COORDINATE-INDEPENDENT DEFINITION OF TIME AND VACUUM IN CURVED SPACE–TIME

Abstract

We propose a definition of time and of the vacuum such that they are intrinsic to a given globally hyperbolic 1 + (n − 1)-dimensional space–time geometry and independent of the choice of coordinates. To arrive at this definition we use the new physical principle that a 1 + 1-dimensional Poincaré algebra, including Killing conditions on the generators, should be valid on the hypersurface of instantaneity. Given a timelike vector at a point (an observer's velocity) we define "an instant of time" to be the spacelike surface of geodesics which pass through that point and are orthogonal to that timelike vector. Gaussian coordinates erected on this surface yield 1 + 1-dimensional subspaces with Poincaré symmetry valid on that surface. The generator associated with time translation now uniquely picks out the direction of time on that surface. This fact permits unambiguous quantization on the surface of a field evolving in this background metric. For flat space–time the corresponding vacuum is always the Minkowski vacuum. We also consider in detail the case of static and Robertson–Walker metrics in 1 + 1 dimensions and find our vacuum to be different from those given before. The vacuum for the de Sitter metric in 1 + 1 dimensions is compared with the results in the literature and found to be different. Our definition of particles, and hence particle production, is consequently different also.