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...
Many black hole spacetimes with a 3-form field exhibit a hidden symmetry encoded in a torsion generalization of the principal Killing-Yano tensor. This tensor determines basic properties of such black holes while also underlying the separability of the Hamilton-Jacobi, Klein-Gordon, and (torsion-modified) Dirac field equations in their background. As a specific example, we consider the Chong-Cvetič-Lü-Pope black hole of D = 5 minimal gauged supergravity and show that the torsion-modified vector field equations can also be separated, with the principal tensor playing a key role in the separability ansatz. For comparison, separability of the Proca field in higher-dimensional Kerr-NUT-AdS spacetimes (including new explicit formulae in odd dimensions) is also presented.
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