Gravitational effect of a plasma on the shadow of Schwarzschild black holes
aa r X i v : . [ g r- q c ] F e b Gravitational effect of a plasma on the shadow of Schwarzschild black holes
Qiang Li and Towe Wang ∗ Department of Physics, East China Normal University,Shanghai 200241, China (Dated: February 4, 2021)Considering a Schwarzschild black hole surrounded by a fully ionized hydrogen plasma, we study the refrac-tive effect and the pure gravitational effect of the plasma on the shadow. The effects are treated in a unifiedformalism but characterized by two different parameters. For semi-realistic values of parameters, we find theircorrections to the shadow radius are both negligible, and the gravitational correction can overtake the refrac-tive correction for active galactic nuclei of masses larger than M ⊙ . Since the refractive effect is inducedby the electromagnetic interaction, this result is in sharp contrast to the textbook knowledge that the ratio ofthe gravitational force to the electromagnetic force is Gm e m p /e = 4 . × − in a hydrogen atom. Withunrealistically large values of parameters, we illustrate the two effects on the light trajectories and the intensitymap. I. INTRODUCTION
Black holes are a miracle of gravity. Although their existence used to be controversial in the last century, they are now believedto play important roles in the life cycle of a massive star, in the center of a galaxy and in the theory of quantum gravity. During therecent decade, more evidence for black holes has been obtained [1–3], including the amazing shadow and ring of M87* imagedby the Event Horizon Telescope (EHT) at a wavelength of . millimeters [3]. In the center of M87, there is a supermassivespinning black hole embedded in a geometrically thick, optically thin accretion disk. The shadow and ring is formed by stronggravitational lensing of synchrotron emission from the hot plasma in the disk. In this scenario, the plasma plays the part of alight source and the central black hole works as a very strong lens.In the future, hopefully images of higher resolution and of other black holes will be generated, then it will become necessaryto study the formation of the shadow and ring in more details. For example, the success of the EHT [3] relied on the conditionthat the accretion disk of M87* is optically thin at millimeter wavelengthes. In this way, the plasma in the disk is treated asa pure emission source, and the small opacity has negligible influence on the fuzzy image. However, for other black holes, inlonger baselines [4] or at smaller wavelengthes [5], the plasma cannot always be transparent, then it will become necessary toconsider the scattering and absorption of photons by the plasma.Besides opacity, the plasma has two other effects on the shadow and ring. First, as a refractive medium the plasma can changethe deflection angle of light rays in the gravitational field of a black hole [6, 7]. Second, the energy and momentum of the plasmacan modify the gravitational field outside the black hole. For a Schwarzschild black hole surrounded by a spherically symmetricplasma, the former refractive effect has been investigated by Ref. [8]. In this paper, we aim to study the latter gravitational effectin a similar toy model. In our model, we will assume that the plasma is static and its density profile falls as r − / outside theevent horizon. This model is far from realistic, but it enables us to assess the gravitational effect of the plasma following themethod introduced in Ref. [8] and make a comparison with the refractive effect.The rest of the paper is organized as follows. In Sec. II, we briefly review some useful formulae in Ref. [8] and establish ourconvention of notations. In Sec. III, we extend the toy model of Ref. [8] by incorporating the plasma’s gravity into the metric.Applying the formulae to the extended toy model, in Sec. IV we work out the corrections of refractive and gravitational effectsto radii of the photon sphere and the shadow. Inserting observational data of some active galactic nuclei (AGNs) [9, 10], we findthese corrections are negligibly small, though the gravitational correction can overtake the refractive correction for AGNs moremassive than about M ⊙ . In Sec. V, we illustrate the influences of refractive and gravitational effects on the trajectories oflight rays and the observed intensity image. This is done by assigning exaggerated values to model parameters. In Sec. VI wesummarize the our results and discuss the implications. ∗ Electronic address: [email protected]
II. PREVIOUS RESULTS AND CONVENTION OF NOTATIONS
We are interested in the shadow of a Schwarzschild black hole surrounded by a spherically symmetric plasma. In this situation,the shadow seen by a distant observer has the shape of a circular disk. To determine the radius of the shadow, one should studythe equations of motion for light rays in the plasma outside the black hole. This has been investigated in Ref. [8] by neglectingthe gravitational field of the plasma particles. In this paper, we will take the gravitational effect of the plasma into account. Inthe current section, as a preparation, we will collect some useful formulae from Ref. [8] which hold for spherical spacetimesgenerally.In general, a static spherical spacetime is described by a metric of the form ds = − A ( r ) c dt + B ( r ) dr + D ( r ) (cid:0) dθ + sin θdφ (cid:1) (1)in which c represents the light velocity. In accordance with the symmetry, the electron number density N ( r ) is a function ofradius only. If we denote the charge and mass of the electron as e and m e respectively, then the plasma frequency is given by ω p ( r ) = 4 πe m e N ( r ) (2)in Gaussian units. It is useful to introduce the function h ( r ) = D ( r ) A ( r ) (cid:20) − A ( r ) ω p ( r ) ω (cid:21) (3)with a constant of motion ω . On the equatorial plane, the obit equation of light rays can be written as drdφ = ± s D ( r ) B ( r ) (cid:20) h ( r ) b − (cid:21) (4)with the parameter b being a constant of motion. In terms of h ( r ) , the radius of the outermost photon sphere r ph is the largestroot of the equation ddr h ( r ) = 0 . (5)For an observer at a distance of r , the opening angle of the shadow is determined by sin α sh = h ( r ph ) h ( r ) , (6)and the radius-squared of the shadow observed is R = r sin α sh = r h ( r ph ) h ( r ) . (7)More generally, it can be verified that in the image seen at a distance of r , photons with the same value of b form a circle ofradius-squared R = r sin α = b r h ( r ) . (8)If we restrict to asymptotically flat spacetimes, ω can be interpreted as the photon frequency at infinity, b can be named as animpact parameter, while R sh will be identical to the impact parameter of the photon sphere.Eqs. (5) and (7) are key formulae for us to compute radii of the photon sphere and the shadow in Sec. IV. Eqs. (4) and (8)are crucial for tracing the trajectory of light rays in Sec. V, where we will assume h ( r ) ≥ and ≤ α ≤ π/ for practicalreasons. To trace the trajectory of light rays, one should integrate the orbit equation (4), usually using a numerical method. Forthe convenience of numerical integration, we will introduce a new variable u = r / g /r / .Throughout this paper, we work in Gaussian units. The mass of the black hole will be denoted as M , and its mass accretionrate will be denoted as ˙ M A . For brevity, we introduce the gravitational radius r g = 2 GM/c with G being the Newtonianconstant and two dimensionless parameters β = e ˙ M A m e m p ω cr g ≈ . × − × (cid:18) λ . (cid:19) (cid:18) M ⊙ M (cid:19) ˙ M A . M ⊙ yr − ! , (9) γ = 4 G ˙ M A c ≈ . × − × ˙ M A . M ⊙ yr − ! . (10)The wavelength of the photon is related to its frequency by λ = 2 πc/ω . III. A TOY MODEL
It is well known that the metric of the Schwarzschild black hole ds = − (cid:16) − r g r (cid:17) c dt + (cid:16) − r g r (cid:17) − dr + r (cid:0) dθ + sin θdφ (cid:1) (11)is a static spherically symmetric solution of the vacuum Einstein equations. In Ref. [8], the refractive effect of a cold plasma onthe shadow of Schwarzschild black holes has been evaluated in a concrete model, where the plasma outside the black hole has amass density ρ ( r ) = ˙ M A πcr / g r / . (12)For a fully ionized hydrogen plasma, the plasma mass density is related to the electron number density by ρ ( r ) = m p N ( r ) .The gravitational field of the plasma has been neglected in Ref. [8]. If we take the plasma’s gravity into consideration, theSchwarzschild metric will be modified into a time-dependent metric. The reasons are as follows. First, owing to the accretion,the mass or equivalently radius of the black hole must increase with time. Second, even if we ignore the accretion, the plasmacannot keep static against the gravitational attraction of the central black hole. Consequently, the modified metric ought to betime-dependent both inside and outside the event horizon. This will prevent us from applying the formulae shown in Sec. II. Inorder to circumvent this obstacle, it is customary to neglect the time dependence and assume [11, 12] A ( r ) = 1 B ( r ) , D ( r ) = r (13)in Eq. (1). Making use of one of the Einstein equations, one can verify that B ( r ) = 1 − Gm ( r ) c r , (14) m ( r ) = (cid:26) M, for r ≤ r g ,M + 4 π R rr g ρ ( r ′ ) r ′ dr ′ , for r > r g . (15)Working out the integral with the mass density (12), we find m ( r ) = ( M, for r ≤ r g ,M + M A cr / g (cid:16) r / − r / g (cid:17) , for r > r g . (16)Corresponding to (16), the spacetime has two horizons. One is the event horizon of black hole located at r = r g , the other isthe cosmological horizon at r = r g (1 − γ )4 γ (cid:16)p − γ + p γ (cid:17) (17)with the small parameter γ defined in (10). Both the photon sphere and the static observer live in the patch of spacetime betweenthe two horizons. Outside the black hole, the above metric coincides with the w = − / subcase in Ref. [13]. However,one should be cautious that the physical interpretations are completely different [14, 15], and here we are studying a differentparameter space to estimate the gravitational effect of the plasma. IV. INFLUENCES ON THE RADIUS
With the model extended, we are now ready to compute the plasma’s refractive and gravitational effects on the shadowaccurately. Before entering on computations, let us make a rough comparison of the two effects. The refractive effect is inducedintrinsically by the electromagnetic interaction. It is textbook knowledge that in a hydrogen atom, the ratio of the gravitationalforce to the electromagnetic force is Gm e m p /e = 4 . × − . Therefore, one would naively expect that the gravitationaleffect is suppressed by this factor in comparison with the refractive effect.Let us take a closer look at the toy model introduced in Sec. III. For this model, we can express the function h ( r ) in terms of β and γ explicitly, h ( r ) = r " − r g r − γr / g r (cid:16) r / − r / g (cid:17) − − r β (cid:16) r g r (cid:17) / . (18)According to Eqs. (5) and (7), radii of the photon sphere and the shadow are determined simply by this function. It is clearthat the refractive effect is controlled by β , while the gravitational effect is dictated by γ . For instance, one can switch off thegravitational effect by setting γ = 0 , then the above expression will reduce to Eq. (50) in Ref. [8]. From this point of view, onewould expect that the gravitational effect is suppressed by a factor γ/β ∼ Gm e m p r g / ( e λ ) in comparison with the refractiveeffect. Intriguingly, here is an enhancement factor r g /λ in competition with Gm e m p /e .As a further step, one can substitute Eq. (18) into Eq. (5) and search for the largest root, but the resulted equation cannot besolved exactly. To get some analytical results, we solve it perturbatively for small β and γ . In this way, we find the second-ordersolution r ph ≈ r + r − r + r − r − r , (19)where r = 32 r g , r = √ βr g , r = − √ ! γr g , (20) r = 75832 β r g , r = − √ ! βγr g , r = √ − ! γ r g . (21)With the help of Eq. (7), we can proceed to compute the radius of shadow observed at a distance r o . To the first order, the resultis R sh ≈ R − R − R (22)in which R = 3 √ r g (cid:18) − r g r o (cid:19) / , (23) R = " √ − √ (cid:18) r g r o (cid:19) / (cid:18) − r g r o (cid:19) βr g (cid:18) − r g r o (cid:19) / , (24) R = √ r o r / g (cid:16) r / + r / g (cid:17) + 98 (cid:16) √ − √ (cid:17) γr g (cid:18) − r g r o (cid:19) / . (25)It is straightforward to find higher order terms. They are very lengthy but negligibly small, and thus not shown here. Remarkably,the gravitational correction to the radius of the black hole shadow is enhanced further by r / /r / g in Eq. (25). In summary, theratios of the gravitational corrections to the refractive corrections are r r ≈ π (cid:16) √ − (cid:17) × Gm e m p r g e λ ≈ . × − × (cid:18) . λ (cid:19) (cid:18) M M ⊙ (cid:19) , (26) R R ≈ √ π × Gm e m p r / g r / e λ ≈ . × (cid:18) . λ (cid:19) (cid:18) M M ⊙ (cid:19) / (cid:18) r o (cid:19) / . (27)To convert the above analytical expressions to numbers, we need the values of λ , r o , M and ˙ M A . Table I is a sample of AGNsselected from Refs. [9, 10]. The data of masses are extracted from Ref. [9], while the data of distances and mass accretion ratesare obtained from Ref. [10]. Making use of the data and ignoring the spins of black holes, we evaluate Eq. (20) for each AGNand draw the results in the left panel of Fig. 1 in logarithmic coordinates, and Eqs. (23), (24), (25) in the right panel. In bothpanels, the uncorrected radii are drawn as asterisks, the refractive corrections are depicted by diamonds, and the gravitationalcorrections are denoted by circles. For all of them, we have set λ = 0 . [3]. This figure can be regarded as a semi-realisticestimation of refractive and gravitational effects of plasma on the radii of the photon sphere and the shadow. We can see clearlythat these effects are negligible despite of the diversity of AGNs in the sample. The right panel is in good agreement with Eq.(27), which shows that the two effects on the radius of shadow are comparable for black holes of mass M ∼ M ⊙ . For SgrA*, the gravitational corrections are most small because of its slowest accretion rate. V. INFLUENCES ON THE INTENSITY
From the conclusion of the previous section, one can infer that the refractive and gravitational effects on the intensity imageshould be undetectable for realistic AGNs. This is due to the smallness of β and γ in Eqs. (9), (10). For this sake, in the currentsection, we will assign unrealistically large values to these parameters. Table I. Masses [9], distances and mass accretion rates [10] for a sample of AGNs.Source Name Distance (Mpc) Mass ( M ⊙ ) Log ( dM/dt ) ( M ⊙ yr − )Cyg A
240 1 × − . NGC1275
75 3 . × − . NGC3227
22 4 . × − . NGC4151
22 4 × − . NGC4261
32 4 × − . NGC4374
20 1 . × − . NGC4486
17 6 . × − . NGC4594
11 1 × − . NGC5548
75 6 . × − . NGC6166
124 3 × − . NGC7469
71 1 . × − . Sgr A* .
008 4 . × − . -20 -15 -10 -5 -15 -10 -5 Figure 1. (color online). Left panel: the radius of the photon sphere and corrections to it according to Eq. (20) and Table I with λ = 0 . .Right panel: the radius of the shadow and corrections to it according to Eqs. (23), (24), (25) and Table I. In both panels, black asterisksrepresent the uncorrected radii, red diamonds denote the refractive corrections, and blue circles mark the gravitational corrections. For the toy model in Sec. III, the obit equation (4) of light rays can be reformed as r g D ( r ) drdφ = ± s r g b (cid:20) − A ω p ( r ) ω (cid:21) − r g AD ( r ) . (28)In terms of a new variable u = r / g /r / , we can write down ω p ( r ) /ω = βu , D ( r ) = r g u − and A = u − (cid:2) u − (1 − γ ) u − γ (cid:3) (29)outside the black hole. For clockwise light rays du/dφ > , the obit equation can be reexpressed in terms of u as u dudφ = r r g b (1 − βu A ) − u A. (30)When performing the numerical integration, the model parameters are assigned exaggerated values r o = 10 r g , β = 0 or . , γ = 0 or . as labeled in Fig. 2.Let us consider an observer located outside the photon sphere. According to the ratio b/h ( r ph ) , the light rays arriving at thisobserver can be classified into three types [16] which are depicted by different line-types in Fig. 2: (i) Light rays with b < h ( r ph ) can travel through the photon sphere. As illustrated by green solid curves in Fig. 2, all orbits of these rays start near the eventhorizon of the black hole. (ii) A critical light ray has b = h ( r ph ) . It propagates in an unstable circular orbit of radius r ph andhas a chance to escape to the observer under radial perturbations. The circle is a great circle of the photon sphere. In every leftpanel of Fig. 2, we plot such a critical light ray as a red dotted curve. (iii) Light rays with b > h ( r ph ) are always outside thephoton sphere. As depicted by blue dashed curves in Fig. 2, each orbit of such rays is symmetric with respect to a diametricalline through its pericenter. The radial coordinate r min of the pericenter is dictated by h ( r min ) = b .It is reasonable to expect that the radiant energy density is proportional to the plasma density. This implies a specific emissivity j ( ν, r ) ∝ r / . (31)For simplicity we assume the emission is uniform in frequency. Then the specific intensity at the point ( x, y ) of the observedimage is [13, 17, 18] I β,γ = Z ray A ( r ) / r / s B ( r ) + r (cid:18) dφdr (cid:19) dr. (32)Here the normalization is unimportant, and the subscripts are used to reminding us the dependence of intensity on β and γ .Numerically it is more convenient to perform this integration in terms of u defined above, I β,γ = Z ray A / s A + u (cid:18) dφdu (cid:19) du. (33)The integration is performed using the backward ray shooting method [16, 17]. For type (i) light rays, we integrate Eq. (33)from the observer to the event horizon of the black hole. For type (iii) light rays, the integration is performed from the observerto the pericenter and then to the cosmological horizon (17). In practice, telescopes do not collect light rays with π/ < α ≤ π ,thus we assume ≤ α ≤ π/ in our simulations.By virtue of the spherical symmetry of the toy model, the image is rotationally invariant, and thus the intensity is dependentsimply on R = (cid:0) x + y (cid:1) / . By definition Eq. (8), it is easy to show R = b A − βu A (34)with u = r / g /r / . By continuously changing the value of b , we have computed the dependence of intensity on R and simulatedthe images with r o = 10 r g , β = 0 or . , γ = 0 or . . The slight differences between these images are not easy to notice,especially near the ring. In the top right panel of Fig. 2, we present the image for I , . In the middle right and bottom rightpanels, we subtract I , from I . , and I , . to illustrate the slight differences. In comparison with the case of β = 0 , γ = 0 ,the intensity outside the ring decreases more significantly in the case of β = 0 , γ = 0 . than the case of β = 0 . , γ = 0 , butthe changes in radius are about the same. VI. DISCUSSION
In this paper, we have investigated the gravitational effect of a plasma on the radius and the intensity of the black hole shadow.We did this in a toy model of a Schwarzschild black hole surrounded by a static plasma of density ρ ( r ) ∝ r − / . Insertingobservational data for a sample of AGNs into our analytical results, we find the gravitational correction R to the shadow radiusis in the range − ∼ − . In contrast, we find the correction R from the plasma’s refractive effect decreases prominentlyas the black hole mass increases, covering the range − ∼ − .Strictly speaking, our analytical results are reliable only for the static toy model, so the numbers above are very rough esti-mations for realistic AGNs which usually have nonnegligible spins [19, 20]. However, as a concrete example, they demonstratethat the gravitational effect of the plasma can exceed the refractive effect under some circumstances. The gravitational effectis more complicated to compute than the refractive effect. If the refractive effect is proven to be detectable in a more realisticmodel, then the gravitational effect should be scrutinized, especially for AGNs of high masses.We did not consider synchrotron self-absorption [5], which is desirable when studying the background emission of the plasma.It would also be interesting to take the black hole spin [21–24] and the QED effect [25] into account. [1] LIGO Scientific, Virgo
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