Birkhoff's Theorem Research Articles

ON TIME DISLOCATION SOLUTION OF EINSTEIN FIELD EQUATIONS

The theory considered here is not Einstein general relativity, but is a Poincaré type gauge theory of gravity, therefore the Birkhoff theorem is not applied and the external solution is not vacuum spherically symmetric and tachyons may exist outside the core defect.

G. Birkhoff's theorems for M -solid varieties

Our aim is to prove two Birkhoff's like theorems for M-solid varieties of algebras, invented in [10], [11], generalizing some results of [5]. They have been presented on the Fifth Mathematical Conference "Workshop 97" at Gronow, Poland on June 27, 1997.

Theorems of Birkhoff type in Finsler spaces

In this paper we give a summary of basic definitions of Finsler geometry as well as a possible extension of General Relativity, in the vacuum case, for these spaces, followed by a brief account of symmetry imposition for Finsler metrics. We then make use of these results in order to generalise, for up to first order departures from the Riemannian setting, the well-known Birkhoff's theorem from General Relativity, thereby establishing a perturbative, first order version of this theorem for Finsler spaces, where some simplifying assumptions have been made. This result has been fully accomplished by means of computer algebra — actually, the very problem of explicitly determining non-Riemannian solutions for some generalised theory of gravity has only been made possible by computer algebra. The process is described in detail within the paper.

Ergodic theorems for individual random sequences

In the framework of the Kolmogorov's approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms we try to formulate probabilistic laws, i.e. statements of the form P{ω¦A(ω)} = 1, where A( ω) is some formula, in “pointwise” form “if ω is random then A( ω) holds”. Nevertheless, not all proofs of such laws can be directly translated into the algorithmic form. In [11] two examples have been distinguished — Birkhoff 's [2] ergodic theorem and Shannon-McMillan-Breiman theorem of information theory [1]. In this paper an analysis of algorithmic effectiveness of these theorems is given. We prove that Birkhoff's ergodic theorem is indeed in some strong sense “nonconstructive”. At the same time the claim to formulate probabilistic laws for algorithmically random sequences is not so restrictive. We present the versions of these laws for individual random sequences.

Discretization of Continuous Markov Chains and Markov Chain Monte Carlo Convergence Assessment

We show that continuous state-space Markov chains can be rigorously discretized into finite Markov chains. The idea is to subsample the continuous chain at renewal times related to small sets that control the discretization. Once a finite Markov chain is derived from the Markov chain Monte Carlo output, general convergence properties on finite state spaces can be exploited for convergence assessment in several directions. Our choice is based on a divergence criterion derived from Kemeny and Snell, which is first evaluated on parallel chains with a stopping time and then implemented, more efficiently, on two parallel chains only, using Birkhoff's pointwise ergodic theorem for stopping rules. The performance of this criterion is illustrated on three standard examples.

Birkhoff's theorem with Λ-term and Bertotti-Kasner space

If a Λ-term is included, the usual generalization of Schwarzschild space is not the only possible spherically symmetric vacuum solution. Another is Bertotti-Kasner space, as has been noted before but not explicitly demonstrated. The purpose of this note is to reformulate the unicity theorem and to discuss the extra solution.

A Birkhoff Theorem for Riemann Surfaces

A Birkhoff Theorem for Riemann Surfaces

A Semantic Approach to Order-sorted Rewriting

Order-sorted rewriting builds a nice framework to handle partially defined functions and subtypes. To be able to prove a critical-pair lemma and Birkhoff's completeness theorem, order-sorted rewriting was restricted to sort decreasing term rewriting systems. However, natural examples show that this approach is too restrictive.To solve this problem, we generalize well-sorted terms to semantically well-sorted terms and well-sorted substitutions to a restricted form of semantically well-sorted substitutions. Semantically well-sorted terms with respect to a set of equationsEare terms that denote well-defined elements in every algebra satisfyingE.We prove a critical-pair lemma and Birkhoff's completeness theorem for so-called range-unique signatures and arbitrary order-sorted rewriting systems. A transformation is given which allows us to obtain an equivalent range-unique signature from each non-range-unique one. We also show decidability and undecidability results.

Quantization of the string inspired dilaton gravity and the Birkhoff theorem

We develop a simple scheme of quantization for the dilaton CGHS model without scalar fields, that uses the Gupta-Bleuler approach for the string fields. This is possible because the constraints can be linearized classically, due to positivity conditions that are present in the model (and not in the general string case). There is no ambiguity nor anomalies in the quantization. The expectation values of the metric and dilaton fields obey the classical requirements, thus exhibiting at the quantum level the Birkhoff theorem.

A TWO-DIMENSIONAL REPRESENTATION OF FOUR-DIMENSIONAL GRAVITATIONAL WAVES

The Einstein equation in D dimensions, if restricted to the class of spacetimes possessing n = D - 2 commuting hypersurface-orthogonal Killing vectors, can be equivalently written as metric-dilaton gravity in two dimensions with n scalar fields. For n = 2, this result reduces to the known reduction of certain four-dimensional metrics which include gravitational waves. Here, we give such a representation which leads to a new proof of the Birkhoff theorem for plane-symmetric spacetimes, and which leads to an explanation, in which sense two (spin zero-) scalar fields in two dimensions may incorporate the (spin two-) gravitational waves in four dimensions. (This result should not be mixed up with well–known analogous statements where, however, the four-dimensional spacetime is supposed to be spherically symmetric, and then, of course, the equivalent two-dimensional picture cannot mimic any gravitational waves.) Finally, remarks on hidden symmetries in two dimensions are made.

A Poincaré–Birkhoff theorem on invariant plane continua

We prove that if $F$ is an orientation-preserving homeomorphism of the plane that leaves invariant an irreducible plane separating continuum $\Delta$, then, with the possible exception of three numbers, if $p/q$ is a reduced rational in the interior of the convex hull of the rotation set of $F\vert_{\Delta}$ (with respect to some lift) there are at least two distinct periodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$. This result also applies to certain nonseparating invariant continua.

Commodious Axiomatization of Quantifiers in Multiple-Valued Logic

We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers.

“Asymptotically non-minkowskian” bubbles with gravitational repulsion

The conventional approach describes the spherical domain walls by the same state equation as the flat ones. In such case they also must be gravitationally repulsive, what is seemingly in contradiction with Birkhoff's theorem. However this theorem is not valid for the solutions which do not display Minkowski geometry in the infinity. In this paper the solution of Einstein equations describing the stable gravitationally repulsive spherical domain wall is considered within the thin-wall formalism for the case of the non-Minkowskian asymptotics.

Distributivity and decomposability on the lattices satisfying the chain conditions

This short note will confirm the converse of Birkhoff's theorem and establish the following: Suppose L is a lattice satisfying the ascending chain condition (resp., the descending chain condition). Then L is distributive if and only if each element of L has a unique finite irredundant decomposition into irreducibles. This conclusion has improved some related results of Dilworth (1961). In addition, some interesting corrollaries are presented.

Doubly stochastic matrices in quantum mechanics

The general set of doubly stochastic matrices of order n corresponding to ordinary nonrelativistic quantum mechanical transition probability matrices is given. Lande's discussion of the nonquantal origin of such matrices is noted. Several concrete examples are presented for elementary and composite angular momentum systems with the focus on the unitary symmetry associated with such systems in the spirit of the recent work of Bohr and Ulfbeck. Birkhoff's theorem on doubly stochastic matrices of order n is reformulated in a geometrical language suitable for application to the subset of quantum mechanical doubly stochastic matrices. Specifically, it is shown that the set of points on the unit sphere in cartesian n!-space is surjective with the set of doubly stochastic matrices of order n. The question is raised, but not answered, as to what is the subset of points of this unit sphere that correspond to the quantum mechanical transition probability matrices, and what is the symmetry group of this subset of matrices.

An Addendum to Birkhoff's Theorem

In spherically symmetric systems, in spite of Birkhoff's Theorem, the rates of clocks in vacuum regions on either side of moving mass shells must be continually readjusted to stay in synchrony. This shows up as time-dependence of the metric in terms of the globally meaningful time coordinate.

A short proof of a theorem of Birkhoff

This paper gives a new proof of a theorem of G. Birkhoff: Every group \( {\goth G} \) can be represented as the automorphism group of a distributive lattice D; if \( {\goth G} \) is finite, D can be chosen to be finite. The new proof is short, and it is easily visualized.

Semilinear Duffing Equations Crossing Resonance Points

In this paper, using a generalized form of the Poincaré–Birkhoff theorem and a fixed point theorem, we prove, under weaker conditions, two theorems for the equationx+g(x)=p(t),p(t)≡p(t+2π), of which one shows the existence of a harmonic solution, the other that the equation may have an infinite number of harmonic solutions in the resonance case. This is an enhancement of the results already obtained.

A Birkhoff theorem for partial algebras via completion

An equational theory (a ‘Birkhoff theorem’) for functorial partial algebras is established via the corresponding theory for functorial total algebras.

Particle dynamics in a class of two-dimensional gravity theories.

We provide a method to determine the motion of a classical massive particle in a background geometry of 2-dimensional gravity theories, for which the Birkhoff theorem holds. In particular, we get the particle trajectory in a continuous class of 2-dimensional dilaton gravity theories that includes the Callan-Giddings-Harvey-Strominger (CGHS) model, the Jackiw-Teitelboim (JT) model, and the $d$-dimensional $s$-wave Einstein gravity. The explicit trajectory expressions for these theories are given along with the discussions on the results.