The Robertson-Walker metrics expressible in static form

It is shown that there are six, and only six, Robertson-Walker metrics which can be expressed in static form. They are precisely those Robertson-Walker metrics whosespacetime curvature is constant. The coordinate transformations which transform these metrics into their static form are also given. An error in Robertson and Noonan's Book [1] is pointed out and corrected.     

Volume matching in Tolman models

Since the equations of general relativity are nonlinear, it is not strictly correct to obtain average values by integrating over spatial volumes. Yet this is really what is done in the attempt to fit our rather lumpy universe to a standard cosmological model of uniform density. Consequently, the fitting problem, raised last year by Ellis and Stoeger [1], asks how accurate the average values derived from observational cosmology can be, even without measurement uncertainties. Do they really describe the best-fit Robertson-Walker model to our universe? One of the alternatives to averaging they suggested was that of volume matching. We try to provide a first estimate of the error due to averaging by fitting a Robertson-Walker model to an inhomogeneous Tolman model using realistic density profiles. Comparing the results from volume matching and from averaging, we find that errors are of the order of 10% or more.     

Finite temperature quantum fields in expanding universes

The thermodynamics of an ideal relativistic quantum gas in expansion is studied. It is found that only for conformally invariant fields in conformally static spacetime can thermal equilibrium be strictly maintained. A finite temperature theory can be defined under the condition of quasi equilibrium when the background expansion is nearly adiabatic. The high temperature expansion of the energy density for massive nonconformal fields in Robertson-Walker universes and for conformal fields in Bianchi Type-I universes are calculated. The importance of these results on phase transition and quantum processes in the early universe is discussed.     

Interpretations of cosmological spectral shifts

It is shown that for Robertson-Walker models with flat or closed space sections, all of the cosmological spectral shift can be attributed to the non-flat connection (and thus indirectly to space-time curvature). For Robertson-Walker models with hyperbolic space sections, it is shown that cosmological spectral shifts uniquely split up into “kinematic” and “gravitational” parts provided that distances are small. For large distances no such unique split-up exists in general. A number of common, but incorrect assertions found in the literature regarding interpretations of cosmological spectral shifts, is pointed out.     

General solution of Liouville's equation in Robertson-Walker space-times

We obtain the general solution of Liouville's equation in Robertson-Walker space-times, based on Killing vector constants of the motion and a new time-dependent first integral. For photons the solution is generated by conformai Killing vectors.     

Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes

A new technique is introduced in order to solve the following question:When is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice? (Generalized Robertson-Walker spacetimes extend classical Robertson-Walker ones to include the cases in which the fiber has not constant sectional curvature.) First, we determine when this hypersurface must be compact. Then, all these compact hypersurfaces in (necessarily spatially closed) spacetimes are shown to be totally umbilical and, except in very exceptional cases, slices. This leads to proof of a new Bernstein-type result. The power of the introduced tools is also shown by reproving and extending several known results.     

Characteristic function of Robertson Walker spaces

The time-like geodesies of Robertson-Walker spaces have recently been considered by Noerdlinger from the Lagrangian point of view. In this paper the Hamiltonian point of view is adopted, to which end the characteristic functionV of an arbitrary Robertson-Walker space is determined by integrating the differential equations which governV. Some particular examples are dealt with explicitly.     

A scalar field influenced Robertson-Walker model

We show that if a scalar field similar to the one mediating inflation is effective even now, then the Robertson-Walker model may have some interesting properties under certain conditions. In this case even a spatially closed universe may expand forever. A dust universe has alwaysq=1/2 for all values of spatial curvature. The critical density is also less than in the standard Robertson-Walker model.     

Spacetime emergence of the robertson-walker universe from a matrix model

Using a novel, string theory-inspired formalism based on a Hamiltonian constraint, we obtain a conformal mechanical system for the spatially flat four-dimensional Robertson-Walker Universe. Depending on parameter choices, this system describes either a relativistic particle in the Robertson-Walker background or metric fluctuations of the Robertson-Walker geometry. Moreover, we derive a tree-level M theory matrix model in this time-dependent background. Imposing the Hamiltonian constraint forces the spacetime geometry to be fuzzy near the big bang, while the classical Robertson-Walker geometry emerges as the Universe expands. From our approach, we also derive the temperature of the Universe interpolating between the radiation and matter dominated eras.     

Hypersurfaces of constant mean extrinsic curvature

The existence of hypersurfaces of constant mean extrinsic curvature is examined. Using techniques developed by Choquet-Bruhat in her work on related subjects and techniques used by D'Eath in his study of perturbed Robertson-Walker universes, theorems are proved about the existence of slices of constant mean extrinsic curvature for spacetimes in a neighbourhood of the open Robertson-Walker Universes. It is shown in particular that those spacetimes which lie in a neighbourhood of Minkowski space or de-Sitter space admit slices of constant mean extrinsic curvature. By modifying the techniques used to prove these theorems, it is shown that asymptotically simple spacetimes which are close to Minkowski space admit slices of constant mean extrinsic curvature. The behaviour of these slices near null infinity is examined and it is shown that a large family of such hypersurfaces exists, indexed by the BMS supertranslations.     

Phase space classification of Kaluza-Klein cosmologies

This work provides an analysis of all possible solutions to the Einstein equations for a class of spatially homogeneous vacuum cosmological models with (Robertson-Walker) × (n-sphere) and (Robertson-Walker) × (n-sphere)×(torus) symmetries. Using qualitative theory of dynamical systems we show that classical evolution of the models can lead to contraction of the extra dimensions.     

On the existence of perturbed robertson-walker universes

Solutions of the full nonlinear field equations of general relativity near the Robertson-Walker universes are examined, together with their relation to linearized perturbations. A method due to Choquet-Bruhat and Deser is used to prove existence theorems for solutions near Robertson-Walker constraint data of the constraint equations on a spacelike hypersurface. These theorems allow one to regard the matter fluctuations as independent quantities, ranging over certain function spaces. In the k = ?1 case the existence theory describes perturbations which may vary within uniform bounds throughout space. When k = +1 a modification of the method leads to a theorem which clarifies some unusual features of these constraint perturbations. The k = 0 existence theorem refers only to perturbations which die away at large distances. The connection between linearized constraint solutions and solutions of the full constraints is discussed. For k = ±1 backgrounds, solutions of the linearized constraints are analyzed using transverse-traceless decompositions of symmetric tensors. Finally the time-evolution of perturbed constraint data and the validity of linearized perturbation theory for Robertson-Walker universes are considered.     

Inflation in Brans-Dicke theory with torsion

We show that an inflationary phase may occur at a sufficiently early epoch in the Robertson-Walker universe model in Brans-Dicke theory with torsion. Some features of this inflationary scenario are briefly discussed.     

Dirac equation in Robertson-Walker metric

We recast the Dirac relativistic equation within the theoretical framework of the Robertson-Walker metric, using spatial hypersurfaces that are essentially curved, and hence more general, as compared to the flat ones employed by Barut and Duru.     

Relative velocities for radial motion in expanding Robertson-Walker spacetimes

The expansion of space, and other geometric properties of cosmological models, can be studied using geometrically defined notions of relative velocity. In this paper, we consider test particles undergoing radial motion relative to comoving (geodesic) observers in Robertson-Walker cosmologies, whose scale factors are increasing functions of cosmological time. Analytical and numerical comparisons of the Fermi, kinematic, astrometric, and the spectroscopic relative velocities of test particles are given under general circumstances. Examples include recessional comoving test particles in the de Sitter universe, the radiation-dominated universe, and the matter-dominated universe. Three distinct coordinate charts, each with different notions of simultaneity, are employed in the calculations. It is shown that the astrometric relative velocity of a radially receding test particle cannot be superluminal in any expanding Robertson-Walker spacetime. However, necessary and sufficient conditions are given for the existence of superluminal Fermi speeds, and it is shown how the four concepts of relative velocity determine geometric properties of the spacetime.     

Invariant solutions of Liouville's equation in Robertson-Walker space-times

We find all solutions of Liouville's equation in Robertson-Walker space-times that are either spatially homogeneous or isotropic or both. Some of these solutions depend on constants of motion that are not generated by Killing vectors. We indicate how these solutions may be used to find Einstein-Liouville solutions.     

Inflation in Poincaré gauge gravitation theory

Exponential inflation and power-law inflation are studied in the Poincaré gauge gravitation theory for a Robertson-Walker metric.     

The Invariant-Related Unitary Transformation Method and the Evolution of Quantum Scalar Field in Robertson-Walker Flat spacetime

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is developed and used to study the evolution of a quantum scalar field in Robertson-Walker flat spacetime. We first solve the functional Schrödinger equation for a free scalar field and obtain the exact solutions, of which the 'ground-state' solution possesses isotropy and homogeneity automatically We then investigate the way of extending the method to treat the case in which there is a high-order perturbative self-in teraction.     

Note on equivalence of cosmological space-times

It is shown that only de Sitter space and Minkowski space can be expressed in more than one of the standard forms for Robertson-Walker space-times.     

Causal dissipation in Robertson-Walker cosmological models

Some of the Robertson-Walker cosmological models filled with a fluid with bulk viscosity have been derived which are consistent with causal thermodynamics. The models are discussed briefly.