Birkhoff's Theorem Research Articles

Spherical collapse in modified gravity with the Birkhoff theorem

We study structure formation in a phenomenological model of modified gravity which interpolates between LambdaCDM and phenomenological DGP-gravity. Generalisation of spherical collapse by using the Birkhoff-theorem along with the modified growth equation shows that the overdensity for spherical collapse delta_c in these models is significantly lowered compared to LambdaCDM, leading to enhanced number densities of massive clusters and enhanced cluster merging rates. We find that delta_c(z) is well fitted by a function of the form delta_c(z) = a - b\exp(-cz). We examine the sensitivity of PLANCK's and SPT's Sunyaev-Zel'dovich survey to constrain the modified gravity parameterisation and find that these experiments can easily distinguish between models with a cosmological constant and modified gravity, if prior constraints from CMB temperature and polarisation anisotropies are included.

A constructive version of Birkhoff's theorem

AbstractA version of Birkhoff's theorem is proved by constructive, predicative, methods. The version we prove has two conditions more than the classical one. First, the class considered is assumed to contain a generic family, which is defined to be a set‐indexed family of algebras such that if an identity is valid in every algebra of this family, it is valid in every algebra of the class. Secondly, the class is assumed to be closed under inductive limits. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Young subgroups for reversible computers

We consider the symmetric group $S_n$ in the special case where $n$ is composite: $n = pq$ (both $p$ and $q$ being integer). Applying Birkhoff's theorem, we prove that an arbitrary element of $S$ pq can be decomposed into a product of three permutations, the first and the third being elements of the Young subgroup $S_p^q$, whereas the second one is an element of the dual Young subgroup $S_q^p$. This leads to synthesis methods for arbitrary reversible logic circuits of logic width $w$. These circuits form a group isomorphic to $S$2w. A particularly efficient synthesis is found by choosing $p=2$ and thus $q=2$w−1. The approach illustrates a direct link between combinatorics, group theory, and reversible computing.

The cosmic horizon

The cosmological principle, promoting the view that the universe is homogeneous and isotropic, is embodied within the mathematical structure of the Robertson-Walker (RW) metric. The equations derived from an application of this metric to the Einstein Field Equations describe the expansion of the universe in terms of comoving coordinates, from which physical distances may be derived using a time-dependent expansion factor. These coordinates, however, do not explicitly reveal properties of the cosmic spacetime manifested in Birkhoff's theorem and its corollary. In this paper, we compare two forms of the metric--written in (the traditional) comoving coordinates, and a set of observer-dependent coordinates--first for the well-known de Sitter universe containing only dark energy, and then for a newly derived form of the RW metric, for a universe with dark energy and matter. We show that Rindler's event horizon--evident in the co-moving system--coincides with what one might call the "curvature horizon" appearing in the observer-dependent frame. The advantage of this dual prescription of the cosmic spacetime is that with the latest WMAP results, we now have a much better determination of the universe's mass-energy content, which permits us to calculate this curvature with unprecedented accuracy. We use it here to demonstrate that our observations have probed the limit beyond which the cosmic curvature prevents any signal from having ever reached us. In the case of de Sitter, where the mass-energy density is a constant, this limit is fixed for all time. For a universe with a changing density, this horizon expands until de Sitter is reached asymptotically, and then it too ceases to change.

Non-metric gravity: II. Spherically symmetric solution, missing mass and redshifts of quasars

We continue the study of the non-metric theory of gravity introduced by Krasnov (2006 Preprint hep-th/0611182) and obtain its general spherically symmetric vacuum solution. It respects the analog of the Birkhoff theorem, i.e. the vacuum spherically symmetric solution is necessarily static. As in general relativity, the spherically symmetric solution is seen to describe a black hole. The exterior geometry is essentially the same as in the Schwarzschild case, with power-law corrections to the Newtonian potential. The behaviour inside the black-hole region is different from the Schwarzschild case in that the usual spacetime singularity gets replaced by a singular surface of a new type, where all basic fields of the theory remain finite but metric ceases to exist. The theory does not admit arbitrarily small black holes: for small objects, the curvature on the would-be horizon is so strong that non-metric modifications prevent the horizon from being formed. The theory allows for modifications of gravity of a very interesting nature. We discuss three physical effects, namely (i) correction to Newton's law in the neighborhood of the source, (ii) renormalization of effective gravitational and cosmological constants at large distances from the source and (iii) additional redshift factor between spatial regions of different curvature. The first two effects can be responsible, respectively, for the observed anomaly in the acceleration of the Pioneer spacecraft and for the alleged missing mass in spiral galaxies and other astrophysical objects. The third effect can be used to propose a non-cosmological explanation of high redshifts of quasars and gamma-ray bursts.

Newtonian limit off(R)gravity

A general analytic procedure is developed to deal with the Newtonian limit of $f(R)$ gravity. A discussion comparing the Newtonian and the post-Newtonian limit of these models is proposed in order to point out the differences between the two approaches. We calculate the post-Newtonian parameters of such theories without any redefinition of the degrees of freedom, in particular, without adopting some scalar fields and without any change from Jordan to Einstein frame. Considering the Taylor expansion of a generic $f(R)$ theory, it is possible to obtain general solutions in terms of the metric coefficients up to the third order of approximation. In particular, the solution relative to the ${g}_{tt}$ component gives a gravitational potential always corrected with respect to the Newtonian one of the linear theory $f(R)=R$. Furthermore, we show that the Birkhoff theorem is not a general result for $f(R)$ gravity since time-dependent evolution for spherically symmetric solutions can be achieved depending on the order of perturbations. Finally, we discuss the post-Minkowskian limit and the emergence of massive gravitational wave solutions.

Cosmological perturbations in Palatini-modified gravity

Two approaches to the study of cosmological density perturbations in modified theories of Palatini gravity have recently been discussed. These utilize, respectively, a generalization of Birkhoff's theorem and a direct linearization of the gravitational field equations. In this paper these approaches are compared and contrasted. The general form of the gravitational Lagrangian for which the two frameworks yield identical results in the long-wavelength limit is derived. This class of models includes the case where the Lagrangian is a power-law of the Ricci curvature scalar. The evolution of density perturbations in theories of the type f(R) = R − c/Rb is investigated numerically. It is found that the results obtained by the two methods are in good agreement on sufficiently large scales when the values of the parameters (b, c) are consistent with current observational constraints. However, this agreement becomes progressively poorer for models that differ significantly from the standard concordance model and as smaller scales are considered.

Exact solutions in five-dimensional axi-dilaton gravity with Euler–Poincaré term

We examine the effective field equations that are obtained from the axi-dilaton gravity action with a second-order Euler–Poincaré term and a cosmological constant in all higher dimensions. We solve these equations for five-dimensional spacetimes possessing homogeneity and isotropy in their three-dimensional subspaces. For a number of interesting special cases, we show that the solutions fall into two main classes: the first class consists of time-dependent solutions with spherical or hyperboloidal symmetry which require certain fine-tuning relations between the coupling constants of the model and the cosmological constant. Solutions in the second class are locally static and prove the validity of Birkhoff's staticity theorem in the axi-dilaton gravity. We also give a class of static solutions, among them the well known charged black hole solutions with a cosmological constant in which the usual electric charge is superseded by an axion charge.

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincare–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)xγ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded.

Analogue of the theorem of Birkhoff in a metric nonsymmetric theory of the gravitational-electromagnetic field

We prove that a spherically symmetric exterior solution of the field equations of a metric nonsymmetric theory of gravitation coupled with the electromagnetic field is necessarily static.

A VACUUM-LIKE CONFIGURATION IN GENERAL RELATIVITY AS A MANIFESTATION OF A LORENTZ-INVARIANT MODE OF FIVE-DIMENSIONAL GRAVITY

A Lorentz-invariant cosmological model is constructed within the framework of five-dimensional gravity. The five-dimensional theorem which is analogical to the generalized Birkhoff theorem is proved, that corresponds to the Kaluza's "cylinder condition." The five-dimensional vacuum Einstein equations have an integral of motion corresponding to this symmetry, the integral of motion is similar to the mass function in general relativity (GR). Space closure with respect to the extra dimensionality follows from the requirement of the absence of a conical singularity. Thus, the Kaluza–Klein (KK) model is realized dynamically as a Lorentz-invariant mode of five-dimensional general relativity. After the dimensional reduction and conformal mapping the model is reduced to the GR configuration. It contains a scalar field with a vanishing conformally invariant energy–momentum tensor on the flat space–time background. This zero mode can be interpreted as a vacuum configuration in GR. As a result the vacuum-like configuration in GR can be considered as a manifestation of the Lorentz-invariant empty five-dimensional space.

Exterior spacetime for stellar models in five-dimensional Kaluza–Klein gravity

It is well known that Birkhoff's theorem is no longer valid in theories with more than four dimensions. Thus, in these theories the effective four-dimensional picture allows the existence of different possible, non-Schwarzschild, scenarios for the description of the spacetime outside of a spherical star, contrary to four-dimensional general relativity. We investigate the exterior spacetime of a spherically symmetric star in the context of Kaluza–Klein gravity. We take a well-known family of static spherically symmetric solutions of the Einstein equations in an empty five-dimensional universe, and analyse possible stellar exteriors that are conformal to the metric induced on four-dimensional hypersurfaces orthogonal to the extra dimension. We show that all these non-Schwarzschild exteriors can continuously be matched with the interior of the star, indicating that the matching conditions at the boundary of a star do not require an unique exterior. Then, without making any assumptions about the interior solution, we prove the following statement: the condition that in the weak-field limit we recover the usual Newtonian physics singles out an unique exterior. This exterior is ‘similar’ to a Scharzschild vacuum in the sense that it has no effect on gravitational interactions. However, it is more realistic because instead of being absolutely empty, it is consistent with the existence of quantum zero-point fields (Wesson 1992 Phys. Essays, Orion 5 591). We also examine the question of how the deviation from the Schwarzschild vacuum exterior would affect the parameters of a neutron star. In the context of a model star of uniform density, we show that the general relativity upper limit M/R < 4/9 is significantly increased as we go away from the Schwarzschild vacuum exterior. We find that, in principle, the compactness limit of a star can be larger than 1/2, without being a black hole. The generality of our approach is also discussed.

Birkhoff’s Theorem, Many-Body Response Functions, and the Ergodic Condition

The ergodic hypothesis is viewed as one that is physically measurable through scattering by an external probe. Linear response theory is used to derive a general ergodic condition on the response functions. It is shown that the same condition is also implied by Birkhoff's theorem. This coincidence allows us to shed light on the abstract terms of the theorem via classical many-body models.

HOW TO BYPASS BIRKHOFF THROUGH EXTRA DIMENSIONS: A SIMPLE FRAMEWORK FOR INVESTIGATING THE GRAVITATIONAL COLLAPSE IN VACUUM

It is fair to say that our current mathematical understanding of the dynamics of gravitational collapse to a black hole is limited to the spherically symmetric situation and, in fact, even in this case much remains to be learned. The reason is that Einstein's equations become tractable only if they are reduced to a (1 + 1)-dimensional system of partial differential equations. Owing to this technical obstacle, very little is known about the collapse of pure gravitational waves because by Birkhoff's theorem there is no spherical collapse in vacuum. In this essay, we describe a new cohomogeneity-two symmetry reduction of the vacuum Einstein equations in five and higher odd dimensions which evades Birkhoff's theorem and admits time-dependent asymptotically flat solutions. We argue that this model provides an attractive (1 + 1)-dimensional geometric setting for investigating the dynamics of gravitational collapse in vacuum.

Metric nonsymmetric theory of gravitation: the analogue of the theorem of Birkhoff

We prove that a spherically symmetric solution of the field equations of the metric nonsymmetric theory of gravitation developed previously is necessarily static.This is the analogue of the well known Birkhoff theorem in general relativity.

Spherically symmetric solutions to fourth-order theories of gravity

Solutions to the field equations generated from Lagrangians of the form f(R) are considered. The spherically symmetric solutions to these equations are discussed, paying particular attention to features that differ from the standard Schwarzschild solution. The asymptotic form of solutions will be described, as will the lack of validity of Birkhoff's theorem. Exact solutions are presented which illustrate these points and the stability and equations of motion of particles in these spacetimes are investigated.

Erratum to “Generalizations of the Poincaré–Birkhoff theorem”

Other results of [1] which rely on Theorem (2.1) are correct as stated and follow from the result above. A disk U in the covering space A is called positively returning for f if f(U)∩U = ∅ and fn(U)∩T k(U) = ∅ for some n, k > 0, where T is the generator of the infinite cyclic group of covering translations. Negatively returning disks are defined similarly. The difference in the statement of the theorem above and the statement of Theorem (2.1) of [1] is the additional requirement in item 3 that the positively and negatively returning disks in the covering space are lifts of disks in the annulus. Equivalently the positively and negatively returning disks in the covering space must be disjoint from their image under T k, k = 0 (standard arguments show it suffices that T (U) ∩ U = ∅). This additional hypothesis is satisfied in other results from [1] which use Theorem (2.1). I do not know whether Theorem (2.1) of [1], as originally stated, is true or not.

Birkhoff’s theorem for a family of probability spaces

The paper extends Birkhoff's theorem on doubly stochastic matrices to some countable families of discrete probability spaces with nonempty intersections. We join every two elements lying in the same probability space by an edge and formulate our results in terms of the obtained graph.

Jordan–Hölder, modularity and distributivity in non-commutative algebra

A study of lattices of subgroups or subrings adequate for non-commutative homological algebra can be pursued in a setting of weakly exact categories, which extend the Puppe-exact ones [D. Puppe, Korrespondenzen in abelschen Kategorien, Math. Ann. 148 (1962) 1–30; B. Mitchell, Theory of Categories, Academic Press, New York, 1965; P. Freyd, A. Scedrov, Categories, Allegories, North-Holland Publishing Co., Amsterdam, 1990] and the semi-abelian ones [G. Janelidze, L. Márki, W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367–386; F. Borceux, A survey of semi-abelian categories, in: Galois Theory, Hopf Algebras, and Semiabelian Categories, in: Fields Inst. Commun., vol. 43, Amer. Math. Soc., Providence, RI, 2004, pp. 27–60; F. Borceux, D. Bourn, Mal’cev, Protomodular, homological and semi-abelian categories, in: Mathematics and its Applications, vol. 566, Kluwer Academic Publishers, Dordrecht, 2004], and are essentially based on a notion of γ -category introduced by Burgin [M.S. Burgin, Categories with involution and correspondences in γ -categories, Tr. Mosk. Mat. Obs. 22 (1970) 161–228; Trans. Moscow Math. Soc. 22 (1970) 181–257]. In this context, subobjects form w-modular w-lattices, equipped with a normality relation. The free w-modular w-lattice generated by two chains with normality conditions is determined and proved to be weakly distributive, by a construction inspired by the well-known Birkhoff theorem for free modular lattices [G. Birkhoff, Lattice Theory, 3rd ed., in: Amer. Math. Soc. Coll. Publ., vol. 25, 1973]. We show that this theorem is relevant for the study of double filtrations, much in the same way as the Birkhoff theorem in the commutative case; similarly, it should be of use in the study of spectral sequences.

Algorithmic combinatorics based on slicing posets

A combinatorial problem usually requires enumerating, counting or ascertaining existence of structures that satisfy a given property B in a set of structures L. This paper describes a technique based on a generalization of Birkhoff's theorem of representation of finite distributive lattices that can be used for solving such problems mechanically and efficiently. Specifically, we give an efficient (polynomial time) algorithm to enumerate, count or detect structures that satisfy B when the total set of structures is large but the set of structures satisfying B is small. We illustrate our techniques by analyzing problems in integer partitions, set families, and set of permutations.