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Date: 5-2-2017
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Date: 2-2-2017
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Motion off the equatorial plane
We consider only a special type of motion off the equatorial plane when particles are moving quasiradially along the trajectories on which the value of the polar angle θ remains constant, θ = θ0. For this motion
(1.1)
If we exclude trivial solutions θ0 = 0, θ0 = π, and θ0 = π/2, the relations between the integrals of motion can be written in the form
(1.2)
(1.3)
Hence, motion with constant θ = θ0 is possible only when Ẽ > 1 (infinite motion).
Non-relativistic particles moving at parabolic velocity (v∞ = 0) and with zero angular momentum (lz = 0) represent a special limiting case. Such particles fall at constant θ and are dragged into the rotation around the black hole.
Another important limiting case is the falling of ultrarelativistic particles (photons) which move at infinity at θ = constant. In this limit, Ẽ → ∞and lz →∞ while their ratio b = lz/Ẽ remains finite and equal to b = a sin2 θ. The null vector nμ tangent to a null geodesic representing the motion of the in-coming photon is
(1.4)
If one substitutes 1 instead of −1 into the right-hand side of this expression one obtains a congruence of outgoing photons. These two null congruences are known as the principal null congruences of the Kerr metric. They are geodesic and shear free. They satisfy the following relations:
(1.5)
where Cαβγ δ is the Weyl tensor. Since the Kerr metric is a vacuum solution, the Weyl tensor is equal to the Riemann tensor. The principal null vectors in the Kerr geometry also obey the relation
(1.6)
where F = -gt t = 1 - 2Mr/Σ.
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