In this paper, we will solve the Hamiltonian constraint describing a curved general relativistic spacetime to find initial data describing how a black hole exists in vacuum. This has been done before by other researchers [Ansorg, 2004], and we will be adapting our own methods to an existing pseudo spectral Poisson solver [Gourgoulhon, 2001]. The need for this adaptation arises from improper numerical handling, done by pseudo spectral-methods, of a large part the Hamiltonian constraint equation due to the presence of the black hole singularity. To resolve a portion of this issue up to a given order, we will determine irregularities by executing a polynomial expansion on the Hamiltonian constraint, analytically solving the troublesome components of the equation and subtracting those out of the numerical process. This technique will increase the equation's differentiability and allow the numerical solver to run more efficiently.
We will cover all the calculations needed to describe one black hole with arbitrary spin and linear momentum. Our process is easily expanded into cases with n black holes [Brandt,1997], which we will show in chapter 2. We will implement a spherical harmonic decomposition of the black hole conformal factor, using them as basis functions by which to further expand and dissect the Hamiltonian Constraint equation. In the end, the expansion and subtraction method will be done out to the order of r4, where r is the spherical radius assuming the black hole is at the coordinate origin, making the Hamiltonian equation, which, unaltered, is a C2 equation, become a C7 equation. Smoothing the Hamiltonian improves numerical precision, especially near the BH where the most interesting physics occurs. The method used in this paper can be further implemented to higher orders of r to yield even smoother conditions. We will test the numerical results of using this method against the existing solver that uses the publicly available Lorene numerical libraries and programmed in C.
Library of Congress Subject Headings
Black holes (Astronomy)--Mathematics; Scalar field theory; Spectral theory (Mathematics); Hamiltonian systems
Applied and Computational Mathematics (MS)
Department, Program, or Center
School of Mathematical Sciences (COS)
Josh A. Faber
Tony A. Harkin
Munro, Eugene, "Using Analytic Techniques to Resolve Numerical Issues in a Pseudo Spectral Solver for a Black Hole Scalar Field" (2013). Thesis. Rochester Institute of Technology. Accessed from
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