Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.     

Integral geometry on families of surfaces in the space

A complex {B_ξ} of submanifolds is called admissible in the sense of integral geometry if there are such densities μ_ξ on B_ξ, that the integral transformation

has a local inversion formula. We prove that compact smooth surfaces of an admissible complex in P³ has degree 3 and obtain complete classification of admissible complexes of quadrics. Several general theorems and conjectures about admissible complex of k-dimensional submanifolds are stated.     

A spin-less particle on a rotating curved surface in Minkowski space

In Minkowski space ${ \mathcal M }$, we derive the effective Schrödinger equation describing a spin-less particle confined to a rotating curved surface ${ \mathcal S }$. Using the thin-layer quantization formalism to constrain the particle on ${ \mathcal S }$, we obtain the relativity-corrected geometric potential ${V}_{g}^{{\prime} }$, and a novel effective potential ${\tilde{V}}_{g}$ related to both the Gaussian curvature and the geodesic curvature of the rotating surface. The Coriolis effect and the centrifugal potential also appear in the equation. Subsequently, we apply the surface Schrödinger equation to a rotating cylinder, sphere and torus surfaces, in which we find that the interplays between the rotation and surface geometry can contribute to the energy spectrum based on the potentials they offer.     

On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

We prove that an isometric immersion of a simply connected Riemannian surface $M$ in four-dimensional Minkowski space, with given normal bundle $E$ and given mean curvature vector $\overrightarrow{H}\in \Gamma \left(E\right)$, is equivalent to a normalized spinor field $\phi \in \Gamma (\Sigma E\otimes \Sigma M)$ solution of a Dirac equation $D\phi =\overrightarrow{H}\cdot \phi$ on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Gálvez, A. Martínez and F. Milán.     

Ruled surfaces with lightlike ruling in 3-Minkowski space

In this paper we discuss ruled surfaces with lightlike ruling in 3-Minkowski space and give some characterizations and examples of so called B-scroll surfaces.     

On time-like Willmore surfaces in Minkowski space

We study time-like surfaces in Minkowski space, which are critical points of the Willmore energy. Transforming the fourth order Willmore equation into a quasi-linear, second order hyperbolic system, we prove existence, uniqueness and symmetry properties of such surfaces, subject to geometric initial conditions.     

On the geometry of null hypersurfaces in Minkowski space

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space ${\mathbb{R}}_1^{n+2}$, classifying all rotation null hypersurfaces in ${\mathbb{R}}_1^{n+2}$. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the $(n+1)$-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the $3$-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of ${\mathbb{R}}_1^4$ and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.     

On the Reality of Minkowski Space

Should physicists deal with the question of the reality of Minkowski space (or any relativistic spacetime)? It is argued that they should since this is a question about the dimensionality of the world at the macroscopic level and it is physics that should answer it.     

Boosts in an Arbitrary Direction and Maximal Causal Velocities in a Deformed Minkowski Space

We discuss boosts in a deformed Minkowski space, i.e., a four-dimensional spacetime with metric coefficients depending on nonmetric coordinates (in particular on the energy). The general form of a boost in an arbitrary direction is derived in the case of space anisotropy. Two maximal trivector velocities are mathematically possible, an isotropic and an anisotropic one. However, only the anisotropic velocity has physical meaning, being invariant indeed under deformed boosts.     

Killing Symmetries of Generalized Minkowski Spaces, 3: Spacetime Translations in Four Dimensions

In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e., a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the translations in such spaces, by confining ourselves (without loss of generality) to the four-dimensional case. In particular, the results obtained are specialized to the case of a deformed Minkowski space (i.e. a pseudoeuclidean space with metric coefficients depending on energy).     

等势面族的条件及例子

介绍静电场中等势面的条件以及几个例子.     

The dynamical properties of a Rydberg hydrogen atom between two parallel metal surfaces

This paper presents the dynamical properties of a Rydberg hydrogen atom between two metal surfaces using phase space analysis methods. The dynamical behaviour of the excited hydrogen atom depends sensitively on the atom--surface distance d. There exists a critical atom--surface distance d_c = 1586 a.u. When the atom--surface distance d is larger than the critical distance d_c, the image charge potential is less important than the Coulomb potential, the system is near-integrable and the electron motion is regular. As the distance d decreases, the system will tend to be non-integrable and unstable, and the electron might be captured by the metal surfaces.     

Solving Bethe-Salpeter equation in Minkowski space

We develop a new method of solving the Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is valid for any kernel given by the irreducible Feynman graphs. For massless ladder exchange, our approach reproduces analytically the Wick-Cutkosky equation. For massive ladder exchange, the numerical results coincide with the ones obtained by Wick rotation.     

The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space

In this paper and its sequel we shall prove the local and then the global existence of solutions of the classical Yang-Mills-Higgs equations in the temporal gauge. This paper proves local existence uniqueness and smoothness properties and improves, by essentially one order of differentiability, previous local existence results. Our results apply to any compact gauge group and to any invariant Higgs self-coupling which is positive and of no higher than quartic degree.Supported in part by the National Science Foundation (Grant No. PHY79-16482 at Yale)     

The geometry of Ingarden spaces

The Randers spaces RFⁿ were introduced by R. S. Ingarden. They are considered as Finsler spaces Fⁿ = (M, α + β) equipped with the Cartan nonlinear connection. In the present paper we define and study what we call the Ingarden spaces, I Fⁿ, as Finsler spaces I Fⁿ = (M, α + β) equipped with the Lorentz nonlinear connection. The spaces R Fⁿ and I Fⁿ are completely different. For I Fⁿ we discuss: the variational problem, Lorentz nonlinear connection, canonical N-metrical connection and its structure equations, the Cartan 1-form ω, the electromagnetic 2-form tF and the almost symplectic 2-form 0. The formula dω = F+θ is established. It has as a consequence the generalized Maxwell equations. Finally, the almost Hermitian model of I Fⁿ is constructed.     

The calculation of the surface energy of high-index surfaces in metals at zero temperature

We used the molecular dynamics simulation with interatomic potentials of the embedded atom method to calculate the high-index surface energies of the surfaces containing the 〈0 0 1〉 axis or 〈−1 1 0〉 axis in f.c.c. metal Al, Cu and Ni at zero temperature. We generalized an empirical formula based on structural unit model for high-index surfaces and present some new formulas that can be used to estimate the surface energy and structural feature of high-index surfaces very well. The results show that the closest surfaces have the lowest surface energy and the surface energies of the closest (1 1 1) surface and the next closest (1 1 0), (1 0 0) surfaces are the extremum on the curve of surface energy versus orientation angle. We also calculated the b.c.c. metal Fe and obtained a similar result. The difference is that in the b.c.c. metal the surface energies of the closest (1 1 0) surface and the next closest (1 0 0), (1 1 2) surfaces are the extremum on the curve of surface energy versus orientation angle. The results of theoretical simulation and the empirical formula consist well with the experiment data.     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

Central configurations of the collinear three-body problem and singular surfaces in the mass space

This Letter is to provide a new approach to study the phenomena of degeneracy of the number of the collinear central configurations under geometric equivalence. A direct and simple explicit parametric expression of the singular surface H₃ is constructed in the mass space (m₁,m₂,m₃)∈³(R⁺). The construction of H₃ is from an inverse respective, that is, by specifying positions for the bodies and then determining the masses that are possible to yield a central configuration. It reveals the relationship between the phenomena of degeneracy and the inverse problem of central configurations. We prove that the number of central configurations is decreased to 3!/2−1=2, m₁, m₂, and m₃ are mutually distinct if m∈H₃. Moreover, we know not only the number of central configurations but also what the nonequivalent central configurations are.