Wormholes in the f(R,L,T) theory of gravity

Morris and Thorne developed wormhole solutions in the late 1980s when they discovered a recipe that wormholes must follow for travelers to cross them safely. They describe exotic matter as satisfying $-p_{r}>\rho$, where $p_{r}$ is the radial pressure and $\rho$ is the energy density of the wormhole. This is a notable characteristic of the General Relativity Theory. The current article discusses traversable wormhole solutions in $f(R, L, T)=R+\alpha L+\beta T$, with $\alpha$ and $\beta$ are model parameters. The wormhole solutions presented here satisfy the metric constraints of traversability while remarkably avoiding the exotic matter condition, indicating that $f(R, L, T)$ gravity wormholes can be filled with ordinary matter. The derived solutions for the shape function of the wormhole meet the required metric conditions. They exhibit behavior that is comparable to that of wormholes reported in earlier references, which is also the case for our solutions for the energy density of such objects.


Introduction
Our current model of cosmology, named ΛCDM model, is replete of flaws, inconsistencies or incompleteness.Ironically, what baptizes the model are exactly among its greatest shortcomings.According to ΛCDM model (also referred to as standard model), the cosmological constant Λ is the responsible for the observed acceleration of the recent universe expansion (Riess et al., 1998;Perlmutter et al., 1999), but it carries "the worst theoretical prediction in the history of physics" (Hobson et al., 2006), namely, the cosmological constant problem (Weinberg, 1989), which is the huge discrepancy between the theoretical and observational values of Λ.On the other hand, cold dark matter (referred to as CDM) is an eccentric kind of matter that does not interact electromagnetically and therefore cannot be seen, but is necessary to fill galaxies and clusters of galaxies in order to fit observations (Salucci, 2019;Ashman, 1992;Limousin et al., 2022;Montes and Trujillo, 2019).Despite several and costly attempts, no particle associated to dark matter was ever detected (Smith and Lewin, 1990;Boehm et al., 2004;Muñoz, 2004;Mayet, 2016).Considering that according to the latest results by the Planck Satellite (Planck Collaboration, 2020), ∼ 95% of the universe composition is in the form of dark energy and dark matter, the above issues regarding Λ, which plays the role of "dark energy" in ΛCDM model, and the persistent non-detection of dark matter, are critical to the perenniality of the model.
Not only do the aforementioned problems persist, ΛCDM model also includes the missing satellites problem (Klypin et al., 1999;Moore et al., 1999) and the core/cusp problem (Dubinski and Carlberg, 1991;Walker and Peñarrubia, 2011), which are, respectively, the over-abundance of the predicted number of halo substructures compared to the number of ob-served satellite galaxies and the difference between the model theoretical prediction that halos should follow a universal, centrally peaked (cuspy) density profile, and observations of dwarf and low surface brightness galaxies, that indicate shallower (cored) profiles.For a review on the fundamental problems of the ΛCDM model, which include the aforementioned ones, but are not restricted to them, we recommend (Bull et al., 2016).
A promising possibility to evade the above shortcomings is to work with Extended Gravity Theories (EGTs).The main motivation for considering EGTs is the possibility of describing, for instance, the dark sector of the universe, from extra gravitational terms that are not contained in Einstein's General Theory of Relativity (GR), which sustains ΛCDM model.Below we present and discuss some EGT examples.
The most popular EGT nowadays is the so-called f (R) gravity theory, in which R stands for the Ricci scalar.An important review on the subject is Reference (De Felice and Tsujikawa, 2010).Effectively, the f (R) theory replaces R in the Einstein-Hilbert action in which g is the metric determinant, by a function of R, namely f (R) (in units such that the speed of light c and Newton's gravitational constant G N are taken both as 1).Naturally, by applying the variational principle in the f (R) action, the resulting field equations will present extra terms when compared to GR field equations, namely, with G µν being the Einstein tensor and T µν the energymomentum tensor.The extra terms may, in principle, play the role of dark energy and dark matter in a different perspective; not as exotic fluids permeating the universe, but as modifying gravity terms (Amendola et al., 2007;Nojiri and Odintsov, 2006;Capozziello et al., 2005;Capozziello and De Laurentis, 2012;Böhmer et al., 2008).Nevertheless, a number of f (R) gravity flaws and shortcomings has already been reported, as one can check (Jorás, 2011) and the very recent reference (Casado-Turrión et al., 2023), for instance, among others.Wormhole solutions in f (R) gravity theory (Mishra et al., 2022;Agrawal et al., 2022) and in unimodular gravity theory (Agrawal et al., 2023) obtained recently.
A common alternative to the f (R) theory shortcomings is the f (R, T ) theory (Harko et al., 2011), for which T stands for the trace of the energy-momentum tensor.While the f (R) gravity is mathematically motivated by the possibility of higher order terms on R in (1), the f (R, T ) gravity maintains this consideration, but also considers material terms to appear in (1).
Some important applications of the f (R, T ) gravity are the following.Wormhole solutions in f (R, T ) gravity were obtained in (Sahoo et al., 2018;Moraes and Sahoo, 2019).A cosmological model was constructed from the simplest non-minimal geometry-matter coupling in f (R, T ) gravity in (Moraes and Sahoo, 2017).The quantum version of f (R, T ) cosmology was constructed by (Xu et al., 2016).
Another interesting form of coupling geometry and matter in a gravity theory is through the f (R, L) gravity (Harko and Lobo, 2010), with L being the matter lagrangian density1 .Some applications of the f (R, L) gravity are the following: the hydrostatic equilibrium configurations of compact stars were first derived in the theory by (Carvalho et al., 2020).Later, quark stars with 2.6M ⊙ were attained in (Carvalho et al., 2022).A cosmological model evading the Big-Bang singularity was obtained within f (R, L) gravity in (Gonc ¸alves et al., 2023).
Further geometry-matter coupling theory approaches are the following: a Palatini formulation of EGT with non-minimal geometry-matter coupling was proposed by (Harko et al., 2011b).A thermodynamic interpretation of geometry-matter coupling gravity models was given by (Harko, 2011).Galactic rotation curves within non-minimal geometry-matter coupling gravity were constructed by (Harko, 2010) and matter density perturbations were studied in (Nesseris, 2009).
In a recent paper by (Haghani and Harko, 2021), a generalized geometry-matter coupling theory of gravity was proposed, namely the f (R, L, T ) gravity.As the name would suggest, this theory proposes an f (R, L, T ) function to substitute R in Eq.( 1).
Here we are going to construct, for the first time in the literature, traversable wormhole solutions in the generalized geometry-matter coupling gravity.Traversable wormholes are a class of GR solutions that allow to rapid interstellar travel (Morris and Thorne, 1988).Such traversable wormholes must have no horizons and, according to GR, the material that generates the wormhole curvature must disobey the energy conditions.
Wormholes have not been observed so far, but attempts to do so have been proposed.For instance, (Paul et al., 2020) numerically constructed images of thin accretion disks in rotating wormhole backgrounds.The results show dramatic differences between such images and those obtained for black hole backgrounds.A method for detecting wormholes surrounded by optically thin dust was proposed by (Ohgami and Sakai, 2015).The ring-down waveform of gravitational waves and gravitational lensing were proposed as a tool for distinguishing between wormholes and black holes by (Nandi et al., 2017).
There is a particular interest in studying traversable wormholes in modified theories of gravity, which is the fact that the extra degrees of freedom of such theories can allow for wormholes to be filled by non-exotic matter, i.e., matter satisfying the energy conditions.This has been attained in braneworld models (Sengupta et al., 2022), Born-Infeld gravity (Shaikh, 2018), Einstein-Cartan theory (Bronnikov and Galiakhmetov, 2015), and f (R, T ) theory (Moraes and Sahoo, 2018).

Static wormholes
The static wormhole metric is given by the Morris-Thorne solution (Morris and Thorne, 1988) as with Φ being the redshift function and b the shape function.
In order for the wormhole spatial geometry to tend to the appropriate asymptotically flat limit, Φ must be such that lim r→∞ Φ < ∞. (4) The traversable wormhole must obey b(r 0 ) = r 0 , (5) at the wormhole throat r 0 , with a prime indicating radial derivative throughout the paper.One must also have and away from r 0 , b < r (8) must be satisfied.Moreover, the energy-momentum tensor of the wormhole is in which ρ is the wormhole matter-energy density, p r is its radial pressure and p t its tangential pressure.

The f (R, L, T) gravity
The f (R, L, T ) gravity formalism starts from the action (Haghani and Harko, 2021) in units such that 8πG N = c = 1, which will be assumed.The variation of ( 10) with respect to metric tensor g µν yields as the field equations of the theory, for which R µν is the Ricci tensor ∂g µν ∂g αβ .In the simple case of an additive structure of the gravitational Lagrangian of the form f (R, L, T ) = R + αL + βT , with α and β being constants, and by assuming L = −ρ, the field equations ( 11) take the form 4. Wormholes in the f (R, L, T) gravity Below, we present the non-null components of the field equations above for the wormhole metric (3) and energy-momentum tensor (9): where we have assumed Φ(r) = constant and η ≡ 8π + α 2 + β.By manipulating the above equations, we can write the matter content of the wormhole as for which the superscript "eff" stands for effective.

Energy Conditions
The existence of wormhole solutions in GR depends on the violation of the null energy condition (NEC) (Raychaudhuri, 1955;Morris and Thorne, 1988;Curiel, 2017).
The NEC states that (Visser, 1995) for any null vector k µ .In terms of the principle pressure, The weak energy condition (WEC) states that for any time-like vector V µ .In terms of the principle pressure, WEC ⇔ ρ (eff) ≥ 0, and ∀ j, ρ (eff) + p (eff) The strong energy condition (SEC) states that for any time-like vector.In terms of principle pressures, SEC ⇔ ∀ j, ρ (eff) + p (eff) j ≥ 0, and ρ (eff) + p (eff) j ≥ 0 (24) Accordingly, T (eff) µν k µ k ν < 0, which represents the violation of NEC requirement, is given by In the context of GR, the flaring-out condition and the NEC are incompatible.To satisfy the flaring-out condition, G µν k µ k ν < 0 must hold, in which k µ is an arbitrary null vector.This implies, from Einstein's field equations (2), that T µν k µ k ν < 0. Choosing a stress-energy tensor T µν of the form given in Eq. ( 9) leads to the violation of NEC.
However, the situation is different in modified theories of gravity, where T e f f µν includes contributions not only from the matter sector but also from the extra gravitational degrees of freedom.Thus, although the effective stress-energy tensor must satisfy T e f f µν k µ k ν < 0 to fulfill the flaring-out condition, it is still possible for the matter stress-energy tensor to satisfy the NEC, i.e., T µν k µ k ν > 0, provided that the extra gravitational contributions compensate the positive matter contributions (Luis Rosa and Kull, 2022).
Using the flaring out condition b ′ r−b b 2 < 0 and imposing the WEC, given by ρ > 0 and ρ + p r > 0, one can find that the coupling parameter η is limited to η < 0, and write where p r = ωρ.
With the condition b ′ < b/r, the allowed region for the parameter ω is restricted to 1 ≥ ω ≥ 0, in which we are also taking the causality limit into account.Moreover, we can write , from which we see that an isotropic traversable wormhole would have to be filled by ω = −1/3 matter, which is out of the acceptable range for our solution (and ω < 0 matter is out of the scope of the present letter as we are dealing with modified gravity).
In Figure 1 below, it can be seem that for the considered values of the model parameters, traversable wormhole conditions (7) and ( 8   Our solutions have shown that wormholes can be filled by dust, radiation and even stiff matter (ω = 1) (Carr, 2010;Bronnikov et al., 2020) and still be traversable.

Conclusions
The traversable wormhole solution exists in GR and violates the NEC due to some form of exotic matter.But within the EGT context, the situation might be different.
Here, the wormhole solution has been discussed in the f (R, L, T ) modified gravity.We have derived the gravitational field equations for this functional form.The shape function of the wormhole was then derived by specifying an equation of state for the matter threading it and applying the flaring out condition at the throat.We demonstrated that the stress-energy tensor of the matter threading the wormhole meets the NEC requirement in certain subspaces of the model parameter space.
We have shown here that for the chosen functional form of the f (R, L, T ) function, it is possible to obtain traversable wormhole solutions satisfying energy conditions.
Our present work can be extended in different fronts.Some other functional forms for the f (R, L, T ) function can be used to construct wormholes and check if those solutions are capable of describing traversable wormholes without exotic matter.
As the f (R, L, T ) gravity formalism was recently proposed, it still lacks several applications, such as those motivated in the Introduction.For instance, the same functional form used here can be extended to construct a cosmological model and check if for a range of values of α and β, it is possible to describe an accelerated expanding universe.