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 3-D vortexes and tendexes (regions where vorticities or tendicities are large). 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 mode indexes (n,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 non-spinning 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 3-D 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.

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