Symplectic Structure in Brane Mechanics

This article treats the generalisation to brane dynamics of the covariant canonical variational procedure leading to the construction of a conserved bilinear symplectic current in the manner originally developped by Witten, Zuckerman and others in the context of field theory. After a general presentation, including a review of the relationships between the various (Lagrangian, Eulerian and other) relevant kinds of variation, the procedure is illustrated by application to the particularly simple case of branes of the Dirac-Goto-Nambu type, in which internal fields are absent.Comment: 13 pages Latex; Contribution to 2002 Peyresq - 7 worksho

Essentials of Classical Brane Dynamics

This article provides a self contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4 dimensional spacetime background. This category also includes ``brane world'' models that treat the observed universe as a 3-brane in 5 or higher dimensional background. The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $bar T^{\mu\nu}$ with the (geometrically defined) second fundamental tensor $K_{\mu\nu}{^\rho}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.Comment: 37 pages, Latex, revised to update references. Based on talk on 5 June, 2000 at Meudon Colloquium ``Motivations for and Implications of Extra Cosmological Dimensions'' (transparencies visible on line at http://darc.obspm.fr/col/M00.html) Contribution to proc. Peyresq 5 meeting, June 2000, ``Quantum spacetime, Brane Cosmology, and Stochastic Effective Theories'

Half century of black-hole theory: from physicists' purgatory to mathematicians' paradise

Although implicit in the discovery of the Schwarzschild solution 40 years earlier, the issues raised by the theory of what are now known as black holes were so unsettling to physicists of Einstein's generation that the subject remained in a state of semiclandestine gestation until his demise. That turning point -- just half a century after Einstein's original foundation of relativity theory, and just half a century ago today -- can be considered to mark the birth of black hole theory as a subject of systematic development by physicists of a new and less inhibited generation, whose enthusastic investigations have revealed structures of unforeseen mathematical beauty, even though questions about the physical significance of the concomitant singularities remain controversial.Comment: 30 pages latex. Contrib. to Encuentros Relativistas Espanoles: A Century of Relativity Theory, Oviedo, 2005 (ed. L. Mornas

Micro-Anthropic Principle for Quantum theory

Probabilistic models (developped by workers such as Boltzmann, on foundations due to pioneers such as Bayes) were commonly regarded merely as approximations to a deterministic reality before the roles were reversed by the quantum revolution (under the leadership of Heisenberg and Dirac) whereby it was the deterministic description that was reduced to the status of an approximation, while the role of the observer became particularly prominent. The concomitant problem of lack of objectivity in the original Copenhagen interpretation has not been satisfactorily resolved in newer approaches of the kind pioneered by Everett. The deficiency of such interpretations is attributable to failure to allow for the anthropic aspect of the problem, meaning {\it a priori} uncertainty about the identity of the observer. The required reconciliation of subjectivity with objectivity is achieved here by distinguishing the concept of an observer from that of a perceptor, whose chances of identification with a particular observer need to be prescribed by a suitable anthropic principle. It is proposed that this should be done by an entropy ansatz according to which the relevant micro-anthropic weighting is taken to be proportional to the logarithm of the relevant number of Everett type branch-channels.Comment: 29 pages Latex, 1 figure. Contribution to `Universe or Multiverse?' ed. B.J. Carr, for Cambridge U.