Abstract
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.
Highlights
Quantities whose value is manifestly independent of coordinates are useful in general relativity [1]
One can construct fourteen such quantities in four-dimensional spacetime, since the Riemann curvature tensor has twenty independent components subject to six conditions on the metric [2]. These quantities have proved useful in, for example, classifying metrics [3,4] and deciding whether or not they are equivalent [5,6]. They have been applied to speed up the estimation of gravitational-wave signatures from black-hole collisions [7], to distinguish between “gravito-electrically” versus “gravito-magnetically dominated” regions of spacetime [8,9,10], to measure the mass and spin and locate the horizons of black holes [11,12,13,14], and to study perturbations of the Kerr metric [15], Lorentzian wormholes [16], and others [17]
We apply the ZM formalism to the Kerr–Newman metric to obtain for the first time explicit algebraic expressions for all seventeen invariants for black holes of the most general kind
Summary
Quantities whose value is manifestly independent of coordinates are useful in general relativity [1]. One can construct fourteen such quantities in four-dimensional spacetime, since the Riemann curvature tensor has twenty independent components subject to six conditions on the metric [2]. These quantities have proved useful in, for example, classifying metrics [3,4] and deciding whether or not they are equivalent [5,6]. Our results will be of most astrophysical interest in the case of nonzero spin and zero charge (because all real black holes rotate, but few possess significant charge due to the preferential infall of opposite-charged matter)
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