An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure

Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession

Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry

AbstractHyperbolic geometry is a fundamental aspect of modern physics. We explore in this paper the use of Einstein's velocity addition as a model of vector addition in hyperbolic geometry. Guided by analogies with ordinary vector addition, we develop hyperbolic vector spaces, called gyrovector spaces, which provide the setting for hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. The resulting gyrovector spaces enable Euclidean trigonometry to be extended to hyperbolic trigonometry. In particular, we present the hyperbolic law of cosines and sines and the Hyperbolic Pythagorean Theorem emerges when the common vector addition is replaced by the Einstein velocity addition

Degree of entanglement for two qubits

In this paper, we present a measure to quantify the degree of entanglement for two qubits in a pure state.Comment: 5 page

On the relation of Thomas rotation and angular velocity of reference frames

In the extensive literature dealing with the relativistic phenomenon of Thomas rotation several methods have been developed for calculating the Thomas rotation angle of a gyroscope along a circular world line. One of the most appealing concepts, introduced in \cite{rindler}, is to consider a rotating reference frame co-moving with the gyroscope, and relate the precession of the gyroscope to the angular velocity of the reference frame. A recent paper \cite{herrera}, however, applies this principle to three different co-moving rotating reference frames and arrives at three different Thomas rotation angles. The reason for this apparent paradox is that the principle of \cite{rindler} is used for a situation to which it does not apply. In this paper we rigorously examine the theoretical background and limitations of applicability of the principle of \cite{rindler}. Along the way we also establish some general properties of {\it rotating reference frames}, which may be of independent interest.Comment: 14 pages, 2 figure

A Derivation of Three-Dimensional Inertial Transformations

The derivation of the transformations between inertial frames made by Mansouri and Sexl is generalised to three dimensions for an arbitrary direction of the velocity. Assuming lenght contraction and time dilation to have their relativistic values, a set of transformations kinematically equivalent to special relativity is obtained. The ``clock hypothesis'' allows the derivation to be extended to accelerated systems. A theory of inertial transformations maintaining an absolute simultaneity is shown to be the only one logically consistent with accelerated movements. Algebraic properties of these transformations are discussed. Keywords: special relativity, synchronization, one-way velocity of light, ether, clock hypothesis.Comment: 16 pages (A5), Latex, one figure, to be published in Found. Phys. Lett. (1997

Seeing the Möbius disc-transformation group like never before

AbstractThe introduction of the gyration notion into nonassociative algebra, hyperbolic geometry, and relatively physics is motivated in this article by the emergence of the gyrogroup notion in the theory of the Möbius transformation group of the complex open unit disc. It suggests the prefix “gyro” that we use to emphasize analogies. Thus, for instance, gyrogroups are classified into gyrocommutative and nongyrocommutative gyrogroups in full analogy with the classification of groups into commutative and noncommutative groups. The road from the Thomas precession of the special theory of relativity to the Thomas gyration as well as the resulting new theory is presented in the author's book: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1]. The main result of this article is a theorem that allows the validity of some gyration identities to be extended from gyrocommutative gyrogroups into gyrogroups that need not be gyrocommutative. To set the stage for the main result, the Möbius disc-transformation group is studied in a novel way that suggests the notion of the gyrogroup and its gyrations