Weak deflection angle of black-bounce traversable wormholes using Gauss-Bonnet theorem in the dark matter medium
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Weak deflection angle of black-bounce traversable wormholes using Gauss-Bonnettheorem in the dark matter medium
AL˙I ¨OVG ¨UN ∗ Physics Department, Eastern Mediterranean University, Famagusta, North Cyprus99628 via Mersin 10, Turkey
Received: .201 • Accepted/Published Online: .201 • Final Version: ..201
Abstract:
In this paper, we first use the optical metrics of black-bounce traversable wormholes to calculate the Gaussiancurvature. Then we use the Gauss-Bonnet theorem to obtain the weak deflection angle of light from the black-bouncetraversable wormholes. Then we investigate the effect of dark matter medium on weak deflection angle using the Gauss-Bonnet theorem. We show how weak deflection angle of wormhole is affected by the bounce parameter a . Using theGauss-bonnet theorem for calculating weak deflection angle shows us that light bending can be thought as a global andtopological effect. Key words:
Relativity and gravitation; Gravitational lensing; Deflection angle; Wormholes; Gauss-Bonnet theorem.
1. Introduction
Root of gravitational lensing is the deflection of light by gravitational fields such as a planet, a black hole, ordark matter predicted by Einstein’s general relativity, in the weak-field limit [1, 2]. Weak deflection is used todetect dark matter filaments, and it is important topic because it helps to understand the large-scale structureof the universe [1, 3].One of the important method to calculate the weak deflection angle using optical geometry is proposedby Gibbons and Werner (GW), which is known as Gauss-Bonnet theorem (GBT) [4, 5]. In the method of GW,deflection angle is considered as a partially topological effect and can be found by integrating the Gaussianoptical curvature of the black hole space using: [4]ˆ α = − Z Z D ∞ K d S, (1.1)where ˆ α is a deflection angle, K is a Gaussian optical curvature, dS is an optical surface and the D ∞ standsfor the infinite domain bounded by the light ray, excluding the lens. Since the GW method provides a uniqueperspective, it has been applied to various types of black hole spacetime or wormhole spacetime [6]-[55].In this paper, our main motivation is to explore weak deflection angle of black-bounce traversablewormholes [56, 57] using the GBT and then extend our motivation of this research is to shed light on theeffect of dark matter medium on the weak deflection angle of black-bounce traversable wormhole using theGBT. Note that the refractive index of the medium is supposed that it is spatially non-uniform but one can ∗ Correspondence: [email protected] work is licensed under a Creative Commons Attribution 4.0 International License. L˙I ¨OVG ¨UN/Turk J Phys consider that it is uniform at large distances [58]-[66]. To do so, the photons are thought that may be deflectedthrough dark matter due to the dispersive effects, where the index of refractive n ( ω ) which is for the scatteringamplitude of the light and dark-matter in the forward.
2. Calculation of weak deflection angle from black-bounce traversable wormholes using the Gauss-Bonnet Theorem
The “black-bounce form” for the spacetime metric of traversable wormhole is: [56] ds = − (cid:18) − M √ r + a (cid:19) dt + (cid:18) − M √ r + a (cid:19) − dr + (cid:0) r + a (cid:1) (cid:0) dθ + sin θdϕ (cid:1) . (2.1)It is noted that the parameter a stands for the bounce length scale and when a = 0 , it reduces tothe Schwarzschild solution. We restrict ourselves to the equatorial coordinate plane ( θ = π ), so that theblack-bounce traversable wormhole spacetime becomes ds = − (cid:18) − M √ r + a (cid:19) dt + (cid:18) − M √ r + a (cid:19) − dr + (cid:0) r + a (cid:1) dϕ . (2.2)Then the optical geometry of the black-bounce traversable wormhole spacetime is found by using g opt αβ = g αβ − g , (2.3)d t = d r (cid:16) − M √ r + a (cid:17) + ( r + a )d ϕ (cid:16) − M √ r + a (cid:17) . (2.4)The Gaussian optical curvature K for the black-bounce traversable wormhole optical space is calculated asfollows: K ≃ m r − mr − a m r + 10 a mr − a r . (2.5)Then we should use the Gaussian optical curvature in GBT to find deflection angle because the GBTis a theory which links the intrinsic geometry of the 2 dimensional space with its topology ( D R in M , withboundary ∂D R = γ ˜ g ∪ C R ) [4]: Z D R K d S + I ∂D R κ d t + X i ǫ i = 2 πχ ( D R ) , (2.6)in which κ is defined as the geodesic curvature ( κ = ˜ g ( ∇ ˙ γ ˙ γ, ¨ γ )), so that ˜ g ( ˙ γ, ˙ γ ) = 1 . It is noted that ¨ γ is anunit acceleration vector and ǫ i is for the exterior angles at the i th vertex. Jump angles are obtained as π/ r → ∞ . Then we find that θ O + θ S → π . If D R is a non-singular, the Euler characteristic becomes χ ( D R ) = 1 ,hence GBT becomes Z Z D R K d S + I ∂D R κ d t + θ i = 2 πχ ( D R ) . (2.7)2 L˙I ¨OVG ¨UN/Turk J Phys
The Euler characteristic number χ is 1, then the remaining part yields κ ( C R ) = |∇ ˙ C R ˙ C R | as r → ∞ .The radial component of the geodesic curvature is given by (cid:16) ∇ ˙ C R ˙ C R (cid:17) r = ˙ C ϕR ∂ ϕ ˙ C rR + Γ rϕϕ (cid:16) ˙ C ϕR (cid:17) . (2.8)For large limits of R , C R := r ( ϕ ) = r = const. we obtain (cid:16) ∇ ˙ C rR ˙ C rR (cid:17) r → − r , (2.9)so that κ ( C R ) → r − . After that it is not hard to see that d t = r d ϕ , where κ ( C R )d t = d ϕ. The GBT reducesto this form
Z Z D R K d S + I C R κ d t r →∞ = Z Z S ∞ K d S + π +ˆ α Z d ϕ. (2.10)The light ray follows the straight line so that, we can assume that r γ = b/ sin ϕ at zeroth order. Theweak deflection angle can be calculated using the formula:ˆ α = − π Z ∞ Z r γ K r d r d ϕ. (2.11)Using the Gaussian optical curvature (2.5), we calculate the weak deflection angle of black-bouncetraversable wormholes up to second order terms:ˆ α ≈ mb + a π b . (2.12)Note that it is in well agreement with the [57] in leading order terms.
3. Deflection angles of photon through dark matter medium from black-bounce traversable worm-holes
In this section, we investigate the effect of dark matter medium on the weak deflection angle. To do so, we usethe refractive index for the dark matter medium [58]: n ( ω ) = 1 + βA + A ω . (3.1)Note that β = ρ m ω , ρ is the mass density of the scattered dark matter particles, A = − ε e and A j ≥ O (cid:0) ω (cid:1) and higher terms are related to the polarizability of the dark-matter candidate.The order of ω − is for the charged dark matter candidate and ω is for a neutral dark matter candidate.In addition, the linear term in ω occurs when parity and charge-parity asymmetries are present. The 2dimensional optical geometry of the wormhole is: dσ = n (cid:18) dr (cid:16) − M √ r + a (cid:17) + ( r + a ) (cid:16) − M √ r + a (cid:17) dϕ (cid:19) , (3.2)3 L˙I ¨OVG ¨UN/Turk J Phys and dσdϕ (cid:12)(cid:12)(cid:12)(cid:12) C R = n (cid:18) r + a (cid:16) − M √ r + a (cid:17) (cid:19) / . (3.3)Using the GBT within optical geometry of black-bounce traversable wormhole, we obtain the weakdeflection angle in a dark matter medium:lim R →∞ Z π + α (cid:20) κ g dσdϕ (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) C R dϕ = π − lim R →∞ Z Z D R K dS. (3.4)First we calculate the Gaussian optical curvature at linear order of M : K ≈ csgn ( r ) a M ( A ω + β A + 1) r − csgn ( r ) M ( A ω + β A + 1) r − a ( A ω + β A + 1) r . (3.5)After that we find lim R →∞ κ g dσdϕ (cid:12)(cid:12)(cid:12)(cid:12) C R = 1 . (3.6)Then for the limit of R → ∞ , the deflection angle in dark matter medium can be calculated using theGBT as follows: α = − lim R →∞ Z π Z R b sin ϕ K dS. (3.7)Hence, we obtain the weak deflection angle in dark matter medium as follows: α = 4 Mb Ψ + a π b Ψ , (3.8)where Ψ = A ω + 2 A A β ω + A β + 2 A ω + 2 β A + 1 , (3.9)which agrees with the known expression found using another method. Of course, in the absence of the darkmatter medium (Ψ = 0 ), this expression reduces to the known vacuum formula α ≈ mb + a π b . Hence, we findthat the deflected photon through the dark matter around the black-bounce traversable wormhole has largedeflection angle compared to black-bounce traversable wormhole without dark matter medium.
4. Conclusion
In this paper, we have studied the weak deflection angle of black-bounce traversable wormholes using the GBT.Then we have investigated the effect of dark matter medium on the weak deflection angle of black-bouncetraversable wormholes. Note that refractive index is taken spatially non-uniform, and it is uniform at largedistances. Hence it is concluded that the deflection angle by black-bounce traversable wormholes increases withincreasing the bounce parameter a , on the other hand the deflection angle decreases in a increasing mediumof dark matter. It is showed that how weak deflection angle of wormhole is affected by the bounce parameter a . Moreover we use the Gauss-bonnet theorem for calculating weak deflection angle which proves us that lightbending can be thought as a global and topological effect.4 L˙I ¨OVG ¨UN/Turk J Phys
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