Yukawa International Seminar

John Friedman, University of Wisconsin, Milwaukee

About 150 people gathered under cloudy skies in the old imperial capital, Kyoto, June 28 - July 2, to listen to six days of talks and view an extraordinarily good set of posters, all summarizing theoretical, observational, and experimental work on gravitational waves, black-hole physics, and numerical gravity during the Yukawa International Symposium (YKIS99) on black holes and gravitational waves. Unfortunately, a summary of posters would make this already long review too long to be useful.

In a first session on black holes in a quantum context, Ted Jacobson summarized his work with Corley and Mattingly (e.g., hep-th/9908099). In the usual semiclassical computation of Hawking radiation, the late-time flux arises from modes that, prior to the black hole's formation, are vastly shorter than the Planck length; and one can worry that black-hole radiance would not survive if the universe's small-scale structure does not allow one to make sense of such ultra-high frequency fields. Jacobson and his coworkers address the problem by placing a lattice on a black-hole background, and looking at field theory on the lattice - considering, in effect, quantum field theory on a discrete spacetime. Satisfyingly, the lattice reproduces the Hawking effect with an accuracy that depends on the ratio of the lattice spacing to the black hole radius. A lattice regular at the horizon is not static, and the lattice used is falling inward. The scattering by the lattice of a ingoing to outgoing wave is formally analogous to the Bloch oscillation of an electron in in a crystal with a uniform electric field.

Following this discussion, Gary Gibbons spoke on black holes in unified theories. He noted that classical solutions are important in quantum theories if their quantum corrections vanish, and that commonly requires supersymmetry to cancel the fluctuations of bosons against those of fermions. The Breckenridge-Myers-Peet-Vafa solution provides an example of a BPS black hole with nonvanishing angular momentum, a solution that was used to count states using D-brane techniques (and gave agreement with black-hole entropy). Gibbons and Herdeiro (hep-th/9906098) have completed a substantial study of the solution, finding its geodesics and computing the scattering of a scalar field off this extreme black hole. The solution includes an example of a ``naked stable time machine,'' with spatially unbound geodesics that can travel back in time; but Gibbons and Herdeiro argue that chronology protection may be enforced by the third law of thermodynamics, preventing the formation of an extreme black hole by means of a finite process.

Describing the work led by Israel's group on the nature of the interiors of black holes formed in collapse, Patrick Brady reviewed the linear instability of the Cauchy horizon and the spherical models of black-hole interiors and then turned to more recent work that appears to confirm key features of the earlier models. In particular, Ori and Flanagan have used the Cauchy-Kovalevsky theorem to show that ``there exist functionally generic solutions of Einstein's equations containing a null and weak scalar curvature singularity,'' and work by Barack and Ori and by Israel, Brady, Chambers, Droz and Morsink characterizes more precisely the Weyl curvature near these null Cauchy-horizon singularities.

Andrzej Krolak continued the discussion of the nature of singularities in gravitational collapse, summarizing theorems that characterize Cauchy horizons or restrict the occurence of naked singularities. Here are a few.
Chrusciel and Galloway and Budzynski, Kondracki, and Krolak have shown the existence of a large class of nowhere differentiable Cauchy horizons. Harada, Iguchi, and Nakao showed that generic counterrotation prevents central shell-focusing formation. This is consistent with Rendall's result with cylindrical symmetry that a regular distribution function in phase space prevents naked singularities of the kind apparently seen by Shapiro and Teukolsky, using a singular distribution function for collisionless matter; and consistent with the conjecture that matter described by smooth distribution functions obeys cosmic censorship - that, as in the Newtonian theory, velocity dispersion dissolves naked singularities.
Matt Choptuik summarized work by about 30 people on critical phenomena in gravitational collapse that has led to a coherent picture. Critical solutions are unstable by construction, lying on the boundary between two distinct stable endstates of collapse - black hole or no black hole. Underlying the key features of near-critical collapse is the fact that the critical solutions are ``minimally unstable intermediate attractors,'' solutions whose linear perturbations have a single unstable mode. Critical solutions exhibit discrete self-similarity (an oscillation within a scaling envelope) characterized by a rescaling exponent, for massless scalar, gravitational, and SU(2) Yang-Mills fields; while perfect fluids and multiple-scalar field systems are continuously self-similar. The transition to collapse studied earlier, of, say neutron stars pushed over their upper mass limit by an addition of an arbitrarily small mass, exhibits a mass gap, and Choptuik calls these type I transitions. Examples are massive scalar fields, and colored black holes (variants with horizons of the Bartnik-Mckinnon Y-M Einstein solutions); the latter have overlapping regions in parameter space that correspond to type I and type II solutions. Choptuik's transparencies, with names and details suppressed here are at http://laplace.physics.ubc.ca/People/matt/Doc/ykis99.ps.

Jeffery Winicour described the characteristic treatment of black holes, and the current status of the The Pitt null code, developed by Welling, Isaacson, Gomez, Papadopoulos, Lehner, Bishop, Maharaj, Szylagyi, and Husa. In the vacuum case, a 3-D code has been implemented and tested in a variety of contexts. More recent work ( gr-qc/9901056) incorporates a perfect fluid with a 1-parameter equation of state. It has so far passed tests involving localized distributions of matter around a Schwarzschild black hole, and the code is found to be stable and convergent. Modifications are needed to handle shocks, and problems of astrophysical interest remain to be tackled.

Next morning, Kip Thorne began his talk by prodding the theorists to intensify their effort to keep up with the rapid progress of the LIGO experimentalists, and provide an accurate template for inspiral, with no drift in phase over the time of detection. Theorists particularly lag in the NS/BH problem, where spin-induced precession is important. The construction of the two LIGO observatories is essentially complete, and LIGO I sensitivity is to be reached by November, 2001. A LISA date of 2008-2010 is likely (i.e., US support is likely). Thorne emphasized that already for LIGO II with signal recycling, the standard quantum limit may be reached. To reach greater sensitivity, one cannot rely on standard position detectors that ignore correlations, and Thorne reviewed work with Braginsky, Gorodetsky, Matsko, Vyatchanin, Khalili, Levin, and Kimble on measurements beyond the SQL. Methods rely on correlations between the photon shot noise and the back-action noise induced by radiation pressure on the test masses. One can modify the input or output optics of current interferometers so that measurements at different times commute and state reduction has no influence on the noise. Thorne's current estimates for LIGO II event rates:
NS/NS, a few/yr; NS-BH a few/month (!); BH-BH, unknown.
Other sources mentioned were r-modes, strongly accreting LMXB's, and accretion induced collapse of white dwarfs.

Seiji Kawamura spoke on the current status of the Japanese detector, TAMA300: a mode cleaner was locked successfully 4 months ago, and a noise spectrum has been obtained. He summarized the recycling arrangement, and ongoing efforts to reduce shot, seismic and thermal noise. Great luck would be needed if there were to be a source strong enough for TAMA300 to detect, but plans are to step up to a much larger, 3-km cryogenic detector in the Kamgoka tunnel.
Next, two talks, by Ed Seidel and Jorge Pullin, on colliding black holes. Seidel noted both the long distance to go before inspiral can be computed and the progress made by the NSF black-hole grand challenge project, in developing a code that handles a variety of initial data sets, giving, in particular a stable 3-D evolution of a set of distorted black holes. The development of the cactus code was outlined, and a general PDE solver, due for public release in August, with an open source code, may be of wide interest to readers of MOG. 10 groups are now using it. Seidel emphasized recent methods (initially due to Shibata and Nakamura) of that give significantly more stable evolution, by promoting badly behaved quantities to independent variables. Although evolutions can be very accurate, by t = 50 M, every code crashes.

Pullin described work by a number of people on colliding black holes in the close limit ( gr-qc/9905081). From a computational viewpoint, binary black hole coalescence has three stages: a post-Newtian inspiral, of about 104 orbits, ending at roughly 10-12 M; the plunge; and the ringdown. The key to a perturbative treatment of the ringdown stage is that realization that it is is the ringdown stage, that a common, distorted horizon surrounds what were two black holes well before their apparent horizons merge. The evolution of the distorted horizon is what Pullin and his collaborators have treated with remarkable success as a perturbation of a Schwarzschild black hole. The work has included the development of a second-order perturbation formalism and its application to the ringdown problem. The approach serves as a code check for numerical relativity and a way to allow dying codes to run longer; and the ringdown serves in its own right as a source for LIGO. An analogous close limit for neutron stars, with the merged stars regarded as a perturbation of a spherical star has also been recently developed ( gr-qc/9903100).

Yasufumi Kojima and I then gave two talks on the recently discovered r-mode variant of the nonaxisymmetric instability that besets rapidly rotating relativistic stars. First noticed by Nils Andersson, in a numerical study, the instability of these axial-parity modes may dominate the spin-down of neutron stars that are rapidly rotating at birth, and the gravitational waves they produce may be detectable by LIGO II with narrow banding from sources out to somewhere between 4 and 20 Mpc. These dramatic implications have led to papers by about 35 authors in the past two years, but because my talk is now on the Web in a version written with Keith Lockitch ( gr-qc/9808083), and each of our talks at the ITP can be seen and heard live at http://www.itp.ucsb.edu/online/gravity_c99/, I'll leave it at that. Caveats are the assumption that nonlinear effects will allow the mode to grow until perturbed velocities are of order of the background velocity - that the mode does not transfer its energy to turbulence or to a magnetic field, while its amplitude is small.

Kojima emphasized research he has done, partly with Hosonuma, on the r-modes of of relativistic stars, reporting work that showed a continuous spectrum for nonisentropic stars in a slow-rotation approximation, and Beyer and Kokkotas make the claim precise. In addition, Kojima and Hosonuma have studied the mixing of axial and polar perturbations to order $\Omega^2$ in rotating relativistic stars, again finding a continuous mode spectrum. Kojima obtained a single, second-order equation for the radial behavior of the modes. If the continuous spectrum is a genuine feature of relativistic stars, it would be remarkable, but it may well be an artifact of approximations that force the frequency to be real: In the slow-motion approximation, the continuous spectrum arises from the vanishing of the highest derivative term in the ODE, found by Kojima, that describes the axial-parity modes.

A session devoted to the post-Newtonian computations reported progress on calculations that must provide the highly accurate templates for binary inspiral that are needed to use gravitational-wave detectors for astronomy. Both Cliff Will and Luc Blanchet spoke on the post-Newtonian description of binary inspiral. Luc Blanchet spoke on the post-Newtonian description of gravitational radiation, developed by Damour and colleagues. He emphasized the use of the Hadamard expansion to regularize the infinities that arise at the position of point-particles, when one expresses the multipole moments of the source as integrals extending over the distribution of stress-energy. One's confidence in using this renormalization method relies on a combination of its success and its elegance.

Cliff Will summarized a method called DIRE (Direct Integration of the Relaxed Einstein Equations), based on a framework developed by Epstein and Wagoner and extended by Will, Wiseman and Pati. Like the Blanchet-Damour-Iyer approach, DIRE begins with integrals over source and field. The integrals are finite when one restricts the use of the slow-motion approximation to the near zone and observes that the far-zone integral is bounded for a source that is well-behaved in the distant past. Equations up to 3.5 PN order are obtained, within the assumption that the orbiting bodies are sufficiently small, by isolating terms that neither vanish nor blow up as the size D of a body shrinks to zero, with the remaining terms absorbed into renormalized masses of the bodies. The resulting procedure is well-defined, although the assumption has been checked completely only at 1PN order.

In an enthusiastic update on the prospects for GEO600, Bernard Schutz emphasized the expected performance with signal recycling in both a narrow-band and broad-band mode: a maximum sensitivity of h < 10-22 at minimum noise, for frequencies between 100 and 1000 Hz. The state of the project: The vacuum system is complete, and the first mode cleaner is locked and working; interferometry and test optics are expected to be ready by mid-2000, and full sensitivity is to be reached by mid-2001. There has been close collaboration with LIGO, and a memorandum of understanding has been signed for full data exchange between LIGO I and GEO600.

The next day saw two talks on black-hole astrophysics. Nils Andersson summarized work on oscillations of rotating black holes, mentioning his work with Krivan, Laguna, and Papadopoulos on a 2-D code that evolves perturbations on a Kerr background and produces waveforms for black-hole ringing. Andersson emphasized two regimes of outgoing-mode ringing for a given value of l, say l=2, corresponding to the different imaginary parts of the frequencies for $m = \pm l$. It remiains to be seen whether, in the extreme Kerr limit both the retrograde, more quickly dying mode and the prograde, slowly dying mode both have comparable amplitudes (Mashoon and Ferrari suggested that the prograde mode was suppressed). Preliminary work suggests a curious feature of near-extreme Kerr: a sum over harmonics appears to give an oscillation whose damping is described by an envelope with a power-law fall-off that is slower than the Price tail. If true, one might never see the Price power law (for $m\neq 0$) in the late-time ringing of a near-extreme black hole.

Shin Mineshige spoke on the dramatic success of the ADAF (advection-dominated accretion flow) model of accretion disks that started with the 1977 work of Ichimaru. In this model, heat is dominantly transported by radial gas motion, and the spectrum is broad-band. Mineshige emphasized the fact that three models of accretion - the standard thin and thick disk models and the ADAF model are all solutions to the same set of equations with different values of optical depth. He reiterated the evidence for a central black hole in ADAF disks, arising from the fact that the emitted power is much smaller (as a fraction of the mass accretion rate) than it should be if matter were falling on a solid surface. And he discussed the observational tests of disk models and a recent model for X-ray novae.

Interspersed with these talks, and continuing through the next morning's sessions were a series of talks on the binary coalescence problem for neutron stars. Eriguchi and Gourgoulhon began these with a discussion of numerical work on Darwin Riemann problems in Newtonian gravity and in GR. The classical Roche, Darwin, and Riemann problems refer respectively to (Roche) the tidal forces on a finite mass orbiting a point mass with its spin and orbital frequencies identical; (Darwin) both masses are finite perfect fluids; (Riemann) the masses can have internal vorticity. The Darwin-Riemann problem that Eriguchi and Gourgoulhon consider has two masses whose spin frequencies are arbitrary, but for which the planes of rotation are aligned with the orbital plane. A series of papers by Uryu and Eriguchi have numerically solved the exact problem for Newtonian polytropes; and Usui, Eriguchi, and Uryu have begun a program to construct quasi-equilibrium models in GR, starting with spacetimes having an exact Killing vector of the form $t^\alpha + \Omega\phi^\alpha$, using a truncated set of field equations that allows a non-radiative field and a smaller set of potentials than one would need for the exact binary system. They thus replace the approximation of spatial conformal flatness by one in which the metric has only a $t-\phi$ off-diagonal term. Bonazzola, Gourgoulhon, and Marck repeat the Mathews-Maronetti-Wilson computation for an n=1 polytrope (adiabatic index $\gamma = 2$), using a multi-spectral method and obtaining agreement with the new MMW code to better than 2%. No innermost stable circular orbit is found for this value of $\gamma$. A useful summary of our knowledge of the ISCO was given (see gr-qc/9904040 and Uryu-Eriguchi).

Wai-Mo Suen and Ken-ichi Oohara summarized progress on the numerical simulation of coalescing neutron stars by the NASA Neutron Star Grand-Challenge group and by the Japanese group. Suen's discussion emphasized the extensive code testing that is underway and successes in meeting the Grand-Challenge milestones. A long enough time evolution to model coalescence is still in the future, but Suen reported a computation of head-on collisions using the coupled Einstein and hydrodynamic equations. These runs tested a conjecture of Shapiro that the shock-heating generated by infall, at least for stars falling from infinity, is enough to support the star until neutino cooling sets in (a time long compared to the dynamical timescale). If true, this would give an early cutoff to gravitational-wave emission. However, Suen argued that the dynamical time scale of infall was so short that the shock heating effect might not be important. He showed a simulation of the head-on collision of two 1.4 solar mass neutron stars, and reported finding an apparent horizon in the infalling time scale. Oohara summarized the longer history of the Japanese program, with a full GR code completed in '94. Test runs of on the order of one revolution have been run on a 2013 grid, with a 10-hr CPU time on a VPP300. Finally, Masaru Shibata has completed and tested on sample problems a related 3-D code for binary coalescence of neutron stars (gr-qc/9908027).

Following the binary-coalescence series, Max Ruffert and Peter Mezaros presented different views on the possible origins of $\gamma$-ray bursts. BATSE has revealed about one burst/galaxy/106 years, distributed isotropically at distances of order 1028-1029 cm., implying luminosities of order 1053 erg/s. The duration of bursts varies greatly, ranging from milliseconds to hours. Evidence for a fireball as a common source is good, but fireballs may be produced by mergers of NS-NS, NS-BH, WD-NS, WD-BH, or by a collapsar, a rotating, collapsing ``failed'' supernova (or possibly a neutron star pushed over its upper mass limit by accretion). Evidence for the collapsar comes from the identification of $\gamma$-ray bursts with galaxies that suggest the bursts probably occur in star-forming regions more often than would be expected for the old NS-NS or NS-BH systems. Ruffert and his collaborators (Janka, Eberl, and Fryer, astro-ph/9908290) have run a series of Newtonian simulations of NS-BH and NS-NS mergers incorporating back-reaction of gravitational waves. Using a Lattimer-Swesty equation of state and carefully taking account of neutrino sources and sinks, they confirm BH-NS mergers as a possible source of $\gamma$-ray bursts, but find an energy of 1051 erg requiring Lorentz beaming at the upper end of the possible. Mezaros' even-handed summary reviewed evidence for the fireball model he developed with Rees and others. Fireballs from all of the mechanisms have similar energies, with 1054 erg possible via MHD, while less than 10erg is likely if only neutrino annihilation is used (as in the simulations discussed by Ruffert). This leaves coalescence clearly still in the game.

The final two talks, by Bernard Carr and Jun'ichi Yokoyama, concerned primordial black holes. Carr delineated ways that PBH's can be used as a probe of the early universe; in particular, the limit on PBH's set by the absence of observed evaporating black-holes limits the spectral index during inflation. Both Carr and Yokoyama emphasized the possibility that MACHOS are PBH's of mass 0.5 $M_\odot$, the right mass for their having been created in a quark-hadron transition. Should the LMC MACHOS be PBH's, work by Yokoyama and collaborators sharply constrains parameters of inflationary models; and nearby BH-BH coalescence would be frequent, with possibly observable gravitational waves (Nakamura, Sasaki, Tanaka, Thorne). Carr considered a PBH-related test of whether G varies.

Jorge Pullin