Free evolution of the hyperboloidal initial value problem in spherical symmetry

Abstract

The hyperboloidal initial value problem is addressed in the context of Numerical Relativity, motivated by its use of hyperboloidal slices - smooth spacelike slices that reach future null infinity, the "place" in spacetime where radiation is to be extracted. This is beneficial for studying the global properties of isolated systems and unambiguously extracting their gravitational radiation. The present approach implements the Einstein equations as a free evolution, using the BSSN and Z4 formulations (standard in current codes) expressed in terms of a conformally rescaled metric as suggested by Penrose, and with a time-independent conformal factor. The main difficulty is that the resulting system of PDEs includes formally divergent terms at null infinity that require a special treatment. The numerical simulations in this thesis are restricted to spherical symmetry, although the regularization in the radial direction is expected to also apply to the full 3D case up to some extent. A critical ingredient are the gauge conditions, which rely on well-chosen source functions and damping terms and control the treatment of future null infinity by means of the scri-fixing condition. Once the numerical implementation was stabilized, stable numerical evolutions of a massless scalar field coupled to the Einstein equations could be performed with regular and black hole trumpet initial data on a hyperboloidal slice. The signal of the scalar field has been successfully extracted at future null infinity. Small perturbations of regular initial data give stationary data that are stable forever, while larger scalar field perturbations result in the formation of a black hole. Schwarzschild trumpet initial data have been found to slowly drift away from the expected stationary values, but for small perturbations the effect is slow enough to allow the observation of the power-law decay tails of the scalar field.

185 pages, PhD thesis

References in corpus (11)

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• Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing
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• Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions
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• Tetrad formalism for numerical relativity on conformally compactified constant mean curvature hypersurfaces
• Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators
• Black hole initial data on hyperboloidal slices
• Stationary hyperboloidal slicings with evolved gauge conditions
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Cited by in corpus (9)

• Spherical symmetry as a test case for unconstrained hyperboloidal evolution II: gauge conditions
• Hyperbolicity of General Relativity in Bondi-like gauges
• Global simulations of Minkowski space-time including space-like infinity
• Evolution of scalar fields surrounding black holes on compactified constant mean curvature hypersurfaces
• Peeling in Generalized Harmonic Gauge
• Conformal diagrams for stationary and dynamical strong-field hyperboloidal slices
• High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation
• Linearised conformal Einstein field equations
• Free Hyperboloidal Evolution of the Einstein-Maxwell-Klein-Gordon System