Cosmological principle and primordial spinor fields

An exact solution of Dirac's equation in an open Robertson-Walker universe is constructed such that the corresponding energy-momentum densityT _µv obeys the cosmological principle. However, in anopen universe, this principlemust always be violated by the other physical densities of the cosmological solution (e.g. pseudoscalar, (axial-) current and polarization). If the current densityj _µ = isrequired to obey the cosmological principle, then the universe must beflat.     

Anisotropic Bianchi type V perfect fluid cosmological models in Lyra’s geometry

The law of variation for mean Hubble’s parameter with average scale factor, in an anisotropic Bianchi type V cosmological space–time, is discussed within the frame work of Lyra’s manifold. The variation of Hubble’s parameter, which gives a constant value of deceleration parameter, generates two types of solutions for the average scale factor; one is the power-law and the other one is of exponential form. Using these two forms, new classes of exact solutions of the field equations have been found for a Bianchi type V space–time filled with perfect fluid in Lyra’s geometry by considering a time-dependent displacement field. The physical and kinematical behaviors of the singular and non-singular models of the universe are examined. Exact expressions for look-back time, luminosity distance and event horizon versus redshift are also derived and their significance are discussed in detail. It has been observed that the solutions are compatible with the results of recent observations.     

Singularity formation in general relativistic dynamics of homogeneous scalar fields

Collapsing dynamics of a wide class of self-interacting, self-gravitating homogeneous scalar field models is analyzed. The assumptions made on the potential satisfy some general conditions allowing to show that the generic evolution is divergent in a finite time. Combining results shown here with the ones from [R. Giambó, F. Giannoni, G. Magli, J. Math. Phys. 49 (2008) 042504], dealing with sub-exponential growing potentials, allows us to obtain the same results of singularity formation for more general potentials. Moreover it turns out that these models can be completed to find radiating collapsing star models of the Vaidya type, where blackholes are generically formed.     

Spéculation sur la cosmologie quantique primordiale

An old-fashioned axiomatic formalism for the quantization of the gravitational field is adapted to a recently introduced notion of cosmological gravitational wave. Dynamical equations for the quantum field are proposed. An outline for the primordial cosmological scenario is suggested.

     

Oblique warped products

We define the oblique warped products and prove their existence. In addition to the Levi-Civita connection we use both the Schouten–Van Kampen and Vr?nceanu connections to study the foliation and curvatures of an oblique warped product. As an application to cosmology we introduce the oblique Robertson–Walker spacetime and give its basic properties.     

Imploding-exploding gravitational waves

A solution of Einstein's vacuum field equations is constructed describing an imploding spherical impulsive gravitational wave followed by an exploding similar wave. The two waves propagate in Minkowskian spacetime and the history of the process is the past and future sheets of the null-cone of an event (taken as origin) in the spacetime. The solution is a superposition of two of Penrose's impulsive wave solutions and is described in a single coordinate system in which the metric tensor components are continuous across the histories of the wave fronts.     

Soliton geometry and the vacuum gravitational field equations

A soliton geometry is introduced on manifolds with arbitrary dimensions. The usual soliton connection 1-form defined by Crampin et al. is recovered when the soldering form is a 0-form. It is shown that Einstein's vacuum field equations admit a soliton connection and a soldering 1-form. An associated linear equation with a spectral parameter of Einstein's vacuum field equations are found and some properties of this equation are explored. An example of a Bäcklund transformation is also given.     

Symmetries of the Einstein Hamiltonian

The vacuum Einstein equations for metrics that have two commuting spacelike Killing vector fields are studied from a Hamiltonian point of view using the Ashtekar variables. It is shown that the evolution equations are equivalent to those of a modified SL(2) principal chiral model with a time dependent coupling constant. This fact is used to extract an infinite set of symmetries of the Einstein Hamiltonian via a generalized zero-curvature formulation. These symmetries give evolving observables explicitly on the phase space, and may be viewed as providing an infinite set of solutions of the Hamiltonian Einstein equations. The possibility of quantization using these observables is briefly discussed.     

Matching stationary spacetimes

Using the quasi-Maxwell formalism, we derive the necessary and sufficient conditions for the matching of two stationary spacetimes along a stationary time-like hypersurface, expressed in terms of the gravitational and gravitomagnetic fields and the 2-dimensional matching surface on the space manifold. We prove existence and uniqueness results to the matching problem for stationary perfect fluid spacetimes with spherical, planar, hyperbolic and cylindrical symmetry. Finally, we find an explicit interior for the cylindrical analogue of the NUT spacetime.     

Analytic solutions in the dyon black hole with a cosmic string: Scalar fields,Hawking radiation and energy flux

Charged massive scalar fields are considered in the gravitational and electromagnetic field produced by a dyonic black hole with a cosmic string along its axis of symmetry. Exact solutions of both angular and radial parts of the covariant Klein–Gordon equation in this background are obtained, and are given in terms of the confluent Heun functions. The role of the presence of the cosmic string in these solutions is showed up. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation spectrum and the energy flux.     

Confluent Heun functions and the physics of black holes: Resonant frequencies,Hawking radiation and scattering of scalar waves

We apply the confluent Heun functions to study the resonant frequencies (quasispectrum), the Hawking radiation and the scattering process of scalar waves, in a class of spacetimes, namely, the ones generated by a Kerr–Newman–Kasuya spacetime (dyon black hole) and a Reissner–Nordström black hole surrounded by a magnetic field (Ernst spacetime). In both spacetimes, the solutions for the angular and radial parts of the corresponding Klein–Gordon equations are obtained exactly, for massive and massless fields, respectively. The special cases of Kerr and Schwarzschild black holes are analyzed and the solutions obtained, as well as in the case of a Schwarzschild black hole surrounded by a magnetic field. In all these special situations, the resonant frequencies, Hawking radiation and scattering are studied.     

Exact solutions of the Klein–Gordon equation in the Kerr–Newman background and Hawking radiation

This work considers the influence of the gravitational field produced by a charged and rotating black hole (Kerr–Newman spacetime) on a charged massive scalar field. We obtain exact solutions of both angular and radial parts of the Klein–Gordon equation in this spacetime, which are given in terms of the confluent Heun functions. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation of charged massive scalar particles.     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Killing spinor initial data sets

A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is made using the space-spinor formalism. Conditions on initial data sets for the Einstein vacuum equations are given so that their developments contain solutions to the twistor and/or Killing equations. These lead to the notions of twistor and Killing spinor initial data. These notions are used to obtain a characterisation of initial data sets whose developments are of Petrov type N or D.     

Nonlinear Grassmannσ-models,Toda equations,and self-dual Einstein equations: supplements to previous papers

We review recent results on the relations between classical solutions of nonlinear-models and Toda equations, and point out some relations among Toda equations (two-dimensions), the continuum Toda equation (three-dimensions), and self-dual Einstein equations (four-dimensions).Partially supported by the Grant-in-Aid for Scientific Research, No. 04640088.     

SL(4, R)-invariant chiral fields

Using a method developed before a set of exact solutions of the chiral equations , wheregSL(4,R) are presented.Work supported in part by CONACYT, México.     

Cosmological models with rotation

Nonstationary and stationary cosmological models with rotation and the Bianchi IX metric are constructed within the general relativity theory. A comoving anisotropic liquid and non-comoving “dust-like liquid” are the sources of gravitation in one case, while a comoving anisotropic, non-comoving “dust-like liquid”, and pure radiation - in the other. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 33–37, August, 2008.     

Gravitational waves without gravitons

The radiation connection induced on an isotropic hypersurface of a Lorentz manifold is described. Consequences for the energy tensor in Einstein field equation are analyzed. A cosmological interpretation is proposed.     

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.     

Killing initial data on totally umbilical & compact hypersurfaces

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry non-trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces (Mⁿ,g,cg)$(M^n,g,cg)$ endowed with a Riemannian metric g$g$ and a second fundamental form cg$cg$ (where c∈C^∞(M)$c\in C^{\infty }\left(M\right)$ a priori) have constant mean curvature and are isometric to one of the following manifolds:

(i)

Sⁿ${\mathbb{S}}^n$ the standard sphere,