r_{0})=0$. Then it is evident from these expressions that all
the above energy conditions are violated in the vicinity of the throat.

###  III.2 Comparison with ordinary neutron stars

Figure 1: Left panels: the ADM mass for ordinary neutron stars and mixed
systems as a function of the radius. Right panels: the radius as a function of
the ADM mass. The vertical dashed lines correspond to the limiting mixed
system with $r_{0}\to R_{s}$.

In this subsection, we construct mass-radius relations for the system under
consideration and compare them with those of ordinary neutron stars with the
same EoSs. To do this, it is necessary to choose appropriate values of the
parameters appearing in the polytropic EoSs (15) and (16). For our purposes,
we take for the EoS1 $n_{1}=1$ and $K_{1}=100\,\text{km}^{2}$ (in units
$c=G=1$) [47]. In turn, for the EoS2, we take the following set of parameters
for the neutron fluid (in CGS units): $m_{\mathrm{b}}=1.66\times
10^{-24}\,\text{g}$, $n_{\mathrm{b}}^{(\mathrm{ch})}=0.1\,\text{fm}^{-3}$,
$k=0.1$, and $n_{2}=1$ [48].

In general, the total mass of the system under consideration is given by [5,
6, 7, 8]

$\displaystyle
M(r)=\frac{r_{0}}{2}+4\pi\int_{r_{0}}^{r}\rho_{\mathrm{tot}}(r^{\prime})r^{\prime
2}dr^{\prime}\,.$ (32)

For the choice of the metric function $\lambda$ in the form (21),
$\rho_{\mathrm{tot}}\equiv 0$ [see Eq. (31)]. Then only the first term on the
right-hand side of (32) is present, which agrees with (26).

Usually, the mass-radius curves for ordinary neutron stars are constructed by
varying the central density of the fluid $\rho_{c}$. As a result, for every
particular value of $\rho_{c}$, one has some particular values of $M$ and
$R_{s}$.

In the case of the mixed system “neutron star plus wormhole” under
consideration, we have the mass-radius relation in the form (26). In order to
compare it with the relations for ordinary neutron stars, we may proceed as
follows: we take some $\rho_{c}$ and assume that the radius of the fluid
$R_{s}$ is equal to the radius of an ordinary neutron star obtained for the
same $\rho_{c}$. The corresponding results of calculations are given in the
left panels of Fig. 1 for both EoSs.

Alternatively, we can take some $\rho_{c}$ and assume that the mass of the
mixed system is equal to the mass of an ordinary neutron star obtained for the
same $\rho_{c}$. The corresponding results of calculations are given in the
right panels of Fig. 1 for both EoSs.

According to the observational data (see, e.g., Ref. [46]), the typical values
of the mass of ordinary neutrons stars lie in the range
$M\sim(1-2)\,M_{\odot}$, while they radii are of the order of $R\sim
8-12\,\text{km}$. As seen from Fig. 1, for the mixed systems under
consideration, such characteristics are accessible both for the EoS1 (see the
top left panel) and for the EoS2 (see the bottom right panel).

##  IV Wormhole inside a black hole

First of all, we should note that the standard Reissner-Nordström spacetime
can be regarded as a wormhole because the geometry connects many universes, as
can be seen from the corresponding Penrose diagram of the spacetime.

We now consider an object that describes a wormhole inside a black hole whose
geometry differs from the Reissner-Nordström spacetime; that is, we consider a
wormhole inside the event horizon. Such a wormhole could have a timelike
throat. Then the object falling into the black hole via the event horizon goes
first through the wormhole, then through the horizon, and eventually appears
in another universe, where the black hole may behave as a white hole.

In this section, we neglect the contribution of the matter. As an example, let
us consider the ansatz

$\displaystyle\mathrm{e}^{-2\lambda}=\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}\,,\quad\mathrm{e}^{2\nu}=1-\frac{r_{\mathrm{horizon}}}{r}\,.$
(33)

Here we assume that the radius $r_{0}$ of the throat is smaller than the
Schwarzschild radius $r_{\mathrm{horizon}}$, $r_{0}<r_{\mathrm{horizon}}$. The
behavior for large $r$ tells us that the ADM mass $M$ is given by
$M=\frac{r_{\mathrm{horizon}}}{2}$. Compared with Eq. (10), here we have

$\displaystyle\mathrm{e}^{2\nu_{0}}=1-\frac{r_{\mathrm{horizon}}}{r_{0}}\,,\quad\nu_{1}=\frac{r_{\mathrm{horizon}}}{{r_{0}}^{2}}\,,\quad\mathrm{e}^{2\lambda}\sim\frac{{r_{0}}^{3}}{2r_{0}\left(r_{0}-r_{\mathrm{horizon}}\right)\left(r-r_{0}\right)}\,.$

Therefore the metric given by Eq. (33) has a wormhole throat at $r=r_{0}$,
although the region around the throat is timelike.

In the limit $r_{0}\to 0$, the metric given by Eq. (33) reduces to the one of
the Schwarzschild spacetime: $\mathrm{e}^{-2\lambda},\,\mathrm{e}^{2\nu}\to
1-\frac{r_{0}}{r}=1-\frac{2M}{r}$. In turn, in this limit, the functions $A$,
$B$, $C$, and $V$ from Eq. (II.2) vanish. In fact, we find

$\displaystyle A\to$
$\displaystyle\,\frac{\mathrm{e}^{2\nu}}{\kappa^{2}}\left[\mathrm{e}^{-2\lambda}\left(\frac{\nu^{\prime}+\lambda^{\prime}}{r}+\nu^{\prime\prime}+\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)\right]$
$\displaystyle\to$
$\displaystyle\,\frac{\mathrm{e}^{4\nu}}{\kappa^{2}}\left[\frac{-\frac{2r_{0}}{r^{3}}\left(1-\frac{r_{0}}{r}\right)-\frac{{r_{0}}^{2}}{r^{4}}}{2\left(1-\frac{r_{0}}{r}\right)^{2}}+\frac{\frac{{r_{0}}^{2}}{r^{4}}}{2\left(1-\frac{r_{0}}{r}\right)^{2}}+\frac{\frac{r_{0}}{r^{3}}}{1-\frac{r_{0}}{r}}\right]=0\,,$
$\displaystyle B=$ $\displaystyle\,0\,,$ $\displaystyle C\to$
$\displaystyle\,\frac{\mathrm{e}^{2\lambda}}{\kappa^{2}}\left[-\mathrm{e}^{-2\lambda}\left(-\frac{\nu^{\prime}+\lambda^{\prime}}{r}+\nu^{\prime\prime}+\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)\right]\to\mathrm{e}^{-4\nu}A\to
0\,,$ $\displaystyle V\to$
$\displaystyle\frac{\mathrm{e}^{-2\lambda}}{\kappa^{2}}\left(\frac{\lambda^{\prime}-\nu^{\prime}}{r}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)\to\frac{\mathrm{e}^{-2\lambda}}{\kappa^{2}}\left(-\frac{\frac{r_{0}}{r^{3}}}{1-\frac{r_{0}}{r}}+\frac{\frac{r_{0}}{r^{3}}}{1-\frac{r_{0}}{r}}\right)=0\,.$

Therefore, in the limit $r_{0}\to 0$, the scalar fields $\phi$ and $\chi$
decouple and the model reduces to the vacuum Einstein gravity without a
cosmological constant.

We now consider the orbit of a test particle to check the motion of the
particle after it penetrates the horizon. The motion can be described by the
corresponding geodesic equation obtained from the Lagrangian

$\displaystyle
L=m\sqrt{-g_{\mu\nu}\dot{x}^{\mu}(\tau)\dot{x}^{\nu}(\tau)}=m\sqrt{\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\left(\frac{dt}{d\tau}\right)^{2}-\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}\left(\frac{dr}{d\tau}\right)^{2}}\,.$
(34)

Here $\tau$ is an affine parameter which parametrizes the orbit of the test
particle. For simplicity, we do not consider here the motion in the angular
directions. The Euler-Lagrange equations derived from the Lagrangian (34) are
as follows:

$\displaystyle 0=$
$\displaystyle\,\frac{d}{d\tau}\left[\frac{m\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\frac{dt}{d\tau}}{\sqrt{\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\left(\frac{dt}{d\tau}\right)^{2}-\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}\left(\frac{dr}{d\tau}\right)^{2}}}\right]\,,$
(35) $\displaystyle 0=$
$\displaystyle\,\frac{d}{d\tau}\left[\frac{\frac{mr_{\mathrm{horizon}}\left(r-r_{\mathrm{horizon}}\right)\left(r-r_{0}\right)}{r^{3}}\frac{dr}{d\tau}}{\sqrt{\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\left(\frac{dt}{d\tau}\right)^{2}-\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}\left(\frac{dr}{d\tau}\right)^{2}}}\right]$
$\displaystyle\,-\frac{m\left[\frac{r_{\mathrm{horizon}}}{r^{2}}\left(\frac{dt}{d\tau}\right)^{2}-\left(-\frac{3r_{\mathrm{horizon}}{r_{0}}^{2}}{r^{4}}+\frac{2{r_{0}}^{2}}{r^{3}}+\frac{r_{\mathrm{horizon}}}{r^{2}}\right)\left(\frac{dr}{d\tau}\right)^{2}\right]}{2\sqrt{\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\left(\frac{dt}{d\tau}\right)^{2}-\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}\left(\frac{dr}{d\tau}\right)^{2}}}\,.$
(36)

The equation (35) yields the conserved energy of the particle $E$,

$\displaystyle\frac{m\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\frac{dt}{d\tau}}{\sqrt{\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\left(\frac{dt}{d\tau}\right)^{2}-\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}\left(\frac{dr}{d\tau}\right)^{2}}}=E\,.$
(37)

We now choose $\tau$ to be the proper time of the test particle, which is
defined by

$\displaystyle-d\tau^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)dt^{2}+\frac{\left(r-r_{\mathrm{horizon}}\right)\left(r^{2}-{r_{0}}^{2}\right)}{r^{3}}dr^{2}\,.$

This gives

$\displaystyle\left(\frac{dt}{d\tau}\right)^{2}=\frac{r^{2}-{r_{0}}^{2}}{r^{2}}\left(\frac{dr}{d\tau}\right)^{2}+\frac{r}{r-r_{\mathrm{horizon}}}\,.$
(38)

Then Eq. (37) is simplified to be

$\displaystyle
m\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\frac{dt}{d\tau}=E\,.$ (39)

Expressing $\frac{dt}{d\tau}$ from Eq. (39) and substituting it into Eq. (38),
we find

$\displaystyle\frac{r^{2}-{r_{0}}^{2}}{r^{2}}\left(\frac{dr}{d\tau}\right)^{2}=\frac{E^{2}r^{2}}{m^{2}\left(r-r_{\mathrm{horizon}}\right)^{2}}-\frac{r}{r-r_{\mathrm{horizon}}}=\frac{E^{2}r^{2}}{m^{2}\left(r-r_{\mathrm{horizon}}\right)^{2}}\left[1-\frac{m^{2}}{E^{2}}\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\right]\,,$
(40)

which is consistent with Eq. (IV). We should note that
$\left[1-\frac{m^{2}}{E^{2}}\left(1-\frac{r_{\mathrm{horizon}}}{r}\right)\right]>0$,
as long as $\frac{m^{2}}{E^{2}}<1$, as usually assumed. Therefore,
$\frac{dr}{d\tau}$ does not vanish as long as $r>r_{0}$. Thus, if
$\frac{dr}{d\tau}<0$ at a certain instant of time, we find
$\frac{dr}{d\tau}<0$ until the particle reaches the throat as $r\to r_{0}$.
Furthermore, when $r\gtrsim r_{0}$, Eq. (40) has the following form:

$\displaystyle\left(r-r_{0}\right)\left(\frac{dr}{d\tau}\right)^{2}\sim\left(\frac{2C}{3}\right)^{2}\equiv\frac{E^{2}{r_{0}}^{2}}{m^{2}\left(r_{0}-r_{\mathrm{horizon}}\right)^{2}}\left[1-\frac{m^{2}}{E^{2}}\left(1-\frac{r_{\mathrm{horizon}}}{r_{0}}\right)\right]\,,$

which gives

$\displaystyle\left(r-r_{0}\right)^{\frac{3}{2}}\sim
C\left(\tau_{0}-\tau\right)\quad\textrm{or}\quad r\sim
r_{0}+C^{\frac{2}{3}}\left(\tau_{0}-\tau\right)^{\frac{2}{3}}\,.$

Therefore, the test particle reaches the throat in a finite proper time and
after that, it appears in another universe and goes through the horizon.
Therefore, instead of a black hole, there is a white hole in another universe.
We may consider the inverse process, that is, the particle falling into the
black hole in another universe may appear in our universe from a white hole.
The wormholes inside the black holes connect an infinite number of universes,
as in the Reissner-Nordström black hole, although the object considered in
this paper has only event (outer) horizons. In Fig. 2, the Penrose diagram of
the wormholes inside the black holes is given.

Figure 2: The Penrose diagram of wormholes inside black holes. The thick lines
correspond to horizons, the thin lines denote infinities, and the dashed lines
represent the throat $r=r_{0}$.

We now consider the Hawking temperature of the geometry given by Eq. (33).
Near the horizon $r\sim r_{\mathrm{horizon}}$, the metric has the following
form:

$\displaystyle
ds^{2}\sim-\frac{r-r_{\mathrm{horizon}}}{r_{\mathrm{horizon}}}dt^{2}+\frac{{r_{\mathrm{horizon}}}^{3}}{\left(r-r_{\mathrm{horizon}}\right)\left({r_{\mathrm{horizon}}}^{2}-{r_{0}}^{2}\right)}dr^{2}+r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\,.$ (41)

We Wick-rotate the time coordinate $t$ by changing $t=\imath p$ and define a
new variable $q$ as

$\displaystyle q\equiv
2\sqrt{\frac{{r_{\mathrm{horizon}}}^{3}\left(r-r_{\mathrm{horizon}}\right)}{{r_{\mathrm{horizon}}}^{2}-{r_{0}}^{2}}}\,,$

using which we rewrite the metric (41) in the form

$\displaystyle
ds^{2}\sim\frac{{r_{\mathrm{horizon}}}^{2}-{r_{0}}^{2}}{4{r_{\mathrm{horizon}}}^{4}}q^{2}dp^{2}+dq^{2}+{r(q)}^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\,.$

In order to avoid the conical singularity at $q=0$, the Euclidean time $p$
should have a periodicity $p\sim
p+\frac{4\pi{r_{\mathrm{horizon}}}^{2}}{\sqrt{{r_{\mathrm{horizon}}}^{2}-{r_{0}}^{2}}}$,
which gives the Hawking temperature $T_{\mathrm{H}}$,

$\displaystyle
T_{\mathrm{H}}=\frac{\sqrt{{r_{\mathrm{horizon}}}^{2}-{r_{0}}^{2}}}{4\pi{r_{\mathrm{horizon}}}^{2}}=\frac{\sqrt{4M^{2}-{r_{0}}^{2}}}{16\pi
M^{2}}\,.$ (42)

In the case of the standard Schwarzschild black hole, which corresponds to the
limit $r_{0}\to 0$, we find the usual result, $T_{\mathrm{H}}\to\frac{1}{8\pi
M}$. Therefore, Eq. (42) tells us that the temperature of the black hole with
a wormhole considered in this paper is lower than that of the standard
Schwarzschild black hole with the identical ADM mass $M$. Especially, in the
limit $r_{0}\to r_{\mathrm{horizon}}$, the Hawking temperature vanishes,
$T_{\mathrm{H}}\to 0$, which is similar to the extremal limit of the Reissner-
Nordström black hole.

We may assume that the thermodynamical entropy $\mathcal{S}$ is given by the
area of the horizon,

$\displaystyle\mathcal{S}=4\pi{r_{\mathrm{horizon}}}^{2}\,.$

Then Eq. (42) yields

$\displaystyle
T_{\mathrm{H}}=\frac{\sqrt{\frac{\mathcal{S}}{4\pi}-{r_{0}}^{2}}}{\mathcal{S}}\,,$

which gives

$\displaystyle\frac{d\mathcal{S}}{dT_{\mathrm{H}}}=\frac{8\pi\mathcal{S}^{2}\sqrt{\frac{\mathcal{S}}{4\pi}-{r_{0}}^{2}}}{8\pi{r_{0}}^{2}-\mathcal{S}}\,.$

By using the thermodynamic identity, the specific heat $C$ is given by

$\displaystyle
C\equiv\frac{dQ}{dT_{\mathrm{H}}}=T_{\mathrm{H}}\frac{d\mathcal{S}}{dT_{\mathrm{H}}}=-\frac{2\mathcal{S}\left(\mathcal{S}-4\pi{r_{0}}^{2}\right)}{\mathcal{S}-8\pi{r_{0}}^{2}}=-\frac{8\pi{r_{\mathrm{horizon}}}^{2}\left({r_{\mathrm{horizon}}}^{2}-{r_{0}}^{2}\right)}{{r_{\mathrm{horizon}}}^{2}-2{r_{0}}^{2}}\,,$
(43)

where $Q$ is the heat. For the Schwarzschild black hole, i.e., when $r_{0}\to
0$, we find $C\to-8\pi{r_{\mathrm{horizon}}}^{2}=-32\pi M^{2}$, which is
always negative. The specific heat $C$ in Eq. (43) is also negative when the
radius of the black hole is large, $r_{\mathrm{horizon}}\gg r_{0}$, as for the
standard Schwarzschild black hole. In turn, the specific heat $C$ vanishes in
the extremal limit $r_{\mathrm{horizon}}\to r_{0}$. One can also see that the
specific heat diverges when $r_{\mathrm{horizon}}=\sqrt{2}\,r_{0}$, which
implies that there is a maximum $T_{\mathrm{H}}^{\mathrm{max}}$ in the Hawking
temperature $T_{\mathrm{H}}$, which is also clear from Eq. (42),

$\displaystyle T_{\mathrm{H}}^{\mathrm{max}}=\frac{1}{8\pi r_{0}}\,.$

The above properties are very similar to the properties of the Reissner-
Nordström black hole, whose specific heat and temperature vanish in the
extremal limit where the radius of the outer horizon coincides with the radius
of the inner horizon, and there is a maximum in the Hawking temperature in the
Reissner-Nordström black hole, where the specific heat diverges. In our model,
the radius $r_{0}$ of the throat plays the role of the radius of the inner
horizon in the Reissner-Nordström black hole.

##  V Summary and Discussion

In this paper, we have considered two exotic objects: (i) a wormhole whose
throat is hidden by a stellar object like a neutron star; and (ii) a wormhole
inside a black hole. These objects can be realized within the models which
include two scalar fields by using the formulation of Ref. [34]. Within the
formulation, we can construct models that realize an arbitrarily given
spherically symmetric and static/time-dependent geometry. In the original
formulation of Ref. [34], however, the models include ghosts, which make the
model physically inconsistent both as classical theory and as quantum theory.
In the works [36, 37, 38, 39], however, it has been shown that the ghosts can
be eliminated by the corresponding constraints. The latter are given by adding
terms including the Lagrange multiplier fields to the action. These
constraints are similar to the mimetic constraint of Ref. [40]. In the
original mimetic model, non-dynamical dark matter effectively appears. Our
model could be regarded as an extension of the mimetic theory and we realized
effectively more general but non-dynamical matter, including exotic matter
like a phantom. We also compared the mass-radius relations of the mixed
configurations “neutron star plus wormhole” with those of ordinary neutron
stars and demonstrated that it is possible to get masses and sizes of the
mixed systems comparable to those typical of neutron stars.

We have also investigated the structure of the exotic objects. For the
wormhole whose throat is hidden inside the star, the ADM mass is proportional
to the radius of the star, as in the case of the Schwarzschild black hole,
although the mass of ordinary stars constructed with some ad hoc profile of
matter similar to that of mixed systems is proportional to the third power of
the radius. We have also found that the null energy condition and therefore
all the energy conditions are violated near the throat. The violation does
not, however, generate any instability because the effective fluid is given by
the non-dynamical scalar fields. The scalar fields do not propagate and do not
fluctuate, and therefore there does not appear sound generated by the
oscillation of the effective fluid. In this sense, the effective fluid is
frozen.

For the wormhole, the throat is timelike and we have clarified the causal
structure. This wormhole has properties similar to those of the Reissner-
Nordström black hole, which has an electric charge and two horizons. Both in
the case of the Reissner-Nordström black hole and for the wormhole inside the
black hole, an infinite number of universes are connected via horizons and/or
throats. Furthermore, in the case of the Reissner-Nordström black hole, there
appears an extremal limit where the radii of the two horizons coincide with
each other, and, in the limit, the Hawking temperature vanishes. In the case
of the wormhole inside the black hole considered in this paper, the extremal
limit appears when the radius of the throat coincides with the radius of the
horizon. In this limit, the Hawking temperature vanishes, again.

We may speculate how one can describe the creation of such exotic objects. As
is clear from the arguments given in the subsection II.2, we may consider the
time evolution of the spacetime. For this case, by starting from some
appropriate initial configuration of the spacetime, of the scalar fields, and
of matter, we may describe the formation of exotic objects. For example, we
may start with an almost flat background and a spherically symmetric
distribution of low-density matter. Alternatively, we may consider as the
initial condition that there are a wormhole and spherically symmetric low-
density matter. Then, for the time-dependent and spherically symmetric
spacetime, instead of (13), the conservation law yields the equations

$\displaystyle 0=\frac{\partial T_{t}^{t}}{\partial t}+\frac{\partial
T_{t}^{r}}{\partial
r}+\left(T_{t}^{t}-T_{r}^{r}\right)\frac{\partial\lambda}{\partial
t}+T^{r}_{t}\left[\frac{\partial}{\partial
r}\left(\nu+\lambda\right)+\frac{2}{r}\right]\,,$ (44) $\displaystyle
0=\frac{\partial T_{r}^{t}}{\partial t}+\frac{\partial T_{r}^{r}}{\partial
r}+T_{r}^{t}\frac{\partial}{\partial
t}\left(\nu+\lambda\right)+\left(T_{r}^{r}-T_{t}^{t}\right)\frac{\partial\nu}{\partial
r}+\frac{2}{r}\left(T^{r}_{r}-T_{\vartheta}^{\vartheta}\right)\,,$ (45)

where $T^{\mu}_{\nu}$ is the total energy-momentum tensor (the scalar fields
plus the fluid). If for an initial distribution of matter we choose a perfect
fluid whose motion is considered to be slow (i.e., the ordinary three-
dimensional fluid velocity $v\ll 1$), the componets $T_{t}^{t}$ and
$T_{r}^{r}$ will contain the leading terms of zeroth order in $v$, while in
the component $T^{r}_{t}$ the leading term is of first order in $v$. Then, as
a rough estimate, one can neglect the terms with $T^{r}_{t}$ and $T_{r}^{t}$
in Eqs. (44) and (45), and the resulting approximate equations will be

$\displaystyle\dot{\rho}+\dot{\lambda}\left(\rho+p\right)\approx 0\,,\quad
p^{\prime}+\nu^{\prime}\left(\rho+p\right)\approx 0\,.$

By analogy with Eq. (14), we can integrate the above equations as follows:

$\displaystyle\nu(t,r)=-\int^{p(t,r)}\frac{dp}{\rho(p)+p}+\nu_{0}(t)\,,\quad\lambda(t,r)=-\int^{\rho(t,r)}\frac{d\rho}{\rho+p(\rho)}+\lambda_{0}(r)\,.$

Here $\nu_{0}(t)$ and $\lambda_{0}(r)$ are arbitrary functions of $t$ and $r$,
respectively. Therefore we may consider them as a proper initial distribution
of the matter.

It might be interesting to consider the fusion of two black holes or neutron
stars. If one or two of the black holes or neutron stars have wormhole(s)
inside, during the fusion, there might appear an energy flow between our
universe and another universe. If the flux goes to another universe, the
radiation after the fusion might become smaller. Conversely, if the flux comes
from another universe, the radiation might be enhanced. Anyway, such fusions
may provide us with some clue concerning the wormhole and another universe.

## Acknowledgements

This work was partially supported by MICINN (Spain), project
PID2019-104397GB-I00 and by the program Unidad de Excelencia Maria de Maeztu
CEX2020-001058-M, Spain (S.D.O). S.N. was partly supported by MdM Core
visiting professorship at ICE-CSIC, Barcelona. This research was funded by the
Committee of Science of the Ministry of Science and Higher Education of the
Republic of Kazakhstan (Grant No. BR21881941).

##  Appendix A Deriving the field equations (II) from the Einstein equations
(II)

In this appendix, we show that the field equations (II) for the scalar fields
$\phi$ and $\chi$ can be obtained from the Einstein equations (II) by using
the Bianchi identity and the conservation law.

Multiplying Eqs. (II) by $\nabla^{\nu}$ and using the Bianchi identity
$\nabla^{\nu}\left(-R_{\mu\nu}+\frac{1}{2}g_{\mu\nu}R\right)=0$ and the
conservation law $\nabla^{\nu}T_{\mathrm{matter}\,\mu\nu}=0$, we obtain

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

Taking into account our assumptions (4) and (5), the $\mu=t$ and $\mu=r$
components become

$\displaystyle 0=$
$\displaystyle\,\frac{1}{2}\left\\{\frac{1}{2}A_{\phi}\mathrm{e}^{-2\nu}-A\mathrm{e}^{-2\nu}\dot{\nu}-B\left(-\mathrm{e}^{-2\lambda}\nu^{\prime}+\mathrm{e}^{-2\lambda}\nu^{\prime}\right)-\frac{1}{2}C_{\phi}\mathrm{e}^{-2\lambda}+C\mathrm{e}^{-2\lambda}\dot{\lambda}-V_{\phi}\right\\}$
$\displaystyle\,+\frac{1}{2}\left\\{-A_{\phi}\mathrm{e}^{-2\nu}+A\left(\mathrm{e}^{-2\nu}\dot{\nu}+\mathrm{e}^{-2\nu}\dot{\nu}-\mathrm{e}^{-2\nu}\dot{\lambda}\right)+B_{\chi}\mathrm{e}^{-2\lambda}\right.$
$\displaystyle\,\left.+B\left(-\mathrm{e}^{-2\lambda}\nu^{\prime}+\frac{2\mathrm{e}^{-2\lambda}}{r}+\mathrm{e}^{-2\lambda}\nu^{\prime}-\mathrm{e}^{-2\lambda}\lambda^{\prime}+\mathrm{e}^{-2\lambda}\nu^{\prime}\right)-C\mathrm{e}^{-2\lambda}\dot{\lambda}\right\\}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\left\\{-\frac{1}{2}A_{\phi}\mathrm{e}^{-2\nu}+A\mathrm{e}^{-2\nu}\left(\dot{\nu}-\dot{\lambda}\right)+B_{\chi}\mathrm{e}^{-2\lambda}-\frac{1}{2}C_{\phi}\mathrm{e}^{-2\lambda}+B\left(\frac{2\mathrm{e}^{-2\lambda}}{r}+\mathrm{e}^{-2\lambda}\nu^{\prime}-\mathrm{e}^{-2\lambda}\lambda^{\prime}\right)-V_{\phi}\right\\}\,,$
$\displaystyle 0=$
$\displaystyle\,\frac{1}{2}\left\\{\frac{1}{2}A_{\chi}\mathrm{e}^{-2\nu}-A\mathrm{e}^{-2\nu}\nu^{\prime}-B\left(-\mathrm{e}^{-2\nu}\dot{\lambda}+\mathrm{e}^{-2\nu}\dot{\lambda}\right)-\frac{1}{2}C_{\chi}\mathrm{e}^{-2\lambda}+C\mathrm{e}^{-2\lambda}\lambda^{\prime}-V_{\chi}\right\\}$
$\displaystyle\,+\frac{1}{2}\left\\{A\mathrm{e}^{-2\nu}\nu^{\prime}-B_{\phi}\mathrm{e}^{-2\nu}+B\left(-\mathrm{e}^{-2\nu}\dot{\lambda}+\mathrm{e}^{-2\nu}\dot{\nu}-\mathrm{e}^{-2\nu}\dot{\lambda}+\mathrm{e}^{-2\nu}\dot{\lambda}\right)+C_{\chi}\mathrm{e}^{-2\lambda}\right.$
$\displaystyle\,\left.+C\left(-\mathrm{e}^{-2\lambda}\lambda^{\prime}+\frac{2\mathrm{e}^{-2\lambda}}{r}+\mathrm{e}^{-2\lambda}\nu^{\prime}-\mathrm{e}^{-2\lambda}\lambda^{\prime}\right)\right\\}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\left\\{\frac{1}{2}A_{\chi}\mathrm{e}^{-2\nu}-B_{\phi}\mathrm{e}^{-2\nu}+B\left(\mathrm{e}^{-2\nu}\dot{\nu}-\mathrm{e}^{-2\nu}\dot{\lambda}\right)+\frac{1}{2}C_{\chi}\mathrm{e}^{-2\lambda}+C\left(\frac{2\mathrm{e}^{-2\lambda}}{r}+\mathrm{e}^{-2\lambda}\nu^{\prime}-\mathrm{e}^{-2\lambda}\lambda^{\prime}\right)-V_{\chi}\right\\}\,,$
(46)

while other components vanish trivially. Here we have used the Christoffel
symbols

$\displaystyle\Gamma^{t}_{tt}=\dot{\nu}\,,\quad\Gamma^{r}_{tt}=\mathrm{e}^{-2(\lambda-\nu)}\nu^{\prime}\,,\quad\Gamma^{t}_{tr}=\Gamma^{t}_{rt}=\nu^{\prime}\,,\quad\Gamma^{t}_{rr}=\mathrm{e}^{2\lambda-2\nu}\dot{\lambda}\,,\quad\Gamma^{r}_{tr}=\Gamma^{r}_{rt}=\dot{\lambda}\,,$
$\displaystyle\Gamma^{r}_{rr}=\lambda^{\prime}\,,\quad\Gamma^{i}_{jk}=\bar{\Gamma}^{i}_{jk}\,,\quad\Gamma^{r}_{ij}=-\mathrm{e}^{-2\lambda}r\bar{g}_{ij}\,,\quad\Gamma^{i}_{rj}=\Gamma^{i}_{jr}=\frac{1}{r}\,\delta^{i}_{\
j}\,.$

On the other hand, the field equations (II) for the scalar fields $\phi$ and
$\chi$ have the following form:

$\displaystyle 0=$
$\displaystyle\,-\frac{1}{2}A_{\phi}\mathrm{e}^{-2\nu}+A\mathrm{e}^{-2\nu}\left(\dot{\nu}-\dot{\lambda}\right)+B_{\chi}\mathrm{e}^{-2\lambda}-\frac{1}{2}C_{\phi}\mathrm{e}^{-2\lambda}+B\mathrm{e}^{-2\lambda}\left(\frac{2}{r}+\nu^{\prime}-\lambda^{\prime}\right)-V_{\phi}\,,$
$\displaystyle 0=$
$\displaystyle\,\frac{1}{2}A_{\chi}\mathrm{e}^{-2\nu}-B_{\phi}\mathrm{e}^{-2\nu}+B\mathrm{e}^{-2\nu}\left(\dot{\nu}-\dot{\lambda}\right)+\frac{1}{2}C_{\chi}\mathrm{e}^{-2\lambda}+C\mathrm{e}^{-2\lambda}\left(\frac{2}{r}+\nu^{\prime}-\lambda^{\prime}\right)-V_{\chi}\,.$
(47)

The above expressions completely coincide with the equations in (A). Therefore
the field equations (II) can be surely obtained from the Einstein equations
(II).

Especially, in the static case, we have from Eq. (II.2),

$\displaystyle A=$
$\displaystyle\,\frac{\mathrm{e}^{2\nu}}{\kappa^{2}}\left[\mathrm{e}^{-2\lambda}\left(\frac{\nu^{\prime}+\lambda^{\prime}}{r}+\nu^{\prime\prime}+\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)\right]-\mathrm{e}^{2\nu}\left(\rho+p\right)\,,$
$\displaystyle B=$ $\displaystyle\,0\,,$ $\displaystyle C=$
$\displaystyle\,\frac{\mathrm{e}^{2\lambda}}{\kappa^{2}}\left[-\mathrm{e}^{-2\lambda}\left(-\frac{\nu^{\prime}+\lambda^{\prime}}{r}+\nu^{\prime\prime}+\left(\nu^{\prime}-\lambda^{\prime}\right)\nu^{\prime}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)\right]\,,$
$\displaystyle V=$
$\displaystyle\frac{\mathrm{e}^{-2\lambda}}{\kappa^{2}}\left(\frac{\lambda^{\prime}-\nu^{\prime}}{r}+\frac{\mathrm{e}^{2\lambda}-1}{r^{2}}\right)-\frac{1}{2}\left(\rho-p\right)\,.$

Then the first equation in (A) is trivially satisfied, and the second equation
gives

$\displaystyle\frac{1}{2}A_{\chi}\mathrm{e}^{-2\nu}-B_{\phi}\mathrm{e}^{-2\nu}+B\mathrm{e}^{-2\nu}\left(\dot{\nu}-\dot{\lambda}\right)+\frac{1}{2}C_{\chi}\mathrm{e}^{-2\lambda}+C\mathrm{e}^{-2\lambda}\left(\frac{2}{r}+\nu^{\prime}-\lambda^{\prime}\right)-V_{\chi}$
$\displaystyle=$
$\displaystyle\,-\nu^{\prime}\left(\rho+p\right)-p^{\prime}=0\,,$

which is nothing but the conservation law (13).

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