TL;DR
This paper introduces a method to find solutions for a real scalar field on manifolds by using harmonic generalized coordinates, applicable to various potentials and providing insights into bulk solutions.
Contribution
It presents a novel approach to obtain scalar field solutions on manifolds through harmonic coordinate substitution, independent of the potential form.
Findings
Solutions for Klein-Gordon and quartic potentials are demonstrated.
Manifolds and solutions are characterized by simple constraints.
Method may inform properties of exact bulk solutions.
Abstract
A generic theory of a single real scalar field is considered, and a simple method is presented for obtaining a class of solutions to the equation of motion. These solutions are obtained from a simpler equation of motion that is generated by replacing a set of the original coordinates by a set of generalized coordinates, which are harmonic functions in the spacetime. These ansatz solutions solve the original equation of motion on manifolds that are defined by simple constraints. These manifolds, and their dynamics, are independent of the form of the scalar potential. Some scalar field solutions, and manifolds upon which they exist, are presented for Klein-Gordon and quartic potentials as examples. Solutions existing on leaves of a foliated space may allow inferences of the characteristics expected of exact bulk solutions.
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