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.

Figure 1

(a) Some magnetic field lines in the equatorial plane for the (1,1) quasinormal mode of the electromagnetic field around a Schwarzschild black hole, with Eddington-Finklestein slicing. The horizon is color coded by the sign of the normal component of the magnetic field. The configuration rotates counterclockwise in time. (b) Some magnetic field lines for this same quasinormal mode, in three dimensions.

Figure 2

Some vortex lines (solid, red lines) and contours of vorticity (shaded regions) in the equatorial plane for the (2,2) magnetic-parity quasinormal mode of a nonrotating, Schwarzschild black hole, with complex eigenfrequency $\omega =(0.37367-0.08896i)/M$ where $M$ is the hole’s mass. The horizon (central circle) is color coded by the horizon vorticity ${\mathcal{B}}_{\mathrm{NN}}$ as seen by someone looking down on the black hole; this vorticity is entirely perturbative. The thick, solid red curves are one set of vortex lines in the equatorial plane—the set with negative vorticity. These lines include some that emerge from the horizon in the negative-vorticity (red) regions and some that never reach the horizon. The other, positive-vorticity, equatorial vortex lines are orthogonal to the ones shown and are identical to those shown but rotated through 90 degrees around the hole so some of them emerge from the horizon in the positive-vorticity (blue) regions. The contours represent the vorticity of the red (negative-vorticity) vortex lines, with largest magnitude of vorticity white and smallest purple (dark gray); the contours mark where the vorticity has fallen to 50%, 25%, 10%, and 5% of the maximum value attained at the center of the horizon vortex. The two dotted circles are drawn at Schwarzschild radii $r=ƛ$ and $r=\pi ƛ=\lambda /2$. They mark the approximate outer edge of the near zone and the approximate inner edge of the wave zone. The arrow marks the direction of rotation of the perturbation.

Figure 3

The vortex lines (solid black for clockwise; dashed black for counterclockwise) and color-coded vorticities in the equatorial plane for the same magnetic-parity (2,2) mode as in Fig. 2. This figure differs from Fig. 2 in ways designed to give information about the emitted gravitational waves: (i) It extends rather far out into the wave zone. (ii) It shows the angular structure of the orticity for the dominant vortex lines in each region of the equatorial plane. More specifically: the color at each point represents the vorticity of the equatorial vortex line there which has the largest magnitude of vorticity, with radial variations of vorticity normalized away (so the linear color code on the left indicates vorticity relative to the maximum at any given radius). The regions of large positive vorticity [blue (dark gray)] are clockwise vortexes; those of large negative vorticity [red (light gray)] are counterclockwise vortexes.

Figure 4

Some three-dimensional clockwise vortex lines (shown black) and regions of large vorticity [vortexes, shown blue (dark gray) and red (light gray)] and small vorticity (shown off-white), for the same magnetic-parity mode of a Schwarzschild black hole as in Figs. 2, 3. More specifically: the inner sphere is the horizon, color coded by its vorticity. The blue region is a clockwise vortex in which one vortex line has vorticity at least 85% of the maximum value at that radius, and similarly for the counterclockwise red region. The four off-white regions are locations where no vortex line has magnitude of vorticity in excess of 25% of the maximum at that radius.

Figure 5

Three-dimensional vortexes for the magnetic-parity, (2,1) mode of a Schwarzschild black hole. The colored surfaces enclose the region where, for each radius, the vorticity of at least one vortex line exceeds 90% of the maximum for that radius. In the blue and red (dark and light gray) regions, the clockwise and counterclockwise vortex lines, respectively, have the larger vorticity.

Figure 6

Vortex lines and vorticities for magnetic-parity (2,0) mode of Schwarzschild in a surface ${\mathcal{S}}_{\varphi }$ of constant $\varphi$. The line and coloring conventions are the same as in Fig. 3 (solid lines for clockwise, dashed for counterclockwise; color shows vorticity of the vortex line with largest magnitude of vorticity, with radial variations removed and intensity of color as in the key on right edge of Fig. 3). The central circle is the horizon, color coded by the horizon vorticity.

Figure 7

Positive-tendicity (blue) and negative-tendicity (red) perturbative tendex lines of a (2,0) magnetic-parity perturbation of a Schwarzschild black hole. These lines spiral around deformed tori of progressively larger diameter. The viewpoint is looking down onto the equatorial plane from the positive symmetry axis. Upper right inset: The negative tendex line spiraling around the outermost torus, viewed in cross section from the equatorial plane. Lower right inset: The negative tendex line spiraling around the small torus third from the center, viewed in cross section from the equatorial plane. Also shown, in black in the main drawing and the large inset, are two of this mode’s vortex lines, one from Fig. 6 wrapping around the outermost torus in a ${\mathcal{S}}_{\varphi }$ plane; the other an azimuthal circle wrapping around that torus in the $\varphi$ direction. This figure was actually drawn depicting vortex lines of the electric-parity mode discussed in Sec. 5d; but by duality (which is excellent in the wave zone), it also represents the tendexes of the magnetic-parity mode discussed in this section.

Figure 8

Equatorial vortex structure of the superposed (2,2) and $(2,-2)$, magnetic-parity, fundamental modes of a Schwarzschild black hole. The colors encode the vorticity of the dashed vortex lines. The vorticity of the solid lines is not shown but can be inferred from the fact that under a 90° rotation, the dashed lines map into solid and the solid into dashed.

Figure 9

Perturbative horizon tendicities ${\delta \mathcal{E}}_{\mathrm{NN}}$ and vorticities ${\delta \mathcal{B}}_{\mathrm{NN}}$ for the (2,2) quasinormal modes with electric and magnetic parities (see column labels at the top). The top row is for a Schwarzschild black hole, $a=0$; the bottom for a rapidly spinning Kerr black hole, $a/M=0.945$. The color intensity is proportional to the magnitude of the tendicity or vorticity, with blue (dark gray) for positive and red (light gray) for negative. For discussion, see Sec. 3a of the text.

Figure 10

Three representations of the vortex lines and vortexes in the equatorial plane of a Schwarzschild black hole perturbed by a magnetic-parity (2,2) quasinormal mode. The bottom panels span a region $56M$ on each side, and the top panels are a zoom-in of the lower panels, $30M$ on each side. All panels show positive-vorticity lines as solid and negative-vorticity lines as dashed. In all panels, blue (dark gray) corresponds to positive and red (light gray) to negative; the intensity of the color indicates the strength of the vorticity at that point normalized by the maximum of the vorticity at that radius (darker shading indicates a larger strength and lighter, weaker). Similarly, in all panels, the central circle surrounded by a narrow white line is the horizon colored by its vorticity as described above. Left column: Vortex lines colored and shaded by their scaled vorticity. Middle column: Negative vorticity coloring the plane with black vortex lines. Right column: Vorticity with the larger absolute value coloring the plane and black vortex lines. For discussion of this figure, see Sec. 3b1.

Figure 11

The equatorial-plane, electric-parity tendexes and tendex lines of a (2,2) perturbation of a Schwarzschild black hole in RWZ gauge (left panel) and IR gauge (right panel). The conventions for the lines, the coloring and the shading are identical to those in the right panels of Fig. 10.

Figure 12

Vortexes and tendexes and their field lines in the equatorial plane for (2,2) modes of Schwarzschild and Kerr black holes. The lines, the coloring and the shading are identical to those in the right panels of Fig. 10. The upper row is for a Schwarzschild black hole ($a=0$); the lower row, for a rapidly spinning Kerr black hole ($a=0.945$); see labels on the left. The left column shows the vortex lines and vorticities for magnetic parity (which are gauge invariant for the perturbations of a Schwarzschild hole); the middle and right columns show the tendex lines and tendicities for the electric-parity mode in IR gauge; see labels at the top. In the right column, the equatorial plane is isometrically embedded in three-dimensional Euclidean space. The top panels are $24M$ across; the bottom, $14M$. This figure elucidates duality and the influence of black-hole spin; see the discussion in Sec. 3b3.

Figure 13

For an electric-parity (2,2) mode of a Schwarzschild black hole: the vorticity of the counterclockwise vortex lines that pass through the equatorial plane at a 45 degree angle. The clockwise vortex lines that pass through the plane have equal and opposite vorticity. By near duality, this figure also depicts the perturbative tendex structure for the magnetic-parity (2,2) mode. The conventions for coloring and shading are the same as in Fig. 10. Because the horizon vorticity is exactly zero for this mode, the horizon is shown as a white disk. The left panel, a region $30M$ across, is a zoom in of the right panel, which is $56M$ across.

Figure 14

Three-dimensional vortexes and tendexes of the same four modes as are shown in Fig. 12. As there, so here, the top row is for a Schwarzschild black hole, and the bottom for fast-spin Kerr, $a=0.945$; the left two columns (one in Fig. 12) are for a magnetic-parity (2,2) mode (with the vortexes in IR coordinates), and the right two columns (one in Fig. 12) are for electric parity in IR gauge. For each parity, the first column shows structures of the field that generates the waves ($\delta \mathcal{B}$ for magnetic parity; $\delta \mathcal{E}$ for electric parity) and the second column shows structures of the other field (not included in the equatorial-plane drawings of Fig. 12). In each panel, the colored surfaces show the outer faces of vortexes (for $\delta \mathcal{B}$) or tendexes (for $\delta \mathcal{E}$), defined as the locations, for a given radius, where the largest-in-magnitude eigenvalue of the field being plotted ($\delta \mathcal{B}$ or $\delta \mathcal{E}$) has dropped to a certain percentage (90, 85, 80 or 75) of its maximum for that radius; that percentage is shown alongside the colored surfaces. As in previous figures, the surface is red (light gray) if that largest-in-magnitude eigenvalue is negative and blue (dark gray) if positive. The off-white regions are surfaces where the largest-in-magnitude eigenvalue has dropped to 15%, 20% or 25% of the maximum at that radius. In each panel the black lines are a few of the vortex lines (for $\delta \mathcal{B}$ panels) or tendex lines (for $\delta \mathcal{E}$ panels) that become transverse when they reach large radii, and thereby produce the tidal or frame-drag force of an emitted gravitational wave. For discussion of this figure, see Sec. 3c.

Figure 15

Left: Vortex lines of a $a/M=0.945$ Kerr black hole perturbed by an electric-parity (2,2) quasinormal-mode in IR gauge. Right: Vortex lines from a $a/M=0.945$ ringing down Kerr black hole obtained from a numerical simulation [19] of two identical merging black holes with spins of magnitude 0.97 aligned parallel to the orbital angular momentum. In the simulation, we chose a late enough time that the common apparent horizon is essentially that of a single, perturbed black hole, and we computed the vortex lines using methods summarized in Ref. 1.

Figure 16

The vorticities and vortex lines in the equatorial plane of a Schwarzschild black hole, for the fundamental magnetic-parity (2,2) mode superposed on the fundamental magnetic-parity $(2,-2)$ mode, depicted using the same three visualization techniques as in Fig. 10. Here, however, we do not scale the vorticity by any function, but the numbers on the vorticity scale on the left of the panels are equal to $\sqrt{\mathrm{\text{vorticity}}\times r}$ (where $r$ is radius), in units of the maximum value of this quantity, which occurs on the horizon at $\varphi =3\pi /4$ and $\varphi =7\pi /4$. The top panels cover a region $30M$ across, and the bottom panels are a zoom-out of the upper panels, $56M$ across. The central circle in all panels is the horizon as viewed from the polar axis, colored by its vorticity. Left column: The two families of vortex lines (one shown dashed, the other solid) with each line colored, at each point, by the sign of its vorticity [blue (dark gray) for positive, i.e., clockwise; red (light gray) for negative, i.e., counterclockwise], and each line has an intensity proportional to the magnitude of its vorticity. Center column: The same vortex lines are colored black, and the equatorial plane is colored by the vorticity of the dashed family of lines. Right column: The same as the center column, but the equatorial plane is colored by the vorticity with the larger magnitude.

Figure 17

For the electric-parity, superposed (2,2) and $(2,-2)$ fundamental modes of Schwarzschild: the vorticity of the counterclockwise vortex lines that pass through the equatorial plane at 45° angles. By near duality, this figure also depicts the perturbative tendex structure for the magnetic-parity superposed mode. The intensity scale of the red color (left edge of figure) is the same as that in the center column of Fig. 16. The left panel, a region $30M$ across, is a zoom-in of the right panel, which is $56M$ across. This figure is the superposed-mode analog of Fig. 13.

Figure 18

Time evolution of the equatorial vortexes (top and middle rows) and equatorial perturbative tendexes (bottom row) for the superposed (2,2) and $(2,-2)$ magnetic-parity mode of Schwarzschild in RWZ gauge. The color scale is the same as the center column of Fig. 16, and the gravitational-wave-induced exponential decay of the vorticity and tendicity has been removed. Top row: Equatorial vortex lines and their vorticity plotted in a region near the horizon ($16M$ across) followed over time $t$. The real part of the eigenfrequency is denoted $\sigma$, so the successive panels, left to right cover half a cycle of the mode’s oscillation. Middle row: The vorticity of the equatorial vortex lines in the near, intermediate and beginning of wave zone ($30M$ across) at the same time steps as the top row. Bottom row: Tendicity of the counterclockwise tendex lines passing through the equatorial plane (which is dual to the left panel of Fig. 17), plotted at the same time steps as the top row.

Figure 19

The (2,1) magnetic-parity horizon vorticity and vortexes. Left panel: The horizon vorticity for the (2,1) magnetic-parity perturbation of Schwarzschild, colored as in Fig. 9. There are four horizon vortexes, two clockwise [blue (dark gray)] and two counterclockwise [red (light gray)]. The horizon vorticity vanishes at the equator and the poles. Middle panel: Vorticity of the counterclockwise vortex lines passing through the equatorial plane, colored and normalized as in the middle and right hand columns of Fig. 10 and plotted in a region $24M$ across. The [blue (dark gray)] vorticity of the clockwise vortex lines has precisely this same pattern, because the two families of lines pass through the equatorial plane with the same magnitude of vorticity at each point. Right panel: Three dimensional vortexes colored and labeled as in Fig. 14. By near duality, this figure also represents (to good accuracy) the tendicity and tendex structure of the (2,1) electric-parity mode.

Figure 20

The (2,1) electric-parity vortex lines, vorticities and vortexes in the equatorial plane. For this mode the horizon’s vorticity vanishes, so the horizon is plotted as a white disk or sphere. Left panel: Counterclockwise vortex lines (dashed) and their vorticity [red (light gray) color] normalized as in Fig. 10, and plotted in a region $24M$ across. Middle panel: Both clockwise (solid) and counterclockwise (dashed) vortex lines, and the vorticity (color) of the line with the larger magnitude of vorticity, in a region $24M$ across. Right Panel: Three-dimensional vortexes colored and labeled as in Fig. 14. By near duality, this figure also represents to good accuracy the perturbative tendex lines, tendicities and tendexes for the magnetic-parity (2,1) mode.

Figure 21

The horizon vorticities ($\delta {\mathcal{B}}_{\mathrm{NN}}$) of the quadrupolar, (2,0), magnetic-parity mode. As in Fig. 9, the color intensity is proportional to the magnitude of the vorticity with blue (dark gray) for positive and red (light gray) for negative. The arrow points along the polar axis. The vorticity oscillates sinusoidally in time, causing $\delta {\mathcal{B}}_{\mathrm{NN}}$ first to vanish and then to change sign.

Figure 22

Left and middle panels: Near- and transition-zone vortex lines and their vorticities in an ${\mathcal{S}}_{\varphi }$ plane of constant $\varphi$, for the axisymmetric (2,0) magnetic-parity mode of Schwarzschild in the near and transition zones ($30M$ across). Left panel: The predominantly counterclockwise family of vortex lines. Middle panel: The predominantly clockwise family of vortex lines. Right panel: Vorticity of the axial lines normal to an ${\mathcal{S}}_{\varphi }$ plane, in near and wave zones ($56M$ across). The color intensity in each panel gives the vorticity of the lines, scaled as in Fig. 16.

Figure 23

For the (2,0) electric-parity mode of Schwarzschild in RWZ gauge: the vorticity of the counterclockwise vortex lines that pass through the plane ${\mathcal{S}}_{\varphi }$ of constant $\varphi$. The color intensity (scale on left) is scaled as in Fig. 16. Left panel: $30M$ across, showing the near and intermediate zones and beginning of the wave zone. Right panel: $56M$ across.

Figure 24

Vortex lines for the (2,0) electric-parity mode of Schwarzschild in RWZ gauge. Left panel: Two vortex lines of each sign in the near zone and innermost part of transition zone. The positive [blue (dark gray) lines] and negative [red (light gray)] lines are identical but wind their tori in opposite directions. Right panel: Two counterclockwise vortex lines in the transition zone, with the vorticity shown in a semitransparent slice ${\mathcal{S}}_{\varphi }$ of constant $\varphi$ as a density plot. The vortex lines are plotted in black rather than red in this panel to aid the eye.

Figure 25

Plot illustrating the contributions to the amplitude $E_{1(e)}{e}^{-i\omega \tilde{t}}$ [Eq. (a25)] of the perturbed horizon tendicity ${\mathcal{E}}_{\mathrm{NN}}^{(1)}$, in RWZ gauge for the electric-parity, (2,2) perturbation. Plotted against $\tilde{t}$ are the amplitude contributions from ${\mathcal{E}}_{\widehat{r}\widehat{r}}^{(1)}$ (dashed line), and from the perturbative shift of the horizon generators [dot-dashed line; see Eqs. (a21, a24)]. The time-dependent amplitude of the total perturbed horizon tendicity is the solid line.

Figure 26

Tendex and vortex lines of Schwarzschild and Kerr black holes (of spin $a/M=0.945$) perturbed by a (2,2) mode of either electric or magnetic parity, without removing the background tidal or frame-drag fields. The tendex lines and vortex lines are colored by the signs of their respective tendicities and vorticities [blue (dark gray) for positive and red (light gray) for negative]. The horizons are colored and shaded by their vorticities or tendicities, and the transparent spheres have no physical significance, but they help to add perspective to the figures. The top panels are electric-parity perturbations and the bottom panels are magnetic-parity ones. Left column: Tendex lines of Schwarzschild black holes. Middle column: Tendex lines of Kerr black holes. Right column: Vortex lines of Kerr black holes.