Instability of the massive Klein-Gordon field on the Kerr spacetime
We investigate the instability of the massive scalar field in the vicinity of a rotating black hole. The instability arises from amplification caused by the classical superradiance effect. The instability affects bound states: solutions to the massive Klein-Gordon equation which tend to zero at infinity. We calculate the spectrum of bound state frequencies on the Kerr background using a continued fraction method, adapted from studies of quasinormal modes. We demonstrate that the instability is most significant for the , state, for . For a fast rotating hole () we find a maximum growth rate of , at . The physical implications are discussed.
The issue of black hole stability was first addressed over fifty years ago. In an influential study, Regge and Wheeler Regge-1957 showed that the Schwarzschild solution is stable. If a Schwarzschild black hole is perturbed slightly, then the perturbation will oscillate and die away, rather than grow, over time Vishveshwara-1970 . Some fraction of the initial perturbation is absorbed through the event horizon, and the remainder is radiated away to infinity. The demonstration of stability contributed to a growing belief that black holes could be more than mere mathematical curiosities.
When Kerr Kerr-1963 published a new black hole solution, describing the gravitational field induced by a rotating point mass, the stability question was addressed anew. Press and Teukolsky Teukolsky-1972 ; Press-1973 showed that the Kerr solution remains stable under gravitational perturbations. However, the situation is complicated somewhat by an effect known as superradiance. Certain perturbations are enhanced by the rotation of the hole. The energy radiated away to infinity may actually exceed the energy present in the initial perturbation. In effect, perturbations may extract rotational energy from the hole. Since superradiance affects classical fields, it cannot be dismissed as a purely quantum phenomenon.
Various authors have shown that, for superradiance to occur, the oscillation frequency of the perturbation must be less than a critical value , given by
Here, is the black hole mass, is the rotation rate of the hole, is the azimuthal number of the perturbation, and is the radius of the outer event horizon.
Press and Teukolsky Press-1972 postulated that, if the superradiance emerging from a perturbed hole were reflected back onto the hole, then an initially small perturbation could be made to grow without bound. This is the so-called “black hole bomb” idea Cardoso-2004-bomb . To reflect the radiation, Press and Teukolsky suggested using a special arrangement of mirrors. However, reflection will also occur naturally if the perturbing field has a rest mass Damour-1976 ; Zouros-1979 ; Detweiler-1980 ; Furuhashi-2004 ; Cardoso-2005-superr . In this case, a spectrum of ‘bound states’ is present Lasenby-2005-bs (also referred to in the literature as ‘resonances’ Damour-1976 or ‘quasi-stationary levels’ Gaina-1993 ). Bound states are localised in the black hole potential well and tend to zero at spatial infinity. Inevitably, bound states have complex frequencies, as flux passes one way through the (outer) event horizon. The imaginary part of the frequency determines the rate at which the perturbation decays (or grows) with time. The bound states idea has been explored by a number of authors Deruelle-1974 ; Damour-1976 ; Ternov-1978 ; Ternov-1980 ; Gal'tsov-1983 ; Gaina-1992 ; Gaina-1993 ; Lasenby-2005-bs ; Grain-2007 ; Laptev-2006 over the years.
The nature of the bound state spectrum and the superradiant instability depend on two parameters. The first is the rotation speed of the hole . The second is the (dimensionless) product of the black hole mass and the field mass . The product is equivalent to the ratio of the event horizon size to the perturbing field’s Compton wavelength,
Some years ago, Detweiler Detweiler-1980 studied the growth rate of bound states in the limit . He estimated an e-folding time for the first co-rotating state (, ) to be
Zouros and Eardley Zouros-1979 computed an approximation for the growth rate in the opposite limit, , using the JWKB approximation. They found
Given these results, when might the instability be significant? For a pion around a solar-mass black hole, . Clearly, from (4), the instability is insignificant for astrophysical black holes, unless there exists an unknown particle with a tiny but non-zero rest mass. However, the instability may be important for primordial black holes Zouros-1979 . Thus, an investigation of the instability in the regime is well-motivated. A numerical study of the Kerr-Newman instability in this regime was recently conducted Furuhashi-2004 . In this paper we develop alternative methods to study the uncharged case.
This paper has four broad aims. First, to show the existence of a spectrum of bound states of the massive Klein-Gordon field on the Kerr background. Second, to show that these bound states decay with time if , but grow with time if . Third, to show that bound state frequency spectra may be calculated numerically using a simple continued-fraction method Leaver-1985 , previously used to compute quasinormal modes (QNMs). Fourth, to compute accurate upper limits on the bound state growth rates, and estimate the physical implications.
The remainder of the paper is organised as follows. Section II examines the analytic properties of the massive Klein-Gordon field on the Kerr background. To begin, we introduce two alternative coordinate systems, and separate the field into radial and angular parts (II.1). Next, by considering the flux of stress-energy across the outer horizon, we derive a simple expression for the time-evolution of the field (II.2). Then, we set up the wave equation on the Kerr background (II.3) and define boundary conditions for bound state modes (II.4). Section III outlines the continued-fraction method Leaver-1985 which we employ to compute bound state spectra numerically. In section IV we present the results of this numerical approach. Bound state spectra are presented in IV.2 and IV.3, and the Kerr instability is examined in detail in IV.4. We conclude in section V by discussing the physical relevance of our results, and some prospects for further work.
Note that throughout this paper we adopt the spacetime signature , and natural units, .
Ii The Klein-Gordon field on the Kerr background
In this section we examine the (classical) Klein-Gordon field on the Kerr black hole background. First, we introduce two alternative coordinate systems to describe the Kerr spacetime, and discuss the separation of variables in these systems. Next, we look at the field’s stress-energy tensor, which satisfies a simple conservation law. By applying Gauss’ theorem, we show that the time-evolution of the field is directly related to a superradiance condition Zouros-1979 . Then, we formulate the wave equation on the Kerr background, and examine the asymptotic behaviour of the field as and . Finally, we define boundary conditions for bound states, and show that in the non-relativistic limit, the spectrum of frequencies is hydrogenic Lasenby-2005-bs .
ii.1 Coordinate systems
The Kerr spacetime is described by a number of coordinate systems. The most commonly used are Boyer-Lindquist coordinates, , for which the line element takes the form
The Kerr solution has two event horizons, at , and two stationary limit surfaces, at . An advantage of Boyer-Lindquist coordinates is that the metric has only one off-diagonal term, , whereas a disadvantage is that it takes an infinite coordinate time for ingoing geodesics to cross the outer horizon at .
Perhaps of more interest from a physical point of view are ingoing Kerr coordinates, , in which the contravariant metric tensor takes the form
In this coordinate system, the ingoing principal null geodesics are straight lines, , and ingoing geodesics pass through the outer horizon in a finite coordinate time.
The and coordinates are the identical in both coordinate systems, and and are related to and by
Following Brill-1972 , we assume that the Klein-Gordon field may be separated into a product of one-dimensional functions,
The frequency is permitted to be complex. The sign of determines whether the solution is decaying () or growing () in time.
Since is a scalar, it is unchanged by a (passive) coordinate transformation. However, the separation of variables is modified, so that
Thus, the radial functions are related by
ii.2 Stress-Energy Conservation and Time-Evolution
Here we seek to relate the time-evolution of the field to the superradiance condition. To do so, we integrate the stress-energy of the field over a four-volume with its boundary at the (outer) event horizon. A similar argument was outlined in Zouros-1979 .
The Lagrangian density of a minimally-coupled (complex) scalar field is
where . The symmetric stress-energy tensor is
The stress-energy tensor satisfies the conservation law . Since the Kerr spacetime is stationary, there exists a Killing vector conjugate to the time coordinate with components . The Killing vector is timelike outside the stationary limit surface at , but spacelike within. Contracting the stress-energy tensor with this Killing vector and using Killing’s equation yields
We now seek to relate the flux of stress-energy crossing the horizon to the time-evolution of the field. For this, we use the ingoing-Kerr coordinate system. Let us construct a semi-infinite four-volume: an infinitessimal time-slice of width exterior to the outer event horizon. This is bounded by upper and lower hypersurfaces ( and ), and a null hypersurface at the horizon (). Applying Gauss’s theorem to (16), and taking the limit , leads to
where the surfaces and , and normal vectors and are defined by
To proceed further, we introduce the separation of variables (11). The time-derivative results in an overall factor of on the right-hand side of (17). The integral over the horizon is straightforward to compute:
Here we have assumed the angular functions are normalised so that . Hence, the imaginary part of is
Therefore, unstable (exponentially-growing) states are possible if
The validity of this argument rests on three conditions. First, that the radial function is finite at the horizon. Second, that the radial solution is normalisable over . Third, that the integral in the denominator of (21) is non-zero and positive. In section II.4, we show that the boundary conditions imposed on the bound states ensure that the first two conditions are met. The third condition requires further analysis. In the ingoing-Kerr frame, the “energy” component of the stress-energy is
where is a radial current given by
is positive everywhere outside . However, can be negative inside the ergosphere (where the Killing vector is spacelike) if there is sufficient coupling between radial and azimuthal currents. If were to pass through zero then the growth rate given by (21) would tend to infinity! However, this is not observed in the numerical results of section IV.
ii.3 The Wave Equation
In this section we separate the Klein-Gordon field equation in Boyer-Lindquist coordinates on the Kerr background Brill-1972 , and examine its asymptotic behaviour as and .
The wave equation for a scalar field of mass is
where . Employing the Boyer-Lindquist metric (5),
Decomposing the field with the ansatz (11) leads to ordinary differential equations for the radial function
and the angular function
From here on, we often set , so that and are measured in units of , and and in units of .
The angular solutions are spheroidal harmonics Abramowitz-1972 ; Berti-2005 , . The degree of spheroidicity depends on the parameter . In the non-rotating limit, the spheroidal harmonics reduce to spherical harmonics, , and . Numerical methods for computing the angular separation constant are discussed in section III.
Close to the outer horizon the radial solutions go as
The choice of sign in the exponent determines the behaviour at the horizon. The negative sign is the correct choice for an ingoing wave (as measured by a comoving observer).
Towards spatial infinity, the radial function has the asymptotic behaviour
The sign of the real part of determines the behaviour of the wavefunction as . If , the solution diverges, whereas if , the solution tends to zero. The general solution is a linear sum of solutions with both types of behaviour, so is also divergent.
ii.4 Bound States
By definition, bound state solutions are ingoing at the horizon, and tend to zero at infinity (). On the other hand, the well-known quasinormal modes Chandrasekhar-1983 are ingoing at the horizon and purely outgoing (and divergent) at infinity (). In both cases, imposing a pair of boundary conditions leads to a discrete spectrum of (complex) frequencies.
One feature of the ingoing boundary condition ( as ) is that decaying states with have a radial function which is divergent at the horizon. This is due to the fact that Boyer-Lindquist coordinates are only valid in the exterior region, . They fail to describe classical geodesics that cross the outer horizon: it takes an infinite coordinate time to cross from to .
To study states that pass through the horizon, and to interpret the physical content of the wavefunction, we may use ingoing-Kerr coordinates. By applying (13) to (30), it is straightforward to verify that the ingoing radial function is regular at the outer horizon. However, the outgoing solution remains divergent. Physically, this implies that an infalling observer measures the local probability density of an ingoing solution to be finite and well-defined. On the other hand, the probability density of the outgoing solution is undefined at .
As mentioned above, a pair of boundary conditions gives rise to a discrete spectrum of frequencies. The spectrum of the (classical) Dirac equation on the Schwarzschild background has been studied in depth by Lasenby et al. Lasenby-2005-bs . In the limit , the bound state spectrum of the massive Dirac field resembles that of the hydrogen atom. That is,
where is the principal quantum number of the state. In Appendix A we show that the scalar field also has a hydrogenic spectrum in this limit. To lowest order in , the spectrum is not dependent on the rotation of the hole or the spin of the field.
Iii Numerical Method
In this section, we show that both the quasinormal modes (QNMs) and the bound states of the massive scalar field can be found by numerically solving a three-term recurrence relation. In a classic study, Leaver Leaver-1985 was the first to use this approach to find the QNMs of the gravitational field. More recently, Konoplya and Zhidenko Konoplya-2006 applied the same method to find the QNMs of the massive scalar field. Cardoso and Yoshida Cardoso-2005-superr have shown that massive scalar bound states can be found by solving a five-term recurrence relation. Here, we instead derive a three-term relation and apply the continued-fraction method.
The radial equation (28) has received much attention. Equations of this form first arose in a study of the electronic spectrum of the hydrogen molecule Jaffe-1934 ; Baber-1935 , over seventy years ago. Equation (28) may be transformed to a singly-confluent Heun equation Fiziev-2006 by a suitable substitution. Heun equations have four regular singular points. In the singly-confluent case, two of these points are merged together at . The other singular points lie at the horizons, and . The singly-confluent equation is related to the “generalised spheroidal equation” Leaver-1986 . In the critically-rotating case (), both horizons are combined at , and (28) is related to the doubly-confluent Heun equation.
In section II.4, we discussed appropriate boundary conditions at and . With this in mind, let us now look for a solution of the form
The choice of the sign of the real part of determines the behaviour of the wavefunction as . If , the solution diverges towards infinity, whereas if the solution tends to zero. Therefore, the same method can be applied to look for both quasinormal modes (by choosing ) and the bound state modes (by choosing ).
The constants , , , and are somewhat more complicated than in the massless case Leaver-1985 . Explicitly,
The angular eigenvalue (appearing in the expression for ) may be expanded as a power series
Seidel Seidel-1989 lists the series expansion coefficients up to . The power series provides an adequate approximation for up to . Beyond this regime, other methods are available. These include Leaver’s continued-fraction method Leaver-1985 or Hughes’ Hughes-2000 spectral decomposition method.
The ratio of successive coefficients is given by an infinite continued fraction
Substituting into the above expression and comparing with leads to the implicit condition
This condition is only satisfied for particular values of corresponding to the bound state (or QNM) frequencies. To find these values we employed a simple 2D minimisation algorithm.
In this section we present frequency spectra determined numerically via the continued-fraction approach. First, to validate the code, we compute some QNM frequencies and compare with values in the literature. Next, we present bound state spectrum in the non-rotating and rotating cases. Finally, we verify the existence of unstable states (with ) and compute their growth rates.
iv.1 Quasinormal Mode Spectra
The continued-fraction method was first developed to find QNM frequencies Leaver-1985 . QNMs have been extensively studied through a variety of methods Leaver-1985 ; Andersson-1992 ; Simone-1992 . Hence, we begin by validating our continued-fraction code by computing QNM frequencies.
Recently, Konoplya and Zhidenko Konoplya-2006 conducted a survey of the massive scalar QNMs using the continued fraction method. Their paper includes tables of QNM frequencies quoted to six decimal places, which we used to validate our numerical code. We found agreement to six decimal places with the frequencies listed in Tables I to V in Konoplya-2006 , apart from a few anomalies (probably introduced at the typesetting stage). In Appendix B we include tables of QNM frequencies for the , , mode (Table 2) and the , , mode (Table 3). These tables are intended to help with any future numerical validation.
Figure 1 shows how the complex frequencies of the lowest , states vary with the field mass and black hole rotation speed . The red (right) lines are for the co-rotating state, , the green (middle) lines are for , and the blue (left) lines are for . In the non-rotating case , the lines meet, since in this case the frequency does not depend on .
The four sets of lines in Fig. 1 correspond to the field masses , , and . As shown by Simone and Will Simone-1992 , larger field mass increases the oscillation frequency but decreases the damping . Rotation also decreases the damping, particularly of the co-rotating state. Faster rotation increases the oscillation frequency of the state, but decreases the oscillation frequency of the state.
iv.2 Bound State Frequencies: Schwarzschild
From the analysis of section II.2, we expect all bound states on the Schwarzschild background to decay with time, since by equation (21). Furthermore, in the non-relativistic regime , we expect the frequencies to be approximated by (32). Our numerical results conform to both expectations.
Figure 2 compares the spectra of the massive Klein-Gordon and Dirac fields (studied in Lasenby-2005-bs ) on the Schwarzschild background. In the limit , the spectra follow equation (32). As is increased, the frequency develops a non-negligible negative imaginary component. At higher couplings, the spin has a significant effect on the frequency levels. In the spin-half case, the degeneracy is split by the black hole interaction. For couplings , the (negative) imaginary part of the energy is comparable to the field mass. This means that decay is extremely fast, similar to the Compton time. To put it another way, if , the state lasts only a few multiples of the light-crossing time for the black hole.
Figure 3 shows the frequency spectrum for scalar states of higher angular momentum, up to . Again, the levels follow the hydrogenic () spectrum (32) in the regime . At low couplings, the states are quasi-stable. Decay dominates beyond about . At around the real part of the energy reaches a minimum. The maximum ‘binding energy’ offered by this minimum increases with , to around of the rest mass energy for . It seems unlikely that this energy could be extracted from the black hole, since the state decays very rapidly (with a lifetime similar to the black hole light-crossing time).
iv.3 Bound State Frequencies: Kerr
As expected, the introduction of black hole rotation breaks the azimuthal degeneracy. That is, bound states with different azimuthal numbers occur at different frequencies. Figure 4 shows the effect of rotation () on the spectrum of states. Clearly, the co-rotating state () behaves very differently to the counter-rotating state (), and at low its decay is heavily suppressed. At higher couplings, the maximum binding energy is significantly greater for the state than the and states ( vs ).
Maximally co-rotating states (with ) are of particular interest because of the influence of superradiance. Figure 5 shows the frequency levels of the , state as a function of , for range of rotation speeds. Faster rotation has two effects on the co-rotating spectrum. First, the real energy minimum moves lower, increasing the maximum binding energy from for to for . Second, faster rotation leads to less damping, at least at low couplings.
iv.4 The Kerr Instability
“Zooming in” on the lower plot of Fig. 5 reveals the Kerr instability: the imaginary part of the frequency is actually positive at low couplings ! This effect was predicted by the analysis of section II.2. Figure 6 shows the imaginary component as a function of coupling . The figure shows that all bound states with are unstable. The imaginary component of the frequency reaches a maximum at a coupling just below the superradiant cutoff, . As expected, faster rotation creates a greater instability.
Figure 7 compares the growth rate of the state with the co-rotating modes of higher angular momentum (, and , ). As predicted Zouros-1979 ; Detweiler-1980 , the , mode proves to be the most unstable. The dotted line in Fig. 7 shows the high-coupling approximation (4).
Table 1 lists some maximum growth rates () for the , mode, for various . Although it is not proved here, we believe that the bound states represent the most unstable solutions to the massive wave equations, since no superradiant radiation escapes to infinity. If this is correct, the values in Table 1 represent absolute upper bounds on the growth rate of the scalar field instability.
V Discussion and Conclusions
In this study, we have shown that the instability in the massive scalar field on the Kerr spacetime is greatest when the gravitational coupling is . The , state is the most unstable, with a maximum growth rate of at . Accurate upper bounds on the growth rate are presented in Table 1, for a range of .
Our numerical results are consistent with earlier studies Zouros-1979 ; Detweiler-1980 ; Furuhashi-2004 ; Cardoso-2005-superr . For example, Furuhashi and Nambu Furuhashi-2004 studied the massive scalar instability for the charged, rotating (Kerr-Newman) black hole, using matching methods. For the special case , they found that the growth rate reaches a maximum value at (see Fig. 6 in Furuhashi-2004 ). By comparison, we find at , for .
In recent years, a number of groups Krivan-1997 ; Andersson-1998 ; Andersson-2000 ; Burko-2004 ; Strafuss-2005 have studied the massive scalar field in the time domain, using numerical finite-difference codes. For example, Strafuss and Khanna Strafuss-2005 try perturbing the Kerr hole with a ‘nearly monochromatic’ wave pulse, and study the late-time tail for signs of instability. They report an e-folding time of for the parameter values , , , . On the other hand, our results (Fig. 6) suggest a minimum e-folding time of around for these values. Without further details, it is difficult to explain this large discrepancy.
Before concluding, it is worth attempting to assess the observable consequences of the instability, if any. That is, does the massive boson instability actually manifest itself in any physical system? For astrophysical systems, the short answer appears to be: almost certainly not! This is because, for any known massive boson coupled to an astrophysical black hole (), the gravitational coupling would be very large, . Hence by (4) the instability is negligible. Nevertheless, this prospect cannot be entirely discounted. If apparently massless bosons turn out to have a small but non-zero rest mass, then the instability could play a role in slowing the rotation of large black holes. For example, for a supermassive black hole (), the instability would be significant if there exists a stable boson with mass . For comparison, the (lab-based) experimental upper bound on the photon’s mass is currently around Tu-2005 .
For the superradiant instability to be significant for small ‘primordial’ black holes (), two conditions must be met Zouros-1979 . First, the e-folding time should be significantly shorter than the Hawking evaporation lifetime. Second, the instability growth rate should exceed the spontaneous decay rate of the particle.
Let us consider the first condition: is the Kerr instability able to compete with Hawking radiation for small black holes? It is thought that over the age of the universe (), primordial black holes smaller than will evaporate through Hawking radiation Page-1976b . The light-crossing time for a primordial black hole would be short (), so the dimensionless growth rate need be no bigger than to be of some significance. Using approximations (3) and (4), this corresponds to a coupling in the range . For this implies that the instability could arise if the field mass is in the range . Repeating this analysis for smaller black holes, kg and kg, leads to mass ranges and , respectively. These mass ranges span the known particle spectrum. It is therefore highly likely that, at least at some stage during the black hole’s evolution, the first condition will be met.
Now we come to the second condition: does instability growth outstrip natural radioactive decay? As a first example, let us consider the neutral pion . The pion has a mass and a lifetime . The instability grows fastest when , which would correspond to a black hole of mass . If we take the maximum growth rate of at , this implies a lower bound on the e-folding time of . It is a curious coincidence that this the instability growth is of the same order as the rate of radioactive decay! Thus, exponential growth in the neutral pion field is only possible in a very narrow parameter range. Though charged pions are potentially much more stable () than the neutral versions, any charged pair of particles in the vicinity of the hole would annihilate quickly, thus quenching the instability growth.
A second possible candidate particle is the spin-1 boson, which mediates the weak force. The boson is about a thousand times more massive than the pion, but decays times faster! Neglecting the effects of spin leads to an estimate of for the shortest possible e-folding time. This is far too slow for any instability to arise. Nevertheless, it would be interesting to repeat the frequency-domain analysis for the massive spin-1 field. As is well-known, the superradiance effect is enhanced by the spin of the field Starobinskii-1973 . For example, the maximum reflection coefficient is for the massless scalar wave, but increases to for electromagnetic waves Press-1972 . It is natural to suppose that the Kerr instability is similarly enhanced.
Finally, some recently-proposed theories Randall-1999b invoke “large” () extra dimensions to explain why gravity is so much weaker than the other three forces. These theories raise the possibility that higher-dimensional gravitational objects may be created through high energy particle collisions. With additional dimensions, possibilities include black strings (with event horizon topology ) and black -branes (), as well as black holes () Kanti-2004 . The stability properties of higher-dimensional black objects have attracted much interest Gregory-1993 ; Berti-2003 ; Cardoso-2005-new . Black strings (branes) are thought to suffer from the Gregory-Laflamme instability Gregory-1993 which breaks the black string (brane) into smaller segments. On the other hand, higher-dimensional black holes are not affected by the Gregory-Laflamme instability. A recent study Cardoso-2005-superr suggests that higher-dimensional rotating black holes do not suffer a superradiant instability either. Though they still exhibit superradiance, it seems that bound states are prohibited. This conclusion is supported by the apparent absence of stable orbits on higher-dimensional rotating backgrounds Frolov-2003 .
Appendix A Non-Relativistic Frequency Spectrum
Here we show that, in the non-relativistic limit, the frequency spectrum of the scalar field bound to a Schwarzschild black hole is given by (32). Let us begin by considering the Schwarzschild spacetime described by Painlevé-Gullstrand (PG) coordinates Martel-2001 ,
The contravariant metric tensor has components , and . It is straightforward to show that the massive Klein-Gordon equation in PG coordinates can be written
where is the 3D Laplacian operator.
To effect a non-relativistic reduction, we split the field into two components and , defined by
This decomposition leads to the pair of coupled equations,
In the non-relativistic limit, we make the assumption that and the approximation . Equally well, we could make the assumption that to recover the non-relativistic antiparticle spectrum. This assumption leads to the Schrödinger equation
where . With a simple substitution, , equation (56) can be transformed to the familiar form
This is the hydrogenic Schrödinger equation, but with the fine-structure constant replaced by the gravitational coupling . Hence the non-relativistic wavefunctions are hydrogenic, and the energy levels are given by (32).
Appendix B Quasinormal Mode Frequencies
Acknowledgements.SD would like to thank Calvin Smith, Vitor Cardoso, Elizabeth Winstanley and Marc Casals for helpful discussions and proof-reading, and the Astrophysics Group at Cambridge University for the continued use of their computing facilities.
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