General reference frames and their associated space manifolds

We propose a formal definition of a general reference frame in a general spacetime, as an equivalence class of charts. This formal definition corresponds with the notion of a reference frame as being a (fictitious) deformable body, but we assume, moreover, that the time coordinate is fixed. This is necessary for quantum mechanics, because the Hamiltonian operator depends on the choice of the time coordinate. Our definition allows us to associate rigorously with each reference frame F, a unique "space" (a three-dimensional differentiable manifold), which is the set of the world lines bound to F. This also is very useful for quantum mechanics. We briefly discuss the application of these concepts to G\"odel's universe.Comment: 14 pages in standard 12pt format. v2: Discussion Section 4 reinforced, now includes an application to G\"odel's universe

On reference frames and the definition of space in a general spacetime

First, we review local concepts defined previously. A (local) reference frame $\mathrm{F}$ can be defined as an equivalence class of admissible spacetime charts (coordinate systems) having a common domain $\mathrm{U}$ and exchanging by a spatial coordinate change. The associated (local) physical space is made of the world lines having constant space coordinates in any chart of the class. Second, we introduce new, global concepts. The data of a non-vanishing global vector field $\,v\,$ defines a global "reference fluid". The associated global physical space is made of the maximal integral curves of that vector field. Assume that, in any of the charts which make some reference frame $\mathrm{F}$: (i) any of those integral curves $l$ has constant space coordinates $x^j$, and (ii) the mapping $l\mapsto (x^j)$ is one-to-one. In that case, the local space can be identified with a part (an open subset) of the global space.Comment: 10 pages. Text of a talk given at the Third International Conference on Theoretical Physics "Theoretical Physics and its Applications", Moscow, June 24-28, 201

Target Space Duality I: General Theory

We develop a systematic framework for studying target space duality at the classical level. We show that target space duality between manifolds M and Mtilde arises because of the existence of a very special symplectic manifold. This manifold locally looks like M x Mtilde and admits a double fibration. We analyze the local geometric requirements necessary for target space duality and prove that both manifolds must admit flat orthogonal connections. We show how abelian duality, nonabelian duality and Poisson-Lie duality are all special cases of a more general framework. As an example we exhibit new (nonlinear) dualities in the case M = Mtilde = R^n.Comment: LaTeX, 29 pages, 1 eps figure. Added a couple of references and corrected a couple of typos. An FAQ that discusses some subtle points may be found at <http://www.physics.miami.edu/~alvarez/papers/duality/

Space-Time Intervals Underlie Human Conscious Experience, Gravity, and a Theory of Everything

Space-time intervals are the fundamental components of conscious experience, gravity, and a Theory of Everything. Space-time intervals are relationships that arise naturally between events. They have a general covariance (independence of coordinate systems, scale invariance), a physical constancy, that encompasses all frames of reference. There are three basic types of space-time intervals (light-like, time-like, space-like) which interact to create space-time and its properties. Human conscious experience is a four-dimensional space-time continuum created through the processing of space-time intervals by the brain; space-time intervals are the source of conscious experience (observed physical reality). Human conscious experience is modeled by Einstein’s special theory of relativity, a theory designed specifically from the general covariance of space-time intervals (for inertial frames of reference). General relativity is our most accurate description of gravity. In general relativity, the general covariance of space-time intervals is extended to all frames of reference (inertial and non-inertial), including gravitational reference frames; space-time intervals are the source of gravity in general relativity. The general covariance of space-time intervals is further extended to quantum mechanics; space-time intervals are the source of quantum gravity. The general covariance of space-time intervals seamlessly merges general relativity with quantum field theory (the two grand theories of the universe). Space-time intervals consequently are the basis of a Theory of Everything (a single all-encompassing coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe). This theoretical framework encompasses our observed physical reality (conscious experience) as well; space-time intervals link observed physical reality to actual physical reality. This provides an accurate and reliable match between observed physical reality and the physical universe by which we can carry on our activity. The Minkowski metric, which defines generally covariant space-time intervals, may be considered an axiom (premise, postulate) for the Theory of Everything

A two-point boundary value problem on a Lorentz manifold arising in A. Poltorak's concept of reference frame

In A. Poltorak's concept, the reference frame in General Relativity is a certain manifold equipped with a connection. The question under consideration here is whether it is possible to join two events in the space-time by a time-like geodesic if they are joined by a geodesic of the reference frame connection that has a time-like initial vector. This question is interpreted as whether an event belongs to the proper future of another event in the space-time in case it is so in the reference frame. For reference frames of two special types some geometric conditions are found under which the answer is positive.Comment: 11 page

Semi-Teleparallel Theories of Gravitation

A class of theories of gravitation that naturally incorporates preferred frames of reference is presented. The underlying space-time geometry consists of a partial parallelization of space-time and has properties of Riemann-Cartan as well as teleparallel geometry. Within this geometry, the kinematic quantities of preferred frames are associated with torsion fields. Using a variational method, it is shown in which way action functionals for this geometry can be constructed. For a special action the field equations are derived and the coupling to spinor fields is discussed.Comment: 14 pages, LaTe

Reference frames and rigid motions in relativity: Applications

The concept of rigid reference frame and of constricted spatial metric, given in the previous work [\emph{Class. Quantum Grav.} {\bf 21}, 3067,(2004)] are here applied to some specific space-times: In particular, the rigid rotating disc with constant angular velocity in Minkowski space-time is analyzed, a new approach to the Ehrenfest paradox is given as well as a new explanation of the Sagnac effect. Finally the anisotropy of the speed of light and its measurable consequences in a reference frame co-moving with the Earth are discussed.Comment: 13 pages, 1 figur