Generalized Birkhoff theorems and 2+2 direct pruduct spacetimes in Weyl conformal gravity

The aim of this paper is to present a framework that integrates optimisation into algebraic structures in Cartesian Closed Categories (CCCs). Traditional mathematical methods treated optimisation as an external process, which limited its foundational role in mathematics. Inspired by the finite nature hypothesis and Hilbert's sixth problem, which calls for the axiomatisation of physical principles, this study formalises optimisation as an intrinsic algebraic axiom in a way that aligns with Hilbert's vision of uniting mathematical and physical laws. The framework builds on Lawvere's categorical treatment of metric spaces and Birkhoff's HSP theorem, which the study uses to define an optimisation algebra. The study then provides proof that the class of such algebras forms a variety in universal algebra and demonstrates categorical soundness within CCCs. The proposed approach guarantees that optimisation is inherent within algebraic systems, establishing natural substructures of optimal elements and facilitating compositional reasoning in computational models. Applying this framework in dataflow networks demonstrates convergence to optimal steady states, enhancing resource utilisation and system efficiency. Future research includes using the framework in enriched categories, distributed systems and incorporating the operator in tools that can be used to solve real-world problems.

This work reveals a fundamental link between general covariance and Birkhoff's theorem. We extend Birkhoff's theorem from general relativity to a broad class of generally covariant gravity theories formulated in the Hamiltonian framework. Conversely, we show that each one-parameter family of static, spherically symmetric spacetimes determines a class of covariant theories, each of which has that family of spacetimes as its entire vacuum solution space. Our systematic and model-independent framework applies to a wide range of spacetimes, including observationally inferred, quantum-gravity-inspired, and regular black holes. It provides a universal tool for probing their dynamical origins and enables the reconstruction of the underlying covariant theories from observational data, including gravitational-wave and black-hole-shadow measurements.

Abstract As argued in [78], introducing a non-minimally coupled scalar field, three-dimensional Einstein gravity can be extended by infinite families of theories which admit simple analytic generalizations of the charged BTZ black hole. Depending on the gravitational couplings, the solutions may describe black holes with one or several horizons and with curvature or BTZ-like singularities. In other cases, the metric function behaves as f (r) $$ \overset{\left(r\to 0\right)}{\sim } $$ ∼ r → 0 $$ \mathcal{O}\left({r}^{2s}\right) $$ O r 2 s with s ≥ 1, and the black holes are completely regular — a feature unique to three dimensions. Regularity arises generically i.e., without requiring any fine-tuning of parameters. In this paper we show that all these theories satisfy Birkhoff theorems, so that the most general spherically-symmetric solutions are given by the corresponding static black holes. We perform a thorough characterization of the Penrose diagrams of the solutions, finding a rich structure which includes, in particular, cases which tessellate the plane and others in which the diagrams cannot be drawn in a single plane. We also study the motion of probe particles on the black holes, finding that observers falling to regular black holes reach the center after a finite proper time. Contrary to the singular cases, the particles are not torn apart by tidal forces, so they oscillate between antipodal points describing many-universe orbits. We argue that in those cases the region r = 0 can be interpreted as a horizon with vanishing surface gravity, giving rise to generic inner-extremal regular black hole solutions. We also analyze the deep interior region of the solutions identifying the presence of Kasner eons and the conditions under which they take place. Finally, we construct new black hole solutions in the case in which infinite towers of terms are included in the action.

In this note, we study existence of infinitely many trajectories bi-normal (i.e. normal at initial and final times) to the xz-plane in the Spatial Circular Restricted Three-Body problem, in the convexity range and near the primaries, under the assumption of the twist condition as defined by Moreno–van-Koert in [10]. Modulo our assumptions, this is an expected application of the relative Poincaré–Birkhoff theorem for Lagrangians in Liouville domains, as proven by the authors in [8].

We establish that regular black holes can form from gravitational collapse in all D≥5. Our model builds on a recent construction that realized regular black holes as exact solutions to purely gravitational theories that incorporate an infinite tower of higher curvature corrections [P. Bueno , Regular black holes from pure gravity, .]. We identify a two-dimensional Horndeski theory that captures the spherically symmetric dynamics of the theories in question and use this to prove a Birkhoff theorem and obtain the generalized Israel junction conditions. Armed with these tools, we consider the collapse of thin shells of pressureless matter, showing that this leads generically to the formation of regular black holes. The interior dynamics we uncover is intricate, consisting of shell bounces and white hole explosions into a new universe. The result is that regular black holes are the unique spherically symmetric solutions of the corresponding theories and also the end point of gravitational collapse of matter. Along the way, we establish evidence for a solution-independent upper bound on the curvature, suggestive of Markov’s limiting curvature hypothesis.

Some Theorems of Birkhoff in Tonoid Matrices

The method of integrating factors is used to study the conservation laws of the Herglotz type Birkhoffian systems in this paper. Firstly, the definition of the integrating factors of the Herglotz type Birkhoffian systems is given. Secondly, the relationship between the integrating factors and conservation laws is studied, and the conservation theorems of Herglotz type Birkhoff's equations and their inverse theorems are established. Thirdly, two types of generalized Killing equations for calculating integrating factors are given. Finally, as an example, a linear damped oscillator is taken. This example can be transformed into a Herglotz type Birkhoffian system. The resulting conservation theorems are used to find the conserved quantities for this example.

A Poincaré–Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems

Particular Kottler spacetimes are analytically investigated. The investigated spacetimes are spherically symmetric nonrotating spacetimes endowed with a Schwarzschild solid-angle element. SchwarzschildNairiai spacetimes, Schwarzschild spacetimes with a linear term, and Schwarzschild spacetimes with a linear term and a cosmological constant are studied. The infinite-redshift surfaces are analytically written. To this aim, the parameter spaces of the models are analytically investigated, and the conditions for which the analytical radii are reconducted to the physical horizons are used to set and to constrain the parameter spaces. The coordinate-singularity-avoiding coordinate extensions are newly written. Schwarzschild spacetimes with a linear term and a cosmological constant termare analytically studied, and the new singularity-avoiding coordinate extensions are detailed. The new roles of the linear term and of the cosmological constant term in characterizing the Schwarzschild radius are traced. The generalized Schwarzschild–deSitter case and generalized Schwarzschild–anti-deSitter case are characterized in a different manner. The weak field limit is newly recalled. The embeddings are newly provided. The quantum implementation is newly envisaged. The geometrical objects are newly calculated. As a result, for the Einstein field equations, the presence of quintessence is newly excluded. The Birkhoff theorem is newly proven to be obeyed.

Filtered colimit elimination from Birkhoff's variety theorem

Abstract We provide multiplicity results for the periodic problem associated with Hamiltonian systems coupling a system having a Poincaré–Birkhoff twist-type structure with a system presenting some asymmetric nonlinearities, with possible one-sided superlinear growth. We investigate nonresonance, simple resonance and double resonance situations, by implementing some kind of Landesman–Lazer conditions.

Is Birkhoff's theorem valid in Einstein-Aether theory?

Abstract A simple proof using Birkhoff theorem is given for the result that the spectral norm of a doubly stochastic matrix is 1. We also show that the result generalizes the results of İpek, Bozkurt, and Jiang and Zhou on circulant matrices and r r -circulant matrices. Spectral norm of level- k k circulant matrix and applications are given.

In this paper, we considered non-static vacuum spherically symmetric solutions of the Einstein equations and harmonicity conditions in the coordinate system with a non-zero space-time component in the metric. For the case of the weak field, a particular solution of the approximate equations was obtained, which corresponds to a nonstatic source whose boundary moves with a constant speed. For the exact Einstein’s equations we obtained a wave-type solution, determined by two implicitly specified functions, depending on the retarded argument and on the radial coordinate, respectively. The connection between these solutions and the Birkhoff theorem is discussed.

Exact solutions of spherically symmetric black hole and gravitational wave are explored in f(R) gravity in arbitrary dimension. We find two exact solutions for the radiation and absorption of null dust. In the framework of general relativity, the Birkhoff theorem strictly forbids the existence of spherical gravitational waves in vacuum space. We find spherical non-perturbative gravitational waves, which are shear-free, twist-free, but expanding.

In this paper, we will focus on the topic of congruences and factorizations in universal algebra. We will first provide an introduction to universal algebra by defining lattices, algebras, congruences and other core structures. Next, we will explore the congruence and factorization properties of an algebra. Then, Birkhoff theorem indicates that every algebra can be embedded into a product of subdirectly irreducible algebras. Based on these fundamental concepts, Heyting algebra will be discussed as a typical example in universal algebra. The one-to-one correspondence between filters and congruences of a Heyting algebra will be proved. Lastly, we will show a specific way to justify the subdirectly irreducibility of a Heyting algebra.

Periodic solutions of Hamiltonian systems coupling twist with generalized lower/upper solutions

We apply, in the context of semigroups, the main theorem from the authors’ paper “Algebras defined by equations” (Higgins and Jackson in J Algebra 555:131–156, 2020) that an elementary class {mathscr {C}} of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of {mathscr {C}} is free of the forall quantifier. We also observe the decidability of the class of equation systems satisfied by semigroups, via a link to systems of rationally constrained equations on free semigroups. Examples are given of EHP-classes for which neither (forall cdots )(exists cdots ) equation systems nor (exists cdots )(forall cdots ) systems suffice.

An extension of the Poincaré–Birkhoff Theorem coupling twist with lower and upper solutions