Abstract
We review some properties of the Kerr–Newman–(anti-)de Sitter solution. We present admissible extremal configurations, but the main focus of this work is charged test particle motion in the equatorial plane and along the spacetime’s axis of rotation, with emphasis on static positions and effective potentials.
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Notes
- 1.
The orientation of the ϕ coordinate is chosen in such a way that angular momentum a is positive.
- 2.
A static observer in the area with negative r would be, among other things, always subjected to a naked singularity.
- 3.
We generally prefer to keep Λ as a free parameter because, from the astrophysical point of view, its value is known and fixed for all black holes.
- 4.
In order to avoid confusion, let us point out that positive values of r are not “above” and negative “below” the black hole on its axis—that is determined by θ. Negative r represent the analytically extended part of the spacetime.
- 5.
Take note that the particle may even be uncharged.
- 6.
The quadratic polynomial q 2r 2 + 2aq 2r + 4a 4 + 4a 2q 2 has a negative discriminant −4(4a 4q 2 + 3a 2q 4), barring real roots.
- 7.
For the Kerr solution this observation can be done directly from (4.20), which for q = 0 reduces into m < a, the known condition for the singularity’s nakedness.
- 8.
Note that according to (4.29) photons on the axis must have E > 0, otherwise they would move backwards in time or not move in time at all.
- 9.
Recall that 4a 2 − q 2 > 0, (4.18). Then indeed sgn(V ″) = sgn(W″).
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Acknowledgements
J.V. was supported by Charles University, project GA UK No. 80918. M.Ž. acknowledges support by GACR 17-13525S.
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Veselý, J., Žofka, M. (2019). Electrogeodesics and Extremal Horizons in Kerr–Newman–(anti-)de Sitter. In: Cacciatori, S., Güneysu, B., Pigola, S. (eds) Einstein Equations: Physical and Mathematical Aspects of General Relativity. DOMOSCHOOL 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-18061-4_11
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