Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing
EEquivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing
Zonghai Li
1, 2 and Tao Zhou ∗ School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China Center for Theoretical Physics, School of Physics and Technology, Wuhan University, Wuhan 430072, China (Dated: February 12, 2020)The Gibbons-Werner method where the Gauss-Bonnet theorem is applied to study the gravitational deflectionangle has received much attention recently. In this paper, we study the equivalence of the Gibbons-Wernermethod to the standard geodesics method, and it is shown that the geodesics method can be derived with theGibbons-Werner method, for asymptotically flat case. In the geodesics method, the gravitational deflectionangle of particle depends entirely on the geodesic curvature of the particle ray in the Euclidean space. Thegravitational deflection of light in Kerr-Newman spacetime is calculated by different technologies under theGibbons-Werner framework, as an intuitive example to show the equivalence.
I. INTRODUCTION
Gravitational lensing plays an important role in gravita-tional theory. In theoretical physics, it is used to test the fun-damental theory of gravity, where a famous example is thatEddington et al. [1, 2] verified Einstein’s general relativity bymeans of the deflection experiment of light in the solar grav-itational field 100 years ago. In astrophysics and cosmology,it is used to measure the mass of galaxies and clusters [3–5],and to detect dark matter and dark energy [6–10]. In mathe-matics, it is related to singularity theory, topology and Finslergeometry [11–15].Recently, Gibbons and Werner [11] introduced an elegantgeometrical method of deriving the bending angle of lightin a static and spherically symmetric spacetime. They usedthe famous Gauss-Bonnet (GB) theorem to a surface de-fined by the corresponding optical metric. Later, Werner [14]extended this method to the rotating and stationary space-times. In stationary spacetimes, the optical geometry is de-fined by the Randers-Finsler metric. Thus, Werner appliedNazım’s method to construct an osculating Riemannian man-ifold where one can easily use the GB theorem. The work byGibbons and Werner promotes the study of light deflection.On one hand, Jusufi et al. [16–31] studied the gravitationallensing not only in asymptotically flat spacetime but also innonasymptotically flat spacetime such as a spacetime withcosmic string. Similar works can also be found in Refs. [32–39]. On the other hand, Ishihara et al. [40–44] studied thefinite-distance corrections for gravitational deflection of lightboth for the weak and the strong deflection limit, where thesource and observer are no longer assumed to be infinitely farapart from a lens. For a review on finite-distance corrections,we refer the reader to Ref. [45].It is well known that there are many massive particles inour Universe, such as massive neutrinos. The study of gravi-tational deflection of massive particles allows one to under-stand the properties of the sources and these particles. Infact, the study of the massive particles lensing using tradi-tional methods can be found in Refs. [46–54]. Moreover, two ∗ [email protected] other routes have been established by applying the GB theo-rem to study the gravitational deflection of massive particles.The first route is related to the Jacobi metric of curved space-time. To be precise, one can calculate the deflection angle ofmassive particles via applying the GB theorem to the surfacedefined by the Jacobi metric [55, 56] for static spacetime andby the Jacobi-Maupertuis Randers-Finsler metric [57] for sta-tionary spacetime. The second route is related to the opticalmedia method. For static and spherically symmetric space-time, Crisnejo and Gallo [58] used the GB theorem to studythe gravitational deflections of light in a plasma medium andthe deflection angle of massive particles. The finite-distancecorrections of light with a plasma medium and the gravita-tional deflection of charged massive particles were studiedquite recently [59, 60]. For rotating and stationary spacetimes,Jusufi [61] used the GB theorem to study the deflection anglesof massive particles by the Kerr black hole and the Teo worm-hole, respectively, based on the corresponding isotropic typemetrics, the refractive index of the corresponding optical me-dia. Furthermore, the method in Ref. [61] was extended todistinguish naked singularities and Kerr-like wormholes [62],and to study the gravitational deflection of charged particlesin Kerr-Newman spacetime [63].In this paper, the method with the GB theorem to studythe deflection angle shall be called the Gibbons-Wernermethod. It is worth investigating whether the Gibbons-Werner method [11] is equivalent to the standard geodesicsmethod [64]. In fact, this topic has been discussed by someresearchers. The first-order equivalence has been shown inRefs. [18, 19, 30, 62], and the second-order equivalence hasbeen shown in Refs. [56, 58]. From a conceptual point ofview, however, the two methods seem to be completely dif-ferent. The Gibbons-Werner method shows that the deflectionof particles (photon and massive particles) is determined by aquantity outside of itself relative to the lens [14, 61], and thusthe gravitational deflection angle can be regarded as a globaltopological effect, whereas the geodesics method is usuallyassociated within a region from particles ray to lens. In thepresent paper, we will demonstrate the equivalence betweenthe Gibbons-Werner method and the geodesics method forasymptotically flat spacetime, in terms of results and concepts.More specifically, the weak gravitational deflection of light inKerr-Newman spacetime will be taken as a simple example. a r X i v : . [ g r- q c ] F e b This paper is organized as follows. In Sec. II, we review theGB theorem and use the theorem to the lens geometry. Then,we show that the equivalence of the Gibbons-Werner methodto geodesics method. In Sec. III, we give the Kerr-Newmanspacetime as an example to show the equivalence. Finally,we summarize our results in Sec. IV. Throughout this paper,we use the natural units where G = c = 1 and the metricsignature ( − , + , + , +) . II. THE EQUIVALENCE BETWEEN THEGIBBONS-WERNER METHOD AND GEODESICSMETHODA. The Gauss-Bonnet theorem
Let D be a compact oriented two-dimensional Riemannianmanifold with the Euler characteristic χ ( D ) and Gaussian cur-vature K , and its boundary ∂D is a piecewise smooth curvewith geodesic curvature k g . Then, the GB theorem statesthat [11, 65]: (cid:90) (cid:90) D KdS + (cid:73) ∂D k g dσ + (cid:88) i =1 θ i = 2 πχ ( D ) , (1)where dS is the area element of the surface, dσ is the lineelement along ∂D , and θ i is the exterior angle defined for the i th vertex in the positive sense. B. Application the Gauss-Bonnet theorem to the lens geometry
Assume M be a two-dimensional smooth manifold withcoordinates ( x, y ) and a Riemannian metric ˆ g ij . Now onecan apply the GB theorem to the lens geometry in a region D ⊂ ( M, ˆ g ij ) . For convenience, D is required to be asymp-totically Euclidean and thus both the particle source S and theobserver O are in the asymptotically Euclidean region. Let ∂D = γ g (cid:83) C i ( i = 1 , , with the particle ray γ g and threecurves C i . γ g is described by the impact parameter b , and thecurves C i are defined by C : x = − R,C : y = − R,C : x = R, with the constant R > . Since the lens L is excluded in thedomain D , χ ( D ) = 1 . Additionally, as R → ∞ , boundarycurve intersections S , A , B and O are in the asymptoticallyEuclidean region, and thus one can have k g ( C i ) = 0 , θ S + θ A + θ B = 3 π/ , and θ O = π/ α with the deflection angle α . Then the GB theorem becomes lim R →∞ (cid:32)(cid:90) (cid:90) D KdS − (cid:90) OS k g ( γ g ) dσ (cid:33) + (cid:18) π π α (cid:19) = 2 π. (2)Thus, the gravitational deflection angle can be written as α = lim R →∞ (cid:32) − (cid:90) (cid:90) D KdS + (cid:90) OS k g ( γ g ) dσ (cid:33) , (3) S x yA B b O D g L C C C FIG. 1. The region D ⊂ ( M, ˆ g ij ) with boundary ∂D = γ g (cid:83) C i ( i = 1 , , . Particle ray γ g is a spatial curve and C i arethree curves defined by C : x = − R , C : y = − R and C : x = R with the constant R > . As R → ∞ , the pointsof intersection S, A, B, and O are in the asymptotically Euclideanregion, where S and O denote the particle source and the observer,respectively. L is the lens, b is the impact parameter and α is thedeflection angle. as shown in Fig. 1. C. The equivalence between the Gibbons-Werner method andgeodesics method
In the discussion above, the Riemannian space ( M, ˆ g ij ) issomewhat arbitrary, which is asymptotically Euclidean andthe condition of using the GB theorem is required. In thefollowing, three cases will be discussed to show the equiva-lence between the Gibbons-Werner method and the geodesicsmethod.
1. Case 1: K (cid:54) = 0 , and k g ( γ g ) = 0 In this case, the particle ray γ g is a spatial geodesic in ( M, ˆ g ij ) , and Eq. (3) becomes α = − lim R →∞ (cid:90) (cid:90) D KdS. (4)Indeed, this is the original consideration of Gibbons andWerner [11, 14] and for convenience we shall call it the narrowGibbons-Werner method. In fact, many studies fall into thiscategory. For light deflection, one has ( M, ˆ g ij ) = ( M, g optij ) ,where g optij is the corresponding optical metric of curvedspacetime. For massive particles, ( M, ˆ g ij ) = ( M, j ij ) , where j ij is the corresponding Jacobi metric of curved spacetime. Instationary spacetime, the optical metric (or Jacobi metric) is aRanders-Finsler metric. However, in these cases one can usethe osculating Riemannian metric by Werner’s method [14] oruse Jusufi’s method to avoid the Finsler metric [61].
2. Case 2: K (cid:54) = 0 , and k g ( γ g ) (cid:54) = 0 Now, the particle ray is not geodesic in a curved space, andEq. (3) can be written as α = α Gauss + α geod , (5)where α Gauss = − lim R →∞ (cid:90) (cid:90) D KdS,α geod = lim R →∞ (cid:90) OS k g ( γ g ) dσ. In Refs. [42–44], Ono et al. considered the so-called general-ized optical metric space as the lens background, and usedEq. (5) to study the deflection angle of light in stationaryspacetimes.
3. Case 3: K = 0 , and k g ( γ g ) (cid:54) = 0 In this case, we assume that M is Euclidean space, andEq. (3) arrives at α = lim R →∞ (cid:90) OS k g ( γ g ) dσ. (6)To our best knowledge, Eq. (6) has not been considered yet,and next it will be proved that this result is the same with theexpression in the geodesics method.The line element of a three-dimensional Euclidean space is dl = dx + dy + dz , (7)and a unit vector normal to the x − y plane is nnn = (0 , , .The particle ray γ g can be denoted by y = y ( x ) , and one candefine its unit tangent vector as TTT = 1 (cid:112) y (cid:48) (1 , y (cid:48) , , (8)where (cid:48) denotes derivative with respect to x . Therefore, ˙ TTT ≡ dTTTdl = y (cid:48)(cid:48) (1 + y (cid:48) ) ( − y (cid:48) , , , (9)and one can obtain the geodesic curvature of γ g in the x − y plane as follows [65]: k g ( γ g ) ≡ ˙ TTT · ( nnn × TTT ) = y (cid:48)(cid:48) (1 + y (cid:48) ) / . (10)Then, one can calculate the deflection angle by α = lim R →∞ (cid:90) OS k g ( γ g ) dl = lim R →∞ (cid:90) OS y (cid:48)(cid:48) (1 + y (cid:48) ) dx = (cid:20) arctan (cid:18) dydx (cid:19)(cid:21) | x →∞ x →−∞ , (11) which is nothing but the formula of calculating deflection an-gle with geodesics method in Refs. [50–52].In short, the geodesics method just corresponds to specialcases for the Gibbons-Werner method, where the GB theo-rem is used to Euclidean space. In other words, the geodesicsmethod categorizes the deflection angle into the influence ofgeodesic curvature of particles moving in Euclidean space.Therefore, the geodesics method also has geometric meaningfrom the perspective of curvature. III. AN EXAMPLE: THE DEFLECTION OF LIGHT INKERR-NEWMAN SPACETIME
For the second-order post-Minkowskian approximation, thecomponents of the metric of the Kerr-Newman spacetime inthe harmonic coordinates ( t, x, y, z ) can be written as [66, 67] g = − mr − m + q r + O ( ε ) ,g i = ζ i + O ( ε ) ,g ij = (cid:18) mr + m r (cid:19) δ ij + ( m − q ) x i x j r + O ( ε ) , (12)where m and q are the mass and electric charge of the Kerr-Newman black hole, respectively. xxx = ( x, y, z ) , r = (cid:112) x + y + z and ζ i is the i th component of the gravita-tional vector potential ζζζ ≡ mar ( y, − x, , where a is the an-gular momentum per unit mass. δ ij is the Kronecker symboland the expanding parameter ε represents the black hole pa-rameters m , a or q . The above metric is expanded as the powerseries of the parameters m , a and q , and O ( ε ) is the serieswith order greater than , such as m , a , q , m a, ma , ... .For stationary spacetime, its optical geometry defined bythe Randers-Finsler metric takes the form [14, 68] F ( x i , dx i ) = dt = (cid:113) ˆ α ij dx i dx j + β i dx i , (13)where ˆ α ij is a Riemannian metric and β i is a one-form satisfy-ing ˆ α ij β i β j < . Consider a null curve in the Kerr-Newmanspacetime, ds = 0 , and one can find a Randers-Finsler met-ric, ˆ α ij = (cid:20) mr + 7 m − q r (cid:21) δ ij + (cid:0) m − q (cid:1) x i x j r H ij + O ( ε ) ,β i dX i = 2 ma ( ydx − xdy ) r + O ( ε ) , (14)where H = 2 xy, H = H = y − x ,H = − xy, H = H = yz,H = 0 , H = H = xz. A. Werner’s method: K (cid:54) = 0 , k g ( γ g ) = 0 In this subsection, we will apply Werner’s method [14]to calculate the gravitational deflection angle of light. Thelight ray is geodesic in Randers-Finsler space, and therefore,Eq. (4) can be considered. To simplify, one can study the nullgeodesic in the equatorial plane. Chose z = 0 as the equato-rial plane, and one can find the Kerr-Newman-Randers blackhole optical metric as follows F (cid:18) x i , dx i dt (cid:19) = (cid:114) ˆ α ij dx i dt dx j dt + β i dx i dt , (15)where ˆ α ij and β i are the same as those in Eq. (14) except that i and j only run in { , } here.The Randers-Finsler metric is characterized by the Hes-sian [14, 68] g ij ( x, vvv ) = 12 ∂ F ( x, vvv ) ∂v i ∂v j , (16)where x ∈ M , and vvv ∈ T x M with T x M the tangent spaceat a given point. In order to obtain a Remannian metric ¯ g ¯ g ¯ g ,one can choose a smooth nonzero vector field VVV over M thatcontains the tangent vectors along the geodesic γ F such that VVV ( γ F ) = vvv , defining ¯ g ij ( x ) = g ij ( x, VVV ( x )) . (17)In this construction, we can obtain a crucial result that thegeodesic γ F of ( M, F ) is also a geodesic γ ¯ g of ( M, ¯ g ) , i.e., γ F = γ ¯ g [14].Following Werner [14], the osculating Riemannian mani-fold ( M, ¯ g ij ) can be used to calculate the gravitational de-fection angle of light. Near the undeflected light rays y = − b [50, 51], one can choose the vector field as V x = dxdt = 1 + O ( ε ) ,V y = dydt = 0 + O ( ε ) . (18)Using Eqs. (16), (17), and (18), finally the osculating Rieman-nian metric can be obtained as follows: ¯ g xx = 1 + 4 mr + 7 m − q r + (cid:0) m − q (cid:1) x r + 4 mayr + O ( ε ) , (19) ¯ g xy = ¯ g yx = (cid:0) m − q (cid:1) xyr − mayr + O ( ε ) , (20) ¯ g yy = 1 + 4 mr + 7 m − q r + (cid:0) m − q (cid:1) y r + 2 mayr + O ( ε ) , (21)with the determinant up to second order det ¯ g = 1 + 8 mr + 6 amyr + 31 m − q r + O ( ε ) , (22) and the Gaussian curvature ¯ K = ¯ R xyxy det ¯ g = − mr − amy (6 x + y ) r + 3(3 m + q ) r + O ( ε ) . (23)In harmonic coordinates, Eq. (4) can be written as α = − (cid:90) ∞−∞ (cid:90) ∞ y ( x ) ¯ K (cid:112) det ¯ gdydx. (24)Here y ( x ) denotes the light ray up to first order (see the Ap-pendix A) y ( x ) = − b + 2 (cid:0) x + √ b + x (cid:1) mb + O ( ε ) . (25)Substituting Eqs. (22), (23) and (25) into Eq. (24), one can getthe second-order deflection angle of light as follows: α = 4 mb − amb + 3 π (cid:0) m − q (cid:1) b + O ( ε ) , (26)which is consistent with the results in Ref. [51]. B. The generalized optical metric method: K (cid:54) = 0 , and k g ( γ g ) (cid:54) = 0 In this section we consider the Riemannian space (3) M de-fined by ˆ α ij . The line element of (3) M is given by dλ = ˆ α ij dx i dx j . (27)The light ray is the spatial curve in (3) M and following Fer-mat’s principle, the motion equation of light ray is [42] de i dλ + (3) Γ ijk e j e k = ˆ α ij (cid:0) β k | j − β j | k (cid:1) e k , (28)where e i ≡ dx i dλ , (3) Γ ijk denotes the Christoffel symbol as-sociated with ˆ α ij , and | denotes the covariant derivative with ˆ α ij . The existence of β i illustrates that the orbit of light is notthe geodesic in (3) M . Naturally, the contribution of geodesiccurvature k g should be considered and we will use Eq. (5) tocalculate the deflection angle. We focus on the motion of thelight in the equatorial plane ( z = 0) . Then the geodesic cur-vature of curve γ g is given by [42] k g ( γ g ) = − (cid:15) ijk N i β j | k , (29)where (cid:15) ijk is the Levi-Civita tensor and NNN is a unit normalvector for the equatorial plane. Then, choose the unit normalvector as N p = − √ ˆ α zz δ zp , and one can obtain k g ( γ g ) = 1 √ det ˆ α ˆ α zz ( β x,y − β y,x ) , (30)where (cid:15) zxy = − (cid:15) zyx = 1 / √ det ˆ α has been used and thecomma denotes the partial derivative. With Eqs. (14) and (30),one can have k g ( γ g ) = − amr + O ( ε ) , (31)where the first-order light ray in Eq. (25) has been used.According to Eq. (5), the deflection angle of the light canbe divided into two parts. First, the Gauss curvature of ˆ α ij is K ˆ α = − mr + 3(3 m + q ) r + O ( ε ) , (32)and one can calculate the part associated with Gauss curvature α Gauss = − (cid:90) + ∞−∞ (cid:90) − b + m ( x + √ x b ) b −∞ K ˆ α √ det ˆ α dy dx = 4 mb + 3 π (cid:0) m − q (cid:1) b + O ( ε ) . (33)Second, from Eqs. (25) and (31), the part associated withgeodesic curvature is α geod = lim R →∞ (cid:90) OS k g ( γ g ) dλ = (cid:90) + ∞−∞ k g ( γ g ) (cid:112) ˆ α xx dx = (cid:90) + ∞−∞ (cid:34) − am ( b + x ) (cid:35) dx = − amb + O ( ε ) . (34)Finally, the total deflection angle can be obtained as follows: α = α Gauss + α geod = 4 mb − amb + 3 π (cid:0) m − q (cid:1) b + O ( ε ) , (35)which is consistent with the result in Eq. (26). C. The geodesics method: K = 0 , k g ( γ g ) (cid:54) = 0 From second-order light ray in Eq. (A3), the following re-lation can be obtained dydx = 2 m ( b − a ) (cid:0) x + √ b + x (cid:1) b √ b + x + 3 (cid:0) m − q (cid:1) b (cid:18) π xb + bxb + x (cid:19) − bm ( b + x ) + b (cid:0) m − q (cid:1) x b + x ) + O ( ε ) . (36)The deflection angle can be obtained by Eq. (6) α = (cid:20) arctan (cid:18) dydx (cid:19)(cid:21) | x →∞ x →−∞ = 4 mb − amb + 3 π (cid:0) m − q (cid:1) b + O ( ε ) . (37) Certainly, this expression is the same as the result obtained byWerner’s method in Eq. (26) and by the generalized opticalmetric method in Eq. (35). IV. CONCLUSION
In this work, we investigate the equivalence of the Gibbons-Werner method to the geodesics method in the study of grav-itational lensing. It is shown that the geodesics method canbe derived with the Gibbons-Werner method for asymptoti-cally flat spacetime. In the Gibbons-Werner procedure, onecan choose the Euclidean space as the lens background andthe deflection effect is completely determined by the geodesiccurvature of the particle’s trajectory. Thus, one can choose ar-bitrary asymptotically Euclidean space as the lens backgroundand the deflection angle can be written as α = α Gauss + α geod .The difference between these different background spaces isthat the contribution on α Gauss and α geod is different. How-ever, the total deflection angle is always constant. In prac-tice, it is more convenient to use the geodesics method or thenarrow Gibbons-Werner method. We can illustrate these twomethods using the following formula (cid:34)(cid:90) OS k g ( γ g ) dσ (cid:35) | Euclidean = (cid:20) − (cid:90) (cid:90) D KdS (cid:21) | Optical . The left side of the equation represents the geodesic method ( α Gauss = 0 , α = α geod ) , while the right side represents thenarrow Gibbons-Werner method ( α geod = 0 , α = α Gauss ) .As an example to show the equivalence, we calculate thesecond-order gravitational deflection angle of light in Kerr-Newman spacetime, for three options with the Gibbons-Werner method, in the harmonic coordinates. More, the har-monic coordinates bring a lot of simplicity and overcome thecumbersome iterative in Ref. [56]. ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-ence Foundation of China under Grants No. 11405136 andNo. 11847307, and the Fundamental Research Funds for theCentral Universities under Grant No. 2682019LK11.
Appendix A: Second-order light orbit
In this Appendix, we calculate the second-order light ray inKerr-Newman spacetime. For the photon, the velocity w = 1 ,and thus Eq. (11) in the literature [51] reads dydp = 2 m ( b − a ) xb √ b + x − bm ( b + x ) + b (cid:0) m − q (cid:1) x b + x ) + 3 (cid:0) m − q (cid:1) b (cid:18) arctan xb + bxb + x (cid:19) + O ( ε ) , (A1)where p is the affine parameter in Kerr-Newman spacetime.With the boundary conditions ˙ y | p →∞ = ˙ y | x →∞ = 0 [51],one can get dydp = 2 m ( b − a ) (cid:0) x + √ b + x (cid:1) b √ b + x + 3 (cid:0) m − q (cid:1) b (cid:18) π xb + bxb + x (cid:19) − bm ( b + x ) + b (cid:0) m − q (cid:1) x b + x ) + O ( ε ) . (A2) Finally, with the first-order parameter transformation dp = dx [51] and integrating y , one can get the second-order lightray as follows: y = − b + 2 (cid:0) x + √ b + x (cid:1) mb − am (cid:0) x + √ b + x (cid:1) b − m + 3 q b + (cid:0) q − m (cid:1) b b + x ) − xm b √ b + x + 3 (cid:0) m − q (cid:1) x (cid:0) π + arctan xb (cid:1) b + O ( ε ) , (A3)where we have considered the boundary conditions y | p →∞ = y | x →∞ = − b [51]. [1] F. W. Dyson, A. S. Eddington, and C. Davidson, Phil. Trans. R.Soc. A , 291 (1920).[2] C. M. Will, Classical Quantum Gravity , 124001 (2015).[3] H. Hoekstra, M. Bartelmann, H. Dahle, H. Israel, M. Limousin,and M. Meneghetti, Space Sci. Rev. , 75 (2013).[4] M. M. Brouwer et al. , Mon. Not. R. Astron. Soc. , 5189(2018).[5] F. Bellagamba et al. , Mon. Not. R. Astron. Soc. , 1598(2019).[6] R. A. Vanderveld, M. J. Mortonson, W. Hu, and T. Eifler, Phys.Rev. D , 103518 (2012).[7] H. J. He and Z. Zhang, J. Cosmol. Astropart. Phys. (2017)036.[8] S. Cao, G. Covone, and Z. H. Zhu, Astrophys. J. , 31 (2012).[9] D. Huterer and D. L. Shafer, Rep. Prog. Phys. , 016901(2018).[10] S. Jung and C. S. Shin, Phys. Rev. Lett. , 041103 (2019).[11] G. W. Gibbons and M. C. Werner, Classical Quantum Gravity , 235009 (2008).[12] G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, and M. C.Werner, Phys. Rev. D , 044022 (2009).[13] G. W. Gibbons and C. M. Warnick, Phys. Rev. D , 064031(2009).[14] M. C. Werner, Gen. Relativ. Gravit. , 3047 (2012).[15] E. Caponio, A. V. Germinario, and M. Sanchez, J. Geom. Anal. , 791 (2016).[16] K. Jusufi, Eur. Phys. J. C , 332 (2016).[17] K. Jusufi, Int. J. Geom. Methods Mod. Phys. , 1750137(2017).[18] K. Jusufi, A. ¨Ovg¨un, and A. Banerjee, Phys. Rev. D , 084036(2017).[19] K. Jusufi, Int. J. Geom. Methods Mod. Phys. , 1750179(2017).[20] K. Jusufi, ˙I. Sakallı, and A. ¨Ovg¨un, Phys. Rev. D , 024040(2017).[21] K. Jusufi, M. C. Werner, A. Banerjee, and A. ¨Ovg¨un, Phys. Rev.D , 104012 (2017).[22] K. Jusufi, A. ¨Ovg¨un, J. Saavedra, Y. Vasquez, and P. A. Gonza-lez, Phys. Rev. D , 124024 (2018).[23] K. Jusufi, N. Sarkar, F. Rahaman, A. Banerjee, and S. Hansraj,Eur. Phys. J. C , 349 (2018).[24] K. Jusufi and A. ¨Ovg¨un, Phys. Rev. D , 024042 (2018).[25] K. Jusufi, Phys. Rev. D , 044016 (2018).[26] K. Jusufi and A. ¨Ovg¨un, Phys. Rev. D , 064030 (2018). [27] K. Jusufi, F. Rahaman, and A. Banerjee, Ann. Phys. (Amster-dam) , 219 (2018).[28] A. ¨Ovg¨un, K. Jusufi, and ˙I. Sakallı, Ann. Phys. (Amsterdam) , 193 (2018).[29] A. ¨Ovg¨un, K. Jusufi, and ˙I. Sakallı, Phys. Rev. D , 024042(2019).[30] K. Jusufi and A. ¨Ovg¨un, Int. J. Geom. Methods Mod. Phys. ,1950116 (2019).[31] T. Zhu, Q. Wu, M. Jamil, and K. Jusufi, Phys. Rev. D ,044055 (2019)[32] ˙I. Sakallı and A. ¨Ovg¨un, Europhys. Lett. , 60006 (2017).[33] A. ¨Ovg¨un, ˙I. Sakallı, and J. Saavedra, J. Cosmol. Astropart.Phys. (2018) 041.[34] H. Arakida, Gen. Relativ. Gravit. , 48 (2018).[35] A. ¨Ovg¨un, Phys. Rev. D , 044033 (2018).[36] P. Goulart, Classical Quantum Gravity , 025012 (2018).[37] W. Javed, R. Babar, and A. ¨Ovg¨un, Phys. Rev. D , 084012(2019).[38] A. ¨Ovg¨un, Phys. Rev. D , 104075 (2019).[39] K. de Leon and I. Vega, Phys. Rev. D , 124007 (2019).[40] A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, and H. Asada,Phys. Rev. D , 084015 (2016).[41] A. Ishihara, Y. Suzuki, T. Ono, and H. Asada, Phys. Rev. D ,044017 (2017).[42] T. Ono, A. Ishihara, and H. Asada, Phys. Rev. D , 104037(2017).[43] T. Ono, A. Ishihara, and H. Asada, Phys. Rev. D , 044047(2018).[44] T. Ono, A. Ishihara, and H. Asada, Phys. Rev. D , 124030(2019).[45] T. Ono and H. Asada, Universe , 218 (2019).[46] A. Accioly and S. Ragusa, Classical Quantum Gravity , 5429(2002).[47] A. Accioly and R. Paszko, Phys. Rev. D , 107501 (2004).[48] A. Bhadra, K. Sarkar, and K. K. Nandi, Phys. Rev. D ,123004 (2007).[49] O. Yu. Tsupko, Phys. Rev. D , 084075 (2014).[50] G. He and W. Lin, Classical Quantum Gravity , 095007(2016).[51] G. He and W. Lin, Classical Quantum Gravity , 029401(2017).[52] G. He and W. Lin, Classical Quantum Gravity , 105006(2017).[53] X. Liu, N. Yang, and J. Jia, Classical Quantum Gravity ,175014 (2016). [54] X. Pang and J. Jia, Classical Quantum Gravity , 065012(2019).[55] G. W. Gibbons, Classical Quantum Gravity , 025004 (2016).[56] Z. Li, G. He, and T. Zhou, Phys. Rev. D , 044001 (2020).[57] S. Chanda, G. W. Gibbons, P. Guha, P. Maraner, and M. C.Werner, J. Math. Phys.(N.Y.) , 122501 (2019).[58] G. Crisnejo and E. Gallo, Phys. Rev. D , 124016 (2018).[59] G. Crisnejo, E. Gallo, and A. Rogers, Phys. Rev. D , 124001(2019).[60] G. Crisnejo, E. Gallo, and J. R. Villanueva, Phys. Rev. D ,044006 (2019).[61] K. Jusufi, Phys. Rev. D , 064017 (2018).[62] K. Jusufi, A. Banerjee, G. Gyulchev, and M. Amir, Eur. Phys.J. C , 28 (2019). [63] K. Jusufi, arXiv:1906.12186.[64] S. Weinberg, Gravitation and Cosmology: Principles and Ap-plications of the General Theory of Relativity (Wiley, NewYork, 1972).[65] M. P. Do Carmo,