Wave Effect in Gravitational Lensing by the Ellis Wormhole
aa r X i v : . [ g r- q c ] M a r Yukawa Institute for Theoretical PhysicsKyoto UniversityDepartment of PhysicsRikkyo University
YITP-13-15RUP-13-3
Wave Effect in Gravitational Lensing by the Ellis Wormhole
Chul-Moon Yoo, ∗ Tomohiro Harada, and Naoki Tsukamoto Yukawa Institute for Theoretical Physics,Kyoto University Kyoto 606-8502, Japan Department of Physics, Rikkyo University, Tokyo 171-8501, Japan
We propose the use of modulated spectra of astronomical sources due to gravita-tional lensing to probe Ellis wormholes. The modulation factor due to gravitationallensing by the Ellis wormhole is calculated. Within the geometrical optics approx-imation, the normal point mass lens and the Ellis wormhole are indistinguishableunless we know the source’s unlensed luminosity. This degeneracy is resolved withthe significant wave effect in the low frequency domain if we take the deviation fromthe geometrical optics into account. We can roughly estimate the upper bound forthe number density of Ellis wormholes as n . − AU − with throat radius a ∼ ∗ Electronic address: [email protected]
I. INTRODUCTION
In a variety of cosmological models based on fundamental theory, exotic astrophysicalobjects which have not been observed are often predicted. Conversely, an observationalevidence for an exotic object would stimulate creative theoretical discussions. Probing theseexotic objects and detecting them will give us significant progress of research in fundamentalphysics. Even if we cannot detect it, giving a constraint on the abundance of the exoticobjects is one of powerful means of investigating the nature of our universe.Generally, the interaction between such unobserved exotic objects and well known mattersis very weak or not well established. Thus, only the gravitational interaction would causereliable observational phenomena. One of the most direct measurements of gravitationaleffects of an exotic object is gravitational lensing. For instance, massive compact haloobjects are probed by using micro-lensing[1–3]. Cosmic strings are also targets for probingby using gravitational lensing phenomena[4–12]. In this paper, we propose a way to probeEllis wormholes[13] by using lensed spectra of astronomical sources.The Ellis wormhole was first introduced by Ellis as a spherically symmetric solution ofEinstein equations with a ghost massless scalar field. The dynamical stability of the Elliswormhole is discussed in Ref. [14] and the possible source to support the Ellis geometrywas proposed in Ref. [15]. Gravitational lensing by the Ellis wormhole was studied inRefs. [16, 17] and recently revisited by several authors[18, 19]. So far, it has been suggestedthat Ellis wormholes can be probed by using light curves of gamma-ray bursts [20], micro-lensing [21–23] (see also Refs. [24, 25]) and imaging observations [26, 27], while our proposalis the use of spectroscopic observations to probe Ellis wormholes.In order to fully investigate the lensed spectrum of a point source, the wave effect ingravitational lensing must be taken into account. The wave effect for the point mass lensis discussed in Refs. [28, 29]. Wave effects in gravitational lensing by the rotating massiveobject[30], binary system[31], singular isothermal sphere[32] and the cosmic string[33, 34]have been considered. In Sec. II, we calculate the amplification factor of gravitational lensingby the Ellis wormhole taking the wave effect into account. The geometrical optics limit isanalytically presented in Sec. III. The difference in the amplification factor between thepoint mass lens and the Ellis wormhole lens is discussed in Sec. IV based on observables.In Sec. V, possible observations to probe Ellis wormholes are listed. Sec. VI is devoted to asummary.In this paper, we use the geometrized units in which the speed of light and Newton’sgravitational constant are both unity.
II. A DERIVATION OF THE LENSED WAVE FORM
The line element in the Ellis wormhole spacetime can be written by the following isotropicform: d s = − d t + (cid:18) a R (cid:19) (cid:0) d R + R dΩ (cid:1) , (1)where R = a corresponds to the throat and we simply call a the throat radius in this paper. Assuming the thin lens approximation is valid, we consider the wormhole lens systemshown in Fig. 1. We use the position vector ~X = ( X, Y, Z ) in the flat space. Then, the
FIG. 1: Lens system with thin lens approximation. S, L, and O represent the source, lens, andobserver positions, respectively. The path SAB is a ray trajectory which is specified with the vector ~ξ on the lens plane Σ A . B ′ is the intersection of the line AO and the plane Σ B . ~ξ ′ is the positionvector of the point B on the plane Σ B . coordinate R is given by R = | ~X − ~X L | , where | ~X L | is the lens position. We set Z -axis asthe perpendicular direction to the lens plane and the source plane. D S , D L , and D LS denotethe distances from the observer plane to the source plane, from the observer plane to thelens plane, and from the lens plane to the source plane, respectively.In the geometrical optics limit, we consider light rays emanated from the source. Thevector ~ξ on the lens plane Σ A in Fig. 1 specifies the light ray which is deflected once at ~X = ~X L + ~ξ . Since ξ := | ~ξ | can be regarded as the closest approach of the light ray, as isshown in Ref. [16], the deflection angle α is given by α ( ξ ) = π (cid:18) aξ (cid:19) + O (cid:18) aξ (cid:19) . (2)As a result, the Einstein radius ξ for the Ellis wormhole is given by ξ = (cid:18) πa D (cid:19) / , (3) Since the throat surface area is given by 16 πa , our definition of the throat radius is half the areal radiusof the throat. where D = 4 D L D LS D S . (4)Since we are interested in the wave effect, that is, the deviation from the geometricaloptics limit, we need to treat the wave equation rather than light rays. Neglecting thepolarization effect, we consider the scalar wave equation with the frequency ω . The waveequation for the monochromatic wave e iωt φ ( ~X ) is given by ω φ + (cid:18) a R (cid:19) − ∂ i (cid:20)(cid:18) a R (cid:19) δ ij ∂ j φ (cid:21) = − πA δ ( ~X − ~X S ) , (5)where ~X S is the position vector of the point source and A in the source term is a constantwhich specifies the amplitude. Without the wormhole, we obtain the wave form ¯ φ O at theobserver O as follows:¯ φ O = A p D + η exp (cid:20) iω q D + η (cid:21) = AD S exp " iωD S η D + O (cid:18) ηD S (cid:19) !! , (6)where η = | ~η | and ~η = ( X S − X L , Y S − Y L ,
0) and we consider the case η ≪ D S in this paper.Our assumptions to calculate the wave form at O are summarized as follows(see Ref. [29]):(a) The geometrical optics approximation is valid between the source plane and the planeΣ B in Fig. 1.(b) Thin lens approximation is valid and a ray from the source is deflected once on thelens plane Σ A .(c) Assuming D S ∼ D L ∼ D LS ∼ D , we use a non-dimensional parameter ǫ defined by ǫ := ξ /D , which gives the typical scale of the deflection angle. Then, we assume1 / ( ωD ) ≪ ǫ ≪ η/D = O ( ǫ ).(d) On the plane Σ B , the gravitational potential of the lens object is negligible and δD/D = O ( ǫ ), where δD is the distance between the planes Σ A and Σ B .On the assumptions made above, we calculate the wave form on the plane Σ B up to theleading order for the amplitude and next leading terms for the phase part. Applying theKirchhoff integral theorem between the plane Σ B and the observer, we calculate the approx-imate wave form at O.The vector ~ξ specifies the light ray which is deflected once on the plane Σ A at ~X = ~X L + ~ξ and reaches the plane Σ B . The deflection angle is fixed by ~ξ and the background geometry.The point A in Fig. 1 denotes the deflected point. We label the intersection of the deflectedlight ray and the plane Σ B as B, while B ′ in the Fig. 1 denotes the intersection of the line AOand the plane Σ B . As will be mentioned at the end of this section, the dominant contributionto the wave form at O comes from rays which satisfy ξ ∼ ǫD , where ξ = | ~ξ | . Therefore weconsider ~ξ/D as O ( ǫ ) hereafter.First, we consider the following ansatz for φ in the region between the source plane andthe plane Σ B : φ = f ( ~X ) e iS ( ~X ) . (7)On the plane Σ B , the amplitude f ( ~X ) is given by f ( ~X ) (cid:12)(cid:12)(cid:12) Σ B = AD LS (1 + O ( ǫ )) . (8)In the geometrical optics approximation, the phase S ( ~X ) satisfies the eikonal equation givenby δ ij ∂ i S∂ j S = ω (cid:18) a R (cid:19) . (9)At the point B, the phase based on the source position S is given by the following integral S | B = Z BS d x i d l ∂ i Sdl, (10)where we have introduced the optical path length l defined as δ ij d x i d l d x j d l = 1 . (11)Since, in the geometrical optics approximation, we find ∂ j S = ω (cid:18) a R (cid:19) δ ij d x i d l , (12)the integral (10) is given by S | B = ω Z BS dl + ωa Z BS R dl. (13)After the calculations explicitly shown in Appendix A, we finally obtain the following ex-pression: S | B ≃ ω D S (cid:18) η D (cid:19) + D L D S D LS ~ξD L − ~ηD S ! − r + πa ξ . (14)This expression for the phase and Eq. (8) for the amplitude can be used for any value of ~ξ ,that is, we have obtained an approximate wave form on the plane Σ B .Applying the Kirchhoff integral theorem[35] and neglecting the contribution from theinfinity, we express the wave form φ O at O by the following integral: φ O = − π Z Σ B d ξ (cid:26) φ B ∂∂Z (cid:18) e iωr r (cid:19) − e iωr r ∂φ B ∂Z (cid:27) , (15)where φ B is the waveform at B. Since we are interested in only the leading order of theamplitude, we obtain φ O ≃ − iωA πD L D LS exp (cid:20) iωD S (cid:18) η D (cid:19)(cid:21) Z d ξ exp iω + D L D S D LS ~ξD L − ~ηD S ! + πa ξ , (16)where we have used the following approximations: ∂∂Z (cid:18) e iωr r (cid:19) ≃ iωr e iωr ≃ iωD L e iωr , (17) ∂φ B ∂Z ≃ − iωAD LS e i S | B . (18)Defining the amplification factor F by F := φ O / ¯ φ O , we obtain F ≃ ωdπi Z d x exp (cid:20) iωd (cid:26) ( ~x − ~y ) + 2 x (cid:27)(cid:21) , (19)where d = ξ D S D LS D L = 2 ξ D = 2 ξ ǫ, (20) ~x = ~ξξ , (21) ~y = ~ηD L ξ D S . (22) d gives the optical path difference between the lensed trajectory and the unlensed one in thegeometrical optics limit for ~η = 0.Introducing a polar coordinate, we rewrite this integral as F ≃ ωdπi e iωdy Z ∞ d xx exp (cid:20) iωd (cid:18) x + 2 x (cid:19)(cid:21) Z π d ϕ exp [ − iωdxy cos ϕ ] (23)= − iωd e iωdy Z ∞ d xx exp (cid:20) iωd (cid:18) x + 2 x (cid:19)(cid:21) J (2 ωdxy ) . (24)The integrand is divergent at the infinity on the real axis. This is caused by our approx-imation associated with ǫ , and not real. If we write down the integrand in a precise formwithout any approximation, we do not have any divergence. Actually, in the precise form,the contribution from the integral in the region x ≫ x ≪ x ≪ /ǫ , we can obtain a reliable result by neglecting the contribution from theregion x ≫ FIG. 2: The path of the integral (24) taken in the numerical integration.
III. GEOMETRICAL OPTICS APPROXIMATION
In this section, we derive an approximate form of the amplification factor F . In theexpression (23), we apply the stationary phase approximation to the integral with respectto x . Then we obtain F ≃ r ωdπi Z π d ϕ x p /x exp [ iωdh ( x )] , (25)where h ( x ) = x − xy cos ϕ + 2 x (26)and x = x ( ϕ ) > h ′ ( x ) = 0 ⇔ x − x y cos ϕ − . (27)Note that Eq. (27) has only one positive root as a function of ϕ .In Eq. (25), we again perform the stationary phase approximation in the integral withrespect to ϕ , and we obtain F ≃ F geo = x p ( x + 2)( x −
1) exp (cid:20) iωd − x + 4 x + (cid:21) + x − p ( x − + 2)(1 − x − ) exp (cid:20) iωd − x − + 4 x − − iπ (cid:21)! , (28)where x ± satisfies x ± ∓ x ± y − . (29)Note that 1 < x + and 0 < x − <
1. If we define µ ± and θ ± as µ ± = x ± ( x ± + 2)( x ± − , (30) θ ± = ωd − x ± + 4 x ± − π ± π , (31) F geo can be expressed as F geo = X ± p | µ ± | e iθ ± . (32) µ ± is the magnification factor for each image in the geometrical optics approximation.As an observable, we focus on | F | in this paper. In the geometrical optics approximation,we obtain | F geo | = | µ + | + | µ − | + 2 p | µ + µ − | sin(2 ωdτ ( y )) (33)with τ ( y ) being the following: τ ( y ) := θ − − θ + + π/ ωd , (34)where note that the definition of τ can be written in terms of x ± , which is a function of y . | F | and | F geo | are depicted as functions of ω for each value of y in Fig. 3. Ω @ (cid:144) T D È F y = È F geo 2 È F Ω @ (cid:144) T D È F y = È F geo 2 È F Ω @ (cid:144) T D È F y = (cid:144) È F geo 2 È F Ω @ (cid:144) T D È F y = (cid:144) È F geo 2 È F FIG. 3: | F | and | F geo | for the wormhole lens, where T = d τ . IV. COMPARISON WITH THE POINT MASS LENS BASED ONOBSERVABLES
In this paper, we assume the following situations for the observation: • We can observe the spectrum of a source. • The unlensed spectrum shape is well known.Note that, in our analysis, knowledge about the luminosity is not necessary. The amplifica-tion factor for the point mass lens is summarized in Appendix B. For both point mass andwormhole cases, in the geometrical optics approximation, we obtain the form of Eq. (33) orequivalently Eq. (B3).In the frequency region where the geometrical optics approximation is valid, there arebasically three observables which characterize the form of the amplification factor. The firstis the frequency ω , the second is the period of the oscillation of the spectrum as a functionof ω , and the third is the ratio κ between the amplitude of the oscillation and the meanvalue. The period of the oscillation of the spectrum makes T := τ ( y ) d an observable. κ isgiven by κ = 2 p | µ + µ − || µ + | + | µ − | (35)from Eq. (33) and plotted as a function of y as is shown in Fig. 4. Since κ is observable, y y Κ point masswormhole FIG. 4: κ as a function of y for the Ellis wormhole case and the point mass lens case. and hence τ ( y ) can be determined if we have enough accuracy of the observation. As shownin Fig. 5, τ ( y ) is a monotonically increasing function of y and close to 2 y in the region y < T = τ ( y ) d we can obtain the value of d .The situation for the point mass lens case is the same as for the Ellis wormhole case.That is, the three observables ω , T , and κ can be regarded as gravitational lensing by apoint mass as well as a wormhole. This fact indicates that we cannot distinguish which is thelens object only by using these three observables in the geometrical optics approximation.This degeneracy is resolved in the small frequency region in which the wave effect becomessignificant as is explicitly shown in Fig. 6.0 y Τ y point masswormhole FIG. 5: τ ( y ) for the Ellis wormhole case and the point mass lens case. The two cases are almostindistinguishable from each other in the region depicted here. Ω @ (cid:144) T D È F (cid:144) Μ t o t Κ= (cid:144) È F p 2 (cid:144) Μ totp È F (cid:144) Μ tot Ω @ (cid:144) T D È F (cid:144) Μ t o t Κ= (cid:144) È F p 2 (cid:144) Μ totp È F (cid:144) Μ tot FIG. 6: Comparison between | F | and | F p | . To simply see this resolution of the degeneracy, we consider the small frequency limit, i.e., ω →
0. In this limit, we have F → ω → | F | lim ω →∞ < | F | > = 1 /µ tot := 1 / ( | µ + | + | µ − | ) , (36)where the bracket < > denotes the average through several periods. We depict 1 /µ tot asa function of y for both the wormhole case and the point mass lens case in Fig. 7. In bothcases, 1 /µ tot approaches to 0 and 1 in the limits y → y → ∞ , respectively. While1 /µ tot < y for the point mass lens, 1 /µ tot can exceed 1 for thewormhole case due to the demagnification effect originated from the negative mass density1 y (cid:144) Μ t o t point masswormhole FIG. 7: 1 /µ tot surrounding the Ellis wormhole. The behaviours of 1 /µ tot are totally different from eachother. This fact shows that they are, in principle, distinguishable. V. OBSERVATIONAL CONSTRAINT
One of the possible sources is gamma-ray bursts, which have been proposed to be usedfor probing small mass primordial black holes[36–39] and low tension cosmic strings[34].Recently, the femto-lensing effects caused by compact objects were searched by using gamma-ray bursts with known redshifts detected by the Fermi Gamma-ray Burst Monitor[40]. Fromnon-detection of the femto-lensing event, a constraint on the number density of compactobjects have been obtained. For a fixed value of the mass M of each compact object,the constraint can be translated into a constraint on Ω CO , where Ω CO is the average energydensity of compact objects in the unit of the critical density ρ cr . Then, the abundance of darkcompact objects is constrained as Ω CO < .
15 at the 95% confidence level for M ∼ × g.Since the number density n of the compact objects is given by n = Ω CO ρ cr M ∼ × − AU − (cid:18) Ω CO . (cid:19) (cid:18) M × g (cid:19) − (cid:18) ρ cr − g / cm (cid:19) , (37)we obtain the constraint for the number dnsity of compact objects with M ∼ × g as n < × − AU − . Then, we can expect a similar constraint on the number density of Elliswormholes with the throat radius which gives the same value of d as that for the compactobject. The mass of 3 × g gives d ∼ × − cm from Eq. (B2) and, from Eq. (20), the2corresponding throat radius a is given by a ∼ . (cid:18) d × − cm (cid:19) / (cid:18) D cm (cid:19) / . (38)Therefore, the number density of Ellis wormholes with a ∼ n . − AU − .Note that this constraint comes from the wave form in the geometrical optics approximationand hence we do not distinguish between the point mass lenses and Ellis wormholes.Another possible observation to probe Ellis wormholes is the observation of gravitationalwaves from compact object binaries. The unlensed wave form of the gravitational waves froma compact object binary is well known. From the chirp signal in the inspiral phase, we canobtain the spectrum of the gravitational waves. In order to distinguish the Ellis wormholefrom a point mass lens, we need to observe not only the typical interference pattern but alsothe wave effect in the lensed spectrum. Hence, the spectrum in d ∼ λ := 2 π/ω is necessaryto probe the Ellis wormhole. Assuming d ∼ λ , the typical throat radius of Ellis wormholeswhich can be probed by using gravitational waves is estimated as follows: a = d / (cid:18) Dπ (cid:19) / ∼ λ / (cid:18) Dπ (cid:19) / ∼ × cm (cid:18) λ cm (cid:19) / (cid:18) D cm (cid:19) / . (39)The same estimate is applicable for galactic sources of electro-magnetic waves. We obtain a ∼ cm for galactic radio sources with λ ∼ a ∼
1m for galactic optical or infra-redsources and a ∼ y dependence of the phase in the amplification factor (33). Since τ ( y )is roughly approximated by 2 y , the period δy for one cycle is given by δy ∼ π/ ( ωd ). Thecorresponding length scale δη on the source plane is given by δη = δy D S D L ξ ∼ πω r D LS D S dD L ∼ r λD LS D S D L ∼ × cm (cid:18) λ (cid:19) / (cid:18) D LS D S /D L (cid:19) / , (40)where we have assumed d ∼ λ . If the source radius is larger than δη , the interferencepattern will be smeared out. Observation of compact galactic sources such as pulsars andwhite dwarfs might be useful to probe not only dark compact objects but also exotic compactobjects such as the Ellis wormhole. VI. SUMMARY
In this paper, we have proposed the probe of Ellis wormholes by using spectroscopicobservations. We have assumed that the spectrum of the target source can be measured in3enough accuracy and the spectrum shape is well known without lensing, but the luminosityis not necessarily observable. Then, we have discussed the distinguishability of the lensedspectrum from the case of the point mass lens.We have derived the wave form after the scattering by the Ellis wormhole including thewave effect in the low frequency domain. The geometrical optics limit of the wave formhas been also analytically derived. Then, we have found that the Ellis wormhole cannot bedistinguished from the point mass lens by using only the high frequency domain in whichthe geometrical optics approximation is valid. We have also found that this degeneracy isresolved in the low frequency domain in which the wave effect is significant. Possible obser-vational constraints are also discussed and we estimated the upper bound for the numberdensity of Ellis wormholes as n . − AU − with throat radius a ∼ Acknowledgements
CY is supported by a Grant-in-Aid through the Japan Society for the Promotion ofScience (JSPS). The work of NT was supported in part by Rikkyo University Special Fund forResearch. TH was supported by the Grant-in-Aid for Young Scientists (B) (No. 21740190)and the Grant-in-Aid for Challenging Exploratory Research (No. 23654082) for ScientificResearch Fund of the Ministry of Education, Culture, Sports, Science and Technology, Japan.
Appendix A: Derivation of Eq. (14)
The first term in Eq. (13) can be evaluated as follows. Z BS dl = |−→ SA | + |−→ AB | = |−→ SA | + |−→ OA − −→ OB | = |−→ SA | + q |−→ OA | + |−→ OB | − −→ OA · −→ OB . (A1) −→ OB and −−→ OB ′ can be written as −→ OB = "(cid:18) − δDD L (cid:19) −→ OL + ~η + (cid:16) ~ξ − ~η (cid:17) D LS + δDD LS − αδD ~ξξ (1 + O ( ǫ ))4= "(cid:18) − δDD L (cid:19) −→ OL + ~ξ − α ( ξ ) δD ~ξξ + (cid:16) ~ξ − ~η (cid:17) δDD LS (1 + O ( ǫ )) , (A2) −−→ OB ′ = (cid:18) − δDD L (cid:19) −→ OA . (A3)From these expressions, we can find |−→ OB | = |−−→ OB ′ | (cid:0) O ( ǫ ) (cid:1) , (A4) −→ OA · −→ OB = −→ OA · −−→ OB ′ (cid:0) O ( ǫ ) (cid:1) . (A5)Therefore we obtain Z BS dl = (cid:16) |−→ SA | + |−→ OA | − |−−→ OB ′ | (cid:17) (cid:0) O ( ǫ ) (cid:1) . (A6)Since we find |−→ SA | = q | ~ξ − ~η | + D = D LS | ~ξ − ~η | D + O ( ǫ ) ! , (A7) |−→ AO | = D L (cid:18) ξ D + O ( ǫ ) (cid:19) , (A8)we obtain the following expression: Z BS dl = D S (cid:18) η D (cid:19) + D L D S D LS ~ξD L − ~ηD S ! − r (cid:0) O ( ǫ ) (cid:1) , (A9)where r = |−−→ OB ′ | .In order to evaluate the second term in Eq. (13), we first consider the integral between Sand A. Letting P be a point on the segment SA, we obtain |−→ LP | = | ~ξ + −→ AP | = (cid:12)(cid:12)(cid:12)(cid:12) ~ξ + (cid:18) − ll SA (cid:19) −→ AS (cid:12)(cid:12)(cid:12)(cid:12) = ξ + ( l SA − l ) + 2 (cid:18) − ll SA (cid:19) ~ξ · −→ AS , (A10)where l SA = |−→ SA | and l = |−→ SP | . Since ~ξ · −→ AS = − ~ξ · ( ~ξ − ~η ), we obtain the following expression: |−→ LP | = ξ + ( l SA − l ) − (cid:18) − ll SA (cid:19) ~ξ · ( ~ξ − ~η ) . (A11)Substituting the above expression of |−→ LP | into R of the second integral in Eq. (13) withthe integral region being from S to A, we obtain Z AS R dl = Z l SA dlξ + ( l SA − l ) − − ll SA ) ~ξ · ( ~ξ − ~η ) . (A12)5Then the integral (A12) can be performed and evaluated as Z l SA dl ( l SA − l ) + ξ − − ll SA ) ~ξ · ( ~ξ − ~η ) = l SA ξD LS " arctan − l + ~ξ · ( ~ξ − ~η ) + ll SA ξD LS ! l SA = π ξ − ~ξ · ~ηξ D LS ! (cid:0) O ( ǫ ) (cid:1) . (A13)The contribution from the integral between A and B can be also evaluated by the similarintegral. Finally, we obtain the expression (14). Appendix B: Point Mass Lens
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