Showing papers in "Annalen der Physik in 2005"

Brownian motion: a paradigm of soft matter and biological physics

Abstract: This is a pedagogical introduction to Brownian motion on the occasion of the 100th anniversary of Einstein's 1905 paper on the subject. After briefly reviewing Einstein's work in a contemporary context, we pursue several lines of further developments and applications to soft condensed matter and biology. Over the last century Brownian motion has been promoted from an odd curiosity of marginal scientific interest to a guiding theme pervading all of the modern life sciences.

The variable viscoelasticity oscillator

Abstract: The complex dynamics of a variable viscoelasticity oscillator is studied using the novel concept of Variable-Order (VO) Calculus. The damping force in the oscillator varies continuously between the elastic and viscous regimes depending on the position of the mass. The oscillator considered here is composed of a linear spring of stiffness k that inputs a restitutive force Fk = -k x, a VO damper of order q(x(t)) that generates a damping force Fq = -cq q(x(t))x, and a mass m. A modified Runge-Kutta method is used in conjunction with a trapezoidal numerical integration technique to yield a second-order accurate method for the solution of the resulting VO Differential Equation (VODE). The VO oscillator is also modelled using a Constant Order (CO) formulation where a number of CO fractional order differentials are weighted to simulate the VO behavior. The CO formulation asymptotically approaches the VO results when a relatively large number of weights is used. For the viscoelastic range of 0 ≤ q ≤ 1, the dynamics of the oscillator is well approximated by the CO formulation when 5 or more fractional terms are included (e.g., 0, 1/4, 1/2, 3/4, and 1).

Of pots and holes: Einstein's bumpy road to general relativity

Abstract: Readers of this volume will notice that it contains only a few papers on general relativity. This is because most papers documenting the genesis and early development of general relativity were not published in Annalen der Physik.After Einstein took up his new prestigious position at the PrussianAcademy of Sciences in the spring of 1914, the Sitzungsberichte of the Berlin academy almost by default became the main outlet for his scientific production. Two of the more important papers on general relativity, however, did find their way into the pages of the Annalen [35,41].Although I shall discuss both papers in this essay, the main focus will be on [35], the first systematic exposition of general relativity, submitted in March 1916 and published in May of that year. Einstein’s first paper on a metric theory of gravity, co-authored with his mathematician friend Marcel Grossmann, was published as a separatum in early 1913 and was reprinted the following year in Zeitschrift fur Mathematik und Physik [50,51]. Their second (and last) joint paper on the theory also appeared in this journal [52]. Most of the formalism of general relativity as we know it today was already in place in this Einstein-Grossmann theory. Still missing were the generally-covariant Einstein field equations. As is clear from research notes on gravitation from the winter of 1912–1913 preserved in the so-called “Zurich Notebook,” Einstein had considered candidate field equations of broad if not general covariance, but had found all such candidates wanting on physical grounds. In the end he had settled on equations constructed specifically to be compatible with energy-momentum conservation and with Newtonian theory in the limit of weak static fields, even though it remained unclear whether these equations would be invariant under any non-linear transformations. In view of this uncertainty, Einstein and Grossmann chose a fairly modest title for their paper: “Outline (“Entwurf”) of a Generalized Theory of Relativity and of a Theory of Gravitation.” The Einstein-Grossmann theory and its fields equations are therefore also known as the “Entwurf” theory and the “Entwurf” field equations. Much of Einstein’s subsequent work on the “Entwurf” theory went into clarifying the covariance properties of its field equations. By the following year he had convinced himself of three things. First, generallycovariant field equations are physically inadmissible since they cannot determine the metric field uniquely. This was the upshot of the so-called “hole argument” (“Lochbetrachtung”) first published in an appendix to [51]. Second, the class of transformations leaving the “Entwurf” field equations invariant was as broad ∗ E-mail: janss011@tc.umn.edu 1 An annotated transcription of the gravitational portion of the “Zurich Notebook” is published as Doc.10 in [11]. For facsimile reproductions of these pages, a new transcription, and a running commentary, see [89]. 2 See Sect. 2 for further discussion of the hole argument.

On the interaction of mesoscopic quantum systems with gravity

Abstract: We review the different aspects of the interaction of mesoscopic quantum systems with gravitational fields. We first discuss briefly the foundations of general relativity and quantum mechanics. Then, we consider the non-relativistic expansions of the Klein-Gordon and Dirac equations in the post-Newtonian approximation. After a short overview of classical gravitational waves, we discuss two proposed interaction mechanisms: (i) the use of quantum fluids as generator and/or detector of gravitational waves in the laboratory, and (ii) the inclusion of gravitomagnetic fields in the study of the properties of rotating superconductors. The foundations of the proposed experiments are explained and evaluated.

Einstein's invention of Brownian motion

Abstract: Einstein’s 1905 paper on Brownian motion was an essential contribution to the foundation of modern atomism [20]. Atomism as understood in science today presupposes, like its predecessor rooted in the theories of nature from Greek antiquity and from early modern times, that matter is constituted by small entities. But it no longer assumes that the properties and the behavior of these entities can simply be inferred from the familiar physical laws governing our macroscopic environment, nor that a description of matter in terms of its atomistic constituents can be exhaustive. Einstein succeeded in interpreting the irregular movements of small particles suspended in a liquid as visible evidence for themolecularmotions constituting the heat of a ponderable body according to the kinetic theory of heat. But he did so by radically changing the understanding of these irregular motions which he no longer conceived as being characterized by a velocity in the classical sense but as a stochastic process that can only be described with the help of statistical methods. It is therefore not surprising that Einstein’s work on Brownian motion also became one of the pillars of modern statistical thermodynamics and, more generally, of the physics of stochastic processes. In the sequel to his groundbreaking work, Einstein published several other related articles, extending the subject to Brownian motion in condensers and the fluctuations of heat radiation. His work aroused widespread interest among physicists and chemists, as indicated by Einstein’s correspondence with other scientists interested in the subject, in particular Conrad Rontgen, Richard Lorenz, Marian von Smoluchowski, and The Svedberg. In 1906 the Polish physicist von Smoluchowski submitted a paper on the kinetic theory of Brownian motion to the Annalen that was stimulated by Einstein’s papers but represented results which he had derived independently. While Smoluchowski’s argument was different from Einstein’s, his results were – apart from a numerical factor – essentially equivalent. Einstein’s interpretation of Brownian motion soon also received striking experimental confirmation by Jean Perrin and others. This success furthered the general acceptance of atomism and helped to convert the then still numerous skeptics. Indeed, while in the nineteenth century atomismwas widely employed as a working hypothesis in numerous fields of physics and chemistry, it was accepted as a physical reality only after the impressive accumulation

The gravitational energy‐momentum tensor and the gravitational pressure

Abstract: In the framework of the teleparallel equivalent of general relativity it is possible to establish the energy-momentum tensor of the gravitational field. This tensor has the following essential properties: (1) it is identified directly in Einstein's field equations; (2) it is conserved and traceless; (3) it yields expressions for the energy and momentum of the gravitational field; (4) is is free of second (and highest) derivatives of the field variables; (5) the gravitational and matter energy-momentum tensors take place in the field equations on the same footing; (6) it is unique. However it is not symmetric. We show that the spatial components of this tensor yield a consistent definition of the gravitational pressure.

Cosmological constant and vacuum energy

Abstract: The general thermodynamic analysis of the quantum vacuum, which is based on our knowledge of the vacua in condensed-matter systems, is consistent with the Einstein earlier view on the cosmological constant. In the equilibrium Universes the value of the cosmological constant is regulated by matter. In the empty Universe, the vacuum energy is exactly zero, λ=0. The huge contribution of the zero point motion of the quantum fields to the vacuum energy is exactly cancelled by the higher-energy degrees of freedom of the quantum vacuum. In the equilibrium Universes homogeneously filled by matter, the vacuum is disturbed, and the energy density of the vacuum becomes proportional to that of matter, λ=ρvac∼ρmatter. This consideration applies to any vacuum in equilibrium irrespective of whether the vacuum is false or true, and is valid both in Einstein's general theory of relativity and within the special theory of relativity, i.e. in a world without gravity.

The Bose-Hubbard Model: From Josephson Junction Arrays to Optical Lattices

Abstract: The Bose-Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We review some of its properties with emphasis on the implications on quantum phase transitions of Josephson junction arrays and quantum dynamics of topological excitations as well as the properties of ultra-cold atoms in optical lattices.

Gravity and the Quantum Vacuum Inertia Hypothesis

Abstract: In previous work it has been shown that the electromagnetic quantum vacuum, or electromagnetic zero-point field, makes a contribution to the inertial reaction force on an accelerated object. We show that the result for inertial mass can be extended to passive gravitational mass. As a consequence the weak equivalence principle, which equates inertial to passive gravitational mass, appears to be explainable. This in turn leads to a straightforward derivation of the classical Newtonian gravitational force. We call the inertia and gravitation connection with the vacuum fields the quantum vacuum inertia hypothesis. To date only the electromagnetic field has been considered. It remains to extend the hypothesis to the effects of the vacuum fields of the other interactions. We propose an idealized experiment involving a cavity resonator which, in principle, would test the hypothesis for the simple case in which only electromagnetic interactions are involved. This test also suggests a basis for the free parameter η(ν) which we have previously defined to parametrize the interaction between charge and the electromagnetic zero-point field contributing to the inertial mass of a particle or object.

Quadratic metric-affine gravity

Abstract: We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp-wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the neutrino. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Spin equation and its solutions

Abstract: The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0 + 1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.

Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes [AdP 35, 898 (1911)]

Abstract: The two English translations appear to have been done independently, although much of the Princeton text is quite similar to the Dover text. Both just used “cut and paste” to produce the Figures. Neither translation seemed to me to be sufficiently accurate to fully convey what Einstein wrote. For these reasons I felt that a new, and freely available, translation would be helpful. In doing the translation I made substantial use of the Dover text, which seems somewhat better than the Princeton one. Where problems occurred I also checked the Princeton text. For anyone with reasonable familiarity with German it is, I think, still a good idea to download a copy of the original. Einstein was, as is well-known, a quite original writer. His exact choice of phrases and individual wording is often important. The purely technical content is, in any case, in the equations. Both translations almost always correctly transcribed the equations, although they both just transcribed typos of the math symbols in the text.

Semiclassical quantum gravity: statistics of combinatorial Riemannian geometries

Abstract: This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at “quantum scales” and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a “semiclassical” state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity.

Tidal Dynamics of Relativistic Flows Near Black Holes

Abstract: We point out novel consequences of general relativity involving tidal dynamics of ultrarelativistic relative motion Specifically, we use the generalized Jacobi equation and its extension to study the force-free dynamics of relativistic flows near a massive rotating source We show that along the rotation axis of the gravitational source, relativistic tidal effects strongly decelerate an initially ultrarelativistic flow with respect to the ambient medium, contrary to Newtonian expectations Moreover, an initially ultrarelativistic flow perpendicular to the axis of rotation is strongly accelerated by the relativistic tidal forces The astrophysical implications of these results for jets and ultrahigh energy cosmic rays are briefly mentioned

On the axioms of topological electromagnetism

Abstract: The axioms of topological electromagnetism that were given by Hehl, Obukhov, and Rubilar are refined by the use of geometrical and topological notions that are found on orientable manifolds. The central problem of defining the spacetime electromagnetic constitutive law in terms of the geometrical and topological structure of the spacetime manifold is elaborated upon in the linear and nonlinear cases. The manner by which the spacetime metric might follow from the electromagnetic constitutive law is examined in the linear case. The possibility that the intersection form of the spacetime manifold might play a role in defining a topological basis for a nonlinear electromagnetic constitutive law is explored. The manner by which electromagnetic wave motion relates to the geometric structure is also discussed.

Special Relativity and Lorentz Invariance

Abstract: Lorentz Invariance, the main ingredient of Special Relativity, is one of the pillars of modern physics. Though Special Relativity has been replaced by General Relativity, Lorentz Invariance is still valid locally. All physical fields have to obey the laws of local Lorentz Invariance. This is also the reason why gravity within the theory of General Relativity has to be described by the metric tensor. Here we give a short introduction into the early experiments and show that they disproved the exact validity of the Galilean framework for the description of classical mechanics. After a short summary of Special Relativity, the procedure of synchronization is analyzed. It is emphasized that no experiment should depend on the synchronization. Otherwise it might be possible to simulate or compensate effects by choosing another synchronization. Accordingly, the requirement of synchronization independence is a guideline for the choice of appropriate measurable quantities which then reveal relativistic physics in an unambiguous manner. Examples are given. In a subsequent article the modern experiments implementing this kind of notions will be discussed. Also some remarks are made on the importance of Lorentz Invariance in daily life. Finally we comment on possible violations of Lorentz Invariance and their measurability.

Green functions for nearest- and next-nearest-neighbor hopping on the Bethe lattice

Abstract: We calculate the local Green function for a quantum-mechanical particle with hopping between nearest and next-nearest neighbors on the Bethe lattice, where the on-site energies may alternate on sublattices. For infinite connectivity the renormalized perturbation expansion is carried out by counting all non-self-intersecting paths, leading to an implicit equation for the local Green function. By integrating out branches of the Bethe lattice the same equation is obtained from a path integral approach for the partition function. This also provides the local Green function for finite connectivity. Finally, a recently developed topological approach is extended to derive an operator identity which maps the problem onto the case of only nearest-neighbor hopping. We find in particular that hopping between next-nearest neighbors leads to an asymmetric spectrum with additional van-Hove singularities.

Quantum phase diagram for homogeneous Bose-Einstein condensate

Abstract: We calculate the quantum phase transition for a homogeneous Bose gas in the plane of s-wave scattering length as and temperature T. This is done by improving a one-loop result near the interaction-free Bose-Einstein critical temperature Tc(0) with the help of recent high-loop results on the shift of the critical temperature due to a weak atomic repulsion based on variational perturbation theory. The quantum phase diagram shows a nose above Tc(0), so that we predict the existence of a reentrant transition above Tc(0), where an increasing repulsion leads to the formation of a condensate.