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Register a new userComplete stationary spacelike surfaces in an $n$-dimensional Generalized Robertson-Walker spacetime
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Several uniqueness results for non-compact complete stationary spacelike surfaces in an $n(geq 3)$-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz-Minkowski spacetime are exposed to show that each of our assumptions is needed.
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Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimotos theorem to complete space-like stationary surfaces in Minkowski spacetime, but also estimate the upper bound of the number of exceptional values when the Gauss image lies in the graph of a rational function f of degree m, showing a sharp contrast to Bernstein type results for minimal surfaces in 4-dimensional Euclidean space. Moreover, we introduce the conception of conjugate similarity on the special linear group to classify all degenerate stationary surfaces (i.e. m=0 or 1), and establish several structure theorems for complete stationary graphs in Minkowski spacetime from the viewpoint of the degeneracy of Gauss maps.
The integral of the energy density function $mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $Lambda$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this functional, subject to volume-preserving variations (critical RW spacetimes). A complete classification of critical RW spacetimes is given and explicit solutions in terms of Weierstrass elliptic functions and their degenerate forms are computed. The standard energy conditions (weak, dominant, and strong) as well as the cyclic property of critical RW spacetimes are discussed.
In a recent paper [arXiv:1206.4916] by T. Padmanabhan, it was argued that our universe provides an ideal setup to stress the issue that cosmic space is emergent as cosmic time progresses and that the expansion of the universe is due to the difference between the number of degrees of freedom on a holographic surface and the one in the emerged bulk. In this note following this proposal we obtain the Friedmann equation of a higher dimensional Friedmann-Robertson-Walker universe. By properly modifying the volume increase and the number of degrees of freedom on the holographic surface from the entropy formulas of black hole in the Gauss-Bonnet gravity and more general Lovelock gravity, we also get corresponding dynamical equations of the universe in those gravity theories.
The equation of the spin-$frac{1}{2}$ particles in the Friedmann-Lema^itre-Robertson-Walker spacetime is investigated. The retarded and advanced fundamental solutions to the Dirac operator and generalized Dirac operator as well as the fundamental solutions to the Cauchy problem are written in explicit form via the fundamental solution of the wave equation in the Minkowski spacetime.
In this paper we study the effects of quantum scalar field vacuum fluctuations on scalar test particles in an analog model for the Friedmann-Robertson-Walker spatially flat geometry. In this scenario, the cases with one and two perfectly reflecting plane boundaries are considered as well the case without boundary. We find that the particles can undergo Brownian motion with a nonzero mean squared velocity induced by the quantum vacuum fluctuations due to the time dependent background and the presence of the boundaries. Typical singularities which appears due to the presence of the boundaries in flat spacetime can be naturally regularized for an asymptotically bounded expanding scale function. Thus, shifts in the velocity could be, at least in principle, detectable experimentally. The possibility to implement this observation in an analog cosmological model by the use of a Bose-Einstein condensate is also discussed.
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