Particle dynamics,black hole shadow and weak gravitational lensing in the f (Q) theory of gravity

Allah Ditta ,Xia Tiecheng ,Farruh Atamurotov ,Ibrar Hussain and G Mustafa

1 Department of Mathematics,Shanghai University and Newtouch Center for Mathematics of Shanghai University,Shanghai,200444,Shanghai,China

2 New Uzbekistan University,Movarounnahr Street 1,Tashkent 100000,Uzbekistan

3 Central Asian University,Milliy Bog’ Street 264,Tashkent 111221,Uzbekistan

4 Tashkent University of Applied Sciences,Str.Gavhar 1,Tashkent 100149,Uzbekistan

5 Institute of Theoretical Physics,National University of Uzbekistan,Tashkent 100174,Uzbekistan

6 School of Electrical Engineering and Computer Science,National University of Sciences and Technology,H-12,Islamabad,Pakistan

7 Department of Physics,Zhejiang Normal University,Jinhua 321004,China

Abstract We study the particle dynamics around a black hole (BH) in f(Q) gravity.First,we investigate the influence of the parameters of f(Q)gravity on the horizon structure of the BH,photon orbits and the radius of the innermost stable circular orbit (ISCO) of massive particles.We further study the effects of the parameters of f(Q)gravity on the shadow cast by the BH.Moreover,we consider weak gravitational lensing using the general method,where we also explore the deflection angle of light rays around the BH in f(Q)gravity in uniform and nonuniform plasma mediums.

Keywords: f(Q) gravity,black holes,photon motion,shadow and lensing

1.Introduction

General relativity(GR)was proposed by Einstein in 1915 as a basic theory that explains the nature of the fundamental force of gravity.The Einstein theory of gravity has successfully passed the test based on different observations and experiments on a macroscopic scale in the Universe.Applying GR can be counted as an examination in the weak field regime to test it within the solar system [1],and it has already been done.The current scrutiny about gravitational waves [2] and the shadows of M87*and SgrA*[3,4] can be counted as an examination of GR in the dexterous field regime.The theory of GR during gravitational collapse has some limitations in explaining the features of the singularity.Other issues that cannot be well explained in the framework of GR include the rotation curve of galaxies,cosmic acceleration,dark energy/matter and the quantum theory of gravity.To resolve these issues,it has been suggested by different authors that the theory of GR should be amended.

Symmetric teleparallelism(ST)is a generalization of GR,which is differentiated from GR based on different sets of geometric postulates.The affine connection,,plays a vital role in differentiating the theory of GR from ST.In GR,the connection is supposed to be torsion-free and metric-compatible,which means it is genuinely based on the Levi-Civita connection.Metric-compatibility hypothesis in ST is excluded and in place of itbecomes torsion-free and results into a vanishing Riemann curvature tensor.As for how this connection fulfills these hypotheses,it can be understood as becoming independent of the metric in a specific and arbitrary manner.Along with the curvature and torsion oftaken as zero,only the non-metricity tensorQαμνis the non-trivial object in ST which is responsible for defining the affine geometry.Non-metricity scalar Q defines the action of ST and can be calculated by using the non-metricity tensor,Qαμν.Famously,the action of ST to a boundary term has a close approximation to the Einstein–Hilbert action of GR [5–9].Therefore,the geometric description of symmetric teleparallel gravity (STG) is different from GR.Specifically,it can be shown that the appearance of only affine connection as a boundary term in the action is non-physical.In short,the field equations are totally independent of the choice of connection,and any connection having compatibility with the STG can be chosen in the field equations.Thus the physical degree of freedom is purely metric-dependent and any sort of connection does not hold any form of physical information.

Changes may occur while taking the generalizations to the more generic theories which are quadratic in the nonmetricity tensor [10],or an expansion of STG [11].In this manuscript,we are interested in the non-linear extension defined by the action:[5],where f(Q) shows the prior arbitrary function of Q.This theory not only possesses non-equivalence to f(R) gravity,but also considers the degree of freedom based on the affine connection.The dependence onshould not be further absorbed into a boundary term in the action.It can be expected that the connection will influence the metric defining the gravity.Against the claim present in the literature [12,13],we work with the realization of this expectation [14] by studying the most common version of static and spherically symmetric spacetimes incorporated in the f(Q) theory of gravity.

Test particle motion can be studied as a beneficial tool to examine the metric-based theories defining gravity as a spacetime structure.The impacts of the spacetime curvature and gravitational field parameters on the particle motion have been extensively studied in the literature [15–30].Analyzing the motion of test particles having non-vanishing electric and magnetic charge may lead towards a direct understanding of the essence of the gravitational and electromagnetic field around the gravitating compact object.Knowledge of photon motion in the vicinity of compact objects,like BHs,is essential for studying the gravitational lensing and shadows cast by BHs and can enhance our understanding about different structures,like distant galaxies,in the Universe.Gravitational lensing can be divided into two categories: (a)strong gravitational lensing with the gravitational deflection angle considerably larger than one and(b)weak gravitational lensing with the gravitational deflection angle considerably less than one.It is assumed that weak gravitational lensing can be considered in investigating the cause of the current accelerated expansion of the cosmos [31].Astrophysically,the observed bending of light and gravitational lensing fall in the limit of weak gravitational lensing,i.e.with the deflection angles very much less than one [32].Therefore,the study of weak gravitational lensing is even more fascinating.Furthermore,it is of much interest to study the gravitational lensing in a plasma medium as it is assumed that the light rays in space always travel through such mediums (see more examples[33–50]and other works[51–54]).In a field having nonuniform plasma,photons travel along curved trajectories,since plasma is a medium possessing a dispersive property,having a permittivity tensor based on its density[55].Photon trajectory in a dispersive nonuniform plasma medium is based on the frequency of the photon without any relation to the gravity.The photon deflection in a nonuniform plasma with the presence of gravity has been discussed in the literature[56,57].This study considers a linear approximation with two effects independently: (a) the deflection caused by the gravitation in the vacuum and (b) the deflection caused as a result of the non-homogeneity of the medium.The first effect is neutral and the second effect is based upon the frequency of photon in a dispersive medium and it approaches to zero in a homogeneous medium.The study of the shadow cast by a BH may provide useful insight on estimation of the value of the spin parameter of a rotating BH [58–60].The BH shadow with and without plasma has been analysed for different BHs in the literature [61–88].

Almost one hundred years after the initial observation of gravitational light deflection,in 2019 a significant breakthrough happened,when the Event Horizon Telescope(EHT)Collaboration [4] accomplished a momentous feat by predicting an image of a BH.This achievement stemmed from our understanding that,when light closely approaches a BH,it shows observable deflection,occasionally traversing circular orbits.This considerable light deflection,combined with the intrinsic property that no light escapes from a BH,leads to the hypothesis of a dark disk-like region inside the celestial sphere,known as the BH shadow.The concept of observing this shadow was first hypothesized in the year 2000[60],which was later acknowledged by the authors in reference [89],through continuous numerical simulations,suggesting the conclusion that successful observations could be made at wavelengths near 1 mm using the technique of very long baseline interferometry (VLBI).The recent remarkable achievement of the EHT Collaboration motivated us to deepen our understanding and provide explanations for both what we can observe and what remains concealed [90].

In this work,we discuss the motion of particles,the BH shadow and weak gravitational lensing in the presence of plasma in f(Q)gravity to look at the effects of the parameters present in f(Q) gravity and get some new insights about this alternative theory of gravity.In section 2 we discuss the basics of f(Q) gravity and particle dynamics.In section 2.1,we study massive particle motion,and section 2.2 consists of the discussion about the motion of massless particles around a BH in f(Q) gravity.Section 3 contains a discussion of BH shadows in f(Q) gravity.In section 4,we investigate weak gravitational lensing (section 4.1 for uniform plasma,and section 4.2 for nonuniform plasma) in the f(Q) theory of gravity.In section 5,we present the conclusion for our study.Throughout we use a system of units in which G=1=c.

Figure 1.Dependence of the horizon structure on f(Q) parameters.

2.Particle motion around a black hole in f(Q)gravity

In this study we consider the action for f(Q) gravity [14],given by

where the determinant of gμνis denoted by g,f(Q) is the function of non-metricity Q,is the multiplier for the Lagrangian,and Lmdenotes the matter Lagrangian density.The metric components in their complete format can be put in the following form [14]

where Mrenis the renormalized mass which is given by

where the above equation satisfies the field equations of the theory under consideration.The scale r1can be initiated by the change in the constant likec6→c6-48M2c7ln(r1),in a desire to have a dimension-free argument in the logarithm function.Here,we describe a new scale

whose strength is characterized beyond the GR correction,which is another new‘BH charge’,also known as connection hair.It is worth taking into consideration that the correction terms could accelerate deviations from the Schwarzschild solution for larger values of r.The correction term for the logarithm will be determined by the command for radii upon the renormalized Schwarzschild term,satisfying the relation

All the f(Q) setup is based upon the model f(Q)=Q+αQ2,so α is a constant parameter with real values.In order to produce results very close to the theory of GR,as an ansatz we set |α|≪1.ciis a real integration constant.The above equation (5) expresses the breakdown of the perturbation theory at larger values of r.Metric perturbations are small exclusively at the smaller values of r in comparison to the background of the Schwarzschild spacetime.The increasing and decreasing behavior of the horizon structure of the BH in f(Q) gravity can be seen in figure 1 and is calculated using the condition gtt=f(r)=0.It can be noticed that one can retrieve the Schwarzschild case by replacing α=0 in equation (2).From figure 1,one can notice that the horizon radius for the f(Q) BH considered in the present study is bigger than that of the Schwarzschild BH.

Figure 2.Veff along radial coordinate r of the massive particle.

For the next two subsections we will discuss the massive and massless particles motion in the vicinity of a BH in f(Q)gravity:

2.1.Massive particle motion around a black hole in f(Q)gravity

The trajectory of the test particle can be found by taking into account the Lagrangian for the test particle having mass m in the form given below

where τ represents an affine parameter,xμdenotes the coordinates and uμexpresses the four-velocity of the test particle.The conserved quantities responsible for the motion of the test particle,such as the energyε and the angular momentumL,can be put in the form below:

If we put equation (7) into the normalization condition gμνuμuν=−∊,it is very convenient to find the equations of motion of a test particle in the equatorial plane in which we haveas [91]:

where parameter ∊is described as given below [91]:

The equation expressing the radial motion reduces into the following specific form:

Here,Veff(r) represents the effective potential for the motion of the test particles.For a massive test particle,the effective potential depends on the radial coordinate around the BH.For some different values of the parameters α,c2,c3,c6,c7and r1which can be fixed for the correct depiction of results,is shown in figure 2.It shows that the stable circular orbits move with respect to the central compact object.Moreover,Veff(r) for the current study is less than the Schwarzschild case,where α=0.

One can use the conditionsr˙=0 and dr˙=0 to discuss the circular motion of a neutral particle around the BH in f(Q)gravity.These conditions permit one to get an expression for the energyε and also for the angular momentumL of the test particle in the equations

Figure 3.ε along the radial coordinate r of the massive particle.

and

where

Further detailed information regarding the conserved quantities can be obtained from the graphs presented in figures 3 and 4.These graphs show the shift of the curves with respect to the central BH for parameters α,c2,c3,c6,c7and r1of f(Q)gravity.Also,ε andL in our present study are more than the GR case,where α=0.

Now,we may take into consideration the radius of the ISCO,rISCO.In order to calculate rISCO,we have to utilize the following conditions [91]:

Due to the complexity of the system we cannot deal with rISCOanalytically.The detailed behaviour of the ISCO depending upon the parameters c2,c3,c6,c7and r1is shown in figure 5.In particualr,one can see that rISCOis bigger than the GR case(α=0),and increases with the parameters c2,c6and α,and decreases with the parameter c7.

2.2.Massless particle motion around a black hole in f(Q)gravity

In this subsection,we study the massless particle (photon)motion in a BH spacetime in f(Q) gravity.By utilizing the metric Lagrangian of the f(Q)gravity BH spacetime,one can obtain the equation of photon motion around the BH in the f(Q)theory of gravity by taking ∊=0 in equation(11).In the equatorial plane the equation of motion can be represented as follows:

By utilizing equation (17),one can easily get the expression for effective potential Veffof the motion of the photon as[91]:

Effective potential dependence on the radial coordinate for the motion of photons is given in figure 6.The shift of photon orbit towards and outwards from the central object for different parameters can be seen in figure 6.The photon’s circular orbit radius rpharound the BH in f(Q) gravity can be obtain from the solution of the second equation given in equation (16).Being complex,we deal with it numerically.The increasing and decreasing behaviour of the photon circular orbits rphwith the f(Q)gravity parameters c2,c3,c6,c7,r1and α is expressed in figure 7.Notice that the increasing and decreasing trend of the radius of the photon orbits is similar to that for the massive particle discussed above.

3.Black hole shadow in f(Q) gravity

This section consists of the study of BH shadows in the f(Q)theory of gravity.The angular radius of the BH shadow[61,83] consists of the following expression:

where αshdenotes the angular radius of the BH shadow,robsrepresents the observed distance and rphstands for the radius of the photon sphere.By combining equations (21) and (22),we obtain the following relation

By utilizing equation (23),the radius of the BH shadow at a large distance for an observer can conveniently be found[61]:

Finally,equation (24) is the non-rotating case of the shadow of the BH.Figure 8 is the graphical depiction of the radius of the BH shadow in f(Q)gravity for various values of the parameters involved.It is important to notice that the BH shadow decreases as compared to the Schwarzschild BH shadow,indicated by the black solid line (curve) along the increasing values of c6.Furthermore,we notice that the shadow of the BH in f(Q)gravity increases with the increase in the values of the parameters c2,c3and α.

Figure 5.rISCO with parameters c2,c6,c7 and α for different values of α and r1.

Figure 6.Veff and the radial coordinate r of the massless particle.

Figure 7.Radius of the photon orbit rph,with different f(Q) gravity parameters for some values of α and r1.

Figure 8.Rsh for the BH shadow with different f(Q) gravity parameters for some values of α and r1.

Figure 9.Uniform plasma effect with different f(Q) and plasma parameters.

Figure 10.Nonuniform plasma effects with different f(Q) and plasma parameters.

Figure 11.Difference between the different effects of uniform and nonuniform plasma in f(Q) gravity.

4.Weak gravitational lensing in f(Q) gravity

In this section,we study an optical characteristic of the BH in f(Q) gravity by analysing the effect of weak gravitational lensing.For an approximation to the weak field,the following form of the metric tensor [33,34] can be used:

where ηαβand hαβdenote the Minkowski spacetime and the perturbed gravitational field describes the f(Q) theory of gravity.The following properties are essential for the two terms ηαβand hαβ

The general equation of the angle of deflection in the presence of a plasma medium can be written in the following form[33,34]

where N(xi)is the density of the plasma particles around the BH,Ke=4πe2/meis a constant,and ω and ωe=4πe2N(xi)/m=KeN(xi) are photon and plasma frequencies,respectively [33].By the use of the basic equations (27)–(26),we arrive at the expression for the angle of deflection around the BH in f(Q)gravity,as given below [33]:

The metric element can be rewritten in the below given form:

It is convenient to find the components hαβof the metric in the form of Cartesian coordinates:

Differentiation of h00and h33with respect to the radial coordinate is described by:

The following is the relation representing the deflection angle[40]

Now we make a schematic plan to analyse and calculate the deflection angle for different forms of the plasma density distributions.

4.1.Uniform plasma distribution

The gravitational deflection angle around the BH in f(Q)gravity containing uniform plasma is written in the form of the following relation [40]:

By using equations(32),(35)and(36),it is easy to obtain the expression for the deflection angle around the BH in f(Q)gravity in a medium consisting of uniform plasma,as given below:

We can plot the dependence of the angle of deflection on the impact parameterand c7for various values of other parameters of f(Q) gravity,such asetc in the BH spacetime of f(Q)gravity.This behavior is represented graphically in figure 9.It can be seen that the deflection anglefor f(Q) gravity is less than for the GR case (α=0).

4.2.Nonuniform plasma distribution

This part of our analysis consists of a non-singular isothermal sphere (SIS),which is the most advantageous model to understand specific properties of the photon sphere around a BH in weak gravitational lensing.In general,an SIS is a spherical gas cloud having singularity which is detected at its centre where the density leads to infinity.The distribution of density for an SIS is as follows [33]:

where mpdenotes the mass and k denotes the dimension-free constant coefficient of the dark-matter–dominated Universe.The plasma frequency in its expressive form is given below:

Here,we explain the effects of nonuniform plasma on the deflection angle in the f(Q)gravity BH spacetime.Expression of the deflection angle in f(Q)gravity around the BH may be written as [40]:

By the combination of equations(32),(36)and(42),one can get the deflection angle in the following form:

These calculations lead to a supplementary plasma constantwhich is given in the form of the following analytic expression [34]:

Here,equation (43) allows us to plot the dependence of the angle of deflection with different parameters of f(Q)gravity,i.e.and c7,for different values of other parameters of f(Q) gravity,such asc2,c3,for the nonuniform plasma,shown in figure 10.The deflection angleαˆSISin f(Q)gravity is less than the GR case (α=0).

Moreover,we compare the different effects of uniform and nonuniform plasma on the angle of deflection around the BH in f(Q) gravity,represented in figure 11.

5.Conclusions

We have discussed the motion of massive and massless particles around a BH in f(Q)gravity and have investigated the effects of the different spacetime parameters on the motion of these particles.We have also analysed the BH shadow and weak gravitational lensing in a plasma medium in the realm backed by the f(Q) gravity spacetime.It is important to mention that we have studied the f(Q) BH features making comparison with the standard Schwarzschild BH of GR.Subsequently,we have fixed other small values of α to look at the impacts of f(Q)gravity.We have observed that all the f(Q)gravity parameters α,c2,c3,c6,c7,r1,plasma impact parameter b,plasma distribution(SIS)have influence on the study of the motion of particles around the BH in f(Q) gravity and on the associated phenomena of gravitational lensing and shadow formation.The effects of all these f(Q) gravity parameters on plasma distributions can readily be seen from the graphs in the figures.The discussion above of the dynamics of particles and the graphical analysis of the BH spacetime in f(Q) gravity leads to the following concluding remarks.

• We have explored the BH horizons in the paradigm of f(Q) gravity.It can be observed from figure 1 that the horizon radius increases with c2,c3and α (for smaller values of α) and decreases with c6.

• For the massive particle motion,we have studied the dependency of the effective potential on the radius r for different values of the f(Q)gravity parameters as plotted in figure 2.The effective potential decreases with increasing values of α,c2,c3,c7and r1and increases with c6.We have also plotted the energyε and angular momentumL in figures 3 and 4.These figures show how the energy and angular momentum of the particles change with the f(Q)gravity parameters.

• We have also plotted the ISCO radius with different parameters of f(Q) gravity for some values of α=0,0.02,0.04,0.06,as shown in figure 5.We have seen that the radius of the ISCO increases with c2and r1and decreases with c6,and c7.

• The photon radius has been obtained in the generic way using the effective potential plotted in figure 6.We have also discussed the photon motion as plotted in figure 7.It is important to note the rphincreases with c2,c3,r1and α,while it decreases with c6,and c7.

• Figure 8 shows the behavior of the radius of the BH shadow in f(Q) gravity,which is to increase with c2,c3and α but to decrease with c6.

• We have described the characteristics of the weak gravitational lensing of the BH on a light ray in f(Q)gravity in uniform and nonuniform plasma concentrations,and plotted in figures 9 and 10 their increasing and decreasing effects.It is worth noting that in all the cases the concentration of uniform plasma from angle of deflection is more than the nonuniform plasma,which can be seen from figure 11.

Here we mention that this study may be of use in future related works as it contains significant new findings about the effects of the f(Q) theory of gravity parameters on particle motion and associated phenomena in the BH spacetime.

Acknowledgments

This work is funded by the National Natural Science Foundation of China 11 975 145.This research is partly supported by Research Grant FZ-20200929344 and F-FA-2021-510 of the Uzbekistan Ministry for Innovative Development.G.Mustafa is very thankful to Prof.Gao Xianlong from the Department of Physics,Zhejiang Normal University,for his kind support and help during this research.Further to this,G.Mustafa acknowledges Grant No.ZC304022919 to support his Postdoctoral Fellowship at Zhejiang Normal University.

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