Abstract
It is explained how to derive analytical formulas for the boundary curve of the shadow as seen by an observer at given position in the domain of outer communication. The formulas are used to analyze the dependency of the shadow of a black hole on the motion of the observer. Furthermore, the horizontal and vertical angular diameters of the shadow are calculated. Although explicit formulas are given for the Kerr space-time only, the method holds true for the general Plebański–Demiański class. After all, the angular diameters for the black holes at the centers of our Galaxy and of M87 are estimated.
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- 1.
- 2.
Because of the symmetry of the Plebański–Demiański space-time, it is enough to specify the r and \(\vartheta \) coordinate to define a fixed position in space-time.
- 3.
Gram–Schmidt orthonormalization: \(\widetilde{e}_{3} \propto e_{3} + g(e_{3},\widetilde{e}_{0})\widetilde{e}_{0}\), \(\widetilde{e}_{1} \propto e_{1} + g(e_{1},\widetilde{e}_{0})\widetilde{e}_{0} - g(e_{1},\widetilde{e}_{3})\widetilde{e}_{3}\), \(\widetilde{e}_{2} \propto e_{2} + g(e_{2},\widetilde{e}_{0})\widetilde{e}_{0} - g(e_{2},\widetilde{e}_{1})\widetilde{e}_{1} - g(e_{2},\widetilde{e}_{3})\widetilde{e}_{3}\).
- 4.
For \(a=0\), one finds \(\zeta =\arg (-m^3)=-\pi \) and \(r_{h_{1,2}}=3\) m. Then \(T^2(3m)\) reproduces (4.34).
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Grenzebach, A. (2016). The Shadow of Black Holes. In: The Shadow of Black Holes. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-30066-5_4
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