Shadow of rotating Hořava-Lifshitz black hole

Abstract

The shadow of rotating Hořava-Lifshitz black hole has been studied and it was shown that in addition to the specific angular momentum a, parameters of Hořava-Lifshitz spacetime essentially deform the shape of the black hole shadow. For a given value of the black hole spin parameter a, the presence of a parameter Λ _W and KS parameter ω enlarges the shadow and reduces its deformation with respect to the one in the Kerr spacetime. We have found a dependence of radius of the shadow R _s and distortion parameter δ _s from parameter Λ _W and KS parameter ω both. Optical features of the rotating Hořava-Lifshitz black hole solutions are treated as emphasizing the rotation of the polarization vector along null congruences. A comparison of the obtained theoretical results on polarization angle with the observational data on Faraday rotation measurements provides the upper limit for the δ parameter as δ≤2.1⋅10⁻³.

Appendix: Quantities in the locally non-rotating frame

All physical quantities are indicated by parenthesis around the Greek indices in the locally non-rotating frame (LNRF). The components of $k^{\mu}=dx^{\mu}/d\mu=\dot{x}^{\mu}$ which is tangent to the null congruence and its projections k ^(μ) on the LNRF are

$$ \begin{aligned} &k^{(0)} = k^{(t)} = e^{\nu}k^{0}=e^{\nu} \dot{t} , \\ &k^{(1)} = k^{(r)} = e^{\lambda}k^{1}=e^{\lambda} \dot{r} , \\ &k^{(2)} = k^{(\theta)} = e^{\mu}k^{2}=e^{\mu} \dot{\theta} , \\ &k^{(3)} = k^{(\phi)} = e^{\psi} \bigl(k^{3}- \varOmega k^{0} \bigr) = e^{\psi} (\dot{\phi} - \varOmega \dot{t} ). \end{aligned} $$

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The functions ν,λ,μ and ψ are listed as following

$$ \begin{aligned} &e^{2\nu}=\frac{\rho^2 \Delta_{\rm r} \Delta_{\rm \theta}}{\varSigma^2 \varXi^2}, \\ &e^{\lambda}=\frac{\rho^2}{\Delta_{\rm r}}, \\ &e^{\mu}=\frac{\rho^2}{\Delta_{\rm \theta}}, \\ &e^{\psi}=\frac{\varXi^2 \sin^2 \theta}{\varSigma^2 \rho^2} . \end{aligned} $$

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The nonzero components of the connection projected on the LNRF (Bardeen et al. 1972) are

$$ \begin{aligned} &\varGamma_{(r)(t)}^{(t)} = \varGamma_{(t)(t)}^{(r)}= \partial_{r}\nu e^{-\lambda} , \\ &\varGamma_{(\theta)(t)}^{(t)} = \varGamma_{(t)(t)}^{(\theta)}= \partial_{\theta}\nu e^{-\mu} , \\ &\varGamma_{(\theta)(r)}^{(r)} = -\varGamma_{(r)(r)}^{(\theta)}= \partial_{\theta}\lambda e^{-\mu} , \\ &\varGamma_{(\theta)(\theta)}^{(r)} = -\varGamma_{(r)(\theta)}^{(\theta)}=- \partial_{r}\mu e^{-\lambda} , \\ &\varGamma_{(\phi)(\phi)}^{(r)} = -\varGamma_{(r)(\phi)}^{(\phi)}=- \partial_{r}\psi e^{-\lambda} , \\ &\varGamma_{(\phi)(\phi)}^{(\theta)} = -\varGamma_{(\theta)(\phi)}^{(\phi)}=- \partial_{\theta}\psi e^{-\mu} , \\ &\varGamma_{(r)(\phi)}^{(t)} = \varGamma_{(t)(\phi)}^{(r)}= \varGamma_{(\phi)(r)}^{(t)} = \varGamma_{(\phi)(t)}^{(r)} \\ &\hphantom{\varGamma_{(r)(\phi)}^{(t)}}=-\varGamma_{(t)(r)}^{(\phi)}=-\varGamma_{(r)(t)}^{(\phi)}= \frac{1}{2}\partial_{r}\varOmega e^{\psi-\nu-\lambda} , \\ &\varGamma_{(\theta)(\phi)}^{(t)}=\varGamma_{(t)(\phi)}^{(\theta)}= \varGamma_{(\phi)(\theta)}^{(t)} = \varGamma_{(\phi)(t)}^{(\theta)} \\ &\hphantom{\varGamma_{(\theta)(\phi)}^{(t)}}= -\varGamma_{(t)(\theta)}^{(\phi)}=-\varGamma_{(\theta)(t)}^{(\phi)}= \frac{1}{2}\partial_{\theta}\varOmega e^{\psi-\nu-\mu} . \end{aligned} $$

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