Motion of Particles and Gravitational Lensing Around (2+1)-dimensional BTZ blackholes in Gauss-Bonnet Gravity

Bakhtiyor Narzilloev,1, 2, 3, ∗ Sanjar Shaymatov,3, 2, 4, 5, 6, † Ibrar Hussain,7, ‡

Ahmadjon Abdujabbarov,3, 8, 5, 6, 9, § Bobomurat Ahmedov,3, 5, 6, ¶ and Cosimo Bambi1, ∗∗

1Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200438 Shanghai, China2Akfa University, Kichik Halqa Yuli Street 17, Tashkent 100095, Uzbekistan

3Ulugh Beg Astronomical Institute, Astronomy St 33, Tashkent 100052, Uzbekistan4Institute for Theoretical Physics and Cosmology,

Zheijiang University of Technology, Hangzhou 310023, China5National University of Uzbekistan, Tashkent 100174, Uzbekistan

6Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Kori Niyoziy 39, Tashkent 100000, Uzbekistan7School of Electrical Engineering and Computer Science,

National University of Sciences and Technology, H-12, Islamabad, Pakistan8Institute of Nuclear Physics, Ulugbek 1, Tashkent 100214, Uzbekistan

9Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai 200030, P. R. China(Dated: September 8, 2021)

We study motion of test particles and photons in the vicinity of (2+1) dimensional Gauss-Bonnet(GB) BTZ black hole. We find that the presence of the coupling constant serves as an attractivegravitational charge, shifting the innermost stable circular orbits outward with respect to the onefor this theory in 4 dimensions. Further we consider the gravitational lensing, to test the GB gravityin (2+1) dimensions and show that the presence of GB parameter causes the bending angle to growup first with the increase of the inverse of closest approach distance, u0, then have its maximumvalue for specific u∗0, and then reduce until zero. We also show that increase in the value of the GBparameter makes the bending angle smaller and the increase in the absolute value of the negativecosmological constant produces opposite effect on this angle.

PACS numbers: 04.70.Bw, 04.20.Dw

I. INTRODUCTION

The existing large amount of data from observations ofastrophysical processes around compact objects, such asgravitational waves [1], black hole shadows [2, 3], etc., to-gether with gravity theories can provide useful tool to un-derstand the nature of the gravitational interaction [4–6].On the other hand testing and constraining the param-eters of the gravity models is a step forward to discoverthe unified field theory.

Avoiding the Lovelock’s theorem indicating that in lessthan 5 dimensions the cosmological constant can appearonly within the general relativity [7] recently there hasbeen proposed a new approach to obtain the solutionin the Einstein-Gauss-Bonnet (EGB) gravity in 4 and 3dimensional (D = 4, D = 3) spaceimes [8]. The approachbased on using the rescaling of the Gauss-Bonnet termin such a way that the limit D → 4 (D → 3 ) does notdiverge.

A lot of work has been done in the literature on EGBgravity in D = 4. Particularly, authors of Ref. [9] have

∗nbakhtiyor18@fudan.edu.cn†sanjar@astrin.uz‡ibrar.hussain@seecs.nust.edu.pk§ahmadjon@astrin.uz¶ahmedov@astrin.uz∗∗bambi@fudan.edu.cn

studied the gravitational collapse in 4-D EGB gravity andhave showed the similarity of spherical dust collapse toone in Einstein’s gravity. The effect of the GB couplingconstant on the superradiance in black hole spacetimeshave been studied in [10]. The stability of linearizedequations of motion has been analyzed in Ref. [11], ex-ploring the perturbations of the black hole event horizonin the GB gravity. The Ref. [12] is devoted to studythe classical spinning test particle motion around non-rotating black hole in a 4-D EGB gravity. Authors ofRef. [13] have analyzed the motion of test particle alongthe geodesic around 4-D EGB black hole. Charged par-ticle and epicyclic motions around 4-D EGB black holeimmersed in an external magnetic field has also been con-sidered in Ref. [14].

The rotating analogue of static black hole solution in4-D EGB gravity has been obtained in [15]. Analyzingthe scalar and electromagnetic perturbation around 4-DEGB rotating black hole, the authors of Ref. [16] havechecked the strong cosmic censorship conjecture. Otherproperties of the rotating spacetime in 4-D EGB blackhole have been explored in Refs. [17–21]. The energeticsand shadow of higher dimensional i.e 6-D EGB black holehave been studied in [22]. Also the authors of Refs. [23,24] have investigated overspinning of 6-D EGB black holeand circular orbits around higher dimensional Einsteinand Gauss-Bonnet rotating black holes.

The thermodynamics, phase transition and Joule-Thomson expansion of 4-D Gauss-Bonnet AdS black

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hole [25], particle acceleration [26], thermodynamic ge-ometry of the 4-D EGB AdS black hole [27], the emer-gent universe scenario in the 4-D EGB gravity [28],extended thermodynamics and microstructures of 4-Dcharged EGB black hole [29], the shadow of rotating 4-D EGB black hole [30] and gravitational lensing of 4-DEGB black hole [31, 32] have been widely studied in theliterature.

At the same time there has been a discussion of the va-lidity of the model. Particularly, Ref. [33] has discussedthe problem of existance of EGB theory in 4-D space-time. The main conclusion of Ref. [34] was that 4-D EGBis not well defined. Other authors [35, 36] have also ques-tioned the validity of the field equations in the case of 4-DEGB gravity. At the same time in the limiting case whenD → 4 the higher-dimensional scattering amplitudes ofthe GB theory differs from the general relativity and maybe caused by additional scalar-tensor field [37]. Here weplan to study the spacetime properties around the 3-DGauss-Bonnet BTZ black hole using the analysis of testparticles and photon motion. This study may be usefulfor developing new tests of the 3-D Gauss-Bonnet the-ory and get corresponding constraints on parameters ofthe theory. Another interesting aspect of studying blackholes in dimension D < 4 is that they may refuse what istrue for black holes in dimension D = 4 [38]. We use thesolution obtained in Ref. [39] describing the 3-Ds BTZblack hole in the EGB theory gravity.

The test particle motion as well as photon trajecto-ries are useful tool to explore the spacetime propertiesand its structure in various gravity theories [40–61]. Par-ticularly the charged and magnetized particles motionaround black hole in the presence of external magneticfield have been widely studied in Refs. [59, 62–81].

Photon motion and its deflection in the gravitationalfield is one of the main features of the metric theoriesof gravity. One may read the review of the effects ofgravitational lensing in Refs. [82–85]. A number of workshave been devoted to explore the gravitational lensingin the weak and strong field regimes [86–104]. One ofthe consequences of the exploration of photon motionleads to shadow of the black holes. The discovery of theimage of the black hole shadow by EHT team [2, 3] hasbeen conducted with the theoretical studies of this effectby various authors [105–133]. In the present study weare keen to explore the gravitational lensing in the 3-DEGB gravity. In particular we examine the effects of theGB parameter and of the cosmological constant on thegravitational lensing in the the 3-D EGB gravity.

The paper is organized as follows: The Sec. II is de-voted to the review of the 3-D BTZ black hole solu-tion in EGB theory. We study the test particle motionaround the 3-D BTZ black hole in EGB theory in Sec. III.We explore the photon motion and gravitational lens-ing in Sect. IV. We conclude our discussion in Sec. V.Throughout this work we use a system of units in whichG = c = 1. Greek indices are taken to run from 0 to 2,while Latin ones from 1 to 2.

II. 3D GAUSS-BONNET BTZ BLACK HOLEMETRIC

In D dimensions, the action for the GB theory withscalar field φ is given by

S =1

16πG

∫ √−gdDx [R− 2Λ + α (φLGB

+ 4Gµν∂µφ∂νφ− 4 (∂φ)2�φ+ 2

((Oφ)

2 )2)],(1)

where α refers to the the the GB coupling constant andLGB the GB term is given by

LGB = RµνλδRµνλδ − 4RµνR

µν +R2 , (2)

with R being the scalar curvature. Recently [8] proposednew approach that it is possible to obtain GB contribu-tion in D = 4 by rescaling the coupling constant. It hasalso been shown [39], that it is possible to obtain D = 3case of the theory with φ = ln( r` ), where ` is the constantof integration. It is worth noting that [39, 134] for D = 3the GB term LGB vanishes.

The form of the GB theory in D = 3 is given by

ds2 = −F (r)dt2 +dr2

F (r)+ r2dϕ2, (3)

with

F (r) = − r2

2α

(1±

√1 +

4α

r2fE

), (4)

where

fE =r2

l2−m, (5)

where m and l respectively refer to the integration con-stants with Λ = −1/l2. Note that this 3−D GB theory

has two separate black hole solutions, i.e. ±√

1 + 4αr2 fE .

It is worth noticing that in the spacetime of 3D BTZblack hole in GB gravity one can calculate the effectiveAdS radius from the condition F (r) = 0 that reads as

rAdS =

√−m

Λ, (6)

which is independent of the GB coupling parameter, α,while in higher-curvature theories this radius is a func-tion of the coupling parameter [13–15, 20, 21, 30]. Fromthe above equation, it is clearly seen that the integrationconstant m is positive quantity as Λ takes negative val-ues only. Note that this AdS radius coincides with onefor 3D BTZ black hole in Einstein gravity in the limitJ → 0 [135]. It can be also seen that the spacetime met-ric allows the cosmological constant to take only negativevalues to have real AdS radius corresponding to the cos-mological horizon which tends to infinity in the case inwhich Λ→ 0.

3

Here we restrict ourselves to the ’minus’ branch of so-lution as it leads to a well-defined BTZ solution only inthe limiting case of small α, i.e., when the metric functionF (r) is expanded in the form [39]

limα→0

F (r) =r2

l2−m− α

r2

(r2

l2−m

)2

+O(α2) , (7)

which coincides with the one for standard BTZ solutionfor small α in the Einstein gravity. However, in the lim-iting case of large r the metric function F (r) yields

F (r) =r2

2α

(√1 +

4α

l2− 1

)− m√

1 + 4αl2

+O(1/r2) . (8)

From the above expression, the coupling parameter cantake negative values, i.e. α > −l2/4 at the large distances(see for example [39] ). However, this would not be thecase in close vicinity of the 3D BTZ black hole. There-fore, we further consider positive values of α to explorethe properties of 3D BTZ black hole in GB gravity.

III. TEST PARTICLES MOTION

Here we consider a test particle motion in the gravi-tational field of the (2+1)-dimensional BTZ black holein the GB gravity. To study the motion of test parti-cles in the vicinity of the BTZ black hole we explore theHamilton-Jacobi equation [136]

H ≡ 1

2gµν

(∂S

∂xµ

)(∂S

∂xν

), (9)

with the action S and the coordinate three-vector xµ.The Hamiltonian is a constant that can be set toH = k/2with k = −m′2 (where m′ is the mass of the test particle).

Then from the symmetry of the system one can writethe action S for the motion of test particles around theblack hole in separable form as

S =1

2kλ− Et+ Lϕ+ Sr(r) . (10)

Here E and L refer to the energy and angular momentumof the particle, respectively, and the parameter λ is anaffine parameter. From Eq. (10), we rewrite Hamilton-Jacobi equation in the following form

k = −F (r)−1E2 + F (r)

(∂Sr∂r

)2

+L2

r2, (11)

In the (2+1)-dimensional system there appear threeindependent constants of motion, i.e. E, L and k whichhave been specified in [136]. In this case the correspond-ing constant related to the latitudinal motion of parti-cles is not available as that of the properties of the BTZ

spacetime. From Eq. (11) we obtain the radial equationof motion for particles as

1

2r2 + Veff (r;L, α,Λ) = E2 , (12)

where the dot denotes derivative with respect to theproper time τ and the radial function Veff (r;L, α, β)refers to the effective potential of the system which isgiven by

Veff (r;L, α,Λ) =r2

2α

(√1− 4α (Λr2 +m)

r2− 1

)

×(

1 +L2

r2

), (13)

with the conserved constants per unit mass m′ given byE = E/m′ and L = L/m′ and k/m′2 = −1.

In Fig. 1 we demonstrate the radial dependence of theeffective potential (13) for different values of GB param-eter α and cosmological constant Λ. From Fig. 1, withincreasing α the curves start coming down. However,we can see that the negative cosmological constant, i.e.Λ < 0 has the opposite effect with respect to the GB pa-rameter α, thereby suggesting that the effect of the cos-mological constant can prevent test particles from escap-ing or falling into the black hole. Since the situation getsaltered for cosmological constant test particles under theeffect of α and Λ can have stable circular orbits around(2+1)-dimensional BTZ black hole in the GB gravity.

Following effective potential (13), we turn to the studyof stable circular orbits of test particles around BTZblack hole in the GB gravity. For test particles to beon stable circular orbits we shall focus on the requiredconditions for which

Veff (r;L, α,Λ) = E2 , (14)

∂Veff (r;L, α,Λ)

∂r= 0 . (15)

From the above equations the corresponding energy andangular momentum for test particles on stable circularorbits can be obtained. Further we show the dependenceon the GB parameter and cosmological constant of theangular momentum required for the test particles to beon stable circular orbit around the black hole, see Fig. 2.It is clearly shown in Fig. 2 that stable circular orbit ofparticles shifts toward the large radii as a result of an in-crease in the value of α. On the other hand, large valuesof α give rise to the increase in the value of L, showingthat particle to stay on stable circular orbit needs moreangular momentum for biger α. The situation howevergets altered as the cosmological parameter has the op-posite effect, thereby reducing the value of the angularmomentum for the particle to be on stable circular or-bit around (2+1)-dimensional BTZ black hole in the GBgravity.

4

α = 0

α = 0.2

α = 0.5

α = 1.0

� � � � ��

-���

-���

-���

-���

-���

-���

-���

-���

�/�

����

Λ = -0.0005

Λ = -0.0010

Λ = -0.0020

� � �� �� ���-�

-�

�

�

�

�/�

����

FIG. 1: Radial dependence of the effective potential for massive particles moving around GB (2+1)-dimensional BTZ blackhole. Veff is plotted for different values of GB parameter α for given Λ = 0 in the left panel while for different values of Λ forgiven α = 0.1 in the right panel.

α = 0.100

α = 0.102

α = 0.105

2 3 4 5 6

0.100

0.102

0.104

0.106

0.108

0.110

0.112

0.114

r/M

L2

Λ = - 0.0001

Λ = - 0.0005

Λ = - 0.0010

2 3 4 5 6 7 8

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

r/M

L2

FIG. 2: The radial dependence of the specific angular momentum for test particles around GB (2+1) -dimensional BTZ blackhole. Left panel: for the different values of parameter α in the case of vanishing cosmological constant, i.e. Λ = 0. Right panel:for different values of cosmological constant in the case of fixed α = 0.5.

TABLE I: The values of the ISCO radius risco are tabulatedin the case of massive particles moving around 3−D GB blackhole for different values of GB parameter α and cosmologicalconstant Λ.

Λα −0.0001 −0.0005 −0.0010 −0.0050

0.01 1.01000 0.77779 0.695958 0.540149

0.05 1.73920 1.34584 1.20781 0.94661

0.1 2.20810 1.70810 1.53567 1.21051

0.5 3.81943 2.99347 2.70679 2.17044

Further we explore the innermost stable circular orbits(ISCO), which is given by the auxiliary condition on theeffective potential

∂2Veff (r;L, α,Λ)

∂r2= 0 . (16)

In the case of massive particles the radius of the ISCO riis obtained from the minimum value of the angular mo-mentum L determined by V ′eff (r;L, α,Λ) = 0. For theexistence of innermost stable circular orbits the abovecondition must always be satisfied. It is difficult to solveand analyse Eq. (16) analytically, and hence we explorethe ISCO radius numerically (see Table I). As can be seenfrom the Table I, the ISCO radius increases with increas-ing the GB parameter α while it decreases with increas-ing the cosmological parameter Λ. This behaviour of theISCO radius is also shown very clearly in Fig. 3. FromEq. (6) it is noticeable that the AdS radius correspond-ing to the cosmological horizon is always greater than theradius of stable circular orbits of the test particles for thechosen values of the cosmological constant. It turns outthat the resultant gravitational force becomes stronger aswe increase GB parameter, thereby indicating that it actsas an attractive charge, whereas an increase in the valueof the cosmological constant weakens the gravity. Thus,for massive particle to be under the combined effect ofGB parameter and cosmological constant it makes sense

5

Λ = - 0.0001

Λ = - 0.0005

Λ = - 0.0010

0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

α

r(ISCO)

FIG. 3: The dependence of the ISCO radius on the GBparameter α for different values of cosmological constant Λ.

so as to move at stable circular orbits around (2+1)-dimensional BTZ black hole in the GB gravity. Thisleads to an interesting question what happens to massiveparticle in case the effect of cosmological parameter isswitched off –could it be fallen into the black hole underthe effect of GB parameter? To settle this question weshall analyse Eq. (16), whether it is always negative ornot. With this aim, we plot the radial profile of V ′′eff (r)for various values of GB parameter, see Fig. 4. We notethat in the case of Λ = 0, V ′′eff (r) < 0 always at all val-ues of r, thereby indicating that there occurs no stablecircular orbits around black hole having GB parameteronly. The situations however gets overturned once theeffect arising from the cosmological constant is included,showing that V ′′eff (r) > 0 always at all r, see (Fig. 4 right

panel). Or, in other words, the particles can be on stablecircular orbits around (2+1)-dimensional BTZ black holein the GB gravity. It is interesting that this behaviourfor stable circular orbits exhibits striking difference fromits Einstein counterpart for which there exists no occur-rence of stable orbits [135]. However, in the GB gravitycase the cosmological constant plays a crucial role in get-ting information of stable circular orbits of test particlesaround BTZ black hole and its properties as well.

IV. PHOTON MOTION AND GRAVITATIONALLENSING

In this part we consider the motion of photon in theGB (2+1)-dimensional BTZ black hole spacetime. Fromthe usual relation for the 3-momentum (note that we aredealing with 3D spacetime) pµp

µ = k we have that for

photons one has to set k = 0. From the Hamilton-Jacobiformalism we obtain the action S in the form

S = −Et+ Lϕ+ Sr(r) , (17)

where E and L are the usual conserved quantities for theenergy and angular momentum of the photon, respec-tively and Sr is function of r only. Now it is straight-forward to obtain the Hamilton-Jacobi equation in thefollowing form

k = − E2

F (r)+ F (r)

(∂Sr∂r

)2

+L2

r2.

(18)

Then from the separability of the action given in Eq. (18)we obtain the radial component of equations of motionfor photons in the following form

r2 = E2 − L2

r2F (r) . (19)

To find the radii of circular orbits for given values of Eand L we can then solve simultaneously r = r = 0, i.e.,

Veff (r, E, L) = 0,∂Veff (r, E, L)

∂r= 0 , (20)

where the function Veff (r, E, L) is defined as

Veff (r, E, L) = E2 − L2

r2F (r) . (21)

In (3+1)-dimensional spacetime, for photon sphere one

needs to solve V ′eff = 0, but interestingly it is not suffi-

cient to find the photon sphere around (2+1)-dimensionalBTZ black hole in the GB gravity. Therefore one maydetermine it using additional condition V ′′eff = 0. Here,this condition gives rph implicitly as

3(4αΛ− 1)r2 + 8αm = 0 . (22)

In the limit of αΛ � 1 we can write the approximateexpressions for the photon orbit rph as

rph =2√

6

3(1 + 2αΛ)

√αm+O(Λ2) . (23)

This clearly shows the parameter α increases the ra-dius of the photon orbits. From the equation of motionfor the mass-less particle one can write uµ = ∂S

∂xµ (anduµ = gµνuν) that allows to write explicit form of thecomponents of the ”three”-velocity as

t =2α

r2(√−4αΛ− 4αm

r2 + 1− 1)E , (24)

φ =L

r2, (25)

r2 = E2 +L2(

1−√

1− 4αΛ− 4αmr2

)2α

, (26)

6

α = 0.1

α = 0.5

α = 1.0

2 3 4 5 6 7 8 9

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

r

∂2Veff/∂r2

α = 0.1

α = 0.5

α = 1.0

2 3 4 5 6 7 8 9

0.000

0.005

0.010

0.015

r

∂2Veff/∂r2

FIG. 4: Radial profile of ∂2Veff/∂r2 for test particles making circular motion around (2+1) -dimensional BTZ black hole in

the GB gravity is plotted for different values of GB parameter α. Left/right panels refer to Λ = 0/ − 0.001. Whereas, in thecase of (2+1) -dimensional BTZ black hole in Einstein gravity the situation gets totally altered from the one in GB gravitywhere V ′′eff does not depend upon r, ( V ′′eff = −8Λ) and hence there occur no stable circular orbits for massive particles (seefor example [135]).

where differentiation is made over the affine parameter.Let us introduce so called the distance of closest approachr0 defined as the minimum distance between the centralgravitating object and the trajectory of photons. Fromthe equation (26) one can obtain the value for this dis-tance that reads

r0 =L2√m√

αE4 + ΛL4 + E2L2. (27)

The dependence of the distance of closest approach on theimpact parameter b = L/E is plotted in Fig. 5. It is ap-parent from the figure that the increase of the GB param-eter decreases the distance of closest approach slightlywhile the increase of the absolute value of the cosmolog-ical constant shows opposite behaviour. It can be inter-preted as follows, since the increase of the negative valueof cosmological constant increases the repulsive force be-tween the central object and any particle with positiveenergy then the increase of such repulsive force should inturn increase the closest distance between the trajectoryof a particle and the central object for the given impact

parameter b which is shown in the right panel of Fig. 5.

Let us now investigate the bending angle of a photonmoving in the spacetime of the BTZ black hole in theGB gravity. To do so one can introduce new variableas u = 1/r that simplifies our calculation to find thebending angle. One can write the equation of motion interms of new variable as

dφ

du=

L√E2 +

L2(1−√

1−4αΛ−4αmu2)2α

(28)

The resultant bending angle then takes the followingform

δ = 2

∫ u0

0

L√E2 +

L2(1−√

1−4αΛ−4αmu2)2α

du− π . (29)

One can explore the case when the GB parameter αis small (i.e. αΛ � 1) and use the linear approximationin it. In this approximation the integral above can bewritten as

δ = 2

∫ u0

0

L√ΛL2 + L2mu2 + E2

−αL3

(Λ +mu2

)22 (ΛL2 + L2mu2 + E2)

3/2du− π . (30)

After integration the bending angle becomes

δ =

(L2(4−αΛ)+3αE2) log

(√m√L2(Λ+mu2

0)+E2+Lmu0

)√m

− αLu0(E2L2(2Λ+mu20)+ΛL4(Λ+mu2

0)+3E4)(ΛL2+E2)

√L2(Λ+mu2

0)+E2

2L2(31)

−(L2(4− αΛ) + 3αE2

)log(√m√

ΛL2 + E2)

2L2√m

− π .

7

α = 0.1

α = 0.2

α = 0.3

0.05 0.10 0.50 10.0

0.2

0.4

0.6

0.8

1.0

b

r 0Λ = - 0.1

Λ = 0

Λ = - 1

Λ = - 2

0.05 0.10 0.50 10.0

0.2

0.4

0.6

0.8

1.0

b

r 0

α = 0.2

FIG. 5: The dependence of the distance of closest approach of photon on the impact parameter b.

The dependence of such bending angle on the param-eter u0 is plotted in Fig. 6 for the chosen energy E = 1and angular momentum L = 3 of the photon and dif-ferent values of the GB parameter α with cosmologicalconstant Λ.

One can easily see from the left figure that in the ab-sence of GB parameter the bending angle increases mono-tonically when one increases the inverse of closest ap-proach distance, u0. When this parameter comes to thegame we see that the bending angle increases until itspeak first and then starts to decrease and becomes zerofor some value of u0. In this case the maximum deflec-tion angle can be easily found by setting equal to zero thederivative of the deflection angle over u0 which in turnreduces to the equation

2E2L− L3(Λ +mu2

0

) (α(Λ +mu2

0

)− 2)

(L2 (Λ +mu20) + E2)

3/2= 0 . (32)

The maximum point then becomes

u∗0 =

[√2αE2 + L2

αLm+

1

αm− Λ

m

] 12

. (33)

V. CONCLUSIONS

In the present study we have seen that the GB parame-ter has opposite effect on the radii of stable circular orbitswith respect to the cosmological constant. The GB pa-rameter allows particles not to be on stable circular orbitswhile the cosmological constant restores the stable orbitsfor particles around the BTZ black hole in the 3-D EGBgravity. It is interesting that in the case of Λ = 0 thereoccurs no stable circular orbits around the BTZ black

hole in the 3-D EGB gravity. In other words the particleunder the effect of α alone either can escapes to infinityor falls into the black hole. The situations however getsoverturned once the effect arising from the cosmologicalconstant is taken into consideration. It thus appears thatcosmological constant in (2+1)-dimensional BTZ blackhole spacetime in the GB gravity plays a crucial role forthe existence of stable circular orbits for particles aroundthe black hole. Also note that the ISCO radius increaseswith increasing the GB parameter while it gets decreasedwith increasing the cosmological constant.

Investigation of photon motion around the BTZ blackhole in the 3-D GB gravity shows that the radius of thephoton orbit increases with the increase in the GB pa-rameter while the cosmological constant with its nega-tive value decreases it. Study of the bending angle ofthe photon approaching the central object from infinityshows that in the presence of the cosmological constantits value increases monotonically in the absence of theGB parameter while in the presence of latter it reachesits peak corresponding to the specific value of the pa-rameter u∗0 and then goes down. It also has been shownthat the increase of the GB parameter reduces the bend-ing angle while the increase in the absolute value of thenegative cosmological constant makes such angle bigger.

In a recent work Hennigar et. al [137], have obtainedthe charged and rotating black hole solutions in the novel3D GB theory of gravity which is generalization of theBTZ solution. Their charged metric is obtained in theMaxwell and Born–Infeld theories. In a separate work wewill analyse the motion of charged particles and photonsin these newly obtained charged and rotating spacetimesin the 3D GB gravity, to look at the effects of the chargeand rotation on the motion of particles.

8

α = 0

α = 0.1

α = 0.2

0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

2.5

u0

δ

π

Λ = - 0.1

Λ = 0

Λ = - 0.05

Λ = - 0.1

0 2 4 6 8 10 120.0

0.5

1.0

1.5

u0

δ

π

α = 0.1

FIG. 6: The dependence of the bending angle of photon on the inverse of the distance of closest approach u0.

Acknowledgments

B.N. acknowledges support from the China Scholar-ship Council (CSC), grant No. 2018DFH009013. This re-

search is supported by Grants of the Uzbekistan Ministryfor Innovative Development and by the Abdus Salam In-ternational Centre for Theoretical Physics under GrantNo. OEA-NT-01.

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