Conformally symmetric wormhole solutions supported by non-commutative geometryin f（Q,T）gravity

Chaitra Chooda Chalavadi ,V Venkatesha ,N S Kavya and S V Divya Rashmi

1 Department of P.G.Studies and Research in Mathematics,Kuvempu University,Shankaraghatta,Shivamogga 577451,Karnataka,India

2 Department of Mathematics,Vidyavardhaka College of Engineering,Mysuru—570002,India

Abstract This paper investigates wormhole solutions within the framework of extended symmetric teleparallel gravity,incorporating non-commutative geometry,and conformal symmetries.To achieve this,we examine the linear wormhole model with anisotropic fluid under Gaussian and Lorentzian distributions.The primary objective is to derive wormhole solutions while considering the influence of the shape function on model parameters under Gaussian and Lorentzian distributions.The resulting shape function satisfies all the necessary conditions for a traversable wormhole.Furthermore,we analyze the characteristics of the energy conditions and provide a detailed graphical discussion of the matter contents via energy conditions.Additionally,we explore the effect of anisotropy under Gaussian and Lorentzian distributions.Finally,we present our conclusions based on the obtained results.

Keywords: traversable wormhole,f(Q,T) gravity,energy conditions,non-commutative geometry,conformal motion

1.Introduction

Wormholes are dual-mouthed hypothetical structures connecting distinct sectors in the same universe or different universes.Initially,Flamm [1] introduced the notion of a wormhole by constructing the isometric embedding of the Schwarzschild solution.Einstein and Rosen [2] employed Flamm’s concept to create a bridge,commonly known as the Einstein–Rosen bridge.Later,Thorne and his student Morris[3] conducted pioneering research on the concept of traversable wormholes.They meticulously examined static and spherically symmetric wormholes,revealing that the exotic matter inside them possesses negative energy,thus violating the null energy condition.Furthermore,in order to establish a physically feasible model,it is necessary to repudiate the existence of the hypothetical matter.Although within the framework of general relativity [4,5],it was not possible to definitively rule out the presence of such a substance,an alternative approach was supported to reduce or eliminate the reliance on exotic matter [6–8].Numerous studies have been conducted to explore wormhole solutions within the background of modified theories [9–36].

Figure 1.The graphical behavior of shape function Ψ* for Gaussian non-commutative geometry with M*=7.25,α=0.45,C2=2 and

In the context of string theory,non-commutative geometry is one of the most intriguing concepts.The idea of noncommutativity arises from the notion that coordinates on a D-brane can be treated as non-commutative operators.This property holds great significance in mathematically explored fundamental concepts of quantum gravity [37–39].Noncommutative geometry aims to unify space-time gravitational forces with weak and strong forces on a single platform.Within this framework,it becomes possible to replace pointlike structures with smeared objects,leading to the discretization of space-time.This discretization arises from the commutator [xa,xb]=iθabwhere θabis an antisymmetric second-order matrix [40–42].To simulate this smearing effect,the Gaussian distribution and Lorentzian distribution with a minimum length ofare incorporated instead of the Dirac delta function.This non-commutative geometry is an intrinsic property of space-time and independent of the behavior of curvature.

Non-commutative geometry plays a crucial role in examining the properties of space-time geometry under different conditions.Jamil et al,[43] explored some new exact solutions of static wormholes under non-commutative geometry.They utilized the power-law approach to analyze these solutions and discuss their properties.Rahaman et al[44–46]conducted an extensive investigation into various studies in non-commutative geometry.They studied fluids in different dimensions influenced by non-commutative geometry,which exhibited conformal symmetry.Additionally,they derived specific solutions of a wormhole within the context off(R)gravity.In the realm of non-commutative geometry,Zubair et al[47]examined wormhole solutions that permit conformal motion within the context off(R ,T)theory.The study employed conformal killing vectors to analyze the properties and characteristics of these wormhole solutions.Kuhfitting[48] investigated the stable wormhole solutions utilizing conformal killing vectors within the framework of a noncommutative geometry that incorporates a minimal length.The study focused on exploring the properties and characteristics of these stable wormholes within this specific theoretical framework.In [49],the authors studied the noncommutative wormhole solution inf(R) gravity.Moreover,the concept of non-commutative geometry has been gaining attention from researchers,and numerous intriguing aspects of this theory have been extensively explored and deliberated upon in the literature[50–64].Inspired by the aforementioned attempts in modified gravity and non-commutative geometry,we now delve into the study of wormhole solutions inf(Q ,T)gravity.We consider Gaussian and Lorentzian noncommutative geometries with conformal killing vectors to explore their implications.

The paper is structured following the subsequent pattern:In section 2,we discuss the traversability condition for a wormhole.We shall construct the mathematical formalism off(Q ,T)gravity in 3.In the same section,we briefly explain the energy condition and the basic formalism of conformal killing vectors.In section 4,we conduct a detailed analysis of the wormhole model under Gaussian and Lorentzian distributions.Within this section,we derive the shape function and explore the impact of model parameters on these functions,as well as the energy conditions.In section 5,we investigate the effect of anisotropy on both distributions.Finally,in section 6,we finalize the conclusive remarks and summarize the key findings of the study.

2.Traversability conditions for wormhole

The Morris–Thorne metric for the traversable wormhole is described as

In this scenario,we have two functions,namely Φ(r)and Ψ(r)which are referred to as the redshift and shape functions respectively.Both of these functions depend on the radial coordinate r.

1.Redshift function: The redshift function Φ(r) needs to have a finite value across the entire space-time.Additionally,the redshift function must adhere to the constraint of having no event horizon,which allows for a two-way journey through the wormhole.

2.Shape function:The shape function Ψ(r)characterizes the geometry of the traversable wormhole.Therefore,Ψ(r)must satisfy the following conditions:

• Throat condition:The value of the function Ψ(r)at the throat is r0and hence10for r>r0.

• Flaring-out condition: The radial differential of the shape function,Ψ′(r) at the throat should satisfy,Ψ′ (r0) <1.

• Asymptotic Flatness condition: As r →∞,

3.Proper radial distance function: This function should be finite everywhere in the domain.In magnitude,it decreases from the upper universe to the throat and then increases from the throat to the lower universe.The proper radial distance function is expressed as,

3.Mathematical formulations of f（Q,T ）gravity

In this article,we are particularly interested inf(Q ,T)gravity,where the Lagrangian is an arbitrary function of nonmetricity scalar and the trace of the energy-momentum tensor.Yixin et al,[15] introducedf(Q ,T)gravity,which is referred to as extended symmetric teleparallel gravity.This was developed within the metric-affine formalism framework.f(Q ,T)gravity theory has been employed to explain both matter-antimatter asymmetry and late-time acceleration.Furthermore,recent investigations suggest thatf(Q ,T)gravity may provide a feasible explanation of various cosmological and astrophysical phenomena [26,65–67].Nevertheless,no further studies on wormholes were conducted based on this theory,which is still in its early stages of development.These considerations motivate us to selectf(Q ,T)gravity to derive wormhole solutions.

The Einstein–Hilbert action forf(Q ,T)gravity is given by

wheref(Q ,T)is an arbitrary function that couples the nonmetricityQ and the traceT of the energy momentum tensor,mL is the Lagrangian density corresponding to matter and g denotes the determinant of the metric gμν.

The non-metricity tensor is defined as

and its traces are

Further,we can define a super-potential associated with the non-metricity tensor as

The non-metricity scalar is represented as

Besides,the energy-momentum tensor for the fluid depiction of space-time can be expressed as

The variation of the action (3) with respect to the fundamental metric,gives the metric field equation

We presume that the matter distribution is an anisotropic stress-energy tensor,which can be written as

where ρ,pr,ptare the energy density,radial and tangential pressures respectively.Here,ημrefers to a four-velocity vector with a magnitude of one,whileμrepresents a spacelike unit vector.Additionally,in this scenario,the tangential pressure will be orthogonal to the unit vector,and the radial pressure will be along the four-velocity vector.

The expression for the trace of the energy-momentum tensor is determined as T=ρ-pr-2ptand equation (9)can be read as

Using the wormhole metric (1),the trace of the nonmetricity scalarQ can be written as,

Now,substituting the wormhole metric (1) and anisotropic matter distribution (11) into the motion equation (10),we found the following expressions:

3.1.Energy condition

Energy conditions provide interpretations for the physical phenomena associated with the motion of energy and matter,which are derived from the Raychaudhuri equation.To evaluate the geodesic behavior,we shall consider the criterion for different energy conditions.With the anisotropic matter distribution for ρ,prand ptbeing energy density,radial pressure and tangential pressure,we have the following:

• Null Energy Conditions: ρ+pt≥0 and ρ+pr≥0.

• Weak Energy Conditions: ρ ≥0 ⇒ρ+pt≥0 and ρ+pr≥0.

• Strong Energy Conditions:ρ+pj≥0 ⇒ρ+Σjpj≥0 ∀j.

• Dominant Energy Conditions: ρ ≥0 ⇒ρ-|pr|≥0 and ρ-|pt|≥0.

3.2.Conformal killing vectors

Conformal killing vectors play a significant role in establishing the mathematical connection between the geometry of space-time and the matter it contains through Einstein’s field equations.These vectors are derived from the killing equations,utilizing the principles of Lie algebra [47,68].Conformal killing vectors are an essential tool for reducing the non-linearity order of field equations in various modified theories.In the context of general relativity,conformal killing vectors find numerous applications in geometric configurations,kinematics,and dynamics based on the structure theory.We employ an inheritance symmetry of space-time characterized by conformal killing vectors,which are defined as[44,69]

where ζ,ηkand gijrepresent the conformal factor,conformal killing vectors and metric tensor respectively.It is supposed that the vector η generates the conformal symmetry and the metric g is conformally mapped onto itself along η.The conformal factor,which characterizes the scaling of the metric,influences the geometry of the wormhole.By inserting the equation Lηgij=ζ(r)gijfrom equation (17) into equation (1),we get the following equations:

On solving the aforementioned expressions,we obtain the following two relationships for the metric components:

where1C and C2are the integrating constants.For the simplification,we assume A(r)=ζ2(r).Consequently,the expression for the shape function can be obtained as

4.Wormhole model in f（Q,T）gravity

In this section,we shall consider a feasible model to study the properties of wormhole geometry.In particular,we suppose the linear form given by

where α and β are the model parameters.For α=1,β=0,one can retain general relativity.By utilizing equations (21),(22) and adopting dimensionless parameters,the field equations (14)–(16) can be solved to obtain the following equations:

Here,the subscript ‘*’ denotes corresponding adimensional quantities and the overhead dot is the derivative of the function with respect toFurther,non-dimensionalization is a powerful tool in theoretical physics.It enables researchers to simplify equations,comprehend the scaling behavior of physical systems,and gain insights into the essential features of complex phenomena such as wormholes.

Now,we shall discuss the physical analysis of wormhole solutions with the help of equations (24–26) under noncommutative distributions.For this purpose,we consider the Gaussian and Lorentzian energy densities of the static and spherically symmetric particle-like gravitational source with a total mass of the form [58,70]

4.1.Gaussian energy density

In this subsection,our attention will be directed towards exploring non-commutative geometry under Gaussian distribution.When we substitute the Gaussian energy density(27) into equation (24),we obtain the resulting differential equation:

We can easily verify the satisfaction of the throat condition by performing a simple calculation ofFurthermore,by evaluating the derivative of the shape function (31) at the throat,we derive the following relation:

In our study,the behavior of energy density and energy conditions are illustrated in figure 2.Both dominant energy conditions,radial null energy condition and strong energy condition are violated.However,the tangential null energy condition is satisfied.

4.2.Lorentzian energy density

In this subsection,we focus on the scenario involving noncommutative geometry with the Lorentzian distribution.By substituting the Lorentzian energy density (28) into (24),we get

Solving the aforementioned differential equation while imposing the throat condition on the shape function,we can derive the following expression:

Figure 2.Gaussian Source: The profile of energy density and energy conditions with respect to for different values of β with fixed parameters M*=7.25,α=0.45,C2=2and =1.6.

Figure 3.The graphical behavior of shape function Ψ* for Lorentzian non-commutative geometry with M*=7.25,α=0.45,C2=2 and 1.6.

where2F1(a,b;c;z) is the hypergeometric function.Hence,the resulting shape function can be expressed as follows:

From the above expression,the derivative of the shape function is given by

Now,substituting function(36)into(25)and(26),we get the pressure elements as

where Γ(a,z) is the gamma function.

Figure 4 illustrates the characteristics of the energy conditions and the corresponding energy density profile for Lorentzian distribution.It shows that in this scenario,the radial null energy condition[figure 4(b)]and dominant energy conditions [figure 4(d)] are violated.But,the tangential null energy condition [figure 4(c)] and strong energy condition[figure 4(f)] are obeyed.

Figure 4.Lorentzian Source: The profile of energy density and energy conditions with respect to for different values of β with fixed parameters M*=7.25,α=0.45,C2=2and =1.6.

Moreover,by investigating the existence of wormhole solutions and analyzing energy conditions in the late-time universe,we explore exotic matter and energy distributions that could enable the formation and stability of wormholes.The presence or absence of these solutions has significant implications for our understanding of the late-time universe’s evolution and the nature of exotic matter needed to support such structures.

5.Effect of anisotropy

In this section,we explore the anisotropy of Gaussian and Lorentzian non-commutative geometry in order to understand the characteristics of the anisotropic pressure.The quantification of anisotropy plays a crucial role in revealing the internal geometry of a relativistic wormhole configuration.It is well known that the level of anisotropy within a wormhole can be measured using the following formula [20,49,62,71–73]:

We can determine the geometry of the wormhole based on anisotropic factor.When the tangential pressure is greater than the radial pressure,it results in Δ>0.This signifies that the structure of the wormhole is repulsive and anisotropic force is acting in an outward direction.Conversely,if the radial pressure is greater than the tangential pressure,it yields Δ<0.This indicates an attractive geometry of the wormhole and force is directed inward.The anisotropy for both the Gaussian (ΔG) and Lorentzian (ΔL) distributions with the linear model is calculated as

Figure 5 depicts the effect of anisotropy for a viable wormhole model under Gaussian and Lorentzian distributions.The investigation reveals that our anisotropy factor Δ is positiveand the structure of the wormhole is repulsive in Gaussian distribution [figure 5(a)],whereas Δ is negativewhich indicates an attractive geometry of the wormhole in Lorentzian distribution[figure 5(b)].

Figure 5.The graphical representation of anisotropy for both distributions.

6.Results and concluding remarks

In this article,we have explored the conformal symmetric wormhole solutions under non-commutative geometry in the background off(Q ,T)gravity.To achieve this,we have considered the presence of an anisotropic fluid in a spherically symmetric space-time.The concept of conformal symmetry and non-commutative geometry have already been used in literature within various contexts of modified theories of gravity [50–60,62–64].Non-commutative geometry is used to replace the particle-like structure to smeared objects in string theory.Furthermore,conformal killing vectors are derived from the killing equation,which is based on the Lie algebra.These vectors are used to reduce the nonlinearity order of the field equation.Conformal symmetry has proved to be effective in describing relativistic stellar-type objects.Furthermore,it has led to new solutions and provided insights into geometry and kinematics[74].It influences the geometry and dynamics of the space-time,impacting key parameters such as throat size and stability.

In the framework of extended symmetric teleparallel gravity,we have derived some new exact solutions for wormholes by using both Gaussian and Lorentzian energy densities of non-commutative geometry.For this object,we presumed the linear wormhole model asf(Q ,T)=αQ +βT,where α and β are model parameters.In both cases,we examined the wormhole scenario using Gaussian and Lorentzian distributions.By applying the throat condition in two distributions,we obtained different shape functions that obey all the criteria for a traversable wormhole.A similar result was presented in [63] where the authors explored wormhole solutions in curvature-matter coupling gravity supported by non-commutative geometry and conformal symmetry.Furthermore,we investigated the impact of model parameters on these two shape functions.Due to the conformal symmetry,the redshift function does not approach zero as r>r0[8,59,60,75].

Figures 1 and 3 show the graphical nature of the obtained shape functions with β ∊[0,0.5).Notably,a slight variation in the value of β can impact the nature of the shape function.Moreover,the graphical behavior of the energy conditions is shown in figures 2 and 4.The energy density is positive throughout the space-time.For all the wormhole solutions,the violation of the null energy conditions indicates the presence of hypothetical matter.Here,this nature of hypothetical fluid is presented in references [63,76,77].Next,we studied the effect of anisotropy for both distributions.The geometry of the wormhole is repulsive in the Gaussian distribution,whereas it is attractive in the Lorentzian distribution[figure 5].

To conclude,this work validates the conformal symmetric wormhole solutions inf(Q ,T)gravity under noncommutative geometry.The authors [78] have identified the possibility of a generalized wormhole formation in the galactic halo due to dark matter using observational data within the matter coupling gravity formalism.In the near future,we plan to investigate various wormhole scenarios in alternative theories of gravity,as discussed in references[79–82].

Acknowledgments

CCC,VV and NSK acknowledge DST,New Delhi,India,for its financial support for research facilities under DSTFIST-2019.

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