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Published November 15, 2012 | Published
Journal Article Open

Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. III. Quasinormal pulsations of Schwarzschild and Kerr black holes


In recent papers, we and colleagues have introduced a way to visualize the full vacuum Riemann curvature tensor using frame-drag vortex lines and their vorticities, and tidal tendex lines and their tendicities. We have also introduced the concepts of horizon vortexes and tendexes and three-dimensional vortexes and tendexes (regions on or outside the horizon where vorticities or tendicities are large). In this paper, using these concepts, we discover a number of previously unknown features of quasinormal modes of Schwarzschild and Kerr black holes. These modes can be classified by a radial quantum number n, spheroidal harmonic orders (l,m), and parity, which can be electric [(-1)^l] or magnetic [(-1)^(l+1)]. Among our discoveries are these: (i) There is a near duality between modes of the same (n,l,m): a duality in which the tendex and vortex structures of electric-parity modes are interchanged with the vortex and tendex structures (respectively) of magnetic-parity modes. (ii) This near duality is perfect for the modes' complex eigenfrequencies (which are well known to be identical) and perfect on the horizon; it is slightly broken in the equatorial plane of a nonspinning hole, and the breaking becomes greater out of the equatorial plane, and greater as the hole is spun up; but even out of the plane for fast-spinning holes, the duality is surprisingly good. (iii) Electric-parity modes can be regarded as generated by three-dimensional tendexes that stick radially out of the horizon. As these "longitudinal," near-zone tendexes rotate or oscillate, they generate longitudinal-transverse near-zone vortexes and tendexes and outgoing and ingoing gravitational waves. The ingoing waves act back on the longitudinal tendexes, driving them to slide off the horizon, which results in decay of the mode's strength. (iv) By duality, magnetic-parity modes are driven in this same manner by longitudinal, near-zone vortexes that stick out of the horizon. (v) When visualized, the three-dimensional vortexes and tendexes of a (l,m)=(2,2) mode, and also a (2,1) mode, spiral outward and backward like water from a whirling sprinkler, becoming outgoing gravitational waves. By contrast, a (2,2) mode superposed on a (2,-2) mode, has oscillating horizon vortexes or tendexes that eject three-dimensional vortexes and tendexes, which propagate outward becoming gravitational waves; and so does a (2,0) mode. (vi) For magnetic-parity modes of a Schwarzschild black hole, the perturbative frame-drag field, and hence also the perturbative vortexes and vortex lines, are strictly gauge invariant (unaffected by infinitesimal magnetic-parity changes of time slicing and spatial coordinates). (vii) We have computed the vortex and tendex structures of electric-parity modes of Schwarzschild in two very different gauges and find essentially no discernible differences in their pictorial visualizations. (viii) We have compared the vortex lines, from a numerical-relativity simulation of a black hole binary in its final ringdown stage, with the vortex lines of a (2,2) electric-parity mode of a Kerr black hole with the same spin (a/M=0.945) and find remarkably good agreement.

Additional Information

© 2012 American Physical Society. Received 15 August 2012; published 12 November 2012. We thank John Belcher, Jeandrew Brink, and Richard Price for helpful discussions. We thank Jeff Kaplan for helpful discussions and web assistance. We thank Mark Scheel for helpful discussions and for version control assistance. A. Z. would also like to thank the National Institute for Theoretical Physics of South Africa for hosting him during a portion of this work. Some calculations in this paper were performed using the Spectral Einstein Code (SPEC) [45]. This research was supported by NSF Grants No. PHY-0960291, No. PHY-1068881 and CAREER Grant No. PHY-0956189 at Caltech, by NSF Grants No. PHY-0969111 and No. PHY-1005426 at Cornell, by NASA Grant No. NNX09AF97G at Caltech, by NASA grant No. NNX09AF96G at Cornell, and by the Sherman Fairchild Foundation at Caltech and Cornell, the Brinson Foundation at Caltech, and the David and Barbara Groce fund at Caltech.

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