Gravitational lensing by wormholes supported by electromagnetic ...

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Feb 21, 2018 - 1Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, MACEDONIA ...... [32] M. Sharif and I. Nawazish, Annals Phys.
Gravitational lensing by wormholes supported by electromagnetic, scalar, and quantum effects ¨ un,3, 4, 5, 6, † Ayan Banerjee,7, 8, ‡ and Izzet ˙ Kimet Jusufi,1, 2, ∗ Ali Ovg¨ Sakallı6, §

arXiv:1802.07680v1 [gr-qc] 21 Feb 2018

1

Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, MACEDONIA 2 Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, MACEDONIA 3 Instituto de F´ısica, Pontificia Universidad Cat´ olica de Valpara´ıso, Casilla 4950, Valpara´ıso, CHILE 4 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, CANADA 5 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, CANADA 6 Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, TURKEY 7 Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, INDIA 8 Astrophysics and Cosmology Research Unit, University of KwaZulu Natal, Private Bag X54001, Durban 4000, SOUTH AFRICA (Dated: February 22, 2018) Wormholes are one of the most interesting topological features in spacetime, offering a rat run between two vastly separated regions of the universe. In this paper, we study the deflection angle of light by wormholes, which are supported by electric charge, magnetic charge, and scalar fields in the weak field limit approximation. To this end, we apply new geometric methods – the Gauss-Bonnet theorem and the optical geometry – to compute the deflection angles. We also verify our findings by using the well-known geodesics method. There exists a similarity between the charge and the quantum corrections on a black hole solution, which has been recently discussed in the context of the relativistic Bohmian quantum mechanics. By replacing classical geodesics with Bohmian trajectories, we introduce a new wormhole solution, whose having matter sources and anisotropic pressure supported by Bohmian quantum effects. The problem of fulfillment of the energy conditions of the Morris-Thorne traversable wormhole is also discussed.

References

CONTENTS

I. Introduction II. Deflection of light by Wormhole with electric charge A. Geodesics of Wormhole with Electric Charge

1 I.

2 4

III. Deflection of Light by a Non-gravitational Wormhole A. Geodesics of a Non-gravitational Wormhole

4 5

IV. Deflection angle of Wormhole with Scalar Field A. Geodesics of Wormhole with Scalar Field

5 6

V. Quantum Corrected Wormhole A. Einstein-Rosen type wormhole B. Morris-Thorne traversable wormhole 1. Gravitational lensing 2. Energy Conditions 3. Volume Integral Quantifier VI. Conclusions Acknowledgments

∗ † ‡ §

[email protected] [email protected]; http://www.aovgun.com ayan [email protected] [email protected]

10

6 6 7 8 8 8 9 9

INTRODUCTION

During the last few decades, wormholes have become one of the most popular and intensively studied topics. They are tunnel-like connections made out of spacetime, offering a shorter distance between two vastly separated regions of spacetime (of the different universes or widely separated areas of our own Universe). In 1988, the seminal works on traversable wormholes, as hypothetical shortcuts in spacetime, was proposed by Morris and Thorne [1]. The wormhole solution culminated in the Visser’s book with title ”Lorentzian Wormholes: From Einstein to Hawking” [2], which reviews the subject up to the year 1995, as well as new notions are presented and hinted at. However, wormhole physics dates back to 1935, when Einstein and Rosen [3] constructed an elementary particle model represented by a bridge linking two identical sheets: Einstein-Rosen bridge (ERB). But, ERB was an unsuccessful particle model. In 1962, Fuller and Wheeler [4] also showed ERB is unstable if it connects two parts of the same universe. At this juncture, it is worth noting that terminology of “wormhole” was first coined by Wheeler [5]. In fact, Wheeler considered wormholes as quantum foams (at the Planck scale), which link separate regions of spacetime [6, 7]. In fact, ERB is a non-traversable wormhole, even by a photon, since the throat pinches off in a finite time. These entities

ployed for resolving the singularities in cosmology [59] and in black holes [60]. In particular, in the study of [60] modified Schwarzschild metric has been derived and, in sequel, the singularity structure and thermodynamical features of the obtained black hole have been studied in detail. Remarkably, it was shown that quantum corrections change the picture of Hawking radiation when the size of BH approaches the Planck scale. Therefore our preliminary aim of this study is to calculate the deflection angle of wormhole with electric charge, magnetic charge and next, scalar fields as well as the quantum effects in the weak field limit approximation using the GWM to do so we use the GBT. The main importance of the paper is to find the difference between the deflection angles which may shed light from an experimental point of view in the near future. This paper has the following organization. In Sec. I, we consider a charged wormhole and study the deflection angle of light from it. Section II is devoted to the deflection angle computation for a wormhole spacetime supported by nonlinear electromagnetic theory. In Sec. III, we perform the same analysis in the case of a scalar field wormhole. In Sec. IV, we introduce a new wormhole solution based on the analogy of charge and quantum effects in the framework of the relativistic Bohmian quantum mechanics. Furthermore, we investigate the deflection angle as well as the energy conditions. We draw our conclusions in Sec. V.

were further explored by Ellis in 1973 [8, 9], by producing a non-singular wormhole solutions using a wrong-sign scalar field. Later, Thorne together with his students came up with a time-machine model using wormholes [10]. Morris and Thorne studied the “traversable wormholes” for human travel [1]. They modeled the wormholes as the objects possessing two mouths and a throat. It was understood that for fabricating such a wormhole solutions, two spacetimes that do not posses any event horizon or physical singularities should be invoked. But the wormhole throat is threaded by exotic matter, which violates the null energy condition (NEC) [2]. The existence of the so-called exotic matter sounds to be unusual in general relativity (GR), but such a matter appears in quantum field theory as well as on scalar-tensor theories. A first example of rotating axially symmetric solution describing a traversable wormhole was given by Teo [11]. However, Teo only showed the generic form (together with some physical properties) of such a rotating wormhole without representing the solution to the associated Einstein-field equations. In a similar way Khatsymovsky [12] discussed some general features of a slowly rotating wormhole. Meanwhile, traversable Lorentzian wormhole in the presence of a generic cosmological constant Λ was studied by Lemos et al. [13], and further wormhole in anti-de Sitter background was found by [14]. On the other hand, wormhole solutions were studied in the framework of modified and alternative theories of gravity such as the braneworld scenario [15, 16], Einstein GaussBonnet Gravity [17–30], f (R) gravity [31–33], f (R, φ) gravity [34], f (R, T ) gravity [35–38], non-commutative gravity [39–42]. A decade ago, new global topology method was suggested by Gibbons and Werner [43, 44] to calculate the deflection angle for static black holes, in the weak limits, by using the Gauss-Bonnet theorem (GBT). Then, Werner [45] also extended the latter result to the stationary black holes. Afterwards, Gibbons and Werner method (GWM) has become popular and is still used in many studies [46–57]. Different than the ordinary methods, in GWM the deflection angle is computed by taking the integral over a domain A∞ outside the light ray: Z Z α ˆ=− Kdσ. (1)

II.

DEFLECTION OF LIGHT BY WORMHOLE WITH ELECTRIC CHARGE

Recently, Kim and Lee [61] have found an exact solution for a charged Lorentzian wormhole, whose spacetime is described by the following metric    −1 Q2 b(r) Q2 ds2 = − 1 + 2 dt2 + 1 − + 2 dr2 +r2 dΩ22 . r r r (2) Throughout the paper, we shall consider the special case when b(r) = b20 /r. Thus, the charged wormhole metric (2) recasts [61]

A∞

   −1 Q2 b2 Q2 ds2 = − 1 + 2 dt2 + 1 − 02 + 2 dr2 +r2 dΩ22 . r r r (3) To find the charged wormhole optical metric, we simply let ds2 = 0. Furthermore, it is quite convenient to simplify the problem by considering the deflection of light in the equatorial plane; hence we set θ = π/2. The optical metric reads

Note that K stands for the Gaussian optical curvature, dσ is the elementary surface area of the optical geometry, and α ˆ is the deflection angle. Despite the success of GR, quantum gravity has become dramatically urgent necessity in order to resolve several problems in cosmos like the missing matter problem or singularities of spacetime. In an recent approach, Das [58] showed that by replacing the classical geodesics with quantal trajectories arising from Bohmian mechanics, one obtains the quantum corrections to the Raychaudhuri equation, which is the so-called quantum Raychaudhuri equation (QRE). QRE has recently been em-

dt2 = 

2

dr2   2 1+ Q 1− r2

b20 r2

r2    dϕ2 . + 2 Q2 +Q 1 + r2 r2

(4)

Let us now compute the geodesic curvature κ. To this end, we first point out that κ(γh ) = 0, since γh is a geodesic. Thus, we end up with

At this point, one may introduce a new radial coordinate u, and a new function ζ(u) as follows du = r

dr   2 1+ Q 1− r2

b20 r2

 , ζ(u) = q 2

Q r2

+

r 1+

Q2 r2

.

κ(CR ) = |∇C˙ R C˙ R |,

(5)

in which one can choose CR := r(ϕ) = R = const. The radial part is evaluated as follows

Thus, our metric can be rewritten as dt2 = hab dλa dλb = du2 + ζ 2 (u)dϕ2 .



(6)

Note that (a, b = r, ϕ) and h = det hab . The Gaussian optical curvature K, is defined by: "    2 2 # 1 dr d dr dζ dr d ζ K=− + . (7) ζ(u) du dr du dr du dr2

K=

3Q − r4



4 b20 Q2 r6

R→∞

and 

κ = h (∇γ˙ γ, ˙ γ¨ ) ,

(9)

AR

CR

Q2 R2

 dϕ

→ R dϕ.

(16)

If we combine Eqs. (15) and (16), we find κ(CR )dt = dϕ. For getting the deflection angle, it is convenient to approximate the boundary curve of A∞ to a notional undeflected ray, which is the line of r(ϕ) = b/ sin ϕ. Then, the deflection angle (1) reads Zπ Z∞ α ˆ=−

Kdσ. 0

(17)

b sin ϕ

If we substitute Eq. (9) into (17), we find out the following integral Zπ Z∞  α ˆ=− 0

R→∞

K dσ +

κ dt =

D∞

p

det hab drdϕ.

b sin ϕ

α ˆ'

πb20 3πQ2 − + O(Q2 , b20 ) 2 4b 4 b2

(19)

Thus, the total deflection angle consists by two leading terms, the first terms is a direct result of the geometry of the spacetime, while the second term a consequence of the electric field.

π+ Z αˆ

ZZ

4 b2 Q2 3 Q2 − b20 − 06 4 r r

(18) It is worth noting that using the relation: du ≈ dr, which is valid in the limit R → ∞, one can easily solve (18) in the leading order terms. Thus, we have

(11)

where the unit speed condition h(γ, ˙ γ) ˙ = 1, holds. If we let R → ∞, our two jump angles (θO , θS ) become π/2, or in other words, the sum of jump angles to the source S, and observer O, satisfies θO + θS → π [43]. Hence, we can rewrite the GBT as K dσ +



R lim dt = lim  q R→∞ 1+

Note that κ gives the geodesic curvature, K stands for the Gaussian optical curvature, while θi is the exterior angle at the ith vertex. We can choose a non-singular domain outside of the light ray with the Euler characteristic number χ(AR ) = 1. The geodesic curvature is defined as

I

(15)

i

∂AR

ZZ

1 . R

R→∞

+ O(Q4 , b40 ).

(14)

R→∞



This result is going to be used to evaluate the deflection angle. The key point behind the GWM is the GBT, which involves the wormhole optical geometry in our case. More precisely, we can choose a non-singular domain outside the light ray, let us say AR , with boundary ∂AR = γh ∪ CR , from where the GBT can be expressed as ZZ I X K dσ + κ dt + θi = 2πχ(AR ). (10) AR

   2 ϕ ϕ r = C˙ R ∂ϕ C˙ R + Γr(op) C˙ R . ϕϕ r(op)

2Q6 + 5Q4 r2 − 2Q4 b20 + 3Q2 r4 − 5Q2 r2 b20 − r4 b20 . K= r6 (Q2 + r2 ) (8) We can approximate the above result in leading orders. Thus, one gets b20

r

In the last equation Γϕϕ gives the Christoffel symbol which is associated to the optical geometry. From the above equation, it is obvious that the first term vanishes, while the second term is calculated using Eq. (13) and the unit speed condition. As R → ∞, the geodesic curvature and dt approximates to lim κ(CR ) = lim ∇C˙ R C˙ R ,

Applying Eq. (7) to our metric (3), we find the charged wormhole Gaussian optical curvature, which is given by

2

∇C˙ R C˙ R

(13)

dϕ = π. (12) 0

3

A.

where α ˆ is the deflection angle to be derived. Following the same arguments given in Ref. [64], the deflection angle can be calculated as

Geodesics of Wormhole with Electric Charge

Lagrangian of metric (2) is expressed as follows   Q2 r(s) ˙ 2 2L = − 1 + t˙2 (s) + 2 2 2 b r(s) 1− 02 + Q 2 r(s)

α ˆ = 2|ϕu=1/b − ϕu=0 | − π.

As expected, the deflection angle in the weak limit approximation is found to be the same result found by GBT. Namely, we have

r(s)

+ r(s)2 θ˙2 (s) + r(s)2 sin2 θϕ˙ 2 (s).

(20)

As it is well known, the geodesic equation has to be supplemented by the normalization (or the so-called metric) condition 2L =  where for massive particles  = 1 and for photon  = 0. Here, we consider the deflection of planar photons i.e., θ = π/2. After using the spacetime symmetries, one obtains two constants of motion l and γ, which are given by [62]: ∂L = 2r2 (s)ϕ(s) ˙ = G, ∂ ϕ˙   ∂L Q2 ˙ = −E. t(s) = −2 1 + pt = r(s)2 ∂ t˙

pϕ =

α ˆ'

Let us now introduce a new variable u(ϕ), which is related to our old radial coordinate as r = 1/u(ϕ). Thus, we obtain the following identity:

ds2 = −dt2 +

(23)

dr2 1−

b(r) r

dr2

dt2 = 1−

2rth 4 −r 4 +r 2 r 4 +r 2

Φ(Q, u, b0 ) = −Q u +

− 2Q u +

b20 u4





r 4 −rth 4

+ r2 dϕ2 .

(31)

r 4 −rth 4

 √ √ 2 r2 rth 4 − r6 + r4 r4 − rth 4 + rth 4 r4 − rth 4 √ √ K=− . r4 r2 + r4 − rth 4 r4 − rth 4 (32) In the following section, we shall use result (32) within the concept of GBT to find the deflection angle. Next, we can approximate this solution as follows

(25)

2

−u .

3rth 4 5rth 8 − 2 r6 8r10

(33)

Substituting Eq. (33) into Eq. (17), we get

It is well known that the solution of the differential equation (45) is given by the following relation [62, 64] ∆ϕ = π + α ˆ,

(30)

Using Eq. (7) the Gaussian optical curvature for wormhole (31) becomes

K'− 2 4

(29)

by which rth is the throat location. The corresponding optical metric of Eq. (29) can be found to be

where Q2 b20 u6

 + r2 dθ2 + sin2 θ dϕ2 ,

where the shape function b(r) is given by √ 2rth 4 − r4 + r2 r4 − rth 4 √ b(r) = , r3 + r r4 − rth 4

Without loss of generality, we can normalize the affine parameter along the light ( = 0) rays by taking E = 1 [62] and approximate the distance of closest approach with the impact parameter i.e., umax = 1/rmin = 1/b, since we shall consider only leading order terms [63]. In this case, one can choose the second constant of motion: G = b. Finally using the metric condition ( = 0) with Eqs. (20-23) in the associated Lagrange equation of u(ϕ), we obtain the following equation:  2 du 1 Q2 − dϕ (Q2 u2 − b20 u2 + 1)u4 u6 u2 (Q2 + u12 )2 1 1 (24) − 8 2 2 1 2 + u2 = 0. u b (Q + u2 )

4 6

(28)

Recently, using a non-linear electromagnetic theory in the framework of Born-Infeld electromagnetism, it was shown that a specific field configuration in flat metric can be viewed as a spherically symmetric wormhole. In particular, the following effective spacetime metric was found by [65] as follows

(22)

From here, it follows that p dϕ b Q2 u2 + 1 , = ±p du b2 Φ(Q, u, b0 ) + Q2 u2 − b20 u2 + 1

3πQ2 πb20 − + O(Q2 , b20 ). 2 4b 4 b2

III. DEFLECTION OF LIGHT BY A NON-GRAVITATIONAL WORMHOLE

(21)

r˙ dr 1 du = =− 2 . ϕ˙ dϕ u dϕ

(27)

 Zπ Z∞  3rth 4 5rth 8 p α ˆ=− − 6 − det hab drdϕ. (34) 2r 8r10

(26)

0

4

b sin ϕ

One can easily solve this integral in the leading order terms as 175πrth 8 9πrth 4 − . 4 64 b 8192b8

α ˆ'

dϕ = ±q du

(35)

∆=

(36)

α ˆ'

IV.

2rth 4 −r(s)4 +r(s) r(s)4 +r(s)2



r(s)4 −rth 4

r(s)4 −rth 4

+ r(s)2 θ˙2 (s) + r(s)2 sin2 θϕ˙ 2 (s).

(39) (40)

Ξu

Ξ=

4 u4 1 − rth . u4

2

dr2 1−

b20 r2

+

α r2

 + r2 dϕ2 .

(49)

The Gaussian optical curvature (7) of metric (49) is then found to be −b20 + α . (50) r4 If we substitute this equation into the deflection angle formula (17), we obtain

Ξu

p

DEFLECTION ANGLE OF WORMHOLE WITH SCALAR FIELD

dt2 = 

K=

where 1+

(46)

 −1  b20 α ds = −dt + 1 − 2 + 2 dr2 +r2 dθ2 + sin2 θdϕ2 . r r (48) From here on in, one can immediately derive the optical metric as follows 2

In sequel, without loss of generality, we normalize the affine parameter along the light rays by taking E = 1 [62] and approximate the distance to the closest point by an impact parameter i.e., umax = 1/rmin = 1/b (recall that we consider only the leading order terms [63]). In this case, one can choose the second constant of motion as G = b. Finally using the metric condition ( = 0) together with Eqs. (38-40), one can derive the following Lagrange equation:  2 1 1 du   + 2 = 0, (41) 4 4 u4 1−r 2r 1 dϕ u th u4 1 − th + 4 − 4 Ξ

9πα2 . b4 b20

Here we take the spacetime geometry to be that of a wormhole with scalar field with metric [61]  −1  b(r) α 2 2 ds = −dt + 1 − + 2 dr2 +r2 dθ2 + sin2 θdϕ2 . r r (47) We will restrict our attention to the case b(r) = b20 /r, which converts the metric (47) to the following form

(38)

Following Sec. (II-A), we first derive the two constants of motion of the geodesics in the non-gravitational wormhole: G = 2r2 (s)ϕ(s), ˙ ˙ E = 2t(s).

(45)

One can easily observe that Eqs. (45) and (46) are nothing but the deflection angles obtained in the weak limit approximation of GBT.

For the geodesics analysis, one can write down the Lagrangian of metric (29) as follows

1−

9πβ 2 . 16 b4 b20

(37)

Geodesics of a Non-gravitational Wormhole

√ 2

(44)

If we compute the deflection angle of the nongravitational wormhole possessing pure electric field, we get

9πα2 + O(α4 , b8 ). b4 b20

r(s) ˙ 2

(43)

After evaluating the integral, the leading order terms lead to the following result for the magnetic field case:

and for electric field case:

2 L = −t˙2 (s) +

b20 − 4β 2 u4 . b20

α ˆ'

9πβ 2 α ˆ' + O(β 4 , b8 ), 16 b4 b20

A.

, √ 2 (4β 2 u4 − b20 ) (b2 u2 − 1) ( ∆ + 1)

where

In the case pof magnetic wormhole α = 0 and β 6= 0, we have rth = 2β|/b0 . On the other hand, in the p case of electric wormhole (β = 0 and α 6= 0): rth = 2 2|α|/b0 . Thus, we remark that in leading order terms the deflection angle is affected by the non-linear electromagnetic fields. For magnetic field case:

α ˆ'

√ b b0 ( ∆ + 1)

(42)

Zπ Z∞  α ˆ=−

After some manipulation, Eq. (41) admits the following differential equation

0

5

b sin ϕ

−b20 + α r4

p

det hab drdϕ.

(51)

After approximating u ≈ r and evaluating the integration the leading order terms yield

as [60] α

ψ(xα ) = Rei S(x ) , α ˆ'

π b20 4b2



πα . 4b2

where ψ(xα ) is a normalizable wave function, R(xα ) and S(xα ) are some real continuous functions associ~ ∂α S, in which ated with the four velocity field uα = m (α = 0, 1, 2, 3). In a recent work, due to implications of quantum corrections, Ali and Khalil [60] presented an interesting model for a black hole solution known as modified Schwarzschild black hole, when they replaced quantal (Bohmian) geodesics with classical geodesics like the metric:   2M dr2 ~η 2  +r2 dΩ22 , ds = − 1 − + 2 dt2 +  ~η 2M r r 1 − r + r2

(52)

Hence we conclude the effect of the scalar field is to bend the light outward the wormhole, in fact similar to the effect of electric charge. A.

Geodesics of Wormhole with Scalar Field

With the following Lagrangian, one can begin to perform the geodesics analysis of the wormhole metric with scalar field (48). r˙ 2

2L = −t˙2 + 1−

b20 r2

+

α r2

+ r2 θ˙2 + r2 sin2 θϕ˙ 2 .

(60) where η is a dimensionless constant and ~ has a dimension of (length)2 in geometric units. When we consider M = 0 and introducing a new coordinate transformation r2 = u2 + 2 , where 2 = −~η, the above metric reduces to the Einstein-Rosen wormhole type solution:

(53)

Considering the planar photons (θ = π/2) and taking cognizance of the two constants of motion l and γ: ˙ G = 2r2 (s)ϕ(s), ˙ and E = 2t(s),

(54)

 u2 dt2 + du2 + u2 − ~η (dθ2 + sin2 θdϕ2 ). − ~η (61) This metric represents a massless quantum version of ERB wormhole with the radius of the throat: Rthro. = √ ~η. It is worth noting that ERB wormhole is nonsingular in the interval u ∈ (−∞, ∞). However this wormhole is a non-traversable wormhole. Its optical metric can be written as ds2 = −

we can normalize the affine parameter along the light rays by taking E =1 (metric condition) [62]. Again, we consider the nearest approach distance with the impact parameter i.e., umax ' 1/b and use the second constant of motion G = b to focus on the leading order terms [63]. In sequel, we use the metric condition with Eqs. (53) and (54) and employ the Lagrange equations. Thus, we obtain the following equation of motion for u(ϕ):  2 du 1 1 1 − 4 2 + 2 = 0. (55) 2 2 2 4 dϕ (αu − b0 u + 1)u u b u

K'

(63)

~ 3~2 π − . b 8b2

(64)

One can infer from the above result that since the deflection of light is negative, it indicates that light rays always bend outward the wormhole due to the quantum correction effects. Of course, for the resulting expression to the deflection angle we need to take the absolute value of the above equation

(58)

QUANTUM CORRECTED WORMHOLE

|ˆ α| ' A.

~ 3~2 + 4. 3 2r 2r

α ˆ'−

(57)

where α ˆ is the deflection angle to be calculated. The deflection angle results in

V.

(62)

After some manipulation, one can compute the deflection angle (17) as

Solution of the differential equation (56) can be expressed as follows [62]

πα π b20 − 2. 4b2 4b

u2 − ~η 2 (u2 − ~η)2 2 du + dϕ . u2 u2

The corresponding Gaussian optical curvature reads

dϕ b = ±p . (56) 2 2 4 4 du b (b0 u − αu − u2 ) − b20 u2 + αu2 + 1

α ˆ'

u2

dt2 =

From here, it follows that

∆ϕ = π + α ˆ,

(59)

Einstein-Rosen type wormhole

~ 3~2 π + . b 8b2

(65)

It is well known that the ER wormholes are not traversable wormholes, hence a static traversable MorrisThorne wormholes have been introduced [1].

As it is well-known, we can introduce a quantum velocity field uα with a wave function of a quantum fluid 6

B.

using the wave function solution one can show that the geodesic motion is modified due to the relativistic quantum potential given by

Morris-Thorne traversable wormhole

The spacetime ansatz for seeking static spherically symmetric traversable (without event horizon) wormhole spacetime can be written in the Schwarzschild coordinates as [1] ds2 = −e2Φ(r) dt2 +

dr2 1−

b(r) r

+r2

VQ = ~2

where (t, r, θ, ϕ) are the standard Schwarzschild coordinates, with Φ(r) and b(r) are the redshift and shape functions, respectively. Since we seek a wormhole solution, the redshift function Φ(r) should be finite everywhere, in order to avoid the presence of an event horizon. Another essential characteristics of a wormhole geometry is about the shape function b(r), which should satisfy the condition of b(r0 ) = r0 , in which r0 is the radius of the wormhole throat where the two spacetimes (upper/lower spacetimes) are joined. Consequently, the shape function must satisfy the flaring-out condition:

(corr.)

ρ(corr.) = Pr(corr.) = Pθ

(67)

θ Gϕ ϕ = Gθ .

(68)

(69)

For an anisotropic fluid Tµν , the energy-momentum tensor is defined as follows T

µ

ν

= (−ρ, Pr , Pθ , Pϕ ) ,

(74)

(75)

where P = Pθ = Pϕ is the pressure measured in the tangential directions. Meanwhile, one can observe that the charged wormhole solution simply by letting Q2 → ~η [61]. Moreover, the geometry encoded in the Einstein equations due to the quantum effect can be obtained by shifting the shape function in the following way

(70)

where ρ is the energy density with Pr , Pθ and Pϕ are nonzero components of the diagonal terms, as measured in the orthogonal direction, respectively. The second term of Eq. (69) has some modifications when the quantum corrections are taken into consideration:   (corr.) T µ ν (corr.) = −ρ(corr.) , Pr(corr.) , Pθ , Pϕ(corr.) . (71)

b → bef f = b −

~η . r

(77)

With more preciously, the quantum effect on the wormhole reveals when shifting the shape function b into (b − ~η/r), after we make use of the following choose   1 ~η Φ(r) → Υ(r) = ln 1 + 2 . (78) 2 r

Using the definition of the 4-momentum given as follows pα = ~ ∂α S,

~η , 8πr4

Using (68) and the Morris-Thorne metric (66), we obtain the following set of field equations   1 ~η 0 ρ(r) = b (r) − 2 , 8πr2 r     1 b(r) Φ0 b(r) ~η Pr (r) = 2 1− − 3 − 4 , 8π r r r r  h b(r) b0 r − b 0 1 1− Φ00 + (Φ0 )2 − Φ P(r) = 8π r 2r(r − b) b0 r − b Φ0 ~η i − 2 Φ0 + − 4 . (76) 2r (r − b) r r

ef f where Gµν and Tµν are the Einstein tensor and effective energy-momentum tensor, respectively. The totalderivative of the source term in the field equations is written as ef f corr. Tµν = Tµν + Tµν .

= Pϕ(corr.) =

in which η is dimensionless constant. It just happened that the effect of this quantum potential to be similar to the effect of the electric field described by a similar energy-momentum tensor components simply by letting Q2 → ~η. On the other hand, the only non-zero components form the Einstein tensor Gµν of the metric (66) are b0 (r) Gtt = − 2 , r   b(r) b(r) Φ0 r Gr = − 3 + 2 1 − , r r r   b(r) h 00 b0 r − b 0 Gθθ = 1 − Φ + (Φ0 )2 − Φ r 2r(r − b) Φ0 i b0 r − b Φ0 + , − 2 2r (r − b) r

db < 1 must hold at the throat of the in which b0 (r) = dr wormhole. In addition to this, the shape function also satisfy the asymptotically flat limit i.e., b(r)/r → 0 as r → ∞. From now on, we focus on the effect of quantum corrections. Einstein equations remain the same with the effect of quantum corrections in the energy-momentum tensor:

1 ef f Gµν = Rµν − gµν R = 8πTµν , 2

(73)

In Ref. [60] it was shown that for a general and static spherically symmetric spacetime metric in Schwarzschild coordinates, the stress-energy tensor has the following non-zero components

 dθ2 + sin2 θdϕ2 , (66)

b(r) − rb0 (r) > 0, b2 (r)

R . R

(72) 7

We can see this from Einstein’s field equations written in terms of the effective shape function

2.

Energy Conditions

In this section, we argue the issue of energy conditions and make some regional plots to check the validity of all energy conditions. WEC is defined by Tµν U µ U ν ≥ 0 i.e., ρ ≥ 0 and ρ(r) + Pr (r) ≥ 0, where Tµν is the energy momentum tensor and U µ denotes the timelike vector. This means that local energy density is positive and it gives rise to the continuity of NEC, which is defined by Tµν k µ k ν ≥ 0 i.e., ρ(r) + Pr (r) ≥ 0, where k µ is a null vector. In this regard, we have the following energy condition at the throat region:

b0ef f , 8πr2    1 bef f (r) Υ0 bef f (r) Pr (r) = , 2 1− − 8π r r r3  h b0ef f r − bef f 0 1 bef f (r) P(r) = 1− Υ00 + (Υ0 )2 − Υ 8π r 2r(r − bef f ) b0ef f r − bef f 0 Υ0 i − 2 , (79) Υ + 2r (r − bef f ) r ρ(r) =

where −1



ρ(r0 ) =

~η ~η 1+ 2 , r3 r  −1 "  −1 # ~η 2~η ~η ~η 00 Υ = 4 1+ 2 3− 2 1+ 2 . (80) r r r r Υ0 = −

ρ(r) + Pr (r) =

  1 h bef f (r)  00 Υ + (Υ0 )2 1− 8π r b0ef f r − bef f 0 b0ef f r − bef f 0 Υ − 2 Υ − 2r(r − bef f ) 2r (r − bef f ) b0ef f Υ0 i + 2 + . (88) r r

ρ(r) + P(r) =

For an observation one can see from Eq.(77), the solub (r) tions are asymptotically flat, i.e., ef rf → 0 as r → ∞. With the above solution one also verifies that the form  function obeys b0ef f (r0 ) = ~η − b20 /r2 < 1, for flaring out condition at the throat. Taking into account the condition we should impose the restriction ~η < 2r02 . As when we consider the wormhole throat, Eq.(87) reduces to i 2(~η)2 1 h 2 − + 2(~η − r ) (ρ + Pr ) |r=r0 = 0 .(89) 8πr04 r02 + ~η

Gravitational lensing

Let us now continue our study to investigate the deflection of light in the quantum corrected spacetime geometry given by Eq.(81). Choosing b(r) = b20 /r, metric (81) reduces to   ~η dr2 ds2 = − 1 + 2 dt2 + + r2 dΩ22 . (82) b20 ~η r 1 − r2 + r2

Taking into account the condition ~η < 2r02 , one verifies that matter configuration violates NEC at the throat i.e. (ρ + Pr ) |r=r0 < 0. In Fig. (1), we show the behavior of the null energy condition.

The quantum corrected optical Gaussian curvature therefore yields 3~η − b20 4b2 ~η − 06 . 4 r r

(83)

Thus, the asymptotic form of the geodesic curvature becomes lim κ(CR ) →

R→∞

1 , R

3.

πb20 3π~η 3 π~η b20 − + . 2 4b 4 b2 8b4

Volume Integral Quantifier

We now consider the “volume integral quantifier,” which provides information about measure of the amount of exotic matter required for wormhole maintenance and is related only to ρ and Pr , not to the transverse components, with the following definite integrals R IV = (ρ(r) + Pr (r)) dV , where dV = r2 sin θdrdθdφ.

(84)

which leads the following deflection angle: α ˆ'

    rb0ef f − bef f 1 bef f Υ0 + 2 1− , 8π r r r3 (87)

and

which is a combination of Morris-Thorne type wormhole with quantum corrections.

K'

(86)

Now, using the field equations (75, 76, 77, 79, 80), one finds the following relations

Our obtained solution refurbishes the work of charged wormhole solution in which it was assumed a probable charged metric in a combination of MT type static spherically symmetric wormhole and Reissner-Nordstr¨om(RN) spacetime. We now demonstrate that our result, due to the quantum effects, is similar to that procedure. The key point is that we can recast the spacetime metric as   ~η dr2 ds2 = − 1 + 2 dt2 + + r2 dΩ22 , (81) b(r) ~η r 1 − r + r2

1.

b0ef f (r0 ) . 8πr02

(85) 8

quantum effects on the deflection angle in the various wormhole geometries. To this end, by considering the optical geometries of the wormholes, we have employed the GWM in the context of GBT. Then, the deflection angles are computed via the surface integrals of the associated Gaussian optical curvatures on a domain outside the light ray, which reveals the effect of gravitational lensing as a global effect of the spacetime. For the wormholes having electric/scalar fields, the deflection angles are found out to be proportional to the throat of the wormhole that bands the light rays towards itself. On the other hand, it is seen that electric/scalar fields play an important role to band the light rays outward the wormhole center. In the case of a non-linear electromagnetic theory in which the wormhole geometry is constructed purely by non-linear magnetic and electric fields, we have shown that deflection angle is proportional to the electromagnetic fields.

FIG. 1. The figure shows the behavior of the NEC i.e., ρ+Pr as a function of r0 and b0 , for chosen ~ = 1 and η = 1. We see that NEC is violated.

We have also constructed the Einstein-Rosen type wormhole solutions as well as the Morris-Thorne wormhole, which are both supported by the quantum effects coming from the Bohmian quantum mechanics. In particular, it has been shown that the exotic matter concentrated on the throat of the wormhole violates the null energy conditions. It is worth noting that the quantum effects on the wormhole geometry are correlated with the shape function: shifting of the shape function is in complete analogy with the charge wormhole solution.

Taking into account Eq.(87) and performing the integration by parts, which is  I  1 + ~η/r2 ∞ [ρ + Pr ] dV = (r − bef f ) ln (90) 1 − bef f /r r0   Z ∞  1 + ~η/r2  dr. (r − b0ef f ) ln − 1 − bef f /r r0 For the construction of wormhole the boundary term at r0 vanishes, and due to asymptotic behavior the boundary term at infinity also vanishes. Thus the above expression reduces to I Z ∞  1 + ~η/r2  [ρ + Pr ] dV = − dr. (r − b0ef f ) ln 1 − bef f /r r0 (91) The value of this volume-integral provides information about the “total amount” of ANEC violating matter in the spacetime, and we are going to solve the integral for our particular choice of shape function b(r) = b20 /r. Here, we suppose that wormhole extends form r0 to a radius situated at ‘a0 and then we get the very simple result      b2 r0 a IV = √ 0 arctan √ − arctan √ . (92) ~η ~η ~η

Finally, we want to emphasize that although our results show consistency in the first-order terms in the deflection angles, the harmonized results of the methods disappear with the consideration of the second-order correction terms. This is simply due to the straight line approximation used in the integration domain. In near future, we plan to commit a study that will show fit in the results of the second-order correction terms. The latter aim is going to be possible when we manage to produce a suitable equation for the light ray or to modify the integration domain associated to the optical geometry.

For an interesting observation when a → r0 then R (ρ + Pr ) → 0, and thus one may interpret that wormhole can be contracted for with arbitrarily small quantities of ANEC violating matter.

VI.

ACKNOWLEDGMENTS

This work was supported by the Chilean FONDECYT ¨ AO ¨ is grateful to the Waterloo Grant No. 3170035 (AO). University, Department of Physics and Astronomy and Perimeter Institute for Theoretical Physics for hosting him as a research visitor where part of this work was done.

CONCLUSIONS

In this paper, we have pointed out in detail the effect of electric charge, magnetic charge, scalar fields, and 9

[1] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988). [2] M. Visser, Lorentzian Wormholes (AIP Press, New York, 1996). [3] A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935). [4] R. W. Fuller and J. A. Wheeler, Phys. Rev. 128, 919 (1962). [5] J. Wheeler, Ann. Phys. 2, (6) 604 (1957). [6] J. A. Wheeler, Phys. Rev. 97, 511 (1955). [7] J. A. Wheeler, Geometrodynamics, (Academic Press, New York, 1962). [8] H. G. Ellis, J. Math. Phys. 14, 104 (1973). [9] H. G. Ellis, J. Math. Phys. 15, 520 (1974)(Erratum). [10] M. Morris, K. S. Thorne and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988). [11] E. Teo, Phys. Rev. D 58, 024014 (1998). [12] V. M. Khatsymovsky, Phys. Lett. B 429, 254 (1998). [13] J. P. S. Lemos, F. S. N. Lobo and S. Q. de Oliveira, Phys. Rev. D 68, 064004 (2003). [14] J. P. S. Lemos and F. S. N. Lobo, Phys. Rev. D 69, 104007 (2004). [15] Francisco S.N. Lobo, Phys.Rev. D 75, 064027 (2007). [16] S. Chakraborty and T. Bandyopadhyay, Astrophys.Space Sci. 317, 209-212 (2008). [17] M. R. Mehdizadeh, M. K. Zangeneh and F. S. N. Lobo Phys. Rev. D 91, 8, 084004 (2015). [18] S. Rani and A. Jawad, Adv.High Energy Phys. 2016, 7815242 (2016). [19] M. G. Richarte, I. G. Salako, J. P. Morais Graa, ¨ un, Phys. Rev. D 96, 084022 H. Moradpour and A. Ovg¨ (2017). [20] A. Ovgun and M. Halilsoy, Astrophys. Space Sci. 361, 214 (2016). [21] M. Halilsoy, A. Ovgun and S. H. Mazharimousavi, Eur. Phys. J. C 74, 2796 (2014). [22] P. Goulart, Class. Quant. Grav. 35, 025012 (2018). [23] G. Clement, D. Gal’tsov and M. Guenouche, Phys. Rev. D 93, 024048 (2016). [24] G. Clement, Phys. Rev. D 50, R7119 (1994). [25] G. Clement, Phys. Rev. D 51, 6803 (1995). [26] M. Azreg-Ainou and G. Clement, Gen. Rel. Grav. 25, 881 (1993). [27] G. W. Gibbons and M. S. Volkov, Phys. Rev. D 96, 024053 (2017). [28] S. W. Hawking, Phys. Rev. D 37, 904 (1988). [29] E. J. Son and W. Kim, Phys. Rev. D 83, 124012 (2011). [30] W. T. Kim, J. J. Oh and M. S. Yoon, Phys. Rev. D 70, 044006 (2004). [31] F. S. N. Lobo and M. A. Oliveira, Phys. Rev. D 80, 104012 (2009). [32] M. Sharif and I. Nawazish, Annals Phys. 389, 283 (2018). [33] S. Bahamonde et al., Phys. Rev. D 94, 4, 044041 (2016). [34] M. Zubair, F. Kousar and S. Bahamonde, arXiv:1712.05699. [35] P. H. R. S. Moraes and P. K. Sahoo Phys. Rev. D 96, 4, 044038 (2017). [36] Z. Yousaf, M. Ilyas and M. Zaeem-ul-Haq Bhatti, Eur. Phys. J. Plus 132, 6, 268 (2017). [37] P. H. R. S. Moraes, R. A. C. Correa and R. V. Lobato, JCAP 1707, 07, 029 (2017). [38] M. Zubair, S. Waheed and Y. Ahmad, Eur. Phys. J. C

76, 8, 444 (2016). [39] P. K. F. Kuhfittig, Int. J. Mod. Phys. D 24, 03, 1550023 (2015). [40] M. Sharif and S. Rani, Phys. Rev. D 88, 12, 123501 (2013). [41] F. Rahaman et al., Int. J. Theor. Phys. 53, 1910 (2014). [42] P. Bhar and F. Rahaman, Eur. Phys. J. C 74, 12, 3213 (2014). [43] G. W. Gibbons and M. C. Werner, Class. Quant. Grav. 25, 235009 (2008). [44] G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick and M. C. Werner, Phys. Rev. D 79, 044022 (2009). [45] M. C. Werner, Gen. Rel. Grav. 44, 3047 (2012). [46] K. Jusufi, Astrophys. Space Sci. 361, no. 1, 24 (2016). [47] K. Jusufi, Eur. Phys. J. C 76, no. 6, 332 (2016). [48] I. Sakalli and A. Ovgun, Europhys. Lett. 118, no. 6, 60006 (2017). ¨ un, [49] K. Jusufi, M. C. Werner, A. Banerjee and A. Ovg¨ Phys. Rev. D 95, no. 10, 104012 (2017). ¨ un, Phys. Rev. D 96, no. [50] K. Jusufi, I. Sakall and A. Ovg¨ 2, 024040 (2017). ¨ un and A. Banerjee, Phys. Rev. D 96, [51] K. Jusufi, A. Ovg¨ no. 8, 084036 (2017). [52] K. Jusufi, Int. J. Geom. Meth. Mod. Phys. 14, no. 12, 1750179 (2017). [53] T. Ono, A. Ishihara and H. Asada, Phys. Rev. D 96, no. 10, 104037 (2017). ¨ un, Phys. Rev. D 97, no. 2, 024042 [54] K. Jusufi and A. Ovg¨ (2018). ¨ un, arXiv:1707.02824. [55] K. Jusufi and A. Ovg¨ [56] K. Jusufi, F. Rahaman and A. Banerjee, Annals Phys. 389, 219 (2018). [57] K. Jusufi, N. Sarkar, F. Rahaman, A. Banerjee and S. Hansraj, arXiv:1712.10175 [gr-qc]. [58] S. Das, Phys. Rev. D 89, 084068 (2014). [59] A. F. Ali and S. Das, Phys. Lett. B 741, 276 (2015). [60] A. F. Ali and M. M. Khalil, Nucl. Phys. B 909, 173 (2016). [61] S. W. Kim and H. Lee, Phys. Rev. D 63, 064014 (2001). [62] R. H. Boyer and R. W. Lindquist, J. Math. Phys. (N.Y.) 8, 265 (1967). [63] L. Iorio, Nuovo Cimento Soc. Ital. Fis. 118B, 249 (2003). [64] S. Weinberg, Gravitation and Cosmology (John Wiley &Sons, New York, 1972). [65] F. Baldovin, M. Novello, S. E. Perez Bergliaffa and J. M. Salim, Class. Quantum Grav. 17, 3265 (2000).

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