Birkhoff's Theorem Research Articles

Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems

The paper investigates estimates which relate two equivalent phenomena: the power-type rate of convergence in von Neumann's ergodic theorem and the power-type singularity at zero (with the same exponent) exhibited by the spectral measure of the function being averaged with respect to the corresponding dynamical system. The same rate of convergence is also estimated in terms of the rate of decrease of the correlation coefficients. Also, constants are found in analogous estimates for the power-type convergence in Birkhoff's ergodic theorem. All the results have exact analogues for wide-sense stationary stochastic processes.Bibliography: 15 titles.

Birkhoff's theorem in higher derivative theories of gravity

In this paper, we present a class of higher derivative theories of gravity which admit Birkhoff's theorem. In particular, we explicitly show that in this class of theories, although generically the field equations are of fourth order, under spherical (plane or hyperbolic) symmetry, all the field equations reduce to second order and have exactly the same or similar structure to those of Lovelock theories, depending on the spacetime dimensions and the order of the Lagrangian.

Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem

In the general setting of a planar first order system (0.1) u ′ = G ( t , u ) , u ∈ R 2 , with G : [ 0 , T ] × R 2 → R 2 , we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G ( t , 0 ) ≡ 0 , at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T -periodic solutions of unforced planar Hamiltonian systems J u ′ = ∇ u H ( t , u ) and unforced undamped scalar second order equations x ″ + g ( t , x ) = 0 . In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

Almost Birkhoff theorem in general relativity

We extend Birkhoff's theorem for almost LRS-II vacuum spacetimes to show that the rigidity of spherical vacuum solutions of Einstein's field equations continues even in the perturbed scenario.

Shifts and periodicity in algebraic analysis

Abstract Shifts and periodicity for functional-differential equations and their generalizations have been studied by the author in various aspects (cf. for instance, [17]–[20] and following papers). Here we would like to give a comprehensive survey of some of these results (without proofs) in order to recall the most important properties of the considered shifts. In particular, it is shown that the so-called true shifts in complete linear metric spaces are hypercyclic and that a necessary and sufficient condition for true shifts in commutative algebras to be multiplicative is that the generating operator D satisfies the Leibniz condition. A consequence of this fact is that in commutative Leibniz algebras with logarithms the operator D is uniquely determined by an isomorphism acting on $$ \frac{d} {{dt}} $$. There are also studied generalized periodic and exponential-periodic solutions of linear and some nonlinear equations with shifts and generalizations of the classical Birkhoff theorem and Floquet theorem. These results are obtained by means of tools given by Algebraic Analysis (cf. the author [17]). A generalization of binomial formula of Umbral Calculus is shown in Section 7 (cf. Roman and Rota [36]). Section 11 contains a perturbation theorem for linear differential-difference equations with non-commensurable deviations and some its consequences.

Quantum Birkhoff–Gustavson normal form in some completely resonant cases

The goal of this paper is to present an explicit computation of the Birkhoff–Gustavson normal form in the 1 : 1 , 1 : 2 and 1 : 3 resonances, by using the concept of Weyl quantization, especially when it is about the first terms. We use in particular the Weyl symbolic calculus and the abstract theorem of the quantum Birkhoff–Gustavson normal form in infinite dimension.

Shock waves and Birkhoff’s theorem in Lovelock gravity

Spherically symmetric shock waves are shown to exist in Lovelock gravity. They amount to a change of branch of the spherically symmetric solutions across a null hypersurface. The implications of their existence for the status of Birkhoff's theorem in the theory is discussed.

A simple proof of Birkhoff's theorem for cosmological constant

We provide a simple, unified proof of Birkhoff’s theorem for the vacuum and cosmological constant case, emphasizing its local nature. We discuss its implications for the maximal analytic extensions of Schwarzschild, Schwarzschild(–anti)-de Sitter, and Nariai spacetimes. In particular, we note that the maximal analytic extensions of extremal and overextremal Schwarzschild–de Sitter spacetimes exhibit no static region. Hence the common belief that Birkhoff’s theorem implies staticity is false for the case of positive cosmological constant. Instead, the correct point of view is that generalized Birkhoff’s theorems are local uniqueness theorems whose corollary is that locally spherically symmetric solutions of Einstein’s equations exhibit an additional local Killing vector field.

BOLTZMANN'S ERGODIC HYPOTHESIS: A MEETING PLACE FOR TWO CULTURES

The physical theory for the ergodic hypothesis is premised on the idea that the hypothesis is measurable by scattering experiment. Therewith it proves the hypothesis by measurable properties of the response function. Birkhoff's theorem proves the ergodic hypothesis by abstract properties of phase space, the measure and transitivity. In this work we apply the two independent approaches to a particular many-body model (spin-1/2 XY model) to see whether they arrive at the same conclusions on the ergodicity question. The two approaches are compared to gain insight into their goals.

A new cubic theory of gravity in five dimensions: black hole, Birkhoff's theorem and C-function

We present a new cubic theory of gravity in five dimensions which has second-order traced field equations, analogous to BHT new massive gravity in three dimensions. Moreover, for static spherically symmetric spacetimes all the field equations are of second order, and the theory admits a new asymptotically locally flat black hole. Furthermore, we prove the uniqueness of this solution, study its thermodynamical properties and show the existence of a C-function for the theory following the arguments of Anber and Kastor (2008 J. High Energy Phys. JHEP05(2008)061 (arXiv:0802.1290 [hep-th])) in pure Lovelock theories. Finally, we include the Einstein–Gauss–Bonnet and cosmological terms and find new asymptotically AdS black holes at the point where the three maximally symmetric solutions of the theory coincide. These black holes may also possess a Cauchy horizon.

Post-Newtonian limit off(R)gravity in the harmonic gauge

A general analytic procedure is developed for the post-Newtonian limit of $f(R)$-gravity with metric approach in the Jordan frame by using the harmonic gauge condition. In a pure perturbative framework and by using the Green function method a general scheme of solutions up to $(v/c)^4$ order is shown. Considering the Taylor expansion of a generic function $f$ it is possible to parameterize the solutions by derivatives of $f$. At Newtonian order, $(v/c)^2$, all more important topics about the Gauss and Birkhoff theorem are discussed. The corrections to "standard" gravitational potential ($tt$-component of metric tensor) generated by an extended uniform mass ball-like source are calculated up to $(v/c)^4$ order. The corrections, Yukawa and oscillating-like, are found inside and outside the mass distribution. At last when the limit $f\rightarrow R$ is considered the $f(R)$-gravity converges in General Relativity at level of Lagrangian, field equations and their solutions.

Quantum corrected spherical collapse: A phenomenological framework

A phenomenological framework is presented for incorporating quantum gravity motivated corrections into the dynamics of spherically symmetric collapse. The effective equations are derived from a variational principle that guarantees energy conservation and the existence of a Birkhoff theorem. The gravitational potential can be chosen as a function of the areal radius to yield specific non-singular static spherically symmetric solutions that generically have two horizons. For a specific choice of potential the effective stress energy tensor violates only the dominant energy condition. The violations are maximum near the inner horizon and die off rapidly. A numerical study of the quantum corrected collapse of a spherically symmetric scalar field in this case reveals that the modified gravitational potential prevents the formation of a central singularity and ultimately yields a static, mostly vacuum, spacetime with two horizons. The matter "piles up" on the inner horizon giving rise to mass inflation at late times. The Cauchy horizon is transformed into a null, weak singularity, but in contrast to Einstein gravity, the absence of a central singularity renders this null singularity stable.

Periodic orbits, invariant tori, and cylinders of Hamiltonian systems near integrable ones having a return map equal to the identity

Generically the return map of an integrable Hamiltonian system with two degrees of freedom in a Hamiltonian level foliated by invariant tori is a twist map. If we perturb such integrable Hamiltonian system inside the class of Hamiltonian systems with two degrees of freedom, then the Poincaré–Birkhoff theorem allows to determine which periodic orbits of the integrable can be prolonged to the perturbed one, and the KAM theory provides sufficient conditions in order that some invariant tori persist under sufficiently small perturbations. If some power of this return map is the identity, then in general for these degenerate Hamiltonian systems we cannot study which periodic orbits of the integrable can be prolonged to the perturbed one, or if some invariant tori persist. This paper studies the perturbation of integrable Hamiltonian systems with two degrees of freedom having some power of the return map equal to the identity. We show with two different models a way to study the prolongation of periodic orbits and of invariant tori or cylinders filled with periodic orbits for such kind of Hamiltonian systems. The main tool in this study is the averaging theory.

WPO, COV and IIA bargaining solutions for non-convex bargaining problems

We characterize all n-person multi-valued bargaining solutions, defined on the domain of all finite bargaining problems, and satisfying Weak Pareto Optimality (WPO), Covariance (COV), and Independence of Irrelevant Alternatives (IIA). We show that these solutions are obtained by iteratively maximizing nonsymmetric Nash products and determining the final set of points by so-called LDR decompositions. If, next, we assume the (set-theoretic) Axiom of Determinacy, then this class coincides with the class of iterated Nash bargaining solutions; but if we assume the Axiom of Choice then we are able to construct an additional large set of discontinuous and even nonmeasurable solutions. We show however that none of these nonmeasurable solutions can be defined in terms of set theoretic formulae. We next show that a number of existing results in the literature as well as some new results are implied by our approach. These include a characterization of all WPO, COV and IIA solutions—including single-valued ones—on the domain of all compact bargaining problems, and an extension of a theorem of Birkhoff characterizing translation invariant and homogeneous orderings.

Hamilton–Jacobi theory for continuation of magnetic field across a toroidal surface supporting a plasma pressure discontinuity

The vanishing of the divergence of the total stress tensor (magnetic plus kinetic) in a neighborhood of an equilibrium plasma containing a toroidal surface of discontinuity gives boundary and jump conditions that strongly constrain allowable continuations of the magnetic field across the surface. The boundary conditions allow the magnetic fields on either side of the discontinuity surface to be described by surface magnetic potentials, reducing the continuation problem to that of solving a Hamilton–Jacobi equation. The characteristics of this equation obey Hamiltonian equations of motion, and a necessary condition for the existence of a continued field across a general toroidal surface is that there exist invariant tori in the phase space of this Hamiltonian system. It is argued from the Birkhoff theorem that existence of such an invariant torus is also, in general, sufficient for continuation to be possible. An important corollary is that the rotational transform of the continued field on a surface of discontinuity must, generically, be irrational.

The Ergodic Theory Carnival

SummaryThe Birkhoff ergodic theorem, proved by George David Birkhoff in 1931, allows us to investigate the long-term behavior of certain dynamical systems. In this article, we explain what it means for a function to be ergodic, and we present Birkhoff's theorem. We construct models of activities typically found at carnivals and compare and contrast them by analyzing their ergodic theory properties. We use these carnival models to show how Birkhoff's ergodic theorem can be used to help a photographer set up her equipment to take pictures of all children on a carousel and to aid a magician in finding a lost jewel in a sticky mess of taffy.

New framework for studying spherically symmetric static solutions inf(R)gravity

We develop a new covariant formalism to treat spherically symmetric spacetimes in metric} f(R) theories of gravity. Using this formalism we derive the general equations for a static and spherically symmetric metric in a general f(R)-gravity. These equations are used to determine the conditions for which the Schwarzschild metric is the only vacuum solution with vanishing Ricci scalar. We also show that our general framework provides a clear way of showing that the Schwarzschild solution is not a unique static spherically symmetric solution, providing some incite on how the current form of Birkhoff's theorem breaks down for these theories.

Extensions of Birkhoff's theorem in 6D Gauss-Bonnet Gravity

We present a generalization of Birkhoff's theorem in the context of six-dimensional Gauss-Bonnet theory. Contrary to what we encounter in ordinary General Relativity, the presence of the higher curvature terms leads to novel constraints on the geometry of the admissible black hole horizons. As a result, we can have static solutions which depart from spherical symmetry, but must satisfy specific constraints involving the Weyl tensor of the four-dimensional horizon. After obtaining the general profile of the static solutions in the transverse space, we give explicit examples of such black hole horizons, which are in general of an anisotropic nature.

Cosmological constant and local gravity

We discuss the linearization of Einstein equations in the presence of a cosmological constant, by expanding the solution for the metric around a flat Minkowski space-time. We demonstrate that one can find consistent solutions to the linearized set of equations for the metric perturbations, in the Lorentz gauge, which are not spherically symmetric, but they rather exhibit a cylindrical symmetry. We find that the components of the gravitational field satisfying the appropriate Poisson equations have the property of ensuring that a scalar potential can be constructed, in which both contributions, from ordinary matter and $\Lambda > 0$, are attractive. In addition, there is a novel tensor potential, induced by the pressure density, in which the effect of the cosmological constant is repulsive. We also linearize the Schwarzschild-de Sitter exact solution of Einstein's equations (due to a generalization of Birkhoff's theorem) in the domain between the two horizons. We manage to transform it first to a gauge in which the 3-space metric is conformally flat and, then, make an additional coordinate transformation leading to the Lorentz gauge conditions. We compare our non-spherically symmetric solution with the linearized Schwarzschild-de Sitter metric, when the latter is transformed to the Lorentz gauge, and we find agreement. The resulting metric, however, does not acquire a proper Newtonian form in terms of the unique scalar potential that solves the corresponding Poisson equation. Nevertheless, our solution is stable, in the sense that the physical energy density is positive.

Spherical collapse and the halo model in braneworld gravity

We present a detailed study of the collapse of a spherical perturbation in DGP braneworld gravity for the purpose of modeling simulation results for the halo mass function, bias and matter power spectrum. The presence of evolving modifications to the gravitational force in form of the scalar brane-bending mode lead to qualitative differences to the collapse in ordinary gravity. In particular, differences in the energetics of the collapse necessitate a new, generalized method for defining the virial radius which does not rely on strict energy conservation. These differences and techniques apply to smooth dark energy models with w unequal -1 as well. We also discuss the impact of the exterior of the perturbation on collapse quantities due to the lack of a Birkhoff theorem in DGP. The resulting predictions for the mass function, halo bias and power spectrum are in good overall agreement with DGP N-body simulations on both the self-accelerating and normal branch. In particular, the impact of the Vainshtein mechanism as measured in the full simulations is matched well. The model and techniques introduced here can serve as practical tools for placing consistent constraints on braneworld models using observations of large scale structure.