Ewa Czuchry

Dynamics of a self-gravitating lightlike matter shell: A gauge-invariant Lagrangian and Hamiltonian description

A complete Lagrangian and Hamiltonian description of the theory of self-gravitating lightlike matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. The energymomentum tensor density of a lightlike matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. The Noether identity and Belinfante-Rosenfeld theorem for such a tensor density are proved. Finally, the Hamiltonian dynamics of the interacting ‘‘gravity1matter’’ system is derived from the total Lagrangian, the latter being an invariant scalar density.

Geometry of null-like surfaces in general relativity and its application to dynamics of gravitating matter

Abstract Geometric tools describing the structure of a null-like surface S (wave front) are constructed. They are applied to the analysis of interaction between a light-like matter shell and the surrounding gravitational field. It is proved that the Einstein tensor Gab describing such a situation may be written in terms of external curvature of S. Conservation laws (Bianchi identities) for G are proved. Also geometry of isolated horizons (surfaces surrounding black holes) is analyzed in terms of the constructed tools. The possibility of application of these results to the problem of motion of isolated objects in general relativity is discussed.

Singularity avoidance in a quantum model of the Mixmaster universe

We present a quantum model of the vacuum Bianchi-IX dynamics. It is based on four main elements. First, we use a compound quantization procedure: an affine coherent state quantization for isotropic variables and a Weyl quantization for anisotropic ones. Second, inspired by standard approaches in molecular physics, we make an adiabatic approximation (Born-Oppenheimer-like approximation). Third, we expand the anisotropy potential about its minimum in order to deal with its harmonic approximation. Fourth, we develop an analytical treatment on the semiclassical level. The resolution of the classical singularity occurs due to a repulsive potential generated by the affine quantization. This procedure shows that during contraction the quantum energy of anisotropic degrees of freedom grows much slower than the classical one. Furthermore, far from the quantum bounce, the classical recollapse is reproduced. Our treatment is put in the general context of methods of molecular physics, which can include both adiabatic and nonadiabatic approximations.

Smooth quantum dynamics of the mixmaster universe

We present a quantum version of the vacuum Bianchi IX model by implementing ane coherent state quantization combined with a Born-Oppenheimer-like adiabatic approximation. The analytical treatment is carried out on both quantum and semiclassical levels. The resolution of the classical singularity occurs by means of a repulsive potential generated by our quantization procedure. The quantization of the oscillatory degrees of freedom produces a radiation energy density term in the semiclassical constraint equation. The Friedmann-like lowest energy eigenstates of the system are found to be dynamically stable.

Vibronic framework for quantum mixmaster universe

Following our previous papers concerning the quantization of the vacuum Bianchi-IX model within or beyond the Born-Oppenheimer and adiabatic approximation, we develop a more elaborate analysis of the dynamical properties of the model based on the vibronic approach utilized in molecular physics. As in the previous papers, we restrict our approach to the harmonic approximation of the anisotropy potential in order to obtain resoluble analytical expressions.

Dynamics of a self-gravitating shell of matter

The dynamics of a self-gravitating shell of matter is derived from the Hilbert variational principle and then described as an (infinite-dimensional, constrained) Hamiltonian system. The method used here enables us to define a singular Riemann tensor of a noncontinuous connection via standard formulas of differential geometry, with derivatives understood in the sense of distributions. Bianchi identities for the singular curvature are proved. They match the conservation laws for the singular energy-momentum tensor of matter. The Rosenfed-Belinfante and Noether theorems are proved to be valid still in the case of these singular objects. The assumption about the continuity of the four-dimensional space-time metric is widely discussed.

Bianchi IX model: Reducing phase space

The mathematical structure of higher-dimensional physical phase spaces of the nondiagonal Bianchi IX model is analyzed in the neighborhood of the cosmological singularity by using dynamical system methods. Critical points of the Hamiltonian equations appear at infinities and are of a nonhyperbolic type, which is a generic feature of the considered singular dynamics. The reduction of the kinematical symplectic 2-form to the constraint surface enables the determination of the physical Hamiltonian. This procedure lowers the dimensionality of the dynamics arena. The presented analysis of the phase space is based on canonical transformations. We test our method for the specific subspace of the physical phase space. The obtained results encourage further examination of the dynamics within our approach.

Spectral properties of the quantum Mixmaster universe

We study the spectral properties of the anisotropic part of Hamiltonian entering the quantum dynamics of the Mixmaster universe. We derive the explicit asymptotic expressions for the energy spectrum in the limit of large and small volumes of the universe. Then we study the threshold condition between both regimes. Finally we prove that the spectrum is purely discrete for any volume of the universe. Our results validate and improve the known approximations to the anisotropy potential. They should be useful for any approach to the quantization of the Mixmaster universe.

Coherent States Quantization and Affine Symmetry in Quantum Models of Gravitational Singularities

We employ the framework of affine covariant quantization and associated semiclassical portrait to address two main issues in the domain of quantum gravitational systems: (i) the fate of singularities and (ii) the lack of external time. Our discussion is based on finite-dimensional, symmetry-reduced cosmological models. We show that the affine quantization of the cosmological dynamics removes the classical singularity and univocally establishes a unitary evolution. The semiclassical portrait based on the affine coherent states exhibits a big bounce replacing the big-bang singularity. As a particularly interesting application, we derive and study a unitary quantum dynamics of the spatially homogenous, closed model, the Mixmaster universe. At the classical level it undergoes an infinite number of oscillations before collapsing into a big-crunch singularity. At the quantum level the singularity is shown to be replaced by adiabatic and nonadiabatic bounces. As another application, we consider the problem of time. We derive semiclassical portraits of quantum dynamics of the Friedman universe with respect to various internal degrees of freedom. Next we compare them and discuss the nature of quantum evolution of the gravitational field.

Multipole moments in Kaluza-Klein theories

The problem of the motion of extended, i.e. non-point, test bodies in multidimensional space is discussed. Extended bodies are described in terms of so-called multipole moments. Using an approximate form of the equations of motion for extended bodies, deviation from geodesic motion is derived. The results are applied to a special form of spacetime.