On Pythagoras Theorem for Products of Spectral Triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K-homology.     

Exact Solutions for Einstein＇s Hyperbolic Geometric Flow

In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein＇s hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.     

A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least ?6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter–Schwarzschild metrics.     

Von Neumann Uniqueness Theorem doesn't Hold in Hyperbolic Quantum Mechanics

It is shown that Von Neumann Uniqueness Theorem doesn't hold in Hyperbolic Quantum Mechanics.     

q-Newton Binomial: From Euler To Gauss

Abstract

A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter.     

Gravitational energy-momentum: The Einstein pseudotensor reexamined

By using a suitable two-point scalar field, a covariant formulation of the Einstein pseudotensor is given. A unique choice of scalar field is made possible by examining the role of linear and angular momentum in their correct geometric context. It is shown that, contrary to many text-book statements, linear momentum is not generated by infinitesimal coordinate transformations on space-time. Use is made of the nonintersecting lifted geodesies on the tangent bundle,T M, to space-time, to define a globally regular three-dimensional Lagrangian submanifold ofT M, relative to an observer at some pointz in space-time. By integrating over this submanifold rather than a necessarily singular spacelike hypersurface, gravitational linear and angular momentum, relative toz, are defined, and shown to have sensible physical properties.This essay received an honorable mention from the Gravity Research Foundation for the year 1979-Ed.     

A Large Class of Non-Constant Mean Curvature Solutions of the Einstein Constraint Equations on an Asymptotically Hyperbolic Manifold

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent tend to its true value. We prove that the solutions of the sub-critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a non-trivial solution. Finally, we give conditions which ensure that the limit equation admits no non-trivial solution.     

爱因斯坦的双重奇迹

一 今年是国际物理年，也是“爱因斯坦奇迹年”100周年，德国政府干脆称它为“爱因斯坦年”．“爱因斯坦奇迹年”指的是1905年，年仅26岁的爱因斯坦，一个在伯尔尼瑞士专利局当三级技术员，利用每天8小时公务以外的业余时间，写出了6篇物理学论文，在三个不同领域做出了四项划时代的贡献，创造了科学史（也是人类历史）上的奇迹．而这些贡献都是在短短7个月（1905年3月至9月）内完成的．     

Einstein and ensembles: Response

This article reexamines Einstein's views concerning ensembles and the quantum state function, by way of responding to criticism on this topic. The response calls attention to the range of interpretations found in Einstein's writings, and their function, and emphasizes the nonspecificity of his discussions. It also offers some guidelines for scholarship and criticism in this area.     

The 1922 Einstein Film: Cinematic Innovation and Public Controversy

In 1922 Hanns Walter Kornblum produced a long and comprehensive educational film on Albert Einstein’s theory of relativity that made extensive use of trick shots. His film was considered to be a milestone in the history of film and also in the popularization of science. Although the original film has been lost, I have reconstructed its content and its reception by members of the German film industry and cultural sector, laymen, scientists and academics, and politically motivated opponents based upon the large collection of newspaper clippings that was assembled by the antirelativist physicist Ernst Gehrcke.     

Einstein as Engineer:The Case of the Little Machine

I first discuss Albert Einstein’s practical and educational background in engineering and then his invention of his “little machine,” an electrostatic induction machine, while working in the Patent Office in Bern, Switzerland, between 1902 and 1909. He believed that it could be used as a voltage or potential multiplier in experiments to test his new theory of Brownian motion of 1905. I then discuss Einstein’s search for collaborators to produce it and the work that his friends Conrad and Paul Habicht, in particular, did in designing and testing it. Although the initial response to it was promising, it never became a success after Paul Habicht manufactured a few specimens of it beginning in 1912.Today only three specimens are known to exist; these are preserved at the Zürcher Hochschule Winterthur, Switzerland, in the Physics Institute of the University of Tübingen, Germany, and in the Museum Boerhaave in Leiden,The Netherlands.     

A Definition of Total Energy-Momenta and the Positive Mass Theorem on Asymptotically Hyperbolic 3-Manifolds. I

Two sets of asymptotically hyperbolic initial data are defined, which correspond to the spatial infinity in asymptotically AdS spacetimes and to the null infinity in asymptotically Minkowski spacetimes respectively. The positive mass theorem involving the total energy, the total linear momentum and the total angular momentum is established for these initial data sets.Research partially supported by National Natural Science Foundation of China under grant 10231050 and the innovation project of the Chinese Academy of Sciences.     

The Fluctuation Theorem

The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in thermostatted dissipative non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy 'thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been in the last few of years that the generality of the Theorem has been appreciated. We now know that the Second Law of Thermodynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quantifies the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observation that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversed 'anti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been confirmed experimentally, both using molecular dynamics computer simulation and very recently in laboratory experiments.     

Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as \(\,k^{-5/3}\) .     

Quantum fractionalism: The Born rule as a consequence of the complex Pythagorean theorem

Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces.Postulating the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds, we show that projection factors give relative amounts of worlds. These appear as relative frequencies of results in quantum experiments, and play the role of probabilities in decisions and inference. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability.     

The Bifurcation Approach to Hyperbolic Geometry

The Thomas precession of relativity physics gives rise to important isometries in hyperbolic geometry that expose analogies with Euclidean geometry. These, in turn, suggest our bifurcation approach to hyperbolic geometry, according to which Euclidean geometry bifurcates into two mutually dual branches of hyperbolic geometry in its transition to non-Euclidean geometry. One of the two resulting branches turns out to be the standard hyperbolic geometry of Bolyai and Lobachevsky. The corresponding bifurcation of Newtonian mechanics in the transition to Einsteinian mechanics indicates that there are two, mutually dual, kinds of uniform accelerations. Furthermore, while current hyperbolic geometry does not use the notion of vector at all (I. M. Yaglom, Geometric Transformations III, p. 135, trans. by Abe Shenitzer, Random House, New York, 1973), our bifurcation approach exposes the elusive hyperbolic vectors, that we call gyrovectors.     

Einstein: Distant parallelism and electromagnetism

Einstein's approach to unified field theories based on the geometry of distant parallelism is discussed. The simplest theory of this type, describing gravitation and electromagnetism, is investigated. It is found that there is a charge-current density vector associated with the geometry. However, in the static spherically symmetric case no singularity-free solutions for this vector exist.     

The Einstein shift in Einstein's box experiment

The question of whether the Einstein shift in clock rates has a bearing on the validity of the fourth Heisenberg uncertainty relation is discussed. It is shown that, even if one would accept all the relevant assumptions and conclusions of Bohr and Rosenfeld, this uncertainty relation could not be saved by an Einstein shift in the case of an electrostatic weighing. This means that the Einstein shift does not play any role in determining the validity of the fourth Heisenberg relation.