Abstract:
It is shown that only the maximally-symmetric spacetimes can be expressed in both the Robertson-Walker form and in static form - there are no other static forms of the Robertson-Walker spacetimes. All possible static forms of the metric of the maximally-symmetric spacetimes are presented as a table. The findings are generalized to apply to functionally more general spacetimes: it is shown that the maximally symmetric spacetimes are also the only spacetimes that can be written in both orthogonal-time isotropic form and in static form.

Abstract:
It is a well known fact that quantum fields on Minkowski spacetime are correlated for each pair of spacetime regions. In Robertson-Walker spacetimes there are spacelike separated regions with disjoint past horizons but the absence of correlations in that case was never proved. We derive in this paper formulae for correlations of quantum fields on Robertson-Walker spacetimes. Such correlations could have reasonably influenced the formation of structure in the early universe. We use methods of algebraic and constructive quantum field theory.

Abstract:
We construct a new class of physical states of the free Klein-Gordon field in Robertson-Walker spacetimes. This is done by minimizing the expectation value of smeared stress-energy. We get an explicit expression for the state depending on the smearing function. We call it a state of low energy. States of low energy are an improvement of the concept of adiabatic vacua on Robertson-Walker spacetimes. The latter are approximations of the former. It is shown that states of low energy are Hadamard states.

Abstract:
A generalized Robertson-Walker spacetime is the warped product with base an open interval of the real line endowed with the opposite of its metric and base any Riemannian manifold. The family of generalized Robertson-Walker spacetimes widely extends the one of classical Robertson-Walker spacetimes. In this article we prove a very simple characterization of generalized Robertson-Walker spacetimes; namely, a Lorentzian manifold is a generalized Robertson-Walker spacetime if and only if it admits a timelike concircular vector field.

Abstract:
We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle. 1. Introduction The mathematical interests for the study of spacelike hypersurfaces in spacetimes began in the seventies with the works of Cheng and Yau [1], Brill and Flaherty [2], Choquet-Bruhat [3], and later on with some other authors, such as in [4–6]. Moreover, the study of such hypersurfaces is also of interest from a physical point of view, because of its relation to several problems in general relativity. More recently, there has been an increasing interest in the study of spacelike hypersurfaces with constant higher order mean curvature, such as in [7–9]. At the same time, in [10, 11] there are system research works about the lightlike hypersurface of semi-Riemannian manifolds. In this paper, based on the previous result, we want to make an attempt to study the lightlike hypersurfaces of spacetimes. First of all, we are interested in the study of lightlike hypersurfaces in conformally stationary spacetimes, and the screen bundle with constant higher order mean curvature. First, we use the Newton transformations as the main analytical tool and study the Minkowski-type integral formulas of the lightlike hypersurface. The use of these kinds of formulas in the Lorentzian manifold was first started by Montiel in [12] for the spacelike hypersurface with constant mean curvature in de Sitter spacetimes, and it was continued by Alías et al. [13, 14] for more general spacetimes. Higher-order Minkowski formula for hypersurface was first obtained by Hsiung in [15] in Euclidean space, and by Bivens in [16] in the Euclidean sphere and hyperbolic space. In this paper, we obtain two Minkowski-type integral formulas about the higher-order mean curvature of the lightlike hypersurface as follows, which are called Minkowski formulas I and II. The Minkowski Formula I. Let be a compact lightlike hypersurface of a conformally stationary spacetime of constant sectional curvature, on which the screen bundle is integrable and Ricci tensor of the induced connection is symmetric; then the following conclusion holds: The Minkowski Formula II. Let be a compact lightlike hypersurface immersed into a conformally stationary

Abstract:
In this letter we summarize our analysis of Bose-Einstein condensation on closed Robertson-Walker spacetimes. In a previous work we defined an adiabatic KMS state on the Weyl-algebra of the free massive Klein-Gordon field. This state describes a free Bose gas on Robertson-Walker spacetimes. We use this state to analyze the possibility of Bose-Einstein condensation on closed Robertson-Walker spacetimes. We take into account the effects due to the finiteness of the spatial volume and show that they are not relevant in the early universe. Furthermore we show that a critical radius can be defined. The condensate disappears above the critical radius.

Abstract:
The Robertson-Walker spacetimes are conformally flat and so are conformally invariant under the action of the Lie group SO(4,2), the conformal group of Minkowski spacetime. We find a local coordinate transformation allowing the Robertson-Walker metric to be written in a manifestly conformally flat form for all values of the curvature parameter k continuously and use this to obtain the conformal Killing vectors of the Robertson-Walker spacetimes directly from those of the Minkowski spacetime. The map between the Minkowski and Robertson-Walker spacetimes preserves the structure of the Lie algebra so(4,2). Thus the conformal Killing vector basis obtained does not depend upon k, but has the disadvantage that it does not contain explicitly a basis for the Killing vector subalgebra. We present an alternative set of bases that depend (continuously) on k and contain the Killing vector basis as a sub-basis (these are compared with a previously published basis). In particular, bases are presented which include the Killing vectors for all Robertson-Walker spacetimes with additional symmetry, including the Einstein static spacetimes and the de Sitter family of spacetimes, where the basis depends on the Ricci scalar R.

Abstract:
We give necessary and sufficient conditions for warped product manifolds with 1-dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. We also construct suitable examples of such manifolds. They are quasi-Einstein or not.

Abstract:
We study the non-linear dynamics of quantum fields in matter and radiation dominated universes, using the non-equilibrium field theory approach combined with the non-perturbative Hartree and the large N approximations. We examine the phenomenon of explosive particle production due to spinodal instabilities and parametric amplification in expanding universes with and without symmetry breaking. For a variety of initial conditions, we compute the evolution of the inflaton, its quantum fluctuations, and the equation of state. We find explosive growth of quantum fluctuations, although particle production is somewhat sensitive to the expansion of the universe. In the large N limit for symmetry breaking scenarios, we determine generic late time solutions for any flat Friedman-Robertson-Walker cosmology. We also present a complete and numerically implementable renormalization scheme for the equation of motion and the energy momentum tensor in flat FRW cosmologies. In this scheme the renormalization constants are independent of time and of the initial conditions.

Abstract:
Quantum inequality restrictions on the stress-energy tensor for negative energy are developed for three and four-dimensional static spacetimes. We derive a general inequality in terms of a sum of mode functions which constrains the magnitude and duration of negative energy seen by an observer at rest in a static spacetime. This inequality is evaluated explicitly for a minimally coupled scalar field in three and four-dimensional static Robertson-Walker universes. In the limit of vanishing curvature, the flat spacetime inequalities are recovered. More generally, these inequalities contain the effects of spacetime curvature. In the limit of short sampling times, they take the flat space form plus subdominant curvature-dependent corrections.