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1. Chaotic behaviour of gravitational field near cosmological singularity
(oscillatory regime, onset of chaos, statistical description)

One of the main our activity is the study of the problem of influence of inhomogenity on the structure of the general oscillatory regime of gravitational field near cosmological singularity. It was shown that an unavoidable and unlimited development of small-scale exitations in asymptotic vicinity to the singularity make the picture very similar to the highly developed turbulence.

The problems under the current research are: an exact mathematical formulation of the state of gravitational turbulence in terms of the charachteristic functional of the random gravitational field in the vicinity of cosmological singularity and construction of appropriate averaging procedure in order to avoid too detailed description of the complicated small-scale structure of such gravitational turbulent regime.

Another direction of the research is the study of the possible compactification mechanisms in multidimensional inhomogeneous Cosmology.

We developed a detailed study about the influence of different kinds of "matter" on the initial singularity present in the Friedmann-Robertson-Walker, quasi-isotropic and generic inhomogenous solution. In particular we considered the effects of ultrarelativistic matter, particles creation, scalar and vector fields [5].

The most recent results obtained during year 2001-2002 are as follows:

1. It has been presented (G. Imponente and G. Montani) a detailed analysis of the stochastic properties characterizing, near the ''Big-Bang'' the Bianchi type VIII and IX cosmological models. In particular we have shown the time independence of their dynamics, as well as of their invariant measure, the choice of a particular temporal gauge [Ref. 2,3,4]; the achievements of these results is based on an Arnowitt-Deser-Misner hamiltonian approach, as written in terms of Misner-Chitrè-like variables. In connection with these subjects, it was provided a straightforward relation between the classical chaoticity of these two Bianchi models and their indeterministic quantum behavior, performed during the Planckian era; The physical ground of this result is based on a WKB approximation, able to reduce the semiclassical quantum probability distribution to the classical microcanonical one [Ref. 5,6]. For a general review on these investigations, see [Ref 7].
2. On the base of a standard hamiltonian approach, developed adopting Misner-Chitrè-like variables, is constructed (G. Montani) a solution to the continuity equation for the Bianchi VIII and IX cosmological models, as written in the asymptotic limit to the initial "Big-Bang''; the knowledge of this solutions provides the asymptotic nonstationary corrections to the invariant measure for the system, which result to decay exponentially [Ref. 8].

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2. Influence of the scalar fields on the cosmological evolution 

By the investigation of a ''generic inhomogeneous'' cosmological solution in the presence of a real self-interaction scalar field (expected responsible for a spontaneous symmetry breaking configuration), it is constructed (A.A. Kirillov and G. Montani) an interpolating "generic'' solution, able to connect a "generic'' Kasner-like regime with an inflationary scenario. This result has the merit to individualize a mechanism for the quasi-isotropization of our universe, i.e. it allows to understand how it is possible to put in the same dynamical picture the necessity of a "generic Big-Bang'' (we recall that the FRW model is unstable backward in time and should admit a spectrum of initial perturbations) and the high homogeneity and isotropy required by the Standard Cosmological Model [Ref. 10].

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3.  Macroscopic gravity
(development of the theory of the smeared gravitational and matter fields)

We developed a study of the gravitational polarization phenomenon in the limit of a perturbative quadratic theory based on a joint Isaacson-Szekeres approach. In particular we individualized a material relation connecting the Isaacson energy-momentum tensor with the trace-free component of the quadrupole one.

Results during period 2000-2002 regard are based as follows.

A study of the polarization phenomenon, based on a macroscopic gravity theory has been developed (G. Montani, R. Ruffini and R Zalaletdinov), in a perturbative approach, to characterize the correlations existing between the matter quadrupole and the high frequency radiation terms, both appearing on a second order averaged expansion (the first order propagative effects vanish on average). The main result of this work is to be regarded as the individualization of the appropriate "material relations'' for the background dynamics [Ref. 1].

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4. Quantum effects in external gravitational field and Quantum Gravity

1. On the base of criticism to the nature of the Wheeler-Dewitt approach, it is provided (G. Montani) a reformulation of the quantum geometrodynamics within the (3+1)-slicing representation of the space-time. The fundamental statement of this analysis is the necessity to include the so-called "kinematical action'' even in the case of a quantum space-time, so getting a completely consistent formulation for a "gauge-fixing'' quantization of the 3-geometries [Ref. 9].
2. In order to obtain a vacuum theory of the torsion field of the second order, as that one of any other field (including the space-time geometry), it is proposed (G. Aprea, G. Montani, R. Ruffini) a Lagrangian formulation for the space-time geometry, characterized by a non-riemannian connection (we retain only its totally antisymmetric and trace components) which is expressed by a scalar and a tensor potential. The obtained field equations predicts the existence of torsion waves, having interesting implications on the motion of the test particles, as described by the self-parallel lines [Ref. 11].
3. By applying, in the line of the idea proposed by I.Prigogine, the theory of open thermodynamical systems to an expanding isotropic universe, is provided (G.Montani) a phenomenolgical approach to the particles creation mechanisms in the very early cosmology; the main result of this work consists in showing how the horizon paradox finds a natural solution upon taking into account the matter creation near the big Bang [Ref. 12].
4. It was derived (D.Oriti, R.M. Williams) the the Barrett-Crane spin foam model for Euclidean 4-dimensional quantum gravity from a discretized BF theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown [Ref. 13].
5. E.R. Livine and D.Oriti study a generalized action for gravity as a constrained BF theory, and its relationship with the Plebanski action. We analyse the discretization of the constraints and the spin foam quantization of the theory, showing that it leads naturally to the Barrett-Crane spin foam model for quantum gravity. Our analysis holds true in both the Euclidean and Lorentzian formulation [Ref. 14].
6. D.Oriti made an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice gauge theory, matrix models, category theory, statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian case. We focus in particular on the Barrett-Crane model for 4-dimensional quantum gravity [Ref. 15].
7. D.Oriti extended the lattice gauge theory-type derivation of the Barrett-Crane spin foam model for quantum gravity to other choices of boundary conditions, resulting in different boundary terms, and re-analyze the gluing of 4-simplices in this context. This provides a consistency check of the previous derivation. Moreover we study and discuss some possible alternatives and variations that can be made to it, and the problems they present [Ref. 16].

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5.  Quantum Field Theory in accelerated frames

By considering quantum theory of the free field in Minkowski and Rindler spacetimes we show that conventional derivation of the Unruh effect is not correct since boundary conditions for the fields in these spacetimes are different. We also show that algebraic approach to this problem leads to the same conclusion. These results are important because there are serious arguments to think that the Unruh effect is closely related to the effect of quantum evaporation of black holes.

Results during period 2000-2002 regard are based as follows.

It was accomplished the final analysis of the so called Unruh effect both from the point of view of canonical and algebraic approach to the quantum field theory (N.B. Narozhny, A.M. Fedotov, B.M. Karnakov, V.D. Mur and V.A. Belinski). It was shown that the quantization procedure proposed by Unruh implies setting a boundary condition for the quantum field operator and this changes drastically the topological properties and symmetry group of the spacetime which lead to the field theory in two disconnected left and right Rindler spacetimes instead of Minkowski spacetime. Thus in spite of the work over last 25 years, there still remain serious gaps in grounding of the Unruh effect, and as of now there is no compelling evidence for the universal behaviour attributrd to all uniformly accelerated detectors [Ref. 17].

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6. Collapse of stellar clusters and mass run away effect

1. It was investigated a model of ballistic ejection effect of matter from spherically symmetric stellar clusters (M. V. Barkov, V.A. Belinski and G.S. Bisnovatyi-Kogan). The problem was solved in newtonian gravity but with cutoff fixing the minimal radius of selfgravitating matter shell by its relativistic gravitational radus. It was shown that during the motion of two initially gravitationally bound spherical shells, consisting of point particles moving along ballistic trajectories, one of the shell may be expelled to infinity at subrelativistic expelling velocity of the order of 0,25c. Also it was shown that the motion of two intersecting shells in the case when they do not runaway shows a chaotic behaviour [Ref. 18].
2. It was found (M. V. Barkov, V.A. Belinski and G.S. Bisnovatyi-Kogan) the complete exact solution in the General Relativity for the intersection process of two massive selfgravitating spherically symmetric shells (in general with tangential pressure). It was shown how one can calculate all shell’s parameters after intersection in terms of the parameters before the intersection. The result is quite new, the solution of this kind was known only for the massless shells (Dray and t’Hooft, 1985). The solution was applied to the analysis of matter ejection effect from relativistic stellar clusters and to the chaotic motion of the shells in relativistic regime. It was shown that in relativistic case the matter ejection effect is stronger than in newtonian gravity [Ref. 19].

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7. Gravitational and Electromagnetogravitational solitons

This  research line is the further development of the theory of gravitational solitons. Here the research is going in the following three drections. First, it was already established the existence of the gravisolitonic topological charge. However the exact mathematical formulation of this phenomenon (especially the exact expression for the topological current) is still remain to be seen. Second, the possibility of the quantum creation of gravisolitons near the cosmological singularity is under investigation. Third, we are intending to construct the well defined energetics of gravisolitons.

Results during period 2000-2002 regard are based as follows.

1. It was accomplished the final version of the book "Gravitational Solitons" (V. Belinski and E. Verdaguer). Here is a self-contained exposition of the theory of gravitational solitons and provides a comprehensive review of exact soliton solutions to Einstein's equations. The text begins with a detailed discussion of the extension of the Inverse Scattering Method to the theory of gravitation, starting with pure gravity and then extending it to the coupling of gravity with the electromagnetic field. There follows a systematic review of the gravitational soliton solutions based on their symmetries. These solutions include some of the most interesting in gravitational physics such as those describing inhomogeneous cosmological models, cylindrical waves, the collision of exact gravity waves,and the Schwarzschild and Kerr black holes [Ref. 20].
2. The Alexeev approach for the construction of Electromagnetogravisolitons was elaborated and translated to the conventional Belinski-Zakharov method. The results was included as one of the chapter to the book "Gravitational Solitons" [Ref. 20].

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SELECTED PUBLICATIONS (1970-2000):

1. V.Belinski, I.Khalatnikov and E.Lifshitz: "Oscillatory Approach to a Singular Point in the Relativistic Cosmology", Advances in Physics, 19, 525, 1970;
2. V.Belinski, I.Khalatnikov and E.Lifshitz: "A General Solution of the Einstein Equations with a Time Singularity", Advances in Physics, 31, 639, 1982;
3. V.Belinski and V.Zakharov: "Integration of the Einstein Equations by means of the inverse scattering problem technique and construction of exact soliton solutions", Sov.Phys. JETP, 48, 985, 1978;
4. V.Belinski and V.Zakharov: "Stationary gravitational solitons with axial symmetry", Sov.Phys. JETP, 50, 1, 1979;
5. V.Belinski: "Gravitational breather and topological properties of gravisolitons" Phys. Rev. D, 44, 3109, 1991;
6. V.Belinski and R.Ruffini: "Radiation from a relativistic Magnetized Star", Astrophys.Journal Letters, 401, L27, 1992;
7. V.Belinski, F. de Paolis, H.W.Lee and R.Ruffini: "Radiation from a Relativistic Rotanting Magnetic Dipole. Magnetic Synchroton Effect", Astron.Astrophys., 283, 1018, 1994;
8. V.Belinski: "On the development of gravitational turbolence near the cosmological singularity", JETP Letters, 56, 421, 1982;
9. V.Belinski and G.Montani: "A Scenario of Dimensional Compactification near the Cosmological Singularity", preprint I.C.R.A., July 1993;
10. V.Belinski: "On the existence of black hole evaporation", 1) Phys.Lett.A, submitted, 1994; 2) The talk given at Proc.of the seventh Marcel Grossmann Meeting, Stanford, USA, 1994; 3) The short version submitted to the Proc.of the seventh Marcel Grossmann Meeting, Stanford, USA, 1994;
11. G.Montani: "On the General Behaviour of the Universe Near Cosmological Singularity", Classical and Quantum Gravity, 12, 2503, (1995).
12. V.Belinski, B.M.Karnakov, V.D. Mur, N.B. Narozhnyi:"Does the Unruh effect exist?", JETP Letters, 65, 902 (1997).
13. G. Montani "On the singularity problem in cosmology", Nuovo Cimento, 112B, 459, (1997)
14. A. Kirillov and G. Montani "Description of statistical properties of the mixmaster universe", Phys. Rev. D56, 6225, (1997)
15. A. Kirillov and G. Montani "Origin of a classical space in quantum inhomogeneous models", JETP Lett. 66, 475, (1997)
16. G. Montani "On the quasi-isotropic solution in the presence of ultrarelativistic matter and a scalar field", Classical and Quantum Gravity, 16, 723, (1999)
17. A.Fedotov, V.Mur, N. Narozhny, V. Belinski, B. Karnakov, "Quantum field aspect of the Unruh problem", Phys. Lett. A254, 126, (1999)
18. N. Narozhny, A. Fedotov, B. Karnakov, V. Mur and V. Belinski, "Boundary conditions in the Unruh problem", hep-th/9906181, (1999)
19. Bini D., Gemelli G., Ruffini R., Spinning test particles in general relativity: nongeodesic motion in the Reissner-Nordstr"om spacetime, Physical Review D, vol. 61, 064013, 2000.
20. Bini D., Jantzen R.T., Circular orbits in Kerr spacetime: equatorial plane embedding diagrams, Classical and Quantum Gravity, vol 17, 1-11, 2000.
21. Bini D., de Felice F., Gyroscopes and gravitational waves, Classical and Quantum Gravity, vol 17, 4627-4635, 2000.
22. Bini D., Jantzen R.T. Gravitoelectromagnetism: a tool for observer-dependent interpretation of spacetime physics,. Il Nuovo Cimento, vol 115B, n. 070809, Luglio-Settembre 2000, pag.713.
23. Bini D., de Felice F. Gyroscopes and Gravitational Waves, in Gravitational Waves, Ed. by I. Ciufolini, V. Gorini, U. Moschella, P. Fre, IOP, Pub., 2000. Cap. 15, pag 268-279.

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1. G. Montani, R. Ruffini and R.M. Zalaletdinov "Gravitating macroscopic media in general relativity and 2001, 115 B, N. 11, 1343.
2. G.Imponente and G. Montani, " Covariance of the mixmaster chaoticity", Physical Review D., 2001, 63,103501.
3. G.Imponente and G. Montani, "Covariant Formulation of the Invariant Measure for the Mixmaster Dynamics", submitted to Physics Letters A, on October 2001
4. G.Imponente and G. Montani, "Covariant Mixmaster Dynamics", in "Similarities and Universality in Relativistic Flows", Ed. by Logos Verlag, Berlin (2001).
5. G.Imponente and G. Montani, " On the Quantum Origin of the Mixmaster Chaos Covariance", 2002, Nuclear Phys. B Proc. Suppl. 104, 193-196
6. G.Imponente and G. Montani, "Mixmaster Chaoticity as Semiclassical Limit of the Canonical Quantum Dynamics", submitted to Classical and Quantum Gravity, on December 2001
7. G.Imponente and G.Montani, "Mixmaster Chaos and Quantum Aspects of the Statistical Probability Distribution", 2001, International Journal of the Korean Physical Society, in press.
8. G. Montani, "Nonstationary correction to the mixmaster model invariant measure'', accepted by Nuovo Cimento B, January 2002, in press.
9. G. Montani, "Canonical quantization of gravity without "frozen formalism'', submitted to Nuclear Physics B, January 2002.
10. A.A. Kirillov and G. Montani, "Quasi-isotropization of the inhomogeneous mixmaster Universe induced by an inflationary process'', submitted to Physics Review D, January 2002.
11. G. Aprea, G. Montani and R. Ruffini, "Lagrangian formulation of a geometrical theory with torsion'', in preparation.
12. G. Montani, "Influence of the particle creation on the flat and negative curved FLRW universes" Class.and Quantum Grav., 2001, 18, 193.
13. D.Oriti, R.M. Williams, "Gluing 4-simplices: a derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity", Phys. Rev. D 63, 024022 (2001); gr-qc/0010031;
14. E.R. Livine, D. Oriti, "Barrett-Crane spin foam model from generalized BF-type action for gravity'', to appear in Phys. Rev. D; gr-qc/0104043
15. D. Oriti, "Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity'', Rep. Prog. Phys. 64, 1489, (2001), gr-qc/0106091;
16. D. Oriti, "Boundary terms in the Barrett-Crane spin foam model and consistent gluing'', submitted for publication; gr-qc/0201xxx (to appear in the e-archives in these days)
17. N.B. Narozhny, A.M. Fedotov, B.M. Karnakov, V.D. Mur and V.A. Belinski, "Boundary conditions in the Unruh problem", Phys. Rev. D65, 025004, 2002.
18. M. V. Barkov, V.A. Belinski and G.S. Bisnovatyi-Kogan, "Chaotic motion and ballistic ejection of gravitating shells", astro-ph/0107051, submitted to "Monthly Notices".
19. M. V. Barkov, V.A. Belinski and G.S. Bisnovatyi-Kogan, "The exact solution in General Relativity for the motion and intersections of the selfgravitating shells in the external field of the massive black hole", submitted to Sov. Phys. JETP.
20. V. Belinski and E. Verdaguer, "Gravitational Solitons", Cambridge University Press, Cambridge Monographs on Mathematical Physics, 2001.

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