Petr Hajicek

Gauge invariant Hamiltonian formalism for spherically symmetric gravitating shells

The dynamics of a spherically symmetric thin shell with arbitrary rest mass and surface tension interacting with a central black hole is studied. A careful investigation of all classical solutions reveals that the value of the radius of the shell and of the radial velocity as an initial datum does not determine the motion of the shell; another configuration space must, therefore, be found. A different problem is that the shell Hamiltonians used in the literature are complicated functions of momenta (nonlocal) and they are gauge dependent. To solve these problems, the existence is proved of a gauge-invariant super-Hamiltonian that is quadratic in momenta and that generates the shell equations of motion. The true Hamiltonians are shown to follow from the super-Hamiltonian by a reduction procedure including a choice of gauge and solution of constraint; one important step in the proof is a lemma stating that the true Hamiltonians are uniquely determined (up to a canonical transformation) by the equations of motion of the shell, the value of the total energy of the system, and the choice of time coordinate along the shell. As an example, the Kraus-Wilczek Hamiltonian is rederived from the super-Hamiltonian. The super-Hamiltonian coincides with that of a fictitious particle moving in a fixed two-dimensional Kruskal spacetime under the influence of two effective potentials. The pair consisting of a point of this spacetime and a unit timelike vector at the point, considered as an initial datum, determines a unique motion of the shell.

Time evolution and observables in constrained systems

The investigation of constrained systems is limited to first-class parametrized systems, where the definition of time evolution and observables is not trivial, and to finite-dimensional systems in order that technicalities do not obscure the conceptual framework. The existence of reasonable true, or physical, degrees of freedom is rigorously defined and called local reducibility. A proof is given that any locally reducible system admits a complete set of perennials. For locally reducible systems, the most general construction of time evolution in the Schrodinger and Heisenberg form that uses only the geometry of the phase space is described. The time shifts are not required to be symmetries. A relation between perennials and observables of the Schrodinger or Heisenberg type results: such observables can be identified with certain classes of perennials and the structure of the classes depends on the time evolution. The time evolution between two non-global transversal surfaces is studied. The problem is posed and solved within the framework of ordinary quantum mechanics. The resulting non-unitarity is different from that known in field theory (Hawking effect): state norms need not be preserved so that the system can be lost during this kind of evolution.

SINGULARITY AVOIDANCE BY COLLAPSING SHELLS IN QUANTUM GRAVITY

We discuss a model describing exactly a thin spherically symmetric shell of matter with zero rest mass. We derive the reduced formulation of this system in which the variables are embeddings, their conjugate momenta, and Dirac observables. A nonperturbative quantum theory of this model is then constructed, leading to a unitary dynamics. As a consequence of unitarity, the classical singularity is fully avoided in the quantum theory.

Quantum Theory of Gravitational Collapse (Lecture Notes on Quantum Conchology)

These notes consist of three parts. The first one contains the review of previous work on a gauge-invariant Hamiltonian dynamics of generally covariant models. The method is based on the exclusive use of gauge-invariant variables, the so-called Dirac observables, and on privileged dynamical symmetries such as the asymptotic time translation. The second part applies the method to the model of spherically symmetric thin shell of light-like substance in its own gravitational field following a paper by C. Kiefer and myself. A natural set of Dirac observables is chosen and the Hamiltonian defined by the time translation symmetry is calculated. In the third part, my construction of a version of quantum mechanics for the model is reviewed. The quantum evolution is unitary in spite of the classical theory containing black and white holes and singularities. The wave packet describing the quantum shell contracts, bounces and reexpands. The state of the quantum horizon is a linear combination of the “white” and “black” (past and future) apparent horizons.

Covariant gauge fixing and Kuchař decomposition

The symplectic geometry of a broad class of generally covariant models is studied. The class is restricted so that the gauge group of the models coincides with the Bergmann-Komar group and the analysis can focus on the general covariance. A geometrical definition of gauge fixing at the constraint manifold is given; it is equivalent to a definition of a background (spacetime) manifold for each topological sector of a model. Every gauge fixing defines a decomposition of the constraint manifold into the physical phase space and the space of embeddings of the Cauchy manifold into the background manifold (Kucha\ifmmode \check{r}\else \v{r}\fi{} decomposition). Extensions of every gauge fixing and the associated Kucha\ifmmode \check{r}\else \v{r}\fi{} decomposition to a neighborhood of the constraint manifold are shown to exist.

Transversal Affine Connection and Quantization of Constrained Systems

The Dirac quantization of a finite‐dimensional relativistic system with a quadratic super‐Hamiltonian and linear supermomenta is investigated. In a previous work, the operator constraints were consistently factor‐ordered in such a way that the resulting quantum theory was invariant under all relevant transformations of the classical theory. The method was based on a special choice of coordinates and gauge. Here, coordinate‐independent methods are worked out and a quite general gauge is used. A new mathematical concept, the so‐called ‘‘transversal affine connection,’’ is introduced. This connection is not a linear connection and is associated with a degenerate metric. The corresponding curvature tensor is defined and its components are calculated. The formalism is used to reconstruct the operator constraints, clarify their geometric meaning, and calculate their commutators.

Can outside fields destroy black holes

Stationary, axisymmetric, asymptotically flat space‐times with a black hole surrounded by matter rings, disks, or shells are considered. A certain set of invariant functions, called local invariants, is defined which contains full information about the metric and electromagnetic field in a small electrovacuum neighborhood of the horizon. The local invariants are shown to satisfy an inequality, which is a generalization of the well‐known Kerr‐Newman inequality m2>a2+e2, and which places an upper bound on the gravimagnetic, electric, and magnetic field strengths as measured at the surface of the black hole, independently of whether the fields are produced by the black hole itself or originate in outside sources.

Embedding variables in the canonical theory of gravitating shells

Abstract A thin shell of light-like dust with its own gravitational field is studied in the special case of spherical symmetry. The action functional for this system due to Louko, Whiting, and Friedman is reduced to Kuchař form: the new variables are embeddings, their conjugate momenta, and Dirac observables. The concepts of background manifold and covariant gauge fixing, that underlie these variables, are reformulated in a way that implies the uniqueness and gauge invariance of the background manifold. The reduced dynamics describes motion on this background manifold.

Perennials and the group‐theoretical quantization of a parametrized scalar field on a curved background

The perennial formalism is applied to the real, massive Klein–Gordon field on a globally‐hyperbolic background space–time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras Scan and Sloc of elementary perennials are constructed. The elements of Scan correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of Sloc are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space–time, and by the time shifts that are generated by space–time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly. The constructi...

Quantum superposition principle and gravitational collapse: Scattering times for spherical shells

A quantum theory of spherically symmetric thin shells of null dust and their gravitational field is studied. In Nucl. Phys. B603, 555 (2001), it has been shown how superpositions of quantum states with different geometries can lead to a solution of the singularity problem and black hole information paradox: the shells bounce and re-expand and the evolution is unitary. The corresponding scattering times will be defined in the present paper. To this aim, a spherical mirror of radius R{sub m} is introduced. The classical formula for scattering times of the shell reflected from the mirror is extended to quantum theory. The scattering times and their spreads are calculated. They have a regular limit for R{sub m}{yields}0 and they reveal a resonance at E{sub m}=c{sup 4}R{sub m}/2G. Except for the resonance, they are roughly of the order of the time the light needs to cross the flat space distance between the observer and the mirror. Some ideas are discussed of how the construction of the quantum theory could be changed so that the scattering times become considerably longer.