Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime

When one splits spacetime into space plus time, the spacetime curvature (Weyl tensor) gets split into an “electric” part ${\mathcal{E}}_{jk}$ that describes tidal gravity and a “magnetic” part ${\mathcal{B}}_{jk}$ that describes differential dragging of inertial frames. We introduce tools for visualizing ${\mathcal{B}}_{jk}$ (frame-drag vortex lines, their vorticity, and vortexes) and ${\mathcal{E}}_{jk}$ (tidal tendex lines, their tendicity, and tendexes) and also visualizations of a black-hole horizon’s (scalar) vorticity and tendicity. We use these tools to elucidate the nonlinear dynamics of curved spacetime in merging black-hole binaries.

Figure 1

Vortexes (with positive vorticity blue, negative vorticity red) on the 2D event horizons of spinning, colliding black holes, just before and just after the merger (from the simulation reported in Ref. 4).

Figure 2

Four different black holes, with horizons colored by their tendicity (upper two panels) or vorticity (lower two panels), ranging from most negative (red) to most positive (blue), and with a Kerr-Schild horizon-penetrating foliation (exercise 33.8 of Ref. 18). (a) A nonrotating black hole and its tendex lines; negative-tendicity lines are red, and positive blue. (b) A rapidly rotating (Kerr) black hole, with spin $a/M=0.95$, and its tendex lines. (c) The same Kerr black hole and its vortex lines. (d) Equatorial plane of a nonrotating black hole that is oscillating in an odd-parity $l=m=2$ quasinormal mode, with negative-vorticity vortex lines emerging from red horizon vortexes. The lines’ vorticities are indicated by contours and colors; the contour lines, in units $(2M{)}^{-2}$ and going outward from the hole, are $-10$, $-8$, $-6$, $-4$, and $-2$.

Figure 3

Head-on, transverse-spin simulation: (a) Shortly after merger, vortex lines link horizon vortexes of the same polarity (red to red, blue to blue). Lines are color coded by vorticity (different scale from horizon). (b) Sloshing of near-zone vortexes generates vortex loops traveling outward as gravitational waves; thick and thin lines are orthogonal vortex lines.

Figure 4

Insets: Snapshots of the common apparent horizon for the $a/M=0.95$ antialigned simulation, color coded with the horizon vorticity $B_{\mathrm{NN}}$. Graphs: $B_{\mathrm{NN}}$ as a function of polar angle $\theta$ at the azimuthal angle $\varphi$ that bisects the four vortexes (along the black curves in snapshots).

Figure 5

Bottom inset: Tendex and vortex lines for a plane gravitational wave; $\mathit{E}\times \mathit{B}$ is in the propagation direction. Upper two insets: For the extreme-kick simulation, as seen looking down the merged hole’s rotation axis ($-z$ direction): the apparent horizon color coded with the horizon tendicity (left inset) and vorticity (right inset) and with 3D vortex lines and tendex lines emerging from the horizon. The tendexes with the most positive tendicity (blue, $\mathit{E}$) lead the positive-vorticity vortexes (blue, $\mathit{B}$) by about 45° as they rotate counterclockwise. This 45° lead is verified in the oscillating curves, which show the rotating ${\mathcal{B}}_{\mathrm{NN}}$ and ${\mathcal{E}}_{\mathrm{NN}}$ projected onto a nonrotating $\ell =2$, $m=2$ spherical harmonic.