Processing math: 100%

Shadow and weak gravitational lensing for Ellis-Bronnikov wormhole

Cited by

1. Han, X., Yang, W., Li, F. FPGA implementation and measurement of cusp-flattop hybrid filter shaping method for nuclear instruments[J]. Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2024. doi: 10.1016/j.nima.2024.169572
2. Zhou, H., Xu, F., Xie, Y. et al. Study on gamma spectrum measurement of high-dose-rate radiation[J]. Journal of Physics: Conference Series, 2024, 2770(1): 012015. doi: 10.1088/1742-6596/2770/1/012015
3. Wang, M., Zhou, J., Wang, H. et al. Pile-up Pulse Recognition Method Based on Ballistic Deficit Shape Feature | [基于弹道亏损形状特征的堆积脉冲识别方法][J]. Yuanzineng Kexue Jishu/Atomic Energy Science and Technology, 2024, 58(1): 231-238. doi: 10.7538/yzk.2023.youxian.0167
4. Zhou, H., Xu, F., Wu, F. et al. Study on Pulse Polarity Self-Adapting of Digital MCA[J]. 2023. doi: 10.1145/3654446.3654457
5. Tang, L., Li, Y., Tang, Y. et al. Application of an LSTM model based on deep learning through X-ray fluorescence spectroscopy | [基于深度学习的 LSTM 模型在 X 荧光光谱中的应用][J]. He Jishu/Nuclear Techniques, 2023, 46(7): 070502. doi: 10.11889/j.0253-3219.2023.hjs.46.070502
6. Kumar Paul, R., Das, A., Dhara, P. et al. Implementation of FPGA based real-time digital DAQ for high resolution, and high count rate nuclear spectroscopy application[J]. Journal of Instrumentation, 2023, 18(7): P07042. doi: 10.1088/1748-0221/18/07/P07042
7. Ma, X.-K., Huang, H.-Q., Huang, B.-R. et al. X-ray spectra correction based on deep learning CNN-LSTM model[J]. Measurement: Journal of the International Measurement Confederation, 2022. doi: 10.1016/j.measurement.2022.111510
8. Wang, Q., Zhang, X., Meng, X. et al. Multi-channel analyzer based on a novel pulse fitting analysis method[J]. Nuclear Engineering and Technology, 2022, 54(6): 2023-2030. doi: 10.1016/j.net.2021.12.019
9. Xiao-feng, Y., Hong-Quan, H., Guo-Qiang, Z. et al. Pulse Pile-up Correction by Particle Swarm Optimization with Double-layer Parameter Identification Model in X-ray Spectroscopy[J]. Journal of Signal Processing Systems, 2022, 94(4): 377-386. doi: 10.1007/s11265-021-01698-4
10. Hao, J., Li, F., Wang, Q. et al. Quantitative analysis of trace elements of silver disturbed by pulse pile up based on energy dispersive X-ray fluorescence (EDXRF) technique[J]. Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2021. doi: 10.1016/j.nima.2021.165672
11. Tang, L., Zhang, J., Shi, K. et al. Application of an improved seeds local averaging algorithm in x-ray spectrum[J]. Mathematical Problems in Engineering, 2021. doi: 10.1155/2021/5545818
12. Wang, X., Li, Z.H., Liu, Z. et al. An effective digital pulse processing method for pile-up pulses in decay studies of short-lived nuclei[J]. Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2020. doi: 10.1016/j.nima.2020.164068
13. Wang, M., Hong, X., Zhou, J.-B. et al. Rising time restoration for nuclear pulse using a mathematic model[J]. Applied Radiation and Isotopes, 2018. doi: 10.1016/j.apradiso.2018.01.018
14. Hong, X., Zhou, J., Ni, S. et al. Counting-loss correction for X-ray spectroscopy using unit impulse pulse shaping[J]. Journal of Synchrotron Radiation, 2018, 25(2): 505-513. doi: 10.1107/S1600577518000322
15. Hong, X., Zhou, J.-B., Zhao, X. et al. Digital on-line uranium concentration determination system design[J]. Hedianzixue Yu Tance Jishu/Nuclear Electronics and Detection Technology, 2016, 36(10): 1004-1007.
16. Hong, X., Ni, S.-J., Zhou, J.-B. et al. Study on the relationship between the shaping parameters of trapezoidal pulse shaping algorithm and the trapezoidal pulse shape[J]. Hedianzixue Yu Tance Jishu/Nuclear Electronics and Detection Technology, 2016, 36(2): 150-153 and 158.

Figures(14) / Tables(1)

Get Citation
Mirzabek Alloqulov, Farruh Atamurotov, Ahmadjon Abdujabbarov, Bobomurat Ahmedov and Vokhid Khamidov. Shadow and weak gravitational lensing for Ellis-Bronnikov wormhole[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad1677
Mirzabek Alloqulov, Farruh Atamurotov, Ahmadjon Abdujabbarov, Bobomurat Ahmedov and Vokhid Khamidov. Shadow and weak gravitational lensing for Ellis-Bronnikov wormhole[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad1677 shu
Milestone
Received: 2023-10-31
Article Metric

Article Views(1076)
PDF Downloads(20)
Cited by(16)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Shadow and weak gravitational lensing for Ellis-Bronnikov wormhole

  • 1. New Uzbekistan University, Movarounnahr street 1, Tashkent 100000, Uzbekistan
  • 2. Institute of Fundamental and Applied Research, National Research University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan
  • 3. Ulugh Beg Astronomical Institute, Astronomy St 33, Tashkent 100052, Uzbekistan
  • 4. Central Asian University, Milliy Bog’ Street 264, Tashkent 111221, Uzbekistan
  • 5. University of Public Safety of the Republic of Uzbekistan, Tashkent Region 100109, Uzbekistan
  • 6. Institute of Theoretical Physics, National University of Uzbekistan, Tashkent 100174, Uzbekistan
  • 7. University of Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent 100149, Uzbekistan
  • 8. Tashkent University of Information Technologies named after Muhammad al Khwarizmi, Amir Temur 108, Tashkent 100014, Uzbekistan

Abstract: In this study, we investigated the gravitational weak lensing and shadow of the Ellis-Bronnikov wormhole. First, we studied the photon motion in a plasma medium and a wormhole shadow. It was shown that the radius of the photon sphere of the Ellis-Bronnikov wormhole and the size of the wormhole shadow become larger under the influence of the parameter a. The upper limit of the parameter a in the Ellis-Bronnikov wormhole spacetime was obtained. Second, we investigated the weak gravitational lensing for the Ellis-Bronnikov wormhole and calculated the deflection angle for uniform and non uniform plasma cases. The value of the deflection angle for uniform plasma increased with the increase in plasma parameter value, and vice versa for non uniform plasma. We found that, under the influence of the parameter a, the values of the deflection angles for two cases decreased. Finally, we investigated the magnification of image brightness using the deflection angle of the light rays around the wormhole in the Ellis-Bronnikov theory.

    HTML

    I.   INTRODUCTION
    • Wormholes, hypothetical tunnels in spacetime that potentially connect distant regions of the universe, attract considerable attention from astrophysicists. Understanding the properties, stability, and possible existence of wormholes is essential in theoretical astrophysics. As solutions to Einstein's field equations, wormholes represent intriguing spacetime configurations. Studying wormholes helps elucidate the complex topology of spacetime, providing valuable insights into the fundamental structure of the universe. General relativity forms the foundation for such studies, offering a framework to investigate the theoretical possibility of traversable wormholes.

      If traversable, wormholes could enable faster-than-light travel between distant regions of the universe. This has profound implications for space exploration and travel, significantly reducing travel times and expanding our reach in the cosmos. Different astrophysical properties of the traversable wormholes have been studied in Refs. [14].

      In addition, wormholes could act as crucial conduits, linking different regions of the universe and allowing for a deeper understanding of galaxies, black holes, and dark matter. By investigating wormholes and their behavior within various gravitational theories, one can explore their potential role in the large-scale structure of the universe.

      Alternative theories of gravity, which deviate from general relativity, present opportunities to study wormholes in diverse gravitational frameworks. Understanding wormholes within these alternative theories can shed light on the nature of dark energy and dark matter, providing alternative explanations for the enigmatic phenomena that dominate the universe [5, 6].

      Overall, theoretical considerations of wormholes challenge our entire understanding of the structure of spacetime. General relativity predictions and alternative theories introduce the possibility of wormholes existing as natural features of the universe.

      The study of compact object shadows in astrophysics has emerged as a crucial area of research, offering unique insights into the fundamental properties of compact objects and the nature of spacetime [711]. Observations of the shadow of the compact objects at the center of Sgr A* and M87 have provided new opportunities to test gravity theories and different solutions representing astrophysical objects [12, 13]. Particularly, the shadow of a wormhole holds special significance, presenting a theoretical opportunity to probe the structure of the universe in unconventional ways. The shadow of a wormhole, if distinct and distinguishable from other compact objects, could potentially serve as indirect evidence for the existence of these hypothetical tunnels in spacetime. By characterizing and identifying unique features in the wormhole shadow, one can contribute to the ongoing quest for observational confirmation of these exotic cosmic structures. By carefully examining the shape, size, and other characteristics of the wormhole shadow, one can infer valuable information about the wormhole throat size, spin, and potential accretion processes. This could aid in developing a comprehensive understanding of these speculative constructs. The analysis of the shadow of different objects (black holes and wormholes) in different gravity models may be found with and without plasma in Ref. [1136].

      Gravitational lensing is one of the fascinating consequences of the metric theories of gravity. This effect enabled the first direct test of general relativity [37]. Gravitational lensing may provide information regarding a gravitating object and its distance from the source. There has been a lot of work on gravitational lensing around a wormhole for various models [3848]. Additionally, the study of gravitational weak and strong lensing around compact objects surrounded by plasma can be found in Refs. [4967].

      Weak and strong deflection gravitational lensing and photon motion have been studied around different types of compact objects, e.g., by a charged Horndeski black hole [68], by a renormalization group improved Schwarzschild black hole [69], in Einstein-scalar-Gauss-Bonnet theories [70], by a quantum deformed Schwarzschild black hole [71], in a black-bounce-Reissner-Nordström spacetime [72], for photons coupled to Weyl tensor in a Schwarzschild black hole [73], by a modified Hayward black hole [74], and by a black-bounce, traversable wormhole [75]. The solar system test can also be useful to justify the gravity theories. Such a test has been discussed within f(R) gravity [76], quantum corrected gravity [77], a Yukawa parameterization approach [78], and nonlocal gravity [79]. Furthermore, test particle motion around compact objects may play an important role in testing the gravity theories (see, e.g. [8085]).

      In this study, we investigated the effect of weak gravitational lensing around the Ellis-Bronnikov wormhole in the presence of a plasma medium and its shadow. The paper is organized as follows. In Sect. II, we review the spacetime surrounding the Ellis-Bronnikov wormhole and study its shadow. Weak gravitational lensing is considered in Sect. III. Image source magnification due to weak lensing is studied in Sect. IV. We draw conclusions based on our obtained results in Sect. V.

    II.   WORMHOLE SHADOW
    • The Ellis-Bronnikov wormhole spacetime metric can be expressed as [47]

      ds2=e2u(r)dt2+e2u(r)[dr2+(r2+a2)dΩ2] ,

      (1)

      where dΩ2=dθ2+sin2θdφ2, r is the radial coordinate running from to +, and

      u(r)=ma(arctanraπ2).

      (2)

      m and a are two free parameters. a corresponds to the radius of the wormhole throat located at r=0, and m is the asymptotic mass of the Ellis-Bronnikov wormhole. Eq. (2) accounts for the following asymptotic behavior:

      e2u(r)|r+=(12mr)+O(r2) ,e2u(r)|r=e2πma(1+2m|r|)+O(r2) .

      (3)

      We can see that the spacetime metric in Eq. (1) has two asymptotically flat regions R±:r±. The asymptotical masses of the R+ and R regions are equal to m+=m and m=mexp(πm/a), respectively. Both masses clearly have different signs and values. Therefore, from the perspective of a distant observer in region R, the Bronnikov-Ellis wormhole appears to have a negative mass. Two asymptotical regions R+ and R are connected by the throat, with the radius of the throat being R2(r)=e2u(r)(r2+a2), which is the minimum radius of two-dimensional sphere. The minimum value of R(r) occurs at r=m and is equal to

      R0=exp[ma(arctanmaπ2)](m2+a2)1/2.

      (4)

      If m is equal to zero, then both m± become zero, and the metric in Eq. (1) reduces to the Ellis metric.

    • A.   Photon motion

    • We now consider the photon motion using the Hamilton-Jacobi equation. One can write the Hamiltonian of a photon around a wormhole surrounded by plasma in the following form [86]:

      H(xα,pα)=12[gαβpαpβ(n21)(pβuβ)2] ,

      (5)

      where xα are the spacetime coordinates, and pα and uβ are the four-momentum and four-velocity of the photon, respectively. In Eq. (5), n represents the refractive index (n=ω/k, where k is the wave number). For plasma, one can write the refractive index as [87]

      n2=1ω2pω2 ,

      (6)

      with plasma frequency ω2p(xα)=4πe2N(xα)/me (me and e are the electron mass and charge, respectively, whereas N is the number density of the electrons). The frequency of photon ω is defined by ω2=(pβuβ)2 and

      ω(r)=ω0f(r) ,ω0=const .

      (7)

      The lapse function is such that f(r)1 as r and ω()=ω0=pt, which shows the energy of the photon at spatial infinity [88]. Besides, the plasma frequency can be sufficiently smaller than the photon frequency (ω2pω2), which allows the BH shadow to be differentiated from the vacuum case (ωp=0).

      One can now write the Hamiltonian for the light rays in plasma medium as follows [50, 86]:

      H=12[gαβpαpβ+ω2p] .

      (8)

      The components of the four-velocity for the photons in the equatorial plane (θ=π/2,pθ=0) can be written as

      ˙tdtdλ=pte2u(r),

      (9)

      ˙rdrdλ=pre2u(r),

      (10)

      ˙ϕdϕdλ=pϕe2u(r)r2+a2,

      (11)

      using the relationship ˙xα=H/pα. From Eqs. (10) and (11), one can obtain the following expression for the phase trajectory of light (or photon) in the following form:

      drdϕ=grrprgϕϕpϕ.

      (12)

      Using the constraint H=0, we can rewrite the above expression as [88]

      drdϕ=grrgϕϕγ2(r)ω20p2ϕ1,

      (13)

      where

      γ2(r)gttgϕϕω2pgϕϕω20 .

      (14)

      The circular radius of light, particularly the one which forms the photon sphere of radius rph, can be determined as the solution of the following equation [88]:

      d(γ2(r))dr|r=rph=0 .

      (15)

      Using Eq. (14), we have solved the above equation numerically. The results are presented graphically in Fig. 1, where we have plotted the dependencies of photon orbits on the plasma frequency. The value of the radius of the photon sphere increased with the increase in plasma frequency. Also, there is a slight increase with the increase in parameter a.

      Figure 1.  (color online) The dependence of the radius of the photon sphere on the plasma frequency.

    • B.   Wormhole shadow in plasma

    • Now, we consider the radius of the shadow of the Ellis-Bronnikov wormhole in the presence of plasma. The angular radius αsh of the wormhole is defined as [88, 89]

      sin2αsh=γ2(rph)γ2(ro),

      (16)

      where rph and ro represent the photon sphere and observer locations, respectively. One can now easily find the γ2(rph) and γ2(ro) from Eq. (14). If the observer is located at a sufficiently large distance from the wormhole, the radius of the wormhole shadow can be approximated using Eq. (16) as [88]

      Rshrosinαsh.

      (17)

      Rsh is calculated numerically, and the radius of the wormhole shadow is depicted for different values of the a parameter in Fig. 2 for a homogeneous plasma with fixed plasma frequencies. Fig. 3 demonstrates the dependence of the wormhole shadow radius on plasma frequencies. These figures show that the size of the wormhole shadow radius decreases with increasing plasma frequency. Accordingly, the BH shadow in the presence of a plasma medium would shrink further, as expected.

      Figure 2.  (color online) The dependence of the wormhole shadow radius on the plasma frequency for the different values of parameter a.

      Figure 3.  (color online) The dependence of the wormhole shadow radius on the a parameter for different plasma frequency values.

      Now, we consider the assumption that the compact objects Sgr A* and M87* are static, spherically symmetric objects in the Ellis-Bronnikov gravity, even though the observation obtained by the EHT collaboration does not support the assumption made here. However, we theoretically investigate the upper limits of the parameter a in the Ellis-Bronnikov wormhole spacetime using the data provided by the EHT collaboration project. The spacetime metric comprises only one parameter, a. Hence, we chose the plasma frequency as the second parameter for the constraint. One can use the observational data provided by the EHT collaboration regarding the shadows of the supermassive black holes Sgr A* and M87* to constrain these two parameters, a and ωp/ω. The angular diameter θM87 of the BH shadow, the distance from Earth, and the mass of the BH at the center of the M87* are θM87=42±3μas, D=16.8±0.8 Mpc, and MM87=6.5±0.7×109M [12], respectively. For Sgr A*, the data provided by the EHT collaboration are θSgrA=48.7±7μD=8277±9±33 pc and MSgrA=4.297±0.013×106M (VLTI) [90]. From this information about it, one can calculate the diameter of the shadow caused by the compact object per unit mass as follows

      dsh=DθM .

      (18)

      We know that from the expression dsh=2Rsh, we can easily get the expression for the diameter of the BH shadow. Thus, we obtain the diameters of the BH shadow as dM87sh=(11±1.5)M for M87* and dSgrsh=(9.5±1.4)M for Sgr A*. From observational EHT data, we can find the upper limits on parameters a and ω2p/ω20 for the supermassive BHs at the centers of the galaxies Sgr A* and M87*. Here, we try to theoretically fulfill the constraint for Ellis-Bronnikov wormhole as a BH mimicker. It is demonstrated numerically in Fig. 4. We have demonstrated that the size of the shadow depends on the wormhole and plasma parameters. The upper limits of the a and plasma parameters correspond to the EHT results. Moreover, we estimated the values of the parameter a and the plasma frequency for the supermassive BHs M87* and Sgr A* in Table 1.

      Figure 4.  (color online) The values of parameters a and ω2p/ω20 for the supermassive BHs in galaxies M87* and Sgr A*. We consider m=1 for both panels.

      ω2p/ω2M87SgrA
      aa
      0.202.531.08
      0.252.671.24
      0.302.811.39
      0.352.971.55

      Table 1.  The estimated values of parameters a and ω2p/ω20 for the supermassive BHs: M87* and Sgr A*.

    III.   WEAK GRAVITATIONAL LENSING FOR WORMHOLE
    • In this section, we investigate the weak gravitational lensing around the Ellis-Bronnikov wormhole. We will expand metric (1) in the weak-field approximation as follows [49, 54]:

      gαβ=ηαβ+hαβ ,

      (19)

      where ηαβ and hαβ refer to the expressions for flat spacetime and perturbation due to gravity, respectively. The above expressions require the following properties:

      ηαβ=diag(1,1,1,1) ,hαβ1,hαβ0underxα ,gαβ=ηαβhαβ,hαβ=hαβ.

      (20)

      Now, we will study the effect of the plasma on the deflection angle αk in the gravitational field of the Ellis-Bronnikov wormhole. In this paper, we consider two types of plasma: ωp and ωc, representing the frequencies for uniform and non-uniform plasma, respectively. One can write the equation for the deflection angle around the Ellis-Bronnikov wormhole as [49]

      ˆαb=12br(dh33dr+11ω2p/ω2dh00drKeω2ω2pdNdr)dz,

      (21)

      where ω represents the frequency of the photon. One can write the line element (1) as

      ds2ds20+(1e2u(r))dt2+(1e2u(r))dr2,

      (22)

      where ds20=dt2+dr2+(r2+a2)(dθ2+sin2θdϕ2). Now, we can easily find the components of hαβ of metric tensor perturbations in Cartesian coordinates as

      h00=1e2u(r),

      (23)

      hik=(1e2u(r))nink,

      (24)

      h33=(1e2u(r))cos2χ,

      (25)

      with cos2χ=z2/(b2+z2) and r2=b2+z2. One may easily calculate the derivative of h00 and h33 as follows:

      dh00dr=2me2u(r)a2+r2,

      (26)

      dh33dr=2z2r3(1+(1mra2+r2)e2u(r)).

      (27)

      The expression for the deflection angle can be written as [91]

      ^αb=^α1+^α2+^α3 ,

      (28)

      with

      ^α1=12brdh33drdz ,^α2=12br11ω2p/ω2dh00drdz ,^α3=12br(Keω2ω2pdNdr)dz.

      (29)

      Now, we can examine and assess the deflection angle for various plasma density distributions.

    • A.   Uniform plasma

    • In this subsection, we consider the uniform plasma distribution, which can be expressed as a sum [91]:

      ˆαuni=ˆαuni1+ˆαuni2+ˆαuni3.

      (30)

      From Eqs. (25), (28), and (29), one can find the expression for ˆαuni. Our metric is complicated, so we cannot find the expression for the deflection angle explicitly. Therefore, we use the numerical method and plot the dependence of the deflection angle on the impact parameter b for different values of the a parameter and plasma parameter ω2p/ω2 in the Ellis-Bronnikov wormhole spacetime. The numerical results are represented graphically in Fig. 5. It can be seen from the graphs that the value of the deflection angle αuni decreased with the increase in the impact parameter for fixed values of the plasma frequency and the a parameter. Moreover, we present the effect of the deflection angle on plasma parameters in Fig. 6. Under the influence of the a parameter, the value of the deflection angle αuni decreases, while the effect of the plasma frequency on the deflection angle is the opposite.

      Figure 5.  (color online) The dependence of the deflection angle ˆαuni on the impact parameter b for different values of parameter a (upper panel) and the plasma medium (bottom panel).

      Figure 6.  (color online) The dependence of the deflection angle ˆαuni on plasma parameters for the fixed value of impact parameter b=5M.

    • B.   Non uniform plasma

    • In this part of our work, we consider the non-singular isothermal sphere (SIS), which represents the most favorable model for understanding the unique characteristics of gravitational weak lensing effects on photons around the wormhole. In general, SIS is a spherical gas cloud characterized by a singularity located at its center, where the density tends to infinity. The density distribution of a SIS is described as follows [49]:

      ρ(r)=σ2ν2πr2 ,

      (31)

      where σ2ν denotes a one-dimensional velocity dispersion. The analytical expression for the plasma concentration is given as [49]

      N(r)=ρ(r)kmp ,

      (32)

      where mp represents the mass of a proton, and k is a dimensionless coefficient generally associated with the dark matter universe. The plasma frequency is

      ω2e=KeN(r)=Keσ2ν2πkmpr2 .

      (33)

      Now, we explore the non-uniform plasma (SIS) effect on the deflection angle in the spacetime of the Ellis-Bronnikov wormhole. One can write the expression of deflection angle around the Ellis-Bronnikov wormhole as [91]

      ˆαSIS=ˆαSIS1+ˆαSIS2+ˆαSIS3 .

      (34)

      For uniform plasma, we also use the numerical method. These calculations establish a supplementary plasma constant ω2c with the following analytic expression [54]:

      ω2c=Keσ2ν2πkmpR2S .

      (35)

      We have represented the dependence of the deflection angle on impact parameter b for different values of the a parameter and ω2c/ω2 for a non-uniform plasma medium in the Ellis-Bronnikov wormhole spacetime, as shown in Fig. 7. The graphs show that the value of the deflection angle αsis decreased as the impact parameter increased. Then, we demonstrate the dependence of the deflection angle of a light ray around a wormhole in the presence of a non-uniform plasma in Fig. 8. One can easily see from Fig. 8 that the value of the deflection angle αsis decreased under the influence of the plasma and a parameters. In addition, we compared the different effects of plasma on the Ellis-Bronnikov wormhole deflection angle with gravity, as shown in Fig. 9. By comparison, it can be seen that the deflection angle of the light ray for the uniform plasma is greater than that for the non-uniform plasma.

      Figure 7.  (color online) The dependence of the deflection angle ˆαsis on the impact parameter for different values of parameter a (upper panel) and plasma parameters (lower panel).

      Figure 8.  (color online) The dependence of the deflection angle ˆαsis on plasma for the fixed value of impact parameter b=5M.

      Figure 9.  (color online) The dependence of the deflection angle ˆαb on the impact parameter (upper panel) and plasma parameters (lower panel).

    IV.   MAGNIFICATION OF GRAVITATIONALLY LENSED IMAGE
    • Now, we explore the image brightness in the presence of plasma through the angle of deflection of light rays around the Ellis-Bronnikov wormhole. By employing the lens equation, the combination of light angles around the Ellis-Bronnikov wormhole can be written (ˆα, θ, and β) [51, 54, 92] as

      θDs=βDs+^αbDds ,

      (36)

      where Ds, Dd, and Dds are the distances from the source to the observer, lens to the observer, and source to the lens, respectively. In Eq. (36), θ and β denote the angular position of the image and the source, respectively. One can now rewrite the above equation for β as follows:

      β=θDdsDsξ(θ)Dmd1θ,

      (37)

      where ξ(θ)=|ˆαb|b , with b=Ddθ, as used in [54].

      When the image has a ring-like appearance, it is classified as Einstein's ring, with a ring radius defined as Rs=DdθE. The angular part θE, arising from the spacetime geometry between the source's images in a vacuum, can be expressed as [51]

      θE=2RsDdsDdDs .

      (38)

      Now, we investigate the brightness magnification equation as follows [54]:

      μΣ=ItotI=k|(θkβ)(dθkdβ)|,k=1,2,,j ,

      (39)

      where Itot and I are the the total brightness of all images and the unlensed brightness of the source, respectively. The magnification of the source can be expressed as [49, 54]

      μpl+=14(xx2+4+x2+4x+2) ,

      (40)

      μpl=14(xx2+4+x2+4x2) ,

      (41)

      where x=β/θE is a dimensionless quantity [54] and μpl+ and μpl are the image magnifications. Using equations (40) and (41), we can obtain the expression for the total magnification as follows:

      μpltot=μpl++μpl=x2+2xx2+4 .

      (42)

      Now, we explore the magnification in the presence of plasma with different distributions in the wormhole environment: (i) uniform and (ii) non-uniform cases.

    • A.   Uniform plasma

    • Here, we study the effect of uniform plasma on the magnification image. The total magnification μpltot can be expressed as

      μpltot=μpl++μpl=x2uni+2xunix2uni+4,

      (43)

      The expression for (θplE)uni is complicated; therefore, we use numerical calculations. xuni, (μpl+)uni, and (μpl)uni are defined as

      xuni=β(θplE)uni.

      (44)

      The image magnifications can be written as

      (μpl+)uni=14(xunix2uni+4+x2uni+4xuni+2),

      (45)

      (μpl)uni=14(xunix2uni+4+x2uni+4xuni2).

      (46)

      The total magnification of the image in the presence of a plasma μpltot for the Ellis-Bronnikov wormhole is demonstrated in Fig. 10. Fig. 11 shows the dependence of the total magnification on x0 in the presence of uniform plasma for different values of the a parameter. These figures clearly indicate that the value of the total magnification of the image for uniform plasma increases with the plasma parameter value.

      Figure 10.  (color online) The dependence of the total magnification μtot on the plasma parameter for different values of parameter a corresponding to the fixed value of impact parameter b=5M.

      Figure 11.  (color online) Image magnification in the presence of uniform plasma. The fixed parameters used are M=1, b=3M, and a=1.

    • B.   Non-uniform plasma

    • In this subsection, we explore the effect of non-uniform plasma on the magnification of the image. We may write the expression for total magnification (μpltot)SIS as

      (μpltot)SIS=(μpl+)SIS+(μpl)SIS=x2SIS+2xSISx2SIS+4,

      (47)

      with

      (μpl+)SIS=14(xSISx2SIS+4+x2SIS+4xSIS+2),

      (48)

      (μpl)SIS=14(xSISx2SIS+4+x2SIS+4xSIS2),

      (49)

      xSIS=β(θplE)SIS.

      (50)

      As with (θplE)uni, we use the numerical method for (θplE)SIS.

      We can find the dependence of total magnification on the plasma parameter using Eq. (47). From Fig. 12, we can see that the magnification decreased as the plasma parameter increased. Moreover, the plot of the dependence of total magnification on x0 in the presence of plasma for the fixed values of parameter a is provided in Fig. 13. These figures show that the value of the total magnification of the image for non-uniform plasma decreased as the plasma parameter values increased. Finally, we compared the two cases, i.e., uniform and non-uniform plasma distributions, in Fig. 14.

      Figure 12.  (color online) The total magnification of the images as a function of non-uniform plasma. Here, the impact parameter is set as b=5M.

      Figure 13.  (color online) Image magnification in the presence of SIS. The fixed parameters used are M=1, b=3M, and a=1.

      Figure 14.  (color online) Image magnifications in the presence of uniform plasma and SIS. The fixed parameters used are M=1, b=3M, ω2pω2=0.9, ω2cω2=0.9, and a=1.

      The effect of uniform and non-uniform plasma on the total magnification differs; the former increases the total magnification while the latter decreases the total magnification of the images. Also, the image magnification for uniform plasma is greater than that for non-uniform plasma.

    V.   CONCLUSIONS
    • In this paper, we discussed the optical properties of the Ellis-Bronnikov wormhole. From the performed research, we can summarize our main results as follows:

      ● We investigated photon motion around a wormhole surrounded by plasma. We obtained numerical results regarding the dependence of the photon sphere radius on the plasma frequency (see Fig. 1). The photon sphere radius increased as the plasma frequency increased.

      ● We also studied the shadow of the Ellis-Bronnikov wormhole in plasma. The wormhole shadow radius was calculated using the numerical method. The dependencies of the wormhole shadow radius on the plasma frequency and wormhole parameter are demonstrated in Figs. 2 and 3. The figures show that the wormhole shadow radius increased as the a parameter increased, and vice versa for the plasma frequency, as demonstrated in Figs. 2 and 3.

      ● Since we have demonstrated that the size of the shadow depends on the wormhole and gravity theory parameters, in the future, the obtained results can be applied to the images of Sgr A* and M87* SMBHs to determine the constraints on Ellis-Bronnikov gravity parameters.

      ● Furthermore, weak gravitational lensing for the Ellis-Bronnikov wormhole was investigated. For this, we considered that the wormhole was surrounded by uniform and non-uniform plasma, and we obtained the deflection angle for each case. The obtained results are illustrated in Figs. 5, 6, 7 and 8. The value of the deflection angle ˆαuni decreased as the impact parameter b increased. Under the influence of the uniform plasma frequency, the deflection angle ˆαuni increased, and vice versa for the a parameter. For uniform plasma, the values of the deflection angle ˆαsis decreased as the impact parameter b increased. Also, a slight decrease in the deflection angle ˆαsiswas observed under the influence of the a parameter and the non-uniform plasma frequency.

      ● In addition, we compared the deflection angle of light for uniform and non-uniform plasma in Fig. 9. The figure clearly indicates that the deflection angle of the light ray for uniform plasma is greater than that for non-uniform plasma.

      ● Finally, we studied the total magnification of the images as a function of uniform and non-uniform plasma. The dependencies are plotted in Figs. 10 and 12. From these figures, we can see that the value of the total magnification of the image for uniform plasma increased as the plasma parameter increased, and vice versa for non-uniform plasma. Also, we investigated the image magnification in both cases. The results are demonstrated in Figs. 11 and 13. Moreover, we compared the image magnifications in the presence of uniform plasma and SIS. The image magnification for uniform plasma is greater than that for SIS, as demonstrated in Fig. 14.

      The above results demonstrate that plasma changes the optical properties of the Ellis-Bronnikov wormhole and its parameters. Also, we can see that the type of plasma plays an important role in the deflection angle because the deflection angle increases as the uniform plasma frequency increased, and vice versa for non-uniform plasma. In this work, we focused only on the constraint of the BH shadow. In future studies, the observational data of the gravitational lensing in Refs. [9396] may be further employed to obtain constraints on the spacetime parameters of EB and estimate the plasma characteristics.

Reference (96)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return