A Review of Stable, Traversable Wormholes in f (R)

Gravity Theories

Ramesh Radhakrishnan 1,2 , Patrick Brown1,2 , Jacob Matulevich1,2 , Eric

Davis1,2 , Delaram Mirfendereski1,3 , and Gerald Cleaver1,2

1
Early Universe, Cosmology and Strings (EUCOS) Group, Center for
Astrophysics, Space Physics and Engineering Research (CASPER)
2
Department of Physics, Baylor University, Waco, TX 76798, USA
3
Department of Physics and Astronomy, The University of Texas Rio
Grande Valley (UTRGV), USA

September 3, 2024

Abstract

It has been proven that in standard Einstein gravity, exotic matter (i.e., matter
violating the pointwise and averaged Weak and Null Energy Conditions) is required
to stabilize traversable wormholes. Quantum field theory permits these violations
due to the quantum coherent effects found in any quantum field . Even reasonable
classical scalar fields violate the energy conditions. In the case of the Casimir effect
and squeezed vacuum states, these violations have been experimentally proven. It
is advantageous to investigate methods to minimize the use of exotic matter. One
such area of interest is extended theories of Einstein gravity. It has been claimed
that in some extended theories, stable traversable wormholes solutions can be found
without the use of exotic matter. There are many extended theories of gravity, and
in this review paper, we first explore f (R) theories and then explore some wormhole
solutions in f (R) theories, including Lovelock gravity and Einstein Dilaton Gauss–
Bonnet (EdGB) gravity. For completeness, we have also reviewed ‘Other wormholes’
such as Casimir wormholes, dark matter halo wormholes, thin-shell wormholes, and
Nonlocal Gravity (NLG) wormholes, where alternative techniques are used to either
avoid or reduce the amount of exotic matter that is required.

1
1 Introduction

The first study of wormhole physics was done by Ludwig Flamm in 1916 [1] during his
research into the Schwarzschild solution to the Einstein field equations. The next solution
resembling a wormhole, called the “Einstein–Rosen bridge” [2], was an idea that arose during
the investigation of blackhole spacetimes by Einstein and Rosen in 1935. They discovered
that, at least theoretically, it was possible for a blackhole surface to act as a bridge that
connected to a remote patch of spacetime. The putative wormhole in the Einstein–Rosen
bridge is colocated with the blackhole’s singularity, so it is not traversable. The surface of
the blackhole is the event horizon, which cannot be a wormhole.

A good review of the historic development of research into traversable wormholes can
be found in [3]. A more recent concise summary of wormholes in extended theories of gravity
can be found in [4]. In 1957, Misner and Wheeler [5] first introduced the term “wormhole”
during their analysis of topological issues in General Relativity (GR). They extensively
analyzed the Riemannian geometry of manifolds of nontrivial topology. This is where the
phrase “physics is geometry” arose. Wheeler also suggested that the geometry of spacetime
might be constantly fluctuating, and it may induce a change in topology to form microscopic
wormholes. The first traversable wormhole called the “Ellis drainhole” was proposed by
both Ellis and independently by Bronnikov in 1973 [6–8]. The “Ellis drainhole” spacetime
is a static, spherically symmetric solution of the Einstein field equations in a vacuum, and it
includes a scalar field ϕ minimally coupled to the spacetime geometry.

The concept of traversable wormholes, which allow inter- and intrauniverse travel by
humans, was first introduced by Morris and Thorne in their classic 1988 paper [9]. It is
well known that the Weak and Null Energy Conditions (WECs and NECs) [10] must be
violated by the stress–energy tensor as a minimum for a stable human traversable wormhole.
The other two energy conditions are the Strong Energy Condition (SEC) and the Dominant
Energy Condition (DEC) [11]. The violation of the NEC is needed due to the flare-out
condition requirement, i.e., the throat should open up outward as a human travels through
the wormhole [9, 10]. Quantum field theory permits these violations due to the quantum
coherent effects found in any quantum field [12]. The traversable wormhole solutions have
geometries that allow for closed timelike curves and “effective” superluminal travel without
surpassing the speed of light locally [3]. There is a claim [13] in the published literature about
stable traversable wormholes that can be constructed within the framework of Einstein’s
General Relativity (henceforth called ‘Einstein Gravity’ in this paper) but without the need
for exotic matter. This claim has been refuted by other authors [14]. Some of the suggested
reasons in [14] are the following: (i) possible error in calculations; (ii) failing to check for
violations at the throat, such as divergence of the inverse of the metric; and (iii) failing to
check for discontinuities in the exterior curvature in the vicinity of the throat based on the
thin-shell formalism. Some simple examples of traversable wormholes such as the polyhedral
wormholes that do not require spherical symmetry were given by Matt Visser in 1989 [15, 16].
These wormholes also require the presence of exotic matter.

2
1.1 Motivation and Equation of State

There are many attempts to use various modified gravity theories to check the exis-
tence of stable traversable wormholes in these theories. In many of these studies [17–19],
the Morris–Thorne (MT) traversable wormhole has been used to see the effects of a mod-
ified gravity background on the stability of the wormhole. One of the main motivations
for studying wormholes in modified gravity theories is to resolve the problem of the need
for exotic matter to stabilize wormholes in GR. Modified gravity theories are also used to
construct viable cosmological models of the Universe and to explain singularities encoun-
tered in cosmology. In modified gravity theories, the stress energy tensor is replaced by an
effective stress energy tensor that contains curvature terms of higher order introduced due
to modifications to GR. f (R) gravity [20, 21] is one such leading theory, and the review of
wormholes in f (R) gravity is one of the main goals of this paper.

While evaluating wormholes in GR and various modified theories, the choice of an

equation of state (EOS) becomes critical. The EOS is an equation that relates the radial
pressure pr (r) and tangential pressure pt (r) to the energy density ρ(r). One of the most
common EOSs is p(r) = ωρ(r). The motivation for using such an equation comes from
cosmology. Observation of the accelerated expansion of the Universe can be explained by
an EOS with ω < − 31 [22]. For the case where ω < −1, the EOS is known as the phantom
energy EOS. Phantom fluid violates the NEC and is a good candidate to be used in the
study of wormholes [23] in various modified gravity theories. According to a study [24], if a
source with such phantom fluid dominates the cosmic expansion, the Universe may end up
in a Big Rip singularity in which the phantom energy rips apart the galaxies, solar systems,
planets, and, eventually, the molecules, atoms, nuclei, and nucleons that we are made of,
leading to the death of the Universe.

1.2 Astronomical Observational Signatures of Traversable Worm-

holes

The presence of naturally occurring negative energy regions in space is predicted to

produce a unique signature corresponding to lensing, chromaticity, and intensity effects in
micro- and macrolensing events on galactic and extragalactic/cosmological scales [25–30]. It
has been shown that these effects provide a specific signature that allows for discrimination
between ordinary (positive mass–energy) and negative energy lenses via the spectral analysis
of astronomical lensing events. The theoretical modeling of negative energy lensing effects
has led to intense astronomical searches for naturally occurring traversable wormholes in the
universe. Computer model simulations and comparison of their results with recent satellite
observations of gamma ray bursts (GRBs) have shown that putative negative energy (i.e.,
traversable wormhole) lensing events very closely resemble the main features of some GRBs.

When background light rays strike a negative energy lensing region, they are swept out
of the central region, thus creating an umbra region of zero intensity [25]. At the edges of the
umbra, the rays accumulate and create a rainbow-like caustic with enhanced light intensity.
The lensing of a negative energy region is not analogous to a diverging lens because, in
certain circumstances, it can produce more light enhancement than does the lensing of an
equivalent positive mass–energy region [25]. Real background sources in lensing events can

3
have nonuniform brightness distributions on their surfaces and a dependency of their emission
on the observed frequency. These complications can result in chromaticity effects, i.e., in
spectral changes induced by differential lensing during the event. The quantification of such
effects is quite lengthy, somewhat model dependent, and with recent application only to
astronomical lensing events.

The rest of this paper is organized as follows: In Section 2, we provide an overview of var-
ious modified gravity theories with just enough background necessary to analyze traversable
wormholes in such theories. In Section 3, we review the basics of Morris–Thorne (MT) worm-
hole stabilization in GR. We end Section 3 with a summary of the steps used to analyze MT
wormholes in modified gravity theories. In Section 4, which forms the core of this paper,
we review the analysis of wormhole stabilization in various f(R) theories. In Section 5, we
review some closely related topics such as Casimir wormholes, thin-shell wormholes, natural
dark matter halo wormholes, and wormholes in nonlocal theories of gravity (NLGs). We end
this paper with a summary of key discussion points in this paper.

2 Modified Gravity Theories

There have been efforts to construct stable traversable wormholes in f (R) theories,
including Lovelock gravity. In this section, we give a brief overview of the basics of these
theories to provide us with enough background to explore in the next section the properties
of traversable wormholes.

2.1 An Overview of Modified Gravity Theories

The various modified gravity theories that we survey here are f (R) theories, f (T ) theo-
ries, f (R, T ) theories, f (G) theory, f (R, Lm ) theory, f (Q) theory, f (Q, T ) theory, Lovelock
gravity (a special case of f (R) theory), Einstein–Gauss–Bonnet (EGB) gravity (another
special case of f (R) theory), Brans–Dicke theory, and Kaluza–Klein (KK) theory.

Lovelock gravity and EGB gravity are f (R) theories that include higher-order curvature
terms, and they are also applicable to higher dimensional spacetimes. For example, in EGB
gravity [10], the authors replace the 2-sphere in the MT wormhole with an (n-2)-sphere,
and thus the MT wormhole line element is modified as follows:
dr2
ds2 = −e2ϕ(r) dt2 + + r2 dΩ2n−2 . (1)
1 − b(r)/r

The standard Einstein–Hilbert action of GR is

√
Z
S = R −g d4 x. (2)

In f (R) theories [31], the Ricci scalar R is replaced by a function of the Ricci Scalar
f (R) as follows:
√
Z
S = f (R) −g d4 x, (3)

4
where f (R) is a function of the Ricci scalar, and g is the determinant of the metric. Examples
of commonly used functions f (R) are f (R) = R, f (R) = R + αRn , and f (R) = R + αe(βR) .

In teleparallel gravity [32, 33], the Ricci scalar R is replaced by the torsion scalar T
as follows:
√
Z
S = T −g d4 x. (4)

In GR, we assume spinless particles to follow the geodesic of the underlying spacetime,
and hence, we have only R in the action and no T . In teleparallel gravity, T replaces
R. This interpretation holds true in the case where we see teleparallel gravity as a gauge
theory [34] for the translation group.

The f (T ) gravity [35] is an additional modification to teleparallel gravity, in which T is

replaced by the torsion function—which is a function of T , namely f (T ), as follows:
√
Z
S = f (T ) −g d4 x, (5)

and we can obtain the modified field equations by varying the action S with respect to the
metric in the same way it is done in Section 2.2 of this paper for f (R) theory.

T is a fundamental geometric quantity in the context of theories of gravity that involve

spacetime torsion. Unlike GR, where spacetime curvature plays a central role in describing
gravity, theories that incorporate torsion consider the twisting or nonmetricity of spacetime,
and the torsion scalar is a measure of this twisting. The torsion scalar is given by the equation
T = Sαβµ T αβµ , (6)
where Sαβµ is the contorsion tensor, which is the difference between the affine (Levi–Civita)
connection’s components and the Weitzenböck connection’s components. The contorsion ten-
sor quantifies nonmetricity, and it describes how spacetime is twisted. The indices α, β, and
µ refer to spacetime coordinates. In teleparallel gravity theory, the teleparallel connection
is used instead of the metric affine connection that is used in GR.

In the f (R, T ) theory, T is the trace of the energy–momentum tensor Tµν . Similar to
f (R) theories [17], the presence of f (R, T ) in the action leads to changes in gravitational
dynamics as compared to Einstein gravity. These modifications can have implications on
the behavior of gravitational fields in various contexts. They also have consequences in
cosmology and gravitational lensing. An example of an f (R, T ) function used to stabilize a
wormhole is
f (R, T ) = R + αR2 + λT , (7)
where α and λ are constants. In f (R, T ) theory, wormhole solutions with normal matter
are feasible when appropriate shape functions are used. The coupling parameters α and λ
in the action of f (R, T ) theory play an important role in accommodating the composition
of matter. According to [17], when α < 0, wormholes exist in the presence of exotic matter,
and when α > 0, wormholes exist even in the absence of exotic matter.

In Brans–Dicke theory [36], a scalar field ϕ is introduced to modify the standard

Einstein–Hilbert action as follows [37]:
√
Z
ω
S = [ϕR − ϕ;µ ϕ;µ + Lm ] −g d4 x, (8)
ϕ

5
where ϕ is the Brans–Dicke scalar field, ϕ;µ is the covariant derivative of ϕ, and ω is a
coupling constant that couples the Brans–Dicke scalar field ϕ with the gravitational field.
Lm is the Lagrangian density for the matter field(s).

The field equations in Brans–Dicke theory can be obtained as always by varying the
action. These field equations are modified as compared to Einstein’s field equations in GR,
and they can be expressed in terms of the scalar field ϕ and its derivatives, as well as the
metric tensor, the curvature tensors, and the scalars. The value of the coupling parameter
ω affects the behavior of the theory, and it can influence the stability of wormholes.

Kaluza–Klein (KK) theory [38] extends the usual four-dimensional spacetime of GR to

include one or more extra dimensions. These extra dimensions are compactified or ‘curled
up’ so small that they are not perceptible on macroscopic scales. The total spacetime is a
product of the usual four-dimensional spacetime and the compactified extra dimensions.

In KK theory, the metric tensor describes the geometry of the higher-dimensional space-
time. It has components corresponding to both the usual four dimensions denoted by µ and
ν and the extra dimensions denoted by a and b. The metric tensor can be decomposed into a
four-dimensional part and an extra-dimensional part. The extra-dimensional part manifests
as an additional vector or scalar fields, and these fields are typically associated with elec-
tromagnetic interactions. This process of decomposing the higher dimensional metric tensor
field is called dimensional reduction.

The action of KK theory after dimensional reduction is

√
Z
S = [R(4) − F ab Fab − Lm ] −g d4 x, (9)

where d4 x is the four-dimensional spacetime volume element, R(4) is the four-dimensional

scalar curvature, F ab is the electromagnetic field strength tensor, and Lm is the usual La-
grangian density for the matter field(s).

There are currently three sets of geometric theories of gravity. The first one is the
general theory of relativity based on curvature. The second is teleparallel gravity based
on torsion (T), as we have seen in f (T ) theory. The third set of geometric theories is
based on nonmetricity (Q), as described here. The origin of f (Q) theory is from ‘symmetric
teleparallel gravity’, which is based on the nonmetricity scalar Q. Nonmetricity is defined as
the covariant derivative of the metric tensor gµν , i.e., Qαµν ≡ ∇α gµν . It vanishes in the case
of Riemannian geometry, and it can be used to study non-Riemannian spacetimes. f (Q)
gravity has inspired research in blackholes, wormholes, and cosmology. In cosmology, f (Q)
models can be used to explain phenomena related to both early and late time cosmology [39],
without dark energy, dark matter, or the inflation field. The action of f (Q) gravity is given
by [40]
√
Z
1
S = [− f (Q) + Lm ] −gd4 x. (10)
16πG
Here, as usual, g is the determinant of the metric gµν , and Lm is the matter Lagrangian
density. f (Q) is an arbitrary function of the nonmetricity scalar Q [41] given by
1 1 1 1
Q = − Qαβγ Qαβγ + Qαβγ Qγβα + Qα Qα − Qα Q̃α , (11)
4 2 4 2

6
where Qα ≡ Qµαµ and Q̃α ≡ Qµα
µ are two independent traces of the nonmetricity tensor Qαµν ,
and these are obtained by contracting the nonvanishing tensor Qαµν ≡ ∇α gµν .

In f (Q, T ) theories [42], the nonmetricity is coupled minimally to the trace of the matter
energy–momentum tensor. The coupling between Q and T leads to nonconservation of the
energy–momentum tensor, which has important physical implications such as changes to
the thermodynamics of the Universe [43], the nongeodesic motion of test particles, and the
appearance of an additional force.

In f (R, Lm ) theory [44], f is a function of both the Ricci scalar and the matter La-
grangian. It is possible to have both an additive function such as
R
f (R, Lm ) = + Lm (12)
2
or an exponential function such as
 
1 1
( 2Λ )R+( Λ )(Lm )
f (R, Lm ) = Λe , (13)

where Λ > 0 is an arbitrary constant. This function becomes

R
f (R, Lm ) ≈ Λ + + Lm + ..., (14)
2
 1
)R + ( Λ1 )Lm ≪ 1. The observed late time acceleration of the Universe

in the limit ( 2Λ
can be described by f (R, Lm ) gravity [45, 46]. In [47], the MT wormhole solution was
studied in f (R, Lm ) gravity assuming three different types of EOSs, namely linear barotropic,
anisotropic, and isotropic EOSs. The wormhole solutions obey the flare-out condition for
both the barotropic and the anisotropic cases. For the isotropic case, the shape function
does not follow the flatness condition. The NEC is violated in the vicinity of the throat.
The amount of exotic matter required near the wormhole throat is minimized in f (R, Lm )
gravity as compared to GR.

In the next two subsections, we take a deeper dive into f (R) and Lovelock theories
of gravity, which will set the required background to study wormholes in these theories.
In Section 3, we provide a synopsis of the study of wormholes in each of these f (R) theories
of gravity.

2.2 f(R) Gravity Theories

We started the discussion of f (R) theories in Section 2.1. To recap, in f (R) the-
ories [21, 22, 31, 48] R is replaced by f (R) as follows:
√
Z
1
S= f (R) −g d4 x, (15)
2κ
8πG
where κ ≡ c4
, and c = 1.

In f (R) theories, there are two formalisms [20, 49] to derive the Einstein field equations
from the action. They are (i) the metric formalism, in which a matter term Sm (gµν , ψ) is

7
added to the action, where ψ represents the matter field(s). The action is then varied with
respect to the metric by not treating the connections Γµαβ independently to obtain the field
equations. Then, there is (ii) the Palatini formalism, in which an independent variation
is done with respect to the metric and the connection. The action is the same, but the
curvature tensors and Ricci scalar are constructed with this independent connection.

Here, we will follow [21, 48] and show the derivation of the Einstein field equations in
much more detail using the metric formalism. The connection coefficient Γαβγ and the com-
α
ponents of the Riemann curvature tensor Rβγδ are calculated using the standard equations

1
Γαβγ = g αλ (gλβ,γ + gλγ,β − gβγ,λ ), (16)
2
and
α
Rβγδ = Γαβδ,γ − Γαβγ,δ + Γαλγ Γλβδ − Γαλδ Γλβγ , (17)
where the comma denotes partial derivatives. Before we vary the action, we first vary each
of the quantities in the action. The variation of the determinant is
√ √
δ −g = −(1/2) −ggµν δg µν . (18)

The Ricci Scalar is R = g µν Rµν , and the variation of R with respect to g µν is

δR = Rµν δg µν + g µν δRµν , (19)

= Rµν δg µν + g µν (∇ρ δΓρνµ − ∇ν δΓρρµ ), (20)

where
δΓρµν = (1/2)g ρα (∇µ δgαν + ∇ν δgαµ − ∇α δgµν ), (21)
Γρµν is the Christoffel symbol representing the Levi–Civita connection, and ∇µ is a covariant
derivative. Now, by substituting (21) in (20), we obtain

δR = Rµν δg µν + gµν □δg µν − ∇µ ∇ν δg µν , (22)

where □ ≡ g αβ ∇α ∇β is known as the D’Alembert operator. The variation of f (R) is

df (R)
δf (R) = δR. (23)
dR

df (R)
Let f ′ (R) ≡ dR
. Then,
δf (R) = f ′ (R)δR. (24)

By varying the action (15), we obtain

√ √ 
Z
1
δS = δf (R) −g + f (R)δ −g d4 x. (25)
2κ

√
Substituting δf (R) and δ −g from (24) and (18) into (25), we obtain
√ 1√
Z  
1 ′
δS = f (R)δR −g − −ggµν δg f (R) d4 x.
µν
(26)
2κ 2

8
 Now, we substitute δR from (22) into (26) to obtain
Z √    
−g ′ µν
 
µν 1 µν
δS = f (R) Rµν δg + gµν □ − ∇µ ∇ν δg − gµν f (R)δg d4 x. (27)
2κ 2

After integrating by parts and factoring out δg µν , we obtain

1√
Z  
µν ′ 1  
′
δS = −gδg f (R)Rµν − gµν f (R) + gµν □ − ∇µ ∇ν f (R) d4 x. (28)
2κ 2

Finally, by requiring that the action remain invariant with the variation of the metric,
we obtain the field equations for f (R) modified gravity theory:
1  
f ′ (R)Rµν − gµν f (R) + gµν □ − ∇µ ∇ν f ′ (R) = κTµν , (29)
2
where √

−2 δ −gLm
Tµν =√ , (30)
−g δg µν
and Lm is the Lagrangian for matter.

2.3 Lovelock Gravity Theory

Introduced in 1971, Lovelock’s theory of gravity [50] is considered to be the most gen-
eralized extension to the theory of gravitation in D dimensions, because it satisfies the
requirements of GR that the field equations be covariant and not include more than the
second-order derivatives of the metric.

We consider a generally covariant theory of gravity in D dimensions. The Lagrangian

is a functional of independent variables (g µν , Rµνρσ ), and the action is
√
Z
S = dD x −gL[g µν , Rµνρσ ] + Sm . (31)
v

where Sm is the action of the matter. Note that, depending on the purpose, different pairs
of independent variables might be chosen to describe the system, such as (g µν , Rµνρσ ) or
(g µν , Rµνρσ ). However, the pair of (g µν , Rµνρσ ) is the most appropriate one when deriving the
field equations. For later convenience, we define the following:
 ∂L   ∂L 
µν µνρσ
P := , P := (32)
∂gµν Rρσλγ ∂Rµνρσ gλγ
where, by definition, P µν is symmetric, and the entropy tensor P µνρσ follows the symmetries

P µνρσ = −P νµρσ = −P µνσρ , P µνρσ = P ρσµν , P µ[νρσ] = 0. (33)

We further construct the generalized form of the Ricci tensor as

Rµν = P µρσλ Rνρσλ (34)

9
where it can be shown that [51, 52]
 ∂L 
= Rµν . (35)
∂g µν Rρσλγ

To obtain equations of motion, one needs to vary the action under the variation of the
metric g µν → g µν + δg µν that leads to the following variations

δgµν = −gµρ gνσ δg ρσ (36)

 
δRµνρσ = g µλ ∇ρ ∇ν δgσλ − ∇ρ ∇λ δgσν − ρ ↔ σ .

By varying the action (31), plugging in (36), and doing the partial integration, we obtain
√ n
Z
1 o
D
δS = d x −g Pµν − gµν L + 2∇ ∇ Pµρσν δg µν
ρ σ
(37)
v 2
where we have used the symmetry properties of Pµνρσ as well. Finally, the field equations of
motion read as Eµν = (1/2)Tµν , where

1
Eµν = Rµν − gµν L + 2∇ρ ∇σ Pµρσν , (38)
2
and Tµν is the energy–momentum tensor of matter. Moreover, Pµν is replaced by the gener-
alized Ricci tensor Rµν via (35).

Given the fact that P µνρσ already contains second derivatives of the metric by definition,
one realizes that the last term in the field equations (38) will involve up to fourth derivatives
of gµν . To ensure that we do not include derivatives higher than the second order, we impose
the following condition:
∇λ P µνρσ = 0. (39)

Even though one might think of imposing ∇λ ∇γ P µνρσ = 0 as a more general condition,
it turns out that this condition does not make any difference. Imposing (39) reduces the
field equations to
1 1
Rµν − gµν L = Tµν (40)
2 2
which are nonlinear in second derivatives of the metric. In fact, imposing the linearity
condition leads to the Einstein–Hilbert action. Here, one must be careful, since the form
of the field equations is analogous to Einstein’s field equations. However, Rµν here is the
generalized Ricci tensor defined in (34).

The next step is finding the form of the generalized Lagrangian that leads to field
equations with no derivatives higher than the second order. This task reduces to finding
those scalar functions of the metric and the Riemann tensor that satisfy (39). It can be
shown that, at each order, such functions are uniquely constructed as

(m) 1 1 µ1 ν1 ···µm νm ρ1 σ1
LD = m
δρ1 σ1 ···ρm σm R µ1 ν1 · · · Rρm σµmm νm , (41)
16π 2

10
which determines the order m Lagrangian in D dimensions. It contains m factors of the
···µm νm
Riemann tensor, and δρµ11σν11···ρm σm
is a completely antisymmetric determinant tensor defined as
 
δβα δρα1 ··· δσαm
 
 µ 
 δ 1 
αµ1 ν1 ···µm νm  β
δβρ = det  .  . (42)

1 σ1 ···ρm σm
 .. ···µm νm
δρµ11σν11···ρ 
 m σm 
 νm 
δβ

The entire Lanczos–Lovelock Lagrangian is then given by a sum over various orders of
(m)
LD , where X (m)
LD = cm LD . (43)
m

The expansion coefficients cm are initially arbitrary. The detailed explanation of the
construction procedure of Lanczos–Lovelock (LL) Lagrangian and the proof of its uniqueness
are out of the scope for this review. So, we refer the interested reader to the original papers
by Lanczos [53, 54] and Lovelock [50].

Note that the order m Lagrangian in (41) has been written in terms of Rµνρσ . This
change of variable enables us to use the following identity that simplifies our guesses for a
generalized form of the Lagrangian, where
 ∂L 
= 0. (44)
∂g µν Rρσλγ

This identity implies that the Lagrangian must be independent of the metric tensor,
and hence, indices of the Riemann tensor Rµνρσ must be contracted only by the Kronecker
deltas in the form that we have in (41).

The zeroth order Lagrangian has m = 0, which simply leads to a constant term (e.g.,
cosmological constant). In the first order, we obtain
1 µρ νσ
L(1) = δ R =R. (45)
32π νσ µρ
So, the first order of the LL Lagrangian gives back the Einstein–Hilbert Lagrangian. As ex-
pected, the generalized equations of motion (40) simply reduce to the Einstein fields equa-
tions for L(0) + L(1) , i.e.,
Λgµν + Gµν = κTµν , (46)
where Λ is the cosmological constant, and Gµν = Rµν − 12 Rgµν is the well-known Einstein ten-
sor.

To study the wormhole solutions of generalized gravity discussed in [18, 55], we go up

to the third order in the LL Lagrangian:
√
Z
S = dD x −g α1 L(1) + α2 L(2) + α3 L(3) + Sm


(47)
v

11
where the next two orders are determined as
1  
L(2) =
16π
Rµνγδ Rµνγδ − 4Rµν Rµν + R2 , (48)
1  µνσκ
L(3) = 2R Rµκρτ Rρτ µν + 8Rµνσρ Rσκντ Rρτ µκ + 24Rµνσκ Rσκνρ Rρµ + 3RRµνσκ Rσκµν
16π

+ 24Rµνσκ Rσµ Rκν + 16Rµν Rνσ Rσµ − 12RRµν Rµν + R3 , (49)

and αm are the Lovelock coefficients. The second order term (48) is known as the Gauss–
Bonnet Lagrangian.

The generalized field equation (40) corresponding to various orders of the Lagrangian
can be written separately as well,
X 1
αm R(m) (m)


Λgµν + Gµν + µν − gµν L = κm Tµν , (50)
m≥2
2

and Tµν is the energy–momentum tensor.

The field equations corresponding to the second and third order Lagrangian are respec-
(2) (3)
tively determined by Rµν and Rµν which are
1 
R(2)
µν ≡ R R
µσκτ ν
σκτ
− 2Rµρνσ R ρσ
− 2Rµσ R σ
ν + RR µν , (51)
8π
and
−3  τ ρσκ
R(3)
µν ≡ 4R Rσκλρ Rλντ µ − 8Rτ ρλσ Rσκτ µ Rλνρκ + 2Rντ σκ Rσκλρ Rλρτ µ
16π
− Rτ ρσκ Rσκτ ρ Rνµ + 8Rτνσρ Rσκτ µ Rρκ + 8Rσντ κ Rτ ρσµ Rκρ
+ 4Rντ σκ Rσκµρ Rρτ − 4Rντ σκ Rσκτ ρ Rρµ + 4Rτ ρσκ Rσκτ µ Rνρ + 2RRνκτ ρ Rτ ρκµ (52)
τ ρ σ
+ 8R νµρ R σ R τ − 8Rσντ ρ Rτσ Rρµ − 8Rτ ρσµ Rστ Rνρ
− 4RR τ ρ
νµρ R τ

+4Rτ ρ Rρτ Rνµ − 8Rτν Rτ ρ Rρµ + 4RRνρ Rρµ − R2 Rµν .

3 Fundamentals of Morris-Thorne (MT) Wormhole

In Section 3.1, we give a brief background of how the stability of a wormhole (usually
an MT wormhole) is studied in GR. In Section 3.2, we summarize a general methodology
to study wormholes in f (R) gravity theories based on the various papers we reviewed on
this subject. This background will be useful to follow Section 4, where we review these
calculations in greater detail for general f (R) theories and Lovelock gravity theory.

3.1 MT Wormhole Stabilization in GR

In [9], they explain in great detail why blackholes and Schwarzschild wormholes are
not traversable. However, the MT wormhole is designed to be made traversable if it has
the following properties that ensure wormhole stability for traversability. The MT metric
allows for the realization of faster-than-light interstellar space travel that does not violate

12
the special relativistic light speed limit. The metric should be spherically symmetric and
static (time independent). The following metric has these properties [9]:

ds2 = −e2Φ dt2 + [1 − (b/r)]−1 dr2 + r2 [dθ2 + sin2 θdϕ2 ]. (53)

The solution must obey the Einstein field equations as does (53). The solution must have
a throat that connects two asymptotically flat regions of spacetime. The spatial geometry
must have a wormhole shape consistent with the well-known Flamm diagram for a spherically
symmetric throat. This puts the following constraints on the shape function b(r) and redshift
function Φ(r):

• The throat is at minimum of r, which is specified as r0 .

• b(r) is finite, continuous, and differentiable.
• In this spacetime, (1 − b/r) ≥ 0, which implies b/r ≤ 1, and so b(r) ≤ r.
• Proper radial distance is defined by
Z r
1
l≡ p dr,
r0 1 − (b/r)

and should be real and finite for r > r0 .

• As l →
− ±∞ (asymptotically flat regions of the Universe), b/r → − 0, and so r ∼ |l|.
• There should be no horizons, since it will prevent two-way travel through the wormhole.
There are no singularities. This implies that Φ is finite, continuous, and differentiable
everywhere, as well as the fact that τ measuring proper time in asymptotically flat
regions implies Φ →
− 0 as l → ± ∞.
• The flare-out condition is
(b − b′ r)
> 0,
2b2
and so rb′ < b. That is, the throat of the wormhole must expand outward from the
central point. The throat of the wormhole must open up as one travels through it.

The tidal gravitational forces experienced by a traveler must be ≤g, where g is the acceler-
ation due to Earth’s gravity. This condition is not a rigid requirement.

The procedure to check the stability of this traversable wormhole in GR involves the
following steps:

Compute Curvature tensors: Here, we give as briefly as possible the method to

calculate the curvature tensors. First, using the MT metric written in the form

ds2 = gαβ dxα dxβ , (54)

with x0 = t, x1 = r, x2 = θ, and x3 = ϕ, the connection coefficient Γαβγ and the components

α
of the Riemann curvature tensor Rβγδ are calculated using the standard Equations (16) and
(17). By applying these equations to the metric, we can obtain the 24 nonzero components
of the Riemann tensor, as shown in Equation (5) of [9]. These were obtained using the basis
vectors (et , er , eθ , eϕ ).

To further simplify the calculations, we can switch to the following orthonormal ba-
sis vectors:

13
 • eˆt = e−Φ et ’
• eˆr = (1 − rb )−1/2 er ;
• eˆθ = r−1 eθ ;
• eˆϕ = (r sin θ)−1 eϕ .

In this basis, the metric coefficients become the same as those of flat (Minkowski) space-
time,  
−1 0 0 0
 0 1 0 0
gαβ = eˆα eˆβ = ηαβ = 
 0 0 1 0 ,
 (55)
0 0 0 1
and the 24 nonzero components of the Riemann tensor take a much simpler form, as seen in
Equation (8) of [9].

Contraction: Next, by contracting the Riemann tensor, we obtain the Ricci tensor,
α
Rµν = Rµαν , (56)
and again, by contracting the Ricci tensor, we obtain the Ricci scalar R.

Compute Einstein tensor: We finally obtain the Einstein tensor Gµν from the met-
ric, Ricci tensor, and the Ricci scalar. This forms the left-hand side of the Einstein field
equations, as given in (46). This yields the nonzero components of Gµν in terms of the shape
function and redshift function, namely
b′
Gtt = , (57)
r2
−b b Φ′
Grr = + 2(1 − ) , (58)
r3 r r
and
(b′ r − b) ′ Φ′ (b′ r − b)
 
b ′′ ′ 2
Gθθ = Gϕϕ = (1 − ) Φ − Φ + (Φ ) + − 2 . (59)
r 2r(r − b) r 2r (r − b)

Compute stress–energy tensor: The next step involves computing the stress–energy
tensor (the right-hand side of the Einstein field equations). Based on Birkhoff’s theorem,
which states that “any spherically symmetric solution of the vacuum field equations must
be static and asymptotically flat”, the exterior solution must be given by the Schwarzschild
metric (with certain modifications), which is a spherical wormhole. Therefore, we cannot
have a vacuum solution for a traversable wormhole, which implies that our wormhole must be
threaded by matter with a nonzero stress–energy tensor. Based on Einstein’s field equations,
Gµν = κTµν . (60)
Tµν must have the same algebraic structure as Gµν in the orthonormal basis that we have
chosen. Similar to Gµν , only the four components Ttt , Trr , Tθθ , and Tϕϕ are nonzero. Based
on a remote static observer’s measurement, each of the components have a simple physical
interpretation as follows:
Ttt = ρ(r) (61)
is the total rest energy density that the static observer measures,
Trr = −τ (r) (62)

14
is the radial tension measured per unit area, and

Tθθ = Tϕϕ = p(r) (63)

is the tangential pressure measured per unit area in a direction orthogonal to the radial
tension τ (r). This give us the final stress–energy tensor:
 
ρ(r) 0 0 0
 0 −τ (r) 0 0 
Tµν =  0
. (64)
0 p(r) 0 
0 0 0 p(r)

Engineer the traversable wormhole: In the next step, we need to ‘engineer’ the
traversable wormhole to obtain the properties enumerated earlier in this section. This can
be done by controlling the shape function b(r) and the redshift function Φ(r) based on a
suitable Tµν that we require. We substitute the Gµν and Tµν found in the previous steps into
(60) to solve for ρ(r), τ (r), and p(r) in terms of b(r) and Φ(r) to obtain

b′
ρ(r) = , (65)
r2
b 2(r − b) ′
τ (r) = 3
− Φ, (66)
r r2
r
p(r) = [(ρ − τ )Φ′ − τ ′ ] − τ. (67)
2

Our strategy for the stabilization of the MT wormhole in Einstein gravity will involve
tailoring b(r) and Φ(r) to build a wormhole with the required properties. Our choice of b(r)
gives us ρ(r). Our choice of b(r) and Φ(r) will give us the tangential pressure τ (r). We next
find p(r) using ρ(r), τ (r), and Φ(r). We then ensure that the averaged NEC and WEC are
violated, which means that the source of matter for traversable wormholes must be exotic.

With this process for analyzing the stability of the MT wormhole in GR, many authors
in our review have used similar techniques to analyze the stability of the Morris–Thorne
wormhole in f (R) gravity theories. We will hone into the nuances of these analysis in the
following sections.

3.2 A General Methodology for MT Wormhole Stabilization in

f(R) Gravity Theories

Based on Section 4, we briefly summarize the method that can be used to stabilize the
MT wormhole in f (R) gravity theories. First, we require that matter threading the worm-
hole satisfy the NEC and WEC. We include the required flare-out condition for traversable
wormholes. We delegate the required energy condition violations and sustenance of the non-
standard wormhole geometry to the higher-order curvature terms, including the derivative
(c)
terms (Tµν ). Consider a redshift function (Φ = c, Φ′ = 0), where c is a constant, which
simplifies calculations and still provides physically relevant solutions. This condition defines
an ultrastatic traversable wormhole and is a very narrow and specific subclass of solutions

15
called zero tidal force (ZTF) solutions. We specify b(r). Some examples used in [48] are
r2 √
b(r) = r0 , b(r) = r0 r, and b(r) = r0 + γ 2 r0 (1 − rr0 ), with 0 < r < 1. For each shape
function b(r), we assume an equation of state such as pr = pr (ρ) or pt = pt (ρ). We find F (r)
(see Section 4.1) from the modified gravitational field equations, with the Ricci curvature
scalar R(r) obtained from the MT metric. Finally, we obtain the exact f (R) that we need
from the trace Equation (68).

4 Wormholes in f(R) Gravity Theories

4.1 Wormholes in f(R) Gravity Theory

In this section, we mainly follow the references [21, 22, 31, 48] to analyze the stability of
an MT wormhole in f (R) gravity theories. In Section 2.2, we obtained (29) for f (R) gravity
theory. Contracting this field equation, we obtain the trace of Tµν ,
F R − 2f (R) + 3□F = T, (68)
where F = f ′ (R) is used for convenience, R is the Ricci scalar, and T = Tµµ is the trace of
the stress–energy tensor Tµν . By substituting (68) into (29) and the rearranging terms, we
obtain an updated field equation:
1 eff
Gµν ≡ Rµν − Rgµν = Tµν . (69)
2

We will call (69) the effective field equation for f (R) gravity theory, where
eff (c) (m)
Tµν = Tµν + Tµν . (70)

(c)
Tµν is the curvature stress–energy tensor for a higher-order curvature given by
 
(c) 1 1 
Tµν = ∇µ ∇ν F − gµν RF + □F + T , (71)
F 4
(m)
and Tµν = TFµν . Here, we write the energy–momentum tensor in terms of an anisotropic
distribution of matter as follows:
Tµν = (ρ + pt )uµ uν + pt gµν + (pr − pt )xµ xν , (72)
such that uµ uµ = −1 and xµ xµ = −1, where uµ is a four-velocity vector,
r
b′ (r) µ
xµ = 1 − δ
r r
is a unit spacelike vector in the radial direction, ρ(r) is the rest energy density, pr (r) is the
radial tension, and pt (r) is the tangential pressure orthogonal to xµ . This gives us
 
−ρ(r) 0 0 0
 0 pr (r) 0 0 
Tµν = 
 0
 (73)
0 pr (r) 0 
0 0 0 pt (r)

16
and
T = (Tµµ ) = (−ρ + 2pr + pt ). (74)

By substituting this value of T in the trace form of the field equations for f (R) gravity
(68), we obtain
F R − 2f + 3□F = (−ρ + 2pr + pt ). (75)

We now use the MT wormhole metric (53) just as we did in GR, where Φ(r) is the
redshift function, and b(r) is the shape function as before. The radial coordinate r decreases
from ∞ to a minimum value r0 at the wormhole throat. At the throat, b(r0 ) = r0 , and it
then increases from r0 back to ∞. We recall the flare-out condition, which is important for
traversability:
b − b′ (r)


> 0. (76)
b2

At the throat, b(r0 ) = r = r0 b(r0 ) > b′ (r0 )r or r > b′ (r0 )r, b′ (r0 ) < 1 is the flare-
out condition that we need for a traversable wormhole solution. For the wormhole to be
traversable, no horizon should be present. Horizons are surfaces with e2Φ → − 0, so we
want Φ(r) to be finite everywhere. This implies that we can use the constant redshift
functions Φ(r), Φ = c, and Φ′ = 0. This will help to simplify calculations and avoid fourth-
order differential equations. The effect of using a variable redshift function is discussed in
Section 4.3. The following steps can be used to calculate ρ, pr , and pt in terms of the shape
and redshift functions. Substitute Tµν in the effective field Equation (69). Use the MT
′
metric (53) to obtain R = 2br . Use

b′ r − b 2F ′
 
′′ ′
□F = (1 − b/r) F − 2 F + . (77)
2r (1 − b/r) r

Define
1
H(r) ≡ (F R + □F + T ). (78)
4
′′ ′
d df (R)
Note that F ′ = dR ( dR ), and F = d(FdR(R)) . Thus, we obtain the following simplified
equations from the effective field Equation (69):

F b′
ρ= , (79)
r2
−bF F′ ′ ′′
pr = 3
+ 2
(b r − b) − F (1 − b/r), (80)
r 2r
and
F′ F
pt = − (1 − b/r) + 3 . (81)
r 2r (b − b′ r)

These are the required simplified equations for matter threading the wormhole as a
function of b(r) and F (r). We will use them later in this paper. We can now determine
the matter content by choosing an appropriate shape function and a specific form of F (r).
At this point, let us recap the strategy for stabilizing a wormhole in f (R) gravity. We
choose a shape function b(r). Then, we specify an equation of state such as pr = pr (ρ) or

17
pt = pt (ρ). This will let us compute F (r) from the effective field equation of Equation (69).
We can also find the Ricci scalar R(r) from the MT wormhole metric (53). Then, we obtain
T = Tµµ as a function of r. Finally, we compute the function f (R) from the trace of the field
Equation (68).

4.2 Violation of Energy Conditions

Just as the motion of a single particle is governed by the geodesic equation, the equation
of motion of a family of particles, also known as a congruence, is governed by the Raychaud-
huri equation [56]. From this equation, the following focusing condition in terms of the Ricci
tensor arises:
Rµν k µ k ν ≥ 0,
where k µ is a null vector. If this condition is satisfied, then the geodesic congruences focus
into a finite value of the parameter for labeling points on the geodesics. In GR, this is written
as the NEC, in terms of the stress–energy tensor, as follows:

Tµν k µ k ν ≥ 0.

(m)
In modified gravity, in particular in f (R) theories, we could first assume that Tµν
satisfies this energy condition, and violation of the energy condition can be assumed to come
(c)
from the higher-order curvature terms Tµν . Note that this condition applied to f (R) gravity
theory does not mean geodesics are focused, as required by the Raychaudhuri equation. In
terms of the radial null vector, violation of the NEC requires
eff µ ν
Tµν k k < 0, (82)

and takes the form

ρ + pr 1 h ′′ b′ r − b i
ρeff + peff
r = = (1 − b/r) F − F ′ 2 , (83)
F F 2r (1 − b/r)
where
ρeff + peff
r < 0.

Using the gravitational field equation of Equation (69), we obtain

b′ r − b
ρeff + peff
r = . (84)
r3

By applying the flare-out condition,

b′ r − b
< 0,
b2
ρeff + peff
r < 0.

At the throat, with r = r0 ,

ρ + pr 1 − b′ (r0 ) F ′
ρeff + peff
r |r0 = |r0 + |r < 0. (85)
F 2r0 F 0

18
 At the throat, this gives us

(ρ + pr )
F0′ < −2r0 |r ; F >0 (86)
(1 − b′ ) 0

(ρ + pr )
F0′ > −2r0 |r ; F <0 (87)
(1 − b′ ) 0

We have been highlighting that matter threading the wormhole should obey the WEC as
well, i.e., ρ ≥ 0, and ρ + pr ≥ 0. Applying these WECs to the simplified Equations (79)–(81)
for ρ, pr , and pt , we obtain
F b′
≥0 (88)
r2
(2F + rF ′ )(b′ r − b) ′′

2
− F (1 − b/r) ≥ 0. (89)
2r
The four inequalities above (86)–(89) must be obeyed by the function f (R) for traversable
wormholes in f (R) theories, thus taking into account that matter threading the wormhole
satisfies the NEC and WEC, and the flare-out conditions are satisfied at the wormhole
throat. The task of maintaining the wormhole geometry (violating the NEC) is delegated to
(c)
the higher-order curvature terms Tµν .

4.3 Effect of Using a Variable Redshift Function

The effect of using a variable redshift function for wormholes has been studied by [57] in
κ(R, T ) gravity with Φ(r) = αr , where α is a constant. The authors of [57] obtained solutions
that require exotic matter. Wormholes in
f (R) gravity were studied by [58–62] with a
r0
variable redshift function Φ(r) = ln r + 1 , and the authors in [58–62] found solutions that
require nonexotic matter. The authors in [63] studied the effect of a variable redshift function
Φ(r) = − αr with α ≥ 0, where α is a constant, and [64] used Φ(r) = αr . [21] in particular
studied wormholes in f(R) gravity with a variable redshift function Φ(r) = 1r in great detail.
Here, we discuss the solution obtained by them. [21] used the function f (R) = R − µRc ( RRc )p ,
where µ, Rc , and p are constants, with µ > 0, Rc > 0, and 0 < p < 1. [21] also used the
r
shape function b(r) = er−r 0
. Their motivation for selecting the above f (R) function is related
to obtaining viable dark energy models, and it is also related to finding explanations for the
accelerated expansion of the universe. Based on past studies, this f (R) function seems to
be a good candidate for exploring wormholes in f (R) gravity with both a constant and a
variable redshift function.

With the redshift function Φ = 1r , they found ρ > 0 for r ≥ 1.2, ρ + pr > 0 for r ≥ 1.2,
ρ + pt > 0 for r > 1.8, ρ + pr + 2pt < 0 for r > 1.2, ρ − |pr | > 0 for r ≥ 1.2, and ρ − |pt | > 0
for r > 1.8, and the NEC, WEC, and DEC were satisfied for r > 1.8 with the variable
redshift function. So in conclusion, for wormholes with r0 > 1.8—where r0 is the radius of
the throat—and with variable redshift function 1r , they obtained wormhole geometries free
of exotic matter. They also did similar calculations in GR with the variable redshift function
1
r
and found that there was no solution in GR without exotic matter for any value of r.

19
4.4 The Stability Condition and the Speed of Sound

In [65], the authors analyzed the stability of a wormhole in addition to traversability

in f (R) gravity. The authors in [65] used a power law f (R) = f0 R1+ϵ , which can be written in
the form f (R) = R + ϵRln(R) + O(ϵ2 ), for ϵ ≪ 1, to consider small deviations from Einstein
gravity. Such an approach was used in the study of compact objects such as neutron stars
and blackholes [66, 67], where departure from GR can be useful to explain observations.

For the analysis, the authors used the MT wormhole metric with a redshift function
r0 1+β
2Φ(r) = rr0 and two different shape functions b(r)


r
= r
, where β is a real number, β+1 >
b(r) r0 r0 −1
0, and r = 1+αr —with α ≡ r0 —to ensure that the wormhole is not singular at r0 .
They obtained the lengthy Equations (13)–(16) in [65] for ρ(r), pr (r), pt (r), and the average
pressure p(r) = 13 [pr (r)+2pt (r)], respectively. Next, they used ideas from fluid dynamics [68–
70] and defined the adiabatic speed of sound:
 
2 ∂p
cs = . (90)
∂ρ s

To obtain a stability condition, we need a vanishing speed of sound at the throat:

dp
|r = 0 (91)
dρ 0

By inserting Equations (13) and (14) in [65] for ρ(r) and pr (r) into Equation (91) above,
they obtained the required stability conditions, namely Equations (24) and (25) in [65],
for b(r)
r
= ( rr0 )β+1 and for b(r)
r
r0
= 1+αr , respectively. Their primary finding was that for
a specific range of values of ϵ in the assumed f (R), and using the condition for vanishing
speed of sound, stable wormhole solutions exist in the presence of nonexotic matter (a perfect
fluid).

4.5 Wormholes in Lovelock Gravity Theory

We will now discuss how the modifications in Lovelock gravity help with the stabilization
of a wormhole. In this section, we mainly follow [18, 55]. The second- and third-order
curvature terms in Lovelock gravity are important to be considered, especially in the case of
wormholes with smaller throat diameters, where the curvature is very high. First, the MT
wormhole metric is modified for n dimensions as follows:
 −1  n−2 Y
X i−1 
2 2Φ(r) 2 2 2 2 2 2
ds = −e dt + 1 − b(r)/r dr + r dθ1 + sin θj dθi , (92)
i=2 j=1

where Φ(r) is the redshift function, and b(r) is the shape function. The WEC requires that

matter threading the wormhole have positive energy density ρ(r) and positive ρ(r) − τ (r) ,
where τ (r) is the radial tension, as well as positive ρ(r) + p(r), where p(r) is the tangential
pressure orthogonal to the radius. If the WEC

ρ = Tµν uµ uν ≥ 0

20
is satisfied everywhere, then the wormhole can be constructed with normal matter without
the need for exotic matter. Here, uµ is a timelike velocity of the observer. Note that the
stress tensor Tµν used in the WEC is calculated using (50), which includes all the Lovelock
gravity modifications to Einstein’s field equations.

The NEC is
Tµν k µ k ν ≥ 0

The authors in [18] used the orthonormal basis set

∂
et̂ = e−Φ , (93)
∂t
 b(r) 1/2 ∂
er̂ = 1 − , (94)
r ∂r
∂
el̂ = r−1 , (95)
∂θ1
and
 Yi−1 −1 ∂
eî = r sin θj (96)
j=1
∂θi
to calculate the components of the energy–momentum tensor, which gives us ρ, τ , and p.
That is,
Ttt = ρ, (97)
Trr = −τ, (98)
and
Tii = p. (99)

Then, by setting the Φ(r) = c = 0, where c is a constant, (ρ − τ ) and (ρ + p) are

calculated as follows:
(n − 2) ′
 2α2 b 3α3 b2 
(ρ − τ ) = − (b − rb ) 1 + + (100)
2r3 r3 r6
and
′
15α3 b2  3α3 b2
 
(b − rb )  6α2 b b 2α2 b
(ρ + p) = − 1+ 3 + + 3 (n − 3) + (n − 5) 3 + (n − 7) 6 , (101)
2r3 r r6 r r r

where α2 and α3 are the second-order and third-order Lovelock coefficients. In order to
obtain positive ρ and (ρ + p) values, the rest of the analysis was done by studying three
types of shape functions b(r): the power law, the logarithmic, and the hyperbolic shape
functions. The power law shape function is given by
r0m
b= , (102)
rm−1
with positive m. The functions ρ and (ρ + p) for the power law above are positive for r > r0 ,
provided that r0 > rc , and rc is the largest positive real root of certain equations given
in [18]. For the logarithmic shape function, we have
r ln (r0 )
b(r) = , (103)
ln (r)

21
where ρ and (ρ + p) are positive for r > r0 , with the condition r0 ≥ rc , and rc is the largest
real root of a second set of equations in [18]. For the hyperbolic shape function, we have

r0 tanh (r)
b(r) = , (104)
tanh (r0 )

where ρ and (ρ + p) are positive if r0 > rc , with the condition that rc is the largest root of a
third set of equations in [18].

To ensure that (ρ − τ ) is positive, we need

2α2 b 3α3 b2
1+ + < 0,
r3 r6
where α2 and α3 are the Lovelock coefficients. For certain combinations of the Lovelock
coefficients contributing to the throat radius, with a negative value for either α2 or α3 ,
the above condition is satisfied only in the vicinity of the throat for all three types of shape
functions discussed above. The throat radius must be in the range r− < r0 < r+ with
r q
r− = −α2 − α22 − 3α3 , (105)

and r q
r+ = −α2 + α22 − 3α3 . (106)

The following points summarize why the higher-order curvature terms are needed to sta-
bilize a wormhole, as well as how the GB (second-order) and Lovelock (third-order) curvature
terms actually help with using normal matter as opposed to exotic matter in the vicinity of
the throat. Higher-order curvature corrections are used in wormholes with smaller throat
radii, since the curvature near the throat is very large for such wormholes. The matter
near the throat can be normal for the region r0 ≤ r ≤ rmax , where rmax depends on the
Lovelock coefficients and the shape function. We obtain a larger radius (more region) with
normal matter in the case of third-order Lovelock gravity with z negative coupling constant
α3 as compared to the second-order Gauss–Bonnet wormholes. There is a lower limit on the
throat radius r0 imposed by the positivity of ρ and (ρ + p) in the case of Lovelock gravity
but not in the case of Einstein gravity. The lower limit depends on the Lovelock coefficients,
the dimensionality of the spacetime, and the shape function.

4.6 Wormholes in Einstein Dilaton Gauss–Bonnet (EdGB) Grav-

ity

The authors in [71] reported stable wormhole solutions in EdGB gravity. Other authors
reported in [72] that the same wormhole solution in EdGB gravity was indeed unstable.
Since these two papers are excellent examples of stability analysis, in this section, we will
review their key findings and methods.

In order to avoid using the required exotic matter necessary to support traversable
wormholes, modified gravity theories were used. One such modified gravity theory is EdGB

22
gravity. In this theory, the low-energy heterotic-string-theory-based effective action in four
dimensions is given by [73, 74]
√ 
Z
1 1 
S= d4 x −g R − ∂µ ϕ∂ µ ϕ + αe−γϕ L(2) , (107)
16π 2
where, in addition to the scalar curvature term R, we have a quadratic GB curvature term
L(2) given by (48) and a scalar field term ϕ known as the dilaton field, with a coupling
constant γ. α is a positive numerical constant given in terms of the Regge slope parameter.

In [71], the authors analyzed the stability of a spherically symmetric wormhole solution
in EdGB gravity. With a coordinate transformation r2 = l2 + r02 , where r0 is the radius
at the throat, the spherically symmetric wormhole solution becomes pathology-free and is
given by
ds2 = −e2ν(l) dt2 + f (l)dl2 + (l2 + r02 ) dθ2 + sin2 θdφ2 .


(108)
In terms of the new coordinates, the expansion at the throat (l = 0) gives

f (l) = f0 + f1 l + ... (109)

e2ν(l) = e2ν0 (1 + ν1 l) + ... (110)

ϕ(l) = ϕ0 + ϕ1 l + ..., (111)
where fi , νi , and ϕi are constant coefficients, and all curvature invariants, including the GB
term, remain finite for l → 0. Next, we analyze the stability of the wormhole in terms of
radial perturbations. The metric and dilaton functions depend on both l and t. We then
decompose the functions into unperturbed and perturbed parts. The perturbations are

ν̃(l, t) = ν(l) + ϵδν(l)eiσt , (112)

f˜(l, t) = f (l) + ϵδf (l)eiσt , (113)

ϕ̃(l, t) = ϕ(l) + ϵδϕ(l)eiσt . (114)
σ is defined in [71] as an unspecified eigenvalue that appears in their Ordinary Differential
Equations (ODEs). Substituting these perturbations into the Einstein and Dilaton equations,
as well as linearizing in ϵ, we obtain a system of linear ODEs for the functions δν(l), δf (l),
and δϕ(l). Rearranging the ODEs, the authors arrived at a coupled second-order equation
in δϕ as follows:
(δϕ)′′ + q1 (δϕ)′ + (q0 + qσ σ 2 )δϕ = 0, (115)
where q0 , q1 , and qσ depend on the unperturbed solution. To be able to normalize, δϕ → 0
as l → ∞. At l = 0, qσ is bounded, and q0 and q1 diverge as 1l . To avoid this singularity,
a transformation is made as follows:

δϕ = F (l)ψ(l), (116)

where F (l) satisfies the equation

F′ q1 (l)
= . (117)
F 2

This implies that

′′
ψ + Q0 ψ + σ 2 qσ ψ = 0, (118)

23
 q′ q2
where Q0 = − 21 − 41 + q0 is bounded at l = 0. Then, ψ → 0 as l → ∞ and for l = 0.
Solving these ODEs for several values of rα2 and f0 , a solution exists only for certain values of
0
the eigenvalues σ 2 . The negative modes obtained for families of wormhole solutions indicate
that the wormhole is unstable. In addition, there are positive modes in a region of parameter
space where the EdGB wormhole solutions are linearly stable under radial perturbations.

In [72], the same wormhole was proved to be unstable for the following reason. In [71],
the perturbation function is
δϕ(t, l) = A(l)χ(t, l), (119)
where the factor A(l) diverges at the throat as 1l . Vanishing boundary conditions were
imposed at the throat to obtain finite perturbation at the throat. This disconnects the
regions of space on both sides of the throat for purely radial modes of perturbations. δϕ is not
a gauge invariant quantity. For general spherically symmetric perturbations of the wormhole,
2
the gauge invariant quantity χ(t, l) ∝ γδϕ(t, l) − rl ϕ′ (l)δr(t, l) satisfies the wavelike equation

∂ 2χ ∂ 2χ
− 2 + Vef f (l)χ(t, l) = 0, (120)
∂t2 ∂y
where Vef f is a finite effective potential. When χ is finite at the throat, δϕ does not diverge
unless the throat size is fixed by chosing δr = 0. Therefore, the wormhole is unstable with
perturbation at the throat radius. The behavior of the dilaton field ϕ at the throat is a pure
artifact of the chosen gauge and therefore can be safely ignored.

5 Other Wormhole Studies

5.1 Casimir Wormholes

Traversable wormhole solutions to Einstein’s field equations require the existence of

exotic matter (matter violating the weak, null, and dominant energy conditions). The most
common type of exotic matter proposed is that which has negative energy density ρ < 0.
Classically, negative energy density is thought to be impossible. However, in quantum field
theory, negative energy densities can and do occur. The most famous example of this is the
Casimir vacuum energy. The usual derivation of it follows quantizing the electromagnetic
field between two neutral parallel plates and renormalizing the energy. This results in an
energy density that is negative or less than the energy of the undisturbed vacuum:

−ℏcπ 2
ρ= , (121)
720a4
where a is the separation between the plates. As a result, there has been much speculation
and study undertaken into the possibility of using the Casimir vacuum energy to construct
traversable wormholes.

This section will serve as an introduction to the basic concepts and results of attempting
to use the Casimir vacuum energy to construct a traversable wormhole. Morris, Thorne, and
Yurtsever [9, 75] were the first to publish the idea of utilizing the Casimir effect to support
traversable wormholes. They noted that there were two distinct possibilities for utilizing

24
the Casimir effect. The first is to consider the Casimir vacuum energy as is, i.e., with a
representing the fixed plate separation. The second is to promote the plate separation to
the radial coordinate r.

The case of fixed plate separation was analyzed by Garattini [76], in which he studied
the connection between the Casimir energy and absurdly benign traversable wormholes. He
utilized the semiclassical Einstein field equations:
Gµν = ⟨Tµν ⟩, (122)
where ⟨Tµν ⟩ is the renormalized quantum expectation value of the stress–energy tensor,
and Gµν is the classical Einstein tensor. The justification for this equation was first given
by Hawking [77] in his derivation for particle creation by black holes, in which he argued
that quantum gravity contributions to the Einstein field equations can be assumed negligible
above the Planck length. The same assumption is made here. Upon solving the Einstein field
equations using the metric for a static spherically symmetric traversable wormhole, defined
by Morris and Thorne, the shape function is found to be
π3
 
ℏG
b(r) = r0 − (r3 − r03 ), (123)
720a4 c3
where r0 is the throat radius. It is clear from this expression that this spacetime is not
asymptotically flat, and in fact, it is asymptotically de Sitter. As stated in [76], the Casimir
vacuum energy can then be viewed as a cosmological constant. However, in close proximity
to the throat, the shape function can be approximated as
r0 lp2 π 3
 
b(r) ≈ r0 1 − (r − r0 ) , (124)
90a4
where it has been assumed that r is very close to r0 , and lp is the Planck length. This
shape function has the same form as the shape function for “absurdly benign traversable
wormholes” defined by Morris and Thorne [9], namely
 2
(r − r0 )
b(r) = r0 1 − ; Φ(r) = 0; r0 ≤ r ≤ r0 + d, (125)
d
b(r) = 0; Φ(r) = 0; r > r0 + d, (126)
where d is given by (128), and exotic matter is confined in the region (r0 ≤ r ≤ r0 + d).

In order to make the connection between the Casimir shape function and this shape
function, the zero tidal force (ZTF) condition must be imposed, where Φ(r) = 0. The leading
order for this shape function close to the throat is
 
(r − r0 )
b(r) = r0 1 − 2 , (127)
d
which gives us the required connection between these two shape functions:
90a4
d= . (128)
r0 lp2 π 3

The absurdly benign traversable wormhole is defined so that the exotic matter is confined
close to the throat (d is small). In order to fit this requirement, there are two possibilities: the

25
first is r0 > 1034 m, and the second is a ∼ 10−15 m for a wormhole with r0 ∼ 1010 m. Neither
of these conditions are physical. It is noted that the identification made for the relationship
between d and a is a physically meaningless assumption. Therefore, Garattini attempted
to refine and reduce the throat radius by examining various examples of inhomogeneous
equations of state, and he managed to establish a throat radius of r0 ∼ 1017 m for a plate
separation of 1 nm. However, these results still present a nonphysical result, and they point
to the likelihood that the semiclassical Einstein field equations may be ill suited for the
description of Casimir traversable wormholes.

Promoting the plate separation to radial coordinates was discussed in [78] as follows.
Once again, semiclassical quantum gravity was used but now with the Casimir vacuum
energy density given by
−ℏcπ 2
ρ= (129)
720r4
where r is the radial coordinate. This gives the following shape function

r12 r12
b(r) = r0 − + (130)
r0 r

3
Here, r12 = πl2 . First, observe that this cannot be transformed into an absurdly benign
p
traversable wormhole. In fact, attempting to impose the ZTF condition produces disastrous
results. Instead, we use the Einstein field equation and the shape function to solve the
redshift function. We also use the EOS pr = ωρ. In solving the redshift function, it is
necessary to impose the condition ωr12 − r02 = 0. This condition is required for the existence
of the wormhole, since if ωr12 − r02 > 0, we obtain a black hole. With this condition, we
obtain the following shape function and redshift function, respectively,

r2
 
1
b(r) = 1 − r0 + 0 (131)
ω ωr

and  
1 rω
Φ(r) = (ω − 1) ln . (132)
2 rω + r0

These result in a wormhole of Planckian size.

We now return to the question of using the semiclassical Einstein field equations.
As stated above, Hawking originally justified the use of this approximation in the derivation
of Hawking radiation. In this scenario, it makes sense that quantum gravitational effects
can be ignored in the Einstein tensor, since Hawking never considered Planckian-sized black
holes, as the black holes would be expected to have evaporated before reaching this size. So,
any such effects could be assumed to be hidden by the formation of the event horizon.

When considering the use of the Casimir effect in constructing traversable wormholes,
we find that we might not be able to ignore quantum gravity. First, as we have just seen
above, promoting the plate separation to a radial coordinate results in a Planckian-sized
wormhole negating Hawking’s assumption of not having to deal with any system of this size.
For the fixed plate separation, we encountered solutions that produce wormholes with throat
radii on the order of the Sun’s radius for plate separation on the order of nanometers and

26
smaller. This makes no physical sense, as the Casimir vacuum energy is confined within the
plates and would have no way of interacting with a body of such size. Lastly, we do not have
any event horizon by definition and therefore cannot assume that quantum gravity effects
are hidden away. So far, no one has made an attempt to incorporate any such quantum
gravity effects into the analysis of Casimir wormholes, and it is therefore left as the subject
for future publications. However, other studies have been done on the subject of Casimir
traversable wormholes. For example, Garattini [79] studied the effects of an electric charge
on a Casimir wormhole. He found that the throat radius becomes directly dependent on the
strength of the electric charge, meaning that there exists the possibility of creating a throat
larger than the Planck length. However, it should be noted that, in his analysis, it was found
that there exists a radius at which the energy density becomes positive. It is not discussed
whether this negates the existence of the wormhole.

Casimir wormholes have also been studied in general D dimensions [80]. It has been
shown that, even in these scenarios, the radius is found to be on the order of the Planck
length or a nonphysical size. It is worth reiterating that these studies have been constructed
primarily using the semiclassical Einstein field equations and therefore do not include any
possible effects of quantum gravity, which is a fact that is most likely the source of many of
the nonphysical results.

5.2 Thin-Shell Wormholes in Modified Gravity

There is a set of wormhole solutions in modified gravity theories that use special tech-
niques to satisfy the energy conditions and thus remove the need for exotic matter. We
summarize the analysis of these wormholes here. In [81], the authors derived asymptoti-
cally flat traversable wormhole solutions that satisfy the NEC in a quadratic form of the
gravitational hybrid metric-Palatani gravity theory. Their solution has an interior part with
a nonexotic perfect fluid near the throat, an exterior Schwarzschild solution, and a dou-
ble gravitational layer thin shell at the junction hypersurface Σ between the interior and
exterior solution.

The generalized hybrid metric-Palatani gravity theory has the action

√ √
Z Z
1 4
S= 2 −gf (R, R)d x + −gLm d4 x. (133)
2κ Ω Ω

Here, κ2 ≡ 8πG
c4
, Ω is the spacetime manifold in which a set of coordinates xa are defined,
g is the determinant of the metric gab , and f (R, R) is a well-behaved function of R and
R. R ≡ g ab Rab is the Ricci scalar, R ≡ g ab Rab is the Palatini scalar curvature, and Rab
is the Palatini Ricci tensor written in terms of an independent connection Γcab . Lm is the
matter Lagrangian density minimally coupled to the metric gab . The junction conditions at
the separation hypersurface Σ were defined to be

[hαβ ] = 0, (134)

[K] = 0, (135)
[R] = 0, (136)
[R] = 0, (137)

27
 fRR na [∂a R] + fRR na [∂a R] = 0, (138)
and
2
 
fRR
ϵδαβ nc [∂c R] − (fR + fR )ϵ Kαβ = 8πSαβ .
 
fRR − (139)
fRR
where hαβ = eaα ebβ gab is the induced metric at the hypersurface Σ, Kαβ = eaα ebβ ∇a nb is the
extrinsic curvature, K = Kαα is the trace of the extrinsic curvature, and Sαβ is the stress–
energy tensor of the thin shell arising at the hypersurface Σ. It was shown in [81] that
the general set of junction conditions shown above can be simplified for particular forms of
f (R, R). In particular, they selected an f (R, R) that is quadratic in R and linear in R.
For this f (R, R), the junction conditions [R] = 0 and [R] = 0 are not mandatory conditions
anymore. This gives rise to additional terms in the stress–energy tensor Sab of the thin shell,
which then forms a double layer thin-shell distribution at the junction hypersurface Σ.

Unlike the general case [82], in which the solutions obtained using scalar–tensor repre-
sentation are scarce, the simplified set of junction conditions in [81] implies that it is possible
to obtain numerous solutions for a wide variety of metrics and actions. There is also the
advantage that asymptotic flatness can be preserved in this case, unlike the general case
where the NEC can be guaranteed only for the asymptotically flat AdS spacetime.

In [83], traversable wormholes with double layer thin shells in quadratic gravity were
analyzed, where f (R) ≡ R + αR2 , in which R is the Ricci scalar. The NEC is satisfied in
this case at the throat, as well as the whole wormhole interior. However, the NEC is not
satisfied for the double layer stress–energy distribution component at the thin shell.

In [84], traversable wormhole solutions were analyzed in f (R, T ) gravity theory for a
linear model, i.e., f (R, T ) = R + γT , where T = Tab T ab , and Tab is the energy–momentum
tensor. It was shown that there are a large set of wormhole solutions in which the matter field
satisfies all the energy conditions (NEC, WEC, SEC, and DEC). Since the field equations are
quadratic in the matter quantities ρ, pr , and pt , as well as complex, they used an analytical
recursive algorithm to extract the nonexotic wormhole solutions. The solutions obtained
were not naturally localized, so the junction conditions were derived for this theory. It was
proven that a matching between two spacetimes must always be smooth and does not allow
thin shells at the boundary. Traversable localized static and spherically symmetric wormhole
solutions satisfying all energy conditions were obtained by matching the interior wormhole
spacetime to an exterior vacuum Schwarzschild solution.

5.3 Dark Matter Halo Wormholes

It is well known that an exotic type of matter with negative energy density can stabilize
wormholes. Dark matter halos are vast, invisible regions of space that surround galaxies.
They are composed of dark matter [85], a substance that does not emit, absorb, or reflect
light, so they cannot be directly detected. Their presence is inferred through their gravi-
tational effect on visible matter such as stars and gas clouds. In recent studies of galaxy
formation, it was found that every galaxy forms within a dark matter halo [86]. The for-
mation and growth of galaxies over time is related to the growth of the halos in which
they form.

28
 In recent years, a few research groups have been investigating traversable wormholes
supported by the dark matter halo [87–90]—both in GR and modified theories of grav-
ity such as f(Q,T). Dark matter halo wormholes are usually studied using the Milky Way
galaxy (MWG) halo profiles, pseudothermal, Navarro–Frenk–White (NFW) models I and II,
and Universal Rotation Curves (URCs). A sample investigation of dark matter profiles in
the Milky Way galaxy can be found in [91]. In [90], they used the “Einasto dark matter
density profile” to produce suitable redshift and shape functions. The NEC is violated for
redshift functions Φ = C and Φ = αr , and the shape function satisfies the flare-out condition.
The anisotropic dark matter content within the wormhole creates the appropriate environ-
ment to stabilize the wormhole structure by violating the NEC. The global monopole charge
η [92–94] plays an important role in the violation of the NEC. The probability of violation
of the NEC decreases for an increasing value of η, and so it is important to minimize the
value of η.

In [95], wormholes supported by galactic halos have been investigated in 4D EGB grav-
ity. This analysis was done for three different dark matter profiles, namely URC, NFW,
and Scalar Field Dark Matter (SFDM). The NEC was violated at the neighborhood of the
wormhole throat. The Gauss–Bonnet coefficient α has an influence on the NEC. α > 0
gives negative energy. The contribution to the violation increases with α. The SEC is also
violated for each of the DM profiles.

5.4 Wormholes in Nonlocal Theories of Gravity (NLGs)

There is a class of nonlocal integral kernel theories of gravity where the inverse of the
d’Alembert operator in the gravitational action is taken into account. These are called non-
local theories of gravity (NLGs). A spherically symmetric MT wormhole solution satisfying
NLG field equations was studied in [96]. Linear and exponential NLG correction terms were
selected due to the existence of Noether symmetry [97]. It was found that the nonlocal grav-
ity contributions allow for the stability and traversability of a wormhole without considering
exotic matter.

6 Summary and Discussion

In this review paper, our goal is to review stable, traversable wormholes in f (R)-like
gravity theories. We started with a historic development of ideas about wormholes in the
Introduction section and then provideed some motivation for this paper and the need for
an equation of state (EOS) in Section 1.1. We then concluded the Introduction with a
discussion of the astronomical observational signatures of natural wormholes in Section 1.2.
In Section 2.1, we started with a brief discussion of each modified gravity theory such as
f (R), f (R, T ), Lovelock, EGB, Brans–Dicke, and KK theory, and we followed with a brief
discussion of nonmetricity theories such as f (Q) and f (Q, T ). We ended Section 2.1 with
a review of f (R, Lm ) theory and its applications. In Sections 2.2 and 2.3, we gave a more
detailed treatment of f (R) gravity theory and Lovelock gravity theory by deriving the EOM
from the action for the respective theory. In Section 3, we discussed the MT wormhole in
greater detail, since it is the standard traversable wormhole geometry used in most of the

29
studies of wormhole in modified gravity theories, as seen in the literature. In Section 4, we
reviewed the study of wormholes in f(R) gravity and Lovelock gravity based on the flare-
out condition and energy condition violation requirements. We concluded Section 4 with a
detailed review of stability analysis under a perturbation using EdGB gravity as an example.
In Section 5, we took the reader on a tour of other wormholes such as Casimir wormholes,
thin-shell wormholes in modified gravity, dark matter halo wormholes, and wormholes in
nonlocal theories of gravity.

In the examples that we discuss, it is possible to have stable wormholes without the use
of exotic matter in many of the modified gravity theories. In some cases, it is shown that
the amount of exotic matter that is needed can be minimized. In their analyses, the authors
varied the redshift function, shape function, the type of fluid used, the equation of state,
and the primary function used in the corresponding modified gravity theory.

The key takeaways for wormholes in f (R) gravity theory are as follows: In GR, violation
of the NEC is required for static traversable wormholes. In the f (R) theory of gravity, mod-
ified field equations are obtained by varying the modified action with respect to the metric.
Using these modified field equations, we require that matter (stress–energy tensor) threading
the wormhole satisfy the NEC, and the required violation of the NEC can be enabled by the
total stress–energy tensor, which includes higher-order curvature terms. The higher-order
curvature terms, interpreted as a gravitational fluid, support the nonstandard wormhole ge-
ometries. A constant redshift function is assumed in many of these analyses to reduce the
complexity of the calculations. It is possible to use a variable redshift function as well. Based
on the review of papers related to wormholes in Lovelock gravity, we have the following key
takeaways: Higher-order curvature terms become useful for the analysis of wormholes with
smaller throat radius. There exists a lower limit for the throat radius in Lovelock grav-
ity imposed by the requirement that ρ and (ρ + p) be positive. There is no such limit in
Einstein’s gravity. The radius of the region with normal matter is higher for wormholes in
third-order Lovelock gravity, with a negative coupling constant (α3 ), compared to wormholes
in second-order Gauss–Bonnet gravity.

In Section 5, we discussed “Other Wormholes” such as Casimir, thin-shell, dark matter

halo, and nonlocal gravity wormholes. In Casimir wormholes, the required negative energy
density for a stable traversable wormhole is provided by the Casimir effect. Casimir worm-
holes are currently being studied in several modified gravity theories. Thin-shell wormholes
use special techniques to satisfy the energy conditions and thus remove the need for ex-
otic matter. For example, there are asymptotically flat traversable wormhole solutions that
satisfy the NEC in the quadratic form of the gravitational hybrid metric-Palatani gravity.
These solution have an interior portion with a nonexotic perfect fluid near the throat, an ex-
terior Schwarzschild solution, and a double gravitational layer thin shell at the junction
hypersurface Σ between the interior and exterior solutions. In dark matter halo wormholes,
the dark matter content within the wormhole creates the appropriate environment to sta-
blize the wormhole structure by violating the NEC. In NLG wormholes, it was found that
the nonlocal gravity contributions allow for the stability and traversability of a wormhole,
without considering exotic matter.

30
References
[1] L. Flamm. Beiträge zur Einsteinschen Gravitationstheorie. Phys. Z., 17:448–454, 1916.

[2] A. Einstein and N. Rosen. The particle problem in the general theory of relativity.
Phys. Rev., 48:73–77, Jul 1935. doi:10.1103/PhysRev.48.73.

[3] Francisco S. N. Lobo. From the flamm–einstein–rosen bridge to the modern renaissance
of traversable wormholes. International Journal of Modern Physics D, 25(07):1630017,
June 2016. doi:10.1142/s0218271816300172.

[4] Jitendra Kumar, S.K. Maurya, Sweeti Kiroriwal, and Sourav Chaudhary.
Developing a framework for understanding wormholes in modified grav-
ity: A comprehensive review. New Astronomy Reviews, 98:101695, 2024.
doi:https://doi.org/10.1016/j.newar.2024.101695.

[5] Charles W. Misner and John A. Wheeler. Classical physics as geometry: Gravitation,
electromagnetism, unquantized charge, and mass as properties of curved empty space.
Annals Phys., 2:525–603, 1957. doi:10.1016/0003-4916(57)90049-0.

[6] K. A. Bronnikov. Scalar-tensor theory and scalar charge. Acta Phys. Polon. B, 4:251–
266, 1973.

[7] H. G. Ellis. Ether flow through a drainhole - a particle model in general relativity. J.
Math. Phys., 14:104–118, 1973. doi:10.1063/1.1666161.

[8] H. G. Ellis. The Evolving, Flowless Drain Hole: A Nongravitating Particle Model In
GR. Gen. Rel. Grav., 10:105–123, 1979. doi:10.1007/BF00756794.

[9] M. S. Morris and K. S. Thorne. Wormholes in space-time and their use for inter-
stellar travel: A tool for teaching general relativity. Am. J. Phys., 56:395–412, 1988.
doi:10.1119/1.15620.

[10] Mohammad Reza Mehdizadeh, Mahdi Kord Zangeneh, and Francisco S. N. Lobo.
Einstein-gauss-bonnet traversable wormholes satisfying the weak energy condition.
Phys. Rev. D, 91:084004, Apr 2015. doi:10.1103/PhysRevD.91.084004.

[11] Eleni-Alexandra Kontou and Ko Sanders. Energy conditions in general relativity and
quantum field theory. Classical and Quantum Gravity, 37(19):193001, September 2020.
doi:10.1088/1361-6382/ab8fcf.

[12] Henri Epstein, V. Glaser, and Arthur Jaffe. Nonpositivity of the energy density in
quantized field theories. Il Nuovo Cimento (1955-1965), 36:1016–1022, 1965. URL
https://api.semanticscholar.org/CorpusID:120375522.

[13] F.R. Klinkhamer. Defect wormhole: A traversable wormhole without exotic matter.
Acta Physica Polonica B, 54(5):1, 2023. doi:10.5506/aphyspolb.54.5-a3.

[14] Joshua Baines, Rudeep Gaur, and Matt Visser. Defect wormholes are defective. Uni-
verse, 9(10), 2023. doi:10.3390/universe9100452.

[15] Matt Visser. Traversable wormholes: Some simple examples. Physical Review D,
39(10):3182–3184, May 1989. doi:10.1103/physrevd.39.3182.

31
[16] Matt Visser. Lorentzian wormholes: From Einstein to Hawking. 1995.

[17] A. Chanda, S. Dey, and B. C. Paul. Morris–thorne wormholes in modified f(r, t) gravity.
General Relativity and Gravitation, 53(8), aug 2021. doi:10.1007/s10714-021-02847-7.

[18] M. H. Dehghani and Z. Dayyani. Lorentzian wormholes in lovelock gravity. Phys. Rev.
D, 79:064010, Mar 2009. doi:10.1103/PhysRevD.79.064010.

[19] Mahdi Kord Zangeneh, Francisco S. N. Lobo, and Mohammad Hossein Dehghani.
Traversable wormholes satisfying the weak energy condition in third-order lovelock grav-
ity. Phys. Rev. D, 92:124049, Dec 2015. doi:10.1103/PhysRevD.92.124049.

[20] Thomas P. Sotiriou and Valerio Faraoni. f(r) theories of gravity. Reviews of Modern
Physics, 82(1):451–497, March 2010. doi:10.1103/revmodphys.82.451.

[21] Nisha Godani and Gauranga C. Samanta. Traversable wormholes in f(r) gravity with
constant and variable redshift functions. New Astronomy, 80:101399, October 2020.
doi:10.1016/j.newast.2020.101399.

[22] Foad Parsaei and Sara Rastgoo. Wormhole in f(r) gravity revisited. 2021.
arXiv:2110.07278 [gr-qc].

[23] Mauricio Cataldo and Fabian Orellana. Static phantom wormholes of finite size. Physical
Review D, 96(6), September 2017. doi:10.1103/physrevd.96.064022.

[24] Robert R. Caldwell, Marc Kamionkowski, and Nevin N. Weinberg. Phantom energy:
Dark energy and cosmic doomsday. Physical Review Letters, 91(7), August 2003.
doi:10.1103/physrevlett.91.071301.

[25] John G. Cramer, Robert L. Forward, Michael S. Morris, Matt Visser, Gregory Benford,
and Geoffrey A. Landis. Natural wormholes as gravitational lenses. Phys. Rev. D,
51:3117–3120, Mar 1995. doi:10.1103/PhysRevD.51.3117.

[26] Diego F. Torres, Luis A. Anchordocqui, and Gustavo E. Romero. Wormholes, gamma
ray bursts and the amount of negative mass in the universe. Modern Physics Letters A,
13(19):1575–1581, June 1998. doi:10.1142/s0217732398001650.

[27] Diego F. Torres, Gustavo E. Romero, and Luis A. Anchordoqui. Might some gamma
ray bursts be an observable signature of natural wormholes? Physical Review D, 58(12),
November 1998. doi:10.1103/physrevd.58.123001.

[28] Luis A. Anchordoqui, Diego F. Torres, Gustavo E. Romero, and I. Andruchow. In

search for natural wormholes. Modern Physics Letters A, 14(12):791–797, April 1999.
doi:10.1142/s0217732399000833.

[29] Margarita Safonova, Diego F. Torres, and Gustavo E. Romero. Macrolensing signa-
tures of large-scale violations of the weak energy condition. Modern Physics Letters A,
16(03):153–162, January 2001. doi:10.1142/s0217732301003188.

[30] Erensto Eiroa, Gustavo E. Romero, and Diego F. Torres. Chromaticity effects in
microlensing by wormholes. Modern Physics Letters A, 16(15):973–983, May 2001.
doi:10.1142/s021773230100398x.

32
[31] B Mishra, AS Agrawal, SK Tripathy, and Saibal Ray. Traversable wormhole models in
f(R) gravity. International Journal of Modern Physics A, 37(05):2250010, 2022.

[32] Sebastian Bahamonde, Konstantinos F Dialektopoulos, Celia Escamilla-Rivera, Gabriel

Farrugia, Viktor Gakis, Martin Hendry, Manuel Hohmann, Jackson Levi Said, Jur-
gen Mifsud, and Eleonora Di Valentino. Teleparallel gravity: from theory to cosmol-
ogy. Reports on Progress in Physics, 86(2):026901, February 2023. doi:10.1088/1361-
6633/ac9cef.

[33] Manuel Hohmann. Teleparallel Gravity, page 145–198. Springer International Publish-
ing, 2023.

[34] H. I. Arcos and J. G. Pereira. Torsion gravity: A reappraisal. International Journal of

Modern Physics D, 13(10):2193–2240, December 2004. doi:10.1142/s0218271804006462.

[35] Mubasher Jamil, Davood Momeni, and Ratbay Myrzakulov. Wormholes in a viable f(t)
gravity. The European Physical Journal C, 73(1), jan 2013. doi:10.1140/epjc/s10052-
012-2267-8.

[36] C. Brans and R. H. Dicke. Mach’s principle and a relativistic theory of gravitation.
Phys. Rev., 124:925–935, Nov 1961. doi:10.1103/PhysRev.124.925.

[37] Georgios Kofinas and Minas Tsoukalas. On the action of the complete
brans–dicke theory. The European Physical Journal C, 76(12), December 2016.
doi:10.1140/epjc/s10052-016-4505-y.

[38] Oskar Klein. Quantum Theory and Five-Dimensional Theory of Relativity. (In German
and English). Z. Phys., 37:895–906, 1926. doi:10.1007/BF01397481.

[39] Lavinia Heisenberg. Review on f (q) gravity, 2023.

[40] Wompherdeiki Khyllep, Jibitesh Dutta, Emmanuel N. Saridakis, and Kuralay

Yesmakhanova. Cosmology in gravity: A unified dynamical systems analysis of
the background and perturbations. Physical Review D, 107(4), February 2023.
doi:10.1103/physrevd.107.044022.

[41] Jose Beltrán Jiménez, Lavinia Heisenberg, and Tomi Koivisto. Coincident general rel-
ativity. Physical Review D, 98(4), August 2018. doi:10.1103/physrevd.98.044048.

[42] Yixin Xu, Guangjie Li, Tiberiu Harko, and Shi-Dong Liang. f(q, t) gravity. The Euro-
pean Physical Journal C, 79(8), August 2019. doi:10.1140/epjc/s10052-019-7207-4.

[43] Haomin Rao, Chunhui Liu, and Chao-Qiang Geng. Thermodynamic of the f (q) universe,
2024.

[44] Tiberiu Harko and Francisco S. N. Lobo. f (r, lm ) gravity. The European Physical
Journal C, 70(1–2):373–379, October 2010. doi:10.1140/epjc/s10052-010-1467-3.

[45] Lakhan V. Jaybhaye, Raja Solanki, Sanjay Mandal, and P.K. Sahoo. Cos-
mology in f(r,l) gravity. Physics Letters B, 831:137148, August 2022.
doi:10.1016/j.physletb.2022.137148.

[46] Lakhan V Jaybhaye, Raja Solanki, and P K Sahoo. Bouncing cosmological models in f(r,
lm) gravity. Physica Scripta, 99(6):065031, May 2024. doi:10.1088/1402-4896/ad4838.

33
[47] Raja Solanki, Zinnat Hassan, and P.K. Sahoo. Wormhole solutions in f (r, lm ) gravity.
Chinese Journal of Physics, 85:74–88, October 2023. doi:10.1016/j.cjph.2023.06.005.
[48] Francisco S. N. Lobo and Miguel A. Oliveira. Wormhole geometries in f (r) modified the-
ories of gravity. Phys. Rev. D, 80:104012, Nov 2009. doi:10.1103/PhysRevD.80.104012.
[49] Thomas P Sotiriou. 6+1 lessons fromf(r) gravity. Journal of Physics: Conference Series,
189:012039, October 2009. doi:10.1088/1742-6596/189/1/012039.
[50] D. Lovelock. The Einstein tensor and its generalizations. J. Math. Phys., 12:498–501,
1971. doi:10.1063/1.1665613.
[51] T. Padmanabhan. Some aspects of field equations in generalized theories of gravity.
Physical Review D, 84(12), December 2011. doi:10.1103/physrevd.84.124041.
[52] T. Padmanabhan and D. Kothawala. Lanczos–lovelock models of gravity. Physics
Reports, 531(3):115–171, October 2013. doi:10.1016/j.physrep.2013.05.007.
[53] Cornelius Lanczos. Electricity as a natural property of Riemannian geometry. Rev.
Mod. Phys., 39:716–736, 1932. doi:10.1103/RevModPhys.39.716.
[54] Cornelius Lanczos. A Remarkable property of the Riemann-Christoffel tensor in four
dimensions. Annals Math., 39:842–850, 1938. doi:10.2307/1968467.
[55] Mohammad Reza Mehdizadeh and Amir Hadi Ziaie. Dynamical wormholes in lovelock
gravity. Phys. Rev. D, 104:104050, Nov 2021. doi:10.1103/PhysRevD.104.104050.
[56] Shibendu Gupta Choudhury. Application of the Raychaudhuri Equation in Gravitational
Systems. PhD thesis, IISER, Kolkata, 2022.
[57] Susmita Sarkar, Nayan Sarkar, Farook Rahaman, and Y Aditya. Wormholes in κ(r, t)
gravity, 2022.
[58] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini. Class of
viable modified f (r) gravities describing inflation and the onset of accelerated expansion.
Phys. Rev. D, 77:046009, Feb 2008. doi:10.1103/PhysRevD.77.046009.
[59] E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini. Nonsingular expo-
nential gravity: A simple theory for early- and late-time accelerated expansion. Phys.
Rev. D, 83:086006, Apr 2011. doi:10.1103/PhysRevD.83.086006.
[60] Shinji Tsujikawa. Observational signatures of f (r) dark energy models that sat-
isfy cosmological and local gravity constraints. Phys. Rev. D, 77:023507, Jan 2008.
doi:10.1103/PhysRevD.77.023507.
[61] A. A. Starobinsky. Disappearing cosmological constant in f(r) gravity. JETP Letters,
86(3):157–163, October 2007. doi:10.1134/s0021364007150027.
[62] Luca Amendola, Radouane Gannouji, David Polarski, and Shinji Tsujikawa. Conditions
for the cosmological viability of f (r) dark energy models. Phys. Rev. D, 75:083504, Apr
2007. doi:10.1103/PhysRevD.75.083504.
[63] Luis A. Anchordoqui, Diego F. Torres, Marta L. Trobo, and Santiago E. Perez Bergli-
affa. Evolving wormhole geometries. Physical Review D, 57(2):829–833, January 1998.
doi:10.1103/physrevd.57.829.

34
[64] Farook Rahaman, Susmita Sarkar, Ksh. Newton Singh, and Neeraj Pant. Gen-
erating functions of wormholes. Mod. Phys. Lett. A, 34(01):1950010, 2019.
doi:10.1142/S021773231950010X. arXiv:1901.01757 [physics.gen-ph].

[65] Salvatore Capozziello, Orlando Luongo, and Lorenza Mauro. Traversable wormholes
with vanishing sound speed in f (r) gravity, 2020.

[66] A. Astashenok, Salvatore Capozziello, S. Odintsov, and V. Oikonomou. Extended

gravity description for the gw190814 supermassive neutron star. Physics Letters B,
811:135910, 12 2020. doi:10.1016/j.physletb.2020.135910.

[67] Salvatore Capozziello, Mariafelicia De Laurentis, Ruben Farinelli, and Sergei D.

Odintsov. Mass-radius relation for neutron stars in f (r) gravity. Phys. Rev. D,
93:023501, Jan 2016. doi:10.1103/PhysRevD.93.023501.

[68] Salvatore Capozziello, Mariafelicia De Laurentis, Orlando Luongo, and Alan Rug-
geri. Cosmographic constraints and cosmic fluids. Galaxies, 1, 12 2013.
doi:10.3390/galaxies1030216.

[69] Orlando Luongo and Marco Muccino. Speeding up the universe using dust with pressure.
Phys. Rev. D, 98:103520, Nov 2018. doi:10.1103/PhysRevD.98.103520.

[70] Farbod Hassani, Benjamin L’Huillier, Arman Shafieloo, Martin Kunz, and Julian
Adamek. Parametrising non-linear dark energy perturbations. Journal of Cos-
mology and Astroparticle Physics, 2020(04):039–039, April 2020. doi:10.1088/1475-
7516/2020/04/039.

[71] Panagiota Kanti, Burkhard Kleihaus, and Jutta Kunz. Wormholes in dilatonic
einstein-gauss-bonnet theory. Physical Review Letters, 107(27), December 2011.
doi:10.1103/physrevlett.107.271101.

[72] M. A. Cuyubamba, R. A. Konoplya, and A. Zhidenko. No stable worm-

holes in Einstein-dilaton-Gauss-Bonnet theory. Phys. Rev. D, 98(4):044040, 2018.
doi:10.1103/PhysRevD.98.044040. arXiv:1804.11170 [gr-qc].

[73] David J. Gross and John H. Sloan. The Quartic Effective Action for the Heterotic
String. Nucl. Phys. B, 291:41–89, 1987. doi:10.1016/0550-3213(87)90465-2.

[74] R.R. Metsaev and A.A. Tseytlin. Order (two-loop) equivalence of the string
equations of motion and the -model weyl invariance conditions: Dependence on
the dilaton and the antisymmetric tensor. Nuclear Physics B, 293:385–419, 1987.
doi:https://doi.org/10.1016/0550-3213(87)90077-0.

[75] Michael S. Morris, Kip S. Thorne, and Ulvi Yurtsever. Wormholes, time machines,
and the weak energy condition. Physical review letters, 61 13:1446–1449, 1988. URL
https://api.semanticscholar.org/CorpusID:28563028.

[76] Remo Garattini. Generalized absurdly benign traversable wormholes powered

by casimir energy. The European Physical Journal C, 80(12), December 2020.
doi:10.1140/epjc/s10052-020-08728-8.

[77] S. W. Hawking. Particle Creation by Black Holes. Commun. Math. Phys., 43:199–220,
1975. doi:10.1007/BF02345020. [Erratum: Commun.Math.Phys. 46, 206 (1976)].

35
[78] Remo Garattini. Casimir wormholes. The European Physical Journal C, 79(11), Novem-
ber 2019. doi:10.1140/epjc/s10052-019-7468-y.

[79] Remo Garattini. Effects of an electric charge on casimir wormholes: changing the throat
size. The European Physical Journal C, 83(5), May 2023. doi:10.1140/epjc/s10052-023-
11464-4.

[80] G. Alencar, V. B. Bezerra, and C. R. Muniz. Casimir wormholes in 2 + 1 dimensions

with applications to the graphene. The European Physical Journal C, 81(10), October
2021. doi:10.1140/epjc/s10052-021-09734-0.

[81] João Luı́s Rosa. Double gravitational layer traversable wormholes in hy-
brid metric-palatini gravity. Physical Review D, 104(6), September 2021.
doi:10.1103/physrevd.104.064002.

[82] João Luı́s Rosa, José P.S. Lemos, and Francisco S.N. Lobo. Wormholes in generalized
hybrid metric-palatini gravity obeying the matter null energy condition everywhere.
Physical Review D, 98(6), September 2018. doi:10.1103/physrevd.98.064054.

[83] João Luı́s Rosa, Rui André, and José P. S. Lemos. Traversable wormholes with double
layer thin shells in quadratic gravity. General Relativity and Gravitation, 55(5), May
2023. doi:10.1007/s10714-023-03107-6.

[84] João Luı́s Rosa, Nailya Ganiyeva, and Francisco S. N. Lobo. Non-exotic traversable
wormholes in f (r, tab tab ) gravity. The European Physical Journal C, 83(11), November
2023. doi:10.1140/epjc/s10052-023-12232-0.

[85] Marco Cirelli, Alessandro Strumia, and Jure Zupan. Dark matter, 2024.

[86] Risa H. Wechsler and Jeremy L. Tinker. The connection between galaxies and their dark
matter halos. Annual Review of Astronomy and Astrophysics, 56(1):435–487, September
2018. doi:10.1146/annurev-astro-081817-051756.

[87] Susmita Sarkar, Nayan Sarkar, Somi Aktar, Moumita Sarkar, Farook Rahaman, and
Anil Kumar Yadav. Dark matter supporting traversable wormholes in the Galactic
halo. New Astron., 109:102183, 2024. doi:10.1016/j.newast.2023.102183.

[88] Moreshwar Tayde, Zinnat Hassan, and P.K. Sahoo. Impact of dark matter galactic halo
models on wormhole geometry within f(q,t) gravity. Nuclear Physics B, 1000:116478,
2024. doi:https://doi.org/10.1016/j.nuclphysb.2024.116478.

[89] Zhaoyi Xu. Exact solution of kerr-like traversable wormhole in dark matter halo. 2023.
arXiv:2303.11677 [gr-qc].

[90] Rahaman, Farook, Samanta, Bidisha, Sarkar, Nayan, Raychaudhuri, Biplab, and Sen,
Banashree. Traversable wormholes supported by dark matter and monopoles with semi-
classical effects. Eur. Phys. J. C, 83(5):395, 2023. doi:10.1140/epjc/s10052-023-11456-4.

[91] Hai-Nan Lin and Xin Li. The dark matter profiles in the milky way.
Monthly Notices of the Royal Astronomical Society, 487(4):5679–5684, June 2019.
doi:10.1093/mnras/stz1698.

36
[92] Hongwei Tan, Jinbo Yang, Jingyi Zhang, and Tangmei He. The global monopole
spacetime and its topological charge. Chinese Physics B, 27(3):030401, March 2018.
doi:10.1088/1674-1056/27/3/030401.

[93] Kirill A. Bronnikov, Boris E. Meierovich, and Evgeny R. Podolyak. Global monopole
in general relativity. J. Exp. Theor. Phys., 95:392–403, 2002. doi:10.1134/1.1513811.
arXiv:gr-qc/0212091].

[94] Xin Shi and Xinzhou Li. The gravitational field of a global monopole. Classical and
Quantum Gravity, 8(4):761–767, April 1991. doi:10.1088/0264-9381/8/4/019.

[95] Zinnat Hassan and P. K. Sahoo. Possibility of the Traversable Wormholes in the Galactic
Halos within 4D Einstein-Gauss-Bonnet Gravity. Annalen Phys., 536:2400114, 2024.
doi:10.1002/andp.202400114. arXiv:2406.13224 [gr-qc].

[96] Salvatore Capozziello and Nisha Godani. Non-local gravity wormholes. Physics Letters
B, 835:137572, December 2022. doi:10.1016/j.physletb.2022.137572.

[97] Adriano Acunzo, Francesco Bajardi, and Salvatore Capozziello. Non-local curva-
ture gravity cosmology via Noether symmetries. Phys. Lett. B, 826:136907, 2022.
doi:10.1016/j.physletb.2022.136907. arXiv:2111.07285 [gr-qc].

37