arXiv subjects

Jerzy Lewandowski

Publications and source records attributed to Jerzy Lewandowski.

At least 19 recordsLinked to original sources

arXiv2026

Canonical Clock Sectors and Relational Frame Equivalence in Brans--Dicke Theory

We investigate the equivalence of the Jordan and Einstein frames of the Brans--Dicke theory before and after relational deparametrization of the constrained Hamiltonian system. Although the two conformal formulations are equivalent at the covariant level and related by a canonical transformation on the standard ADM phase space, their equivalence after reduction with respect to an internal clock is nontrivial. Using the Brans--Dicke scalar as a relational clock, we show that the apparent discrepancy between the reduced Hamiltonians does not indicate a physical inequivalence of the two frames. Instead, it arises from a mismatch in the canonical embedding of the clock sector prior to reduction. While the scalar configuration variable is preserved under the conformal transformation, its conjugate momentum is shifted by a contribution involving the gravitational momentum trace. Consequently, relational dynamics are determined not by the clock variable $T$ alone, but by the complete canonical clock pair $(T,P_T)$. We construct a frame-adapted canonical chart in which the clock sector is consistently transformed prior to deparameterization. The resulting reduced Hamiltonian coincides with that of the Einstein frame, restoring the equivalence of the reduced relational dynamics. Our results identify the canonical clock sector as the fundamental structure governing relational evolution in Brans--Dicke theory and provide a general framework for understanding frame dependence in scalar--tensor gravity and reduced phase-space quantization.

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arXiv2025

Extremal isolated horizons of the NUT type

We provide a construction of a new class of axisymmetric extremal isolated horizons admitting a structure of U(1)-principal fiber bundle over a two-sphere. In contrast to the previous examples, the null generators are assumed to be transversal to the bundle fibers. We impose the Einstein equations at the horizon and explicitly derive all intrinsic geometries of the extremal horizon, consisting of a two-sphere metric and a rotation 1-form, in the above class. The 2-geometries turn out to be equivalent to the classification of conically singular horizons with product topology. Both the rotating and non-rotating horizons are then embedded in the Pleba\'nski-Demia\'nski spacetimes, which naturally admit horizons of this type. Furthermore, we compare our results with previously obtained solutions to the Einstein vacuum extremal horizon equation with cosmological constant and the solution of Petrov type D equation with transversal bundle structure.

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arXiv2024

Black holes and covariance in effective quantum gravity: A solution without Cauchy horizons

The issue of general covariance in effective quantum gravity models within the Hamiltonian framework is addressed. The previously proposed equations for the covariance condition in spherically symmetric models are explicitly derived. By solving this equation, a new effective Hamiltonian constraint is obtained, incorporating free functions that can account for quantum gravity effects. The resulting spacetime structure is analyzed by specifying the free functions. Remarkably, in this model, the classical singularity is replaced by a region where the metric asymptotically approaches a Schwarzschild-de Sitter one with negative mass. Thus, this new quantum-corrected black hole model avoids the Cauchy horizons presented typically in previously studied models. The covariant approach is also applicable to matter coupling in the models.

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arXiv2024

Quadrupole formulae with cosmological constant: comparison

We consider three different approaches (by Ashtekar, Bonga and Kesavan; Hoque and Virmani; and Dobkowski-Ry\l{}ko and Lewandowski) to investigate gravitational radiation produced by time changing matter source in de Sitter spacetime. All of them lead to generalizations of the quadrupole formula, however, due to different gauge conditions and choices of the hypersurfaces, across which the energy flux is computed, it is nontrivial to see that they all coincide, as one would expect from the symplectic theory. Each of the expressions for the radiated energy in the form of gravitational waves is expressed in terms of the mass and pressure quadrupole moments and written explicitly up to the linear order in $\sqrt{\Lambda}$, or equvalently in Hubble parameter $H$. It is shown that up to the first order all three of the generalizations of the quadrupole formula agree.

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arXiv2024

Black Holes and Covariance in Effective Quantum Gravity

The longstanding issue of general covariance in effective models of quantum gravity is addressed, which arises when canonical quantum gravity leads to a semiclassical model described by an effective Hamiltonian constraint. In the context of spherically symmetric models, general covariance is precisely formulated into a set of equations, leading to the necessary and sufficient conditions for ensuring covariance. With the aid of these conditions, we derive the equations for the effective Hamiltonian constraint. The equations yield two candidates for effective Hamiltonian constraints dependent on a quantum parameter. The resulting quantum modified black hole spacetimes are analyzed. Our models show improvement by casting off the known limitations of previous works with similar results.

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arXiv2024

Extreme horizon equation

Extremal horizons satisfy an equation induced by the Einstein vacuum equations that determines the shape of the horizon and the manner in which it rotates (the EEH equation). Until recently, however, the classification of solutions required the assumption of axial symmetry. Recently, there has been a breakthrough: Dunajski and Lucietti proved that every non-static solution possesses a one-dimensional symmetry group. The first part of our work is inspired by this result. An identity satisfied by the solutions of the EEH equation has been distilled (Master Identity), which is crucial for studying their properties. It is a bit stronger than the original Dunajski-Lucietti identity and leads directly to the rigidity theorem for any value of the cosmological constant. Master Identity is used for a simple derivation of the local form of the general static solution of the EEH equation with non-positive cosmological constant. All the globally defined compact static solutions are derived. Thus the list of solutions given in the literature is completed. In the two-dimensional case (which corresponds to horizons in four-dimensional spacetime), the Einstein-Maxwell equations of an extremal horizon (EMEH) and the equations of quasi-Einstein spaces are studied. The general solution on a compact surface with non-zero genus is derived. In the case of zero genus, the static solutions are investigated and their axial symmetry is proven. Together with the new results on non-static solutions on sphere of Colling, Katona and Lucietti that leads to the uniqueness of the Reissner-Nordstr\"om-(Anti)de-Sitter extremal horizons. Interestingly, the static rigidity result is also valid for non-compact spaces with a zero first cohomology group.

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arXiv2024

Vacuum Petrov type D horizons of non-trivial $U(1)$ bundle structure over Riemann surfaces with genus $> 0$

We consider isolated horizons (Killing horizons up to the second order) whose null flow has the structure of a U(1) principal fiber bundle over a compact Riemann surface. We impose the vacuum Einstein equations (with the cosmological constant) and the condition that the spacetime Weyl tensor is of Petrov D type on the geometry of the horizons. We derive all the solutions in the case when the genus of the surface is $>1$. By doing so for all the non-trivial bundles, we complete the classification. We construct the embedding spacetimes and show that they are locally isometric to the toroidal or hyperbolic generalization of the Taub-NUT-(anti-) de Sitter spacetimes for horizons of genus $1$ or $>1$ respectively, after performing Misner's identification of the spacetime. The horizon bundle structure can be naturally extended to bundle structure defined on the entire spacetime.

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arXiv2024

Status of Birkhoff's theorem in polymerized semiclassical regime of Loop Quantum Gravity

The collapse of a spherically symmetric ball of dust has been intensively studied in Loop Quantum Gravity (LQG). From a quantum theory, it is possible to recover a semiclassical regime through a polymerization procedure. In this setting, general solutions to the polymerized Einstein field equations (PEFE) will be discussed both for the interior and the exterior of the dust cloud. Exterior solutions are particularly interesting since they may lead to a semiclassical version of the Birkhoff's theorem. It is seen that if time independence of the vacuum is imposed, there exists a unique class of solutions depending on two parameters. Nevertheless, the possibility of more intricate time dependent solutions is not ruled out completely. Ultimately, these results will be compared to a model of spherical collapse obtained independently from the Einstein equations.

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arXiv2023

Isolated horizons of the Hopf bundle structure transversal to the null direction, the horizon equations and embeddability in NUT-like spacetimes

Isolated horizons that admit the Hopf bundle structure $H\rightarrow S_2$ are investigated, however the null direction is allowed not to be tangent to the bundle fibres. The geometry of such horizons is characterised by data set on a topological two-dimensional sphere, singular at its poles. The horizon equations induced by Einstein's equations are imposed. The existence of regular extremal horizons satisfying the vacuum (with cosmological constant) equation of extremality, obtained from singular solutions on the sphere is pointed out. All horizons (with assumed topology and in the generic case) satisfying the $\Lambda$ vacuum type D equation are derived. They are compared to the Killing horizons contained in the accelerated Kerr-NUT-(Anti) de Sitter spacetimes. Both families of horizons have the same dimension, but the problem of mutual correspondence needs to be better understood. If the cosmological constant takes special values determined by the other parameters the bundle fibers become tangent to the null direction. As an additional but also important result, spacetimes of the topology $H\times \mathbb{R}$ locally isometric to the accelerated Kerr-NUT-(Anti) de Sitter spacetimes are constructed for every value of the mass, Kerr, NUT parameters, the cosmological constant and the acceleration. When the acceleration parameter is not zero, the conical singularity can be removed whenever the NUT parameter does not vanish either.

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arXiv2023

Symplectic charges in the Yang-Mills theory of the normal conformal Cartan connection: applications to gravity

It is known that a source-free Yang-Mills theory with the normal conformal Cartan connection used as the gauge potential gives rise to equations of motion equivalent to the vanishing of the Bach tensor. We investigate the conformally invariant presymplectic potential current obtained from this theory and find that on the solutions to the Einstein field equations, it can be decomposed into a topological term derived from the Euler density and a part proportional to the potential of the standard Einstein-Hilbert Lagrangian. The pullback of our potential to the asymptotic boundary of asymptotically de Sitter spacetimes turns out to coincide with the current obtained from the holographically renormalized gravitational action. This provides an alternative derivation of a symplectic structure on scri without resorting to holographic techniques. We also calculate our current at the null infinity of asymptotically flat spacetimes and in particular show that it vanishes for variations induced by the BMS symmetries. In addition, we calculate the Noether currents and charges corresponding to gauge transformations and diffeomorphisms.

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arXiv2023

Emergence of Riemannian Quantum Geometry

In this chapter we take up the quantum Riemannian geometry of a spatial slice of spacetime. While researchers are still facing the challenge of observing quantum gravity, there is a geometrical core to loop quantum gravity that does much to define the approach. This core is the quantum character of its geometrical observables: space and spacetime are built up out of Planck-scale quantum grains. The interrelations between these grains are described by spin networks, graphs whose edges capture the bounding areas of the interconnected nodes, which encode the extent of each grain. We explain how quantum Riemannian geometry emerges from two different approaches: in the first half of the chapter we take the perspective of continuum geometry and explain how quantum geometry emerges from a few principles, such as the general rules of canonical quantization of field theories, a classical formulation of general relativity in which it appears embedded in the phase space of Yang-Mills theory, and general covariance. In the second half of the chapter we show that quantum geometry also emerges from the direct quantization of the finite number of degrees of freedom of the gravitational field encoded in discrete geometries. These two approaches are complimentary and are offered to assist readers with different backgrounds enter the compelling arena of quantum Riemannian geometry.

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arXiv2022

Scalar curvature operator for quantum-reduced loop gravity

In a previous article we have introduced an operator representing the three-dimensional scalar curvature in loop quantum gravity. In this article we examine the new curvature operator in the setting of quantum-reduced loop gravity. We derive the explicit form of the curvature operator as an operator on the Hilbert space of the quantum-reduced model. As a simple practical example, we study the expectation values of the operator with respect to basis states of the reduced Hilbert space.

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arXiv2022

Quantum Oppenheimer-Snyder and Swiss Cheese models

By considering the quantum Oppenheimer-Snyder model in loop quantum cosmology, a new quantum black hole model whose metric tensor is a suitably deformed Schwarzschild one is derived. The quantum effects imply a lower bound on the mass of the black hole produced by the collapsing dust ball. For the case of larger masses where the event horizon does form, the maximal extension of the spacetime and its properties are investigated. By discussing the opposite scenario to the quantum Oppenheimer-Snyder, a quantum Swiss Cheese model is obtained with a bubble surrounded by the quantum universe. This model is analogous to black hole cosmology or fecund universes where the big bang is related to a white hole. Thus our models open a new window to cosmological phenomenology.

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arXiv2022

Symmetries of the asymptotically de Sitter spacetimes

We start a systematic investigation of possible isometries of the asymptotically de Sitter solutions to Einstein equations. We reformulate the Killing equation as conformal equations for the initial data at $\mathcal{I}^+$. This allows for partial classification of possible symmetry algebras. In particular, if they are not maximal, they may be at most $4$-dimensional. We provide several examples. As a simple collorary it is shown that the only spacetime in which the Killing horizon intersects $\mathcal{I}^+$ (after a conformal completion) is locally the de Sitter universe.

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arXiv2022

A generalization of the quadruple formula for the energy of gravitational radiation in de Sitter spacetime

We study gravitational radiation produced by time changing matter source in de Sitter spacetime. We consider a cosmological Killing horizon instead of the conformal boundary used in the radiation theory in the Minkowski spacetime. The energy of the radiation passing through the horizon is derived. Our result takes the form of a generalized quadruple formula expressed in terms of the mass and pressure quadruple moments and is written explicitly up to the first order in $\sqrt{\Lambda}$. The zeroth order term recovers the famous Einstein's quadruple formula obtained for the perturbed Minkowski spacetime, whereas the first order term is a new correction.

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arXiv2022

Loop representation and r-Fock measures for $SU(N)$ gauge theories

In this article we continue the work on translating elements of the perturbative quantum field theory defined on Minkowski spacetime into the background independent framework of Loop Quantum Gravity. We present the construction of r-Fock measures for $SU(N)$ gauge theories and provide a relation between these new r-Fock measures and the difeomorphism invariant measure used in loop quantum gravity.

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arXiv2021

Fermion coupling to loop quantum gravity: canonical formulation

In the model of a fermion field coupled to loop quantum gravity, we consider the Gauss and the Hamiltonian constraints. According to the explicit solutions to the Gauss constraint, the fermion spins and the gravitational spin networks intertwine with each other so that the fermion spins contribute to the volume of the spin network vertices. For the Hamiltonian constraint, the regularization and quantization procedures are presented in detail. By introducing an adapted vertex Hilbert space to remove the regulator, we propose a diffeomorphism covariant graph-changing Hamiltonian constraint operator of the fermion field. This operator shows how fermions move in the loop quantum gravity spacetime and simultaneously influences the background quantum geometry.

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arXiv2021

Charges and Fluxes on (Perturbed) Non-expanding Horizons

In a companion paper we showed that the symmetry group $\mathfrak{G}$ of non-expanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group $\mathfrak{G}$ at $\mathcal{I}^{+}$. For each infinitesimal generator of $\mathfrak{G}$, we now define a charge and a flux on NEHs as well as perturbed NEHs. The procedure uses the covariant phase space framework in presence of internal null boundaries $\mathcal{N}$. However, $\mathcal{N}$ is required to be an NEH or a perturbed NEH. Consequently, charges and fluxes associated with generators of $\mathfrak{G}$ are free of physically unsatisfactory features that can arise if $\mathcal{N}$ is allowed to be a general null boundary. In particular, all fluxes vanish if $\mathcal{N}$ is an NEH, just as one would hope; and fluxes associated with symmetries representing `time-translations' are positive definite on perturbed NEHs. These results hold for zero as well as non-zero cosmological constant. In the asymptotically flat case, as noted in \cite{akkl1}, $\mathcal{I}^\pm$ are NEHs in the conformally completed space-time but with an extra structure that reduces $\mathfrak{G}$ to $\mathfrak{B}$. The flux expressions at $\mathcal{N}$ reflect this synergy between NEHs and $\mathcal{I}^{+}$. In a forthcoming paper, this close relation between NEHs and $\mathcal{I}^{+}$ will be used to develop gravitational wave tomography, enabling one to deduce horizon dynamics directly from the waveforms at $\mathcal{I}^{+}$.

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