Abstract
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 $ \overline{a} $ , $ \overline{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, which is different with those in the usual four-dimensional Kerr black hole spacetime. Moreover, we also find that with the increase of the scale of extra dimension ρ0, the marginally circular photon radius ρ ps 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 Gödel black hole they increase for the smaller global rotation parameter j and decrease for the larger one. In the extremely squashed case ρ 0 = 0, the coefficient a in the rotating squashed KK black hole increases monotonously with the rotation parameter, but in the squashed KK Gödel black hole it is a constant and independent of the global rotation of the Gödel Universe. These information could help us to understand further the effects of the rotation parameter and the scale of extra dimension on the strong gravitational lensing in the black hole spacetimes.
Highlights
JHEP03(2014)089 et al have considered the effect of the charge on the strong gravitational lensing in the squashed KK black holes [47, 48]
We find that with the increase of the scale of extra dimension ρ0, the marginally circular photon radius ρps and the angular position of the relativistic images θ∞ first decreases and 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
As the rotation parameter b → 0, we find that the function H(ρ) → 0, which yields that the impact parameter (3.7) and the equation of circular photon orbits (3.8) reduce to those in the usual Schwarzschild squashed KK black hole spacetime [45]
Summary
Let us review briefly a rotating squashed KK black hole without charge, which can be obtained by applying the squashing transformation techniques to a five-dimensional Kerr black hole with two equal angular momenta [7]. B4 − 4b2ρ0 (ρ0 − ρM ) + 4ρ20 (ρM + ρ0) r′2∞ = b2 + 2ρ0ρM + 2ρ20 + b4 − 4b2ρ0 (ρ0 − ρM ) + 4ρ20 (ρM + ρ0)2, With these quantities, we can find that all of coefficients in the metric (2.19) can be expressed as the functions of the parameters ρ0, ρM and b, which means that we can study the strong gravitational lensing in the rotating squashed KK black hole spacetime (2.6) through the standard form used in [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]
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