Weak deflection angle by asymptotically flat black holes in Horndeski theory using Gauss-Bonnet theorem
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International Journal of Geometric Methods in Modern Physics© World Scientific Publishing Company
Weak deflection angle by asymptotically flat black holes in Horndeski theoryusing Gauss-Bonnet theorem
WAJIHA JAVED
Division of Science and Technology, University of Education,Township, Lahore-54590, [email protected]
JAMEELA ABBAS
Department of Mathematics, University of Education,Township, Lahore-54590, [email protected]
YASHMITHA KUMARAN
Physics Department, Eastern Mediterranean University,Famagusta, 99628 North Cyprus via Mersin 10, [email protected]
AL˙I ¨OVG ¨UN
Physics Department, Eastern Mediterranean University,Famagusta, 99628 North Cyprus via Mersin 10, [email protected]
Received (Day Month Year)Revised (Day Month Year)The principal objective of this project is to investigate the gravitational lensing by asymptot-ically flat black holes in the framework of Horndeski theory in weak field limits. To achievethis objective, we utilize the Gauss-Bonnet theorem to the optical geometry of asymptoticallyflat black holes and applying the Gibbons-Werner technique to achieve the deflection angleof photons in weak field limits. Subsequently, we manifest the influence of plasma mediumon deflection of photons by asymptotically flat black holes in the context of Horndeski the-ory. We also examine the graphical impact of deflection angle on asymptotically flat blackholes in the background of Horndeski theory in plasma as well as non-plasma medium.
Keywords : Weak gravitational lensing; Deflection of light; Black hole; Deflection angle; Horn-deski theory; Gauss-Bonnet theorem
1. Introduction
The anecdote of a falling apple fostering Newtonian gravity has been imparted ongenerations since the year 1666. For centuries from then, physicists have believedthat the gravitational force is related to the ratio of the product of the interact-ing masses and the square of their separation through a proportionality constant. ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Authors’ Names
This ’Gravitational’ constant was given an accurate value via the Cavendish ex-periment. In 1910s, Einstein’s general theory of relativity transpired suggesting afinite, spherical universe in which the Gravitational constant, G was discerned toexhibit a dependence on mass distribution and size of the universe, so as to ac-count for the inertial forces [1].A decade later, Hubble’s observations confirmed the Big Bang Theory, essen-tially establishing that the universe was expanding, and potentially infinite. Theextent of the universe that can be observed is limited to a maximum distance de-termined by the time that light takes to reach Earth from the observable edge. Inother words, mass distribution and the size of the universe changes with time,and hence, so does the gravitational constant, G [2,3]. This discrepancy led to thespeculation that the effect of G could rather be a scalar field, than a constant num-ber. According to Einstein’s formulation, the so-called metric field contains theinfluence of gravity, and mathematically known as a tensor. Therefore, the idea ofconsolidating a scalar field due to mass distribution with the metric is called as thescalar-tensor theory [4,5,6].One such generic gravitational scalar-tensor theory is the Horndeski theory [7].Defined for a four-dimensional spacetime, the scalar field is incorporated as a newdegree of freedom to formulate the Lagrangian of the system, begetting second-order field equations of motion. This notion is inspired from the Lovelock theoryof gravity: relaxing Lovelock’s assumptions not only exercises the scalar-tensortheory, but also extends Einstein’s theory of gravity.The kinetic term of the Lagrangian characterizes the quadratic derivative of thefield and depends on the Lovelock tensor – which is proportional to the Einsteintensor, G µν – encompassing non-minimal coupling between the scalar field, ϕ andcurvature [8]. Hence, the action principle for a 4-D spacetime ( n = 4 ) becomes[9,10]: I [ g µν , ϕ ] = Z √− g d x (cid:20) k ( R − −
12 ( αg µν − ηG µν ) ∇ µ ϕ ∇ ν ϕ (cid:21) , (1)where, k ≡ / πG . Here, the first term accounts for the scalar field with non-minimal coupling for matter owing to the Ricci scalar, R , and the second termaccounts for the Einstein-Hilbert action for gravity owing to the cosmological con-stant, Λ . Values of the parameters α and η are governed by the positive energydensity of matter field.The above equation is the reduced form of the action analyzed with standardmatter, matter fields and non-standard scalar field [10]: implementing assump-tions - the geometry is static, spherically symmetric and homogeneous with thescalar and metric fields obeying this symmetry in an asymptotically flat space-time - renders this equation as the limiting case of Horndeski theory. Eq. 1 is thefoundation of the ensuing work.When the gravitational force of a massive object bends the trajectory of lightthat originated from a distant object behind it, gravitational lensing occurs. Thisebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) is a consequence of general relativity used to understand the universe, galaxies,dark energy and dark matter [1]. Various authors studied the gravitational lensingby black holes, wormholes, cosmic strings and other objects since the first gravita-tional lensing observation by Eddington [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37].In 2008, Gibbons and Werner cleverly showed that there is an alternativeway to obtain the weak deflection angle for asymptotically flat optical space-times using the Gauss-bonnet theorem [38]. Later, Werner managed to derivethe weak deflection angle of stationary spacetimes using the same [39]. Notethat both considered the source and the observer to be placed at asymptotic re-gions. Next, Ishihara et al. showed that it is also possible to use this methodfor finite-distances (large impact parameter cases) [40]. Then, Crisnejo and Galloshowed that the plasma medium deflects photons [41]. For more recent works,one can see [42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82].Multi-messenger astronomy constrained the scalar-tensor theories substan-tially through the detection of GW170817. The arrival times of the gravitationalwaves and it’s electromagnetic counterpart from the NGC 4993 galaxy were ob-served to have fluctuated by less than a minute when two neutron stars spirallingeach other ultimately merged. The speed of the gravitational wave is seen to beaffected when the scalar field is coupled to curvature. To be coherent with theseobservations, quintic and quartic models are neglected restricting our calculationsto linear observables. Note that Horndeski theories have a serious flaw, relatedto their primordial tensor spectrum, namely, the gravitational wave speed is notequal to unity. Theories of this sort are problematic. For example, the detectionof GW170817 eliminates any late-universe application of Horndeski theory [83].These theories can be amended by using a new framework developed in [84],firstly developed in [85] and improved by [88,86,87]. The results inferred by thesestudies are however beyond the scope of this work.In this paper, we intend to study the deflection angle of the black holes gov-erned by Horndeski theory using the Gauss-Bonnet theorem to test the validityof the modified gravity theory. To compare, we consider the idea of the deflectionangle of massive particles in a plasma medium from a black hole. Our main aim isto check the effects of Horndeski theory on weak deflection angle.This paper is organized as follows: section reviews some basics on asymp-totically flat black holes, computes the Gaussian optical curvature and calculatesthe deflection angle using GBT. In section , we calculate the deflection angle inplasma medium, followed by concluding remarks in the last section.
2. Calculation of photon lensing for for Asymptotically flat black holes
When the action comprises of a cosmological term, a new asymptotically locallyflat black hole can be found. Here, the kinetic term (constructed with Einstein ten-ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Authors’ Names sor) of the scalar field alone is considered to yield the matter term taking α = 0 inthe action, which reduces the latter to: I [ g µν , ϕ ] = Z √− gd x [ k ( R − η G µν ∇ µ ϕ ∇ ν ϕ ] . (2)In [84], the authors have obtained an equation applying the slow-roll con-ditions to incorporate the consequences due to the experimental findings ofGW170717, further simplified by [88]: the action appears to acquire an extra termequal to ϕ G where G = R − R µν R µν + R µνρσ R µνρσ .Setting the integration constant that emanated from the first integral of the fieldequation to zero, the following metric defines a solution of the system for K = 0 [9]: ds = − H ( r ) dt + 15(Λ r − K ) K dr H ( r ) + r d Σ K, , (3)where d Σ K, = dθ + sin θdϕ and H ( r ) = (60 K − Kr + 3Λ r ) − µr . (4)If Λ disappears, the scalar field vanishes reducing the solution to that of thetopological Schwarzschild solution in a flat space-time, representing a black holeonly in a spherically symmetrical scenario [9]. By taking Λ = 0 : ds = − H ( r ) dt + 60 KH ( r ) dr + r d Σ K, , (5)and H ( r ) = 60 K − µr . (6)Here, µ is the integration constant and can be explicate as the black hole mass.Now to acquire the null geodesics ( ds = 0 ), the black hole optical spacetime inequatorial plane θ = π/ is written as: dt = 60 KH ( r ) dr + r H ( r ) dϕ , (7)along with optical metric ˆ g optab = g ab ( − g tt ) , according to the Fermat principle, thegeodesics are spatial rays of light. The Gaussian optical curvature can be deter-mined by allowing use of the description for the two-dimensional optical metricgiven in Eq. 7 K = R icciScalar , (8)in which R for optical metric is the Ricciscalar. Following the computation of non-zero Christoffel symbols, we obtain the following equation particularly for theGaussian optical curvature of the optical metric K = − Kµr + µ Kr + O ( r − ) . (9)ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) Let us recall the GBT for a two dimensional manifold. In this regard, we con-sider a regular domain D R aligned by 2-dimensional surface S with Riemannianmetric ˆ g ij , along with its piece-wise smooth boundary ∂D R = γ g ∪ C R , then GBTpermits a connection among the geometry and topology in terms of the subse-quent relation [38] Z Z D R K dS + I ∂D R ˆ kdσ + X j ˜ θ j = 2 π X ( D R ) , (10)where K is the Gaussian optical curvature, ˜ θ j is the exterior angle at the j th vertexand σ is the line element along the boundary D R . Let γ be a smooth curve in thesame domain. Thus, ˙ γ comes to be the unit speed vector [38]. It is well knownthat for regular domain the Euler characteristic X D R = 1 , while ˆ k is termed as ageodesic curvature and is defined as ˆ k = g opt ( ∇ ˙ γ ˙ γ, ¨ γ ) , (11)having the unit speed condition g opt ( ˙ γ, ˙ γ ) = 1 , where ¨ γ is the unit acceleration vec-tor perpendicular to ˙ γ . In the case of R → ∞ , the respective jump angles are takenas π/ (in short, the sum of angles corresponding to the observer and the source: ˜ θ O + ˜ θ S → π ). Using the fact that the geodesic curvature offers no contribution i.e. ˆ k ( γ ˜ g ) = 0 , we shall pursue a contribution by the virtue of the curve C R computedas: ˆ k ( C R ) = | ∇ ˙ C R ˙ C R | . (12)Let us consider C R := r ( ϕ ) = R = const, while R endows the distance from thecoordinate origin. The radial component of the geodesic curvature states as ( ∇ ˙ C R ˙ C R ) r = ˙ C ϕR ( ∂ ϕ ˙ C ϕR ) + Γ rϕϕ ( ˙ C ϕR ) . (13)Using the above equation, we note that the first term vanishes, then the sec-ond term can be obtained using the unit speed condition. Then, ˆ k is calculated as: lim R →∞ ˆ k ( C R ) = lim R →∞ (cid:12)(cid:12)(cid:12) ∇ ˙ C R ˙ C R (cid:12)(cid:12)(cid:12) → R . We take the large limits of the radial dis-tance, and find: lim R →∞ d t → R d ϕ. Hence the deflection angle can be calculatedin the form: [38]
Θ = − Z π Z ∞ b/ sin ϕ K dS, (14)where b is the impact parameter, a dimensionless quantity that endorses thestraight line approximation in which the light ray is assumed to be expressed as r = b/ sin ϕ at zeroth order in the weak deflection limits [38]. This equation in-scribes the global impact on the lensing of particles on account of the fact that onehas to integrate over the optical domain of integration outside the enclosed mass.Now, by using Eq. 9 into Eq. 14, we obtain the weak deflection angle for flat blackholes in Horndeski theory to be: Θ = 2
Kµb + µ π Kb + O ( µ ) . (15)ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Authors’ Names
Graphical Analysis
This segment is dedicated to review the impact of deflection angle Θ on asymptot-ically flat black holes graphically and to illustrate the physical eminence of thesegraphs to examine the influence of curvature constant K and impact parameter b on the deflection angle by analyzing the stable and unstable state of the black hole.2.1.1. Deflection angle versus Impact parameter
For µ = 2 , the deflection angle is Θ plotted against the impact parameter b fordifferent values of the curvature constant K in Figure 1. K =- =- = = = - - (cid:1) ( i ) μ = = = = = =
250 20 40 60 80 100024681012 b Θ ( i ) μ = Figure 1: Θ versus b . • Figure 1 demonstrates the influence of Θ w.r.t b for different values of K . One can examine that for small b deflection angle increases but as b increases, the deflection angle decreases for fixed µ . So for stable behaviorwe choose the domain b ∈ [1 , .Figure (i) illustrates graphically the impact of Θ w.r.t b by varying K .For negative K , we obtain locally hyperbolic behavior but for K = 0 thebehavior is locally flat. If there is small change in the variation of K, de-flection angle is exponentially decreasing. • Figure (ii) shows that with the increase of K , Θ decreases gradually asit tends to positive infinity. We obtain physical stable behavior just for K ≥ . .
3. Photon lensing in a plasma medium
In this section, we investigate the effect of plasma on photon lensing by asymptot-ically flat black hole in Horndeski theory. The refractive index for a flat black holeis stated as follows [41], n ( r ) = s − ω e ω ∞ H ( r ) , (16)ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) then, the corresponding optical metric illustrated as d ˜ σ = g optjk dx j dx k = n ( r ) H ( r ) (cid:20) KH ( r ) dr + r dϕ (cid:21) , (17)where, the metric function H ( r ) in the optical metric is given by: H ( r ) = 60 K − µr . (18)The corresponding optical Gaussian curvature is calculated by using Eq. tobe: K = − µKr + µ Kr + 90 µK r ω e ω ∞ − µ K r ω e ω ∞ − µKr (60 K r − µ ) ω e ω ∞ + µ Kr ω e ω ∞ + 3 µ (60 K r − µ )80 Kr ω e ω ∞ . (19)Then the geodesic curvature approaches for R goes to ∞ as: lim R →∞ ˆ k g d ˜ σdϕ (cid:12)(cid:12)(cid:12)(cid:12) C R = 1 . (20)Using the straight line approximation given by r = b/ sin ϕ as R → ∞ , GBT can bestated as: [41]: lim R →∞ Z π +Θ0 (cid:20) ˆ k g d ˜ σdϕ (cid:21) | C R dϕ = π − lim R →∞ Z π Z Rb/ sin ϕ K dS. (21)After simplification, we obtain Θ ≃ µ π b K + µ b K ω e ω ∞ + 2 µKb − µ Kπ b ω e ω ∞ + 180 µK b ω e ω ∞ + O ( µ , K ) . (22)The above results shows that the photon rays are moving in a medium of homo-geneous plasma. Graphical Analysis
This section is focused to investigate the graphical effect of deflection angle Θ onasymptotically flat black holes in a plasma medium. Further, we exemplify thephysical implications of these graphs to analyze the effect of curvature constant K , ω e ω ∞ and impact parameter b on deflection angle by analyzing the stable andunstable state of black hole.3.1.1. Deflection angle versus Impact parameter b This subsection offers the examination of deflection angle Θ w.r.t impact parameter b for different ranges of curvature constant K , ω e ω ∞ , for fixed µ = 2 . For simplicity,here we suppose ω e ω ∞ = η .ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Authors’ Names η = η (cid:0) η (cid:2) η (cid:3) η (cid:4) Θ ( i ) (cid:5)(cid:6)
2, K (cid:7) η = η = η = η = η = (cid:9)(cid:10) Θ ( i ) (cid:11) =
2, K = Figure 2: Relation between Θ and impact parameter b . • Figure 2 depicts the influence of Θ w.r.t b for varying η and for fixed µ = 2 and K = 1 .(1) In figure (i), represents the behavior of Θ w.r.t b for small variation of η . For η = 0 . the deflection angle decreases, it is observed that thebehavior is same for η = 0 . → . and . → . there is smallchange in deflection angle but greater than . the deflection angleactually increase.(2) In figure (ii), shows that the deflection angle gradually increase byincreasing η .
4. Conclusion
In this paper, we accomplished an extensive analysis of deflection angle of lightby asymptotically flat black hole in the background of Horndeski theory in weakfield approximation. In this regard, we employ the optical geometry of asymptoti-cally flat black hole in Horndeski theory. Thenceforth, we have utilized the GBT byusing straight line approximation and computed the deflection angle procured bythe leading order terms. The obtained deflection angle is evaluated by integratinga domain outside the impact parameter, that depict the globally impact of gravita-tional lensing. Additionally, we have found the deflection angle of photon lensingfor asymptotically flat black hole in a plasma medium. Also, we have analyzedthe influence of the impact parameter, the curvature constant and the plasma termon the deflection angle of the photon lensing by asymptotically flat black hole inthe context of Horndeski theory graphically. We infer that the proposed deflec-tion angle increases by decreasing the impact parameter, the mass term µ is foundto decrease the deflection angle, and increasing the curvature constant is seen todecrease the deflection angle gradually. Moreover, if we disregard the impact ofplasma medium ( ω e ω ∞ → , in the following equation Θ ≃ µ π b K + µ b K ω e ω ∞ + 2 µKb − µ Kπ b ω e ω ∞ + 180 µK b ω e ω ∞ + O ( µ , K ) , (23)we obtain the weak deflection angle for non-plasma medium case:ebruary 8, 2021 1:32 WSPC/INSTRUCTION FILE main Instructions for Typing Manuscripts (Paper’s Title) Θ = 2
Kµb + µ π Kb + O ( µ ) . (24)The observations that follow from the Horndeski theory and its mathematicalimplications include determining observables such as angular positions, separa-tion, magnification and fluxes: a case study of astrophysical applications for Sagit-tarius A* and M87 can be found in [11]. Additionally, distinct researches includelinear Horndeski theories to study dark energy and gravitational waves [89], andHorndeski gravity to study dark matter [90]. Our results can be extrapolated tocorrect for quantum effects while determining the observables, thus, increasingprecision - however, the resulting calculations to attain it are beyond the scope ofthis work. 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