Diffeomorphism invariant eigenvalue problem for metric perturbations in a bounded region

TL;DR

This paper develops a method for constructing diffeomorphism invariant boundary conditions for metric perturbations, analyzing eigenvalue problems on a Euclidean disk and deriving explicit eigenfunctions.

Contribution

It introduces a general approach for boundary conditions invariant under diffeomorphisms and applies it to a specific geometric setting, deriving explicit eigenfunctions.

Findings

Explicit eigenfunctions of the Laplace operator on metric perturbations are derived.

Boundary conditions compatible with hermiticity are characterized.

Eigenvalue problems are reduced to lower-dimensional vector, tensor, and scalar fields.

Abstract

We suggest a method of construction of general diffeomorphism invariant boundary conditions for metric fluctuations. The case of $d+1$ dimensional Euclidean disk is studied in detail. The eigenvalue problem for the Laplace operator on metric perturbations is reduced to that on $d$-dimensional vector, tensor and scalar fields. Explicit form of the eigenfunctions of the Laplace operator is derived. We also study restrictions on boundary conditions which are imposed by hermiticity of the Laplace operator.