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Curvature Invariants for Charged and Rotating Black Holes

Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252, USA
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Department of Physics, University of California, San Diego, CA 92093, USA
Author to whom correspondence should be addressed.
Universe 2020, 6(2), 22;
Received: 25 October 2019 / Revised: 18 January 2020 / Accepted: 21 January 2020 / Published: 24 January 2020
(This article belongs to the Special Issue Probing New Physics with Black Holes)
Riemann curvature invariants are important in general relativity because they encode the geometrical properties of spacetime in a manifestly coordinate-invariant way. Fourteen such invariants are required to characterize four-dimensional spacetime in general, and Zakhary and McIntosh showed that as many as seventeen can be required in certain degenerate cases. We calculate explicit expressions for all seventeen of these Zakhary–McIntosh curvature invariants for the Kerr–Newman metric that describes spacetime around black holes of the most general kind (those with mass, charge, and spin), and confirm that they are related by eight algebraic conditions (dubbed syzygies by Zakhary and McIntosh), which serve as a useful check on our results. Plots of these invariants show richer structure than is suggested by traditional (coordinate-dependent) textbook depictions, and may repay further investigation. View Full-Text
Keywords: black holes; curvature invariants; general relativity black holes; curvature invariants; general relativity
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MDPI and ACS Style

Overduin, J.; Coplan, M.; Wilcomb, K.; Henry, R.C. Curvature Invariants for Charged and Rotating Black Holes. Universe 2020, 6, 22.

AMA Style

Overduin J, Coplan M, Wilcomb K, Henry RC. Curvature Invariants for Charged and Rotating Black Holes. Universe. 2020; 6(2):22.

Chicago/Turabian Style

Overduin, James, Max Coplan, Kielan Wilcomb, and Richard Conn Henry. 2020. "Curvature Invariants for Charged and Rotating Black Holes" Universe 6, no. 2: 22.

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