Hadamard's variational formula and the energy-momentum tensor

TL;DR

This paper reformulates Hadamard's variational formula for Green functions using energy-momentum and strain tensors within a Riemannian manifold, connecting geometric variations to field theory concepts.

Contribution

It introduces a novel formulation of Hadamard's variational formula using energy-momentum tensors, extending it to general Riemannian subdomains and linking it to field theory.

Findings

Formulation of Hadamard's formula in terms of energy-momentum tensor

Generalization to arbitrary-dimensional Riemannian manifolds

Connection between geometric variations and field theory energy-momentum tensors

Abstract

The Hadamard variational formula for the Green function is formulated in terms of a polarized energy-momentum tensor and a strain tensor. This is elaborated in a general setting of subdomains of a Riemannian manifold in arbitrary dimension and linked to the way the energy-momentum tensor in general field theory appears as a result of varying the metric tensor in a Lagrangian function.