Massive vector fields in Kerr--Newman and Kerr--Sen black hole spacetimes

The superradiant instability modes of ultralight massive vector bosons are studied for weakly charged rotating black holes in Einstein--Maxwell gravity (the Kerr--Newman solution) and low-energy heterotic string theory (the Kerr--Sen black hole). We show that in both these cases, the corresponding massive vector (Proca) equations can be fully separated, exploiting the hidden symmetry present in these spacetimes. The resultant ordinary differential equations are solved numerically to find the most unstable modes of the Proca field in the two backgrounds and compared to the vacuum (Kerr black hole) case.

the string frame. The corresponding unstable superradiant modes are studied numerically, and compared to those of the Kerr-Newman spacetime. We do so in an astrophysically viable situation where the black holes are fast spinning (close to extremal) and weakly charged [39,40].
The paper is organized as follows. In Sec. 2 we review and compare the Kerr-Newman and Kerr-Sen spacetimes, and discuss their symmetries and extremality. In Sec. 3 we discus how the Proca equation is modified by the presence of the dilaton Φ and the 3-form H, and present the separated equations (a full derivation of which may be found in App. A). In Sec. 4 we numerically investigate the separated equations and visualize the behaviour of the most unstable superradiant modes. We conclude in Sec. 5.

Kerr-Sen and Kerr-Newman black holes
In this section we present the Kerr-Sen and the Kerr-Newman metrics, the background spacetimes in which we will study the instability modes of the Proca equation. The two metrics we consider both describe rotating and charged black hole spacetimes, however there are some key differences in the Kerr-Sen case due to modifications of general relativity coming from the low energy heterotic string theory effective action.

Kerr-Newman geometry
The Kerr-Newman solution [17] is the most general solution of the Einstein-Maxwell equations for an asymptotically flat, stationary and axisymmetric black hole. Its line element and vector potential read: ds 2 = − ∆ ρ 2 dt − a sin 2 θdφ 2 + ρ 2 ∆ dr 2 + sin 2 θ ρ 2 a dt − (r 2 + a 2 )dφ 2 + ρ 2 dθ 2 , where ρ 2 = r 2 + a 2 cos 2 θ , ∆ = r 2 − 2M r + a 2 + Q 2 . (2. 2) The solution describes a black hole with mass M , charge Q, angular momentum J = M a, and a magnetic dipole moment µ g = Qa. The metric possesses a curvature singularity at ρ 2 = 0, which is protected by an event horizon at r = r + ≡ M + M 2 − a 2 − Q 2 provided that a 2 + Q 2 ≤ M 2 . In the case where the equality holds we have an extremal black hole. The rotation of the black hole causes inertial frame dragging whose extreme manifestation is the existence of the ergosphere, for r + < r < r e ≡ M + M 2 − Q 2 − a 2 cos 2 θ. In this region the time-like vector ∂ t becomes null and therefore any massive particle must rotate. It is this region that leads to superradiant emission and is responsible for the instability modes for perturbations on this spacetime. The black hole horizon rotates with angular velocity Ω H = − g tφ g φφ = a r 2 + + a 2 , (2.3) and can be assigned the following Hawking temperature, entropy, and electrostatic potential: T H = ∆ (r + ) r 2 + + a 2 = r 2 + − a 2 − Q 2 4πr + (r 2 + + a 2 ) , S = π(r 2 + + a 2 ) , φ H = Qr + r 2 + + a 2 .

(2.4)
These quantities satisfy the first law of black hole thermodynamics δM = T H δS + Ω H δJ + φ H δQ , (2.5) as well as the associated Smarr relation, M = 2(T H S + Ω H J) + φ H Q . The Kerr-Newman metric admits a hidden symmetry of the principal tensor (PT), which is a non-degenerate closed conformal Killing-Yano 2-form h, obeying the following equation [18]: It is this tensor that underlies the separation of variables in the Proca equation. It explicitly reads as follows h = r(dt − a sin 2 θ dφ) ∧ dr − a cos θ a dt − (r 2 + a 2 )dφ ∧ d cos θ , (2.7) and gives rise to the associated Killing tensor which generates the generalized Carter's constant for charged geodesics. It also yields the two independent isometries of the spacetime: ξ a in (2.6) and η a = K ab ξ b [18].

Kerr-Sen geometry
The Kerr-Sen black black hole [16] is an exact classical solution of the low-energy effective theory describing heterotic string theory given by the following action: H abc H abc , (2.9) where g ab represents the metric in the string frame, Φ is the dilaton field, F = dA is the Maxwell field strength, and H = dB − 2A ∧ F is a 3-form defined in terms of the vector potential A and a 2-form potential B. 2 The action is invariant under a U (1) transformation A → A+dλ provided we also send B → B +2λF and the corresponding equations of motion for the background fields A and H are, (2.10) These will be important in section 3 where we motivate a generalization of the Proca equation to this background. The full set of equations of motion is supplemented by the Einstein and dilaton equations. Since these will not play any role in the further discussion we do not write them here explicitly and refer the interested reader to for example [19].
2 Note that we have rescaled the vector potential A → 2 √ 2A so that the Maxwell Lagrangian has the canonical prefactor [16].
In any case, the Kerr-Sen metric in the standard Boyer-Lindquist-type coordinates and the string frame reads [16,19,41]: where the metric functions are given by (2.12) The 3-form H reads Note that, the transformation g ab → e Φ g ab can be performed to go from the string frame to the Einstein frame. Our choice for the string frame is guided by the fact that, in the context of separability, the string frame seems to be more fundamental than the Einstein one, as are the hidden symmetries present in the Kerr-Sen spacetime, see [19]. As mentioned in the introduction, the solution describes a black hole with mass M , angular momentum J = M a, and magnetic dipole moment µ g = Qa. When the 'twist parameter' is set to zero, the solution reduces to the Kerr geometry. The horizon of the Kerr-Sen black hole is located at r = r + ≡ M − b + (M − b) 2 − a 2 when the inequality M − b ≥ |a| holds. As in the Kerr-Newmann case the ergosphere is present and responsible for the instability modes but it is now located at r + < r < r e ≡ M − b + (M − b) 2 − a 2 cos 2 θ. Moreover, the Kerr-Sen black hole also obeys the first law, (2.5), where now the (Einstein frame) thermodynamic quantities are given by , S = π(r 2 + + 2br + + a 2 ) . (2.16) The spacetime no longer possesses the hidden symmetry of the principal tensor. However, as shown in [43] a weaker structure of the principal tensor with torsion exists [36]. This is a non-degenerate closed conformal Killing-Yano tensor with torsion, obeying the following generalization of Eq. (2.6): Here, the covariant derivative with torsion is defined as and the torsion is simply identified [43] with the 3-form H, (2.13), More explicitly we have Despite being a weaker structure, the principal tensor with torsion still gives rise to standard Killing tensor, via (2.8). However, the isometries of the spacetime are no longer straightforwardly generated from h [43].

Separability of Proca equations
In this section we present the form of the Proca equations in the Kerr-Newmann geometry and motivate how this changes for the Kerr-Sen case. We then outline our ansatz and resulting separated equations using the (generalized) hidden symmetries of these two spacetimes. The full details of the calculation for the Kerr-Sen spacetime can be found in appendix A.

Proca in Kerr-Newman spacetime
In curved spacetime and in the absence of sources, the standard Proca equation reads where m stands for the mass of the particle and the the field strength F is defined in terms of the massive U (1) vector field P in a standard way, F ab = ∇ a P b − ∇ b P a . Due to the presence of mass term, there is no longer gauge invariance, however (3.1) automatically implies the "Lorenz condition" ∇ a P a = 0 .
The separability of the Proca equation in the Kerr-Newmann background was demonstrated in [31] (the Kerr-Newman metric is a special case of the D = 4 canonical metric for which the separability was shown there). Let us briefly recapitulate this result. The key step is to use the LFKK ansatz [28][29][30][31] for the gauge field P , where B is the polarization tensor (not to be confused with the 2-form potential B appearing in the definition of H), which can be covariantly written in terms of the metric and the principal tensor h, (2.7), as B ab (g bc + iµh bc ) = δ a c , (3.4) where µ is a separation constant. The function Z assumes the standard multiplicative separation form, where m φ and ω are the eigenvalues of i∂ t and −i∂ φ . Note that φ has period 2π, and regularity of the spherical harmonics S(θ) e im φ φ requires that m φ ∈ Z.
With this ansatz, the Proca equation (3.1) reduces to two differential equations in r and θ, respectively, which only couple to each other via their dependence on the Killing parameters {ω, m φ }, the separation constant µ, the Proca mass parameter m, and the black hole parameters {M, Q, a}. 4 These equations take the explicit form The demonstrated separation depends crucially on the existence of the principal tensor in a number of ways. First, the separation occurs in geometrically preferred coordinates determined by the principal tensor -coordinates r and cos θ are related to the eigenvalues of the principal tensor, e.g. [18]. Second, the principal tensor explicitly enters the separation ansatz (3.3) via the polarization tensor (3.4). Third, the principal tensor gives rise to a complete set of mutually commuting operators that guarantee this separability [29,34,43]. Namely, apart from the (trivial) ones connected with Killing vectors, the following two (2nd order) operators directly link to the separation ansatz: where K ab is the Killing tensor (2.8) and V a = ξ b B ba , see App. A for more details.

Generalized Proca in Kerr-Sen spacetime
Test fields in the Kerr-Sen background naturally pick up modifications due to the presence of background fields φ, A, and H, see for example [19,36,44]  equation. To motivate the generalized Proca equation, we assume that the massive vector field P couples to the background fields Φ and H in analogy to the massless Maxwell field already present in the Kerr-Sen action (2.9). As in [38], we also demand that the modified Proca is linear in P , reduces to the Proca equation in the absence of the background fields, and obeys current conservation in the presence of sources. It follows that there are two key modifications to the Proca equation in the Kerr-Sen background. First, in the string frame the dilaton enters the field equation Second, the 3-form H abc = T abc contributes to the field strength in a torsion-like fashion To separate the generalized Proca equation (3.9) in the Kerr-Sen background we exploit the same machinery as for the Kerr-Newman case, with the only difference that the principal tensor (2.7) is now replaced with the principal tensor with torsion (2.19). Upon this, the LFKK ansatz (3.3) continues to work, see App. A, and we recover the following separated equations: where K r = a m φ − (a 2 +r 2 + 2rb)ω , K θ = m φ − a ω sin 2 θ , q r = 1 + µ 2 r 2 , q θ = 1 − µ 2 a 2 cos 2 θ , which are to be compared to the Proca equations in the Kerr-Newman spacetime (3.6), and upon setting b = 0 reduce to the those in the Kerr spacetime. Note that the angular equation in all three cases is exactly the same, while the radial one picks up some small modifications.
5 Note that of the 3 possible generalizations of the 'Maxwell operator' ∇ · dP : it is only the first one which obeys the current conservation equation and (upon including the dilatonic modification) consequently appears in (3.9).
Similar to the Kerr-Newman case, the separability is underlain by a complete set of mutually commuting symmetry operators, one of which is constructed from the generalized principal tensor, Now that we have presented and separated the Proca equations for the Kerr-Newman and Kerr-Sen black holes we can turn to studying the consequences for the instabilities of the massive vector field.

Unstable modes
Numerics: formulation of the problem Let us now present our numerical results for finding the most unstable modes of the Proca equation in the three backgrounds. We need to solve numerically the Proca coupled pair of (radial and angular) ODEs (3.6) for the Kerr-Newman black hole and (3.13) for the Kerr-Sen black hole; the results for the Kerr black hole are obtained by setting Q = 0 in these equations.
Using ∆(r + ) = 0 (for Kerr-Newman) or ∆ b (r + ) = 0 (for Kerr-Sen) we can replace the mass M by the outer horizon radius parameter r + . We then find it convenient to work with a compact radial coordinate y ∈ [0, 1] and with a new polar coordinate x ∈ [0, 1] related to the standard coordinates r, θ of (2.1) and (2.11) by Note that the horizon is located at y = 0 and asymptotic infinity is at y = 1. For numerics it is also convenient to work with the dimensionless quantities { a = a/r + , Q = Q/r + , m = m r + , ω = ω r + , µ = µ r + } (although our final results will be presented in mass units, i.e. in terms of the dimensionless quantities (4.6) below). In this setting, we have two unknown functions R(y) and S(x) whose equations have to be solved subject to appropriate physical boundary conditions 6 . We are particularly interested in searching for unstable modes as these determine the signatures of the Proca fields in gravitational wave signals. These modes have frequencies whose real part is smaller than the potential barrier height set by the Proca field mass, Re(ω) < m. A Frobenius analysis at asymptotic infinity y = 1 then indicates that unstable modes must decay as Here, we have already imposed a boundary condition that eliminates a solution that grows unbounded at infinity as e √ At the horizon, regularity of the perturbation in ingoing Eddington-Finkelstein coordinates requires that we impose the boundary condition, where Ω H and T H are the horizon angular velocity and temperature, given in see (2.3), (2.4) and (2.16), respectively) which excludes outgoing modes, We still need to apply boundary conditions at north and south poles of the S 2 . Here, regularity of the perturbations requires that m φ is quantized to be an integer. We are interested in unstable modes which must co-rotate with the black hole because these will extract energy from the black hole. Thus we must have m φ > 0. Under these conditions, regularity requires that the perturbations behave as which eliminates irregular modes that would diverge as and search numerically for analytical functions q 1 (y) and q 2 (x). Our pair of Proca ODEs are coupled only via the eigenvalues ω and µ. But this is a non-linear eigenvalue problem for ω and µ.

Numerics: technique
We discretize the field equations using a pseudospectral collocation grid on Gauss-Chebyshev-Lobbato points. The eigenvalues and respective eigenvectors are efficiently and accurately computed using a powerful numerical procedure developed in gravitational contexts in [45,46,[48][49][50][51][52][53][54][55][56][57][58][59][60] -discussed in detail in [56] and in section III.C and VI.A of the topical review [47] -which employs the Newton-Raphson root-finding algorithm. These numerical methods are very well tested in different contexts and extremely robust. In particular, they are the same that were used to compute the ultraspinning and bar-mode gravitational instabilities of rapidly spinning Myers-Perry black holes [45,46,[48][49][50][51], the near-horizon scalar condensation and superradiant instabilities of black holes [52][53][54], the gravitational superradiant instability of Kerr-AdS black holes [55,56], the electro-gravitational quasinormal modes of the Kerr-Newman black holes [57] and the modes that violate strong cosmic censorship in de Sitter backgrounds [58][59][60], to mention a few. All our results have the exponential convergence on the number of gridpoints expected for a code that uses pseudospectral collocation (see [47]). In particular, all the results that we present in the next section are accurate at least up to the 11th decimal digit.

Discussion of the parameter space
Our Proca-black hole system has a scaling symmetry which we use to present the dimensionless physical quantities measured in black hole mass units, namely In these conditions one could fix, for example, the rotation of the black hole and plot the frequency and angular eigenvalues as a function of Q/M and mM . Alternatively, one could fix the black hole charge Q/M or the Proca mass mM and generate the corresponding 3-dimensional plots. However, we find that these 3-dimensional plots are not particularly enlightening (especially because we want to compare Kerr-Newman with Kerr-Sen and thus we would have two 2-dimensional surfaces on the same plot). Therefore, we opt to produce 2-dimensional plots that are clear and describe the typical qualitative behaviour of the system. Concretely To localize ourselves in the parameter phase space it is a good idea to consider first a Proca field in the Kerr black hole background. Recall that both the Kerr-Newman and the Kerr-Sen solutions reduce to Kerr when Q/M = 0. The properties of this solution were already studied in [10,31,33] and, as a test of our numerical codes for Kerr-Newman and Kerr-Sen when Q = 0, we have reproduced them. As described in detail in [10,31,33], Proca fields have three sectors of perturbations. Here, we will only discuss the most interesting family, namely the one that is the most unstable. For the same reason, we also only consider its lowest radial overtone case and modes with azimuthal number m φ = 1.

Kerr: unstable modes at fixed black hole rotation
Firstly, in Fig. 1 we fix the the rotation of the Kerr black hole to be and we plot how the frequency ωM and angular eigenvalue µM of the unstable modes change as we change the Proca mass mM . In the upper panels we give the behaviour of the dimensionless frequency ωM while in the lower panels we describe the properties of the angular eigenvalue µM . On the left-upper panel, we analyze the imaginary part of the frequency, Im(ωM ). We see that starting from zero at mM = 0 it first increases until it  Then, increasing mM , Im(ωM ) quickly drops to zero at mM 0.64853159519. The rightupper plot studies the behaviour of the real part of the frequency. More concretely, we take the difference m φ Ω H M − Re(ωM ), with m φ = 1, because the instability of the system is sourced by superradiance and this quantity should go to zero when Im(ωM ) → 0. We find that this is indeed the case, i.e. m φ Ω H −Re(ω) = 0 at the same value mM 0.64853159519 where Im(ωM ) = 0. Not shown in Fig. 1, the plots for other values of fixed J/M 2 are qualitatively similar. The critical mass mM above which the instability ceases to exist decreases when J/M 2 decreases. These behaviours for the superradiant instability of massive Proca fields are similar to those found in massive scalars in the Kerr black hole, as pointed out in [10,31,33]. The plots in the lower panel of Fig. 1 for the angular eigenvalue speak for themselves and need few further explanations. We just highlight that the minimum of Im(µM ) − see (4.8) − occurs at the same mass mM ∼ 0.5391620000 where Im(ωM ) has its maximum and Im(µM ) vanishes at the same mM 0.64853159519 as Im(ωM ).

Kerr: unstable modes at fixed Proca mass
In the plots of Again, if we fix other values of mM , we obtain plots that are qualitatively similar to those of Fig. 2 as long as the Proca mass mM is not too large, in which case there is no regime of J/M 2 where the system is unstable. The maximum of the instability decreases when mM decreases (as shown in Fig. 1). Altogether, the plots of Figs. 1 and 2 allow to anticipate the shape of the 2-dimensional surface describing the system if we did the 3-dimensional plots ωM (or µM ) as a function of J/M 2 and mM , so we do not plot this here (the reader can find these in Figs. 4 and 5 of [10]).

Unstable modes in Kerr-Newman and Kerr-Sen black holes
We are now ready to present our results for the unstable modes of the Kerr-Newman and Kerr-Sen black holes. Recall again, that the brown diamond in Figs The most important plot is the left-upper plot where we display the imaginary part of the frequency. As discussed above, the system is already unstable (Im(ωM ) > 0) in the Q/M = 0 Kerr limit (brown diamond). We then find that when the electric charge The left-upper plot of Fig. 3 also shows that, in the parameter space range where both co-exist, Kerr-Sen black holes are more unstable than Kerr-Newman. 7 The other plots of Fig. 3 are self-explanatory and need no further comments other than that they also have a monotonic behaviour and start at the expected Proca-Kerr solution. These plots 7 A word of caution for interpreting these results is due here regarding the use of the string vs. the Einstein frame. Namely, our calculation for the Kerr-Sen black hole has been performed in the string frame, where the Proca equations decouple and separate. It remains to be seen whether this can be directly compared to the Kerr-Newman case where the dilaton field identically vanishes. illustrate in a clear way the main properties of unstable Proca fields in Kerr-Newman and Kerr-Sen black holes. Indeed if we fix other combinations of J/M 2 and mM (for which the instability is already present in the Kerr limit) we find similar qualitative features as those illustrated in Fig. 3. Thus we conclude our discussion of the most unstable Proca modes in Kerr-Newman and Kerr-Sen black holes.

Conclusions
In this paper we have shown that the LFKK ansatz can be used to separate the Proca equations in the Kerr-Sen black hole background of the low energy heterotic string theory. This happens for a (well motivated) modification of these equations and in the string frame. It is a highly non-trivial result as the Kerr-Sen black hole no longer admits the principal tensor, which is the key object for the LFKK ansatz, and only its much weaker (torsion) generalization is present. We have then used the resulting separated ordinary differential equations to study the corresponding instability modes of the Proca field in the Kerr-Sen background and compared them to the instability modes in the Kerr and Kerr-Newman backgrounds. This is the first study of the Proca instability modes around rotating black holes which considered a possibility of weakly charged solutions. Moreover we have considered an astrophysically viable setting where the black holes are highly spinning (close to extremal) and weakly charged. Our results allow one to compare the prediction of the two theories: the Einstein-Maxwell theory (represented by the Kerr-Newman solution) and the low energy heterotic string theory (with the corresponding Kerr-Sen black hole). Our findings indicate that, at equal asymptotic charges, Kerr-Newman black holes are more stable than Kerr-Sen ones.
Finally, our work not only generalizes the exploitation of black hole superradiance for detecting possible dark matter candidates to include charged black holes, it also opens new horizons for applications of the generalized hidden symmetries (which may not be so weak structure as previously expected).
are arbitrary functions of one variable. Thence in what follows we shall leave X µ (x µ ) arbitrary.
The off-shell metric admits a generalized principal tensor with torsion, which reads [43] h = x 1 dx 1 ∧ A 1 + x 2 dx 2 ∧ A 2 , (A. 8) and obeys the defining equation (2.17) with ξ = e Φ ∂ ψ 0 , (A. 9) and the torsion identified with the 3-form H, The associated irreducible Killing tensor is given by

Separability of Proca equations
Let us now apply the LFKK ansatz to separate the Proca equations in the generalized beckground (A.1). As argued in Sec. 3 the Proca equation in this background reads ∇ n e Φ F na − m 2 e Φ P a = 0 , (A. 12) and implies the corresponding Lorenz condition ∇ a e Φ P a = 0 . (A. 13) In order to separate these equations, we employ the LFKK ansatz [28][29][30][31], P a = B ab ∇ b Z , B ab (g bc + iµh bc ) = δ a c , (A.14) where µ is a complex parameter, h bc in the generalized principal tensor (A.8), and the potential function Z is written in the multiplicative separated form Z = R 1 (x 1 )R 2 (x 2 )e iL 0 ψ 0 e iL 1 ψ 1 .
(A.15) Similar to ref. [29,31] we first concentrate on the Lorenz condition (A.13), for which the ansatz (A.14) yields: where the differential operators are given by The Lorenz condition (A.13) will be satisfied provided the mode functions R ν obey the separated equations Here C 0 and C 1 are two new separation constants. Then expression in (A.16) reduces to ∇ a (e Φ P a ) = e Φ Z q 1 q 2 C 0 + C 1 (−µ 2 ) , (A. 20) and we see that the Lorenz condition holds provided we fix At this stage we are left with one new separation constant C 0 but this will be fixed by solving the full Proca equations. The results of [29] can be also used to find the representation of the Proca equation (A.12) for the ansatz (A.14). Employing the Lorenz condition (A.13) one finds ∇ n e Φ F na − m 2 e Φ P a = e Φ B am ∇ m J . (A.22) Here we have introduced the object At this stage, by employing the LFFK ansatz and enforcing the Lorenz condition, the Proca equation has been reduced to solving a scalar "wave equation". In particular this "wave equation" may be written as an eigenvalue problem gZ = m 2 Z , (A.24) whereĝ = e −Φ ∇ a (e Φ g ab ∇ b ) − 2iµV a g ab ∇ b , V a = ξ b B ba . (A. 25) In this suggestive form, where we consider the metric tensor as the trivial Killing tensor K (0) ab = g ab , one can guess from the Kerr-Newman case that this operator can be generalized to the two commuting operators in 4 dimensions which are enough to guarantee the separability of this equation. Thus we definê where K ab is the Killing tensor generated from the principle tensor (A.11). Then one can explicitly check that these two operators commute, i.e. These operators are just a torsion generalization of those presented in [29,34,43] and we expect that this construction can be generalized to all dimensions. Thus the separability of J (A.23) is guaranteed and in fact J separates in the form where D µ is same the operator defined in (A.17). In the above expression we have used the identity,