Towards Non-Linear Perturbations of Black Holes in Horizon-Penetrating Coordinates

Bhattacharyya, Maitraya Kanta (2021) Towards Non-Linear Perturbations of Black Holes in Horizon-Penetrating Coordinates. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Maitraya Kanta Bhattacharyya (14IP002)) 14IP002.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

The deviations of non-linear perturbations of black holes from the linear case are important in the context of ringdown signals with large signal-to-noise ratio. To facilitate a comparison between the two, in the first part of the thesis, we derive several results of linear perturbation theory in coordinates which may be adopted in numerical work. Specifically, our results are derived in Kerr-Schild coordinates adjusted by a general height function. Firstly, we address the questions: for an initial configuration of a massless scalar field, what is the amplitude of the excited quasinormal mode (QNM) for any observer outside outside the event horizon, and furthermore what is the resulting tail contribution? This is done by constructing the full Green’s function for the problem with exact solutions of the confluent Heun equation satisfying appropriate boundary conditions. We then detail new developments to our pseudospectral numerical relativity code bamps to handle scalar fields. In the linear regime we employ precisely the Kerr-Schild coordinates treated by our previous analysis. In particular, we evolve pure QNM type initial data along with several other types of initial data and report on the presence of overtone modes in the signal. In the second part of the thesis, we present an implementation of the dual foliation generalized harmonic gauge (DF-GHG) formulation within our pseudospectral code. The formalism promises to give greater freedom in the choice of coordinates that can be used in numerical relativity. As a specific application we focus here on the treatment of black holes in spherical symmetry. Existing approaches to black hole excision in numerical relativity are susceptible to failure if the boundary fails to remain outflow. We present a method, called DF-excision, to avoid this failure. Our approach relies on carefully choosing coordinates in which the coordinate light-speeds are under strict control so that the excision boundary must remain outflow. These coordinates are then combined with the DF-GHG formulation. After performing a set of validation tests in a simple setting, we study the accretion of large pulses of scalar field matter on to a spherical black hole. We compare the results of DF-excision with a naive setup. DF-excision proves reliable even when the previous approach fails. In the third and final part of the thesis, we use DF-excision and DF-GHG to perform fully non-linear simulations of a massless scalar field minimally coupled to general relativity using coordinates which are analogous to the Kerr-Schild coordinates from the linear setting. We perform a detailed analysis comparing the results of the non-linear simulations and the linear simulations and show that the scalar QNM frequencies of the black hole inversely scale appropriately with the mass of the remnant leading to the conclusion that most of the ringdown happens after the black hole has already ‘grown’ due to the scalar field. We also perform modeling of the non-linear time series data using several machine learning algorithms and compare their performance against each other.

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