Vacuum stress-energy density and its gravitational implications
In nongravitational physics the local density of energy is often regarded as merely a bookkeeping device; only total energy has an experimental meaning - and it is only modulo a constant term. But in general relativity the local stress-energy tensor is the source term in Einstein's equation. In closed universes, and those with Kaluza-Klein dimensions, theoretical consistency demands that quantum vacuum energy should exist and have gravitational effects, although there are no boundary materials giving rise to that energy by van der Waals interactions. In the lab there are boundaries, and in general the energy density has a nonintegrable singularity as a boundary is approached (for idealized boundary conditions). As pointed out long ago by Candelas and Deutsch, in this situation there is doubt about the viability of the semiclassical Einstein equation. Our goal is to show that the divergences in the linearized Einstein equation can be renormalized to yield a plausible approximation to the finite theory that presumably exists for realistic boundary conditions. For a scalar field with Dirichlet or Neumann boundary conditions inside a rectangular parallelepiped, we have calculated by the method of images all components of the stress tensor, for all values of the conformal coupling parameter and an exponential ultraviolet cutoff parameter. The qualitative features of contributions from various classes of closed classical paths are noted. Then the Estrada-Kanwal distributional theory of asymptotics, particularly the moment expansion, is used to show that the linearized Einstein equation with the stress-energy near a plane boundary as source converges to a consistent theory when the cutoff is removed. © 2008 IOP Publishing Ltd.
Publication Source (Journal or Book title)
Journal of Physics A: Mathematical and Theoretical
Estrada, R., Fulling, S., Kaplan, L., Kirsten, K., Liu, Z., & Milton, K. (2008). Vacuum stress-energy density and its gravitational implications. Journal of Physics A: Mathematical and Theoretical, 41 (16) https://doi.org/10.1088/1751-8113/41/16/164055