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

Abstract

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. black hole with mass M and U (1) charge Q and contains two extra background fields; the scalar dilaton Φ and the 3-form H. In the limit that these two fields vanish the spacetime reduces to the Kerr black hole. On the other hand, this spacetime should be compared to the Kerr-Newman solution [17] which is the unique stationary black hole solution to the Einstein equations with U (1) charge (it can also be understood as a solution of N = 2, D = 4 supergravity). Both Kerr-Newman and Kerr-Sen black holes are stationary and axisymmetric spacetimes, possessing two Killing vectors which aid in understanding the behaviour of test fields in these backgrounds. In the Kerr-Newman case there exists an additional hidden symmetry of the principal Killing-Yano tensor which gives rise to Carter's constant for charged geodesics [18]. For the Kerr-Sen spacetime only a generalized principal tensor with torsion exists which is a weaker but still rather useful structure [19].The aim of this paper is to study the superradiant instabilities of the Kerr-Sen and Kerr-Newman black holes, as triggered by the ultralight massive bosons. These are well understood in the case of massive scalar fields, see [20] and [21], but the corresponding study for massive vectors is currently missing. The reason is simple. Even for vacuum (Kerr) black holes, the corresponding Proca equations are rather complicated partial differential equations whose direct decoupling and separation à la Teukolsky [22,23] does not work due to the presence of the mass term [24,25]. As a consequence the problem was investigated either using approximations [9,24,25] or employing serious numerical analysis [10,26,27].However, a separability renaissance for vector fields has begun in the last couple of years due to a new ansatz by Lunin [28]. Simplified and written in covariant form by Frolov-Krtouš-Kubizňák [29,30], the new ansatz works for the massive vector field case [31] and can be applied in the Kerr-NUT-AdS spacetimes for all dimensions. Importantly the Lunin-Frolov-Krtouš-Kubizňák (LFKK) ansatz exploits the existence of hidden symmetries in these spacetimes which are encoded in the principal tensor [18] and allows the Proca equations to be decoupled and separated into ordinary differential equations [31]. In four dimensions, the separation equally applies to the non-accelerating electro-vacuum type D Plebański-Demiański spacetimes and thence to the Kerr-Newman black holes. 1...

Vector perturbations of Kerr-AdS5 and the Painlevé VI transcendent

“…For a 1 → 1, the angular system is better served by the expansion of the τ function around u 0 = 1, which was given in [24]. Recently, in [34] a similar analysis was performed for four dimensional Kerr-Newman and Kerr-Sen black holes. Nevertheless, our problem presents an extra difficulty related to the presence of µ in the radial and angular equations, which is not the case in four dimensions where the separation parameter µ can be absorbed by a redefinition of the separation constant C m , as can be seen in [13,20].…”

Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions

“…This development has allowed more detailed studies of the Proca superradiant instability -see e.g. [43,44]. The first goal of this paper is to make use of this development to determine and characterize the stationary vector clouds around Kerr BHs in terms of {n, , j, m j }.…”

The Painlevé VI tau-function of Kerr-AdS5

“…For a real scalar field we have 49) which leads to the Klein-Gordon (KG) equation in the given background metric…”

“…From (4.48), any possible imaginary part for the eigenfrequency will then come from the imaginary part of Ξ n, . The latter can be calculated by using the reflexion property of the digamma function 49) or, in terms of m 1 and m 2 ,…”

“…For a 1 → 1, the angular system is better served by the expansion of the τ function around u 0 = 1, which was given in [71]. Recently, in [49] a similar analysis was performed for four dimensional Kerr-Newman and Kerr-Sen black holes. Nevertheless, our problem presents an extra difficulty related to the presence of µ in the radial and angular equations, which is not the case in four dimensions where the separation parameter µ can be absorbed by a redefinition of the separation constant C m , as can be seen in [59,66].…”