Fiaz Hussain ,Murtaza Ali ,Muhammad Ramzan and Sabiha Qazi

1 Department of Mathematics,The Islamia University of Bahawalpur,Pakistan

2 Faculty of Engineering Sciences,GIK Institute of Engineering Sciences and Technology,Topi,Swabi,KPK,Pakistan

Abstract In this paper,we classify static spherically symmetric (SS) perfect fluid space-times via conformal vector fields (CVFs) in f(T) gravity.For this analysis,we first explore static SS solutions by solving the Einstein field equations in f(T)gravity.Secondly,we implement a direct integration technique to classify the resulting solutions.During the classification,there arose 20 cases.Studying each case thoroughly,we came to know that in three cases the space-times under consideration admit proper CVFs in f(T) gravity.In one case,the space-time admits proper homothetic vector fields,whereas in the remaining 16 cases either the space-times become conformally flat or they admit Killing vector fields.

Keywords: spherically symmetric space-times,conformal vector fields,f(T) gravity

1.Introduction

The mathematical form representing the governing equations of general relativity (GR) is the set of Einstein field equations(EFEs)that provide an assembly between geometry and physics.It has been observed that significant nonlinearity exists in these equations,which presents hurdles to identifying their exact solutions.The problem of solving these equations is expected to be resolved if one chooses a suitable space-time geometry and then seeks their solutions.The geometry admitting spherical symmetry provides one of the best frameworks to explore the solutions that in turn help to describe challenges encountered in modern theoretical physics.For instance,the theory of black holes has been greatly supplemented by the spherically symmetric (SS) space-times admitted by the Schwarzschild metric.Other renowned models depicting spherical symmetry include Bertotti–Robinson metrics [1],Kiselev?s black hole space-time and the Reissner–Nordstr?m and Schwarzschild–(anti-)de Sitter solutions[2].Due to having the capability of describing a wider class of physical phenomena,SS space-times have been implemented in studying various astrophysical phenomena including the study of compact stars[3–5],neutron stars[6]and gravastars[7,8].The SS space-times could also be considered as key ingredients that yield a deeper approach to describing stellar models and the mysteries associated with redshifts[9].The above particular classes of the SS solutions belong to some of the early solutions of GR that help confirm its validity.Surely,GR is an effective theory of gravitation as it has been verified in several astrophysical trials [10–12].Some of the important predictions that have been confirmed by GR include the perihelion precession of Mercury,gravitational redshift,and deflection of light by the Sun [13,14].As well as these phenomena there also exist tests involving gravitational waves [15–18] and black hole shadows [19–21] that have been performed to test GR.

On the other hand,it is a hot topic of recent debate that GR is not an ultimate theory of gravitation as it appears that it lacks a way to address certain problems of theoretical physics.The most prominent topic of debate is the unexpected expanding behavior of our universe [22–24] which raises a question about the predictions claimed by GR.The surprisingly inadequate proportion of the mysterious dark matter and dark energy have added further problems which need to be understood[25,26].Thus,at the cosmological and large-scale level,GR also seems to fail in explaining various gravitational singularities.Such inconsistencies have led the scientific community to develop some reasonable tools.In this regard,a number of approaches have been proposed for achieving the desired cosmological results.The modified theories (MTs) of gravitation have appeared in order to address such problems of present day cosmology.Some of the astonishing facts supported by the MTs involve the predictions of non-singular cosmologies [27–29],the paradigm of galaxies?rotation curves,the hierarchy problem in high energy physics and the merging of Grand Unified Theories[30].Plenty of additional capabilities of the MTs of gravitation and their cosmological applications have been well documented in [31].

The first candidate that belongs in the category of MTs of gravitation is termedf(R)gravity,wheref(R)is a function of the Ricci scalarR[32].In the background of SS space-times,f(R)gravity has produced excellent results related to the existence of solutions of the EFEs [33–40].As a generalization off(R)gravity,Harkoet alintroducedf(R,T) gravity by adding the trace of the energy–momentum tensor (EMT) in the existing function of thef(R)gravity[41].Some of the other well-known classes of the MTs of gravitation and their characteristics are listed in[42–49].Some of these theories show their hands based on curvature invariants while others are based on torsion.The teleparallel equivalent (TE) of GR is the most basic torsion tensor theory that is designed based on certain tetrad fields[50].A generalization of the TE of GR is namedf(T)gravity,wheref(T) is a function of the torsion scalar [51].f(T) gravity is extremely important for several reasons.It can designate models of inflation[52,53].Implementation of such viablef(T)gravity models further helps to deepen understanding of the phenomenon of transition redshift[54,55].In cosmography,f(T)gravity shows important assets in the large-scale dynamics of the Universe[56].In light of such useful characteristics admitted byf(T)gravity,various reviews have been made in the literature regarding the existence of SS black hole solutions [57,58] as well as regular black hole solutions [59].In the background of SS space-times,work on solutions of the EFEs inf(T) gravity has been done in[60–68].In[69],the authors have employed a well-known class of symmetry restriction known as Noether symmetry to explore spherically as well as cylindrically symmetric solutions inf(T) gravity.Symmetries are one of the powerful tools available for solving dynamical problems via Lie differentiation.The space-time symmetries also help to reduce partial differential equations to ordinary differential equations that in turn become easy to handle.A very basic symmetry that is defined by the vanishing of the Lie derivative of the metric tensor is Killing symmetry which carries Killing vector fields(KVFs).This sort of symmetry helps to select different conservation laws of physics[70].The next symmetry restriction is called homothetic symmetry associated with homothetic vector fields (HVFs).This sort of symmetry forces the space-time to remain preserved up to a constant scale factor.A generalization of both the KVFs and HVFs are the conformal vector fields(CVFs).A CVF saysYis affiliated with the conformal symmetry (CS) and is defined as [71]

whereL,gab,ξ=ξ(t,r,θ,φ) and comma represent the Lie derivative,metric tensor,conformal factor and partial derivative respectively.In equation(1),ifξ=constant,thenYrepresents an HVF (proper homothetic if ξ ≠0).If ξ=0,thenYcharacterizes a KVF,otherwiseYrepresents a proper CVF.

The CS has been widely implemented to investigate the exact solutions of the EFEs employing certain restrictions on the gravitational field.Geometrically,CS preserves a null cone as it maps null geodesic curves.The CS also gives a deep insight into the space-time geometry that further helps to describe the associated kinematics as well as dynamics.With these properties,special attention has been given to the study of the CVFs in MTs of gravitation in the last few years[72–88].Continuing this stream of work,we are conducting a study to classify the static SS perfect fluid space-times via CVFs in thef(T) theory of gravity.

2.Static SS solutions of the EFEs and their CVFs in f(T) gravity

The SS space-times have great importance in the study of black hole thermodynamics and compact stars,as well as in the investigation of gravitational collapse.The space-times admitting spherical symmetry have generated several physically acceptable results of GR,such as the Schwarzschild,Bertotti–Robinson and Einstein metrics [1].In this paper,we consider a static SS space-time in the usual coordinates(t,r,θ,φ)labeled(y0,y1,y2,y3)respectively with the line element [89]

wherea=a(r),b=b(r)andQ=Q(r)are unknown functions of the radial coordinater.The minimum numbers of KVFs admitted by the above space-times (2) are [71]

The torsion scalarTassuming a diagonal tetrad for the above space-times (2) reads as [90]

where the prime afterQandarepresentsAs mentioned earlier in this paper,we want to explore CVFs of the static SS space-times inf(T)gravity,and therefore we start with the EFEs off(T) gravity which are [51]

where ρ andprepresent the energy density(ED)and pressure of the fluid distribution respectively.The thing which is important to point out here is that in contrast with GR,the equations of motion inf(T)gravity for a diagonal tetrad field associated with the static SS space-times (2) admit an additional equation that comes from the(r,θ)component [90]

To solve equation(10),we are imposing certain constraints on the space-time components as discussed earlier.By implementing this procedure,we find that there exist ten cases that depict the solutions of equation(10).For the sake of clarity,we also focus on determining the constraints admitted by the spacetime components along with their respective solution.It is important to indicate here that we have also deduced the torsion scalar by utilizing the resulting solutions in equation (4).A summary of solutions of equation (10) is given below:

Now,we discuss the second possibility arising from equation (9),that isT′=0,giving

Equation (11) yields solutions having constant torsion scalar as it resulted from the conditionT′=0.Again,adopting a similar classification procedure for equation (11) as we had performed while searching for the solutions of equation (10),we reach the following cases:

An important thing that is to be highlighted here is that because of the above classification,we have obtained various important classes of physical interest.It has been observed that most of the metric potentials under the preview of spherical symmetry admit logarithmic functions.The choice of the SS space-times (2) with exponential scale factors or metric potentials is quite important as the space-times with exponential functions as a scale factor help to get rid of such logarithmic functions and hence produce physically realistic models.One of these includes the Bertotti–Robinson solutions (see cases (viii) and (ix)).From the physical perspective,such classes represent the non-null homogeneous Einstein–Maxwell fields[89,93,94].From the point of view of the existence of black holes,most of the classes,for example cases (i),(iii),(iv),(v),(vi),(vii),(xi),(xii),(xiii),(xiv),(xv),(xvi),(xviii) and (xix),admit singularities at certain values of the radial coordinater,which implies that such solutions might be used to model some sorts of black hole.

Now,we utilize the above solutions (i)–(xx) to identify CVFs that meet the requirements of equation (1).It is important to acknowledge here that the CVFs for cases(i),(ii)and (iii) have previously been determined in [81].So,we overlook these cases and find CVFs of the remaining cases(iv)–(xx).We summarize the results in table 1 given below ignoring the lengthy process of calculations.

In table 1,

3.Summary and discussion

Thef(T)theory of gravity has been proven to be an effective component of the MTs of gravitation.It depends upon the torsion scalar which plays a significant role as it has beenobserved that the torsion scalar is responsible for measuring the intensity of the gravitational field[95].It has the potential to address late-time accelerated expansion as well as certain cosmological and astrophysical phenomena.These physical aspects are purely affiliated with the EFEs that vary by varying thef(T)gravity models.It is important to note that to achieve the desired cosmological results,one has to impose certain constraints appearing in the specificf(T) gravity model.Such constraints could be obtained by fixing the model parameters involved in the functional form off(T)gravity because of observational data.Such practices have been used by the authors in [96],to constrain model parameters of two well-studiedf(T) ansatzes,namely the powerlaw and the exponential,and they found less than one percent divergence from the TE of GR.The detail of such constraints under the preview of some important forms off(T) gravity can be seen in[97–99].Now,the issue is to explore a suitable form off(T) that further helps to deduce the solution for resulting EFEs.In this paper,we have employed a technique to explore the perfect fluid solutions of the EFEs inf(T)gravity by assuming static SS space-times.It is important to mention here that we have solved the governed system of equations (6)–(9) for two possibilities that arose from equation (9).Firstly,we have consideredfTT=0 and have found the solutions.In order to simplify the calculations,we have made some algebraic manipulations to get equation(10)that only involved the space-time components.Implementation of certain constraints on such components led us to select the exact forms of space-times.Secondly,we have considered the possibilityT′=0 and have found the solutions independently.It is necessary to clarify here that equation(10)is a part of the first possibility which is not concerned with the second possibility.In fact,we are treating the problem for each possibility separately.

Table 1.CVFs of obtained static SS metrics.

As an application of our extracted solutions,we have further implemented the CS for studying the CVFs.The reason for the interest in exploring CVFs has been well described in [100] where a wider class of applications of the CVFs in cosmology and astrophysics has been discussed.A subclass of the CVFs that is the KVFs carries a conserved quantity that admits certain conservation laws of physics.The other symmetry restriction that comes within the shadow of conformal motion has been termed homothetic motion.This symmetry restriction forces the metric to remain invariant up to a constant scale factor.The results of this study have been divided into the following categories.In cases(i)and(ii),the space-times have turned out to be conformally flat,so attain maximum dimension i.e.15 [81].Similarly,the CVFs for case(iii)have also been discussed in[81]where the resulting space-time admits four KVFs.The results of the remaining cases after performing the above procedure are given below:

(a) The CVFs in cases (iv),(v),(vi),(viii),(ix),(xi),(xii),(xiii),(xiv),(xv),(xvii),(xviii) and (xix) become the KVFs.Physically,the KVFs produce conservation laws i.e.conservation of energy and linear momentum are related with the translational KVFs ?tand ?φrespectively,whereas conservation of angular momentum has been depicted by the rotational isometriesY2andY3respectively.

(b) In cases (vii),(xvi) and (xx),the space-times admit proper CVFs.The dimension of CVFs for these cases has turned out to be six,of which four are KVFs which are given in equation (3),whereas the remaining two are proper CVFs which areThe CVFs have a wide scope of application in loop quantum cosmology and astrophysics giving models of compact stars,dense stars and gravastars [85].

(c) In case(x),the space-time admits proper HVFs due to the conformal factor ξ being a non-zero constant.The HVFs are quite important from several points of view.First,HVFs have been found useful in discussing the constant of motion that allows for examining particle trajectories in space-time [84].Secondly,the homothetic motion helps to address the singularity issue in GR.It is to be noted that by studying self-similar solutions of the EFEs,the class of HVFs has also played a significant role.

It is valuable to indicate here that to complete the study,we have also evaluated the dynamical parameters like ED(ρ) and pressure(p) of the fluid distribution in each of the cases (i)–(xx)by utilizing values of space-time components along with the functionf(T)in equations(6)–(8).Recall that in cases(i)–(x),f(T) is a linear function i.e.f(T)=d1T+d2,whered1,d2?R,whereas the solutions in cases (xi)–(xx) have been obtained assumingT′=0 indicating that the torsion scalar is a constant.This condition further implies thatf(T)would be a constant function.On utilizing this fact in equations(6)–(8),we have found that for each of the static SS models represented by cases (xi)–(xx),both the ED and pressure are non-zero constants and are related asp=-ρ.The values of the physical parameters for the rest of the cases(i)–(x) are tabulated below.

In table 2,the ED and pressure for cases(i),(iii),(iv),(v),(vi),(viii),(ix) and (x) are related asp=-ρ,which shows that the dominant universes behave as though they contain dark energy or the ED of a vacuum or cosmological constant.In cases (ii) and (vii),the ED and pressure are positive if the constantsd1andd2turn out to be positive.Such types of cases would produce an attractive gravitational effect on our universe’s expansion whereas negative pressure in dark energy has been interpreted as the effect of antigravity[101].

Moreover,it is clear from table 2 that both the ED and pressure have turned out to be constant except in case (vii).Physically,the SS solutions showing such behavior are important,particularly in astrophysics.It has been observed that several constant density solutions are able to model stellar objects and relativistic stars [102].Exterior Schwarzschild–anti-de Sitter solutions and Einstein static metrics are well known examples of SS solutions that admit constant ED.

As already mentioned,we have obtained the values of ED(ρ) and pressure(p) by utilizing the metric components along with the observedf(T) in equations (6)–(8).The solutions deduced in this paper enjoy certain important physical properties that could be judged via energy conditions.It is well known that some sorts of energy conditions exist in theliterature that help to select physically realistic solutions for the EFEs [103].These conditions include the weak energy condition (WEC),strong energy condition (SEC),dominant energy condition (DEC) and null energy condition (NEC).The mathematical representation for a perfect fluid solution to admit the WEC is ρ ≥0,ρ+p≥0 and for a solution to fulfill the SEC it is ρ+p≥0,ρ+3p≥0.Similarly,solutions obeying ρ ≥0,ρ≥∣p∣indicate that they satisfy the DEC whereas those solutions which satisfy only condition ρ+p≥0 are said to satisfy the NEC [104].To check the physical soundness of the obtained solutions,we also classify them according to the energy condition that they may admit.Particularly,we have observed from table 2 that in case (i),the WEC is satisfied ifd1<0 andFor case (ii),we found that the values of ED and pressure satisfy the WEC ifd1,d2>0.Case (iii) satisfies the NEC forAgain,case (iv) satisfies the WEC on account ofd2>0.Cases (v)and (vi) fulfill the WEC ford1,d2>0 andd2>0 respectively.Similarly,case(vii)fulfills the NEC ifd1>0 whereas cases (viii),(ix) and (x) satisfy the WEC ifd2>0.

Table 2.ED and pressure.

It is worth mentioning here that the present study is conducted using static SS space-times admitting a diagonal tetrad with linear or constantf(T) gravity.The analysis may become more impressive if one deals with the non-diagonal tetrad and nonlinearf(T) gravity and then performs this type of classification to explore solutions and study the dynamics as well as the space-time symmetries of the resulting solutions.