arXiv subjects
Publications and source records attributed to Wojciech Kamiński.
Our work clarifies the origin of enhanced symmetry of near horizon geometries. We relate this property to an existence of specific foliation by non-expanding horizons. The existence of this foliation can be shown in any spacetime satisfying null convergence condition. Our proof is geometric and based on analyzing null hypersurfaces in near horizon geometry spacetimes.
In this paper, we introduce a general method to prove the non-degeneracy of the Hessian in the spinfoam vertex amplitude for quantum gravity and apply it to the spinfoam models with a cosmological constant ($Λ$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat ${\rm SL}(2,\mathbb{C})$ connections, we demonstrate that the Hessian is non-degenerate for critical points corresponding to non-degenerate, geometric 4-simplices in de Sitter or anti-de Sitter space. Non-degeneracy of the Hessian is an important necessary condition for the stationary phase method to be applicable. With a non-degenerate Hessian, this method not only confirms the connection of the $Λ$-SF model to semiclassical gravity, but also shows that there are no dominant contributions from exceptional configurations as in the Barrett-Crane model. Given its general nature, we expect our criterion to be applicable to other spinfoam models under mild adjustments.
In this short note, we construct an example of spiraling conformal geodesic in Euclidean signature in dimension $3$, answering the question posed by Helmuth Friedrich and Paul Tod, if such objects exists. Our example is not real analytic, but similar constructions can lead also to real analytic metrics in arbitrary dimensions.
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-Nordström-(Anti)de-Sitter extremal horizons. Interestingly, the static rigidity result is also valid for non-compact spaces with a zero first cohomology group.
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.
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.
Vanishing of the Fefferman-Graham obstruction tensor was used by Andersson and Chru{ś}ciel to show stability of the asymptotically de Sitter spaces in even dimensions. However, existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions Graham-Jenne-Mason-Sparling operators. In particular, we use one of these operators to propagate Gover's condition of being almost conformally Einstein. This allows to study initial data also for Cauchy surfaces which cross conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose conformal factor such that Branson Q-curvature vanishes.
We describe a Lorentzian manifold that is globally hyperbolic and geodesically complete, but such that the (minimally coupled) Klein-Gordon operator with the standard domain is not essentially self-adjoint.
In the Hamiltonian formulation, it is not a priori clear whether a symmetric configuration will keep its symmetry during evolution. In this paper, we give precise requirements of when this is the case and propose a symmetry restriction to the phase space of the symmetric variables. This can often ease computation, especially when transcending from the infinite dimensional phase space of a field theory to a possibly finite dimensional subspace. We will demonstrate this in the case of gravity. A prominent example is the restriction of full Hamiltonian general relativity to the cosmological configurations of Robertson-Walker type. We will demonstrate our procedure in this setting and extend it to examples which appear useful in certain approaches to quantum gravity.
We discuss several approaches to determine the Casimir force in inertial frames of reference in different dimensions. On an example of a simple model involving mirrors in Rindler spacetime we show that Casimir's and Lifschitz's methods are inequivalent and only latter can be generalized to other spacetime geometries. For conformally coupled fields we derive the Casimir force in conformally flat spacetimes utilizing an anomaly and provide explicit examples in the Friedmann-Lemaitre-Robertson-Walker $(k=0)$ models.
It was recently shown that the volume operator of loop quantum cosmology (LQC) and all its positive powers are ill-defined on physical states. In this paper, we investigate how it effects predictions of cosmic microwave background (CMB) power spectra obtained within dressed metric approach for which expectations values of $\hat{a}$ are the key element. We find that almost every step in the procedure is ill-defined and relies heavily upon a (seemingly premature) numerical truncation. Thus, it suggests that more care is needed in making predictions regarding pre-inflationary physics. We propose a new scheme which contains only well-defined quantities. The surprising agreement of the hitherto models with observational data, especially at low angular momenta $l$ is explained.
We will prove in an elementary way that off-diagonal Seeley-DeWitt and Hadamard coefficients are (sesqui-)symmetric for smooth manifolds of arbitrary signature.
The Near Horizon Geometry (NHG) equation with a cosmological constant Λ is considered on compact 2-dimensional manifolds. It is shown that every solution satisfies the Type D equation at every point of the manifold. A similar result known in the literature was valid only for non-degenerate in a suitable way points of a given solution. At the degenerate points the Type D equation was not applicable. In the current paper we prove that the degeneracy is ruled out by the compactness. Using that result we find all the solutions to the NHG equation on compact 2-dimensional manifolds of non-positive Euler characteristics. Some integrability conditions known earlier in the Λ = 0 case are generalized to arbitrary value of Λ. They may be still useful for compact 2-manifolds of positive Euler characteristic.
The Petrov type D equation imposed on the 2-metric tensor and the rotation scalar of a cross-section of an isolated horizon can be used to uniquely distinguish the Kerr - (anti) de Sitter spacetime in the case the topology of the cross-section is that of a sphere. In the current paper we study that equation on closed 2-dimensional surfaces that have genus $>0$. We derive all the solutions assuming the embeddability in 4-dimensional spacetime that satisfies the vacuum Einstein equations with (possibly 0) cosmological constant. We prove all of them have constant Gauss curvature and zero rotation. Consequently, we provide a quazi-local argument for a black hole in 4-dimensional spacetime to have a topologically spherical cross-section.
We show that the von Neumann--Lueders collapse rules in quantum mechanics always select the unique state that maximises the quantum relative entropy with respect to the premeasurement state, subject to the constraint that the postmeasurement state has to be compatible with the knowledge gained in the measurement. This way we provide an information theoretic characterisation of quantum collapse rules by means of the maximum relative entropy principle.
Prominent approaches to quantum gravity struggle when it comes to incorporating a positive cosmological constant in their models. Using quantization of a complex $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory we include a cosmological constant, of either sign, into a model of quantum gravity.
In this addendum we clarify a point which strengthens one of the results from [the original paper]. Namely, we show that the algebra of the observables $F(r,θ)$ is yet simpler then it was described in [the original paper]. This is an important point, because with this simplification an important subalgebra becomes canonical, allowing for a natural reduction of the phase space.
We study the expectation value of a nonplanar Wilson graph operator in SL(2,C) Chern-Simons theory on $S^3$. In particular we analyze its asymptotic behaviour in the double-scaling limit in which both the representation labels and the Chern-Simons coupling are taken to be large, but with fixed ratio. When the Wilson graph operator has a specific form, motivated by loop quantum gravity, the critical point equations obtained in this double-scaling limit describe a very specific class of flat connection on the graph complement manifold. We find that flat connections in this class are in correspondence with the geometries of constant curvature 4-simplices. The result is fully non-perturbative from the perspective of the reconstructed geometry. We also show that the asymptotic behavior of the amplitude contains at the leading order an oscillatory part proportional to the Regge action for the single 4-simplex in the presence of a cosmological constant. In particular, the cosmological term contains the full-fledged curved volume of the 4-simplex. Interestingly, the volume term stems from the asymptotics of the Chern-Simons action. This can be understood as arising from the relation between Chern-Simons theory on the boundary of a region, and a theory defined by an $F^2$ action in the bulk. Another peculiarity of our approach is that the sign of the curvature of the reconstructed geometry, and hence of the cosmological constant in the Regge action, is not fixed a priori, but rather emerges semiclassically and dynamically from the solution of the equations of motion. In other words, this work suggests a relation between 4-dimensional loop quantum gravity with a cosmological constant and SL(2,C) Chern-Simons theory in 3-dimensions with knotted graph defects.