[Paper Review] Free evolution of the hyperboloidal initial value problem in spherical symmetry

Alex Vañó-Viñuales|arXiv (Cornell University)|Nov 30, 2015

Pulsars and Gravitational Waves Research12 citations

TL;DR

This PhD thesis presents a free evolution scheme for the hyperboloidal initial value problem in spherical symmetry using the BSSN and Z4 formulations with a conformally rescaled metric and time-independent conformal factor. It successfully stabilizes the system to extract scalar field radiation at future null infinity, demonstrating stable evolutions with regular and black hole trumpet initial data, including observation of power-law decay tails.

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.

Motivation & Objective

• To develop a free evolution framework for the hyperboloidal initial value problem in numerical relativity.
• To enable unambiguous extraction of gravitational radiation at future null infinity using hyperboloidal slices.
• To address the challenge of formally divergent terms in partial differential equations at null infinity.
• To stabilize the numerical system via tailored gauge conditions and source functions.
• To demonstrate long-term stability and radiation extraction in spherically symmetric spacetimes with scalar field coupling.

Proposed method

• Adopt the BSSN and Z4 formulations of the Einstein equations with a conformally rescaled metric as proposed by Penrose.
• Use a time-independent conformal factor to simplify the evolution system and maintain hyperboloidal slicing.
• Implement a regularization technique in the radial direction to handle divergent terms at future null infinity.
• Apply a scri-fixing condition via carefully chosen source functions and damping terms in the gauge conditions.
• Construct hyperboloidal initial data for massless scalar fields coupled to gravity, including regular and trumpet black hole configurations.
• Perform free evolution simulations in spherical symmetry with adaptive mesh refinement and boundary treatment at null infinity.

Experimental results

Research questions

• RQ1 Can a free evolution scheme be stabilized for the hyperboloidal initial value problem in spherical symmetry despite divergent terms at null infinity?
• RQ2 How effectively can gravitational radiation be extracted at future null infinity using hyperboloidal slices?
• RQ3 What is the long-term stability of hyperboloidal evolutions with regular and black hole trumpet initial data?
• RQ4 To what extent do gauge conditions with source functions and damping terms control the behavior at future null infinity?
• RQ5 Can power-law decay tails of scalar fields be observed in stable hyperboloidal evolutions?

Key findings

• Stable numerical evolutions were achieved for both regular and black hole trumpet initial data on hyperboloidal slices.
• The scalar field signal was successfully extracted at future null infinity, confirming the method's capability for radiation monitoring.
• Small perturbations of regular initial data led to stationary solutions that remained stable indefinitely.
• Larger scalar field perturbations resulted in black hole formation, consistent with physical expectations.
• Schwarzschild trumpet initial data exhibited slow drift from stationary values, but the drift was slow enough to allow observation of power-law decay tails.
• The regularization and gauge conditions effectively managed divergent terms, enabling long-term evolution without blow-up.