Relativistic Dyson Rings and Their Black Hole Limit

In this Letter we investigate uniformly rotating, homogeneous and axisymmetric relativistic fluid bodies with a toroidal shape. The corresponding field equations are solved by means of a multi-domain spectral method, which yields highly accurate numerical solutions. For a prescribed, sufficiently large ratio of inner to outer coordinate radius, the toroids exhibit a continuous transition to the extreme Kerr black hole. Otherwise, the most relativistic configuration rotates at the mass-shedding limit. For a given mass-density, there seems to be no bound to the gravitational mass as one approaches the black-hole limit and a radius ratio of unity.Comment: 13 pages, 1 table, 5 figures, v2: some discussion and two references added, accepted for publication in Astrophys. J. Let

Functions of linear operators: Parameter differentiation

We derive a useful expression for the matrix elements $[\frac{\partial f[A(t)]}{\partial t}]_{i j}$ of the derivative of a function $f[A(t)]$ of a diagonalizable linear operator $A(t)$ with respect to the parameter $t$. The function $f[A(t)]$ is supposed to be an operator acting on the same space as the operator $A(t)$. We use the basis which diagonalizes A(t), i.e., $A_{i j}=\lambda_i \delta_{i j}$, and obtain $[\frac{\partial f[A(t)]}{\partial t}]_{i j}=[\frac{\partial A}{\partial t}]_ {i j}\frac{f(\lambda_j) - f(\lambda_i)} {\lambda_j - \lambda_i}$. In addition to this, we show that further elaboration on the (not necessarily simple) integral expressions given by Wilcox 1967 (who basically considered $f[A(t)]$ of the exponential type) and generalized by Rajagopal 1998 (who extended Wilcox results by considering $f[A(t)]$ of the $q$-exponential type where $\exp_q(x) \equiv [1+(1-q)x]^{1/(1-q)}$ with $q \in {\cal {R}}$; hence, $\exp_1 (x)=\exp(x))$ yields this same expression. Some of the lemmas first established by the above authors are easily recovered.Comment: No figure

Measurement in biological systems from the self-organisation point of view

Measurement in biological systems became a subject of concern as a consequence of numerous reports on limited reproducibility of experimental results. To reveal origins of this inconsistency, we have examined general features of biological systems as dynamical systems far from not only their chemical equilibrium, but, in most cases, also of their Lyapunov stable states. Thus, in biological experiments, we do not observe states, but distinct trajectories followed by the examined organism. If one of the possible sequences is selected, a minute sub-section of the whole problem is obtained, sometimes in a seemingly highly reproducible manner. But the state of the organism is known only if a complete set of possible trajectories is known. And this is often practically impossible. Therefore, we propose a different framework for reporting and analysis of biological experiments, respecting the view of non-linear mathematics. This view should be used to avoid overoptimistic results, which have to be consequently retracted or largely complemented. An increase of specification of experimental procedures is the way for better understanding of the scope of paths, which the biological system may be evolving. And it is hidden in the evolution of experimental protocols.Comment: 13 pages, 5 figure

Late Time Tail of Wave Propagation on Curved Spacetime

The late time behavior of waves propagating on a general curved spacetime is studied. The late time tail is not necessarily an inverse power of time. Our work extends, places in context, and provides understanding for the known results for the Schwarzschild spacetime. Analytic and numerical results are in excellent agreement.Comment: 11 pages, WUGRAV-94-1

Brane Gravitational Extension of Dirac's "Extensible Model of the Electron"

A gravitational extension of Dirac's "Extensible model of the electron" is presented. The Dirac bubble, treated as a 3-dim electrically charged brane, is dynamically embedded within a 4-dim $Z_{2}$-symmetric Reissner-Nordstrom bulk. Crucial to our analysis is the gravitational extension of Dirac's brane variation prescription; its major effect is to induce a novel geometrically originated contribution to the energy-momentum tensor on the brane. In turn, the effective potential which governs the evolution of the bubble exhibits a global minimum, such that the size of the bubble stays finite (Planck scale) even at the limit where the mass approaches zero. This way, without fine-tuning, one avoids the problem so-called 'classical radius of the electron'.Comment: 6 PRD pages, 4 figures; References adde

Plasmarings as dual black rings

We construct solutions to the relativistic Navier-Stokes equations that describe the long wavelength collective dynamics of the deconfined plasma phase of N=4 Yang Mills theory compactified down to d=3 on a Scherk-Schwarz circle and higher dimensional generalisations. Our solutions are stationary, axially symmetric spinning balls and rings of plasma. These solutions, which are dual to (yet to be constructed) rotating black holes and black rings in Scherk-Schwarz compactified AdS(5) and AdS(6), and have properties that are qualitatively similar to those of black holes and black rings in flat five dimensional supergravity.Comment: 40 pages, 40 figures. (v2) Correction to black brane equation of state, additional reference

Finsler and Lagrange Geometries in Einstein and String Gravity

We review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of ''orthodox'' physicists. Although the bulk of former models of Finsler-Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modelled equivalently on Riemann-Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/ Kahler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modelling Lagrange-Finsler structures with solitonic pp-waves and speculate on their physical meaning.Comment: latex 2e, 11pt, 44 pages; accepted to IJGMMP (2008) as a short variant of arXiv:0707.1524v3, on 86 page

Linear Stability of Triangular Equilibrium Points in the Generalized Photogravitational Restricted Three Body Problem with Poynting-Robertson Drag

In this paper we have examined the linear stability of triangular equilibrium points in the generalised photogravitational restricted three body problem with Poynting-Robertson drag. We have found the position of triangular equilibrium points of our problem. The problem is generalised in the sense that smaller primary is supposed to be an oblate spheroid. The bigger primary is considered as radiating. The equations of motion are affected by radiation pressure force, oblateness and P-R drag. All classical results involving photogravitational and oblateness in restricted three body problem may be verified from this result. With the help of characteristic equation, we discussed the stability. Finally we conclude that triangular equilibrium points are unstable.Comment: accepted for publication in Journal of Dynamical Systems & Geometric Theories Vol. 4, Number 1 (2006

Pictorial Representation for Antisymmetric Eigenfunctions of PS-3 Integral Equations

Eigenvalue problem for Poincare-Steklov-3 integral equation is reduced to the solution of three transcendential equations for three unknown numbers, moduli of pants. The complete list of antisymmetric eigenfunctions of integral equation in terms of Kleinian membranes is given.Comment: 33 pages, 13 figures. This paper is an extended version of CV/061173

Integrable matrix equations related to pairs of compatible associative algebras

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we investigate in details the multiplications of the A-type and integrable matrix ODEs and PDEs generated by them.Comment: 12 pages, Late