On the energy-momentum tensor for a scalar field on manifolds with boundaries

TL;DR

This paper investigates the classical and quantum energy-momentum tensor for a scalar field on manifolds with boundaries, emphasizing the importance of boundary contributions for conservation laws and vacuum energy calculations.

Contribution

It derives a general expression for the boundary contribution to the energy-momentum tensor and demonstrates its significance in quantum vacuum energy computations.

Findings

Boundary terms are essential for energy conservation.

Surface energy matches zero-point energy sums.

Zeta function technique evaluates surface energy in spherical shells.

Abstract

We argue that already at classical level the energy-momentum tensor for a scalar field on manifolds with boundaries in addition to the bulk part contains a contribution located on the boundary. Using the standard variational procedure for the action with the boundary term, the expression for the surface energy-momentum tensor is derived for arbitrary bulk and boundary geometries. Integral conservation laws are investigated. The corresponding conserved charges are constructed and their relation to the proper densities is discussed. Further we study the vacuum expectation value of the energy-momentum tensor in the corresponding quantum field theory. It is shown that the surface term in the energy-momentum tensor is essential to obtain the equality between the vacuum energy, evaluated as the sum of the zero-point energies for each normal mode of frequency, and the energy derived by the integration of the corresponding vacuum energy density. As an application, by using the zeta function technique, we evaluate the surface energy for a quantum scalar field confined inside a spherical shell.