Space Of Constant Negative Curvature Research Articles

Linearly and nonlinearly transforming fields on homogeneous spaces of the (4,1)-de Sitter group

Scalar functions on the homogeneous spaces ℋR of the de Sitter group G=SO(4,1) are studied, where the spaces ℋR are of the form G/K with K being a subgroup of the Lorentz group H=SO(3,1) contained in SO(4,1). The spaces ℋR can be regarded as fiber bundles ER=ER(G/H,H/K), with the base V′4 =G/H being a space of constant negative curvature characterized by a fundamental length parameter R[(4,1)-de Sitter space], and the fiber S=H/K being a homogeneous space of the Lorentz group. The action of G on the spaces ER is a linear action on V4 and a nonlinear action on S, where the latter action is defined by a generalized Wigner rotation. A gauge theory based on the (4,1)-de Sitter group is investigated with matter represented in terms of a generalized wave function Φ(x;ξ,ỹ) [with x∈U4 (Riemann–Cartan space-time), ξ∈V′4, and ỹ∈S] which is defined as a map from a cross section on the bundle E=E(U4, F=ER, G=SO(4,1)) over space-time U4 with fiber F=ER =G/K and structural group G=SO(4,1) into the complex numbers. The introduction of purely nonlinearly transforming fields (N)Φ(x;ỹ) is discussed as well as the nonlinear realization of the SO(4,1) symmetry in terms of transformations of the Lorentz subgroup H (generalized Wigner rotations). The geometric implications of symmetry breaking are pointed out.

Atomic fine-structure calculations in a space of constant negative curvature

Previous results concerning the Dirac energies and wave functions in a space of constant positive curvature are used to derive their counterparts in a space of constant negative curvature. In particular, it is shown that, when switching from a spherical closed space to a hyperbolic open space, the sign of the curvature-induced splittings and shifts in the hydrogenic fine-structure spectrum is reversed.

Melting as the transition to a curved space

It is shown that a crystal lattice at high enough temperatures is unstable with respect to the transition into a space of constant negative curvature. The instability is associated with melting of the crystal. The curvature is proportional to the density of disclinations in the real physical space. The temperature of melting is found as the functional of the pair inter-atomic potential. In a good approximation this temperature depends only on the second derivative of the interatomic potential at the point of the maximum, i.e. on the volume bulk module. The Lindemann ratio is expressed through the first zero of the interatomic potential.

Stochastic behavior in quantum scattering

A 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated. It contains an open end, i.e. an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loud-speakers. In the Poincaré model of hyperbolic geometry, the solutions of Schrödinger's equation for the reflection of a particle which enters through the horn are easily constructed. The scattering phase shift as a function of the momentum is essentially given by the phase angle of Riemann's zeta function on the imaginary axis, at a distance of 1 2 from the famous critical line. This phase shift shows all the features of chaos, namely the ability to mimick any given smooth function, and great difficulty in its effective numerical computation. A plot shows the close connection with the zeros of Riemann's zeta function for low values of the momentum (quantum regime) which gets lost only at exceedingly large momenta (classical regime?) Some generalizations of this approach to chaos are mentioned.

A geometrical formulation of electrodynamics

A geometrical formulation of electrodynamics is presented. This formulation is based on a nonlinear constraint for the electromagnetic potential. As a consequence of the constraint, a three-dimensional photon field is associated with the co-ordinates of a symmetric space of constant negative curvature. A formalism covariant under general co-ordinate transformations is constructed and extended, in a manner analogous to that used in general relativity, to include spinor realizations. A generally covariant Lagrangian density is exhibited and shown to be equivalent, by a suitable redefinition of the fields, to the usual electrodynamics. In such an approach a close relationship between Lorentz invariance and the electromagnetic interaction is manifest.

Variation of curvature in binocular visual space estimated by the triangle method

The Luneberg-Blank model of binocular visual space presumes the two hypotheses: (1) Visual space is a Riemannian space of constant negative curvature. (2) Apparent visual angle equals physical visual angle. The first hypothesis was tested by the triangle method on the assumption that any point on a Vieth-Müller circle is perceived equidistant from the observer. The results indicated that the curvature changes from negative toward zero with increasing absolute bipolar latitude of configuration and it also changes from negative to positive with decreasing angular separation of configuration. The second hypothesis was examined by determining exponent values of the power function for visual angle. The resulting value was 1.03 on the average. These results allowed us to modify the Luneberg-Blank model.

On a theorem of Chern

A new proof is given for Chern’s theorem showing that the Laplace operator for differential forms commutes with decomposition of forms associated with G-structures admitting a suitable connection. An analogous result is proved for symmetric tensor fields, and an application is made to determine all harmonic symmetric fields on a compact space of constant negative curvature. Vector-valued forms are also discussed.

Some Uses of Hyperbolic Velocity Space

This paper deals with the definition and properties of the velocity space of special or general relativity. The metric naturally induced by the enveloping space-time is that of a space of constant negative curvature which is referred to here as a 3-dimensional Lobachevsky space. Our purpose is to show how this notion can simplify or reinterpret ideas of relativistic kinematics. Supplementing the more familiar applications, three others are given here which make essential use of the curvature, namely to the derivations of aberration, Thomas precession, and the forms of relativistic collision invariants.

Parallel vector fields

We speak of a parallel vector field if parallel transport is independent of the path of propagation [1, p. 239]. Riemannian spaces which are flat offer a simple example for the existence of such fields [1, p. 239]. We wish to generalize the concept of parallel displacement of a vector in a Riemannian space. In our case, spaces of constant negative curvature are the most conspicuous to admit parallel vector fields. DEFINITION. Let the direction of a vector field at any point be that of the unit vector V. The field is said to be parallel if

Analytical Approximations for Color Metric Coefficients*

On the assumption of a color space of constant negative curvature, von Schelling has given simple formulas for the color metric coefficients. An attempt has been made to fit such formulas to newly available color-discrimination data for 12 normal observers. A least-squares analysis indicates that Y should be replaced with the quantity, Y′=Y−0.085X+0.011Z, in the special role assigned to luminance by von Schelling. Corresponding to this change, coordinates x′=X/(X+Y′+Z), y′=Y′/(X+Y′+Z) yield the simplest formulas for the coefficients of the expression for color difference: ΔS=(g11Δx′2+2g12Δx′Δy′+g22Δy′2+g33ΔY2/Y2)12.Least-squares fitting of the experimental data to von Schelling’s formulas yielded: g11=g33/(10x′2-6.886x′+2.083+2.5y′-16.66x′y′+19.6y′2).g12=g11(0.3443-x′)/y′.g22=g11[0.09+(0.3443-x′)2]/y′2.These formulas are the best approximations obtainable on the assumption of a color space of constant negative or zero curvature. However, they give such poor approximations to the experimental data that it is concluded that the assumption of a color space inherently more complicated than a space of constant curvature would be required to fit the data to a closeness commensurate with the experimental reproducibility.

Experimental Determination of Luneburg's Constants σ and κ

Visual space is, according to Luneburg, a non-Euclidean hyperbolic space of constant negative curvature. Two personal constants enter into the geometrical description of binocular subjective space. One is σ indicating the degree of depth perception characteristic to the observer. The second is K, the curvature; a value which denotes by how much the visual space of an individual departs from Euclidean space. The constants σ and K were determined experimentally for 30 untrained observers by Luneburg's 3- and 4-point tests. All values found for K were negative and support the hypothesis that the geometry of binocular space is hyperbolic. Some discrepancies between the theory and findings were observed, while the sensitivity of the experiments to the lack of training and even a slight measure of aniseikonia is evident.

A Note on the Kepler Problem in a Space of Constant Negative Curvature

A Note on the Kepler Problem in a Space of Constant Negative Curvature