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 ${\xi }^k$ is $\Delta a_j=-{\mathcal{E}}_{jk}{\xi }^k$), 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 ${\xi }^k$, as measured using a local inertial frame at the tail of ${\xi }^k$, is $\Delta {\Omega }_j={\mathcal{B}}_{jk}{\xi }^k$). 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.

Figure 1

Spacetime geometry for computing the precession of a gyroscope at one location ${\mathcal{P}}^{\prime }$, relative to gyroscopic standards at a nearby location $\mathcal{P}$.

Figure 2

Tendex lines outside a spherically symmetric, gravitating body. The lines are colored by the sign of their tendicity: light-colored (red) lines have negative tendicity (they stretch a person oriented along them); dark (blue) lines have positive tendicity (they squeeze).

Figure 3

Vortex lines outside a slowly spinning, spherically symmetric, gravitating body with spin angular momentum $\mathit{S}$. The lines are colored by the sign of their vorticity: light-colored (red) lines have negative vorticity (they produce a counterclockwise differential precession of gyroscopes); dark (blue) lines have positive vorticity (clockwise differential precession).

Figure 4

For a weakly gravitating, slowly rotating body with spin angular momentum $\mathit{S}$, the dipolar frame-dragging angular velocity relative to inertial frames at infinity, ${\mathbf{\Omega }}_{\mathrm{fd}}$. The arrows are all drawn with the same length rather than proportional to the magnitude of ${\mathbf{\Omega }}_{\mathrm{fd}}$.

Figure 5

For two stationary point particles sitting side-by-side with their spins in opposite directions (thick black arrows), two types of streamlines in the plane of reflection symmetry formed by the particles’ spins and their separation vector. (a) The frame-dragging angular velocity ${\mathbf{\Omega }}_{\mathrm{fd}}$ and its streamlines, with the arrows all drawn at the same length rather than proportional to the magnitude of ${\mathbf{\Omega }}_{\mathrm{fd}}$. (b) The two sets of vortex lines of the frame-drag field $\mathcal{B}$. The negative-vorticity vortex lines are solid and colored red, and the positive-vorticity ones are dashed and blue. In this figure, as in preceding figures, the colors are not weighted by the lines’ vorticities, but only by the signs of the vorticities.

Figure 6

Current-quadrupolar streamlines associated with the two stationary spinning particles of Fig. 5, for which the current-quadrupole moment has nonzero components ${\mathcal{S}}_{xz}={\mathcal{S}}_{zx}=Sa$. (a) The frame-dragging angular velocity ${\mathbf{\Omega }}_{\mathrm{fd}}$ and its streamlines, and (b) the two sets of vortex lines, in the $x\mathrm{\text{−}}z$ plane. Figure (b) also describes the tendex lines for a static mass-quadrupolar particle whose only nonzero quadrupole moment components are ${\mathcal{I}}_{xz}={\mathcal{I}}_{zx}$.

Figure 7

The tendex lines (left) and vortex lines (right) of a plane gravitational wave propagating in the $z$ direction (out of the picture). The tendex lines are lines of constant $x$ and $y$, and the vortex lines are rotated by $\pi /4$ (lines of constant $x\pm y$). The blue (dashed) curves correspond to positive tendicity and vorticity (squeezing and clockwise differential precessing, respectively) and the red (solid) curves denote negative tendicity and vorticity (stretching and counterclockwise precessing). The tendicity (vorticity) is constant along a tendex line (vortex line), and the tendicity (vorticity) of a red (solid) line is equal in magnitude but opposite in sign to that of a blue (dashed) line.

Figure 8

Tendex lines (left) and vortex lines (right) for the gravitational waves that would arise from the merger of equal-mass black holes falling together along the z axis. The positive tendicity and vorticity lines are shown in blue (dashed) and the negative lines are depicted in red (solid). Each line’s intensity is proportional to its tendicity (or vorticity), which varies over the sphere as the dominant spin-weighted spherical harmonic, ${}_{-2}Y_{2,0}(\theta ,\varphi )\propto {sin}^2\theta$. Dark red and blue near the equator correspond to large-magnitude tendicity and vorticity, and light nearly white colors at the poles indicate that the tendicity and vorticity are small there.

Figure 9

For a rotating current quadrupole in linearized theory, two families of vortex lines in the plane of reflection symmetry (the $x\mathrm{\text{−}}y$ plane). The red (solid) curves are lines with negative vorticity, and the blue (dashed) curves are lines of positive vorticity. The color intensity of the curves represents the strength of the vorticity, but rescaled by $(kr{)}^5/[1+(kr{)}^4]$ (with $k$ the wave number) to remove the vorticity’s radial decay. We see the quadrupolar near-zone pattern and the transition into the induction zone. In the induction zone, the pattern carries four “triradius” singular points [45] in each family of curves, necessitated for the transition from the static quadrupole pattern to the spiraling radiation pattern. This same figure also describes the tendex lines of a rotating mass quadrupole (see the end of Sec. 6f).

Figure 10

Same as Fig. 9 but zoomed out to show the wave zone. In the wave zone, the lines generically collect into spirals, which form the boundaries of vortexes (regions of concentrated vorticity).

Figure 11

Tendex lines in the equatorial plane for a rotating current quadrupole in linearized theory. The curves shown are lines of identically zero tendicity, enforced by symmetry. The lines are shaded by the absolute value of the tendicity of the other two tendex lines that cross the lines shown, but are not tangent to the plane, and have equal and opposite tendicities.

Figure 12

For the same rotating current quadrupole as in Figs. 9 and 10, the family of vortex lines that pass orthogonally through the $x\mathrm{\text{−}}y$ plane of reflection symmetry, color coded as in Fig. 9. In the wave zone, lines with approximately zero vorticity extend away from the source nearly radially, while lines with significant vorticity are dragged into tangled spirals by the rotation of the source. In the left inset, we see the transition between the near and wave zones. Here, lines with nearly zero vorticity escape to infinity as in the wave zone, but those with significant vorticity are drawn toward the source. The right inset delves down into the near zone, where the lines are approximately those of a stationary current quadrupole. This same figure also describes the tendex lines of a rotating mass quadrupole (see the end of Sec. 6f).

Figure 13

Vortex lines of a time-varying current quadrupole at very large $r$. The lines are colored by the vorticity scaled by $r$, to remove the $1/r$ falloff, but the color coding is the same as in previous figures. At very large distances from the source, the lines are transverse and live on a sphere. The third vortex line not shown is radial and has vanishing eigenvalue.

Figure 14

Vortex lines of a rotating current quadrupole at sufficiently large $r$ that the $1/r^2$ part of the frame-drag field may be thought of as a perturbation to the transverse vortex lines of Fig. 13. The lines are colored by the vorticity as in that figure. We also show a black dotted circle in the equatorial plane to identify this plane. The red solid lines shown here continue to collect on the maximum-vorticity spiral, but they oscillate much more in polar angle than do the similar lines shown in the near wave zone in Fig. 12. The blue dashed lines shown here form closed loops that pass from one positive-vorticity spiral to the next. This family of lines more closely resembles the transverse lines of Fig. 13, though in the limit of infinite radius, the spiraling lines will also close to form transverse lines on the sphere. There are also spiraling positive-vorticity (blue dashed) lines and closed-loop, negative-vorticity (red solid) lines, but to keep the figure from appearing muddled, we do not show them.

Figure 15

For an oscillating current quadrupole in linearized theory, two families of vortex lines in the plane of reflection symmetry (the $x\mathrm{\text{−}}y$ plane). The color coding is the same as for the rotating current quadrupole, Fig. 9. The vortex lines begin, near the origin, like the static quadrupole pattern of Fig. 6. The effects of time retardation cause the pattern to stretch making larger rectangular loops in the transition zone. As time passes and the quadrupole oscillates, these loops detach from the origin and propagate out into the wave zone. This same figure also describes the tendex lines of an oscillating mass quadrupole (see the end of Sec. 6f).

Figure 16

Same as Fig. 15, but zoomed out to show the wave zone. Farther from the source, the loops take on a more regular alternating pattern of gravitational waves. The coloring shows that the vorticity is strongest at the fronts and backs of the loops, where the vortex lines are transverse to the direction of propagation. In the regions of the closed loops that extend radially, the field is weak (as one would expect for a transverse gravitational wave).

Figure 17

Vortex lines of an oscillating current quadrupole at sufficiently large $r$ that the $1/r^2$ part of the frame-drag field may be thought of as a perturbation to the transverse vortex lines of Fig. 13. The lines are colored in the same way as that figure, and the pattern of the lines around the poles is nearly identical to the transverse lines of Fig. 13. Near the equator, the $1/r^2$ perturbation causes the lines to bend and form closed loops that reside in either the northern or the southern hemisphere. The dark (blue) horizontal lines in the blow-up inset should be compared with dense blue (dashed) bundles in Fig. 16, and medium-colored (red) lines with the red (solid) bundles immediately outside of the blue (dashed) ones.

Figure 18

For a weak-gravity binary made of identical nonspinning point particles, in the near zone where retardation is negligible, two families of tendex lines lying in a plane that passes through the two particles (e.g., the orbital plane). The red (solid) curves are lines with negative tendicity, and the blue (dashed) curves have positive tendicity. The color intensity of the curves represents the magnitude of the tendicity, rescaled by $r_A^3{r}_B^3/[M^3(r_A^3+r_B^3)]$, where $r_A$ and $r_B$ are the distances to the particles, to remove the tendicity’s radial die out. Near each particle, the tendex lines resemble those of an isolated spherical body; as one moves closer to the particle’s companion, the lines bend in response to its presence. At radii large compared to the particles’ separation $a$, the binary’s monopole moment comes to dominate, and the tendex lines resemble those of a single isolated spherical body.

Figure 19

Tendex lines in the orbital plane of the same binary as Fig. 18, with separation $a=20M$ (where $M$ is the total mass), focusing on the transition and wave zones $r\gtrsim ƛ=2.24a$. The solid black circle has radius $ƛ$. The colors are fixed by the tendicity weighted by $\omega r$ so as to scale out the $1/r$ falloff in the wave zone (with dark blue, along dashed lines, strongly positive; dark red, along solid lines, strongly negative; and light green, along both solid and dashed lines, near zero). Inside the dotted black curve ($r=\frac{1}{2}{a}^2/M=10a$), the binary’s (nonradiative) monopole moment dominates, $\mathcal{E}\simeq M/r^2$, and the red (stretching) tendex lines are nearly radial. Outside the dotted black curve, the (radiative) quadrupole moment dominates, $\mathcal{E}\simeq 4M^3/a^{4}r$, and the tendex lines are strong (significant tendicity) only where they are approximately transverse to the radial direction.

Figure 20

Tendex lines outside the (central, horizontal) orbital plane, for the same binary and parameters as Fig. 19. In the inner region, the binary’s monopole moment dominates, $\mathcal{E}\simeq M/r^2$, so the light-colored (red, stretching) tendex lines are nearly radial and the dark (blue, squeezing) tendex lines are nearly circular. At larger radii, the (radiative) quadrupole moment begins to be significant and then dominate, so the tendex lines begin to spiral outward as for the rotating quadrupole of Fig. 12.