Quantum effects near a point mass in (2 + 1)-dimensional gravity

Abstract

We investigate the behavior of classical and quantum fields in the conical space-time associated with a point mass in 2 + 1 dimensions. We show that the presence of conical boundary conditions alters the electrostatic field of a point charge leading to the presence of a finite self-force on the charge from the direction of the point mass exactly as if the point mass itself were charged. The conical space-time geometry also affects the zero-point fluctuations of a quantum scalar field leading to the existence of a vacuum polarization ⟨T_μν⟩ in the (2 + 1)-dimensional analogue of the Schwarzschild metric. The resulting linearized semiclassical Einstein equations G_μν=8_πG⟨T_μν⟩ possess a well-defined Newtonian limit, in marked contrast to the classical case for which no Newtonian limit is known to exist. An elegant reformulation of our results in terms of the method of images is also presented. Our analysis also covers the nonstatic de Sitter-Schwarzschild metric in 2 + 1 dimensions, in which in addition to the vacuum polarization, a nonzero vacuum flux of energy ⟨T_rt⟩ is also found to exist. As part of this analysis, we evaluate the scalar field propagator in an n-dimensional de Sitter space; as a result some novel features of quantum field theory in odd dimensions are seen to emerge.