We study the shadow of a rotating squashed Kaluza-Klein (KK) black hole and the shadow is found to possess distinct properties from those of usual rotating black holes. It is shown that the shadow for a rotating squashed KK black hole is heavily influenced by the specific angular momentum of photon from the fifth dimension. Especially, as the parameters lie in a certain special range, there is no any shadow for a black hole, which does not emerge for the usual black holes. In the case where the black hole shadow exists, the shadow shape is a perfect black disk and its radius decreases with the rotation parameter of the black hole. Moreover, the change of the shadow radius with extra dimension parameter also depends on the rotation parameter of black hole. Finally, with the latest observation data, we estimate the angular radius of the shadow for the supermassive black hole Sgr $A^{*}$ at the centre of the Milky Way galaxy and the supermassive black hole in $M87$.
In this paper we investigate the strong gravitational lansing in a charged squashed Kaluza-Klein Gödel black hole. The deflection angle is considered by the logarithmic term proposed by Bozza et al. Then we study the variation of deflection angle and its parameters $\bar{a}$ and $\bar{b}$ . We suppose that the supermassive black hole in the galaxy center can be considered by a charged squashed Kaluza-Klein black hole in a Gödel background and by relation between lensing parameters and observables, we estimate the observables for different values of charge, extra dimension and Gödel parameters.
The stability of squashed Kaluza-Klein black holes is studied. The squashed Kaluza-Klein black hole looks like five dimensional black hole in the vicinity of horizon and 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, $SU(2)\times U(1)\simeq U(2)$, 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 th…
We explore the obsevability of extra dimensions through 5-dimensional squashed Kaluza-Klein black holes residing in the Kaluza-Klein spacetime. With the expectation that the Hawking radiation reflects the 5-dimensional nature of the squashed horizon, we study the Hawking radiation of a scalar field in the squashed black hole background. As a result, we show that the luminosity of Hawking radiation tells us the size of the extra dimension, namely, the squashed Kaluza-Klein black holes open a window to extra dimensions.
We have investigated the strong gravitational lensing in a rotating squashed Kaluza-Klein (KK) black hole spacetime. Our result show that the strong gravitational lensings in the rotating squashed KK black hole spacetime have some distinct behaviors from those in the backgrounds of the four-dimensional Kerr black hole and of the squashed KK Gödel black hole. In the rotating squashed KK black hole spacetime, the marginally circular photon radius $ρ_{ps}$, the coefficient $\bar{a}$, $\bar{b}$, the deflection angle $α(θ)$ in the $ϕ$ direction and the corresponding observational variables are inde…
In this paper we investigate the strong gravitational lensing in a charged squashed Kaluza-Klein black hole. We suppose that the supermassive black hole in the galaxy center can be considered by a charged squashed Kaluza-Klein black hole and then we study the strong gravitational lensing theory and estimate the numerical values for parameters and observables of it. We explore the effects of the scale of extra dimension $ρ_0$ and the charge of black hole $ρ_q$ on these parameters and observables.
We investigate the geodetic precession effect of a parallely transported spin-vector along a circular geodesic in the five-dimensional squashed Kaluza-Klein black hole spacetime. Then we derive the higher-dimensional correction of the precession angle to the general relativity. We find that the correction is proportional to the square of (size of extra dimension)/(gravitational radius of central object).
We discuss Hawking radiation from a five-dimensional squashed Kaluza-Klein black hole on the basis of the tunneling mechanism. A simple manner, which was recently suggested by Umetsu, is possible to extend the original derivation by Parikh and Wilczek to various black holes. That is, we use the two-dimensional effective metric, which is obtained by the dimensional reduction near the horizon, as the background metric. By using same manner, we derive both the desired result of the Hawking temperature and the effect of the back reaction associated with the radiation in the squashed Kaluza-Klein b…
We study motions of photons in an unmagnetized cold homogeneous plasma medium in the five-dimensional charged static squashed Kaluza-Klein black hole spacetime. In this case, a photon behaves as a massive particle in a four-dimensional spherically symmetric spacetime. We consider the light deflection by the squashed Kaluza-Klein black hole surrounded by the plasma in a weak-field limit. We derive corrections of the deflection angle to general relativity, which are related to the size of the extra dimension, the charge of the black hole and the ratio between the plasma and the photon frequencie…
We study the scalar perturbation in the background of the charged Kaluza-Klein black holes with squashed horizons. We find that the position of infinite discontinuities of the heat capacities can be reflected in quasinormal spectrum. This shows the possible non-trivial relation between the thermodynamical and dynamical properties of black holes.
We study the quasinormal modes of scalar perturbation in the background of five-dimensional charged Kaluza-Klein black holes with squashed horizons immersed in the Gödel universe. Besides the influence due to the compactness of the extra dimension, we disclose the cosmological rotational effect in the wave dynamics. The wave behavior affected by the Gödel parameter provides an interesting insight into the Gödel universe.
We present new solutions of the $d=5$ Einstein-Yang-Mills theory describing black holes with squashed horizons. These configurations are asymptotically locally flat and have a boundary topology of a fibre bundle $R\times S^1 \hookrightarrow S^{2}$. In a $d=4$ picture, they describe black hole solutions with both nonabelian and U(1) magnetic charges.
We study the Hawking radiation from the five-dimensional charged static squashed Kaluza-Klein black hole by the tunneling of charged scalar particles and charged fermions. In contrast to the previous studies of Hawking radiation from squashed Kaluza-Klein black holes, we consider the phenomenological quantum gravity effects predicted by the generalized uncertainty principle with the minimal measurable length. We derive corrections of the Hawking temperature to general relativity, which are related to the energy of the emitted particle, the size of the compact extra dimension, the charge of the…
We investigate the strong gravitational lensing in a Kaluza-Klein black hole with squashed horizons. We find the size of the extra dimension imprints in the radius of the photon sphere, the deflection angle, the angular position and magnification of the relativistic images. Supposing that the gravitational field of the supermassive central object of the Galaxy can be described by this metric, we estimated the numerical values of the coefficients and observables for gravitational lensing in the strong field limit.
The Abbott-Deser mass, the Hamiltonian and the Komar mass of the 5-dimensional Kaluza-Klein black hole with squashed horizons take different values. Introducing a new couple of thermodynamic variables for the Komar mass, we show that each mass can be interpreted as a thermodynamic potential with its own natural variables, i.e. all masses are related to each other by the Legendre transformations. It is found that the new variables and the gravitational tension represent the squashing of the outer horizon.
We study geometrical structures of charged static black holes in the five-dimensional Einstein-Maxwell theory. The black holes we study have horizons in the form of squashed $ {\rm S}^3$, and their asymptotic structure consists of a twisted ${\rm S}^1$ bundle over the four-dimensional flat spacetime at the spatial infinity. The spacetime we consider is fully five-dimensional in the vicinity of the black hole and four-dimensional with a compact extra dimension at infinity.
Thermal radiation of scalar particles and Dirac fermions from squashed charged rotating five-dimensional black holes is considered. To obtain temperature of the black holes we use the tunnelling method. In case of scalar particles we make use of the Hamilton-Jacobi equation. To consider tunnelling of fermions the Dirac equation was investigated. The examination shows that radial parts of the action for scalar particles and fermions in quasi-classical limit in the vicinity of horizon are almost the same and as a consequence it gives rise to the identical expressions for the temperature in both …
Recently a five-dimensional Kaluza-Klein black hole solution with squashed horizon has been found in hep-th/0510094. The black hole spacetime is asymptotically locally flat and has a spatial infinity $S^1 \hookrightarrow S^{2}$. By using "boundary counterterm" method and generalized Abbott-Deser method, we calculate the mass of this black hole. When an appropriate background is chosen, the generalized Abbott-Deser method gives the same mass as the "boundary counterterm" method. The mass is found to satisfy the first law of black hole thermodynamics. The thermodynamic properties of the Kaluza-K…
We investigate the strong gravitational lensing in a squashed Kaluza-Klein black hole immersed in the Gödel universe with global rotation. Our result show that the strong gravitational lensing in the squashed Kaluza-Klein Gödel black hole spacetime has some distinct behaviors from that in the Kerr case. In the squashed Kaluza-Klein Gödel black hole spacetime, the photon sphere radius, the minimum impact parameter, the coefficient $\bar{a}$, $\bar{b}$ and the deflection angle $α(θ)$ in the $ϕ$ direction are independent of whether the photon goes with or against the global rotation of the Gödel …
We study the accretion process in the thin disk around a squashed Kaluza-Klein black hole and probe the effects of the extra dimensional scale $ρ_0$ on the physical properties of the disk. Our results show that with the increase of the parameter $ρ_0$, the energy flux, the conversion efficiency, the radiation temperature, the spectra luminosity and the spectra cut-off frequency of the thin accretion disk decrease, but the inner border of the disk increases. This implies that the extra dimension scale imprints in the mass accretion process in the disk.
Recently we suggested a solution generating technique for five-dimensional supergravity with three Abelian vector fields based on the hidden SO(4,4) symmetry of the three-dimensionally reduced theory. This technique generalizes the $G_{2(2)}$ generating technique developed earlier for minimal 5D supergravity (A. Bouchareb, G. Clément, C-M. Chen, D. V. Gal'tsov, N. G. Scherbluk, and Th. Wolf, Phys. Rev. D {\bf 76}, 104032 (2007)) and provides a new matrix representation for cosets forming the corresponding sigma-models in both cases. Here we further improve these methods introducing a matrix-va…
We investigate solutions of the Klein-Gordon equation in a class of five dimensional geometries presenting the same symmetries and asymptotic structure as the Gross-Perry-Sorkin monopole solution. Apart from globally regular metrics, we consider also squashed Kaluza-Klein black holes backgrounds.
We study gravitational and electromagnetic perturbation around the squashed Kaluza-Klein black holes with charge. Since the black hole spacetime focused on this paper have $SU(2) \times U(1) \simeq U(2)$ symmetry, we can separate the variables of the equations for perturbations by using Wigner function $D^{J}_{KM}$ which is the irreducible representation of the symmetry. In this paper, we mainly treat $J=0$ modes which preserve $SU(2)$ symmetry. We derive the master equations for the $J=0$ modes and discuss the stability of these modes. We show that the modes of $J = 0$ and $ K=0,\pm 2$ and th…
We study the shadow produced by a spinning Kaluza-Klein black hole in Einstein gravity coupled to a Maxwell field and a dilaton. The size and the shape of the shadow depend on the mass, the charge, and the angular momentum. We find that, for fixed values of these parameters, the shadow is slightly larger and less deformed than for its Kerr-Newman counterpart.
We investigate thermodynamics constructed on different background reference spacetimes for squashed Kaluza-Klein (SqKK) black hole and electrically charged black string in five-dimensional Einstein-Maxwell system. Two spacetimes are possible to be reference spacetimes giving finite gravitational classical actions: one is four-dimensional Minkowski times a circle and the other is the KK monopole. The boundary of the SqKK black hole can not be matched perfectly to that of the former reference spacetime because of the difference in topology. However, the resultant classical action coincides with …
Using the large $D$ effective theory approach, we construct a static solution of non-extremal and squashed black holes with/without an electric charge, which describes a spherical black hole in a Kaluza-Klein spacetime with a compactified dimension. The asymptotic background with a compactified dimension and near-horizon geometry are analytically solved by the $1/D$ expansion. Particularly, our work demonstrates that the large $D$ limit can be applied to solve the non-trivial background with a compactified direction, which leads to a first-order flow equation. Moreover, we show that the extrem…
We present a new expression for the five-dimensional static Kaluza-Klein black hole solution with squashed $S^3$ horizons and three different charge parameters. This black hole solution belongs to $D = 5$ $N = 2$ supergravity theory, its spacetime is locally asymptotically flat and has a spatial infinity $R \times S^1 \hookrightarrow S^2$. The form of the solution is extraordinary simple and permits us very conveniently to calculate its conserved charges by using the counterterm method. It is further shown that our thermodynamical quantities perfectly obey both the differential and the integra…
In this paper we study the shadow of rotating Kaluza-Klein (KK) black holes and the connection between the shadow radius and the real part of quasi-normal modes (QNMs) in the eiokonal limit. In addition we have explored the quasi-periodic oscillations (QPOs) in the rotating KK black hole.
We examine the shadow of a rotating Kaluza-Klein black hole in Einstein gravity coupled to a Maxwell field and a dilaton. The size and the shape of the shadow depend on the mass, the charge, and the angular momentum of the compact object. For a given mass, the size increases with the rotation parameter and decreases with the electric charge. The distortion with respect to the non rotating case grows with the charge and the rotation parameter. For fixed values of these parameters, the shadow is slightly larger and less deformed than in the Kerr-Newman case.
We examine an exact solution which represents a charged black hole in a Kaluza-Klein universe in the five-dimensional Einstein-Maxwell theory. The spacetime approaches to the five-dimensional Kasner solution that describes expanding three dimensions and shrinking an extra dimension in the far region. The metric is continuous but not smooth at the black hole horizon. There appears a mild curvature singularity that a free-fall observer can traverse the horizon. The horizon is a squashed three-sphere with a constant size, and the metric is approximately static near the horizon.
We establish an explicit correspondence of Einstein gravity on the squashed spheres that are the $U(1)$ bundles over $\mathbb{CP}^m$ to the Kaluza-Klein AdS gravity on the tori. This allows us to map the Ricci-flat Kerr metrics in odd dimensions with all equal angular momenta to charged Kaluza-Klein AdS black holes that can be lifted to become singly rotating M-branes and D3-branes. Furthermore, we find maps between Ricci-flat gravitational instantons to the AdS domain walls. In particular the supersymmetric bolt instantons correspond to domain walls that can be interpreted as distributed M-br…
Rotating Black Holes Surrounded by Massive Vector Fields in Kaluza Klein Gravity
Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；Institute for Physical Problems, Baku State University（巴库国立大学物理问题研究所）；Institute of Physics, Ministry of Science and Education（教育与科学部物理研究所）；National Nanotechnology Laboratory of Open Type（开放式国家纳米技术实验室）；Al-Farabi Kazakh National University（阿尔法拉比哈萨克国立大学）；Helsinki Institute of Physics, University of Helsinki（赫尔辛基大学赫尔辛基物理研究所）；Department of Physics, Faculty of Science, University of Hradec Králové（赫拉德茨克拉洛韦大学理学院物理系）；Khazar University, Department of Physics and Electronics（哈扎尔大学物理与电子学系）
Reducing motion of particles to a two-dimensional potential problem, we show that there are stable circular orbits around a squashed Kaluza-Klein black hole with a spherical horizon and multi-Kaluza-Klein black holes with two spherical horizons in five dimensions. For a single horizon, we show analytically that the radius of an innermost stable circular orbit monotonically depends on the size of an extra dimension. For two horizons, the radius of an innermost stable circular orbit depends on the separation between two black holes besides the size of an extra dimension. More precisely, the set …
The light rings (LRs) topological charge (TC) of a spacetime measures the number of stable LRs minus the number of unstable LRs. It is invariant under smooth spacetime deformations obeying fixed boundary conditions. Asymptotically flat equilibrium black holes (BHs) have, generically, TC=$-1$. In Einstein-Maxwell theory, however, the Schwarzschild-Melvin BH - describing a neutral BH immersed in a strong magnetic field - has TC$=0$. This allows the existence of BHs without LRs and produces remarkable phenomenological features, like panoramic shadows. Here we investigate the generalised Schwarzsc…
We consider the Weyl$-$Yang gauge theory of gravitation in a $(4+3)$-dimensional curved space-time within the scenario of the non-Abelian Kaluza$-$Klein theory for the source and torsion-free limits. The explicit forms of the field equations containing a new spin current term and the energy-momentum tensors in the usual four dimensions are obtained through the well-known dimensional reduction procedure. In this limit, these field equations admit (anti-)dyon and magnetic (anti-)monopole solutions as well as non-Einsteinian solutions in the presence of a generalized Wu$-$Yang ansatz and some spe…
Black holes have their proper oscillations, which are called the quasi-normal modes. The proper oscillations of astrophysical black holes can be observed in the nearest future with the help of gravitational wave detectors. Quasi-normal modes are also very important in the context of testing of the stability of black objects, the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence and in higher dimensional theories, such as the brane-world scenarios and string theory. This dissertation reviews a number of works, which provide a thorough study of the quasi-normal spectrum of a wide cl…
