Observations of Orbiting Hot Spots around Scalarized Reissner-Nordstr\"om Black Holes

This paper investigates the observational signatures of hot spots orbiting scalarized Reissner-Nordstr\"om black holes, which have been reported to possess multiple photon spheres. In contrast to the single-photon sphere case, hot spots orbiting black holes with two photon spheres produce additional image tracks in time integrated images capturing a complete orbit of hot spots. Notably, these newly observed patterns manifest as a distinct second-highest peak in temporal magnitudes when observed at low inclination angles. These findings offer promising observational probes for distinguishing black holes with multiple photon spheres from their single-photon sphere counterparts.

Recent observations and numerical simulations provide compelling evidence for the formation of hot spots surrounding supermassive black holes.These highly energized regions are often associated with magnetic reconnection and flux eruptions within magnetized accretion disks [19][20][21].Notably, recurrent observations have detected hot spots in close proximity to Sgr A* [22][23][24].Additionally, a notable instance is the detection of an orbiting hot spot within unresolved light curve data captured at the EHT's observing frequency [25].Crucially, the origin of these hot spots within the compact region near the Innermost Stable Circular Orbit (ISCO) makes them invaluable for probing black holes in the extreme gravity regime [25,26].
Intriguingly, scalarized RN black holes can harbor multiple photon spheres outside the event horizon within specific parameter ranges [54].This unique feature has sparked intensive research on the optical appearances of various phenomena in their vicinity, including accretion disks [54][55][56], luminous celestial spheres [57] and infalling stars [58].Studies have revealed that an additional photon sphere significantly amplifies observed accretion disk flux, generates beat signals in the visibility amplitude, creates triple higher-order images of a luminous celestial sphere and triggers a cascade of additional flashes from an infalling star.Furthermore, the presence of multiple photon spheres raises concerns about spacetime stability due to the potential for long-lived modes [59][60][61][62][63]. Recent work has shown that these photon spheres can induce superradiance instabilities for charged scalar perturbations [64].Moreover, the existence of two photon spheres outside the event horizon has also been demonstrated in dyonic black holes with a quasi-topological electromagnetic term [65,66], black holes in massive gravity [67,68] and wormholes in the black-bounce spacetime [69][70][71].For a comprehensive analysis of black holes with multiple photon spheres, we refer readers to [72].
This paper explores the observational characteristics of hot spots orbiting scalarized RN black holes, particularly focusing on how the presence of an additional photon sphere impacts these signatures.The subsequent sections of this paper are structured as follows: In Section II, we begin with a concise review of scalarized RN black hole solutions within the EMS framework, discussing geodesic motion and gravitational lensing within these spacetimes.Section III is devoted to the hot spot model, followed by an analysis of time integrated images, temporal fluxes and centroids.
Finally, Section IV presents our conclusions.We adopt the convention G = c = 1 throughout the paper.

II. SCALARIZED RN BLACK HOLES
This section first presents a concise review of the scalarized RN black hole solution within the 4-dimensional EMS model.Following this, we investigate the properties of photon spheres and ISCOs within the black hole spacetime.

A. Black Hole Solution
The EMS model, as outlined in [27], combines a gravity theory with a scalar field ϕ and an electromagnetic field A µ through the action, where R is the Ricci scalar, and In this EMS model, the scalar field ϕ is non-minimally coupled to the electromagnetic field A µ through the coupling function f (ϕ).For the existence of scalar-free black holes like RN black holes, the coupling function must satisfy the condition f ′ (0) ≡ df (ϕ) /dϕ| ϕ=0 = 0 [27,28].This study focuses on the exponential coupling function f (ϕ) = e αϕ 2 with α > 0. Within the RN background, the equation of motion for scalar perturbations δϕ is given by where µ 2 eff = −αQ 2 /r 4 .It is noteworthy that tachyonic instabilities emerge when the effective mass square µ 2 eff becomes negative.As demonstrated in [27,48], these instabilities can be sufficiently strong near the event horizon, inducing the formation of scalarized RN black holes from their scalar-free counterparts.
To find scalarized RN black hole solutions, we employ the following ansatz for the metric and electromagnetic field, In addition, proper boundary conditions are imposed at the event horizon r h and spatial infinity as Here, δ 0 and ϕ 0 can be used to characterize black hole solutions, and Ψ is the electrostatic potential.
By specifying δ 0 and ϕ 0 , we obtain scalarized RN black hole solutions with a non-trivial scalar field ϕ using the shooting method implemented in N DSolve function of W olf ram ®M athematica.
Black hole mass M and charge Q are determined from the asymptotic behavior of the metric functions at infinity For simplicity and generality, we express all physical quantities in units of the black hole mass by setting M = 1 throughout the paper.

B. Photon Spheres
The motion of light in scalarized RN black holes is governed by the geodesic equation, where λ is the affine parameter, and Γ µ ρσ represents the Christoffel symbol.For null geodesics characterized by ds 2 = 0, the radial component simplifies to where L and E are the conserved angular momentum and energy of photons, respectively.In spherically symmetric black holes, unstable circular null geodesics constitute a photon sphere of radius r ph , determined by where b ph is the corresponding critical impact parameter.In other words, a local maximum of the effective potential signifies the presence of a photon sphere.for α = 0.9 and Q = 1.070.This interesting dependence led us, in [72], to delineate the region in the α-Q/M parameter space where double photon spheres emerge.
To investigate the impact of photon spheres on light deflection, we focus on the deflection angle α experienced by a light ray with impact parameter b.For rays traveling from and returning to spatial infinity, the deflection angle is expressed by [73], where r 0 , the turning point, satisfies Notably, previous studies have demonstrated that α diverges logarithmically at critical impact parameters [73][74][75][76].This divergence indicates that light rays undergo multiple circumnavigations of the black hole in the vicinity of photon spheres, essentially trapped in their gravitational influence.
The lower-left panel of FIG. 1

C. Innermost Stable Circular Orbit
Finally, we turn our attention to the ISCO of massive particles in scalarized RN black holes.
The ISCO is widely believed to be the origin of hot spots observed in active galactic nuclei due to synchrotron radiation emitted by matter orbiting it [77][78][79].Analogously to the null case for photons, the metric condition ds 2 = −1/2 for timelike geodesics leads to the following radial equation, where E and L represent the total energy and angular momentum per unit mass of the orbiting particle, respectively.Here, the effective potential for massive particles is defined as Consequently, the energy per unit mass E i , angular momentum per unit mass L i and radius r i of the ISCO are identified by the conditions, Once r i is determined, E i and L i can be obtained from explicit expressions involving the metric functions and their derivatives at r i , , For a hot spot orbiting the black hole at the ISCO on the equatorial plane, its four-velocity takes the form, The corresponding angular velocity and period are Ω e = [N ′ (r i ) − 2δ ′ (r i ) N (r i )] e −2δ(r i ) / (2r i ) and T e = 2π/Ω e , respectively.

III. OBSERVATIONS OF HOT SPOT
This section explores observable signatures of hot spots orbiting scalarized RN black holes.We simplify the analysis by considering an isotropically emitting hot spot as a sphere.Utilizing the computational framework described in [80,81], we place the observer at (r o , φ o ) = (100, π) with an inclination angle of θ o , while the hot spot, with a radius of 0.25, circles counterclockwise on the ISCO at r i .For optimal precision and efficiency, we employ a 1000 × 1000 pixel grid for each snapshot and generate 200 snapshots for a full hot spot orbit.By tracing light rays backward from the observer to the hot spot, we glean observational information for each image plane pixel.
Specifically, at each time t k , each pixel is assigned an intensity I klm , collectively forming lensed images of the hot spot.Our analysis then focuses on the following image properties, as outlined in [79,82,83], • Time integrated image: summarizing the intensity received at each pixel.
• Total temporal flux: representing the total intensity received at time t k .Here, ∆Ω denotes the solid angle per pixel.
• Temporal magnitude: quantifying the relative brightness of each snapshot.
• Temporal centroid: indicating the center of intensity distribution in each snapshot.Here, − → r lm represents the position relative to the image center.This difference arises due to the decreased impact of the Doppler effect at the lower inclination angle.Consequently, the flux becomes less frequency-dependent, enabling the primary image to dominate the total flux contribution for most of the time.Moreover, the influence of higher-order images on the centroids diminishes, causing it to shift toward the center of the primary image's orbit.

IV. CONCLUSIONS
In this paper, we have examined the observable behavior of hot spots orbiting scalarized RN black holes along the ISCOs.Intriguingly, depending on the parameters of the black hole, a scalarized RN black hole may possess either one or two photon spheres [55].Given the substantial influence of photon spheres on black hole imaging, our observations have revealed distinctive characteristics between the single-photon sphere and double-photon sphere cases.
In the single-photon sphere case, the primary observational features align closely with those observed in the Schwarzschild black hole case [81].Specifically, the primary image with n = 0 > traces a closed semicircular track, culminating in a pronounced peak within the temporal flux.
Conversely, in the case of double-photon spheres, photons have the capacity to orbit between the two photon spheres multiple times, resulting in the generation of additional hot spot images situated between the two critical curves.Consequently, when viewed at lower inclinations, these additional images contribute to a distinct secondary peak in the temporal flux.
Through the analysis of these image characteristics, we are able to gain deeper insights into the optical manifestations of hot spots in proximity to black holes.This understanding paves the way for the discrimination between black holes harboring a single photon sphere and those possessing multiple photon spheres.The advent of next-generation Very Long Baseline Interferometry holds promising potential for leveraging our findings as a means of exploring black holes with multiple photon spheres.

FIG. 1 .
FIG. 1.The effective potential of photons (Upper Row) and the deflection angle α as a function of the impact parameter b (Lower Row).Left Column: For α = 0.9 and Q = 1.054, the potential exhibits a single maximum at r ph = 1.610, marking the photon sphere with a critical impact parameter b ph = 3.673.The deflection angle α exhibits a logarithmic divergence at b = b ph .Right Column: With α = 0.9 and Q = 1.070, the potential reveals two maxima at r in = 0.204 and r out = 1.255, corresponding to two distinct photon spheres with critical impact parameters b in = 2.322 and b out = 3.515, respectively.Between b in and b out , the deflection angle α reaches a minimum value greater than 3π, indicating that light rays briefly trapped between the two photon spheres orbit the black hole at least once before escaping.
displays the deflection angle α as a function of b for a scalarized RN black hole with α = 0.9 and Q = 1.054.At large impact parameters, light rays travel far from the photon sphere, resulting in small deflection.However, as b approaches the critical value b ph , α diverges logarithmically, reflecting the capture of light rays by the photon sphere.Note that light rays with b < b ph plunge into the event horizon and are excluded from our analysis.The lower-right panel of FIG. 1 presents the deflection angle α as a function of b for a scalarized RN black hole with α = 0.9 and Q = 1.070.The presence of two photon spheres, marked by dashed red lines at b in and b out (inner and outer, respectively), distinguishes this case from the single-photon sphere case in the lower-left panel.For b > b out , α behaves similarly to the single-photon sphere case, approaching zero with increasing impact parameter.However, due to the combined gravitational lensing effects of both photon spheres, light rays with b in < b < b out can undergo multiple orbital paths around the black hole, trapped between the two photon spheres.This orbital confinement manifests as a minimum in the deflection angle within the range b in < b < b out .

A 3 . 1 ○○ and 2 ○
FIG.2displays the time integrated images of scalarized RN black holes with one or two photon spheres, viewed at inclination angles of 80 • and 50 • .White lines mark critical curves traced by light rays escaping the photon spheres.While the single-photon sphere case shows three prominent image tracks outside the critical curve, the double-photon sphere case exhibits additional tracks between the two critical curves.To decipher the origin of these tracks, we use a numerical count n signifying the number of equatorial plane crossings during a light ray's journey, characterizing its path and resulting image track.Note that, in double-photon sphere cases, photons can orbit between the inner and outer spheres.Therefore, superscripts > and <> denote light rays traveling outside the (outer) photon sphere and orbiting between the inner and outer photon spheres, respectively.Focusing on an observer at θ o = 80 • , the upper-left panel of FIG.2presents the time integrated image of the hot spot for a single-photon sphere scalarized RN black hole with α = 0.9 and Q =

Figure 5
Figure 5 displays the temporal magnitudes and centroids for an inclination angle of θ o = 50 • .In contrast to the θ o = 80 • case, only a single peak is visible in the temporal magnitudes for θ o = 50 • .

5 .
Temporal magnitudes m k (Upper Row) and centroids c k (Lower Row) plotted against t/T e for the single-photon sphere (Left Column) and double-photon sphere (Right Column) cases at an inclination of θ o = 50 • .Due to the reduced influence of the Doppler effect at lower inclinations, both cases display a single peak in the temporal magnitudes, and the centroids are shifted towards the center.