# Singular space-times with bounded algebraic curvature scalars

Renan B. Magalhães<EMAIL_ADDRESS>Programa de Pós-Graduação em
Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.  Gabriel
P. Ribeiro<EMAIL_ADDRESS>Programa de Pós-Graduação em Física,
Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.  Haroldo C. D.
Lima Junior<EMAIL_ADDRESS>Programa de Pós-Graduação em Física,
Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.  Gonzalo J. Olmo
<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto
Universitat de València - CSIC. Universitat de València, Burjassot-46100,
València, Spain.  Universidade Federal do Ceará (UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Luís C. B.
Crispino<EMAIL_ADDRESS>Programa de Pós-Graduação em Física, Universidade
Federal do Pará, 66075-110, Belém, Pará, Brazil.

###### Abstract

We show that the absence of unbounded algebraic curvature invariants
constructed from polynomials of the Riemann tensor cannot guarantee the
absence of strong singularities. As a consequence, it is not sufficient to
rely solely on the analysis of such scalars to assess the regularity of a
given space-time. This conclusion follows from the analysis of incomplete
geodesics within the internal region of asymmetric wormholes supported by
scalar matter which arise in two distinct metric-affine gravity theories.
These wormholes have bounded algebraic curvature scalars everywhere, which
highlights that their finiteness does not prevent the emergence of pathologies
(singularities) in the geodesic structure of space-time. By analyzing the
tidal forces in the internal wormhole region, we find that the angular
components are unbounded along incomplete radial time-like geodesics. The
strength of the singularity is determined by the evolution of Jacobi fields
along such geodesics, finding that it is of strong type, as volume elements
are torn apart as the singularity is approached. Lastly, and for completeness,
we consider the wormhole of the quadratic Palatini theory and present an
analysis of the tidal forces in the entire space-time.

##  I Introduction

According to general relativity (GR), gravitational collapse can lead to the
formation of different compact objects such as neutron stars and black holes.
There is also theoretical evidence that, depending on the distribution of
matter, this final stage may lead to the formation of a naked singularity
(see, for instance, Joshi:1993 ; Mosani:2020 ), in contrast with the widely
accepted belief that singularities should always be hidden behind event
horizons Penrose:1969 . If naked singularities could indeed exist, they would
potentially display distinctive phenomenology, such as their shadow image
Gyulchev:2019 , which would set them apart from other types of compact
objects. Consequently, understanding the nature and properties of space-time
singularities is crucial for both phenomenological and theoretical discussions
and interpretations.

In a first approximation to the problem, space-time singularities could be
described as locations in space-time where the laws of Physics break down
because the curvature or other physical quantities become infinite. This
intuitive notion has undergone refinement and revision over the years.
Remarkable advances in this sense were made in the 1960s with the singularity
theorems Penrose:1965 ; Senovilla:2014gza ; Witten:2020 . According to these
theorems, the classification of whether a space-time is singular (or not) is
not based on the presence of divergent quantities such as curvature scalars or
energy densities. Instead, it relies on the existence of incomplete curves in
those space-times Geroch:1968 . For instance, the story of observers following
time-like incomplete geodesics necessarily ends (or starts) at a finite value
of their proper time, implying that they can be destroyed (or created).
Something similar applies to light-like curves, which transport information.
Incomplete geodesics, therefore, would lead to the creation/destruction of
information and to our inability to perform physical measurements (when
observers are destroyed or unable to exist).

In the classical framework of GR, the singularity theorems do not offer
information on the attributes, positions, or quantities of singularities that
may exist within a given space-time. These theorems guarantee that, provided
reasonable energy conditions are met, the gravitational collapse of matter
distributions will inevitably result in the formation of space-time
singularities if trapped surfaces are produced Hawking:1973 . One would like,
however, to have more information about the nature and signatures of possible
singularities existing in a given space-time. For example, one may be
interested in whether physical or curvature quantities behave well when
approaching these singularities along time-like or null trajectories.
Depending on this behavior, space-time singularities can be regarded as
curvature or non-curvature singularities, respectively Ellis:1979 .

Curvature singularities are commonly associated with unbounded scalar
polynomials of the curvature tensor, such as the Ricci or Kretschmann scalars,
along an incomplete causal curve. However, even if all curvature
invariants111The set of all polynomial curvature invariants includes curvature
invariants of any order. A curvature invariant of order $k$ consists of a
scalar constructed as contractions of the metric, the Riemann tensor and its
covariant derivatives of order $\leq k$. Throughout this paper, we simply call
algebraic curvature scalars the zeroth-order polynomial curvature invariants.
are bounded everywhere, a different type of curvature singularity can occur
Hirschmann:2004 . These non-scalar curvature singularities arise from the
divergence of any component of the curvature tensor in a parallelly propagated
basis attached to an inextensible causal geodesic. For example, one can
investigate the finiteness of the tidal forces along incomplete causal curves.
If at least one component of the tidal tensor is unbounded, then the space-
time is said to have a parallelly propagated curvature singularity Clarke:1975
.

Studying space-times that have bounded curvature scalars but still have
inextensible curves can provide valuable insights into the physical mechanisms
behind space-time singularities. However, finding examples of such objects in
GR is challenging since most known naked singularity solutions exhibit scalar
curvature singularities Janis:1968 ; Wyman:1981 . Extensions of GR may help in
this regard. In fact, it has been shown that free scalar fields coupled with
some Ricci-Based Gravity theories (RBGs) in the Palatini framework may give
rise to geodesically incomplete asymmetric wormhole solutions in which
algebraic curvature invariants are bounded everywhere AORG:2017 ; MCO:2022 .
Despite the behavior of their curvature scalars, observers moving towards the
interior of the wormhole reach the asymptotic internal region in a finite
proper time, which is a clear indication of the existence of physical
pathologies in such space-times. A detailed analysis of what goes wrong in
such geometries is not yet available, and that is precisely what motivates
this work.

These singular asymmetric wormholes arise in two different gravity theories
coupled to free scalar fields, namely, i) Palatini $f({\cal R},{\cal
R}_{(\mu\nu)}{\cal R}^{(\mu\nu)})={\cal R}+a{\cal R}^{2}+b{\cal
R}_{(\mu\nu)}{\cal R}^{(\mu\nu)}$ gravity coupled to a non-linear matter
Lagrangian MCO:2022 , and ii) the Eddington-inspired Born-Infeld (EiBI)
gravity model coupled to a canonical scalar matter Lagrangian AORG:2017 .
Surprisingly, despite the different physical settings (different gravity
theories with different scalar Lagrangians), the asymptotic interior region of
the resulting asymmetric wormholes can be approximated by the same line
element (up to model-dependent constants) MCO:2022 . Here we investigate in
detail the geodesic structure of this asymptotic region, discussing the non-
scalar nature of the singularity, whose presence is manifest from the
existence of radial time-like and null incomplete geodesics.

The analysis of tidal forces around compact objects is important for both
observational and theoretical reasons. Tidal forces play a significant role in
the so-called tidal disruption radius, also known as the Roche radius. The
Roche radius is the radial distance to the center of a compact object such
that a star is tidally disrupted. The tidal disruption of stars by black holes
has been conjectured to be the cause of the flares observed in different
branches of the electromagnetic spectrum, such as optical TDE1 , ultraviolet
TDE2  and X-ray TDE3 . It is important to investigate the tidal forces for
different compact objects, such as naked singularities and wormholes, as they
may give rise to signatures that deviate from those predicted by black holes.
In this work, we perform an analysis of the tidal forces and Jacobi fields in
order to characterize the type and strength of the singularity in the interior
of the family of wormholes considered here. Our analysis shows that the
interior region has a strong parallelly propagated curvature singularity. As a
result, anyone moving towards the asymptotic region would be torn apart due to
the emergence of infinite tidal forces.

The content of this paper is organized as follows. In Sec. II we give an
overview of the asymmetric wormholes that arise in the Ricci-Based Gravity
theories à la Palatini mentioned above. In Sec. III we discuss the
approximations of the interior region of these wormholes, investigating their
curvature scalars and geodesic structure. In Sec. IV we perform an analysis of
the tidal forces to characterize the type of singularity in the interior of
the wormhole. Moreover, we also study the Jacobi fields approaching the
singularity and discuss its strength. For completeness, in Sec. V, we make an
analysis of the tidal forces in the whole space-time of one of the singular
asymmetric wormholes, namely the one associated to the quadratic Palatini
gravity. Finally, we summarize our results and discuss some perspectives in
Sec. VI.

##  II Two gravity models with scalar matter

Extensions of GR in the Palatini formalism, such as $f({\cal R})$ and $f({\cal
R},{\cal R}_{(\mu\nu)}{\cal R}^{(\mu\nu)})$ gravities, have been studied in
the literature, allowing, for instance, to produce modified dynamics without
the introduction of new degrees of freedom Olmo:2011 . Remarkably, in the
cases of $f({\cal R})$ and $f({\cal R},{\cal R}_{(\mu\nu)}{\cal
R}^{(\mu\nu)})$ gravities, the field equations in vacuum reduce to those of
General Relativity with an effective cosmological constant that depends on the
model Ferraris:1992 ; Borowiec:1996 . Other sort of Palatini gravities, with
other contractions of the Ricci tensor, have also been considered in the
literature, such as the inspired by Born-Infeld theories Vollick:2003qp ;
BeltranJimenez:2017 ; Afonso:2021aho . All these theories can be cast as
members of a family of extensions of GR, the so-called RBGs, which can be
described by an action of the form Gonzalo:2022

$\displaystyle S_{RBG}(g_{\mu\nu},\Gamma^{\alpha}_{\beta\gamma},\psi_{m})=$
$\displaystyle\dfrac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}f(g^{\mu\alpha}{\cal
R}_{(\alpha\nu)})$ $\displaystyle+S_{m}(g_{\mu\nu},\psi_{m}),$ (1)

where $\kappa$ is a coupling constant in suitable units; $g_{\mu\nu}$ and $g$
denote the components of the metric tensor and its determinant, respectively;
$\Gamma^{\alpha}_{\beta\gamma}$ is the affine connection, which is independent
of the space-time metric since we are in the Palatini picture, that define the
Riemann tensor ${\cal R}^{\alpha}{{}_{\beta\mu\nu}}$; and $\psi_{m}$ denotes
any sort of matter field present in the space-time. From now on, let us denote
${\cal R}\equiv g^{\mu\nu}{\cal R_{(\mu\nu)}}$ and ${\cal Q}\equiv{\cal
R_{(\mu\nu)}}{\cal R^{(\mu\nu)}}$. By varying the action (II) with respect to
the metric and affine connection, independently, one obtains the field
equations associated to these two fields. The source of the metric field
equation is the so-called energy-momentum tensor,
$T_{\mu\nu}\equiv(-2/\sqrt{-g})\delta S_{m}/\delta g^{\mu\nu}$, while the
source of the connection equation, the so-called hypermomentum tensor, is
identically zero, since we are not assuming couplings between matter fields
and the affine connection Afonso:2017 . Since in action (II) only the
symmetric part of the Ricci tensor appears, it is projective invariant, thus
ensuring the absence of ghosts in the theories BeltranJimenez:2019 ;
BeltranJimenez:2020 .

By introducing an effective metric, $h_{\mu\nu}$, with determinant $h$,
related to the space-time metric by
$h_{\mu\nu}=g_{\mu\alpha}\Omega^{\alpha}{{}_{\nu}}$, where
$\Omega^{\alpha}{{}_{\nu}}$ is the so-called deformation matrix, one obtains
the metric and connection field equations in the Einstein-frame representation
given, respectively, by Gonzalo:2022

$\displaystyle
G^{\mu}{{}_{\nu}}(h)=\dfrac{\kappa^{2}}{\sqrt{|\Omega|}}\left[T^{\mu}{{}_{\nu}}-\dfrac{1}{2\kappa^{2}}(f+\kappa^{2}T)\delta^{\mu}{{}_{\nu}}\right],$
(2) $\displaystyle\nabla_{\alpha}(\sqrt{-h}h^{\mu\nu})=0,$ (3)

where
$G^{\mu}{{}_{\nu}}(h)=h^{\mu\alpha}(R_{\alpha\nu}(h)+h_{\alpha\nu}R(h)/2)$ is
the Einstein tensor of the auxiliary metric $h_{\mu\nu}$, $|\Omega|$ is the
determinant of $\Omega^{\alpha}{{}_{\nu}}$ (from now on, vertical bars also
denote determinant) and $T$ is the trace of the energy-momentum tensor.

Coupled to the RBG theories, let us consider a scalar field distribution with
matter action given by

$S_{m}(X,\phi)=-\dfrac{1}{2}\int d^{4}x\sqrt{-g}P(X,\phi),$ (4)

where $P(X,\phi)$ is some function of the kinetic term $X\equiv
g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ and the scalar field $\phi$.
The energy-momentum tensor related to the scalar action (4) reads

$T^{\mu}{{}_{\nu}}=P_{X}X^{\mu}{{}_{\nu}}-\dfrac{P(X,\phi)}{2}\delta^{\mu}{{}_{\nu}},$
(5)

where $P_{X}\equiv\partial P/\partial X$ and $X^{\mu}{{}_{\nu}}\equiv
g^{\mu\alpha}\partial_{\alpha}\phi\partial_{\nu}\phi$. The evolution equation
of the scalar field comes from the variation of the action (4) with respect to
the scalar field, namely

$\dfrac{1}{\sqrt{-g}}\partial_{\mu}(P_{X}\sqrt{-g}g^{\mu\alpha}\partial_{\alpha}\phi)-\dfrac{P_{\phi}}{2}=0,$
(6)

where $P_{\phi}\equiv\partial P/\partial\phi$.

In the next two sections, we discuss wormhole solutions supported by scalar
fields in two different RBG theories coupled with distinct matter actions but
which, nonetheless, end up producing a similar internal region.

###  II.1 Quadratic Palatini $f({\cal R},{\cal Q})$ gravity

It was shown in Ref. AORG:2018 , that the space of solutions of GR coupled to
a scalar field may be mapped into a wide family of Palatini RBGs. The explicit
mappings to Palatini $f({\cal R})$ and EiBI gravities were implemented in
Refs. AOORG:2018 ; AOORG:2019 , whereas the mapping to Palatini $f({\cal
R},{\cal Q})$ gravity was worked out in Ref. MCO:2022 . In those papers one
finds that starting with a static, spherically symmetric solution of GR
canonically coupled to a free scalar field, one can obtain solutions in RBGs
coupled to a non-linear matter Lagrangian that inherits the nonlinearities of
the gravitational sector.

The key idea behind the mapping procedure relies on a suitable
reinterpretation of Eq. (2). The right-hand side of this equation can be
visualized as an effective energy-momentum tensor, such that Eq. (2) may be
cast as the equations of GR for the auxiliary metric $h_{\mu\nu}$ where the
matter source is a non-linear scalar field minimally coupled to $h_{\mu\nu}$.
By relating the energy-momentum tensor of the original RBG theory (where the
scalar field was minimally coupled to $g_{\mu\nu}$) to the effective one in
the Einstein frame, we can derive the mapping equations and the deformation
matrix that connects the space-time and the auxiliary metrics. Once we have
the mapping equations and the deformation matrix, we can solve the Einstein’s
equations for the auxiliary metric, and use this solution as a seed to obtain
a solution in the corresponding RBG model.

For concreteness, let us now consider the quadratic Palatini (QP) gravity
described by the action

$S_{QP}=\dfrac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}\left({\cal R}+a{\cal
R}^{2}+b{\cal Q}\right),$ (7)

where $a$ and $b$ are constants with dimensions of length squared. It was
shown in Ref. MCO:2022  that one can obtain solutions of the QP model (7)
using the mapping method by coupling the action (7) to a nonlinear scalar
field with action

$S_{m}(X,\phi)=-\dfrac{1}{2}\int
d^{4}x\sqrt{-g}\left(X+(a+b)\kappa^{2}X^{2}\right).$ (8)

The mapping method establishes that the space of solutions of the theory (7)
and (8) is in correspondence with the space of solutions of GR minimally
coupled to a free scalar field of the form

$\tilde{S}_{m}(Z)=-\dfrac{1}{2}\int d^{4}\sqrt{-h}Z,$ (9)

where $Z\equiv h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ is the kinetic
term of the scalar field associated with the auxiliary metric. When one
considers static, spherically symmetric solutions, the GR problem leads to the
Wyman solution Wyman:1981  (also discovered before by other authors Janis:1968
), which describes a naked singularity. By considering the scalar field as a
radial coordinate, one can cast the Wyman solution in the form

$ds^{2}_{GR}=-e^{\nu}\,dt^{2}+\dfrac{e^{\nu}}{W^{4}}\,dy^{2}+\dfrac{1}{W^{2}}\,(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (10)

such that the scalar field satisfies the simple Klein-Gordon equation

$\phi_{yy}=0,$ (11)

that has $\phi=\zeta y$ as a solution, where $\zeta$ is a constant related
with the amplitude of the scalar field. From the requirement of the asymptotic
flatness of the Wyman solution, the functions $\nu$ and $W$ are given by
Wyman:1981

$\displaystyle\nu$ $\displaystyle=-2M\,y,$ (12) $\displaystyle W$
$\displaystyle=\dfrac{e^{-M\,y}\sinh(\eta\,y)}{\eta},$ (13)

where $M$ is the asymptotic ADM mass of the solution and
$\eta\equiv\sqrt{M^{2}+\mu^{2}/2}$, with $\mu^{2}\equiv\kappa^{2}\zeta^{2}$.

From the mapping, one finds that solutions from GR coupled to the action (9)
are disformally related to solutions of Palatini $f({\cal R},{\cal Q})$
coupled to the action (8) via the deformation matrix MCO:2022

$\Omega^{\mu}{{}_{\nu}}=\sqrt{f_{\cal
R}F_{X}}\left(\delta^{\mu}{{}_{\nu}}-\dfrac{2\kappa^{2}f_{\cal
Q}}{F_{X}}{X^{\mu}}_{\nu}\right),$ (14)

where $F_{X}=f_{\cal R}+2\kappa^{2}Xf_{\cal Q}$, with $f_{\cal R}$ and
$f_{\cal Q}$ taking the explicit form

$\displaystyle f_{\cal R}$ $\displaystyle=1+2a\kappa^{2}X$ (15) $\displaystyle
f_{\cal Q}$ $\displaystyle=b\ ,$ (16)

respectively. From the above, it follows that the relation between
$Z=h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ and
$X=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ is given by

$Z=X\sqrt{\dfrac{F_{X}}{f_{\cal{R}}^{3}}}.$ (17)

In order to find $X(Z)$, one must solve a cubic polynomial equation that
generates three roots $X_{i}(Z)$ ($i=1$, 2 or 3), which must be carefully
combined in order to build the right solutions on the physically relevant
interval. Depending on the parameters $a$ and $b$ considered, taking into
account the limit $X\approx Z$ for small values of $Z$, the desired solutions
can be obtained by requiring the continuity and differentiability of the
resulting curve.

Finally, the counterpart of the GR solution (10) obtained via the mapping
method in the QP theory (7) is given by

$\displaystyle ds^{2}$ $\displaystyle=-\dfrac{e^{\nu}}{\sqrt{f_{\cal
R}F_{X}}}\,dt^{2}+\left(\dfrac{e^{\nu}}{\sqrt{f_{\cal
R}F_{X}}W^{4}}+\dfrac{2\mu^{2}b}{F_{X}}\right)\,dy^{2}$
$\displaystyle+\dfrac{1}{\sqrt{f_{\cal
R}F_{X}}W^{2}}\,(d\theta^{2}+\sin^{2}\theta d\varphi^{2}).$ (18)

In the limit that the QP model (7) goes to the Starobinsky model, that is
$b\to 0$, the line element (18) becomes conformally related to (10), and one
obtains the solutions found in Refs. AOORG:2018 ; AOORG:2019 . Here we focus
in the case $b\neq 0$, such that the relation between the metrics is
disformal. The properties of the line element (18) for the positive and
negative branch of $a$ were investigated in Ref. MCO:2022 . Here, in
particular, we are interested in the negative branch of $a$, since it presents
a throat-like structure (a regular minimum in $r^{2}(y)=1/(\sqrt{f_{\cal
R}F_{X}}W^{2})$) connecting two asymmetric universes. The external one is
asymptotically flat, whereas the internal one describes a non-asymptotically
flat space-time. A remarkable feature of this solution is that all its
algebraic curvature invariants are bounded.

###  II.2 Eddington-inspired Born-Infeld gravity

We now consider static and spherically symmetric solutions of a free scalar
field minimally coupled to the Palatini EiBI theory, whose gravitational
sector is given by Vollick:2003qp ; BeltranJimenez:2017

$S_{EiBI}=\dfrac{1}{\epsilon\kappa^{2}}\int
d^{4}x\left[-|g_{\mu\nu}+\epsilon{\cal R}_{(\mu\nu)}|-\lambda\sqrt{-g}\right]\
.$ (19)

Here $\epsilon$ is a parameter with dimensions of length squared. By taking a
perturbative expansion in $\epsilon$ for the fields $|{\cal R}_{(\mu\nu)}|\ll
1/\epsilon$ on the action (19), one obtains the field equations of GR plus an
effective cosmological constant, that is $\text{GR}+\Lambda_{\text{eff}}+{\cal
O}(\epsilon)$. From that expansion in $\epsilon$, it reads that the parameter
$\lambda$ is related to the effective cosmological constant as
$\lambda=1+\epsilon\kappa^{2}\Lambda_{\text{eff}}$. A glance at action (19)
suggests the convenient redefinition $h_{\mu\nu}=g_{\mu\nu}+\epsilon{\cal
R}_{(\mu\nu)}$. From this choice, the metric field equation becomes

$\sqrt{-h}h^{\mu\nu}-\sqrt{-g}\lambda
g^{\mu\nu}=-\kappa^{2}\epsilon\sqrt{-g}T^{\mu\nu},$ (20)

while the connection field equation is given by Eq. (3), which implies that
the affine connection is the Levi-Civita connection of the auxiliary metric
$h_{\mu\nu}$. From Eq. (20) one finds that the deformation matrix also has a
disformal structure,

$\sqrt{|\Omega|}(\Omega^{-1})^{\mu}{{}_{\nu}}=\lambda\delta^{\mu}{{}_{\nu}}-\epsilon\kappa^{2}T^{\mu}{{}_{\nu}}.$
(21)

Coupled to EiBI, let us consider the canonical scalar Lagrangian, that is
$P(X,\phi)=X-2V(\phi)$, where $V(\phi)$ is the potential, for which the
energy-momentum tensor and the dynamical equation of the scalar field,
respectively, reduce to

$\displaystyle
T^{\mu}{{}_{\nu}}=X^{\mu}{{}_{\nu}}-\dfrac{(X-2V(\phi))}{2}\delta^{\mu}{{}_{\nu}},$
(22)
$\displaystyle\dfrac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\alpha}\partial_{\alpha}\phi)-\dfrac{dV}{d\phi}=0.$
(23)

One can follow Ref. AORG:2017  in order to find scalar configurations
generated by a self-gravitating free scalar field ($V(\phi)=0$), and consider
the following line element of a generic spherically symmetric space-time,

$ds^{2}=-A(x)+\dfrac{1}{B(x)}dx^{2}+r^{2}(x)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}).$ (24)

By substituting the line element (24) in the scalar field equation, one can
integrate it to obtain $r^{2}\sqrt{AB}\phi_{x}=C$, where $\phi_{x}=d\phi/dx$
and $C$ is an integration constant. From Eqs. (21) and (22), together with the
line element (24), we see that a suitable ansatz for the deformation matrix is

$\displaystyle\Omega^{\mu}{{}_{\nu}}$
$\displaystyle=\text{diag}(\Omega^{t}{{}_{t}},\Omega^{\theta}{{}_{\theta}},\Omega^{\varphi}{{}_{\varphi}},\Omega^{x}{{}_{x}})$
$\displaystyle=\text{diag}(\Omega_{+},\Omega_{+},\Omega_{+},\Omega_{-}),$ (25)

where $\Omega_{+}=(\lambda^{2}-X_{\epsilon}^{2})^{1/2}$,
$\Omega_{-}=(\lambda+X_{\epsilon})^{3/2}(\lambda-X_{\epsilon})^{-1/2}$ and
$X_{\epsilon}$ is defined by
$X_{\epsilon}\equiv\epsilon\kappa^{2}X/2=\epsilon\kappa^{2}C^{2}/(2r^{4}A)$.
The metric field equations in the Einstein-frame then reads AORG:2017

$\epsilon h^{\mu\alpha}{\cal
R}{{}_{(\alpha\nu)}}(h)=\begin{pmatrix}\left(1-\dfrac{\lambda+X_{\epsilon}}{\sqrt{|\Omega|}}\right)I_{3\times
3}&0\\\ 0&1-\dfrac{\lambda-X_{\epsilon}}{\sqrt{|\Omega|}}\end{pmatrix}.$ (26)

To solve Eq. (26) one can proceed by using the direct integration of the field
equations, just like Wyman did in Ref. Wyman:1981 . The key point of this
approach is to identify the scalar field as a radial coordinate, which can be
done by taking $\phi_{x}$ as a constant, that is $\phi_{x}=\upsilon$. With
this choice, one can write the following line elements related to the space-
time and auxiliary metric, respectively,

$\displaystyle ds^{2}$
$\displaystyle=-e^{\nu}dt^{2}+\dfrac{e^{\nu}}{C_{0}^{2}W^{4}}dx^{2}+\dfrac{1}{W^{2}}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (27) $\displaystyle d\tilde{s}^{2}$
$\displaystyle=-e^{\tilde{\nu}}dt^{2}+\dfrac{e^{\tilde{\nu}}}{C_{0}^{2}\tilde{W}^{4}}dy^{2}+\dfrac{1}{\tilde{W}^{2}}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (28)

where $C_{0}=C/\upsilon$, the metric functions $\nu$ and $W$ depend only on
$x$, while $\tilde{\nu}$ and $\tilde{W}$ depend only on $y$. The relation
between the metric functions and radial coordinates of the space-time and
auxiliary metric are AORG:2017

$\displaystyle
e^{\tilde{\nu}}=\Omega_{+}e^{\nu},\,\,\,\,\,\,\,\tilde{W}^{2}=W^{2}/\Omega_{+},$
(29) $\displaystyle
dy=\dfrac{\Omega_{-}}{\sqrt{|\Omega|}}dx=\dfrac{1}{|\lambda-X_{\epsilon}|}dx.$
(30)

From the line elements (27) and (28), one obtains that the Klein-Gordon
equation (23) reduces to $\phi_{xx}=0$. By substituting the line element (28)
in Eq. (26) and by considering the right-hand side of Eq. (26) as an effective
energy-momentum tensor, $\tau^{\mu}{{}_{\nu}}$, one obtains the following set
of differential equations

$\displaystyle\tilde{\nu}_{yy}$
$\displaystyle=-\dfrac{\kappa_{0}^{2}\Omega_{+}^{3}}{X_{\epsilon}}\left(1-\dfrac{\lambda+X_{\epsilon}}{\sqrt{|\Omega|}}\right),$
(31)
$\displaystyle=\tilde{W}_{yy}-\tilde{\nu}_{y}\tilde{W}_{y}-\dfrac{\kappa_{0}^{2}\Omega_{+}^{3}}{2\sqrt{|\Omega|}}\tilde{W},$
(32) $\displaystyle\tilde{W}\tilde{W}_{yy}-\tilde{W}^{2}_{y}$
$\displaystyle=-\dfrac{e^{\tilde{\nu}}}{C_{0}^{2}}+\dfrac{\kappa_{0}^{2}\Omega_{+}^{3}}{2X_{\epsilon}}\left(1-\dfrac{\lambda+X_{\epsilon}}{\sqrt{|\Omega|}}\right),$
(33)

where $\kappa_{0}^{2}=\kappa^{2}\upsilon^{2}$. There is a technical difficulty
in the above set of equations, namely that $X_{\epsilon}$ depends on $\nu$ and
$W$, instead, of $\tilde{\nu}$ and $\tilde{W}$. In order to rewrite
$X_{\epsilon}$ in terms of the tilde functions, one can introduce
$\tilde{X}_{\epsilon}\equiv\epsilon\kappa_{0}^{2}C^{2}\tilde{W}^{4}e^{-\tilde{\nu}}/2$,
such that

$X_{\epsilon}^{2}(\tilde{X}_{\epsilon})=\lambda^{2}+\dfrac{1}{\sqrt[3]{18}|\tilde{X}_{\epsilon}|}\left(\tilde{K}-\dfrac{\sqrt[3]{12}}{\tilde{K}}\right),$
(34)

with
$\tilde{K}=((12+(9\lambda^{2}|\tilde{X}_{\epsilon}|)^{2})^{1/2}-9\lambda^{2}|\tilde{X}_{\epsilon}|)^{1/3}.$
The negative branch of $\epsilon$ leads to non-singular bouncing solutions in
cosmological models OORG:2014  and non-singular black holes in electrovacuum
scenarios ORGSP:2016 ; Gonzalo:2016 .

Due to the deep nonlinearity present in the above set of metric field
equations, the differential equations (31) and (32) can be solved numerically,
together with Eq. (33), which is a constraint that $\tilde{\nu}$ and
$\tilde{W}$ must satisfy, to obtain the auxiliary metric $h_{\mu\nu}$. Then,
by using the deformation matrix, one may recover the space-time metric. A full
numerical analysis of this problem was performed in Ref. AORG:2017 . The
authors found that the areal radius $r=1/W^{2}(x)$ has a regular minimum,
therefore unveiling a throat structure in the space-time. Far from the object,
$x\rightarrow 0$, the solution has the same asymptotic behavior as
Schwarzschild. The inner region, on the contrary, has a non-asymptotically
flat character, where all the algebraic curvature invariants are bounded. The
numerical solution for the metric confirms that some approximations are
possible in the internal region, that allow to obtain analytical expressions
for the metric functions there. Such approximations and the form of the
resulting line element are presented in the following section.

##  III Interior solution approximation

It was noted in Ref. MCO:2022  that the asymptotic internal region of (18),
obtained from the coupling of a QP gravity to a nonlinear scalar field
Lagrangian, has the same internal asymptotic structure as the solution
discussed in Sec. II.2, which comes from the coupling of EiBI to a canonical
scalar field Lagrangian. Far away from the throat in the internal region, the
line element of these two space-times can be written as

$ds^{2}=-\frac{\alpha}{r^{4}}dt^{2}+\frac{\beta}{r^{2}}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (35)

where $\alpha$ and $\beta$ are constants that depend on the gravitational and
matter sectors. The explicit dependence on the parameters of EiBI and QP
gravities are, respectively AORG:2017 ; MCO:2022

$\displaystyle\alpha_{EiBI}=r_{\epsilon}^{4},$
$\displaystyle\beta_{EiBI}=\dfrac{4\sigma^{2}}{C_{0}^{2}},$ (36)
$\displaystyle\alpha_{QP}=\dfrac{2|a|^{3}\mu^{2}}{b^{2}},$
$\displaystyle\beta_{QP}=\dfrac{18|a|\mu^{2}}{(2M-\eta)^{2}},$ (37)

where $r_{\epsilon}\propto\sqrt{|\epsilon|}$ is a constant with dimension of
length and $\sigma$ is a combination of some constants that arise during the
integration of the field equations.

###  III.1 Curvature scalars

To study some aspects of the asymptotic internal region of the asymmetric
wormholes described by the line element (35), one can compute its curvature
invariants. The non-vanishing components of the Riemann tensor, related with
the line element (35), are

$\displaystyle R_{t\varphi t\varphi}$
$\displaystyle=-\frac{2\alpha\sin^{2}\theta}{\beta r^{2}},$ $\displaystyle
R_{trtr}=\frac{4\alpha}{r^{6}},$ $\displaystyle R_{t\theta t\theta}$
$\displaystyle=-\frac{2\alpha}{\beta r^{2}},$ $\displaystyle R_{r\theta
r\theta}=-1,$ $\displaystyle R_{r\varphi r\varphi}$
$\displaystyle=-\sin^{2}\theta,$ $\displaystyle
R_{\theta\varphi\theta\varphi}=\frac{r^{2}(\beta-r^{2})\sin^{2}\theta}{\beta},$
(38)

where $R_{\mu\nu\rho\delta}$ denotes the Riemann tensor components of the
space-time metric, which should not be confused with the Riemann tensor
associated to the independent affine connection, ${\cal
R}_{\mu\nu\rho\delta}$. As pointed out in Ref. MCO:2022 , the line element
(35) has its algebraic curvature invariants finite everywhere in the internal
region. For instance the Ricci, Ricci-squared and Kretschmann scalars are
given, respectively, by

$\displaystyle g^{\mu\nu}R_{\mu\nu}=\frac{2}{r^{2}}-\frac{6}{\beta},$ (39)
$\displaystyle R_{\mu\nu}R^{\mu\nu}=\frac{2}{r^{4}}+\frac{36}{\beta^{2}},$
(40) $\displaystyle
R_{\mu\nu\rho\delta}R^{\mu\nu\rho\delta}=\frac{4\left(26{r}^{4}+{\left(\beta-{r}^{2}\right)}^{2}\right)}{{\beta}^{2}{r}^{4}},$
(41)

which in the asymptotic internal region of the wormhole are constants and
bounded by $\lim_{r\to\infty}g^{\mu\nu}R_{\mu\nu}=-6/\beta$,
$\lim_{r\to\infty}R_{\mu\nu}R^{\mu\nu}=36/\beta^{2}$ and
$\lim_{r\to\infty}R_{\mu\nu\rho\delta}R^{\mu\nu\rho\delta}=108/\beta^{2}$. One
can also compute the complete set of algebraic curvature invariants given by
Zakhary and McIntosh (ZM), ${\cal K}=(I_{1},I_{2},I_{3},\dots,I_{17})$
Zakhary:1997 . ZM invariants are a basis for any other algebraic curvature
invariant of the Riemann tensor. For example, the Kretschmann scalar in terms
of the ZM invariants, in four dimensions, is
$R_{\mu\nu\rho\delta}R^{\mu\nu\rho\delta}=I_{1}+2I_{6}-I_{5}^{2}/3$. It is
important to point out that, although the ZM set is complete, it has more than
14 elements, therefore not all of them are independent for all the Petrov and
Segre types Zakhary:1997 . The relations between the dependent and independent
elements are called syzygies. We exhibit the non-vanishing ZM invariants of
the metric (35) in Appendix A. In Table 1, we show the asymptotic limit of the
ZM invariants, from where one notices that all the 17 ZM invariants are
bounded in the asymptotic internal region. Hence, any curvature invariant
constructed using the ZM set is also bounded in the asymptotic internal
region.

Table 1: Asymptotic limit of the ZM invariants of the line element (35) in the internal region. ZM invariant |  Degree |  Asymptotic limit ($r\to\infty$)
---|---|---
$I_{1}$ | 2 | $48/\beta^{2}$
$I_{2}$ | 2 | $0$
$I_{3}$ | 3 | $-96/\beta^{3}$
$I_{4}$ | 3 | $0$
$I_{5}$ | 1 | $-6/\beta$
$I_{6}$ | 2 | $36/\beta^{2}$
$I_{7}$ | 3 | $-216/\beta^{3}$
$I_{8}$ | 4 | $1296/\beta^{4}$
$I_{9}$ | 3 | $0$
$I_{10}$ | 3 | $0$
$I_{11}$ | 4 | $216/\beta^{4}$
$I_{12}$ | 4 | $0$
$I_{13}$ | 5 | $0$
$I_{14}$ | 5 | $0$
$I_{15}$ | 4 | $27/2\beta^{4}$
$I_{16}$ | 5 | $-27/2\beta^{5}$
$I_{17}$ | 5 | $0$

###  III.2 Geodesic analysis

To further investigate the geometry of the internal region of the asymmetric
wormholes generated by scalar fields in RBGs with asymptotic line element
(35), it is useful to study the geodesics on it. The motion of free point-like
particles are encoded in the Lagrangian $2{\cal L}=\dot{s}^{2}=k$, where the
overdot denotes a derivative with respect to an affine time $\tau$ and $k=-1$
denotes a massive particle, while $k=0$ corresponds to light-rays. From the
symmetries of the line element (35), the Lagrangian has a time translation
symmetry and also a rotational symmetry. Therefore, there are two conserved
quantities along the geodesics: the energy of the particle, $E$, due to
$\partial{\cal L}/\partial t=0$, and the angular momentum of the particle,
$L$, due to $\partial{\cal L}/\partial\varphi=0$. (For light-rays it is
convenient to define the impact parameter via the ratio $L/E$.) By using the
Euler-Lagrange equation, one obtains that geodesics in the equatorial plane
$(\theta=\frac{\pi}{2})$, in the asymptotic internal region of the asymmetric
wormhole, satisfy

$\dfrac{\alpha\beta}{r^{6}}\dot{r}^{2}=E^{2}-\dfrac{\alpha}{r^{4}}\left(\dfrac{L^{2}}{r^{2}}-k\right).$
(42)

For massless particles, radial geodesics (with $L=0$) in the internal region
become

$\dfrac{\alpha\beta}{4}\dot{u}^{2}=E^{2},$ (43)

while for massive particles we have

$\dfrac{\alpha\beta}{4}\dot{u}^{2}=E^{2}-\alpha u^{2},$ (44)

where we defined $u(\tau)\equiv 1/r(\tau)^{2}$.

By solving Eq. (43) one finds that radial light-rays follow

$u(\tau)=\pm\dfrac{2E\tau}{\sqrt{\alpha\beta}}+u_{0},$ (45)

where the $\pm$ sign denotes ingoing ($-$) and outgoing ($+$) geodesics, with
$u_{0}=1/r_{0}^{2}$ determined by the initial position of the massless
particle. We notice that for radial geodesics propagating towards the
asymptotic internal region (the minus-sign solution of Eq. (45)), $u(\tau)$
vanishes at a finite value of $\tau$, namely,

$\tau_{f}=\dfrac{\sqrt{\alpha\beta}}{2E}u_{0}\ .$ (46)

This means that $r(\tau)$ goes to infinity after a finite affine time,
implying that radial null geodesics are incomplete.

The radial motion of massive particles is found by solving Eq. (44), whose
solutions can be easily obtained by just taking a time derivative of that
equation, finding that

$u(\tau)=\dfrac{E}{\sqrt{\alpha}}\cos\left(\frac{2(\tau-\tau_{0})}{\sqrt{\beta}}\right),$
(47)

where $\tau_{0}$ denotes the instant at which $r_{0}^{2}=\sqrt{\alpha}/E$. The
asymptotic region $r\to\infty$ is obtained at time
$\tau=\tau_{0}+(\pi\sqrt{\beta}/4)$, beyond which the solution becomes
negative and no longer represents a physical situation. Thus, radial time-like
geodesics are also incomplete. From now on, we chose the integration constants
$u_{0}$ and $\tau_{0}$ such that the singularity is reached at $\tau=0$
(therefore, the affine parameter is negative).

It is also worth pointing out that, regardless of $k$ or $L$, Eq. (42) for
large $r$ can be approximated by MCO:2022

$\dot{r}^{2}=\dfrac{E^{2}}{\alpha\beta}r^{6}+{\cal O}(r^{2}),$ (48)

which indicates that all geodesics degenerate into incomplete radial null
geodesics (see Eq. (43)). Therefore, asymmetric wormholes with an internal
region described by the line element (35) are (null- and time-like)
geodesically incomplete. Accordingly, they represent singular space-times
despite having bounded algebraic curvature invariants everywhere.

##  IV Parallelly propagated curvature singularities

Geodesically incomplete space-times can be classified as having curvature or
non-curvature singularities Ellis:1979 . If a space-time has a scalar
polynomial of the Riemann tensor unbounded on an incomplete curve, one says
that the space-time has a scalar (polynomial) curvature singularity. The Wyman
solution of Eq. (10) is an example of this sort of singularity, since its
curvature invariants blow up along incomplete geodesics. If any component of
the curvature tensor in a basis parallelly propagated along an incomplete
curve is unbounded, one says that the space-time has a parallelly propagated
curvature singularity. (For example, if the space-time presents unbounded
tidal forces.) Other types of singularities without curvature nature can also
be found, such as the quasiregular singularities Clarke:1977 ; Vickers:1987 ;
Konkowski:1985 .

A remarkable feature of the class of wormholes studied in this paper is that
all their algebraic curvature invariants are finite, despite the presence of
incomplete causal geodesics. This suggests that the singularities on these
wormholes are of a non-scalar nature. In order to probe the nature of these
singularities, let us investigate how two nearby geodesics behave in these
wormholes, specifically by studying the tidal forces exerted by them.

###  IV.1 Tidal forces in the internal region of non-regular wormholes

In a curved space-time, two nearby geodesics may move toward or away from each
other. Mathematically, the relative acceleration between two nearby particles
is described by the so-called geodesic deviation equation, namely Pirani_rep ;
dinverno

$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}={R^{\mu}}_{\rho\nu\beta}T^{\rho}T^{\nu}\xi^{\beta},$
(49)

where $\xi^{\mu}$ represents the separation vector between (infinitesimally)
nearby geodesics (these vectors are also known as Jacobi fields, and in Sec.
IV.2 we make use of this nomenclature) and $T^{\mu}$ are the components of the
tangent vector to the geodesics. As we saw, both time-like and null-like
radial geodesics are incomplete in the interior region of the asymmetric
wormhole described by Eq. (35). Therefore, let us investigate how two nearby
time-like geodesics propagating radially toward the asymptotic region behave
according to Eq. (49). For this aim, it is convenient to introduce a vierbein,
that is, a set of orthonormal vectors, that follows the radial time-like
geodesics. One of the vectors in the vierbein is chosen to represent the
tangent vector to the geodesics Crispino:2016 ; Goel:2015 , which for time-
like geodesics is given by
$\bm{e}_{\hat{0}}=\dot{t}\,\partial_{t}+\dot{r}\,\partial_{r}$. Here the hat
index labels each vector in the vierbein, $\partial_{t}$ and $\partial_{r}$
together with $\partial_{\theta}$ and $\partial_{\varphi}$ are the basis
vectors associated to the coordinate system, $\dot{t}=Er^{4}/\alpha$ and, from
Eq. (42), the radial component of the four-velocity is

$\dot{r}=\pm r^{3}\sqrt{\dfrac{E^{2}-\dfrac{\alpha}{r^{4}}}{\alpha\beta}},$
(50)

with the positive and negative signs being related to ingoing and outgoing
geodesics, respectively. The other three vectors can be found by applying the
orthonormality condition,
$g_{\mu\nu}e_{\hat{a}}{{}^{\mu}}e_{\hat{b}}{{}^{\nu}}=\eta_{\hat{a}\hat{b}}$,
where $\eta_{\hat{a}\hat{b}}=(-1,1,1,1)$ are the components of the Minkowski
metric in Cartesian coordinates. Therefore, the vectors of the vierbein are

$\displaystyle\bm{e}_{\hat{0}}$
$\displaystyle=\frac{Er^{4}}{\alpha}\,\partial_{t}\pm
r^{3}\sqrt{\frac{E^{2}-\dfrac{\alpha}{r^{4}}}{\alpha\beta}}\,\partial_{r},$
(51) $\displaystyle\bm{e}_{\hat{1}}$
$\displaystyle=\pm\frac{r^{4}}{\alpha}\sqrt{E^{2}-\dfrac{\alpha}{r^{4}}}\,\partial_{t}+\frac{Er^{3}}{\sqrt{\alpha\beta}}\,\partial_{r},$
(52) $\displaystyle\bm{e}_{\hat{2}}$
$\displaystyle=\dfrac{1}{r}\partial_{\theta},$ (53)
$\displaystyle\bm{e}_{\hat{3}}$
$\displaystyle=\dfrac{1}{r\sin\theta}\partial_{\varphi}.$ (54)

In the orthonormal frame, the geodesic deviation equation (49) can be written
as Crispino:2016

$\frac{D^{2}\xi^{\hat{a}}}{D\tau^{2}}={K^{\hat{a}}}_{\hat{b}}\xi^{\hat{b}},$
(55)

where
${K^{\hat{a}}}_{\hat{b}}\equiv{R^{\hat{a}}}_{\hat{0}\hat{0}\hat{b}}={R^{\mu}}_{\rho\nu\beta}e^{\hat{a}}_{\mu}e^{\rho}_{\hat{0}}e^{\nu}_{\hat{0}}e^{\beta}_{\hat{b}}$
are the components of the tidal tensor in the parallelly propagated frame
$\\{\bm{e}_{\hat{a}}\\}$. From the computed components of the Riemann tensor
(38), one finds that the non-vanishing tidal tensor components are given by

$\displaystyle{K^{\hat{1}}}_{\hat{1}}$ $\displaystyle=-\frac{4}{\beta},$ (56)
$\displaystyle{K^{\hat{2}}}_{\hat{2}}$
$\displaystyle={K^{\hat{3}}}_{\hat{3}}=\frac{3r^{4}E^{2}-\alpha}{\alpha\beta}.$
(57)

By substituting the above equations in Eq. (49), one obtains the expressions
for the relative acceleration between two nearby geodesics, that is, the
radial and angular tidal forces in the asymptotic internal region of the
wormhole. One notices that, since $\beta$ is positive in both the EiBI and the
QP models AORG:2017 ; MCO:2022 , the radial tidal force,
${K^{\hat{1}}}_{\hat{1}}$ has a negative constant value in the asymptotic
internal region of the asymmetric wormhole. On the other hand, the angular
tidal forces, ${K^{\hat{2}}}_{\hat{2}}$ and ${K^{\hat{3}}}_{\hat{3}}$, are
unbounded in the internal region of the wormhole, presenting a positive term
proportional to $r^{4}$.

As previously mentioned, if any component of the Riemann tensor in a vierbein
associated to an incomplete curve is unbounded, the space-time singularity is
considered a parallelly propagated curvature singularity. In the case of the
asymmetric wormholes under study, the components
${R^{\hat{2}}}_{\hat{0}\hat{0}\hat{2}}$ and
${R^{\hat{3}}}_{\hat{0}\hat{0}\hat{3}}$ become unbounded along incomplete
radial time-like geodesics. Accordingly, while an observer in the internal
region of the wormhole might measure finite curvature scalars, they would
experience infinite angular stretching.

For the sake of completeness, we also present the tidal tensor components for
a static observer whose vierbein is given by

$\displaystyle\bm{\lambda}_{\hat{0}}$
$\displaystyle=\frac{r^{2}}{\sqrt{\alpha}}\partial_{t},\qquad\bm{\lambda}_{\hat{1}}=\frac{r}{\sqrt{\beta}}\partial_{r},$
(58) $\displaystyle\bm{\lambda}_{\hat{2}}$
$\displaystyle=\dfrac{1}{r}\partial_{\theta},\qquad\ \ \
\bm{\lambda}_{\hat{3}}=\dfrac{1}{r\sin\theta}\partial_{\varphi}.$ (59)

The non-vanishing tidal tensor components for the static frame are:

$\displaystyle{\widetilde{K}^{\hat{1}}}_{\ \ \hat{1}}$
$\displaystyle=-\frac{4}{\beta},$ (60)
$\displaystyle{\widetilde{K}^{\hat{2}}}_{\ \ \hat{2}}$
$\displaystyle=\frac{2}{\beta}.$ (61)

We notice that the tidal forces, as measured by a static observer, are bounded
regardless of the value of $r$. The observation of finite tidal forces by a
static observer is consistent with the presence of a parallelly propagated
singularity, since a static observer in the asymptotic internal region is
subjected to an acceleration, hence it is not parallelly propagated.
Curiously, the radial tidal force is the same in the parallelly propagated and
static frames.

In the following section we investigate how Jacobi fields and volume elements
evolve in the internal region when approaching the singularity, in order to
characterize the strength of the parallelly propagated curvature singularity
in this wormhole scenario.

###  IV.2 Strength of the singularity

In order to classify the various types of singularities, criteria about their
strength and impact on observers have been established in the literature. The
first and more intuitive criterion is the characterization of a strong
curvature singularity as that case in which all objects moving towards it are
compressed to zero volume. This was mathematically formalized by Tipler
Tipler:1977a ; Tipler:1977b  and further refined by Clarke and Królak
Clarke:1985 . One can establish that, according to Tipler’s definition, a
necessary condition for volume elements following incomplete time-like
geodesics to be crushed into zero volume at a point (with affine
parametrization $\tau=0$) is that for some component
${R^{\hat{a}}}_{\hat{0}\hat{b}\hat{0}}$ of the Riemann tensor in the vierbein
basis, the quantity

${I^{\hat{a}}}_{\hat{b}}(\tau)=\int_{\tau_{1}}^{\tau}d\tau^{\prime}\int_{\tau_{1}}^{\tau^{\prime}}d\tau^{\prime\prime}|{R^{\hat{a}}}_{\hat{0}\hat{b}\hat{0}}|$
(62)

does not converge as $\tau\to 0^{-}$, with $\tau_{1}<0$. On the other hand,
the necessary condition for strong curvature singularities according to
Królak’s definition is that

${J^{\hat{a}}}_{\hat{b}}(\tau)=\int_{\tau_{1}}^{\tau}d\tau^{\prime}|{R^{\hat{a}}}_{\hat{0}\hat{b}\hat{0}}|$
(63)

does not converge as $\tau\to 0^{-}$. Sufficient conditions for both
definitions are obtained by replacing
$|{R^{\hat{a}}}_{\hat{0}\hat{b}\hat{0}}|$ by $R_{\mu\nu}T^{\mu}T^{\nu}$ in
Eqs. (62) and (63), together with the time-like or null convergence condition
Clarke:1985 , $R_{\mu\nu}T^{\mu}T^{\nu}\geq 0$, where $R_{\mu\nu}$ are the
components of the Ricci tensor of the space-time metric. In order to make
clearer the physical meaning of a strong singularity, other characterizations
were introduced, in particular, the notion of deformationally strong
singularity was proposed in Refs. nolan ; ori , where it also includes the
case in which any of the Jacobi fields is unbounded. This can lead, for
instance, to a divergent volume element or a finite volume element at the
singularity, which according to Tipler’s definition would be a weak curvature
singularity. It was shown in Ref. ori  that the necessary condition for
Tipler’s strong singularities is also a necessary condition for
deformationally strong singularities. Therefore, for strong singularities, the
integral (62) does not converge as $\tau\to 0^{-}$. One should bear in mind
that any Tipler’s strong singularity is also a deformationally strong
singularity, though the opposite is not true.

Let us now check if the necessary condition for strong singularities is
fulfilled by the singularity in the internal region of the wormhole. By
substituting, for instance,
${R^{\hat{2}}}_{\hat{0}\hat{2}\hat{0}}=-{K^{\hat{2}}}_{\hat{2}}$ in Eq. (62)
and using that near the singularity ($r\to\infty$),
$r\approx(\alpha\beta)^{1/4}/\sqrt{2E|\tau|}$, one obtains

${I^{\hat{a}}}_{\hat{b}}(\tau)=\int_{\tau_{1}}^{\tau}d\tau^{\prime}\int_{\tau_{1}}^{\tau^{\prime}}d\tau^{\prime\prime}\left(\frac{3}{4\tau^{2}}-\frac{1}{\beta}\right),$
(64)

where we are also taking into account the fact that
${R^{\hat{2}}}_{\hat{0}\hat{2}\hat{0}}<0$ near the singularity. Actually, this
is true for any $-\sqrt{3\beta}/2<\tau<0$. One can verify that Eq. (64) does
not converge as $\tau\to 0^{-}$, since the integral diverges with
$-3\ln|\tau|/4$ near the singularity. Therefore, the necessary condition is
satisfied.

By analyzing the component $R_{\hat{0}\hat{0}}$ of the Ricci tensor in the
vierbein, which near the singularity is

$R_{\hat{0}\hat{0}}\approx\dfrac{6}{\beta}-\frac{3}{2\tau^{2}},$ (65)

one notices that the time-like convergence condition, $R_{\hat{0}\hat{0}}\geq
0$, is not fulfilled when approaching the singularity, which means that the
sufficient condition for the singularity to be a Tipler’s strong singularity
does not hold. Therefore, we have to take another approach to identify this
singularity.

In order to characterize the strength of the singularity in the asymptotic
internal region of the asymmetric wormhole, we have to study the solutions of
the geodesic deviation equation. We are interested in investigating if the
Jacobi fields are bounded when they approach the singularity, and how volume
elements behave in its neighborhood. As already mentioned, a Jacobi field
$\bm{\xi}$ is any smooth vector field that satisfies Eq. (49) along a causal
geodesic. Since Eq. (49) is a second order differential equation, there are
six independent Jacobi fields, depending on the values of $\xi^{\hat{a}}$ and
$\dot{\xi}^{\hat{a}}$ at some point $\tau_{i}$. (In this section we are using
the overdot to denote $D/D\tau$.) The components of any Jacobi field may be
expressed in terms of the initial values $\xi^{\hat{a}}(\tau_{i})$ via
$\xi^{\hat{a}}(\tau)={{\cal
A}^{\hat{a}}}_{\hat{b}}(\tau)\xi^{\hat{b}}(\tau_{i})$ if all
$\xi^{\hat{a}}(\tau_{i})\neq 0$, or via $\xi^{\hat{a}}(\tau)={{\cal
B}^{\hat{a}}}_{\hat{b}}(\tau)\dot{\xi}^{\hat{b}}$ if all
$\xi^{\hat{a}}(\tau_{i})=0$. The components of the $3\times 3$ matrices
${{\cal A}^{\hat{a}}}_{\hat{b}}(\tau)$ and ${{\cal
B}^{\hat{a}}}_{\hat{b}}(\tau)$ go to ${\delta^{\hat{a}}}_{\hat{b}}$ and zero,
respectively, at $\tau_{i}$. With three independent Jacobi fields
$\bm{\xi}_{(i)}={\xi_{(i)}}^{\hat{a}}\bm{e}_{\hat{a}}$, with $i=1,2,3$, one
can write the volume 3-form along a given geodesic as nolan

$V(\tau)={\bm{\xi}}_{(1)}\wedge{\bm{\xi}}_{(2)}\wedge{\bm{\xi}}_{(3)}.$ (66)

One can relate this volume element with the determinant of the matrices
${{\cal A}^{\hat{a}}}_{\hat{b}}(\tau)$ and ${{\cal
B}^{\hat{a}}}_{\hat{b}}(\tau)$ and some initial data of the volume at
$\tau_{i}$. In particular, when all $\xi^{\hat{a}}(\tau_{i})=0$, the volume
may be written as $V(\tau)=\det|{{\cal B}^{\hat{a}}}_{\hat{b}}(\tau)|.$

Let us follow Ref. nolan  to compute the functional dependence of the volume
form (66). For this aim, it is convenient to rewrite the line element (35)
using null coordinates, that is,

$ds^{2}=-\dfrac{\beta}{(v-u)^{2}}dvdu+r^{2}(u,v)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (67)

where we defined the null coordinates $v\equiv t+r_{\star}$ and $u\equiv
t-r_{\star}$ in terms of the radial coordinate
$r_{\star}=r^{2}\sqrt{\beta/4\alpha}$. In the above coordinate system, namely
$(u,v,\theta,\varphi)$, one may write the tangent vector of an arbitrary time-
like curve $\gamma(\tau)$ as $\bm{T}=h\partial_{u}+(\beta
h)^{-1}(v-u)^{2}\partial_{v}$, such that $T^{\mu}T_{\mu}=-1$, where $h\equiv
h(u,v)$ is an arbitrary function of the null coordinates $u$ and $v$. If
$\gamma(\tau)$ is a geodesic, its tangent vector must satisfy
$T^{\mu}\nabla_{\mu}T^{\nu}=0$, therefore one finds that $h(u,v)$ satisfies
$\partial_{\nu}h+\beta h^{2}(\partial_{u}h-2h/(u-v))/(u-v)^{2}=0$.

Let us take three orthogonal vectors to the unit tangent vector of a time-like
curve, $\bm{T}$, namely

$\displaystyle\bm{\xi}_{(1)}$
$\displaystyle=a(u,v)\left[h(u,v)\partial_{u}-\dfrac{(v-u)^{2}}{\beta
h(u,v)}\partial_{v}\right],$ (68) $\displaystyle\bm{\xi}_{(2)}$
$\displaystyle=p(u,v)\partial_{\theta},$ (69) $\displaystyle\bm{\xi}_{(3)}$
$\displaystyle=q(u,v)(\sin\theta)^{-1}\partial_{\varphi},$ (70)

as Jacobi fields candidates. The functions $a$, $p$ and $q$ are found by
requiring that those fields satisfy the geodesic deviation equation. In terms
of the affine parameter $\tau$, the function $p$ (and equivalently $q$) must
satisfy the equation

$r\ddot{p}+2\dot{r}\dot{p}=0,$ (71)

whose solution can be cast as

$p(\tau)=p_{1}+p_{0}\int\dfrac{d\tau}{r^{2}(\tau)},$ (72)

where $p_{0}$ and $p_{1}$ are integration constants. The function $a$ must
satisfy the equation

$\ddot{a}+\dfrac{4}{\beta}a=0,$ (73)

that is, $a$ satisfies a differential equation with the same coefficients as
the radial tidal force equation in the vierbein (see Eq. (56)). The general
solution to Eq. (73) is

$a(\tau)=a_{1}\cos\left(2\tau/\sqrt{\beta}\right)+a_{2}\sin\left(2\tau/\sqrt{\beta}\right),$
(74)

where $a_{1}$ and $a_{2}$ are integration constants.

One can evaluate the norm of the Jacobi fields, finding that
$|\bm{\xi}_{(1)}|=|a|$, $|\bm{\xi}_{(2)}|=r|p|$ and $|\bm{\xi}_{(3)}|=r|q|$.
Therefore, one finds that the norm of the volume 3-form is simply given by

$|V(\tau)|=|apq|r^{2}.$ (75)

In order to analyze how the volume evolves along the geodesic, one may
consider two sorts of initial conditions, namely: (IC-I)
$\bm{\xi}_{(i)}(\tau_{1})=0$ and $\dot{\bm{\xi}}_{(i)}(\tau_{1})>0$ or (IC-II)
$\bm{\xi}_{(i)}(\tau_{1})>0$ and $\dot{\bm{\xi}}_{(i)}(\tau_{1})=0$, with
$\tau_{1}<0$. Intuitively, the initial condition IC-I corresponds to a body
(made by dust) exploding from a point $r_{i}=r(\tau_{i})$. The initial
condition IC-II, on the other hand, corresponds to releasing a body initially
at rest.

Let us consider the initial condition IC-I (one can check that by considering
IC-II, the analysis will lead to the same qualitative results), namely

$\displaystyle a(\tau_{1})$ $\displaystyle=0,\,\,\,\dot{a}(\tau_{1})=C_{1},$
(76) $\displaystyle p(\tau_{1})$
$\displaystyle=0,\,\,\,\dot{p}(\tau_{1})=C_{2},$ (77) $\displaystyle
q(\tau_{1})$ $\displaystyle=0,\,\,\,\dot{q}(\tau_{1})=C_{3},$ (78)

with $C_{i}>0$ $(i=1,2,3)$. In the asymptotic internal region, $r\to\infty$,
$r\approx(\alpha\beta)^{1/4}/\sqrt{2E|\tau|}$. Thus, the behaviors of the
functions $a$, $p$ and $q$ near the singularity ($\tau\to 0^{-}$) are,
respectively,

$\displaystyle a(\tau)$
$\displaystyle\approx\dfrac{\tilde{C}_{0}}{2}+\tilde{C}_{1}|\tau|+\dfrac{\tilde{C}_{0}}{\sqrt{\beta}}\tau^{2},$
(79) $\displaystyle p(\tau)$
$\displaystyle\approx\dfrac{C_{2}}{|\tau_{1}|}(\tau_{1}^{2}-\tau^{2}),$ (80)
$\displaystyle q(\tau)$
$\displaystyle\approx\dfrac{C_{3}}{|\tau_{1}|}(\tau_{1}^{2}-\tau^{2}),$ (81)

where $\tilde{C}_{0}$ and $\tilde{C}_{1}$ are constants that depend on
$C_{1}$, $|\tau_{1}|$ and $\beta$. One can see that as the singularity is
approached, $a$, $p$ and $q$ go to constant values, namely
$a_{0}=\tilde{C}_{0}/2$, $p_{0}=C_{2}\tau_{1}^{2}/|\tau_{1}|$ and
$q_{0}=C_{3}\tau_{1}^{2}/|\tau_{1}|$, respectively. From this, one sees that
$\bm{\xi}_{(2)}$ and $\bm{\xi}_{(3)}$ are unbounded, because their norm,
respectively $|p|r$ and $|q|r$, diverges near the singularity. This
characterizes the singularity as a deformationally strong singularity ori .
The behavior of the volume (75) near the singularity is

$|V(\tau)|\approx\dfrac{|a_{0}p_{0}q_{0}|\sqrt{\alpha\beta}}{2E|\tau|}.$ (82)

As the singularity is approached, the volume behaves as
$|V(\tau)|\sim|\tau|^{-1}$ and diverges as $\tau\to 0^{-}$. This divergence is
caused by the infinite positive tidal force experienced in the angular
directions, while the radial tidal force remains constant at large distances
within the internal region. Thus, observers following radial geodesics towards
the asymptotic internal region are ripped apart and their story ends in a
deformationally strong singularity.

##  V Tidal forces in asymmetric wormholes of quadratic Palatini gravity

In the previous section we investigated the tidal forces in the internal
region of the asymmetric wormholes supported by scalar fields in EiBI and QP
gravities. In this section, for completeness, we present a full tidal force
description of the QP gravity wormhole. We have chosen QP instead of EiBI
gravity wormhole due to the fact that, despite being cumbersome, in the former
we have an analytical expression for the space-time metric everywhere.

Let us write the line element (18) as

$ds^{2}=-A(y)dt^{2}+B(y)dy^{2}+r^{2}(y)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (83)

where

$\displaystyle A(y)$ $\displaystyle=\dfrac{e^{\nu}}{\sqrt{f_{\cal R}F_{X}}},$
(84) $\displaystyle B(y)$ $\displaystyle=\dfrac{e^{\nu}W^{-4}}{\sqrt{f_{\cal
R}F_{X}}}+\dfrac{2\mu^{2}b}{F_{X}},$ (85) $\displaystyle r^{2}(y)$
$\displaystyle=\dfrac{W^{-2}}{\sqrt{f_{\cal R}F_{X}}},$ (86)

where the definitions of $\nu$, $W$, $f_{\cal R}$ and $F_{X}$ where provided
in Sec. II.1. In Fig. 1 we show the behavior of the areal radius squared for
this space-time. We are fixing the theory parameter $a=-1$, the mass $M=1$ and
the amplitude of the field $\zeta=0.1$ (adopting $\kappa^{2}=1$, it
corresponds to $\mu^{2}=0.01$), and considering $b=0.05$, $0.2$ and $0.3$. The
minimum in this function identifies the throat of the wormhole MCO:2022  that
connects the two asymptotic regions of the space-time. For $b=0.05$, $0.2$ and
$0.3$, the throats are approximately located at $y_{th}\approx 1.877$, $1.901$
and $1.918$, respectively. The limit $y\to 0$ represents the original
asymptotically flat region, which rapidly converges to the Wyman solution far
from the naked singularity. On the other hand, the limit $y\to\infty$
corresponds to the internal non-asymptotically flat space. In Fig. 2, we plot
some of the curvature invariants of the asymmetric wormhole, from where one
notices that the Ricci, Ricci-squared, and Kretschmann scalars, respectively
$g^{\mu\nu}R_{\mu\nu}$, $R^{\mu\nu}R_{\mu\nu}$ and
$R^{\mu\nu\rho\delta}R_{\mu\nu\rho\delta}$, are finite everywhere. We also
notice that the asymptotic values of the curvature invariants in the inner
region of the wormhole are in very good agreement with the ones found using
the approximation discussed in Sec. III, even for not too large values of $y$.

Figure 1: Behavior of the areal radius squared, $r^{2}(y)$, of the QP
asymmetric wormhole for some values of the theory parameter $b$. The red dots
in the figure represent the location of the wormhole throat, which is
identified by the local minimum of the areal radius.

Figure 2: Behavior of the Ricci (top), Ricci-squared (middle) and Kretschmann
(bottom) scalars of the QP asymmetric wormhole for some choices of the theory
parameter $b$. In the outer region far from the wormhole’s throat, the scalars
all vanish, indicating that the space-time is asymptotically flat. In the
asymptotic internal region these scalars go to finite values, hence they are
bounded everywhere. The red dashed lines are the asymptotic values found using
the approximated line element (35).

In order to compute the tidal forces in the whole space-time, we introduce a
vierbein attached to a radial time-like geodesic, namely Crispino:2016 ;
Goel:2015 ,

$\displaystyle\bm{e}_{\hat{0}}$
$\displaystyle=\frac{E}{A(y)}\,\partial_{t}\pm\sqrt{\frac{E^{2}-A(y)}{A(y)B(y)}}\,\partial_{y},$
(87) $\displaystyle\bm{e}_{\hat{1}}$
$\displaystyle=\pm\frac{1}{A(y)}\sqrt{E^{2}-A(y)}\,\partial_{t}+\frac{E}{\sqrt{A(y)B(y)}}\,\partial_{y},$
(88) $\displaystyle\bm{e}_{\hat{2}}$
$\displaystyle=\dfrac{1}{r(y)}\partial_{\theta},$ (89)
$\displaystyle\bm{e}_{\hat{3}}$
$\displaystyle=\dfrac{1}{r(y)\sin\theta}\partial_{\varphi},$ (90)

where the plus and minus signs refer to ingoing or outgoing geodesics,
respectively. Again, we are identifying the zeroth vector of the basis as the
unit tangent vector to the geodesic (four-velocity), and using the
orthonormality condition,
$g_{\mu\nu}e_{\hat{a}}{{}^{\mu}}e_{\hat{b}}{{}^{\nu}}=\eta_{\hat{a}\hat{b}}$,
to determine the other vectors.

By computing the independent components of the Riemann tensor associated with
the line element (83), one can find the components of the tidal tensor via
${K^{\hat{a}}}_{\hat{b}}={R^{\mu}}_{\nu\alpha\beta}e_{\mu}^{\hat{a}}e^{\nu}_{\hat{0}}e^{\alpha}_{\hat{0}}e^{\beta}_{\hat{b}}$,
namely,

$\displaystyle{K^{\hat{1}}}_{\hat{1}}$
$\displaystyle=\frac{A(y)A^{\prime}(y)B^{\prime}(y)+B(y)\left(A^{\prime}(y)^{2}-2A(y)A^{\prime\prime}(y)\right)}{4A(y)^{2}B(y)^{2}},$
(91) $\displaystyle{K^{\hat{2}}}_{\hat{2}}$
$\displaystyle={K^{\hat{3}}}_{\hat{3}}=\frac{2A(y)B(y)\left(E^{2}-A(y)\right)r^{\prime\prime}(y)}{2A(y)^{2}B(y)^{2}r(y)}+$
$\displaystyle-\dfrac{r^{\prime}(y)\left(E^{2}B(y)A^{\prime}(y)+A(y)\left(E^{2}-A(y)\right)B^{\prime}(y)\right)}{2A(y)^{2}B(y)^{2}r(y)},$
(92)

where the primes denote derivatives with respect to $y$. Far from the throat,
in the outside region ($y\to 0$), $W(y)\approx y$, such that the kinetic term
$\kappa^{2}Z\approx\mu^{2}y^{4}\to 0$. In the limit $Z\to 0$, one recovers
$X\approx Z$ MCO:2022 , such that $f_{\cal R}\approx 1$ and $f_{X}\approx 1$,
hence the line element (83) reduces to the Wyman solution. The tidal forces
for the Wyman space-time are

$\displaystyle{K^{\hat{1}}}_{\hat{1}}$
$\displaystyle=-\frac{2Me^{-2My}\sinh^{4}(\eta y)(M-\eta\coth(\eta
y))}{\eta^{4}},$ (93) $\displaystyle{K^{\hat{2}}}_{\hat{2}}$
$\displaystyle={K^{\hat{3}}}_{\hat{3}}=-\frac{Me^{-2My}\sinh^{3}(\eta
y)\cosh(\eta y)}{\eta^{3}}$ $\displaystyle\frac{e^{-2My}\sinh^{4}(\eta
y)\left(E^{2}(M-\eta)(\eta+M)e^{2My}+\eta^{2}\right)}{\eta^{4}}.$ (94)

Thus, one can check that $y\to 0$ leads to zero tidal forces in the radial and
angular directions, as expected, given that $y\to 0$ corresponds to an
asymptotically flat space.

The complete equations for the tidal forces exerted by the QP wormhole are
cumbersome and unwieldy, which motivates the selection of concrete model
parameters to represent their behavior. In particular, we have chosen several
values of the theory constant $b$ and fixed values of $M$, $\mu^{2}$ and $a$.
In Figs. 3 and 4 we plot, respectively, the radial tidal force,
$\ddot{\xi}^{\hat{1}}={K^{\hat{1}}}_{\hat{1}}\xi^{\hat{1}}$, and the angular
tidal forces, $\ddot{\xi}^{\hat{i}}={K^{\hat{i}}}_{\hat{i}}\xi^{\hat{i}}$
($i=2,3$), as functions of $y$. We also exhibit a better view of the tidal
forces in the outside region, in Fig. 5.

Figure 3: Radial tidal force in the QP asymmetric wormhole as a function of
the radial coordinate $y$ for three values of $b$. We notice that the radial
tidal force may present local maxima and minima depending on the
configuration. In the asymptotic internal region, the radial tidal force goes
to a finite, negative constant value. The red dashed line is the radial tidal
force computed via the approximated line element (35) (cf. Eq. (56)), which
correctly captures the asymptotic behavior. Figure 4: Angular tidal forces in
the QP asymmetric wormhole as as function of the radial coordinate $y$ for
three values of $b$. We notice that inside the wormhole the angular tidal
forces are positive everywhere and increase without any bound as $y\to\infty$.
The red lines represent the angular tidal forces computed via the approximated
line element (35) (cf. Eq. (57)), which are in excellent agreement with the
exact expressions even for not too large values of $y$.

Figure 5: Radial (top) and angular (bottom) tidal forces in the outer region
of the wormhole. The red dots in the top panel represent the throat of each
wormhole. We notice that the radial tidal force has a local maximum, while the
angular tidal forces present a local minimum. Additionally, the tidal forces
may vanish before crossing the throat.

We notice that, in the outside region, the radial tidal force is small, close
to zero, but always positive, and presents a local maximum. Near the throat,
but still in the outside region, the radial tidal force decreases, and may
vanish and become negative before reaching the throat. In the internal region,
depending on the configuration, one may notice that other maxima and minima
may arise, producing a small oscillatory behavior. Far from the throat, the
radial tidal force goes to an asymptotically constant value. We observe
excellent agreement between the asymptotic limit of the radial tidal force and
the value obtained using the approximation outlined in Sec. III. Particularly
noteworthy is the fact that this limit is independent of $b$. For smaller
values of the theory parameter $b$, the local maximum and minimum in the
internal region are more noticeable. In particular, the internal local maximum
may have a positive value (cf. Fig. 3), such that more zeroes of the radial
tidal force may appear inside the throat.

The angular tidal forces, on the other hand, are negative for small values of
$y$, and present a local minimum in the outside region. They vanish just once,
and it happens before reaching the throat. After that, they are always
positive. By moving towards the internal region, the angular tidal forces
rapidly increase without bound. A glance at Fig. 4 shows that in the
asymptotic internal region the angular tidal force has an exponential growth
in terms of the $y$ radial coordinate. Again, the results for the asymptotic
internal region are in excellent agreement with the approximation considered
in Sec. III (cf. Fig. 4). Upon inspecting Eqs. (37) and (57), it becomes
apparent that, for the QP wormhole, the divergent term of the angular tidal
forces is proportional to $b^{2}$. Therefore, the divergent nature is more
pronounced when larger values of the theory parameter $b$ are considered, as
can be seen in Fig. 4.

From the above analysis, one can summarize what happens with observers coming
from the asymptotically flat region and moving toward the asymptotic internal
region of the QP wormhole. Far from the throat in the asymptotically flat
region ($y\approx 0$), the tidal forces are almost zero. As the observers
approach the throat they experience a stretching in the radial direction and
compression in the angular directions. Before crossing the throat, the tidal
forces may change their signs, and the observers begin to be stretched in the
angular directions and compressed in the radial direction. After crossing the
throat, the radial tidal force may present an oscillatory behavior, being able
to vanish again for some configurations in the internal region. Inside the
throat, observers moving toward the asymptotic region face a negative radial
tidal force that tends to a constant value (see Eq. (56)), and at the same
time they suffer an unbounded stretching in the directions perpendicular to
the radial one, caused by the divergent angular tidal forces.

##  VI Final remarks

We have investigated the tidal forces and geometric structure of a family of
asymmetric wormholes engendered by self-gravitating scalar fields in two
Ricci-based gravity models, namely, Palatini $f({\cal R},{\cal
R}_{(\mu\nu)}{\cal R}^{(\mu\nu)})={\cal R}+a{\cal R}^{2}+b{\cal
R}_{(\mu\nu)}{\cal R}^{(\mu\nu)}$, and the EiBI model. The study of these
wormholes is motivated by a nontrivial coincidence, as these two completely
different gravitational settings develop internal regions with the same
functional dependence (up to model dependent constants), which can be obtained
in analytical form. We investigated the geodesic structure and curvature
invariants of this internal region and found that, irrespective of the
behavior of the algebraic curvature scalars, which turn out to be bounded
everywhere, observers and light-rays in radial motion will reach the
asymptotic internal infinity in a finite affine time, which clearly indicates
that the space-time is singular.

By analyzing the tidal forces in the interior region, we found that the radial
component of the tidal tensor is constant and negative in the asymptotic
internal region, while the angular components of the tidal tensor are positive
and grow without bound. This means that, although the curvature scalars are
bounded, there are unbounded components of the curvature tensor in a
parallelly propagated basis along the incomplete curves. As a result, the
singularity in the interior region may be classified as a non-curvature
singularity.

To delve deeper into the physical consequences of the singularity in the
internal wormhole region, we studied the evolution of Jacobi fields as the
singularity is approached. We found that those fields diverge, leading to
unbounded angular tidal forces. According to this, the singularity can be
regarded as a deformationally strong curvature singularity. More precisely,
any volume element following the incomplete geodesic paths diverges as the
singularity is approached, ripping apart its structure because of the infinite
tidal forces perpendicular to the radial motion.

For completeness, we also studied the tidal forces in the entire space-time of
the geodesically incomplete asymmetric wormhole that arises in the quadratic
Palatini model $f({\cal R},{\cal R}_{(\mu\nu)}{\cal R}^{(\mu\nu)})={\cal
R}+a{\cal R}^{2}+b{\cal R}_{(\mu\nu)}{\cal R}^{(\mu\nu)}$. Our analysis shows
that the tidal forces vanish far from the throat in the outer region, where
the space-time is asymptotically flat. When approaching the throat, both
radial and angular tidal forces can change their signs. Depending on the
theory parameters, other zeroes may appear in the radial tidal force inside
the throat. In contrast, the angular tidal forces remain positive and
exponentially growing throughout the interior region. We compared our results
for the tidal forces in the asymptotic internal region with those obtained
using the asymptotic metric approximation, finding excellent agreement between
the two approaches.

Our study shows that the absence of unbounded algebraic curvature invariants
constructed from polynomials of the Riemann tensor cannot guarantee the
absence of strong singularities. Therefore, relying solely on the analysis of
algebraic curvature scalars is not sufficient to assess the regularity of
space-times. In alternative theories of gravity, particularly those based on
the Palatini framework, the lack of correlation between curvature invariants
and the geodesic structure of space-times seems to be more evident
Gonzalo:2016 . Our investigation of the geodesically incomplete asymmetric
wormhole provides a concrete illustration of this lack of correlation.
Examining these non-standard naked singularities may enhance our understanding
of the physical significance of singularities in geometric gravity theories.

###### Acknowledgements.

The authors would like to acknowledge Fundação Amazônia de Amparo a Estudos e
Pesquisas (FAPESPA), Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) – Finance Code 001, from Brazil, for partial financial
support. H.L.J. and L.C. thank University of Aveiro, in Portugal; and R. M.
thanks the University of Valencia, in Spain, for the kind hospitality during
the completion of this work. This work is supported by the Spanish Grant
PID2020-116567GB- C21 funded by MCIN/AEI/10.13039/501100011033, and the
project PROMETEO/2020/079 (Generalitat Valenciana). This research has further
been supported by the European Union’s Horizon 2020 research and innovation
(RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the
European Horizon Europe staff exchange (SE) programme HORIZON-MSCA-2021-SE-01
Grant No. NewFunFiCO-101086251.

##  Appendix A Zakhary and McIntosh invariants

Zakhary and McIntosh introduced the first complete set of algebraic curvature
invariants of the Riemann tensor Zakhary:1997 . This set corresponds to 17
scalars, being 4 Weyl invariants, 4 Ricci invariants and 9 mixed invariants,
namely Zakhary:1997 ; Bittencourt:2014 ; Overduin:2020 ; Kraniotis:2022 :

  * •

Weyl invariants:

$\displaystyle I_{1}$
$\displaystyle={C_{\alpha\beta}}^{\mu\nu}{C_{\mu\nu}}^{\alpha\beta},$
$\displaystyle
I_{2}={C^{\star}_{\alpha\beta}}^{\mu\nu}{C_{\mu\nu}}^{\alpha\beta},$
$\displaystyle I_{3}$
$\displaystyle={C_{\alpha\beta}}^{\mu\nu}{C_{\mu\nu}}^{\lambda\rho}{C_{\lambda\rho}}^{\alpha\beta},$
$\displaystyle
I_{4}={C_{\alpha\beta}}^{\mu\nu}{C^{\star}_{\mu\nu}}^{\lambda\rho}{C_{\lambda\rho}}^{\alpha\beta};$
(95)

  * •

Ricci invariants:

$\displaystyle I_{5}$ $\displaystyle=g_{\alpha\beta}R^{\alpha\beta},$
$\displaystyle I_{6}=R_{\alpha\beta}R^{\alpha\beta},$ $\displaystyle I_{7}$
$\displaystyle={R_{\alpha}}^{\beta}{R_{\beta}}^{\mu}{R_{\mu}}^{\alpha},$
$\displaystyle
I_{8}={R_{\alpha}}^{\beta}{R_{\beta}}^{\mu}{R_{\mu}}^{\nu}{R_{\nu}}^{\alpha};$
(96)

  * •

Mixed invariants:

$\displaystyle I_{9}$
$\displaystyle={C_{\alpha\beta\mu}}^{\nu}R^{\beta\mu}{R_{\nu}}^{\alpha},$
$\displaystyle I_{10}$
$\displaystyle=-{C^{\star}_{\alpha\beta\mu}}^{\nu}R^{\beta\mu}{R_{\nu}}^{\alpha},$
$\displaystyle I_{11}$
$\displaystyle=R^{\alpha\beta}R^{\mu\nu}\left({{C_{\lambda\alpha}}^{\rho}}_{\beta}{{C_{\rho\mu}}^{\lambda}}_{\nu}-{{C^{\star}_{\lambda\alpha}}^{\rho}}_{\beta}{{C^{\star}_{\rho\mu}}^{\lambda}}_{\nu}\right)$
$\displaystyle I_{12}$
$\displaystyle=-R^{\alpha\beta}R^{\mu\nu}\left({{C^{\star}_{\lambda\alpha\beta}}^{\rho}}{{C_{\rho\mu}}^{\lambda}}_{\nu}+{{C_{\lambda\alpha\beta}}^{\rho}}{{C^{\star}_{\rho\mu}}^{\lambda}}_{\nu}\right),$
$\displaystyle I_{13}$
$\displaystyle={C^{\alpha\beta}}_{\mu\nu}{S_{\alpha}}^{\lambda}{S_{\lambda}}^{\mu}{S_{\beta}}^{\rho}{S_{\rho}}^{\nu},$
$\displaystyle I_{14}$
$\displaystyle={{C^{\star}}^{\alpha\beta}}_{\mu\nu}{S_{\alpha}}^{\lambda}{S_{\lambda}}^{\mu}{S_{\beta}}^{\rho}{S_{\rho}}^{\nu},$
$\displaystyle I_{15}$
$\displaystyle=\dfrac{1}{16}R^{\alpha\beta}R^{\mu\nu}\left({C_{\lambda\alpha\beta\rho}}{{C^{\lambda}}_{\mu\nu}}^{\rho}+{C^{\star}_{\lambda\alpha\beta\rho}}{{{C^{\star}}^{\lambda}}_{\mu\nu}}^{\rho}\right),$
$\displaystyle I_{16}$
$\displaystyle=-\dfrac{1}{32}R^{\alpha\beta}R^{\mu\nu}\left({C_{\lambda\kappa\sigma\rho}}{{C^{\lambda}}_{\alpha\beta}}^{\rho}{{C^{\kappa}}_{\mu\nu}}^{\sigma}+{C_{\lambda\kappa\sigma\rho}}{{{C^{\star}}^{\lambda}}_{\alpha\beta}}^{\rho}{{{C^{\star}}^{\kappa}}_{\mu\nu}}^{\sigma}-{{C^{\star}}_{\lambda\kappa\sigma\rho}}{{{C^{\star}}^{\lambda}}_{\alpha\beta}}^{\rho}{{C^{\kappa}}_{\mu\nu}}^{\sigma}+{{C^{\star}}_{\lambda\kappa\sigma\rho}}{{C^{\lambda}}_{\alpha\beta}}^{\rho}{{{C^{\star}}^{\kappa}}_{\mu\nu}}^{\sigma}\right),$
$\displaystyle I_{17}$
$\displaystyle=\dfrac{1}{32}R^{\alpha\beta}R^{\mu\nu}\left({{C^{\star}}_{\lambda\kappa\sigma\rho}}{{C^{\lambda}}_{\alpha\beta}}^{\rho}{{C^{\kappa}}_{\mu\nu}}^{\sigma}+{{C^{\star}}_{\lambda\kappa\sigma\rho}}{{{C^{\star}}^{\lambda}}_{\alpha\beta}}^{\rho}{{{C^{\star}}^{\kappa}}_{\mu\nu}}^{\sigma}-{C_{\lambda\kappa\sigma\rho}}{{{C^{\star}}^{\lambda}}_{\alpha\beta}}^{\rho}{{C^{\kappa}}_{\mu\nu}}^{\sigma}+{C_{\lambda\kappa\sigma\rho}}{{C^{\lambda}}_{\alpha\beta}}^{\rho}{{{C^{\star}}^{\kappa}}_{\mu\nu}}^{\sigma}\right),$

where ${S_{\alpha\beta}}\equiv{R_{\alpha\beta}}-(R/4)g_{\alpha\beta}$ is the
trace-free Ricci tensor, $C_{\alpha\beta\mu\nu}$ is the Weyl tensor and
$C^{\star}_{\alpha\beta\mu\nu}\equiv(1/2)\epsilon_{\alpha\beta\lambda\rho}{C^{\lambda\rho}}_{\mu\nu}$
is the dual of the Weyl tensor.

The non-vanishing ZM invariants of the line element (35) are

$\displaystyle I_{1}$
$\displaystyle=\frac{4\left(\beta-6r^{2}\right)^{2}}{3\beta^{2}r^{4}},$
$\displaystyle I_{3}=\frac{4\left(\beta-6r^{2}\right)^{3}}{9\beta^{3}r^{6}},$
$\displaystyle I_{5}=\frac{2}{r^{2}}-\frac{6}{\beta},$ $\displaystyle I_{6}$
$\displaystyle=\frac{36}{\beta^{2}}+\frac{2}{r^{4}},$ $\displaystyle
I_{7}=\frac{2}{r^{6}}-\frac{216}{\beta^{3}},$ $\displaystyle
I_{8}=2\left(\frac{648}{\beta^{4}}+\frac{1}{r^{8}}\right),$ $\displaystyle
I_{9}$ $\displaystyle=\frac{24}{\beta^{2}r^{2}}-\frac{2}{3r^{6}},$
$\displaystyle
I_{11}=\frac{2\left(\beta-6r^{2}\right)^{2}\left(2\beta^{2}+12\beta
r^{2}+27r^{4}\right)}{9\beta^{4}r^{8}},$ $\displaystyle
I_{13}=-\frac{6\left(\beta^{2}-36r^{4}\right)}{\beta^{4}r^{6}},$
$\displaystyle I_{15}$
$\displaystyle=\frac{\left(\beta-6r^{2}\right)^{2}\left(2\beta^{2}+12\beta
r^{2}+27r^{4}\right)}{72\beta^{4}r^{8}},$ $\displaystyle
I_{16}=\frac{\left(\beta-6r^{2}\right)^{3}\left(2\beta+3r^{2}\right)\left(2\beta+9r^{2}\right)}{432\beta^{5}r^{10}}.$
(98)

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11institutetext:  1Pattern Recognition Lab, Friedrich-Alexander-Universität
Erlangen-Nürnberg (FAU), Erlangen, Germany

# Few-shot Unsupervised Domain Adaptation for Multi-modal Cardiac Image
Segmentation

Mingxuan Gu1(✉)  Sulaiman Vesal1 Ronak Kosti1 Andreas Maier1
<EMAIL_ADDRESS>
###### Abstract

Unsupervised domain adaptation (UDA) methods intend to reduce the gap between
source and target domains by using unlabeled target domain and labeled source
domain data, however, in the medical domain, target domain data may not always
be easily available, and acquiring new samples is generally time-consuming.
This restricts the development of UDA methods for new domains. In this paper,
we explore the potential of UDA in a more challenging while realistic scenario
where only one unlabeled target patient sample is available. We call it Few-
shot Unsupervised Domain adaptation (FUDA). We first generate target-style
images from source images and explore diverse target styles from a single
target patient with Random Adaptive Instance Normalization (RAIN). Then, a
segmentation network is trained in a supervised manner with the generated
target images. Our experiments demonstrate that FUDA improves the segmentation
performance by 0.33 of Dice score on the target domain compared with the
baseline, and it also gives 0.28 of Dice score improvement in a more rigorous
one-shot setting. Our code is available at https://github.com/MingxuanGu/Few-
shot-UDA.

##  1 Introduction

As manual contouring of medical images is tedious and time-consuming,
automatic medical image segmentation is more desirable [1]. While deep
learning methods often suffer from performance degradation when a domain gap
is observed between training (source) and testing (target) data. UDA methods
tackle this problem by reducing the domain gap with a variety of techniques,
for example, discrepancy reduction [2], adversarial learning [3], image
translation [4], etc. These methods are conditioned on the availability of a
large amount of target data which, however, is quite scarce.

In this work, we consider a more realistic and practical scenario where we
still have sufficient labeled source data, while we only have one unlabeled
target data for training. To this end, a style transfer method called Random
Adaptive Instance Normalization (RAIN) [5] is used to generate diverse target-
style images from a single target patient data. Then, a segmentation module
can be trained in a supervised manner with generated images. Our contributions
are: (1) we explore the potential of FUDA for multi-modal cardiac CMR
segmentation and it shows better performance compared with its baseline model
and other recent UDA methods, (2) we extend our method to one-shot learning
and demonstrate the possibility of FUDA with only one slice of target data
available.

Figure 1: Overview of the proposed FUDA segmentation framework. RAIN is first
pre-trained with $x^{content}$ (bSSFP) and $x^{style}$ (T2). Then during
training of the segmentation module, stylized target images will be generated
by RAIN with source images and $\varepsilon$ generated from target image(s).
After that, the segmentation module can be trained using the source and
stylized images. The $\varepsilon$ is iteratively updated to generate more
difficult stylized images.

##  2 Materials and methods

###  2.1 Dataset

We assess the proposed FUDA on MS-CMRSeg [6] 2019 challenge dataset, which
consists of 45 short-axis bSSFP, T2-weighted and LGE scans from patients
diagnosed with cardiomyopathy. The ground-truth contours are generated by two
experts and include right ventricle (RV) cavity, left ventricle (LV) cavity,
and myocardium (Myo) region. Only affine transformation (rotation,
translation, shearing, etc.) is applied and the sequences are normalized using
min-max normalization. Then, the sequences are center-cropped to $224\times
224$ pixels to have only region-of-interest (ROIs) areas.

###  2.2 Problem statement

In UDA for semantic segmentation, a set of labeled data in source domain
$\mathcal{D}^{s}(x^{s},y^{s})$ is given, where $x^{s}$ represents one sample,
and $y^{s}$ the corresponding label in $\mathcal{D}^{s}$. Whereas for target
domain ($\mathcal{D}^{t}$) only unlabeled target images ($x^{t}$) are given.
The goal is to improve the performance of the segmentation by reducing the
distribution gap between the source and target domain. In our case, we
consider only one unlabeled target patient data (${x}^{t}$) being available.

###  2.3 RAIN Module

RAIN is developed on the basis of Adaptive Instance Normalization (AdaIN) [7].
AdaIN has an encoder-decoder architecture. It generates stylized images which
have the appearance of the style image while preserving the structure of the
content images by re-normalizing the features of content images with Eq. 1:

$AdaIN(f_{c},f_{s})=\sigma(f_{s})(\frac{f_{c}-\mu(f_{c})}{\sigma(f_{c})})+\mu(f_{s}),$
(1)

where $f_{c}$, $f_{s}$ are the latent features of the content and style image,
$\mu(.)$ and $\sigma(.)$ denote channel-wise mean and standard deviation. To
achieve realistic image transfer, a content loss $\mathcal{L}_{c}$ and a style
loss $\mathcal{L}_{s}$ [7] is employed.

RAIN takes advantage of the style transfer in AdaIN and involves a style
variational auto-encoder (VAE) in between the encoder and the decoder. The
style-VAE is composed of an encoder $E_{vae}$ and a decoder $D_{vae}$.
$E_{vae}$ encodes $\mu(f_{s})\oplus\sigma(f_{s})$ to $N(\psi,\xi)$, where
$\oplus$ denotes concatenation, and $D_{vae}$ aims to reconstruct the original
style with $\widehat{\mu(f_{s})\oplus\sigma(f_{s})}=D_{vae}(\varepsilon)$,
where $\varepsilon\sim N(\psi,\xi)$. Kullback-Leibler (KL) divergence loss is
applied to enforce $N(\psi,\xi)$ to be normal distributed. Furthermore, L2
loss is applied between $\mu(f_{s})\oplus\sigma(f_{s})$ and
$\widehat{\mu(f_{s})\oplus\sigma(f_{s})}$ to force identity reconstruction
$\mathcal{L}_{Rec}$.

Thus, the overall loss function to train RAIN can be formulated as
$\mathcal{L}_{RAIN}=\mathcal{L}_{c}+\lambda_{s}\mathcal{L}_{s}+\lambda_{kL}\mathcal{L}_{KL}+\lambda_{Rec}\mathcal{L}_{Rec}$
where $\lambda_{s}$, $\lambda_{KL}$ and $\lambda_{Rec}$ denote the weights for
the losses.

Figure 2: Qualitative results of RAIN on bSSFP $\rightarrow$ LGE. The first
column shows a LGE axial slice as the style image. The first row shows the
bSSFP images as the content images, the second row shows the corresponding
stylized LGE images. Figure 3: A visual comparison of segmentation output
results produced by different methods for test LGE images under one and few-
shot learning settings. The first two rows show the results for few-shot
models and the last two rows are for one-shot respectively. LV is shown in
yellow, Myo in brown and RV in light-blue colors.

###  2.4 Few-shot UDA for cardiac MRI segmentation

The proposed FUDA framework is constructed based on ASM [5] implementation and
is shown in Fig. 1. To train the segmentation module and explore styles from a
single target image, we first generate the latent distribution
$\mathcal{N}(\psi,\xi)$ from the input image $x^{t}$. Then we sample an
$\varepsilon_{0}$ from $\mathcal{N}(\psi,\xi)$. After that, we can generate
stylized images $\hat{x^{t}}$ from any source image $x^{s}$. Then the style
transfer can be formulated as:

$\hat{x^{t}}=D(AdaIN(E(x^{s}),D_{vae}(\varepsilon)))$ (2)

Furthermore, the segmentation module can be trained in a supervised manner
with $x^{s}$ and the corresponding $\hat{x^{t}}$.

We train the segmentation module with a combination of cross-entropy loss (CE)
and jaccard distance loss (JD) as
$\mathcal{L}_{seg}=\mathcal{L}_{CE}+\mathcal{L}_{JD}$. To enforce the
segmentation module to produce domain invariant features between $x^{s}$ and
the corresponding $\hat{x^{t}}$, $\mathcal{L}_{con}=||z^{s}-z^{t}||_{2}$ is
applied as a consistency loss, where $z^{s/t}$ are the latent features of the
source and the corresponding generated target image from the bottleneck layer
of the segmentation module. Then the overall loss function for the
segmentation module is:

$\mathcal{L}_{S}=\mathcal{L}_{seg}+\lambda\mathcal{L}_{con}$ (3)

where $\lambda$ is the weight of $\mathcal{L}_{con}$. Finally, to generate
more diverse and increasingly difficult target images , $\varepsilon$ is
updated in the direction that makes the segmentation module perform worse on
segmentation with:

$\varepsilon_{i+1}=\varepsilon_{i}+\alpha\triangledown_{\varepsilon_{i}}\mathcal{L}_{seg}(\hat{x^{t}},y^{s})$
(4)

where $i$ is the iteration number, $y^{s}$ represents the corresponding source
label for $\hat{x^{t}}$ and $\alpha$ denotes the learning rate.

Table 1: Dice coefficient (DC) and Hausdorff distance (HD) measures for the proposed FUDA together with baseline (W/o UDA) method, inter-observer study and the performance of the two successful UDA methods AdaptSeg [8] and ADVENT [3]. Baseline is trained with bSSFP images and tested directly on LGE images. For few-shot learning, we take the whole LGE squence of patient 10 as the training target data. For one-shot learning, we only use slice 13 of patient 10 as the training target data. Best results are shown in bold. |  | DC ($\uparrow$) | HD[mm] ($\downarrow$)
---|---|---|---
Method | Data | Myo | LV | RV | AVG | Myo | LV | RV | AVG
W/o UDA | N/A | 0.24 | 0.40 | 0.27 | 0.30 | 31.7 | 31.0 | 45.0 | 35.9
AdaptSeg | Few | 0.39 | 0.63 | 0.58 | 0.53 | 39.1 | 28.7 | 25.6 | 31.1
ADVENT | Few | 0.39 | 0.59 | 0.52 | 0.50 | 38.4 | 35.3 | 37.9 | 37.2
Proposd | Few | _0.46_ | _0.77_ | _0.65_ | _0.63_ | _24.5_ | _13.7_ | _22.4_ | _20.2_
AdaptSeg | One | _0.45_ | 0.65 | 0.52 | 0.54 | _32.3_ | 34.3 | 36.4 | 34.3
ADVENT | One | 0.37 | 0.61 | 0.51 | 0.50 | 42.7 | 26.9 | 35.0 | 34.9
Proposd | One | 0.39 | _0.73_ | _0.63_ | _0.58_ | 36.2 | _19.0_ | _25.6_ | _26.9_
Observer | N/A | 0.76 | 0.88 | 0.81 | 0.82 | 12.0 | 14.3 | 21.5 | 15.9

###  2.5 Training

The training process has two stages. First, we use bSSFP-MRI as content images
and T2-weighted-MRI as style images to pretrain the RAIN module. Since it does
not involve any target images (LGE), this process makes it convenient to train
RAIN anytime before training the segmentation module. In the second stage, the
RAIN model is frozen, and we employ Dilated-Residual UNet (DR-UNet) [9] as the
segmentation module. We first pretrain the DR-UNet with bSSFP images and the
generated LGE-style images in a supervised manner for 40-50$k$ iterations.
Then the model is trained together with $\varepsilon$ being iteratively
updated for another 40-50$k$ iterations. We used stochastic gradient descent
(SGD) with a momentum of $0.9$ and a weight decay of $5e-4$ as the optimizer.
We empirically set the hyper-parameters of $\lambda_{s}=5,\lambda_{KL}=1$,
$\lambda_{Rec}=5$ and $\lambda=2e-3$. The proposed method is trained and
tested on one Geforce 1080Ti GPU. The training of RAIN and segmentation module
takes 2 hours and 21 hours respectively. Overall inference takes 23 seconds in
average for each patient.

##  3 Results

Fig. 2 shows the qualitative results of the pretrained RAIN. We can observe
that RAIN successfully captures the features of target images. Tab. 1
summarizes the quantitative results of different methods. Baseline method
achieved the lowest average volumetric Dice (0.30) and an HD (35.9mm). With a
few-shot UDA setting, our proposed method achieved the best overall Dice
(0.63) and the best lowest HD (20.2mm). Subsequently, we demonstrate the
results for one-shot UDA, and our proposed method achieved the highest Dice
(0.58) and the lowest HD (26.9mm). Fig. 3 illustrates qualitative examples of
different segmentation approaches. Compared with other methods, the proposed
method is able to produce more complete and precise segmentation maps.

##  4 Discussion

In this work, we presented a few-shot UDA (FUDA) for multi-modal CMR image
segmentation while restricting the experiments in a more challenging yet
realistic scenario where only one target sample is available. By comparing the
proposed method with other approaches under the same settings, we show that
FUDA highly reduced the domain gap with only a few target slices. We also
demonstrated that the proposed few-shot method produces promising results with
the more rigorous one-shot setting. We find for conventional UDA methods like
AdaptSeg and ADVENT, performance on one-shot and few-shot settings only has a
slight difference. This can be attributed to the fact that the slices of one
patient only have a small distribution shift, hence the knowledge learned by
the model from target slices of one patient is limited. While for FUDA, the
model has the ability to explore unseen styles of the target images, hence
more data provided results in more diverse target styles. Consequently, better
segmentation performance could be achieved. Furthermore, we believe there is
still room to improve the quality of the segmentation prediction. As a result,
in the future, we will explore feasible techniques like contrastive learning
and attention to improve the performance of the proposed FUDA.

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