Einstein,de Sitter and the beginning of relativistic cosmology in 1917

In 1917, both Einstein and de Sitter proposed a new interpretation of the universe as a whole: the structure of the universe could be described in terms of relativistic field equations. Their contributions marked the beginning of the modern scientific comprehension of the origin and evolution of the universe. Our aim is to propose a critical review paper, based on references in primary sources, on the formulation in 1917 of Einstein’s and de Sitter’s models of the universe, which represents a fundamental chapter in the history of relativistic Cosmology.     

The Vaidya solution in higher dimensions

The Vaidya metric representing the gravitational field of a radiating star is generalized to spacetimes of dimensions greater than four.     

Dilaton Black Hole Tunneling Radiation in de Sitter Universe

The Hawking radiation via tunneling from the dilaton black hole in de Sitter universe is investigated using Parikh Wilczek＇s method. We show that if the self-gravitational interaction and energy conservation are taken into account, the modified radiation spectrum deviates from exact thermal spectrum and satisfies the unitary theory.     

Radiating Kerr particle in Einstein universe

A generalized Kerr-NUT type metric is considered in connection with Einstein field equations corresponding to perfect fluid plus a pure radiation field. A general scheme for obtaining the exact solutions of these field equations is developed. Two physically meaningful particular cases are investigated in detail. One gives the field of a radiating Kerr particle embedded in the Einstein universe. The other solution may probably represent a deSitter-like universe pervaded by a pure radiation field.     

Letter: Casimir Stress for Cylindrical Shell in de Sitter Space

The Casimir stress on a cylindrical shell in the de Sitter background for a massless scalar field satisfying Dirichlet boundary conditions on the cylinder is calculated. The metric is written in conformally flat form to make maximal use of the Minkowski space calculations. In the framework of a toy model, we have considered the quantum vacuum effect in the evolution of a domain wall between a cylindrical or in fact circle region around the z axis in which vacuum is _in and the remaining part of space where vacuum is _out .     

Gravitational field of a tachyon in a de Sitter universe

An exact solution of Einstein’s equations is interpreted as describing the gravitational field of a tachyon in a de Sitter universe. Switching off the cosmological constant yields the gravitational field of a tachyon in flat spacetime background.     

Kerr metric in the deSitter background

In addition to the Kerr metric with cosmological constant Λ several other metrics are presented giving a Kerr-like solution of Einstein’s equations in the background of deSitter universe. A new metric of what may be termed as rotating deSitter space-time—a space-time devoid of matter but containing null fluid with twisting null rays, has been presented. This metric reduces to the standard deSitter metric when the twist in the rays vanishes. Kerr metric in this background is the immediate generalization of Schwarzschild’s exterior metric with cosmological constant.     

De Sitter universes and the emerging landscape in string theory

We discuss a recent proposal to construct de Sitter vacua in string theory. It is based on flux compactifications in string theory where all the moduli are stabilised and supersymmetry is broken with control. The resulting picture is that of a complicated landscape with many vacua of widely varying values for the cosmological constant.     

Kerr-Newman metric in deSitter background

In addition to the Kerr-Newman metric with cosmological constant several other metrics are presented giving Kerr-Newman type solutions of Einstein-Maxwell field equations in the background of deSitter universe. The electromagnetic field in all the solutions is assumed to be source-free. A new metric of what may be termed as an electrovac rotating de-Sitter space-time—a space-time devoid of matter but containing source-free electromagnetic field and a null fluid with twisting rays—has been presented. In the absence of the electromagnetic field, our solutions reduce to those discussed by Vaidya.     

Doubly special relativity in de Sitter spacetime

We discuss the generalization of Doubly Special Relativity to a curved de Sitter background. The model has three fundamental observer‐independent scales, the velocity of light c, the de Sitter radius α, and the Planck energy κ, and can be realized through a nonlinear action of the de Sitter group on a noncommutative position space. We consider different choices of coordinates on the de Sitter hyperboloid that, although equivalent, may be more suitable for treating different problems. Also the momentum space can be described as a hyperboloid embedded in a five‐dimensional space, but in this case different choices of coordinates lead to inequivalent models. We investigate the kinematics and the Hamiltonian dynamics of some specific models and describe some of their phenomenological consequences. Finally, we show that it is possible to construct a model exhibiting a duality for the interchange of positions and momenta together with the interchange of α and κ.     

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.     

Warped compactification to de Sitter space

We explore in detail the prospects of obtaining a four-dimensional de Sitter universe in classical supergravity models with warped and time-independent extra dimensions, presenting explicit cosmological solutions of the (4+n)$(4+n)$-dimensional Einstein equations with and without a bulk cosmological constant term. For the first time in the literature we show that there may exist a large class of warped supergravity models with a noncompact extra dimension which lead to a finite 4D Newton constant as well as a massless 4D graviton localized on an inflating four-dimensional FLRW universe. This result helps establish that the ‘no-go’ theorem forbidding acceleration in ‘standard’ compactification of string/M-theory on physically compact spaces should not apply to a general class of warped supergravity models that allows at least one noncompact direction. We present solutions for which the size of the radial dimension takes a constant value in the large volume limit, providing an explicit example of spontaneous compactification.     

Time-dependent fluctuations and particle production in cosmological de Sitter and anti-de Sitter spaces

We study the evolution of time-dependent fluctuations and particle production in an expanding dS and contracting AdS universe. Using the functional Schrödinger formalism we are able to probe the time-dependent regime which is out of the reach of the standard approximations like the Bogolyubov method. In both cases, the evolution of fluctuations is governed by the harmonic oscillator equation with time-dependent frequency. In the case of an expanding dS universe we explicitly show that the frequency of fluctuations produced at a certain moment diminish in time, while the distribution of the created particles quickly approaches the thermal radiation of the dS space. In the case of a contracting AdS universe we show that the frequency of fluctuations produced at a certain moment grow in time. Nominally, the temperature of radiation diverges as the Big Crunch is approaching, however, increasing oscillations of the spectrum make the temperature poorly defined, which is in agreement with the fact that AdS space does not have an event horizon which would cause thermal radiation. Unlimited growth of fluctuations indicates that an eventual tunneling into AdS vacuum would have catastrophic consequences for our universe.     

高维de Sitter时空中宇宙视界处的Fermi子隧穿

运用半经典近似理论,本文研究了来自静态高维de Sitter时空和高维Schwarzschild-de Sitter时空宇宙视界处的Fermi子隧穿辐射.在文中,描述1/2自旋粒子行为的Dirac方程被简化为一个简单的形式,接着运用方程组有非平凡解的条件,可以得到了半经典的Hamilton-Jacobi方程,从而使得问题大大得以简化,最终得到了静态de Sitter时空中宇宙视界处的Fermi子隧穿率和Hawking温度.     

第五讲暗能量和德西特时空

最近的天文观测表明，宇宙是在加速膨胀，而不是原来认为的减速膨胀．为解释加速膨胀，必须在宇宙的物质能量中引入暗能量这一成分，文章讨论了暗能量的可能侯选者，特别强调了宇宙常数问题、德西特时空问题以及和德西特时空相关的一些基本问题．     

Q-Stars in Anti de Sitter Spacetime

We study q-stars with various symmetries in anti de Sitter spacetime in 3+1 dimensions. Comparing with the case of flat spacetime, we find that the value of the field at the center of the soliton is larger when the other parameters show a more complicated behavior. We also investigate their phase space when the symmetry is local and the effect of the charge to its stability.     

高维de Sitter时空中宇宙视界处的Fermi子隧穿

运用半经典近似理论,本文研究了来自静态高维deSitter时空和高维Schwarzschild-de Sitter时空宇宙视界处的Fermi子隧穿辐射.在文中,描述1/2自旋粒子行为的Dirac方程被简化为一个简单的形式,接着运用方程组有非平凡解的条件,可以得到了半经典的Hamilton-Jacobi方程,从而使得问题大大得以简化,最终得到了静态de Sitter时空中宇宙视界处的Fermi子隧穿率和Hawking温度.     

Decay of de Sitter Vacua by Thermal Activation

Decay of a de Sitter vacuum may proceed through a static instanton, representing pair creation of critical bubbles separated by a distance comparable to the Hubble radius—a process somewhat analogous to thermal activation in flat space. We compare this with related processes recently discussed in the literature.     

Renormalized entropies in a de Sitter spacetime

The quantum entropies due to the scalar and Dirac fields are investigated in a pure de Sitter spacetime. The leading divergent terms in both cases are regularized by the Pauli-Villars scheme. It is shown that the explosive entropies can be renormalized according to the Bekenstein-Hawking formula.     

Kerr Black Hole in the Background of the Einstein Universe

The Vaidya-Einstein-Kerr (VEK) black hole which represents the spacetime of the Kerr black hole in a non-vacuum, asymptotically non-flat background is investigated. The energy-momentum tensor corresponding to this spacetime satisfies reasonable energy conditions. We study several properties of this black hole and compare and contrast them with those of the Kerr black hole. We investigate the effect of the background on the geometry of the event horizon by computing the equatorial and polar circumferences and determining the oblateness of the horizon. We find that the surface area of the VEK black hole gets nontrivially coupled to rotation in sharp contrast to the Kerr case. We show that the angular velocity of the VEK horizon goes up significantly as the background influence increases. By using the `equatorial tangential velocity' of the VEK horizon we classify the horizon and define the `limiting black hole' a generalization that contains the extreme Kerr black hole as a special case. Finally we investigate the Gaussian curvature and establish conditions for global embedding of the VEK black hole in Euclidean space.