Abstract

In the construction of initial data, by way of describing one or more black holes for Numerical Relativity, solving the Hamiltonian Constraint remains a fundamental challenge, as the presence of coordinate singularities in the so-called "puncture" formalism often complicates traditional grid-based numerical methods. This thesis addresses this challenge by developing a semi-analytical framework based on an iterative spherical harmonic modal expansion. We begin by decomposing the source term of the Hamiltonian Constraint, given in terms of the extrinsic curvature, into a basis of spherical harmonics. To navigate the non-linearity of the governing equation, we implement an iterative scheme derived from a generalized binomial expansion of the conformal factor, which allows us to solve for the potential by applying the Inverse Laplacian Operator within the spherical harmonic subspace, transforming the problem into a hierarchy of linear steps. We investigate the lowest-order behavior of these solutions to derive generalized nth-order analytical expressions for punctures characterized by linear momentum, angular momentum, and the complex coupling of both. Through this iterative process, we demonstrate how higher-order multipole moments naturally emerge to describe the local geometry of the puncture. Finally, the framework is extended to binary black hole systems by incorporating the presence of a second puncture. By expanding the reciprocal distance to the companion hole, we derive the lowest order tidal interaction potential, providing a structured representation of the mutual gravitational influence between the pair. These results provide a "high-order accurate" analytical foundation for the characterization of the spacetime geometry, offering a robust alternative for generating initial data in simulations of black hole mergers.

Publication Date

5-2026

Document Type

Thesis

Student Type

Graduate

Degree Name

Applied and Computational Mathematics (MS)

College

College of Science

Advisor

Joshua Faber

Advisor/Committee Member

Anthony Harkin

Advisor/Committee Member

Yosef Zlochower

Campus

RIT – Main Campus

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