-
Non-rotating spacetime in black bounce gravity has the following linear element in spherical coordinates [83, 84]:
ds2=−f(r)dt2+dr2f(r)+γ(r)(dθ2+sin2θdϕ2) ,
(1) where the metric functions
f(r) andγ(r) are defined asf(r)=1−2Mr2(a2+r2)3/2 ,
(2) γ(r)=r2+a2 ,
(3) where M is the mass of the compact object, and a is a nonnegative constant [83]. To understand the nature of the gravitational potential of the source, we may introduce the effective mass as
Meff=M(r/√r2+a2)3 .Spacetime metric (1) describes a regular black hole when the condition
0<a/m<4√3/9 is satisfied. To obtain more information about the spacetime structure, we can analyze the functionf(r) . The dependence off(r) on radial coordinates is shown in Fig. 1. From the plot, we can easily see that there is an extreme value ofa/M when the graph has oner/M -intercept. The extreme value of the parameteraext=4√3/9 . The values of parameter a that are less than its extreme value correspond to a black hole with two horizons (Fig. 2 shows the existence of two roots for a set of parameters a values, which corresponds to the Cauchy horizon (smaller root) and event horizon (larger root)), whereas the values of parameter a that are greater than its extreme value correspond to a naked singularity. The dependence of the radius of the horizon corresponding to the conditionf(r)=0 is represented in Fig. 2. From the plot, we can see that with an increase in parameter a in black-bounce spacetime, the outer horizon decreases until the value M is reached. -
Consider the motion of photons around a gravitational object in the framework of black-bounce gravity. To describe the dynamics, we can use the Hamilton-Jacobi equation in the following form [85]:
gμνdSdxμdSdxν=0,
(4) where
gμν is the metric tensor of spacetime metric (1), and S is the action. Using the separation of variables method and the symmetry of spacetime metric (1), the action S can be expressed as the following form:S=−Et+Lφ+Sr(r)+Sθ(θ),
(5) where E and L can be described as the energy and angular momentum of a massless particle, respectively. The dynamics of particles with zero rest mass are defined by null geodesics of spacetime. Using Eqs. (4) and (5), we can obtain the equations of motion of massless particles at the equatorial plane (
θ=0.5π ,˙θ=0 ) asdtdλ=Ef(r),
(6) drdλ=±√R(r),
(7) dϕdλ=Lγ(r),
(8) with
R(r)=E2−f(r)γ(r)L2,
(9) where λ is the affine parameter, and we consider
θ=0.5π . Accordingly, the effective potential for the radial motion of photons takes the following form:Veff=f(r)L2γ(r).
(10) The study of circular orbits r reflects significant interest while considering the photon (massless particle) motion. Photon circular orbits
rph are determined using the following conditions:V′eff=0.
(11) Using the above conditions, we can obtain the equation to determine the radius of circular photon orbits
rph as(a2+r2)3/2+2Ma2−3Mr2=0.
(12) We can obtain the radius of the photon sphere (
rph) by solving Eq. (12) with respect to the radial coordinate,rph=√−a2(10M2D+1)+D+9M4D+3M2 ,
(13) where D is defined as
D=(25a4M22−45a2M4+5√52a3M2√5a2−4M2+27M6)1/3.
(14) The dependence of the photon orbit
rph on the parameter a is shown in Fig. 3. From the plot, we can conclude that with an increase in parameter a, the radius of the photon orbitrph decreases. -
The detailed study of photon motion around a black hole leads to a phenomenon known as a black hole shadow [16]. The shadow cast by a black hole is a result of photon capture by the central object, and the observer will see a black spot on the bright background [16]. In this section, we explore the radius (
Rsh ) of the shadow of a black hole in the presence of a plasma medium [55]. The boundary of the shadow is fully defined by the photon (massless particle) trajectory governed by the equation of motion in the plasma medium. To analyze the shadow of the black hole in black-bounce spacetime in a plasma medium, we should first study the photon equation in a plasma environment. -
The Hamiltonian for photons around a black hole in the presence of an electron plasma medium can be expressed as [55, 85]
H=12[gμνpμpν−(n2−1)p2tf(r)]=0,
(15) with
n2=1−ω2eω2(xi),
(16) where
ω(xi) andωe are the photon and plasma frequencies, respectively, and n is the refractive index of the medium. We denote the value of the photon frequency observed at infinity asω0 :ω(∞)=ω0=−pt.
(17) The components of the four-velocity (
˙t ,˙ϕ ,˙r ) for the photons (massless particle) in the equatorial plane (θ=0.5π ,pθ=0 ) are given bydtdλ=−ptf(r),
(18) drdλ=prf(r),
(19) dϕdλ=pϕr2+a2,
(20) where we consider the relationship
dxμ/dλ=∂H/∂pμ . From Eqs. (19) and (20), we can obtain an expression for the phase trajectory of light (photon) rays,drdϕ=grrprgϕϕpϕ
(21) Using the constraint
H=0 , we can define the above equation as [50]drdϕ=√grrgϕϕ√h2(r)ω20p2ϕ−1,
(22) where the following definition is introduced:
h2(r)≡−gttgϕϕ−1gϕϕω2e(r)ω20.
(23) The radius of a circular orbit of photons
rph , particularly the one that forms the photon sphere of radiusrph , is determined by the following equation [50]:d(h2(r))dr|r=rph=0.
(24) Substituting Eq. (24) into Eq. (23), we can define an algebraic expression for
rph in the plasma environment as(a2+r2)f′(r)f(r)+2f(r)ωe(r)ω20[(a2+r2)ω′e(r)+rωe(r)]=2r,
(25) where the prime stands for the derivative with respect to r. In the case of homogeneous plasma (
ω2e(r)=const. ), Eq. (25) can be simplified to(a2+r2)f′(r)f(r)+2rf(r)ω2eω20=2r.
(26) As mentioned above, the solution to this equation with respect to r leads to the radius of the photon sphere
rph of the black hole in a uniform plasma environment.Figure 4 gives information on the dependence of the radius of the photon sphere
rph on the black-bounce parameter a and plasma parameters. It is clear from the plots that the radius of the photon sphererph decreases with an increase in parameter a. Note that the presence of plasma causes an increase in the value of the photon sphere radius, as shown in Fig. 4. -
In this section, we study the radius (
Rsh ) of the shadow of the black hole in the presence of uniform plasma. The angular radiusαsh of the black hole shadow is defined by [7, 50]sin2αsh=h2(rsh)h2(ro)=(a2+r2ph)(1f(rph)−ω2e(rph)ω20)(a2+r2o)(1f(ro)−ω2e(ro)ω20),
(27) where
ro andrph are the radial positions of the observer and photon sphere (introduced in the previous section), respectively. If the observer is located at a sufficiently large distance from the black hole, we can approximate the radius of the black hole shadow (for the case of uniform plasma) using Eq. (27) [50]:Rsh≃rosinαsh=√(a2+r2ph)(1f(rph)−ω2eω20).
(28) For the case of no plasma in the vicinity of the black-bounce black hole, the shadow radius can be represented by
Rsh≃rosinαsh=√a2+r2phf(rph).
(29) Figure 5 describes the dependence of the radius of black hole shadow on parameter a and plasma parameters. In the plots, we define
Rsh in terms ofMeff because it is known from EHT data that the masses of M87* and Sgr A* are given by the gravitational potential. Based on these data, we can conclude that the shadow radius is diminished due to the influence of both plasma and the black-bounce parameter a.Figure 5. (color online) Dependence of the radius of the black hole shadow on parameter a (top) and plasma frequency (bottom).
Now, we consider the rough assumption that the supermassive black holes M87* and Sgr A* are the black-bounce black holes that we are studying. Then, we can theoretically obtain the constraints from the EHT observation results.
The angular diameter of the image of the supermassive black hole M87* is
θ=42±3 μas at the1σ confidence level, and the mass of M87* and the distance from the solar system areM=6.5×109M⊙ andD= 16.8 Mpc, respectively [3, 4]. The same data for Sgr A* are as follows:θ=48.7±7 μas ,M≃4×106M⊙ , andD≃8 kpc [5]. The shadow radius caused by the black hole per unit mass can be expressed as the following equation:dsh=DθMeff,
(30) where
Meff=M(r/√r2+a2)3 from black-bounce spacetime. Here, we must use the effective mass because the masses of M87 and Sgr A* are given by the gravitational potential.Using Eq. (30), we can easily calculate the diameter of the shadow images. The diameter of the black hole shadow is
dsh=(11±1.5)Meff for M87* anddsh=(9.5±1.4)Meff for Sgr A*. From these data, we can now obtain the constraints on the black-bounce and plasma parameters for the supermassive black holes Sgr A* and M87*. The results are presented in Fig. 6 for the case of no plasma around the black hole and in Fig. 7 for the black hole in plasma. -
Now, we can move on to another part of our study: gravitaional lensing in spacetime metric (1) in a weak-field case explicated via [59]
gαβ=ηαβ+ξαβ ,
(31) where
ηαβ is the general metric of Minkowski spacetime, andξαβ is a perturbation of the flat spacetime metric, which can be described by the spacetime metric theory of gravity and is written as [59]ηαβ=diag(−1,1,1,1),ξαβ≪1,ξαβ→0underxi→∞ ,gαβ=ηαβ−ξαβ, ξαβ=ξαβ .
(32) Here, we can study the effects of the plasma medium on the deflection angle
ˆαk in the gravitational weak field of the black hole. The general expression for the deflection angle in plasma is [59, 64]ˆαi=12∫∞−∞(ξ33+ξ00ω2−KeN(xi)ω2−ω2e),idz, i=1,2
(33) where
N(xi) is the number density of particles in a plasma medium around a compact object,Ke=4πe2/me is constant (e andme are the electron charge and mass), and ω andωe are the photon (massless particle) and plasma frequencies, respectively [59]. Using Eqs. (32) and (33), we can rewrite Eq. (33) for the deflection angle in the following form [59]:ˆαb=12∫∞−∞br(dξ33dr+11−ω2e/ω2dξ00dr−Keω2−ω2e dNdr)dz,
(34) where b is the impact parameter of light rays. It is useful to note that the values of
ˆαb can be both negative and positive [59].In the weak-field case for far distances from the black hole, the black-bounce spacetime metric (1) can be written as
ds2=ds20+2Mr2(a2+r2)3/2dt2+2Mr2(a2+r2)3/2dr2+a2(dθ2+sin2θdϕ2),
(35) where
ds20=−dt2+dr2+r2(dθ2+sin2θdϕ2) is the metric element in the Minkowski spacetime metric, and we can use the notationRs=2M for further calculations.To analyze the deflection angle (
ˆαb ) of light rays around the black hole in the plasma medium using Eq. (34), we can rewrite the required components (ξ00 andξ33 ) in Cartesian coordinates asξ00=Rsr2(a2+r2)3/2,ξ33=Rsr2(a2+r2)3/2cos2χ+a2r2sin4χ ,
(36) where
cos2χ=z2/(b2+z2) andr2=b2+z2 are introduced, for example, in [64]. Here, z is the coordinate aligned with the optical axis. We can obtain the derivative ofξ00 andξ33 with respect to the radial coordinate asdξ33dr=−3rRsz2(a2+r2)5/2−2a2(r4−4r2z2+3z4)r7,dξ00dr=Rs(2a2r−r3)(a2+r2)5/2.
(37) For simplicity, the expression for the deflection angle
ˆαb can be expanded as [63]ˆαb=ˆα1+ˆα2+ˆα3,
(38) with
ˆα1=12∫∞−∞brdξ33drdz,ˆα2=12∫∞−∞br(11−ω2e/ω2dξ00dr)dz,ˆα3=12∫∞−∞br(−Keω2−ω2e dNdr)dz,
(39) where the notations
ˆα1 ,ˆα2 , andˆα3 represent the contributions to the deflection angle due to gravity, uniform plasma, and non-uniform plasma, respectively. In this study, we use Eqs. (38) and (39) to investigate the effect of plasma on the deflection angle in gravitational weak lensing. In the following sections, we consider each case in detail. -
Here, we can test the deflection angle of photons (light rays) around the black hole in the presence of a uniform plasma environment using expression (38) written in the following form:
ˆαuni=ˆαuni1+ˆαuni2+ˆαuni3,
(40) We can claim that the third term of this equation will vanish due to the uniform distribution of the plasma environment. Using Eqs. (38), (37), and (40), we can obtain an equation for the deflection angle of light rays (photon) around the black hole in black-bounce spacetime surrounded by uniform plasma:
ˆαuni=bRsa2+b2+3πa216b2−bRs(a2−b2)(a2+b2)2ω2ω2−ω2e.
(41) In this equation, the second term appears owing to the property of black-bounce spacetime itself.
Using expression (41), we can plot the deflection angle against the impact b, plasma frequency, and a. Fig. 8 shows the dependence of the deflection angle
ˆαuni of light rays (photon) around the black hole on the impact parameter b for different values of the parameter a and plasma parameterω2e/ω2 . The dependence of the deflection angleˆαuni plasma frequency and parameter a for fixed values of the impact parameter is shown in Fig. 9. The graphs of the defection angleˆαuni on the impact parameter b and plasma parameter are reliable, as expected. The presence of uniform plasma leads to an increase in the value of the deflection angle of light rays, whereas the angle of deflection decreases dramatically with increasing impact parameter b. However, interestingly, the deflection angle remains almost unchanged with the influence of the black-bounce parameter a. There is only a slight increase in the value of the deflection angle with an increase in parameter a, as shown in Figs. 8 and 9. -
Here, we investigate the deflection angle
ˆαuni of light rays (photons) around the black hole in the presence of non-uniform plasma and in the framework of black-bounce spacetime with a nonzero parameter a. We can consider the singular isothermal sphere (SIS) medium as the non-uniform plasma distribution [59, 64]. The plasma number density of the SIS environment is written in the following form [59, 63]:N(r)=ρ(r)kmp ,
(42) where
mp is the proton mass, and k is a non-dimensional coefficient responsible for the dark matter contribution in the SIS model and withρ(r)=σ2ν2πr2 ,
(43) where
σν andρ(r) are the velocity of dispersion and the density of plasma around the compact object, respectively [59]. Here, Eq. (38) can be written in the non-uniform plasma case as [63]ˆαSIS=ˆαSIS1+ˆαSIS2+ˆαSIS3 ,
(44) where
\hat{\alpha}_{\rm SIS1} ,\hat{\alpha}_{\rm SIS2} , and\hat{\alpha}_{\rm SIS3} represent the deflection angle due to the gravity of the black hole, due to plasma, and due to the density of the non-uniform plasma medium around the compact object in black-bounce spacetime, respectively. From Eqs. (38), (37), and (44), we can obtain an equation for the deflection angle of light rays (photons) around the black hole in black-bounce surrounded by non-uniform plasma:\hat\alpha_{\rm SIS} = \frac{3 \pi a^2}{16 b^2}+\frac{2 b^3 R_s}{\left(a^2+b^2\right)^2} +\frac{R_s^2 \omega_c^2}{2 b^2 \omega^2} + \frac{2 b R_s^3 \omega_c^2 \left(5 b^2-7 a^2\right)}{15 \pi \omega^2 \left(a^2+b^2\right)^3},
(45) where the new notation for the non-uniform case is
\omega_c^2= \frac{\sigma^2_{\nu} K_e}{2 k m_p R^2_{s}}.
(46) In Fig. 10, we show the plasma effect of non-uniform plasma and the parameter a on the deflection angle of photons around the black hole in black-bounce spacetime. The dependence of the deflection angle
\hat{\alpha}_{\rm SIS} on non-uniform plasma and the parameter a for fixed values of the impact parameter b and the non-uniform plasma parameter is clearly shown in Fig. 11. The plots reveal that the deflection angle increases gradually with increasing plasma parameter, whereas there is a small change in the value of the deflection angle due to the influence of parameter a.Figure 10. (color online) Dependence of the deflection angle
\hat \alpha_{\rm SIS} on the impact parameter b for different values of the parametera/M (left panel) and non-uniform plasma parameter (right panel).Figure 11. (color online) Dependence of the deflection angle
\hat\alpha_{\rm SIS} on the parametera/M (left panel) and non-uniform plasma parameter (right panel) for a fixed value of the impact parameterb/M=30 .We can easily compare the two cases for analysis (uniform and non-uniform) with the help of Fig. 12. It is evident that the deflection angle of light rays (photons) in the presence of a uniform plasma medium is larger than that of the non-uniform plasma case.
-
Now, we can study the brightness of an image source as a consequence of gravitational lensing in the presence of uniform plasma using a lens equation in the form [64, 76]
\begin{array}{*{20}{l}} \theta D_s=\beta D_s +\hat{\alpha}_{b} D_{ds}\ , \end{array}
(47) where
D_s andD_{ds} are the distances from the distant source to the observer and the lens object, respectively, and θ and β represent the angular positions of the image and source, respectively. We use the following relation between the impact parameter b and angle θ:\begin{array}{*{20}{l}} b=D_d\theta, \end{array}
(48) where
D_d is the distance from the observer to the lens object. Using these expressions, we can rewrite Eq. (47) as [64, 76]\beta = \theta - \frac{D_{ds}}{D_s}\frac{F(\theta)}{D_d} \frac{1}{\theta} \ ,
(49) with
F(\theta)=|\alpha_b| b=|\alpha_b (\theta)| D_d \theta.
We can write the expression for Einstein's angle
\theta_E^{pl} in the presence of uniform plasma around a black-bounce black hole using Eq. (41),\theta_E^{\rm pl}=\frac{D_{ds}}{D_{s}}\hat\alpha_{\rm uni}(b).
(50) We numerically determine the angle by solving this equation to analyze the dependence of the magnification on the black hole parameter.
It is well-known that the general equation to compute the magnification of an image source can be expressed in the following form [59]:
\mu_\Sigma=\frac{I_{\rm tot}}{I_{*}}=\sum\limits_k \left| \left(\frac{\theta_k}{\beta}\right) \left(\frac{{\rm d}\theta_k}{{\rm d}\beta}\right) \right| \ , \quad k=1,2, ..., s\ ,
(51) where s denotes the total number of images of the source, and
I_{\rm tot} can be explained as the value of the total increased brightness due to multiple images of the source with brightnessI_{*} .From Eq. (51), we can obtain the expression for the magnification of the image source as
\mu_{\rm tot}^{\rm pl}=\mu_+^{\rm pl}+\mu_-^{\rm pl}=\frac{x_{\rm uni}^2+2}{x_{\rm uni}\sqrt{x_{\rm uni}^2+4}}\ ,
(52) where
x_{\rm uni}=\frac{\beta}{(\theta_E^{\rm pl})_{\rm uni}} .
(53) The magnification of the image source is written as
\left(\mu_+^{\rm pl}\right)_{\rm uni}=\frac{1}{4}\Bigg(\frac{x_{\rm uni}}{\sqrt{x_{\rm uni}^2+4}}+\frac{\sqrt{x_{\rm uni}^2+4}}{x_{\rm uni}}+2\Bigg)\ ,
(54) \left(\mu_-^{\rm pl}\right)_{\rm uni}=\frac{1}{4}\Bigg(\frac{x_{\rm uni}}{\sqrt{x_{\rm uni}^2+4}}+\frac{\sqrt{x_{\rm uni}^2+4}}{x_{\rm uni}}-2\Bigg)\ .
(55) Using Eq. (52), we present the dependence of the total magnification on the plasma parameter for different values of parameter a in Fig. 13. From these plots, we can conclude that the total magnification of the images increases owing to the influence of uniform plasma, and there is a slight change in the total magnification with an increase in the parameter a.
Photon motion and weak gravitational lensing in black-bounce spacetime
- Received Date: 2023-05-10
- Available Online: 2023-11-15
Abstract: The effect of spacetime curvature on photon motion may offer an opportunity to propose new tests on gravity theories. In this study, we investigate and focus on the massless (photon) particle motion around black-bounce gravity. We analyze the horizon structure around a gravitational compact object described by black-bounce spacetime. The photon motion and the effect of gravitational weak lensing in vacuum and plasma are discussed, and the shadow radius of the compact object is also studied in black-bounce spacetime. Additionally, the magnification of the image is studied using the deflection angle of light rays.