Circular Orbit Structure and Thin Accretion Disks around Kerr Black Holes with Scalar Hair

The spacetimes we investigate are described from first principles by minimally coupling a complex scalar field to gravity via the action

where R is the Ricci curvature, g is the metric determinant, and Φ is a complex scalar field whose mass and self-interaction are determined by the potential U. Therefore, the scalar field acts solely as a source field of nonbaryonic matter and the underlying theory is still GR. The Einstein field equations are obtained as usual by varying the action with respect to g^{μ ν} ,

where the right-hand side is simply the energy-momentum tensor of the scalar matter. Varying the action with respect to Φ and Φ* yields a pair of Klein–Gordon equations,

to be solved simultaneously with the field equations. The system is invariant under U(1) transformations, Φ → Φe^{i α} , and therefore possesses a conserved Noether current,

which, when projected over the future timelike normal direction n_μ and integrated over the spacelike three-volume bounded by the horizon surface and infinity , gives a conserved charge,

where dV is the volume element.

We are interested in stationary and axisymmetric solutions, and choose a metric form in adapted spherical coordinates {t, r, θ, φ} given by

for which {F₀, F₁, F₂, ω} is the set of functions dependent on both r and θ that we solve for, and , where r_H is the coordinate size of the horizon. In these coordinates, the Killing vector fields are ξ^μ = ∂ _t and χ^μ = ∂ _φ , corresponding respectively to stationarity and axisymmetry. The chosen aesthetics of the line element is not of importance for the work developed here, but the choice of coordinates is. Therefore, we will refer to the metric functions written in a more general way,

The underlying mechanism to give rise to scalarization is a superradiant instability that only occurs for rotating systems. To level the scalar field with a fluid it is necessary to assign to it a four-velocity, and therefore, it must depend on all four spacetime coordinates. Stationarity and axisymmetry, however, require that the field’s dependence on t and φ be given in an explicit way such that

where ω_s is its natural frequency and m is the winding number, belonging to the integer set. A further criterion for scalarization is obtained by evolving the linearized version of Equation (3) on a fixed Kerr background and observing where the threshold between stable and unstable modes lies, i.e., where ω_s is real, which is demanded for stationary solutions. This occurs only when the scalar field’s angular velocity matches that of the horizon, explicitly

As for the potential, we consider noninteracting fields

where μ is the mass of the boson, which scales the equations through the transformations

which are implied in the rest of the text.

The Arnowitt–Deser–Misner mass and angular momentum can be extracted asymptotically from the metric functions’ leading terms at infinity,

and can also be calculated through the Komar integrals defined with the spacetime Killing vector fields, which, after explicitly breaking down to the contributions given by the hole itself (M_H , J_H ) and the scalar hair (M_ϕ , J_ϕ ), give

where is the horizon surface, dA_H its area element, and σ^μ a spacelike vector perpendicular to it such that σ^μ σ_μ = 1.

The angular momentum stored in the hair can be expressed as an integer multiple of the total charge, J_ϕ = mQ. It is useful then to define a normalized charge as a measure of how hairy a particular solution is, q ≡ J_ϕ /mJ_H . The higher the winding number m is, the more angularly excited the solutions become. Hence, we restrict our analysis to, the case m = 1.

In the thin-disk approach, each volume element of the fluid moves freely in a circular orbit on the equatorial plane, while the vertical profile of the disk and the accretion process itself are taken in a phenomenological fashion; e.g., see Section 2.3. Hence we revisit next the equations describing this kind of geodesic motion and how the physical quantities are related. On the equatorial plane θ = π/2, d θ = 0 and the line element reads simply

Each constituent of the fluid is a timelike particle in circular motion whose four-velocity is u^μ = u^t (ξ^μ + Ωχ^μ ), where Ω:=u^φ /u^t is the orbital velocity. Due to stationarity and axisymmetry, the particle’s normalized energy E = − p_t and angular momentum L = p_φ are conserved. Therefore, we have

The four-velocity norm u^μ u_μ = −1 can be used to derive the equation of motion for the radial coordinate, and it gives

where we define the effective potential V_eff in terms of the potentials V_±

This differential equation can be solved with a suitable initial condition for pairs of energy and angular momentum to describe geodesics on the equatorial plane. Circular orbits, characterized by a fixed radius, demand that both V_eff and ∂_r V_eff be zero. Thus, with these two constraints, there is a fixed pair of energy and angular momentum for each circular orbit radius. Using the angular velocity as a constraint, we can write

The ISCO dwells at the smallest radius for which the orbit is marginally stable, meaning . Thus, the equation

together with the aforementioned constraints, is satisfied at r = r_ISCO.

The thin-disk approach was developed in relativistic form in the early 1970s (Novikov & Thorne 1973; Page & Thorne 1974). The equation governing the phenomenon are derived from a series of simplifying, yet fairly reasonable assumptions. For the characteristic timescale of the physical process, the background spacetime is unchanged, being stationary, axisymmetric, asymptotically flat, and even with respect to reflection onto the equatorial plane. The disk has negligible self-gravity, its central plane lies precisely in the equatorial plane, and it is thin, i.e., the height of the accreting disk is negligible compared to its horizontal extension at any given radius, h ≪ r. Furthermore, radiation is only emitted perpendicularly to the disk’s plane, namely, it is optically thick. The particles that constitute the fluid are in Keplerian motion and the infalling matter is therefore treated phenomenologically with a mass-accretion rate and four-velocity radial component u^r placed in an ad hoc manner. The inner edge of the disk is given by the radius of the ISCO, and we impose no restriction on the outer boundary other than convergence for the quantities of interest. In this context, the system is in a steady state, hence is independent of the radial coordinate. The radiation energy F emitted by the disk’s surface is given by

and some of the steps necessary for deriving it are recalled in the Appendix. Because the system is in steady state, it must be in thermodynamical equilibrium emitting as a blackbody whose temperature profile obeys the law F = σ T⁴, where σ is the Stefan–Boltzmann constant. Because for Kerr BHs the flux scales with the BH’s mass and accretion rate, we define a normalized flux to better compare profiles yielded from spacetimes of different spin parameters via

where M_g = GM/c².

The amount of gravitational energy converted into radiation as a particle falls down from infinity all the way to the BH can be quantified in terms of the efficiency of the central body in doing so. If all photons can escape to infinity, the efficiency can be measured as the difference between the normalized energy measured at infinity and at the ISCO,

The efficiency for the Schwarzschild BH is just below 6%, while for Kerr BHs it increases with the spin parameter reaching 42% in the extremal case. The observed luminosity for the frequency ν is

where I is the Planck’s distribution function, h is the Planck constant, k_B is the Boltzmann constant, γ is the disk’s inclination and ν_e is the emitted frequency that gets redshifted on its way to the observer, who measures ν = ν_e /(1 + z), where

The expression between the equality signs in Equation (25) above contains a d² factor, where d is the spatial distance between the source and the observer. Nevertheless, the final expression at the right-hand side of the second equality sign is independent of it for the original element of the solid angle that enters the integral is proportional to d⁻², e.g., , Bhattacharyya et al. (2001). Note, also, that we ignore any bending of light and assume the observer is at the asymptotic flat region. Even though light bending produces large discrepancies with respect to this approach near the ISCO in the general case, the approximation is well justified for very small inclination angles where these effects stop being predominant Luminet (1979). Hence, we shall only work with a face-on configuration, namely γ = 0.

On the equatorial plane of most of the studied stationary spacetimes, especially in general relativity, and constraining the analysis to the regions exterior to a horizon (if there is any), the three metric functions that appear in these equations are monotonic, with the clear exception for manifolds with naked singularities. As a consequence, one might intuitively and naively take for granted that

• 1.  
  For all r > r_ISCO, there exists a circular orbit and it is stable.
• 2.  
  For all circular orbits, Ω₊ > 0 while Ω₋ < 0, and therefore, they correspond to the angular velocities of corotating and counterrotating orbits, respectively.

Complex scalar fields are angularly excited upon rotation, in a very similar fashion to the hydrogen atom with increasing magnetic quantum number from zero to one. Thus, the field distributes itself in a torus, with its energy density behaving the same way. This off-center energy configuration causes the spacetime to warp in unusual ways and, in the presence of a BH at the center, the metric functions might feature different local maxima and minima. The listed items above then fail to be true for a set of solutions, and other peculiar features arise, which we shall discuss in detail below. They include, for example, regions where, for a fixed radius, there are two prograde orbits with different angular velocities and regions where no (inertial) circular orbits are possible at all.

The existence domain of KBHsSH is displayed in Figure 1 in an M versus ω_s diagram. The region is bounded by three qualitatively different sets of solutions. The blue curve corresponds to the clouds, namely Kerr BHs with a marginally bound scalar cloud beyond which the instabilities grow and stationary scalarized solutions appear in the nonlinear regime. On this curve, the scalar cloud is not backreacting, and therefore, the charge is zero, q = 0. The red curve features no horizon, i.e., it is the set of pure solitonic solutions where q = 1. Finally, the green curve represents the extremal hairy solutions and one expects it to join the solitonic curve after swirling around each other, in a point near the region in the graph where both stop (Herdeiro & Radu 2014b, 2015; Collodel et al. 2020a). Accessing it numerically is notoriously difficult, though. Kerr BHs exist from the solid black curve (extremal holes) below, where the parameters obey

Within the scalarized solutions, there are four overlapping shaded regions where circular orbits on the equatorial plane show unusual features. The grayish area encloses the solutions for which g_tt has a local maximum outside an ergoregion, and hence, there is a ring of points where Ω₋ = 0 called the static ring. In the greenish region, which overtakes the whole static ring area, lie the solutions that contain an interval of r with degenerate prograde stable orbits, i.e., BHs that at certain distances of the horizon have Ω₋ > 0 and both orbits are stable. The next shaded region in pink features solutions where not all corotating circular orbits for r > r_ISCO are stable. The last particular set of solutions, in orange, consists of those for which, in certain places (again for r > r_ISCO), an inertial circular trajectory is not possible. These traits can be readily understood from the equatorial profile of the metric functions, as analyzed below.

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3.2.1. Static Ring

On the equatorial plane, if g_tt has a local extreme in a region where it is negative (with the metric signature here adopted), then it is clear from Equation (20) that Ω₋ = 0 at these points. We remark that if g_tt > 0, this is an ergoregion, and static orbits are then only possible for spacelike particles. Indeed, the lower boundary of this region in Figure 1 is characterized by the appearance of this second ergoregion with toroidal topology (in addition to the S² one in the vicinity of the horizon), where the solutions feature an ergo-Saturn as discussed in Herdeiro & Radu (2014a). The transition solutions, i.e., those for which the local maximum of g_tt occurs for g_tt = 0, contain a ring where massless particles of zero energy can be found inertially at rest, the so-called light points (Grandclément 2017). In Figure 2, we depict the profiles of g_tt and g_{t φ} for a solution with such a feature. In these spacetimes, there is both a local minimum and maximum for g_tt , and it is possible for a test particle to remain at rest in these regions with respect to a static observer at infinity.

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3.2.2. Degenerate Prograde Stable Orbits

As discussed above, there are certain KBHsSH for which on the equatorial plane there exists a region with no retrograde orbits, i.e., Ω₋ > 0. According to Equation (20), if the slope of g_tt is positive, then the quantity in the square root will always be less than ∂_r g_tt , because the slope of g_{φ φ} (which is monotonic) is always positive. Observing g_tt in Figure 2, its slope is positive between the two local extrema where the static rings live. Furthermore, it is possible that in some interval of the radial coordinate, both prograde orbits are stable. Therefore, there are degenerate prograde stable orbits, a feature worth discriminating in the parameter space of Figure 1. To better illustrate this behavior, we provide an example in Figure 3 where both Ω_± are plotted (only the positive values) and the stable regions are displayed. In this particular case, most of the radial interval with no retrograde orbits is characterized by stable orbits for both Ω_±.

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3.2.3. Discontinuous Prograde Stable Orbits

3.2.4. Discontinuous Existence of Circular Orbits

The three particular cases investigated above arise from the fact that g_tt is not monotonic as it usually happens in the exterior region of BHs, that is, it can interchange between negative and positive slopes. Things can get even more interesting when the slope is sufficiently positive such that the square root of Equation (20) becomes zero. This equality occurs when the dragging of the spacetime hits a local extreme and at this point Ω_± = ω and locally g_{t φ} ∝ g_{φ φ} . An inertial observer that rotates along with the dragging of the spacetime must have zero angular momentum (L = 0) and therefore is called a zero angular momentum observer (ZAMO). Usually, ZAMOs have nonzero radial velocity and are therefore not in circular motion. In contrast, as can be seen from Equations (17) and (16), in order for them to be in a circular orbit, their energy must be zero. One should also note that ω is the minimum and maximum value that Ω₊ and Ω₋ can achieve, respectively. After this point, Ω_± becomes complex and circular orbits are no longer possible for inertial particles until the slope of g_tt becomes small enough for yet another point of circularly orbiting ZAMOs to appear. Again, this phenomenon is due to the off-center energy density distribution of the scalar hair. There is a gravitational potential well for each hole and hair, and angular momentum only contributes to a centrifugal force that prevents the particle from maintaining itself at a fixed radius without accelerating. Figure 4 depicts such interrupted existence of circular orbits. Regions where retrograde circular orbits cease to exist have also been found for axion BSs (Delgado et al. 2020a) and Kerr BHs with axion hair (Delgado et al. 2020b).

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The nonuniqueness of the scalarized solutions for a pair of mass and angular momentum is reflected on the profile of the radiant energy flux for a fixed value of j. In what follows, we present several cases specified by a certain spin parameter and plot it within the flux for BHs of different normalized charges. Although it is not possible to find a general pattern, some particular traits appear in each set. Near the marginally bound clouds, the maximum of the flux decreases with increasing charge, while the opposite is seen in regions far from the clouds in the M versus ω_s diagram. The solutions span a large interval of j, but a more prominent set of qualitatively different solutions is concentrated at values slightly lower than the Kerr bound j = 1. In Figure 5, we offer a general view of the maximum value of the normalized flux in the same mass versus ω_s diagram we displayed before. Some curves of fixed j are shown for illustration purposes, and we note that for any value j ≤ 1, there will be a small sample of solutions starting at the clouds that are not depicted because they require much higher resolution. Clearly, their sequences are not connected to their respective (same j) sets displayed in different regions above the extremal Kerr line. The exception happens for the particular case of j = 1 that forms two curves connected at the intersection of three points, namely where the lines of extremal Kerr, extremal KBHsSH, and clouds meet. Another interesting case shown here is that of j = 0.8, where far from the clouds the solutions lie somewhat diffusely on a certain region rather than forming a curve. Thus, for a chosen value of the spin parameter, there might be solutions severely different from one another, which can result in radiant energy fluxes of different orders of magnitude. On the lower-left corner of the diagram, the normalized flux for Kerr BHs is shown, and as it is known, it increases monotonically with the spin parameter, e.g., see Equation (27). From the figure, we observe that the maximum normalized flux for KBHsSH can be an order of magnitude higher than that of extremal Kerr BHs.

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The set of solutions starting at the clouds display a profile with decreasing maximum of the flux for increasing charge, and in this case, the bald Kerr BHs feature the strongest radiant energy flux. Sets of lower values of the spin parameter exist only in this region, and an example with j = 0.75 is given in Figure 6, where, as the charge grows larger, the profile flattens out.

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This behavior changes in other regions of the parameter space, as can be seen from Figure 7 where six different examples are given. The top-left panel displays the flux for solutions featuring j = 0.8, which are far from the clouds. As mentioned above, this is a special case where the set does not form a curve crossing a large area of the parameter space in Figure 5 but is concentrated in a more limited region where the solutions are quite similar. Hence, a single behavior is observed in the flux profile as the normalized charge varies, i.e., the larger it is, the higher the peak of the flux gets. The top-right and middle-left panels show solutions for which j = 0.90 and 0.95, respectively. Here we enter the regime where a wider region of the parameter space is crossed by a curve of fixed j. As we notice from the pictures, there is not a unique way of how the flux profile scales with the scalar charge. In the panel for j = 0.90, we display an inset to show the flux corresponding to Kerr, along with those solutions close to the cloud for which the maximum of the flux decreases with increasing charge. More importantly, we notice that the highest flux peak found is over two orders of magnitude greater than the Kerr peak. The middle right panel depicts solutions for which j = 1, a quite special case because here the curve lying near the clouds is connected to the one existing above the extremal Kerr line. Furthermore, we observe that the fluxes are of the same order of magnitude. The bifurcation from the extremal Kerr is made clear: for the solutions lying above this line, the peak of the flux increases with the scalar charge, whereas for those below it, the peak decreases with increasing charge. Nevertheless, we remark that, interestingly, the radius of the ISCO increases in both cases. Finally, in the bottom panel, we show fluxes for solutions with a spin parameter above the Kerr bound, which never approach the clouds but remain mainly in a narrow region of the parameter space for high values of ω_s . In these cases, the peak increases with the charge, and we note that, albeit hard to establish numerically, the higher j > 1 is, the larger is the lowest charge found in the solution set.

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BHs offer the most efficient mechanism for converting rest energy into radiation. As stated earlier, this parameter is given by the difference between the particle’s normalized energy at infinity—which is one—and the energy at the ISCO, see i.e., Equation (24). Figure 8 offers the efficiency of KBHsSH against j, where the limiting cases of Kerr BHs and solitons are highlighted in two different curves. Heeding the peculiarities discussed in Section 3, we consider only regular solutions, those lying outside the shaded region of Figure 1. This parameter space is not unique and several solutions can be found for a given pair of (j, ).

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Given a specific value of j < 1, KBHsSH can always be found below the Kerr curve. At the Kerr bound (and above), all solutions are less efficient than the bald Kerr, in agreement with Figure 7, as the radius of the ISCO increases with the scalar charge. However, for the interval j ∈ (0.8, 1.0), we find a range in the diagram well above the Kerr line, and the highest efficiency found exceeds 90%. This is only possible for BHs with a very small horizon radius and very large scalar charge as they behave more strongly as solitons than BHs, and the ISCO is found at very small radii.

Let us now evaluate the observed luminosity for a few scalarized BHs. We assume the central object to be a supermassive BH of M = 2.5 × 10⁶ M_⊙ and yr⁻¹, with the plane of the disk oriented at zero degrees with the Euclidean line path to the observer. These values are chosen for illustration purposes only, but we stress that they are fairly conservative (much below the Eddington limit) and reasonable for representing supermassive BHs. We also consider four different cases of fixed spin parameters separately, i.e., j ∈ {0.75, 0.9, 1.0, 2.0}. The results are displayed in Figure 9, together with the curves corresponding to the respective Kerr BH (when applied). For the chosen mass scale and accretion rate, the luminosity peaks at the visible–UV interface part of the electromagnetic spectrum, while stellar-mass BHs with similar mass-accretion rate ratio would lie on the X-ray band. In all cases below the Kerr bound, some solutions naturally show negligible differences with the bald BH case (also part of the solution set). As we get farther away from the clouds in the parameter space, though, observable discrepancies arise.

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The particular case of j = 0.75 has a peak in luminosity for q = 0.0 (Kerr BHs). Recall that the efficiency for this spin parameter decreases with the charge, and so does the radiant energy flux. The luminosity curves intersect each other, increasing and decreasing faster for larger charges as the frequency increases. Therefore, at lower frequencies, ν L is higher for larger charges and at higher frequencies, ν L is higher for lower charges. The largest difference in the peak of luminosity between the curves is 36.7% whereas for the peaking frequency, it is 31.6%.

As previously shown, there are many qualitatively different solutions with fixed spin parameters in the interval j ∈ (0.8, 1.0). This is apparent in the luminosity profile as curves can fall either below or above the Kerr case. For j = 0.9 in the examples depicted in the figure, the peak of the luminosity is highest for q = 0.91 and lowest for q = 0.55, with a relative difference of 77.2%, and of 53.3% in the peaking frequency. Extremal Kerr displays the highest luminosity peak among the curves of j = 1.0, but interestingly not the highest-peaking frequency, which is offset by 14.4%. The largest relative difference in the peak of luminosity is of 49.4%. Above the Kerr bound, most solutions are highly scalarized and much closer to a solitonic star than to a bald BH. For j = 2.0, the peak of the luminosity is proportional to the charge, while the peaking frequency decreases with it. In all cases, the differences are much higher at the end of the observed spectrum, and the interval for the cutoff frequencies (where ν L = 0) grows consistently with j, with an interval of 1.25 × 10¹⁶ Hz for j = 0.75 and of 1.5 × 10¹⁷ Hz for j = 2.0.