Showing papers by "Jerzy Lewandowski published in 2018"

Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity

Abstract: Although the physical Hamiltonian operator can be constructed in the deparameterized model of loop quantum gravity coupled to a scalar field, its property is still unknown. This open issue is attacked in this paper by considering an operator $\hat{H}_v$ representing the square of the physical Hamiltonian operator acting nontrivially on two-valent spin networks. The Hilbert space $\mathcal{H}_v$ preserved by the graphing changing operator $\hat{H}_v$ is consist of spin networks with a single two-valent non-degenerate vertex. The matrix element of $\hat{H}_v$ are explicitly worked out in a suitable basis. It turns out that the operator $\hat{H}_v$ is essentially self-adjoint, which implies a well-defined physical Hamiltonian operator in $\mathcal{H}_v$ for the deparameterized model.

Local version of the no-hair theorem

Abstract: In this paper we study nonextremal isolated horizons embeddable in 4-dimensional spacetimes satisfying the vacuum Einstein equations with cosmological constant. The horizons are assumed to be stationary to the second order. The Weyl tensor at the horizon is assumed to be of the Petrov type D. The corresponding equation on the intrinsic horizon geometry is solved in the axisymmetric case. The family of the solutions is 2-dimensional and it is parametrized by the area and the angular momentum. The embeddability in the Kerr-(anti) de Sitter and the near extremal horizon spacetimes obtained by the Horowitz limit from the extremal Kerr-de Sitter and extremal Kerr--anti-de Sitter is discussed. This uniqueness of the axisymmetric type D isolated horizons is a generalization of an earlier similar result valid in the cosmological constant free case.

Axial symmetry of Kerr spacetime without the rigidity theorem

Abstract: Local conditions that imply the no-hair property of black holes are completed. The conditions take the form of constraints on the geometry of the 2-dimensional crossover surface of black hole horizon. They imply also the axial symmetry without the rigidity theorem. This is the new result contained in this paper. The family of the solutions to our constraints is 2-dimensional and can be parametrized by the area and angular momentum. The constraints are induced by our assumption that the horizon is of the Petrov type D. Our result applies to all the bifurcated Killing horizons: inner or outer black hole horizons as well as cosmological horizons. Vacuum spacetimes with a given cosmological constant can be reconstructed from our solutions via Racz's black hole holograph.

The Petrov type D isolated null surfaces

Abstract: Generic black holes in vacuum-de Sitter / Anti-de Sitter spacetimes are studied in quasi-local framework, where the relevant properties are captured in the intrinsic geometry of the null surface (the horizon). Imposing the quasi-local notion of stationarity (null symmetry of the metric up to second order at the horizon only) we perform the complete classification of all the so called special Petrov types of these surfaces defined by the properties (structure of principal null direction) of the Weyl tensor at the surface. The only possible types are: II, D and O. In particular all the geometries of type O are identified. The condition distinguishing type D horizons, taking the form of a second order differential equation on certain complex invariant constructed from the Gaussian curvature and the rotation scalar, is shown to be an integrability condition for the so called near horizon geometry equation. The emergence of the near horizon geometry in this context is equivalent to the hyper-suface orthogonality of both double principal null directions. We further formulate a no-hair theorem for the Petrov type D axisymmetric null surfaces of topologically spherical sections, showing that the space of solutions is uniquely parametrized by the horizon area and angular momentum.

The Petrov type D isolated null surfaces

Abstract: Generic black holes in vacuum-de Sitter / Anti-de Sitter spacetimes are studied in quasi-local framework, where the relevant properties are captured in the intrinsic geometry of the null surface (the horizon). Imposing the quasi-local notion of stationarity (null symmetry of the metric up to second order at the horizon only) we perform the complete classification of all the so called special Petrov types of these surfaces defined by the properties (structure of principal null direction) of the Weyl tensor at the surface. The only possible types are: II, D and O. In particular all the geometries of type O are identified. The condition distinguishing type D horizons, taking the form of a second order differential equation on certain complex invariant constructed from the Gaussian curvature and the rotation scalar, is shown to be an integrability condition for the so called near horizon geometry equation. The emergence of the near horizon geometry in this context is equivalent to the hyper-suface orthogonality of both double principal null directions. We further formulate a no-hair theorem for the Petrov type D axisymmetric null surfaces of topologically spherical sections, showing that the space of solutions is uniquely parametrized by the horizon area and angular momentum.

Spacetime near Kerr isolated horizon

Abstract: The theory of isolated horizon provides a quasi-local framework to study the spacetime geometry in the neighbourhood of the horizon of a black hole in equilibrium without any reference to structures far away from the horizon. While the geometric properties of the Kerr-(A)dS and more general algebraically special solutions have drawn substantial interest recently in the isolated horizon formalism, their horizon metrics have never been written down explicitly in the adapted Bondi-like coordinate system. Following the approach by Krishnan and assuming that the horizon symmetry extends to certain order in the bulk, we present in this note a general method to compute the metric functions order by order radially in Bondi-like coordinates in 4-dimensions from a small set of intrinsic data -- the connection and the Newman-Penrose spin coefficient $\pi$ specified on the horizon cross-section. Applying this general method, we then present the horizon metric of non-extremal Kerr-dS in Bondi-like coordinates. For the pure Kerr case without a cosmological constant, we also show explicitly the metric functions to the first order.

Quantum Reference Frames via Transition Amplitudes in Timeless Quantum Gravity

Abstract: We propose an algorithm of extracting Schrodinger theories under all viable physical time from the Einstein-Hilbert path integral, formulated as the timeless transition amplitudes $\hat{\mathbb{P}}:\mathbb{K} \to \mathbb{K}^*$ between the boundary states in a kinematic Hilbert space $\mathbb{K}$. Each of these Schrodinger theories refers to a certain set of quantum degrees of freedom in $\mathbb{K}$ as a background, with their given values specifying moments of the physical time. Restricted to these specified background values, the relevant elements of $\hat{\mathbb{P}}$ are transformed by the algorithm into the unitary propagator of a corresponding reduced phase space Schrodinger theory. The algorithm embodies the fundamental principle of quantum Cauchy surfaces, such that all the derived Schrodinger theories emerge from one timeless canonical theory defined by $\hat{\mathbb{P}}$ as a rigging map, via the relational Dirac observables referring to the corresponding backgrounds. We demonstrate its application to a FRW loop quantum cosmological model with a massless Klein-Gordon scalar field. Recovering the famous singularity-free quantum gravitational dynamics with the background of the scalar field, we also obtain in another reference frame a modified Klein-Gordon field quantum dynamics with the background of the spatial (quantum) geometry.

Spin-foam model for gravity coupled to massless scalar field

Abstract: A spin-foam model is derived from the canonical model of Loop Quantum Gravity coupled to a massless scalar field. We generalized to the full theory the scheme first proposed in the context of Loop Quantum Cosmology by Ashtekar, Campiglia and Henderson, later developed by Henderson, Rovelli, Vidotto and Wilson-Ewing.