Numerical Integration of Nonlinear Wave Equations for General Relativity

A second-order numerical implementation is given for recently derived nonlinear wave equations for general relativity. The Gowdy T$^3$ cosmology is used as a test bed for studying the accuracy and convergence of simulations of one-dimensional nonlinear waves. The complete freedom in space-time slicing in the present formulation is exploited to compute in the Gowdy line-element. Second-order convergence is found by direct comparison of the results with either analytical solutions for polarized waves, or solutions obtained from Gowdy's reduced wave equations for the more general unpolarized waves. Some directions for extensions are discussed.Comment: 19 pages (LaTex), 3 figures (ps

Past and future gauge in numerical relativity

Numerical relativity describes a discrete initial value problem for general relativity. A choice of gauge involves slicing space-time into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of present time. Here, we propose solving the discretized Einstein equations with a choice of gauge in the future and a dynamical gauge in the past. The method is illustrated on a polarized Gowdy wave.Comment: To appear in Class Quantum Grav, Let

On the origin of exponential growth in induced earthquakes in Groningen

The Groningen gas field shows exponential growth in earthquakes event counts around a magnitude M1 with a doubling time of 6-9 years since 2001. This behavior is identified with dimensionless curvature in land subsidence, which has been evolving at a constant rate over the last few decades {essentially uncorrelated to gas production.} We demonstrate our mechanism by a tabletop crack formation experiment. The observed skewed distribution of event magnitudes is matched by that of maxima of event clusters with a normal distribution. It predicts about one event $<$\,M5 per day in 2025, pointing to increasing stress to human living conditions.Comment: 12 pages, 7 figures, to appear in Earthquakes and Structure

Extended black hole cosmologies in de Sitter space

We generalize the superposition principle for time-symmetric initial data of black hole spacetimes to (anti-)de Sitter cosmologies in terms of an eigenvalue problem $\Delta_g\phi={1/8}(R_g-2\Lambda)\phi$ for a conformal scale $\phi$ applied to a metric $g_{ij}$ with constant three-curvature $R_g$. Here, $R_g=0,2$ in the Brill-Lindquist and, respectively, Misner construction of multihole solutions for $\Lambda=0$. For de Sitter and anti-de Sitter cosmologies, we express the result for $R_g=0$ in incomplete elliptic functions. The topology of a black hole in de Sitter space can be extended into an infinite tower of universes, across the turning points at the black hole and cosmological event horizons. Superposition introduces binary black holes for small separations and binary universes for separations large relative to the cosmological event horizon. The evolution of the metric can be described by a hyperbolic system of equations with curvature-driven lapse function, of alternating sign at successive cosmologies. The computational problem of interacting black hole-universes is conceivably of interest to early cosmology when $\Lambda$ was large and black holes were of mass $<{1/3}\Lambda^{-1/2}$, here facilitated by a metric which is singularity-free and smooth everywhere on real coordinate space.Comment: to appear in Class. Quant. Gra

Vistas in numerical relativity

Upcoming gravitational wave-experiments promise a window for discovering new physics in astronomy. Detection sensitivity of the broadband laser interferometric detectors LIGO/VIRGO may be enhanced by matched filtering with accurate wave-form templates. Where analytic methods break down, we have to resort to numerical relativity, often in Hamiltonian or various hyperbolic formulations. Well-posed numerical relativity requires consistency with the elliptic constraints of energy and momentum conservation. We explore this using a choice of gauge in the future and a dynamical gauge in the past. Applied to a polarized Gowdy wave, this enables solving {\em all} ten vacuum Einstein equations. Evolution of the Schwarzschild metric in 3+1 and, more generally, sufficient conditions for well-posed numerical relativity continue to be open challenges.Comment: invited talk, Asian Pacific CTP Winter School on black hole astrophysics, Pohang, Kore