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Gabor Etesi

Publications and source records attributed to Gabor Etesi.

At least 19 recordsLinked to original sources
arXiv2025

On the emergence of the Lorentzian metric structure of space-time in general relativity

In this short note we argue that, even if, as sometimes remarked, a Lorentzian manifold does not model correctly the structure of the spatio-temporal continuum as it is, yet a Lorentzian manifold should describe its macroscopic structure as we experience it. More precisely, theoretically motivated by von Weizs\"acker's chronological relative frequency interpretation of probability, and taking the Diaconis--Mosteller principle (also called the law of truly large numbers) as an empirical evidence in the macroscopic world, we argue that large collections of physical events appear in a composition of two fundamentally different patterns, termed as progression and sample here, suggesting, in this framework, to use a Lorentzian-type metric on a manifold to describe matter-filled macroscopic regions of the spatio-temporal continuum.

physics.gen-ph↗
arXiv2024

An operator algebraic characterization of the Riemannian vacuum Einstein equation in four dimensions

In this paper, using connected compact oriented smooth 4-manifolds, some representations of the hyperfinite II_1-type factor von Neumann algebra are constructed. The Murray--von Neumann coupling constant of these representations gives rise to a new smooth 4-manifold invariant whose very first properties are investigated. Moreover as a part of this construction, a connected compact oriented smooth 4-manifold admits an embedding into the hyperfinite II_1 factor. This embedding, on the one hand, induces a Riemannian metric on the manifold such that its Riemannian curvature tensor belongs to the von Neumann algebra; on the other hand the metric induces a periodic dynamics on the von Neumann algebra, what we call the Hodge dynamics on the hyperfinite II_1 factor. It is observed that the metric is Einstein i.e., satisfies the (Riemannian) vacuum Einstein equation with a possibly non-zero cosmological constant, if and only if its Riemannian curvature tensor belongs to the fixed-point-subalgebra of the Hodge dynamics. Finally, we make a comprehensive enumeration of all representations of the hyperfinite II_1 factor constructed here, from the viewpoint of thermal equilibrium states and phase transitions in algebraic quantum field theory.

math-ph↗
arXiv2024

Primordial black holes from collapsing antimatter

In this paper a simple (i.e. free of fine-tuning, etc.) new mechanism for primordial black hole formation based on the collapse of large antimatter systems in the early Universe is introduced. A peculiarity of this process is that, compared to their material counterparts, the collapse of large antimatter systems takes much less time due to the reversed thermodynamics of antimatter, an idea which has been proposed in our earlier paper. This model has several testable predictions. The first is that the photon-baryon ratio is roughly computable and is equal to $1.95\times 10^9$ which is quite close to its experimentally confirmed value. The second is that the mass of black holes arising from this mechanism is at least $10^5$-$10^6M_\odot$ hence they contribute to the super- or hypermassive end of the primordial black hole mass spectrum. The third prediction is that these sort of primordial black holes constitute at least $20\%$ of dark matter. Last but not least the observed current asymmetry of matter and antimatter, even if their presence in the Universe was symmetric in the beginning, acquires a natural explanation, too.

hep-th↗
arXiv2024

A set-theoretic analysis of the black hole entropy puzzle

Motivated by the known mathematical and physical problems arising from the current mathematical formalization of the physical spatio-temporal continuum, as a substantial technical clarification of our earlier attempt, the aim in this paper is twofold. Firstly, by interpreting Chaitin's variant of Gödel's first incompleteness theorem as an inherent uncertainty or fuzziness present in the set of real numbers, a set-theoretic entropy is assigned to it using the Kullback--Leibler relative entropy of a pair of Riemannian manifolds. Then exploiting the non-negativity of this relative entropy an abstract Hawking-like area theorem is derived. Secondly, by analyzing Noether's theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a black hole is modeled by the set of real numbers in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole (proportional to the area of its ``instantaneous'' event horizon), indicating that this apparently physical quantity might have a pure set-theoretic origin, too.

physics.gen-ph↗
arXiv2024

The universal von Neumann algebra of smooth four-manifolds revisited

Making use of its smooth structure only, out of a connected oriented smooth $4$-manifold a von Neumann algebra is constructed. It is geometric in the sense that is generated by local operators and as a special four dimensional phenomenon it contains all algebraic (i.e., formal or coming from a metric) curvature tensors of the underlying $4$-manifold. The von Neumann algebra itself is a hyperfinite factor of ${\rm II}_1$-type hence is unique up to abstract isomorphisms of von Neumann algebras. Over a fixed $4$-manifold this universal von Neumann algebra admits a particular representation on a Hilbert space such that its unitary equivalence class is preserved by orientation-preserving diffeomorphisms consequently the Murray--von Neumann coupling constant of this representation is well-defined and gives rise to a new and computable real-valued smooth $4$-manifold invariant. Its link with Jones' subfactor theory is noticed as well as computations in the simply connected closed case are carried out. Application to the cosmological constant problem is also discussed. Namely, the aforementioned mathematical construction allows to reformulate the classical vacuum Einstein equation with cosmological constant over a $4$-manifold as an operator equation over its tracial universal von Neumann algebra such that the trace of a solution is naturally identified with the cosmological constant. This framework permits to use the observed magnitude of the cosmological constant to estimate by topological means the number of primordial black holes about the Planck era. This number turns out to be negligable which is in agreement with known density estimates based on the Press--Schechter mechanism.

math-ph↗
arXiv2023

The four dimensional Yang--Mills partition function in the vicinity of the vacuum

The partition function of four dimensional Euclidean, non-supersymmetric SU(2) Yang--Mills theory is calculated in the perturbative and weak coupling regime i.e. in a small open ball about the flat connection (what we call the vicinity of the vacuum) and when the gauge coupling constant acquires a small but finite value. The computation is based on various known inequalities, valid only in four dimensions, providing two-sided estimates for the exponentiated Yang--Mills action in terms of the $L^2$-norm of the derivative of the gauge potential only; these estimates then give rise to Gaussian-like infinite dimensional integrals involving the Laplacian hence can be formally computed via zeta-function and heat kernel techniques. It then turns out that these integrals give a sharp value for the partition function in the aforementioned perturbative and weak coupling regime of the theory. In the resulting expression for the partition function the original classical coupling constant is shifted to a smaller one which can be interpreted as the manifestation, in this approach, of the existence a non-trivial trivial $β$-function and asymptotic freedom in non-Abelian gauge theories.

hep-th↗
arXiv2021

A proof of the Geroch-Horowitz-Penrose formulation of the strong cosmic censor conjecture motivated by computability theory

In this paper we present a proof of a mathematical version of the strong cosmic censor conjecture attributed to Geroch-Horowitz and Penrose but formulated explicitly by Wald. The proof is based on the existence of future-inextendible causal curves in causal pasts of events on the future Cauchy horizon in a non-globally hyperbolic space-time. By examining explicit non-globally hyperbolic space-times we find that in case of several physically relevant solutions these future-inextendible curves have in fact infinite length. This way we recognize a close relationship between asymptotically flat or anti-de Sitter, physically relevant extendible space-times and the so-called Malament-Hogarth space-times which play a central role in recent investigations in the theory of "gravitational computers". This motivates us to exhibit a more sharp, more geometric formulation of the strong cosmic censor conjecture, namely "all physically relevant, asymptotically flat or anti-de Sitter but non-globally hyperbolic space-times are Malament-Hogarth ones". Our observations may indicate a natural but hidden connection between the strong cosmic censorship scenario and the Church-Turing thesis revealing an unexpected conceptual depth beneath both conjectures.

gr-qc↗
arXiv2021

Strong cosmic censorship and topology change in four dimensional gravity

A physical interpretation of the recently discovered vast class of vacuum space-times, which stably violate the strong cosmic censor conjecture (in its usual broad formulation) in four dimensions, is exhibited. Namely, by elementary Morse theory we demonstrate that these geometries describe transitions between topologically different spacelike hypersurfaces. More precisely, in these four dimensional smooth vacuum space-times there exist three dimensional spacelike hypersurfaces which display an increasingly violent unbounded oscillation between topologically different closed, orientable three-manifolds as one moves towards the asymptotic region in their ambient space-times. Moreover this spatial oscillation appears as a cosmological redshift for late time observers. Therefore these new vacuum solutions shed some light onto the deep dynamic regime of general relativity and the structure of the four dimensional continuum itself. This picture, beyond offering a physical clarification how global hyperbolicity breaks down in these solutions, is also consistent with the fact that the Riemannian counterparts of these Lorentzian vacuum geometries are not only Ricci-flat but even self-dual four-manifolds hence give rise to gravitational instantons. Consequently the role of these Riemannian solutions is similar to that of Yang--Mills instantons in semi-classical non-Abelian gauge theories over Minkowski space-time.

gr-qc↗
arXiv2021

Global solvability of the vacuum Einstein equation and the strong cosmic censorship in four dimensions

Let $M$ be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put $M^\times:=M\setminus\{{\rm point}\}$.In this paper we prove that if $X^\times$ is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to $M^\times$ (more precisely, it carries a smooth structure à la Gompf) then $X^\times$ can be equipped with a complete Ricci-flat Riemannian metric. As a byproduct of the construction it follows that this metric is self-dual as well consequently $X^\times$ with this metric is in fact a hyper-Kähler manifold. In particular we find that the largest member of the Gompf--Taubes radial family of large exotic ${\mathbb R}^4$'s admits a complete Ricci-flat metric (and in fact it is a hyper-Kähler manifold). These Riemannian solutions are then converted into Ricci-flat Lorentzian ones thereby exhibiting lot of new vacuum solutions which are not accessable by the initial vaule formulation. A natural physical interpretation of them in the context of the strong cosmic censor conjecture and topology change is discussed.

math.DG↗
arXiv2018

Exotica and the status of the strong cosmic censor conjecture in four dimensions

An immense class of physical counterexamples to the four dimensional strong cosmic censor conjecture---in its usual broad formulation---is exhibited. More precisely, out of any closed and simply connected 4-manifold an open Ricci-flat Lorentzian 4-manifold is constructed which is not globally hyperbolic and no perturbation of it, in any sense, can be globally hyperbolic. This very stable non-global-hyperbolicity is the consequence of our open spaces having a "creased end" i.e., an end diffeomorphic to an exotic ${\mathbb R}^4$. Open manifolds having an end like this is a typical phenomenon in four dimensions. The construction is based on a collection of results of Gompf and Taubes on exotic and self-dual spaces, respectively, as well as applying Penrose' non-linear graviton construction (i.e., twistor theory) to solve the Riemannian Einstein's equation. These solutions then are converted into stably non-globally-hyperbolic Lorentzian vacuum solutions. It follows that the plethora of vacuum solutions we found cannot be obtained via the initial value formulation of the Einstein's equation because they are "too long" in a certain sense (explained in the text). This different (i.e., not based on the initial value formulation but twistorial) technical background might partially explain why the existence of vacuum solutions of this kind have not been realized so far in spite of the fact that, apparently, their superabundance compared to the well-known globally hyperbolic vacuum solutions is overwhelming.

gr-qc↗
arXiv2017

Gravity as a four dimensional algebraic quantum field theory

Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from $C^*$-algebras generated by local curvature tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth $4$-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in $4$ dimensions naturally embeds into a quantum framework. Several Hilbert space representations of the theory are found. First a "tautological representation" of the limiting global $C^*$-algebra is constructed allowing to associate to any oriented smooth $4$-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan--Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call "positive mass representations" with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical representaions" with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity. Finally we observe that the whole family of "positive mass representations" comprise a $2$ dimensional conformal field theory in the sense of G. Segal.

hep-th↗
arXiv2015

Exotica or the failure of the strong cosmic censorship in four dimensions

In this letter a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely---taking into account that the conjecture lacks any precise formulation yet---first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically stronger concept of a "robust counterexample". Then making use of Penrose' nonlinear graviton construction (i.e., twistor theory) and a Wick rotation trick we construct a smooth Ricci-flat but not flat Lorentzian metric on the largest member of the Gompf--Taubes uncountable radial family of large exotic ${\mathbb R}^4$'s. We observe that this solution of the Lorentzian vacuum Einstein's equations with vanishing cosmological constant provides us with a sort of counterexample which is weaker than a "robust counterexample" but still reasonable to consider as a "generic counterexample". It is interesting that this kind of counterexample exists only in four dimensions.

gr-qc↗
arXiv2015

The complex structure on the six dimensional sphere

Proof of existence of a complex structure on the six-sphere, followed by an explicit computation of its underlying integrable almost complex tensor by the aid of inner automorphisms of the octonions, is exhibited. Both are elementary and self-contained however the size and complexity of the emerging almost complex tensor field on the six-sphere is perplexing.

math.DG↗
arXiv2015

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking

Existence of a complex structure on the $6$ dimensional sphere is proved in this paper. The proof is based on re-interpreting a hypothetical complex structure as a classical ground state of a Yang--Mills--Higgs-like theory on $S^6$. This classical vacuum solution is then constructed by Fourier expansion (dimensional reduction) from the obvious one of a similar theory on the $14$ dimensional exceptional compact Lie group ${\rm G}_2$.

math.DG↗
arXiv2015

S-duality in Abelian gauge theory revisited

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.

math.DG↗
arXiv2013

On the energy spectrum of Yang--Mills instantons over asymptotically locally flat spaces

In this paper we prove that over an asymptotically locally flat (ALF) Riemannian four-manifold the energy of an "admissible" SU(2) Yang--Mills is always integer. This result sharpens the previously known energy identity for such Yang--Mills instantons over ALF geometries. Furthermore we demonstrate that this statement continues to hold for the larger gauge group U(2). Finally we make the observation that there might be a natural relationship between 4 dimensional Yang--Mills theory over an ALF space and 2 dimensional conformal field theory. This would provide a further support for the existence of a similar correspondence investigated by several authors recently.

math.DG↗
arXiv2011

Harmonic functions and instanton moduli spaces on the multi-Taub--NUT space

Explicit construction of the basic SU(2) anti-instantons over the multi-Taub--NUT geometry via the classical conformal rescaling method is exhibited. These anti-instantons satisfiy the so-called weak holonomy condition at infinity with respect to the trivial flat connection and decay rapidly. The resulting unital energy anti-instantons have trivial holonomy at infinity. We also fully describe their unframed moduli space and find that it is a five dimensional space admitting a singular disk-fibration over R^3. On the way, we work out in detail the twistor space of the multi-Taub--NUT geometry together with its real structure and transform our anti-instantons into holomorphic vector bundles over the twistor space. In this picture we are able to demonstrate that our construction is complete in the sense that we have constructed a full connected component of the moduli space of solutions of the above type. We also prove that anti-instantons with arbitrary high integer energy exist on the multi-Taub--NUT space.

math.DG↗