Tendex Lines Research Articles

Visualizing gravitational Bessel waves

We explicitly derive a vortex inspired solution for the metric perturbation within the linearized Einstein general theory of relativity in arbitrary dimensions $D\ensuremath{\ge}4$. We focus on $D=4$ where our solution is the gravitational analog of the well-known electromagnetic (or electron) Bessel vortex beams. Next we visualize the perturbed spacetime via tidal tendexes and frame-drag vortexes. We display and analyze mostly two-dimensional sections of the tendexes and the vortexes for different values of an angular momentum of the wave solution. Corresponding geodesic deviation equations are solved and the results are visualized. We show that the physically most important quadrupolelike case leads to a wave with rotating polarization. We discuss asymptotical features of the found solution. We provide also several 3D plots of tendex lines. One of them concerns a special cylindrical-like case and we utilize the topological classification of singularities of the depicted line fields as an approach to characterize the radiation field.

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 $[(\ensuremath{-}1{)}^{l}]$ or magnetic $[(\ensuremath{-}1{)}^{l+1}]$. 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 $(2,\ensuremath{-}2)$ 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.

Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. II. Stationary black holes

When one splits spacetime into space plus time, the Weyl curvature tensor (which equals the Riemann tensor in vacuum) splits into two spatial, symmetric, traceless tensors: the tidal field $\mathcal{E}$, which produces tidal forces, and the frame-drag field $\mathcal{B}$, which produces differential frame dragging. In recent papers, we and colleagues have introduced ways to visualize these two fields: tidal tendex lines (integral curves of the three eigenvector fields of $\mathcal{E}$) and their tendicities (eigenvalues of these eigenvector fields); and the corresponding entities for the frame-drag field: frame-drag vortex lines and their vorticities. These entities fully characterize the vacuum Riemann tensor. In this paper, we compute and depict the tendex and vortex lines, and their tendicities and vorticities, outside the horizons of stationary (Schwarzschild and Kerr) black holes; and we introduce and depict the black holes' horizon tendicity and vorticity (the normal-normal components of $\mathcal{E}$ and $\mathcal{B}$ on the horizon). For Schwarzschild and Kerr black holes, the horizon tendicity is proportional to the horizon's intrinsic scalar curvature, and the horizon vorticity is proportional to an extrinsic scalar curvature. We show that, for horizon-penetrating time slices, all these entities ($\mathcal{E}$, $\mathcal{B}$, the tendex lines and vortex lines, the lines' tendicities and vorticities, and the horizon tendicities and vorticities) are affected only weakly by changes of slicing and changes of spatial coordinates, within those slicing and coordinate choices that are commonly used for black holes. We also explore how the tendex and vortex lines change as the spin of a black hole is increased, and we find, for example, that as a black hole is spun up through a dimensionless spin $a/M=\sqrt{3}/2$, the horizon tendicity at its poles changes sign, and an observer hovering or falling inward there switches from being stretched radially to being squeezed. At this spin, the tendex lines that stick out from the horizon's poles switch from reaching radially outward toward infinity to emerging from one pole, swinging poloidally around the hole and descending into the other pole.

Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: General theory and weak-gravity applications

When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free tensors: (i) the Weyl tensor's so-called electric part or tidal field ${\mathcal{E}}_{jk}$, which raises tides on the Earth's oceans and drives geodesic deviation (the relative acceleration of two freely falling test particles separated by a spatial vector ${\ensuremath{\xi}}^{k}$ is $\ensuremath{\Delta}{a}_{j}=\ensuremath{-}{\mathcal{E}}_{jk}{\ensuremath{\xi}}^{k}$), and (ii) the Weyl tensor's so-called magnetic part or (as we call it) frame-drag field ${\mathcal{B}}_{jk}$, which drives differential frame dragging (the precessional angular velocity of a gyroscope at the tip of ${\ensuremath{\xi}}^{k}$, as measured using a local inertial frame at the tail of ${\ensuremath{\xi}}^{k}$, is $\ensuremath{\Delta}{\ensuremath{\Omega}}_{j}={\mathcal{B}}_{jk}{\ensuremath{\xi}}^{k}$). Being symmetric and trace-free, ${\mathcal{E}}_{jk}$ and ${\mathcal{B}}_{jk}$ each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of ${\mathcal{E}}_{jk}$'s eigenvectors tidal tendex lines or simply tendex lines, we call each tendex line's eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for ${\mathcal{B}}_{jk}$ are frame-drag vortex lines or simply vortex lines, their vorticities, and their vortexes. These concepts are powerful tools for visualizing spacetime curvature. We build up physical intuition into them by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side-by-side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. We show that a rotating current quadrupole has four rotating vortexes that sweep outward and backward like water streams from a rotating sprinkler. As they sweep, the vortexes acquire accompanying tendexes and thereby become outgoing current-quadrupole gravitational waves. We show similarly that a rotating mass quadrupole has four rotating, outward-and-backward sweeping tendexes that acquire accompanying vortexes as they sweep, and become outgoing mass-quadrupole gravitational waves. We show, further, that an oscillating current quadrupole ejects sequences of vortex loops that acquire accompanying tendex loops as they travel, and become current-quadrupole gravitational waves; and similarly for an oscillating mass quadrupole. And we show how a binary's tendex lines transition, as one moves radially, from those of two static point particles in the deep near zone, to those of a single spherical body in the outer part of the near zone and inner part of the wave zone (where the binary's mass monopole moment dominates), to those of a rotating quadrupole in the far wave zone (where the quadrupolar gravitational waves dominate). In Paper II we will use these vortex and tendex concepts to gain insight into the quasinormal modes of black holes, and in subsequent papers, by combining these concepts with numerical simulations, we will explore the nonlinear dynamics of curved spacetime around colliding black holes. We have published a brief overview of these applications in R. Owen et al., Phys. Rev. Lett. 106, 151101 (2011). We expect these vortex and tendex concepts to become powerful tools for general relativity research in a variety of topics.

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 E(jk) that describes tidal gravity and a "magnetic" part B(jk) that describes differential dragging of inertial frames. We introduce tools for visualizing B(jk) (frame-drag vortex lines, their vorticity, and vortexes) and 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.