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Gravitational Lensing in Rotating and Twisting Universes

O. Gurtug [email protected] T. C. Maltepe University, Faculty of Engineering and Natural Sciences, 34857, Istanbul -Turkey    M. Mangut [email protected] Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey    M. Halilsoy [email protected] Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey
Abstract

Gravitational lensing caused by the gravitational field of massive objects has been studied and acknowledged for a long period of time. In this paper, however, we propose a different mechanism where the bending of light stems from the non-linear interaction of gravitational, electromagnetic and axion waves that creates the high curvature zone in the space-time fabric. The striking distinction in the present study is that in contrast to the convex lensing in the gravitational field of a massive object, hyperbolic nature of the high curvature zone of the background space-time may give rise to concave lensing. Expectedly detection of this kind of lensing becomes possible through satellite detectors.

Gravitational lensing
pacs:
95.30.Sf, 98.62.Sb

I Introduction

The experimental observation of gravitational waves has opened a new window to understand the universe in a more global scale. From the first observation to date, it is understood that our universe contains propagating gravitational waves. According to the Einstein’s theory of general relativity, these gravitational waves, alone or coupled with electromagnetic (em) waves, do not pass through each other without a significant interaction. Their interaction is nonlinear and hence, induces irreversible consequences on the structure of the fabric of space-time. The remarkable influence on the space-time structure is the high curvature zones.

In our earlier studies, we have shown that nonlinear interaction of plane em shock waves accompanied with gravitational waves with different amplitude profiles generates cosmological constant that can be associated with dark energy 1 , which is believed to be the source of the accelerating expansion of our universe. In an another study, we have shown that the nonlinear interaction of plane gravitational waves and shock em waves with second polarization induces Faraday rotation in the polarization vector of em waves which paves the way for an indirect evidence of the gravitational waves 2 . In this paper, we present the gravitational lensing in the high curvature zones which stem from the nonlinear interaction of gravitational waves coupled with em and axion waves such that any massive or massless particle that moves in such a region might be diffused.

Bending of light while passing near a massive object such as a star or black hole is a well-known, one century old problem named as gravitational lensing. This is due to the fact that light has energy which is equivalent to mass and is attracted by another mass as a requirement of gravity. In particular, when light passes near a highly massive star, the deflection angle becomes significant enough to be detected by telescopes.

The purpose of this study is to introduce another mechanism rather than the gravitational field of a massive object that leads to gravitational lensing. This mechanism incorporates with the high curvature zone of space-time. In such space-times there is no definite center, thus any chosen point can act for the purpose and all deflections are computed with respect to that center. The nonlinear interaction ( or collision) of gravitational, em and axion waves produces high curvature regions that are isometrically transformable to the spherical coordinates in which the bending angle calculations become more tractable.

Our first example in this direction is the rotating Bertotti-Robinson (RBR) space-time 3 ; 4 , which is isometric to the collision of two cross polarized em waves 5 ; 6 . It has been known that the collision of two linearly polarized em waves, the so-called Bell-Szekeres (BS) solution 7 , can be transformed into the Bertotti-Robinson (BR) space-time 8 which is isotropic and conformally flat (CF), so that light remains undeflected. However, the nonlinear interaction of the cross polarized em waves is locally isometric to the RBR space-time. The cross polarized nature of the waves that participate in the collision breaks the isotropy and induces a cross term in the metric and naturally deflects the light. In other words, the cross term amounts to a non-zero Weyl curvature which behaves like a mass term. The reason that we take collision of em waves into account can be summarized as follows: If we follow the Big Bang at the end of a period of nearly 400.000400.000 years, em interaction was switched on to allow strong collision of such waves to shape the future of the newly born universe. Further expansion naturally cooled down the em radiation to form the present patterns of cosmic microwave background (CMB) radiation. The energy density of em radiation which is described by the Newman-Penrose (NP) 9 quantity Φ11\Phi_{11} is dependent on the angle that can be plotted to exhibit the non-isotropy as a function of an angular variable. In particular, via an observation by a differential interval of very small angle, such as 0<Δθ<10<\Delta\theta<1 in degrees, the energy distribution can be plotted to show the variation. Such a distribution may be compared with the energy density of the CMB radiation. Note, however, that the general distribution depends on more factors, whereas in our present model we take only the rotational effects into account to distort the isometry.

Our second example consists of the space-time formed by the collision of em waves coupled to an axion field 10 . The axion also induces a cross term in the metric to bend the light. Applying a coordinate transformation to the em-axion problem, we obtain once more a non-isotropic space-time and we find in such a space-time the bending angle of light. It is interesting that light bending in this problem is provided entirely by the axion field. Vanishing of the axion field reduces the space-time once more to the isotropic BR in which the light shows no deflection. This particular example of axion is also important in the sense that axion is considered to contribute to the dark matter. Looking to the problem from the other direction, detection of deflection of light in such a space-time may be helpful in detection of the axion.

The last example that we shall consider within this context consists of the Newman-Unti-Tamburino (NUT) space-time 11 . In a previous study, the NUT parameter was interpreted as the twist of the empty space-time 12 . As the off-diagonal metric component gave rise to light deflection, the twist property of space-time also creates bending of light in the empty space. Let us note that apart from the twist interpretation, the NUT parameter can be interpreted as a gravito magnetic mass 13 , as an extension of the Schwarzschild mass. Naturally, the latter also leads to the effect of empty space bending due to the curvature of the space-time. Expectedly, as the NUT parameter vanishes, one recovers the result of light bending nearby a Schwarzschild black hole, a well-known result.

One important distinction in the present study is that although masses are always attractive and give rise to convex lensing, in our examples, the lensing can also be concave. That is, the null geodesics are deflected in the outward direction by the rotation/twist that induces hyperbolic property to the space-time. For instance, in the case of the NUT space-time, if the NUT parameter is smaller / larger than the angular momentum parameter, then accordingly we encounter with a convex / concave type bending for the light trajectory. Similarly, the rotation parameter of BR may shift the character of light bending due to the presence of axion. It is worth to emphasize at this stage that some researchers interrelated the concave lensing effect with the presence of dark energy 14 ; 15 .

Our method of finding light bending is based on the method developed by Rindler and Ishak 16 , which has been employed previously in different examples 17 ; 18 ; 19 ; 20 . In section II, we give a brief review of this formalism. In analogy with the Kepler’s problem of Newtonian mechanics, we project the orbits onto the θ=π/2\theta=\pi/2 plane and express the u=1ru=\frac{1}{r} as a function of azimuthal angle φ\varphi. Our case is just for the light orbits, as null geodesics which can be developed perturbatively in terms of φ\varphi.

Organization of the paper is as follows. In Section II, we review the general formalism of bending angle calculation in rotating/twisting geometries. Specific examples are given in Sec. III, such as rotating BR, axionic BR and twisting NUT spacetimes. We complete the paper with results and discussions in Section IV.

II General Formalism of the Bending Angle of Light for Rotating Geometries

Among the others, the method proposed by Rindler and Ishak (RI) 15 for calculating the bending angle of light has been found more powerful, especially, when one wishes to display the effect of the background fields to the gravitational lensing. In this section, we extend this method to cover the metrics that describe rotating fields. The form of the metric that we will be interested in for the rotating geometries is defined at the equatorial plane as

ds2=f(r)dt2+2g(r)dtdφh(r)dr2p(r)dφ2.ds^{2}=f(r)dt^{2}+2g(r)dtd\varphi-h(r)dr^{2}-p(r)d\varphi^{2}. (1)

The method of RI incorporates with the inner product of two vectors that remains invariant under the rotation of coordinate systems. As a result, the angle between two coordinate directions dd and δ\delta, as shown in Fig. 1, is given by the invariant formula

cos(ψ)=diδi(didi)(δjδj)=gijdiδj(gijdidj)(gklδkδl).\cos\left(\psi\right)=\frac{d^{i}\delta_{i}}{\sqrt{\left(d^{i}d_{i}\right)\left(\delta^{j}\delta_{j}\right)}}=\frac{g_{ij}d^{i}\delta^{j}}{\sqrt{\left(g_{ij}d^{i}d^{j}\right)\left(g_{kl}\delta^{k}\delta^{l}\right)}}. (2)
Refer to caption
Figure 1: The orbital map of the light rays corresponding to Eq.(6). The one-sided bending angle is defined from this map as ϵ=ψφ.\epsilon=\psi-\varphi. The upper thick straight line represents the undeflected light rays described by the solution of the homogeneous part of the Eq.(6). The function F(φ)F(\varphi) is found to be eωφe^{\omega\varphi} and eφe^{\varphi} for the RBR and em-axion case respectively. In the case of twisting NUT universe, the function F(φ)F(\varphi) for small NUT parameter is sin(δφ)sin(\delta\varphi) and for large NUT parameter it reads eζφe^{\zeta\varphi}.

In this formula, gijg_{ij} stands for the metric tensor of the constant time slice of the metric (1). In this method, the orbital plane of the light rays is defined by introducing a two-dimensional curved (r,φ)(r,\varphi) space, which is defined simultaneously at the equatorial plane (when θ=π/2\theta=\pi/2 ) and a constant time slice,

dl2=h(r)dr2+p(r)dφ2.dl^{2}=h(r)dr^{2}+p(r)d\varphi^{2}. (3)

The constants of motion related to the null geodesics in the considered rotating spacetime are

dtdλ=[p(r)K(r)]E+[g(r)K(r)]l, dφdλ=g(r)Ef(r)lK(r),\frac{dt}{d\lambda}=\left[\frac{-p(r)}{K(r)}\right]E+\left[\frac{g(r)}{K(r)}\right]l,\text{ \ \ \ \ \ }\frac{d\varphi}{d\lambda}=\frac{-g(r)E-f(r)l}{K(r)}, (4)

in which λ\lambda stands for the parameter other than proper time and K(r)=g2(r)+p(r)f(r)K(r)=g^{2}(r)+p(r)f(r). From these conserved quantities, we obtain

(drdφ)2=K(r)h(r)[g(r)E+f(r)l)]2[p(r)E2f(r)l22g(r)lE],\left(\frac{dr}{d\varphi}\right)^{2}=\frac{K(r)}{h(r)\left[g(r)E+f(r)l)\right]^{2}}\left[p(r)E^{2}-f(r)l^{2}-2g(r)lE\right], (5)

where EE and ll represent energy and angular momentum constants, respectively. It has been found more practical to introduce a new variable uu, such that, u=1r.u=\frac{1}{r}.  Using this transformation, Eq.(5) leads to

d2udφ2=2u3κ(u)+u42dκ(u)du,\frac{d^{2}u}{d\varphi^{2}}=2u^{3}\kappa(u)+\frac{u^{4}}{2}\frac{d\kappa(u)}{du}, (6)

in which κ(u)=K(r)h(r)[g(r)E+f(r)l)]2[p(r)E2f(r)l22g(r)lE]\kappa(u)=\frac{K(r)}{h(r)\left[g(r)E+f(r)l)\right]^{2}}\left[p(r)E^{2}-f(r)l^{2}-2g(r)lE\right]. The solution of Eq.(6) will be used to define another equation in the following way

A(r,φ)drdφ.A(r,\varphi)\equiv\frac{dr}{d\varphi}. (7)

If the direction of the orbit is denoted by dd and that of the coordinate line φ=\varphi= constant δ,\delta, we have

d\displaystyle d =\displaystyle= (dr,dφ)=(A,1)dφ dφ<0,\displaystyle\left(dr,d\varphi\right)=\left(A,1\right)d\varphi\text{ \ \ \ \ \ \ \ }d\varphi<0,
δ\displaystyle\delta =\displaystyle= (δr,0)=(1,0)δr.\displaystyle\left(\delta r,0\right)=\left(1,0\right)\delta r. (8)

Using these definitions in (2), we get

tan(ψ)=[h1(r)p(r)]1/2|A(r,φ)|.\tan\left(\psi\right)=\frac{\left[h^{-1}(r)p(r)\right]^{1/2}}{\left|A(r,\varphi)\right|}. (9)

Then, the one-sided bending angle is defined as ϵ=ψφ.\epsilon=\psi-\varphi.

III EXAMPLES of GRAVITATIONAL LENSING in ROTATING GEOMETRIES

III.1 The Rotating Bertotti - Robinson Space-Time

The Bertotti - Robinson space-time is the sole conformally flat solution of the Einstein - Maxwell equations, which describes the universe filled with uniform non-null electromagnetic field. It is non-null as FμνFμν0F_{\mu\nu}F^{\mu\nu}\neq 0 and uniform because the null tetrad component of the Ricci tensor Φ11\Phi_{11} is constant. The rotating version of the BR solution was obtained by Carter 3 , and the related solution is described by the metric 2

ds2=F(θ)r2[dt2dr2r2dθ2r2sin2θF2(θ)(dφqrdt)2],ds^{2}=\frac{F(\theta)}{r^{2}}\left[dt^{2}-dr^{2}-r^{2}d\theta^{2}-\frac{r^{2}\sin^{2}\theta}{F^{2}(\theta)}\left(d\varphi-\frac{q}{r}dt\right)^{2}\right], (10)

where aa is the constant rotation parameter, F(θ)=1+a2(1+cos2(θ))F(\theta)=1+a^{2}(1+cos^{2}(\theta)) and q=2a1+a2q=2a\sqrt{1+a^{2}}.

The physical description of this solution is achieved via the Newman - Penrose (NP) formalism. The set of proper null tetrads 1form1-form is given by

l=F(θ)22r(dtdr),n=2F(θ)2r(dt+dr),m=iF(θ)2+sin(θ)2F(θ)(2ar1+a2dtdφ).\begin{gathered}l=\frac{\sqrt{F(\theta)}}{2\sqrt{2}r}(dt-dr),\\ n=\frac{2\sqrt{F(\theta)}}{\sqrt{2}r}(dt+dr),\\ m=\frac{i\sqrt{F(\theta)}}{\sqrt{2}}+\frac{sin(\theta)}{\sqrt{2F(\theta)}}\left(\frac{2a}{r}\sqrt{1+a^{2}}dt-d\varphi\right).\end{gathered} (11)

The non-zero Weyl and Ricci scalars are

Ψ2=a2F3(θ)[(1+a2)cos(2θ)+a2cos2(θ)ia1+a2cos(θ)(1+2a2+a2sin2(θ))],Φ11=14F2(θ).\begin{gathered}\Psi_{2}=\frac{a^{2}}{F^{3}(\theta)}\left[(1+a^{2})cos(2\theta)+a^{2}cos^{2}(\theta)-\frac{i}{a}\sqrt{1+a^{2}}cos(\theta)\left(1+2a^{2}+a^{2}sin^{2}(\theta)\right)\right],\\ \Phi_{11}=\frac{1}{4F^{2}(\theta)}.\end{gathered} (12)

The contribution of rotation parameter aa to the Weyl and Ricci scalars indicates that the resulting space-time character is of Petrov type - D. Furthermore, the rotation parameter breaks the uniform nature of the em field. This non-uniformity in the em field creates an anisotropy when compared to the non-rotating BR solution. The change in the em energy density of anisotropy between θ=0o\theta=0^{o} and θ=1o\theta=1^{o} directions can be found by

ΔΦ11=Φ11(θ=1o)Φ11(θ=0o)=0.0025a2(1+1.99a2)(1+2a2).\Delta\Phi_{11}=\Phi_{11}(\theta=1^{o})-\Phi_{11}(\theta=0^{o})=\frac{0.0025a^{2}}{(1+1.99a^{2})(1+2a^{2})}. (13)

The variation in the em energy density is plotted in Fig.2, against the rotation paramater aa. From (12), we can easily compute the maximum anisotropy by comparing the distribution of energy density along θ=0\theta=0 and θ=π/2\theta=\pi/2. By doing so, we obtain

Φ11(θ=0)Φ11(θ=π/2)=(1+a21+2a2)2,\frac{\Phi_{11}(\theta=0)}{\Phi_{11}(\theta=\pi/2)}=\left(\frac{1+a^{2}}{1+2a^{2}}\right)^{2}, (14)

which suggests that for fast rotations (a)(a\rightarrow\infty), the maximum anisotropy is of the order 1/41/4. Stated otherwise, if our universe is governed entirely by rotating em radiation, the maximum possible distortion along the zz-axis is 1/41/4 times the distortion along the equatorial direction. This is not unusual, as the rotational effects create bulges in the θ=π/2\theta=\pi/2 plane compared with the θ=0\theta=0 direction in analogy with the flattening of the rotating Earth.

Refer to caption
Figure 2: At a=0a=0 we have complete isotropy and maximum energy density Φ11=1/4\Phi_{11}=1/4. For increasing aa, Φ11\Phi_{11} decreases for aa\rightarrow\infty. Angular dependence is shown for a very narrow angle of ΔΦ11\Delta\Phi_{11}.

This anisotropy in the electromagnetic field influences the light rays that propagates within it. To show this effect, we calculate the bending angle of light, by using the method of RI.

By using Eq.(6), we get

d2udφ2[qE3(1+a2)3γ3(u)]u=[l(1+a2)γ(u)+4aEl(1+a2)3/2γ3(u)]u2[qE(1+l23a2l2)(1+a2)2γ3(u)+1(1+a2)γ2(u)]u3,\frac{d^{2}u}{d\varphi^{2}}-\left[\frac{qE^{3}}{(1+a^{2})^{3}\gamma^{3}(u)}\right]u=-\left[\frac{l}{(1+a^{2})\gamma(u)}+\frac{4aEl}{(1+a^{2})^{3/2}\gamma^{3}(u)}\right]u^{2}-\left[\frac{qE(1+l^{2}-3a^{2}l^{2})}{(1+a^{2})^{2}\gamma^{3}(u)}+\frac{1}{(1+a^{2})\gamma^{2}(u)}\right]u^{3}, (15)

where γ(u)=qE1+a2+l1+a2((1+a2)2q2)u.\gamma(u)=\frac{qE}{1+a^{2}}+\frac{l}{1+a^{2}}\left((1+a^{2})^{2}-q^{2}\right)u.

Since u<<1u<<1, Eq.(15) simplifies to

d2udφ2ω2u=μu2νu3,\frac{d^{2}u}{d\varphi^{2}}-\omega^{2}u=-\mu u^{2}-\nu u^{3}, (16)

where ω=1q\omega=\frac{1}{q}, μ=lqE+4al(1+a2)3/2q3\mu=\frac{l}{qE}+\frac{4al(1+a^{2})^{3/2}}{q^{3}} and ν=(1+a2)(2+l23a2l2)E2q2\nu=\frac{(1+a^{2})(2+l^{2}-3a^{2}l^{2})}{E^{2}q^{2}}.

The first approximate solution, u=eωφR,u=\frac{e^{\omega\varphi}}{R}, is the solution of the homogeneous part of Eq.(16). This solution corresponds to the undeflected line. If this solution is substituted back in Eq.(16), the perturbed solution for uu is obtained as

u(φ)=eωφR+ν8ω2R3e3ωφ+μ3ω2R2e2ωφ,u(\varphi)=\frac{e^{\omega\varphi}}{R}+\frac{\nu}{8\omega^{2}R^{3}}e^{3\omega\varphi}+\frac{\mu}{3\omega^{2}R^{2}}e^{2\omega\varphi}, (17)

and the Eq.(7) becomes

A(r,φ)=r2[ωReωφ+3ν8ωR3e3ωφ+2μ3ωR2e2ωφ].A(r,\varphi)=-r^{2}\left[\frac{\omega}{R}e^{\omega\varphi}+\frac{3\nu}{8\omega R^{3}}e^{3\omega\varphi}+\frac{2\mu}{3\omega R^{2}}e^{2\omega\varphi}\right]. (18)

Here RR is a constant parameter related to the physically meaningful area distance r0r_{0} of the closest approach calculated at φ=π/2\varphi=\pi/2 which is found to be

1r0=eωπ/2Rν8ω2R3e3ωπ/2μ3ω2R2eωπ.\frac{1}{r_{0}}=\frac{e^{\omega\pi/2}}{R}-\frac{\nu}{8\omega^{2}R^{3}}e^{3\omega\pi/2}-\frac{\mu}{3\omega^{2}R^{2}}e^{\omega\pi}. (19)

We calculate the bending angle when φ=0\varphi=0 that corresponds to a large distance from the source. This small angle approximation leads us to set tanψψ0\tan\psi\approx\psi_{0}. For this particular case, we found

rR, A(r,φ=0)ωR,r\approx R,\text{ \ \ \ \ \ }A(r,\varphi=0)\approx-\omega R, (20)

so that, the one - sided bending angle becomes

ϵ=ψ0=1ωR2=qR2=2a1+a2R2.\epsilon=\psi_{0}=\frac{1}{\omega R^{2}}=\frac{q}{R^{2}}=\frac{2a\sqrt{1+a^{2}}}{R^{2}}. (21)

The effect of rotation on the bending angle of light is depicted in Fig.3.

Refer to caption
Figure 3: Variation of the bending angle both as a function of rotation parameter aa and the distance RR. Evidently, rotation of the universe increases the bending angle, which is expected.

III.2 Rotating Axion - coupled Electromagnetic Field

The singularity-free colliding gravitational wave solution coupled with axion field was given in 10 . The interaction region represents the solution of Einstein-Maxwell-dilaton-axion equations in the limit of zero dilaton field. The obtained solution reduces to the Bell-Szekeres (BS) solution, which describes the non-linear interaction of shock em waves in the absence of axion field. The remarkable feature of the BS solution is that the interaction region is locally isometric to the part of the BR solution. The resulting metric that describes the collision of em field coupled with axion is given by

ds2=2dudvΔdy2δ(dx+q0τdy)2,ds^{2}=2dudv-\Delta dy^{2}-\delta(dx+q_{0}\tau dy)^{2}, (22)

in which the used notations represent

Δ=1τ2, δ=1σ2, τ=sin(auθ(u)+bvθ(v)),σ=sin(auθ(u)bvθ(v)) and q0=constant.\Delta=1-\tau^{2},\text{ \ \ \ }\delta=1-\sigma^{2},\text{ \ \ \ }\tau=\sin(au\theta(u)+bv\theta(v)),\\ \sigma=\sin(au\theta(u)-bv\theta(v))\text{ \ \ \ }and\text{ \ \ \ }q_{0}=constant. (23)

Here, (u,v)(u,v) are the double null coordinates, (a,b)(a,b) are the constant electromagnetic parameters, (θ(u),θ(v))(\theta(u),\theta(v)) are the unit step functions and (τ,σ)(\tau,\sigma) are the prolate coordinates. The constant parameter q0q_{0} is related to the axion field and the physically acceptable solution to the field equations are obtained when q0=1q_{0}=1. If we set q0=0q_{0}=0, the axion vanishes and the solution reduces to BS solution. The metric in (τ,σ,x,y)(\tau,\sigma,x,y) coordinates becomes

ds2=12ab(dτ2Δdσ2δ)Δdy2δ(dx+τdy)2.ds^{2}=\frac{1}{2ab}(\frac{d\tau^{2}}{\Delta}-\frac{d\sigma^{2}}{\delta})-\Delta dy^{2}-\delta(dx+\tau dy)^{2}. (24)

The metric in the usual (t,r,θ,φ)(t,r,\theta,\varphi) coordinates is obtained via the following transformations

τ=12r(r2t2+1), σ=cosθ, tanhy=12t(r2t21), x=φ12ln(r+t)21(rt)21,\tau=\frac{1}{2r}(r^{2}-t^{2}+1),\text{ \ \ \ }\sigma=\cos\theta,\text{ \ \ \ }\tanh y=\frac{1}{2t}(r^{2}-t^{2}-1),\text{ \ \ \ }x=\varphi-\frac{1}{2}\ln\mid\frac{(r+t)^{2}-1}{(r-t)^{2}-1}\mid, (25)

from which we have removed the overall constant factor 1/2ab1/2ab by metric scaling. This transformation leads us to express the resulting metric in the BR form as

ds2=1r2[dt2dr2r2dθ2r2sin2θ(dφ1rdt)2].ds^{2}=\frac{1}{r^{2}}\left[dt^{2}-dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta\left(d\varphi-\frac{1}{r}dt\right)^{2}\right]. (26)

This metric represents a space-time filled with em field coupled to an axion field. Next, we calculate the contribution of a mixture of these fields to the bending angle of light. As before, we start our calculation with Eq.(6), which yields

d2udφ2u=αu2βu3.\frac{d^{2}u}{d\varphi^{2}}-u=-\alpha u^{2}-\beta u^{3}. (27)

where α=lE\alpha=\frac{l}{E} and β=(2+l2)E2\beta=\frac{(2+l^{2})}{E^{2}}.

The homogeneous part of Eq.(27) has a solution of the form u=eφRu=\frac{e^{\varphi}}{R}, This solution is substituted back into Eq.(27) and the perturbed resulting solution for uu is obtained as

u(φ)=eφR+β8R3e3φ+α3R2e2φ,u(\varphi)=\frac{e^{\varphi}}{R}+\frac{\beta}{8R^{3}}e^{3\varphi}+\frac{\alpha}{3R^{2}}e^{2\varphi}, (28)

and the Eq.(7) becomes

A(r,φ)=r2[1Reφ+3β8R3e3φ+2α3R2e2φ].A(r,\varphi)=-r^{2}\left[\frac{1}{R}e^{\varphi}+\frac{3\beta}{8R^{3}}e^{3\varphi}+\frac{2\alpha}{3R^{2}}e^{2\varphi}\right]. (29)

The closest approach distance r0r_{0} is calculated at φ=π/2,\varphi=\pi/2, and is given by

1r0=eπ/2R+β8R3e3π/2+α3R2eπ.\frac{1}{r_{0}}=\frac{e^{\pi/2}}{R}+\frac{\beta}{8R^{3}}e^{3\pi/2}+\frac{\alpha}{3R^{2}}e^{\pi}. (30)

We calculate the bending angle when φ=0\varphi=0, which is the bending angle measured for a long distance from the source. For this particular case we have

rR, A(r,φ=0)R,r\approx R,\text{ \ \ \ \ \ }A(r,\varphi=0)\approx-R, (31)

hence, the one - sided bending angle becomes

ϵ=ψ0=1R2.\epsilon=\psi_{0}=\frac{1}{R^{2}}. (32)

It is important to note that the bending arises due to the existence of the axion field that creates rotation in the universe. Since we have chosen q0=1q_{0}=1, its presence in the calculated bending angle is not apparent.

III.3 Lensing in the Twisting NUT Universe

The NUT metric is written as

ds2=f(r)[dt+2lcosθdφ2]2dr2f(r)(r2+l2)[dθ2+sin2θdφ2],ds^{2}=f(r)\left[dt+2lcos\theta d\varphi^{2}\right]^{2}-\frac{dr^{2}}{f(r)}-\left(r^{2}+l^{2}\right)\left[d\theta^{2}+sin^{2}\theta d\varphi^{2}\right], (33)

where f(r)=12(mr+l2)r2+l2.f(r)=1-\frac{2(mr+l^{2})}{r^{2}+l^{2}}.

This is another vacuum metric that generalizes Schwarzschild metric in analogy with the Kerr metric. The NUT parameter is specified by ll and the metric reduces to Schwarzschild for l=0l=0. For the physical interpretation of NUT metric, there are different views 12 ; 13 . We adapt the interpretation that ll corresponds to the twist of the vacuum space-time 12 . We arrive at this interpretation by consideration of a general class of Einstein-Maxwell solutions in the limit when the em field vanishes 13 . The space-time left as a result corresponds exactly to an isometric spacetime of the NUT solution.

The source-free em field gives, in an appropriate NP tetrad 12 ,

Ψ2=mpr6{r3+3m(p1)(r2m)r+2m3(p1)2p+i(p21)1/2[(3mr)r22m2(p1)p]}\Psi_{2}=\frac{-mp}{r^{6}}\left\{r^{3}+3m(p-1)(r-2m)r+2m^{3}\frac{(p-1)^{2}}{p}+i(p^{2}-1)^{1/2}\left[(3m-r)r^{2}-2m^{2}\frac{(p-1)}{p}\right]\right\} (34)

and the NUT parameter

l=±mp1p,l=\pm m\sqrt{p-\frac{1}{p}}, (35)

in which pp is the twist parameter left-over from the twisting spacetime. Using Eq.(6), we get

d2udφ2=2ul2h2+2u3l4h2(u+2l2u3)f(u)[u2+l2u42]df(u)du,\frac{d^{2}u}{d\varphi^{2}}=\frac{2ul^{2}}{h^{2}}+\frac{2u^{3}l^{4}}{h^{2}}-(u+2l^{2}u^{3})f(u)-\left[\frac{u^{2}+l^{2}u^{4}}{2}\right]\frac{df(u)}{du}, (36)

where f(u)=12u2(mu+l2)1+u2l2f(u)=1-\frac{2u^{2}(\frac{m}{u}+l^{2})}{1+u^{2}l^{2}} and hh is the constant measuring the angular momentum of the test particle.

Since u<<1u<<1, Eq.(36) simplifies to

d2udφ2+δ2u=3mu2+ρu3+5ml2u4+6l4u5,\frac{d^{2}u}{d\varphi^{2}}+\delta^{2}u=3mu^{2}+\rho u^{3}+5ml^{2}u^{4}+6l^{4}u^{5}, (37)

where δ2=12l2h2\delta^{2}=1-\frac{2l^{2}}{h^{2}} and ρ=2l4h2+2l2\rho=\frac{2l^{4}}{h^{2}}+2l^{2}. We have two different cases.

III.3.1 The case of 1>2l2h21>\frac{2l^{2}}{h^{2}} (small NUT parameter):

The first approximate solution, u=sin(δφ)R,u=\frac{sin(\delta\varphi)}{R}, is substituted back in Eq.(37) and its resulting solution for uu is obtained as

u(φ)=sin(δφ)R+124δ2R5\displaystyle u(\varphi)=\frac{sin(\delta\varphi)}{R}+\frac{1}{24\delta^{2}R^{5}} {(6l4sin(δφ)8l2mR)cos4(δφ)+cos2(δφ)[(3ρR2+27l4)sin(δφ)+48l2mR+24mR3]\displaystyle\left\{(-6l^{4}sin(\delta\varphi)-8l^{2}mR)cos^{4}(\delta\varphi)+cos^{2}(\delta\varphi)[(3\rho R^{2}+27l^{4})sin(\delta\varphi)+48l^{2}mR+24mR^{3}]\right. (38)
9δφ(ρR2+5l4)cos(δφ)+sin(δφ)(6ρR2+24l4)+24mR(l2+R2)},\displaystyle\left.-9\delta\varphi(\rho R^{2}+5l^{4})cos(\delta\varphi)+sin(\delta\varphi)(6\rho R^{2}+24l^{4})+24mR(l^{2}+R^{2})\right\},

so that Eq.(7) becomes

A(r,φ)\displaystyle A(r,\varphi) =r2{δcos(δφ)R+124δ2R5{6l4δcos5(δφ)4δcos3(δφ)sin(δφ)(6l4sin(δφ)8l2mR)+δcos3(δφ)(3ρR2+27l4)\displaystyle=-r^{2}\left\{\frac{\delta cos(\delta\varphi)}{R}+\frac{1}{24\delta^{2}R^{5}}\left\{-6l^{4}\delta cos^{5}(\delta\varphi)-4\delta cos^{3}(\delta\varphi)sin(\delta\varphi)(-6l^{4}sin(\delta\varphi)-8l^{2}mR)+\delta cos^{3}(\delta\varphi)(3\rho R^{2}+27l^{4})\right.\right. (39)
δsin(2δφ)[(3ρR2+27l4)sin(δφ)+48l2mR+24mR3]9δcos(δφ)(ρR2+5l4)+9δ2φsin(δφ)(ρR2+5l4)\displaystyle\left.\left.-\delta sin(2\delta\varphi)[(3\rho R^{2}+27l^{4})sin(\delta\varphi)+48l^{2}mR+24mR^{3}]-9\delta cos(\delta\varphi)(\rho R^{2}+5l^{4})+9\delta^{2}\varphi sin(\delta\varphi)(\rho R^{2}+5l^{4})\right.\right.
+δcos(δφ)(6ρR2+24l4)}}.\displaystyle\left.\left.+\delta cos(\delta\varphi)(6\rho R^{2}+24l^{4})\right\}\right\}.

The closest approach distance r0r_{0} is calculated at φ=π/2,\varphi=\pi/2, which is found to be

1r0=1R+mδ2R2+mρ24δ2R3+ml2δ2R4+l2δ2R5.\frac{1}{r_{0}}=\frac{1}{R}+\frac{m}{\delta^{2}R^{2}}+\frac{m\rho^{2}}{4\delta^{2}R^{3}}+\frac{ml^{2}}{\delta^{2}R^{4}}+\frac{l^{2}}{\delta^{2}R^{5}}. (40)

We calculate the bending angle when φ=0\varphi=0 which is the bending angle named as the small angle ψ0\psi_{0} and R>>1R>>1. For this particular case we found that

rδ2R22m, A(r,φ=0)r2δR,r\approx\frac{\delta^{2}R^{2}}{2m},\text{ \ \ \ \ \ }A(r,\varphi=0)\approx-r^{2}\frac{\delta}{R}, (41)

and the one-sided bending angle is

ϵ=ψ02mδ3R{14m2δ2R2+8m2l2δ4R4}1/22mδ3R{12m2δ2R2+4m2l2δ4R4}+𝒪(m5l4R9).\epsilon=\psi_{0}\simeq\frac{2m}{\delta^{3}R}\left\{1-\frac{4m^{2}}{\delta^{2}R^{2}}+\frac{8m^{2}l^{2}}{\delta^{4}R^{4}}\right\}^{1/2}\simeq\frac{2m}{\delta^{3}R}\left\{1-\frac{2m^{2}}{\delta^{2}R^{2}}+\frac{4m^{2}l^{2}}{\delta^{4}R^{4}}\right\}+\mathcal{O}\left(\frac{m^{5}l^{4}}{R^{9}}\right). (42)

This particular case implies that the effect of NUT parameter is very small. Hence, the mass term dominates all the others and the obtained bending angle becomes similar to the Schwarzchild case, which is convex lensing.

III.3.2 The case of 1<2l2h21<\frac{2l^{2}}{h^{2}} (large NUT parameter):

In this case Eq.(36) takes the from

d2udφ2ζ2u=3mu2+ρu3+5ml2u4+6l4u5,\frac{d^{2}u}{d\varphi^{2}}-\zeta^{2}u=3mu^{2}+\rho u^{3}+5ml^{2}u^{4}+6l^{4}u^{5}, (43)

where ζ2=2l2h21\zeta^{2}=\frac{2l^{2}}{h^{2}}-1.

The first approximate solution, u=eζφR,u=\frac{e^{\zeta\varphi}}{R}, is substituted back in Eq.(43) and its resulting solution for uu is obtained as

u(φ)=eζφRl4ζ2R5e5ζφ5ml2ζ2R4e4ζφρζ2R3e3ζφ3mζ2R2e2ζφ,u(\varphi)=\frac{e^{\zeta\varphi}}{R}-\frac{l^{4}}{\zeta^{2}R^{5}}e^{5\zeta\varphi}-\frac{5ml^{2}}{\zeta^{2}R^{4}}e^{4\zeta\varphi}-\frac{\rho}{\zeta^{2}R^{3}}e^{3\zeta\varphi}-\frac{3m}{\zeta^{2}R^{2}}e^{2\zeta\varphi}, (44)

and Eq.(7) becomes

A(r,φ)=r2[ζeζφR5l4ζR5e5ζφ20ml2ζR4e4ζφ3ρζR3e3ζφ6mζR2e2ζφ].A(r,\varphi)=-r^{2}\left[\frac{\zeta e^{\zeta\varphi}}{R}-\frac{5l^{4}}{\zeta R^{5}}e^{5\zeta\varphi}-\frac{20ml^{2}}{\zeta R^{4}}e^{4\zeta\varphi}-\frac{3\rho}{\zeta R^{3}}e^{3\zeta\varphi}-\frac{6m}{\zeta R^{2}}e^{2\zeta\varphi}\right]. (45)

The closest approach distance r0r_{0} is calculated at φ=π/2,\varphi=\pi/2, which is found to be

1r0=eζπ/2Rl4ζ2R5e5ζπ/25ml2ζ2R4e2ζπρζ2R3e3ζπ/23mζ2R2eζπ.\frac{1}{r_{0}}=\frac{e^{\zeta\pi/2}}{R}-\frac{l^{4}}{\zeta^{2}R^{5}}e^{5\zeta\pi/2}-\frac{5ml^{2}}{\zeta^{2}R^{4}}e^{2\zeta\pi}-\frac{\rho}{\zeta^{2}R^{3}}e^{3\zeta\pi/2}-\frac{3m}{\zeta^{2}R^{2}}e^{\zeta\pi}. (46)

We calculate the bending angle when φ=0\varphi=0 , which yields for this particular case,

rR, A(r,φ=0)ζR,r\approx R,\text{ \ \ \ \ \ }A(r,\varphi=0)\approx-\zeta R, (47)

so that the one-sided bending angle is

ϵ=ψ01ζR{12mR2+2l2R4}1/21ζR{1mR2+l2R4}+𝒪(l4R9).\epsilon=\psi_{0}\simeq\frac{1}{\zeta R}\left\{1-\frac{2m}{R^{2}}+\frac{2l^{2}}{R^{4}}\right\}^{1/2}\simeq\frac{1}{\zeta R}\left\{1-\frac{m}{R^{2}}+\frac{l^{2}}{R^{4}}\right\}+\mathcal{O}\left(\frac{l^{4}}{R^{9}}\right). (48)

This is the case where the NUT parameter dominates the mass term. The calculated bending angle indicates that there is a drastic decrease in the bending angle. The reason of this is the concave role played by the NUT parameter on the geometry.

IV Results and Discussions

The gravitational lensing in rotating and twisting universes are studied. The considered model of universes are assumed to develop whenever the gravitational waves coupled with em and axion waves interact nonlinearly and give rise to high curvature zone in the fabric of space-time. Hence, high curvature zone space-time structures constitute another mechanism that cause the light to bend. Moreover, the rotation and twisting parameters are the key parameters that take part in gravitational lensing. Let us add that in this study we did not consider the case of extreme curvature zones, as it is in the inner horizon of a black hole where the light bending creates photon spheres.

A striking result is obtained in the NUT space-time. The solution is dependent on two different behaviour of the NUT parameter; thus, we investiged two different bending angles based on the mathematically bounded NUT parameter to angular momentum ratio. We noticed that when the NUT parameter is small, the bending angle comes out to be almost the same as the one for the Schwarzchild case, which is convex lensing. On the other hand, the large NUT parameter generates a diverging (concave) lens effect. We recall that 90%90\% of the universe consists of voids due to dark energy. During the first ten billion years following the Big Bang, dark matter used to dominate and as a result bending of light was convex type. Later on, the mysterious dark energy took over and we have in the present era an accelerating expansion of universe which implies concave lensing from the hyperbolic nature of the space-time. Note that in this description we exclude the local sources that might create regional convex lensing. Our conclusion is that concave nature of the light bending becomes inevitable in an expanding universe of voids.

To sum up, we studied the gravitational lens effect of three different non-singular cosmological models that results as a nonlinear interaction of gravitational waves coupled with em and axion waves. Calculations have revealed that not only the gravitational field of massive objects is effective on the bending angle of light, at the same time, the high curvature zone of the space-time is also effective. It remains to be seen whether deflection of light by voids in absence of massive objects can be considered as an indirect evidence of the mysterious dark energy. Finally let us add that whatever has been done//said for twisting//rotating vacua of space-time are valid also for the mysterious dark matter since its gravitational effect cannot be suppressed.

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