Geometrodynamics: The Nonlinear Dynamics of Curved Spacetime

TL;DR

This paper reviews recent advances in the nonlinear dynamics of curved spacetime, highlighting phenomena like gravitational collapse, black hole interactions, and the potential for gravitational wave observations to reveal new insights.

Contribution

It provides a comprehensive overview of key discoveries in geometrodynamics made through numerical solutions of Einstein's equations, emphasizing phenomena and future observational prospects.

Findings

Critical phenomena and self-similarity in gravitational collapse

Instability of black strings in higher dimensions

Black hole collision dynamics and gravitational wave prospects

Abstract

We review discoveries in the nonlinear dynamics of curved spacetime, largely made possible by numerical solutions of Einstein's equations. We discuss critical phenomena and self-similarity in gravitational collapse, the behavior of spacetime curvature near singularities, the instability of black strings in 5 spacetime dimensions, and the collision of four-dimensional black holes. We also discuss the prospects for further discoveries in geometrodynamics via observation of gravitational waves.

Figures

Equations

$${\mathcal{E}}_{jk}=R_{j0k0}.$$

$$\Delta a_j=-{\mathcal{E}}_{jk}\Delta x_k$$

$$d{s}^2$$

$${\mathcal{B}}_{jk}={ϵ}_{jpq}{R}_{pqk0}.$$

$$\Delta {\Omega }_j={\mathcal{B}}_{jk}\Delta x_k.$$

Full text

I Introduction

In the 1950s and 60s, John Archibald Wheeler Wheeler (1962) speculated that curved, empty spacetime could exhibit rich, nonlinear dynamics — which he called geometrodynamics — analogous to the writhing surface of the ocean in a storm. Wheeler exhorted his students and colleagues to explore geometrodynamics by solving Einstein’s general relativistic field equations.

In 1965, Yakov Borisovich Zel’dovich, with his young protégés Andrei Doroshkevich and Igor Novikov Doroshkevich et al. (1965), gave strong evidence that, when a highly deformed star collapses to form what would later be called a black hole, the hole’s curved spacetime, by its nonlinear dynamics, will somehow shake off all the deformations, thereby becoming a completely smooth, axially symmetric hole.

The Wheeler/Zel’dovich challenge of exploring geometrodynamics in general, and for black holes in particular, has largely resisted analytic techniques. In the 1980s and 90s, that resistance motivated the authors and our colleagues to formulate a two-pronged attack on geometrodynamics: numerical solutions of Einstein’s equations to discover general relativity’s predictions, and observations of gravitational waves from black-hole births and black-hole collisions to test those predictions. The numerical simulations have now reached fruition, and the observations will do so in the near future.

In this article—dedicated to the memory of Zel’dovich and Wheeler (who deeply respected each other despite cold-war barriers, and who were primary mentors for one of us, Kip Thorne)—we shall present an overview of some of the most interesting things we have learned about geometrodynamics from numerical simulations, and a preview of what gravitational-wave observations may bring. More specifically:

We shall describe geometrodynamic discoveries in four venues: gravitational implosion, where a phase transition, discrete self-similarity and critical behavior have been observed (Sec. II); the dynamics of spacetime near singularities, where richly chaotic behavior has been observed (Sec. III); the unstable evolution of a black string in five spacetime dimensions, where a dynamical sequence of strings linking black holes has been observed (Sec. IV); and collisions of black holes in four spacetime dimensions, where dynamical interactions of tidal tendexes and frame-drag vortexes have been observed (Sec. V). Then we shall briefly describe the prospects for observing some of these phenomena via gravitational waves (Sec. VI).

II Gravitational Collapse: Phase Transition and Critical Behavior

The first numerical simulations to exhibit rich geometrodynamics were in 1993, by Matthew Choptuik Choptuik (1993), who was then a postdoc at the University of Texas. Choptuik simulated the spherical implosion (Fig. 1) of a linear, massless scalar field (satisfying $\Box\Psi=0$). The field’s energy, momentum and stress (which are quadratic in the field) generated spacetime curvature, with which the field interacted via the curvature’s influence on its wave operator $\Box$. That interaction produced surprising nonlinear dynamics:

If the wave amplitude $p$, for some chosen ingoing waveform, was larger than some critical value $p_{*}$, the implosion produced a black hole. If $p$ was smaller than $p_{*}$, the imploding waves passed through themselves, travelled back outward, and dispersed. For $p=p_{*}$, the imploding waves interacted with themselves nonlinearly (via their spacetime curvature), producing a sequence of frequency doublings and wavelength halvings, with an intriguing, discretely self-similar pattern that was independent of the initial, ingoing waveform. Waves with ever decreasing wavelength emerged from the “nonlinearly boiling” field, carrying away its energy, and ultimately leaving behind what appeared to be an infinitesimal naked singularity (a region with infinite spacetime curvature, not hidden by a black-hole horizon). One year after Choptuik’s simulations, the mathematician Demetrios Christodoulou Christodoulou (1994) developed a rigorous proof that the endpoint, for $p=p_{*}$, was, indeed, a naked singularity.

The transition of the implosion’s endpoint, from black hole for $p>p_{*}$, to naked singularity for $p=p_{*}$, to wave dispersal for $p<p_{*}$, was a phase transition analogous to those that occur in condensed-matter physics. And, as in condensed matter, so also here, the phase transition exhibited scaling: For $p$ slightly larger than $p_{*}$, the mass of the final black hole scaled as $M_{\rm BH}\propto(p-p_{*})^{\beta}$. For $p$ slightly below $p_{*}$, the radius of curvature of spacetime at the center of the “boiling region” reached a minimum value, before field dispersal, that scaled as $\mathcal{R}_{\rm min}\equiv(R^{\mu\nu\sigma\rho}R_{\mu\nu\sigma\rho})_{\rm max}^{-1/4}\propto(p_{*}-p)^{\beta}$, with the same numerically measured exponent $\beta=0.374$. Here $R_{\mu\nu\sigma\rho}$ is the Riemann curvature tensor.

Choptuik’s discovery triggered many follow-on simulations. Most interesting to us was one by Andrew Abrahams and Charles Evans Abrahams and Evans (1993), later repeated with higher resolution by Evgeny Sorkin Sorkin (2011). In their simulations, the imploding, spherically symmetric scalar field was replaced by an imploding, axially symmetric (quadrupolar) gravitational wave — so they were dealing with pure, vacuum spacetime as envisioned by Wheeler. Once again, there was a critical wave amplitude $p_{*}$; and for $p$ near $p_{*}$ the behavior was similar to the scalar-wave case, to within numerical error: for $p=p_{*}$, strong evidence for discretely self-similar evolution leading to an infinitesimal final singularity; for $p>p_{*}$, the same black-hole mass scaling $M_{\rm BH}\propto(p-p_{*})^{\beta}$; for $p<p_{*}$, the same spacetime-curvature scaling $\mathcal{R}\propto(p_{*}-p)^{\beta}$; and to within numerical accuracy, the scaling exponent was the same: $\beta=0.38$ for the quadrupolar gravitational waves, and $\beta=0.374$ for the spherical scalar wave. This is reminiscent, of course, of the universality one encounters in condensed-matter phase transitions.

For a detailed review of these and many other studies of critical gravitational implosion, see an article by Carsten Gundlach Gundlach (1999).

III Generic Space-Time Singularities

III.1 BKL Singularity

The singular endpoint of the implosions described above is non-generic, in the sense that no singularity occurs if $p_{*}$ is only infinitesimally different from $p$.

However, there are other situations that lead to generic singularities.111Perhaps the most important generic singularity for astrophysics is one that arises when matter with negligible pressure is present. Leonid Petrovich Grishchuk in 1967 Grishchuk (1967) showed that the matter, generically, undergoes gravitational collapse to form flat pancakes with infinite density and curvature; and in 1970 Zel’dovich Zel’dovich (1970) showed that, astrophysically, pressure halts the collapse before the infinities but after the pancake structure has been strongly established. A few years later Zel’dovich realized that these pancakes, seen edge on, are filamentary structures that astronomers observe in the distribution of galaxies on the sky. This was demonstrated analytically in the 1960s by Roger Penrose, Stephen Hawking and others, using tools from differential topology Hawking and Penrose (1969). In 1969–70, Vladimir Belinsky, Isaac Khalatnikov and Evgeny Lifshitz Belinskii et al. (1970) used approximate differential-geometry techniques to deduce the geometrodynamical behavior of spacetime as it nears one generic type of singularity, a type now called BKL.

In the 1970s, 80s and 90s, there was much skepticism in the US and UK about this BKL analysis, because its rigor was much lower than that of the Penrose-Hawking singularity theorems. (This lower rigor was inevitable, because the geometrodynamical approach to a singularity is very complex—see below—and deducing its details is much more difficult than proving that a singularity occurs.) As a result, the BKL geometrodynamics came to be called, in the West, the BKL conjecture.

There was little hope for proving or disproving this “conjecture” by analytical techniques, so the skeptics turned to numerical simulations for probing it. After nearly a decade of code development, David Garfinkle in 2003 Garfinkle (2004) carried out simulations which demonstrated that Belinsky, Khalatnikov and Lifshitz had been correct. The geometrodynamical evolution, approaching a BKL singularity, is as they predicted, though with one additional feature that they had missed: a set of nonlocal spikes in the spacetime curvature Lim et al. (2009).

The BKL geometrodynamics can be described in terms of tidal-gravity observations by observers who fall into the BKL singularity on timelike geodesics; Fig. 2. As two observers, A and B, approach the singularity, they lose causal contact, in the sense that, after passing through A’s particle horizon (at point $P$ in the diagram), B can no longer influence A. This causal decoupling is so strong, in the BKL spacetime, that spatial derivatives cease having significant influence on the geometrodynamics, as the singularity is approached—there is a spatial decoupling—and as a result, it turns out, there is no correlation between the late-time observations of adjacent observers.

Each observer’s experience, when approaching the singularity, can be described in terms of the tidal gravitational field $\mathcal{E}_{jk}$ that he feels. This tidal gravitational field has components, in the observer’s local Lorentz frame, that are equal to the space-time-space-time part of the Riemann curvature tensor.

[TABLE]

The physical manifestation of the tidal field is a relative gravitational acceleration

[TABLE]

of particles with vector separation $\Delta x_{k}$. (The tidal field acquires its name from the fact that the Sun’s and Moon’s tidal field produces the tides on the Earth’s oceans. In the Newtonian limit, the tidal field is $\mathcal{E}_{jk}=\partial^{2}\Phi/\partial x_{j}\partial x_{k}$, where $\Phi$ is the Newtonian gravitational potential.)

Being a symmetric tensor, the tidal field can be described by three orthogonal principal axes (unit vectors ${\mathbf{e}}_{\hat{1}}$, ${\mathbf{e}}_{\hat{2}}$, ${\mathbf{e}}_{\hat{3}}$), and their eigenvalues, $\mathcal{E}_{\hat{j}\hat{j}}\equiv{\mathbf{e}}_{\hat{j}}\cdot\boldsymbol{\mathcal{E}}\cdot{\mathbf{e}}_{\hat{j}}$. If $\mathcal{E}_{\hat{1}\hat{1}}<0$, then an object is tidally stretched along its ${\mathbf{e}}_{\hat{1}}$ principal axis, and similarly for the other principal axes. If $\mathcal{E}_{\hat{1}\hat{1}}>0$, the object is tidally squeezed along ${\mathbf{e}}_{\hat{1}}$. The tidal field, in vacuum, is trace-free, so its eigenvalues must sum to zero, which means there must be a squeeze along at least one principal axis and a stretch along at least one.

Figure 3 shows the pattern of stretches and squeezes experienced by an observer when falling into the BKL singularity. The pattern is divided, in time, into cycles and eras. During a single cycle, there is a stretch along one axis and a squeeze along the other two. Between cycles, the stretch axis switches to squeeze and the more strongly squeezing axis switches to stretch. At the end of each era, the axes rotate in some manner, relative to the observer’s local Lorentz frame, and the pattern begins over. The number of cycles in each era and the details of their dynamics are governed by a continued-fraction map that is chaotic, in the technical sense of chaos (extreme sensitivity to initial conditions). This chaos plays a key role in destroying correlations between the observations of adjacent observers as they approach the singularity.

The full details of this, as worked out by Belinsky, Khalatnikov and Lifshitz Belinskii et al. (1970), occasionally are violated (numerical simulations reveal): A cycle gets skipped, and during that skip, there is an extreme spike in the tidal field that is more sensitive to spatial derivatives than expected, and whose details are not yet fully understood. See Lim et al. (2009) and references therein.

III.2 Singularities Inside a Black Hole

This BKL behavior is speculated to occur in the core of a young black hole. However, numerical simulations are needed to confirm or refute this speculation.

As the black hole ages, the singularity in its core is speculated to break up into three singularities, one of BKL type, and two that are much more more gentle than BKL. This speculation arises from the expectation that, just as the hole’s exterior spacetime geometry settles down into the quiescent, axially symmetric state described by the Kerr metric, so its interior will settle down into the Kerr-metric state, except near two special regions called Cauchy horizons, where the Kerr metric is dynamically unstable. Nonlinear geometrodynamics is thought to convert those Cauchy horizons into null, generic singularities (Fig. 4).

These singularities are null in the sense that ingoing or outgoing photons, in principle, could skim along them, not getting captured. The ingoing singularity (called a mass inflation singularity) is generated, according to approximate analytical analyses Poisson and Israel (1990); Ori (2000), by radiation and material that fall into the black hole and pile up along the ingoing Cauchy horizon, there gravitating intensely. The outgoing singularity (called a shock singularity Marolf and Ori (2012)) is generated by ingoing radiation that backscatters off the hole’s spacetime curvature, and then travels outward, piling up along the outgoing Cauchy horizon, there gravitating intensely. In both cases, the piling-up stuff could be gravitational waves rather than material or nongravitational radiation, in which case we are dealing with pure vacuum spacetime curvature—pure geometrodynamics.

Neither of these null singularities is oscillatory, but at both of them, the spacetime curvature goes to infinity (the radius of curvature of spacetime $\mathcal{R}$ goes to zero). This divergence of curvature happens so quickly at the shock singularity, that objects might be able to pass through it, though with a destructive net compression along two dimensions and net stretch along the third. If so, they will likely then be destroyed in the BKL singularity.

The mass-inflation singularity is expected also to produce only a finite, not infinite, net compression and stretch of objects falling through. If anything survives, its subsequent fate is totally unknown.

These (highly informed) speculations, which arise from extensive, approximate analytically analyses, mostly perturbation theory, will be tested, in the coming few years, by numerical simulations. And just as the BKL conjecture missed an important phenomenon (the curvature spikes), so these speculations about the geometrodynamics of black-hole interiors may fail in some modest, or even spectacular way.

For far greater detail on what we now know and speculate about the interiors of black holes and the bases for that knowledge and speculation, see Marolf and Ori (2012) and references therein.

IV Black String in Five Spacetime Dimensions

A remarkable example of geometrodynamics has been discovered by Luis Lehner and Frans Pretorius Lehner and Pretorius (2010), in numerical simulations carried out in five spacetime dimensions.

Lehner and Pretorius began with a black string in its equilibrium state. This black string is a vacuum solution of Einstein’s equations in five spacetime dimensions, with metric

[TABLE]

This is precisely a four-spacetime-dimensional Schwarzschild black hole, translated along the $z$ axis in the fifth (spatial) dimension. The event horizon is at $r=2M$; at fixed time $t$, it is a cylinder with spherical cross section, $R\times S^{2}$.

In 1993, Ruth Gregory and Raymond LaFlamme Gregory and Laflamme (1993) proved, analytically, that such a black string is unstable against linear, axisymmetric perturbations with wavelength longer than about 1.2 times the string’s circumference. But little definitive was known about the instability’s nonlinear, geometrodynamical evolution until Lehner and Pretorius’s 2010 simulations Lehner and Pretorius (2010). They revealed that the string’s horizon evolves as depicted in Fig. 5:

The string develops a sausage instability—analogous to that of a magnetically confined plasma in a $Z$-pinch, and the Rayleigh-Plateau instability for a cylinder of fluid confined by its surface tension—but with outgoing gravitational waves carrying off energy. This instability gives rise to a chain of five-spacetime-dimensional black holes linked by segments of shrunken black string—segments whose circumferences are far smaller than that of the original string. The instability then repeats on each shrunken string, producing smaller black holes linked by segments of more extremely shrunken string. Each successive sausage instability produces its smaller black holes on an evolution timescale proportional to the string’s circumference. These successively shorter timescales converge: An infinite sequence of instabilities, in finite time, presumably ends in a naked singularity.

Because these simulations assumed 2-sphere symmetry from the outset, we cannot be certain that their predictions are the black string’s true geometrodynamical evolution. To learn the true evolution, we need simulations in five spacetime dimensions, that do not assume any symmetry. Such simulations are beyond current capabilities, but may be possible in a decade or so. In the meantime, it is conjectured that the Lehner-Pretorius evolution (Fig. 5) is the true evolution, because black strings have been proved stable against all linear nonspherical perturbations.

V Black-Hole Collisions

Recent advances in numerical simulations have enabled the study of geometrodynamics in the violent collisions of two black holes—including the generation of gravitational waves, and the relaxation of the remnant to a single, quiescent, spinning black hole (as predicted by Zel’dovich, Doroshkevich and Novikov Doroshkevich et al. (1965)). We and collaborators have developed a new set of so-called vortex/tendex tools for visualizing this black-hole geometrodynamics Owen et al. (2011). We will first describe these tools, and then we will use them to visualize the geometrodynamics of black-hole collisions.

V.1 Vortex-tendex tools

The gravitational field felt by a local observer is described by the Riemann curvature tensor $R_{\mu\nu\sigma\rho}$. Any such observer, freely falling or accelerated, can decompose Riemann into an “electric” part, $\mathcal{E}_{jk}$, defined by Eq. (1), and a “magnetic” part, $\mathcal{B}_{jk}$, defined by

[TABLE]

Here the indices are components on the observer’s local, orthonormal basis, the index 0 refers to the time component (i.e., the component along the observer’s world line), Latin indices refer to the observer’s three spatial components, and $\epsilon_{jpq}$ is the spatial Levi-Civita tensor. Both $\mathcal{E}_{jk}$ and $\mathcal{B}_{jk}$ are symmetric and trace free in vacuum (the situation of interest to us).

As discussed in Section III.1, $\mathcal{E}_{jk}$ is called the tidal field, and describes the tidal stretching and squeezing experienced by objects in the observer’s local reference frame, according to Eq. (2). The “magnetic” quantity $\mathcal{B}_{jk}$ is called the frame-drag field. The physical manifestation of this field is a relative precession, or dragging of inertial frames: two gyroscopes with vector separation $\Delta x_{k}$ will precess relative to each other with angular velocity

[TABLE]

This differential frame dragging is a general relativistic effect with no analogue in Newtonian gravity. Its global (non-differential) analog is one of the two relativistic effects recently measured by Gravity Probe B Everitt et al. (2011).

Note that the decomposition of the Riemann tensor into $\mathcal{E}_{jk}$ and $\mathcal{B}_{jk}$ depends on the observer’s reference frame. Different observers at the same location, moving relative to each other, will measure different tidal fields and frame-drag fields. This is the same situation as for classical electromagnetism, in which the electromagnetic tensor $F_{\mu\nu}$ can be decomposed into the familiar electric and magnetic field vectors, $E_{j}=F_{j0}$ and $B_{j}=\epsilon_{jpq}F_{pq}$, in the same observer-dependent manner. This correspondence between gravitation and electromagnetism motivates the use of the names “electric” and “magnetic” for $\mathcal{E}_{jk}$ and $\mathcal{B}_{jk}$.

The frame-drag field $\mathcal{B}_{jk}$ and the tidal field $\mathcal{E}_{jk}$ are useful in describing geometrodynamics at the surface of a black hole (its event horizon). If ${\mathbf{n}}$ is a unit vector normal to the hole’s horizon, with spatial components $n^{i}$, then we define the horizon tendicity $\mathcal{E}_{nn}\equiv n^{j}n^{k}\mathcal{E}_{jk}$ as the normal-normal component of the tidal field. For positive horizon tendicity, an object is tidally squeezed normal to the horizon, and for negative horizon tendicity, an object is tidally stretched normal to the horizon. This is illustrated in the left panel of Fig. 6. We call a region on the horizon with large tendicity a horizon tendex.

We can similarly define the horizon vorticity $\mathcal{B}_{nn}\equiv n^{j}n^{k}\mathcal{B}_{jk}$ as the normal-normal component of the frame-drag field at the surface of a black hole. For positive horizon vorticity, an object experiences a clockwise twist; for negative horizon vorticity, the twist is counterclockwise. We call a region on the horizon with large vorticity a horizon vortex. The horizon vorticity of a spinning black hole is illustrated in the right panel of Fig. 6.

We now turn to vortex/tendex tools in regions away from the horizon. In Section III.1 we discussed how, at any point, $\mathcal{E}_{jk}$ can be described by three orthogonal eigenvectors (unit vectors ${\mathbf{e}}_{\hat{1}}$, ${\mathbf{e}}_{\hat{2}}$, ${\mathbf{e}}_{\hat{3}}$), and their eigenvalues, $\mathcal{E}_{\hat{j}\hat{j}}\equiv{\mathbf{e}}_{\hat{j}}\cdot\boldsymbol{\mathcal{E}}\cdot{\mathbf{e}}_{\hat{j}}$. We call the eigenvalue $\mathcal{E}_{\hat{j}\hat{j}}$ the tendicity associated with the corresponding eigenvector ${\mathbf{e}}_{\hat{j}}$; the tendicity measures the tidal stretching or squeezing of an object oriented along the eigenvector. In analogy with electric field lines, we define tendex lines as the integral curves along each of the three eigenvectors ${\mathbf{e}}_{\hat{j}}$. Whereas in electromagnetism there is a single electric field line passing through each point, in geometrodynamics there are generically three tendex lines passing through each point: one tendex line for each of the three eigenvectors of $\mathcal{E}_{jk}$. Because (in vacuum) $\mathcal{E}_{jk}$ has vanishing trace, the eigenvalues must sum to zero, so generically there exist both positive and negative tendex lines at each point.

The left panel of Fig. 7 shows the tendex lines outside a rapidly rotating black hole. A collection of tendex lines with particularly large tendicity is called a tendex. The rotating hole has a fan-shaped, stretching (red or dark gray) tendex sticking out of its equatorial horizon tendex, and a poloidal squeezing (blue or light gray) tendex that emerges from its north polar horizon tendex, and reaches around the hole to its south polar horizon tendex.

As with $\mathcal{E}_{jk}$, the frame-drag field $\mathcal{B}_{jk}$ can be described by three orthogonal eigenvectors and their eigenvalues. An integral curve of one of these eigenvectors is called a vortex line, and the corresponding eigenvalue is the vorticity associated with that vortex line. The vorticity of a vortex line describes the twist, or differential frame dragging, experienced by an object oriented along that vortex line: positive vorticity corresponds to a clockwise twist, and negative vorticity corresponds to a counterclockwise twist.

The vortex lines associated with a spinning black hole are shown in the right panel of Fig. 7. A counterclockwise (red or dark gray) vortex (collection of large-negative-vorticity lines) emerges from the horizon’s north polar vortex, reaches around the south polar region and descends back into the horizon’s north polar vortex. Similarly, a clockwise (blue or light gray) vortex emerges from the south polar horizon vortex, reaches around the north polar region and descends back into the south horizon vortex.

The black holes of Figs. 6 and 7 have stationary (time-independent) vortex and tendex structures. Vortex and tendex lines, and their associated vortexes and tendexes, can also behave dynamically. The equations of motion for $\mathcal{E}_{jk}$ and $\mathcal{B}_{jk}$ are similar to Maxwell’s equations. Like Maxwell, they have wavelike solutions—gravitational waves—in which energy is fed back and forth between $\mathcal{E}_{jk}$ and $\mathcal{B}_{jk}$. Figure 8 shows vortex and tendex lines for a plane gravitational wave without any nearby sources. As the wave passes an observer, the tendicities and vorticities oscillate in sign with one oscillation per gravitational wave period, leading to an oscillatory stretch and squeeze in horizontal and vertical directions, and an oscillatory twist in diagonal directions, out of phase with the stretch and squeeze.

V.2 Black-hole collisions

We shall illustrate the geometrodynamical richness of black-hole collisions by describing the vortex/tendex behaviors in several numerical simulations. All these simulations were performed by members of the Collaboration to Simulate Extreme Spacetimes (SXS), which included numerical relativists from Caltech, Cornell, the Canadian Institute for Theoretical Astrophysics, and Washington State University at the time of these simulations, and has since been expanded. We are members of this collaboration, and one of us (Scheel) played a significant role in most of these simulations.

V.2.1 Head-on collision of two black holes with transverse spins

Our first example is a simulation Lovelace et al. (2010) of the head-on collision of two transversely spinning black holes, depicted in Fig. 9.

As the black holes merge, the vortexes retain their individuality. When the four vortexes (one pair from each hole) discover each other, all attached to the same horizon, they begin to interact: each one tries to convert those adjacent to it into a replica of itself. As a result, they exchange vorticities; each oscillates back and forth between clockwise and counterclockwise. At the moment when all are switching direction, so the horizon momentarily has zero vorticity, the vortex lines have popped off the horizon and joined onto each other, creating a toroidal structure, much like a smoke ring, that is beginning to travel outward. Simultaneously, adjacent to the horizon much of the oscillation energy is stored in tendexes, which then regenerate the horizon vortexes, but with twist directions reversed. As the toroidal bundle of vortex lines travels outward, its motion generates toroidal tendex lines, intertwined with the vortex lines in just such a manner as to become, locally, the gravitational-wave structure described in Fig.  8 above.

The process repeats over and over, with successive, toroidal, tendex/vortex structures being ejected and morphing into gravitational waves. The waves carry away oscillation energy, and some oscillation energy goes down the hole, so the oscillations die out, with an exponentiation time of order an oscillation period.

V.2.2 Collision of identical, spinning black holes, in an inspiraling

circular orbit

For collisions of orbiting (i.e. non head-on) black holes, the vortex and tendex lines similarly travel to the wave zone and become gravitational waves.

The left panel of Fig. 10 shows a schematic diagram of the horizon vortexes and the vortex lines for the collision of two orbiting, spinning black holes that are about to merge. Just after merger, the horizon vortexes retain their individuality, and travel around the horizon of the merged black hole, trailing their vortex lines outward and backward in a pattern like water from a spinning sprinkler head (shown schematically on the right of Fig. 10). In the wave zone, the vortex lines acquire tendex lines and become gravitational waves.

Similarly, the near-field tendex lines, attached to the merged hole’s horizon tendexes, trail backward and outward in a spiral pattern, acquiring accompanying vortex lines as they travel, and becoming gravitational waves. These horizon-tendex-generated gravitational waves have the opposite parity from the horizon-vortex-generated waves; and there is a remarkable duality between the two sets of waves Nichols et al. (2012)

Figure 10 is schematic. For a more precise depiction, we focus on the merged hole at late times, when it is weakly perturbed from its final, Kerr-metric state, and its perturbations are predominantly $\ell=2,\;m=2$ quasinormal modes Nichols et al. (2012). Then the perturbations of the frame-drag field, $\delta\mathcal{B}_{ij}$, generated by the horizon vortexes, have the vortex lines and vorticities shown in the left panel of Fig. 11; and the perturbations of the tendex field, $\delta\mathcal{E}_{ij}$, generated by the horizon tendexes, have the tendex lines and tendicities show in the right panel. Notice that the two figures are very nearly identical, aside from sign (interchange of red and blue, or dark and light gray). This is due to the (near) duality between the two sets of perturbations.

V.2.3 Extreme-kick black-hole collision

An interesting example of geometrodynamics, and of the interplay between vortexes and tendexes, is the “extreme-kick” black-hole collision first simulated, not by our SXS collaboration, but by Campanelli et al. Campanelli et al. (2007) and others Gonzalez et al. (2007); Lousto and Zlochower (2011). Our collaboration has repeated their simulations, in order to extract the vortex and tendex structures.

In these simulations, two identical black holes merge from an initially circular orbit, with oppositely directed spins lying in the orbital ($x,y$) plane. The name “extreme-kick” arises because gravitational waves generated during the merger carry linear momentum preferentially in either the $+z$ or $-z$ direction, resulting in a gravitational recoil of the remnant black hole with speeds as high as thousands of km/s. The magnitude and direction of the recoil depends on the angle between holes’ identical spin axes, and the holes separations, at the moment of merger. This angle can be fine-tuned (for instance, to produce maximum recoil in the $+z$ direction) by adjusting the initial conditions in the simulation.

To understand the mechanism of the recoil, consider the remnant black hole just after merger. The left panel of Fig. 12 shows the horizon tendicity and the tendex lines emerging from the remnant black hole at some particular time. The tendex structure is rotating counterclockwise around the hole’s horizon. The rotating tendex lines acquire accompanying vortex lines as they extend into the wave zone in a pattern like that shown in the right panel of Fig. 11, and there they become gravitational waves. During merger, the horizon vortexes (right panel of Fig. 12) lock onto the same rotational angular velocity as the horizon tendexes, and generate gravitational waves in the same manner, with a pattern that looks like the left panel of Fig. 11.

In the wave zone, the gravitational waves produced by the rotating near-zone tendexes and those produced by the rotating near-zone vortexes superpose coherently, and the resulting radiation pattern depends on the angle between the horizon tendex labeled “E” and the vortex labeled “B” in Fig. 12. For the case shown, this angle is 45 degrees, with ${\mathbf{E}}\times{\mathbf{B}}$ in the $-z$ direction (into the page). This is the same as the structure of a gravitational wave propagating in the $-z$ direction (Fig. 8). As a result, the gravitational waves produced by the vortexes and those produced by the tendexes superpose constructively in the $-z$ direction and destructively in the $+z$ direction, resulting in a maximum gravitational-wave momentum flow in the $-z$ direction and a maximum remnant recoil in the $+z$ direction.

V.2.4 Generic black-hole collisions

The geometrodynamic behaviors of more general black hole collisions are now being explored numerically. For example, Fig. 13 shows trajectories of two black holes in a fully generic orbit. Vortexes from the larger, rapidly-spinning black hole pull the orbit of the smaller black hole into a complicated precession pattern. The spin directions of both black holes also precess as the holes orbit each other. Eventually a common apparent horizon222An apparent horizon is a surface of zero outgoing null expansion, and lies inside or on the event horizon. Apparent horizons are local quantities that are much easier to find in numerical simulations than are event horizons, because the location of the event horizon depends on the full future evolution of the spacetime. forms around the two individual apparent horizons, and the two holes merge into one. The Ricci scalar (approximately equal to -2 times the horizon tendicity) is shown on the two individual horizons and on the common horizon, at the moment the common horizon forms.

V.2.5 Tidal disruption of a neutron star by a spinning black hole

Our final example illustrates the interaction of geometrodynamics and matter. Figure 14 shows a simulation of a neutron star orbiting a black hole, a binary system important for gravitational-wave detectors and possibly for high-energy astrophysical phenomena such as gamma-ray bursts. Here the black hole has 3 times the mass of the neutron star, and a dimensionless spin $S/M^{2}=0.5$ in a direction inclined approximately 45 degrees to the orbital angular momentum. When the orbit has shrunk sufficiently, because of energy lost to gravitational radiation, the black hole’s tidal tendexes rip apart the neutron star, and its frame-drag vortexes then pull the stellar debris out of its original orbit and into the black hole’s equatorial plane. If the black hole has a small enough spin or a large enough mass, the neutron star is not disrupted, but instead is swallowed whole by the black hole Foucart (2012).

VI Gravitational-Wave Observations

Geometrodynamics generically produces gravitational waves. We are entering an era in which these waves, generated by sources in the distant universe, will be detected on Earth.

A first generation of interferometric gravitational-wave detectors has operated at sensitivities where it would require luck to see waves. We were not lucky Abadie et al. (2012); Aasi et al. (2013a).

Second generation detectors, much more advanced and complex in their design, are under construction. The first two of these (the advanced LIGO detectors in the U.S.) will begin operating in 2015 and should reach their design sensitivity by 2019, and perhaps sooner Aasi et al. (2013b). They will be joined a bit later by the advanced VIRGO detector in Europe, KAGRA in Japan, and an advanced LIGO detector in India Fairhurst (2012). These advanced detectors will cover a 1000 times larger volume of the universe than the initial detectors did, with estimated event rates for mergers of black-hole and neutron-star binaries ranging from a few per year to a few per week Deller et al. (2009); O’Shaughnessy et al. (2008); Abadie et al. (2010). Plans are being developed for further improvements, which should increase the event rate by another factor ten or more, without major changes in detector design.

The most interesting gravitational-wave sources for these detectors, we think, are the dynamically evolving vortexes and tendexes attached to merging black holes, and to a black hole tearing apart a neutron star — the geometrodynamic phenomena discussed above.

The numerical relativity community is building a catalog of binary simulations and associated gravitational waveforms to underpin the advanced detectors’ searches for these waves. Simulations of binary black holes with several hundred different sets of parameters (mass ratios and initial vectorial spins) have been completed Pekowsky et al. (2013); Mroue et al. (2013); Ajith et al. (2012, 2013); Aasi et al. (2014); Hinder et al. (2014); and many more are underway or planned. A sample of computed waveforms from our SXS collaboration is shown in Fig. 15.

Once waves are being detected, comparison of observed waveforms with those from simulations will be crucial for understanding the waves’ sources. By such comparisons, can we deduce the sources’ geometrodynamics and test general relativity’s predictions.

VII Conclusions

Physicists have barely scratched the surface of geometrodynamics. As numerical simulations continue to improve and are used to explore more complicated and generic situations, we expect to learn more about the geometrodynamics of critical behavior, singularities, dynamical black holes, and other phenomena. We look forward to observations of gravitational waves from strongly gravitating astrophysical sources, which will enable us to test the geometrodynamical predictions of Einstein’s equations for the first time.

Acknowledgements.

We are grateful to Thorne’s dear friend, the late Leonid Petrovich Grishchuk, for urging us to write this review article, and for many stimulating discussions. We thank members of the SXS collaboration and of the Caltech/Cornell Vortex/Tendex research group for data used for figures in Sec V, and for the work that made much of that section possible. We gratefully acknowledge support from the Sherman Fairchild Foundation, the Brinson Foundation, and NSF grants PHY-106881 and AST-1333520.