The Strong Gravitational Lensing In A Rotating Squashed Kaluza-klein Black Hole Spacetime

General relativity tells us that the matter and their motions results in the deformation of a spacetime, which yields that the light ray close to the matter would be deflected. This is so-called gravitational lensing. The gravitational lensing is applied widely in many fields including the detection of dark matter, the constraint to cosmological constant and the observation of the dim star far away from us, and so on. Moreover, it can also be used to examine and identify various theories of gravity including general relativity and the higher dimensional string theory. The strong gravitational lensing is the phenomenon that the light ray is deflected sharply from the original direction as it propagates in the strong gravitational field. The light ray near the photon sphere makes several complete loops around the lens before reaching the observer, which leads to the formation of the relativistic images. Theoretically, it is possible that there are infinite relativistic images at two sides of the black hole. Through observing the positions and the relative magnitudes of these images, we can infer the structure and the parameter of the compacted object. In this thesis, we will study the strong gravitational lensing in the rotating squashed Kaluza-Klein (KK) black hole spacetime.In the first chapter, we introduce in brief the basic principle, the classification and the application of the gravitational lensing. We also review the basic·formulas for the deflection angle, the magnification and the position of the images in gravitational lensing.In the second chapter, we introduce the formulas for the deflection angle of strong gravitational lensing in the static spherical symmetric spacetime, which pro-posed by V. Bozza. It is shown that as the distance of the closest approach of the light ray x0 approaches to the radius of photon sphere xm the deflection angle in-creases and diverges logarithmically. Adopting the strong field approximation, V. Bozza has built the connection between the parameter of spacetime metric and the observables of the strong gravitational lensing, which could provide some theoreti-cal evidence to detect the structure and the parameter of the compacted object by strong gravitational lensing.In the third chapter, we adopt to the method proposed by V. Bozza and study the strong gravitational lensing in the rotating squashed KK black hole space-time. We make a comparison among the strong gravitational lensing in the rotating squashed KK black hole, the four-dimensional Kerr black hole and the squashed KK Godel black hole spacetimes and make a summary about the effects of the different rotation parameter on the strong gravitational lensing. We find:(1) In the rotating squashed KK black hole spacetime, the marginally circular photon radius ρps, the coefficient a, b, the deflection angle α(θ) in the φ direction and the corresponding observational variables are independent of whether the photon goes with or against the rotation of the background. The strong gravitational lensing possesses the sim-ilar properties in the squashed KK Godel black hole spacetime. These properties of strong gravitational lensing in squashed KK black hole spacetimes are different distinctly from those in the four dimensional Kerr black hole spacetime. (2) With the increase of the scale of extra dimension ρ0, the marginally circular photon ra-dius ρps and the angular position of the relativistic images θ∞, first decreases and then increases in the rotating squashed KK black hole for fixed rotation parameter b, but in the squashed KK Godel black hole they increase for the smaller global rotation parameter j and decrease for the larger one. (3) In the extremely squashed case ρ0＝0, the coefficient a in the rotating squashed KK black hole increases monotonously with the rotation parameter, but in the squashed KK Godel black hole it is a constant and independent of the global rotation of the Godel Universe.Finally, we present a summary and make some prospects of the strong gravita-tional lensing in the rotating squashed KK black hole spacetime.