Stability of squashed Kaluza-Klein black holes

Abstract

The stability of squashed Kaluza-Klein black holes is studied. The squashed Kaluza-Klein black hole looks like a five dimensional black hole in the vicinity of horizon and and looks like a four dimensional Minkowski spacetime with a circle at infinity. In this sense, squashed Kaluza-Klein black holes can be regarded as black holes in the Kaluza-Klein spacetimes. Using the symmetry of squashed Kaluza-Klein black holes, SU (2) × U (1) ≃ U (2), we obtain master equations for a part of the metric perturbations relevant to the stability. The analysis based on the master equations gives a strong evidence for the stability of squashed Kaluza-Klein black holes. Hence, the squashed Kaluza-Klein black holes deserve to be taken seriously as realistic black holes in the Kaluza-Klein spacetime. PACS numbers: 04.50.+h, 04.70.BwRecently, higher dimensional black holes have attracted much attention. In particular, many exotic black holes in the asymptotically flat spacetime are found [1][2][3][4][5][6][7]. From a realistic point of view, however, the extra dimensions need to be compactified to reconcile the higher dimensional gravity theory with our apparently four-dimensional world. The higher dimensional spacetimes with compact extra dimensions are called Kaluza-Klein spacetimes. The black holes should reside not in the asymptotically flat spacetimes but the asymptotically Kaluza-Klein spacetimes. We call these 'Kaluza-Klein black holes'. It would be important to study Kaluza-Klein black holes in the general dimensions. In this paper, we will consider five dimensional Kaluza-Klein black holes as a first step.It is well known that the simplest five-dimensional Kaluza-Klein black hole is the black string which is the direct product of four dimensional Schwarzschild black hole and a circle [8]. The topology of the horizon of black strings is S 2 × S 1 . The stability analysis of black strings has been done, and it was shown that black strings are stable when the horizon radius is larger than the scale of compact extra dimension [9]. Because of the stability, black strings are natural candidate of Kaluza-Klein black holes.Interestingly, another possibility has been recognized [10]. It is squashed Kaluza-Klein (SqKK) black holes that could also reside in the Kaluza-Klein spacetime. The topology of the horizon of SqKK black holes is S 3 , while it looks like four dimensional black holes with a circle as an internal space in the asymptotic region. SqKK black holes were originally derived as five-dimensional vacuum solutions in the context of Kaluza-Klein theory [11,12].Recently, much effort has been devoted to reveal the properties of squashed Kaluza-Klein black holes [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Since the horizons of these black holes have the same nature as the fivedimensional black holes, Hawking radiation and quasi-normal modes from SqKK black holes would be different from those seen in four-dimensional black holes even at low energy [22,27,28]. That means that the extra dimension can be o...

Evolution of perturbations of squashed Kaluza-Klein black holes: Escape from instability

“…The quasinormal frequencies of the scalar field are different from those of the Schwarzschild black holes and have smaller real oscillation frequencies and longer lived for not very larger ρ 0 . The obtained evolution of gravitational perturbations shows damped oscillations in time domain, so that no instability is observed in concordance with analytically proved stability in [36]. Let us note that we analyzed here only "zero mode" perturbations, so that, strictly speaking, the opportunity for instability is remained in higher multipole perturbations.…”

“…There are two distinctive features of gravitational perturbations, on the contrary to test scalar field perturbations. First, the K = 1 potential contains the small negative gap near the event horizon, which nevertheless does not induce any instability [36]. Second, K = 1, 2 perturbations for some values of ρ 0 have effective potentials without maximum but with a monotonic growth instead.…”

“…As is shown in [36], the perturbation equations for the |K| = 2 mode can be reduced to the wave equation for h ++ with the effective potential in the form…”

“…Then, it is interesting to study the the quasinormal modes of the squashed Kaluza-Klein black holes. This is even more motivated if we take into account that squashed Kaluza-Klein black holes are gravitationally stable [36].…”

Black Holes in Higher Dimensions

“…The equations governing gravitational perturbations will then reduce to a set of coupled scalar ODEs. These equations have not yet been derived for the Myers-Perry solution (however, this method has been applied to study perturbations of a static Kaluza-Klein black hole with SU (2) × U (1) symmetry [158]). …”