Hyperbolic Angles Research Articles

Mehler–Fock transforms and retarded radiative Green functions in hyperbolic and spherical spaces

We develop the theory of causal radiation Green functions on hyperbolic and hyperspherical spaces using a constructive approach based on generalized Mehler–Fock transforms. This approach focuses for HRd on the kernel of the transformation expressed in terms of hyperbolic angles θ with 0 ≤ θ < ∞. The kernel provides an explicit representation for the generalized delta distribution, which acts as the source term for the radiation, and allows for an easy implementation of the causality or retardation condition and determination of the Green function. We obtain the corresponding kernel distribution on SRd by analytic continuation of the kernel distribution of an appropriate Helmholtz equation on HRd and then show that this construction leads to the proper retarded Green function for the wave equation. That result is then used to establish the validity of a new generalized Mehler–Fock transformation for 0 ≤ θ < π. The present results clarify and extend those obtained recently by Cohl et al. [SIGMA 14, 136 (2018)].

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds.

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399-447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature boundedbelow.

SU(1,1) parity and strong violations of a Bell inequality by entangled Barut–Girardello coherent states

We study the violations of the Bell–Clauser–Horne–Shimony–Holt inequality for entangled SU(1,1) Barut–Girardello coherent states. As in a previous paper where violations of the inequality were studied for entangled SU(1,1) coherent states of the Perelomov form [Phys. Rev. A.93, 042104 (2016)PLRAAN1050-294710.1103/PhysRevA.93.042104], we choose as our observable the SU(1,1) parity operator, though here we discuss the physical meaning of the operator for single-mode and two-mode bosonic realizations of the su(1,1) Lie algebra. We show that the SU(1,1) parity operator is not the same as the usual photon number parity operator but rather is a form of higher-order photon number parity. Distant observers Alice and Bob perform on each of their subsystems non-compact SU(1,1) transformations characterized by hyperbolic angles, followed by measurements of the SU(1,1) parity operators. We find strong violations of the inequality over a wide range of parameters, and these violations are stronger than those that can be obtained by the entangled Perelomov SU(1,1) coherent state.

Synchronization of relativistic particles in the hyperbolic Kuramoto model.

We formulate a noncompact version of the Kuramoto model by replacing the invariance group SO(2) of the plane rotations by the noncompact group SO(1, 1). The N equations of the system are expressed in terms of hyperbolic angles αi and are similar to those of the Kuramoto model, except that the trigonometric functions are replaced by hyperbolic functions. Trajectories are generally unbounded, nevertheless synchronization occurs for any positive couplings κi, arbitrary positive multiplicative parameters λi and arbitrary exponents ωi. There are no critical values for the coupling constants. We measure the onset of synchronization by means of several order and disorder parameters. We show numerically and by means of exact solutions for N = 2 that solutions can develop singularities if the coupling constants are negative, or if the initial values are not suitably restricted. We describe a physical interpretation of the system as a cluster of interacting relativistic particles in 1 + 1 dimensions, subject to linear repulsive forces with space-time trajectories parametrized by the rapidity αi. The trajectories synchronize provided that the particle separations remain predominantly time-like, and the synchronized cluster can be viewed as a bound state of N relativistic particle constituents. We extend the defining equations of the system to higher dimensions by means of vector equations which are covariant with respect to SO(p, q).

Algebraic Representation of the Group of Havrda–Charvat–Daroczy Entropy Vectors in Nonextensive Statistical Mechanics

An algebraic representation of the group of nonextensive, parameterized Havrda–Charvat–Daroczy entropy vectors that depend on three distributions is constructed. The composition law of conformally-generalized hypercomplex numbers is considered, and properties of a commutative, nonassociative algebra are derived. The exponential form of the number and functions of numbers with hyperbolic angles are presented.

On the pair correlation density for hyperbolic angles

Let $\Gamma< \mathrm{PSL}_2(\mathbb{R})$ be a lattice and $\omega\in \mathbb{H}$ a point in the upper half plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between geodesic rays of the lattice $\Gamma \omega$ intersected with increasingly large balls centered at $\omega$, thus proving a conjecture of Boca-Popa-Zaharescu.

Geometric Representation of the Group of Entropy Vectors in Nonextensive Statistical Mechanics

A geometric representation of the group of nonextensive entropy vectors in four-dimensional space is derived. The three-dimensional hyperbolic entropy vector and the vector-parameter of hyperbolic angles are determined. A composition law for hyperbolic entropy vectors with quadratic nonlinearity is given and an interrelationship with three distributions characterizing the system is presented.

Pair correlation of hyperbolic lattice angles

Let ω be a point in the upper half plane, and let Γ be a discrete, finite covolume subgroup of PSL2(ℝ). We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice Γω, intersected with increasingly large balls centered at ω. We prove this conjecture for Γ = PSL 2(ℤ) and ω an elliptic point.

Null scrolls as fluctuating surfaces: a new simple way to construct extrinsic string solutions

We exhibit a surprising phenomenon that happens in the conformal Lorentz Minkowski three space, which has no counterpart in a Riemannian setting. Whenever a curve, no matter its causal character, propagates transversely through a conformal geodesic null vector field, it is generating the worldsheet of an extrinsic Polyakov string solution. Furthermore, the Polyakov extrinsic energy of these solutions only depend on the world- sheet topology and it can be computed not only intrinsically, but also holographically by measuring the hyperbolic angles in the boundary corners. This geometric approach, to provide extrinsic string solutions, can be considered as an alternative to the Pohlmeyer reduced mechanism. Then, we describe how to translate these solutions to the language of Pohlmeyer theory. We will also show that any curve in the conformal boundary of the anti de Sitter 3-space can be viewed as a piece of the generalized Wilson loops associated with an extrinsic string solution obtained by this geometric mechanism.

The Geometry of Minkowski Space in Terms of Hyperbolic Angles

The geometry of Minkowski space is investigated by using concepts such as hyperbolic angles, hyperbolic curves, and hyperbolic arc lengths. The hyperbolic angle between two inertial observers is given by θ = 1 2 log{(1 + v)/(1− v)}. The usual scalar product between any two Lorentz vectors can be written in terms of the hyperbolic angle between them. The scalar product of two timelike vectors A and B, for example, can be written as A ·B = |A||B| coshθ, where |A| and |B| are their Lorentz invariant lengths. This is a natural generalizations of Euclidean geometry.

Hyperbolic Sine and Cosine Rules for Geodesic Triangles on the Hyperbolic Unit Sphere \(H_{0}^{2}\)

In plane Lorentzian geometry it is studied points, timelike, spacelike and lightlike lines, triangles, etc [4]. On the hyperbolic sphere, there are points, but there are no straight lines, at least not in the usual sense. However, straight timelike lines in the Lorentzian plane are characterized by the fact that they are the shortest paths between points. The curves on the hyperbolic sphere with the same property are hyperbolic circles. Thus it is natural to use these circles as replacements for timelike lines. The formulas for the sine and cosine rules are given for the Euclidean sphere 2 S [2, 3, 6] and hyperbolic sphere [5]. In this study, we obtained formulas related with the spacelike angles and hyperbolic angles corresponding to the sides of geodesic triangles on hyperbolic unit sphere \(H_{0}^{2}\).

The new expressions of Gaussion curvature at spacelike congruences

We obtained two fundamental formula in [1] for the geometry of spacelike congruences by means of dual hyperbolic and Central angles. Beside we stated and proved the Manheitn’s, Liouville’s and Hamilton’s formulae. In this study we give the new expressions of the Gaussian curvature for the spacelike congruences by means of fundamental formulae.

Tilted electromagnetic Bianchi type I and type II cosmologies

Abstract We investigate tilting electromagnetic models Bianchi of types I and II. Explicit expressions for the hyperbolic tilt angles are given for the case of “stiff” matter.

Isospin and local space-time rotations

An 8-component spinor field carries with itself a large number of 4-vector currents and invariants, the relationships between which are analysed. The vector densities can be grouped into a number of orthogonal frames, which describe tetrad fields. Two of the tetrads, related to each other by charge conjugation, connect isospin transformations directly with local space-time rotations. The main tetrad planes determine geometrical configurations of considerable symmetry. The bilinear invariants define angles and hyperbolic angles which appear directly in the rotations and Lorentz transformations connecting the tetrads.

Graphical Method of Illustrating Relativistic Conservation Laws

Graphical methods can be utilized to analyze relativistic free-particle interactions by representing the momentum of a free particle as a vector in two-dimensional momentum space. This vector is constrained to lie on the mass hyperbola for all inertial frames of reference. By employing the Lorentz invariance of the difference in the hyperbolic angles for any two vectors, one can analyze graphically free-particle interactions in the most convenient frame of reference. In particular two-particle interactions are studied in the laboratory and center of mass frames of reference.

Origin and meaning of circular and hyperbolic functions in electrical engineering

Sinusoidal currents in time, arising from impressed sinusoidal voltages in time, need no explanation; but circular and hyperbolic functions of unobvious circular and hyperbolic space angles arising apparently spontaneously in electrical circuits call for some explanation. Accordingly, the concept of an angle, adequate both for circular and hyperbolic functions, is first developed geometrically, pointing out that an angle is nothing but the percentage value (more precisely, percentage/100 or per-unit value) of an arc recognized as the vector increment of the radius vector. Naturally, the radius vector itself is used as the per-unit base, hence the name radians (meaning, radii) for this kind of measure numbers. The idea naturally leads to the compound-interest like calculation of the per-unit value of a gross change in the radius vector. The vector concept recognizes as angles those incremental arcs that are parallel to the radius vector just as well as those that are normal. The former are called hyperbolic angles, the latter circular. Thus an arc that represents a circular angle and one that represents a hyperbolic angle are j with respect to each other. Obviously, in the first case, only the direction of the radius vector changes; in the latter, only its magnitude. Applied to a physical variable, this concept of an angle means that a change in the variable expressed in per-unit 2 2 All such reckonings in this paper are to be understood as compound interest calculation. is the angular value of that change in radians. This concept of angles leads to logarithms and exponentials, yielding a simple interpretation of the logarithms of negative, imaginary and complex numbers so that, in its light, by inspection log (− 1) is recognized as πj, log j is recognized as ( π 2 )j , etc. The cosine of an angle θ (whether θ is circular, or hyperbolic, or mixed) is interpreted as the resultant of two variables, initially alike, but one having undergone the vector per-unit change (+ θ), and the other (− θ). A sine is interpreted as the resultant of two variables, initially equal and opposite, and one having undergone the vector per unit change (+ θ), the other (− θ). The origin of such pairing of vectors is found in reflection. In transmission lines, discontinuities like open circuits and short circuits act as perfect reflectors, and provide reflected variables which match the original variable in the indicated manner to yield with the original a resultant corresponding to the sine or cosine (or sinh or cosh) of the per-unit change involved.