On special relativity with cosmological constant

This paper, which is meant to be a tribute to Minkowski's geometrical insight, rests on the idea that the basic observed symmetries of spacetime homogeneity and of isotropy of space, which are displayed by the spacetime manifold in the limiting situation in which the effects of gravity can be neglected, leads to a formulation of special relativity based on the appearance of two universal constants: a limiting speed c and a cosmological constant Λ which measures a residual curvature of the universe, which is not ascribable to the distribution of matter-energy. That these constants should exist is an outcome of the underlying symmetries and is confirmed by experiments and observations, which furnish their actual values. Specifically, it turns out on these foundations that the kinematical group of special relativity is the de Sitter group dS(c, Λ) = SO(1,4). On this basis, we develop at an elementary classical and, hopefully, sufficiently didactical level the main aspects of the theory of special relativity based on SO(1,4) (de Sitter relativity). As an application, we apply the formalism to an intrinsic formulation of point particle kinematics describing both inertial motion and particle collisions and decays.

Special relativity in the 21st century

We focus on the dynamical aspects on Newton-Hooke space-time NH+ mainly from the viewpoint of geometric contraction of the de Sitter spacetime with Beltrami metric. (The term spacetime is used to denote a space with non-degenerate metric, while the term space-time is used to denote a space with degenerate metric.) We first discuss the Newton-Hooke classical mechanics, especially the continuous medium mechanics, in this framework. Then, we establish a consistent theory of gravity on the Newton-Hooke space-time as a kind of Newton-Cartan-like theory, parallel to the Newton's gravity in the Galilei space-time. Finally, we give the Newton-Hooke invariant Schrodinger equation from the geometric contraction, where we can relate the conservative probability in some sense to the mass density in the Newton-Hooke continuous medium mechanics. Similar consideration may apply to the Newton-Hooke space-time NH- contracted from anti-de Sitter spacetime.

Mechanics and Newton-Cartan-like gravity on the Newton-Hooke space-time

Yang's model as triply special relativity and the Snyder's model–de Sitter special relativity duality

http://blog.sciencenet.cn/upload/blog/file/2009/3/200931593216981394.pdf

On principle of inertia in closed universe

Based on some important properties of dS space, we present a Beltrami model BA that may shed light on the observable puzzle of dS space and the paradox between the special relativity principle and cosmological principle. In BA, there are inertial-type coordinates and inertial-type observers. Thus, the classical observables can be defined for test particles and light signals. In addition, by choosing the definition of simultaneity the Beltrami metric is transformed to the Robertson-Walker-like metric. It is of positive spatial curvature of order A. This has already been shown by the CMB power spectrum from WMAP and should be further confirmed by its data in large scale.

On Beltrami model of de Sitter spacetime

Since the special relativity can be viewed as the physics in an inverse Wick rotation of 4-d Euclid space, which is at almost equal footing with the 4-d Riemann/Lobachevski space, there should be important physics in the inverse Wick rotation of 4-d Riemann/Lobachevski space. Thus, there are three kinds of special relativity in de Sitter/Minkowski/anti-de Sitter space at almost equal footing, respectively. There is an instanton tunnelling scenario in the Riemann-de Sitter case that may explain why $\La$ be positive and link with the multiverse.

Three Kinds of Special Relativity via Inverse Wick Rotation

Based on the Beltrami-de Sitter spacetime, we present the Newton-Hooke model under the Newton-Hooke contraction of the $BdS$ spacetime with respect to the transformation group, algebra and geometry. It is shown that in Newton-Hooke space-time, there are inertial-type coordinate systems and inertial-type observers, which move along straight lines with uniform velocity. And they are invariant under the Newton-Hooke group. In order to determine uniquely the Newton-Hooke limit, we propose the Galilei-Hooke's relativity principle as well as the postulate on Newton-Hooke universal time. All results are readily extended to the Newton-Hooke model as a contraction of Beltrami-anti-de Sitter spacetime with negative cosmological constant.

Newton-Hooke Limit of Beltrami-de Sitter Spacetime, Principles of Galilei-Hooke's Relativity and Postulate on Newton-Hooke Universal Time

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, omega) via the highest Euler-Lagrange cohomology. It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in Hamilton mechanics. The ordinary canonical equations are included as a special case with a 2-form 1/(n - 1)Homega, where H is the corresponding Hamiltonian.

Bin Zhou

Papers

On Beltrami model of de Sitter spacetime

Three Kinds of Special Relativity via Inverse Wick Rotation

Newton-Hooke Limit of Beltrami-de Sitter Spacetime, Principles of Galilei-Hooke's Relativity and Postulate on Newton-Hooke Universal Time

General volume-preserving mechanical systems

The euler-lagrange cohomology on symplectic manifolds