229,995 research outputs found
Stochastic Spacetime and Brownian Motion of Test Particles
The operational meaning of spacetime fluctuations is discussed. Classical
spacetime geometry can be viewed as encoding the relations between the motions
of test particles in the geometry. By analogy, quantum fluctuations of
spacetime geometry can be interpreted in terms of the fluctuations of these
motions. Thus one can give meaning to spacetime fluctuations in terms of
observables which describe the Brownian motion of test particles. We will first
discuss some electromagnetic analogies, where quantum fluctuations of the
electromagnetic field induce Brownian motion of test particles. We next discuss
several explicit examples of Brownian motion caused by a fluctuating
gravitational field. These examples include lightcone fluctuations, variations
in the flight times of photons through the fluctuating geometry, and
fluctuations in the expansion parameter given by a Langevin version of the
Raychaudhuri equation. The fluctuations in this parameter lead to variations in
the luminosity of sources. Other phenomena which can be linked to spacetime
fluctuations are spectral line broadening and angular blurring of distant
sources.Comment: 15 pages, 3 figures. Talk given at the 9th Peyresq workshop, June
200
State-space Geometry, Statistical Fluctuations and Black Holes in String Theory
We study the state-space geometry of various extremal and nonextremal black
holes in string theory. From the notion of the intrinsic geometry, we offer a
new perspective of black hole vacuum fluctuations. For a given black hole
entropy, we explicate the intrinsic state-space geometric meaning of the
statistical fluctuations, local and global stability conditions and long range
statistical correlations. We provide a set of physical motivations pertaining
to the extremal and nonextremal black holes, \textit{viz.}, the meaning of the
chemical geometry and physics of correlation. We illustrate the state-space
configurations for general charge extremal black holes. In sequel, we extend
our analysis for various possible charge and anticharge nonextremal black
holes. From the perspective of statistical fluctuation theory, we offer general
remarks, future directions and open issues towards the intrinsic geometric
understanding of the vacuum fluctuations and black holes in string theory.
Keywords: Intrinsic Geometry; String Theory; Physics of black holes;
Classical black holes; Quantum aspects of black holes, evaporation,
thermodynamics; Higher-dimensional black holes, black strings, and related
objects; Statistical Fluctuation; Flow Instability.
PACS: 02.40.Ky; 11.25.-w; 04.70.-s; 04.70.Bw; 04.70.Dy; 04.50.Gh; 5.40.-a;
47.29.KyComment: 28 pages. arXiv admin note: substantial text overlap with
arXiv:1102.239
Toric Hyperkahler Varieties
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric
hyperkahler varieties, which involves toric geometry, matroid theory and convex
polyhedra. The framework is a detailed study of semi-projective toric
varieties, meaning GIT quotients of affine spaces by torus actions, and
specifically, of Lawrence toric varieties, meaning GIT quotients of
even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler
variety is a complete intersection in a Lawrence toric variety. Both varieties
are non-compact, and they share the same cohomology ring, namely, the
Stanley-Reisner ring of a matroid modulo a linear system of parameters.
Familiar applications of toric geometry to combinatorics, including the Hard
Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are
extended to the hyperkahler setting. When the matroid is graphic, our
construction gives the toric quiver varieties, in the sense of Nakajima.Comment: 32 pages, Latex; minor corrections and a reference adde
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