Minkowski Manifold Research Articles

Optics in anisotropic inhomogeneous media and Lagrange geometry

In this paper, the equations of motion associated with a Lagrangian inspired by relativistic optics in inhomogeneous moving medium are considered. The model describes optical effects in the inhomogeneous moving medium with special optical properties given by the self-consistent system. When using the metric restricted to the Minkowski manifold, we have established the Euler–Lagrange equations for corresponding geodesics. We have specified the general model to the special case when the metric coefficient [Formula: see text] linearly increases along the direction [Formula: see text]. The exact analytical solutions of the Euler–Lagrange equations have been constructed. The study of the obtained solutions shows that the light ray bends to the axis [Formula: see text] along which the effective refractive index increases.

Generalized finite difference schemes with higher order Whitney forms

Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first order partial differential equations and by (ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.

Geometrothermodynamics for black holes and de Sitter space

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald’s method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald’s method. The new method is applied to Schwarzschild, Kerr, and Kerr–Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.

The Schrödinger Semigroup on Some Flat and Non Flat Manifolds

In this paper we apply known techniques from semigroup theory to the Schrödinger problem with initial conditions. To this end, we define the regularized Schrödinger semigroup acting on a space-time domain and show that it is strongly continuous and contractive in \(L_p,\) with \(\frac{3}{2}\!<\!p\!<\!3.\) These results can easily be extended to the case of conformal operators acting in the context of differential forms, but they require positiveness conditions on the curvature of the considered Minkowski manifold. For that purpose, we will use a Clifford algebra setting in order to highlight the geometric characteristics of the manifold. We give an application of such methods to the regularized Schrödinger problem with initial condition and we will extended our conclusions to the limit case. For the torus case and a class of non-oriented higher dimensional Möbius strip like domains we also give some explicit formulas for the fundamental solution.

SELF-SIMILAR COSMOLOGIES IN 5D: SPATIALLY FLAT ANISOTROPIC MODELS

In the context of theories of the Kaluza–Klein type, with a large extra dimension, we study self-similar cosmological models in 5D that are homogeneous, anisotropic and spatially flat. The "ladder" to go between the physics in 5D and in 4D is provided by Campbell–Maagard's embedding theorems. We show that the 5D field equations RAB = 0 determine the form of the similarity variable. There are three different possibilities: homothetic, conformal and "wavelike" solutions in 5D. We derive the most general homothetic and conformal solutions to the 5D field equations. They require the extra dimension to be spacelike, and are given in terms of one arbitrary function of the similarity variable and three parameters. The Riemann tensor in 5D is not zero, except in the isotropic limit, which corresponds to the case where the parameters are equal to each other. The solutions can be used as 5D embeddings for a great variety of 4D homogeneous cosmological models, with and without matter, including the Kasner universe. Since the extra dimension is spacelike, the 5D solutions are invariant under the exchange of spatial coordinates. Therefore they also embed a family of spatially inhomogeneous models in 4D. We show that these models can be interpreted as vacuum solutions in braneworld theory. Our work (I) generalizes the 5D embeddings used for FLRW models; (II) shows that anisotropic cosmologies are, in general, curved in 5D, in contrast with FLRW models, which can always be embedded in a 5D Riemann-flat (Minkowski) manifold; and (III) reveals that anisotropic cosmologies can be curved and devoid of matter, both in 5D and in 4D, even when the metric in 5D explicitly depends on the extra coordinate, which is quite different from the isotropic case.

Galilei-invariant gauge symmetries in Fokker–Planck dynamics with logarithmic diffusion and drift terms

A tensorial approach to Galilean invariance is utilized, together with Lie symmetries of differential equations, in order to derive equations of Fokker–Planck type containing a logarithmic diffusion tensor and drift term. The formalism is based on the projection from an extended (by one space-like dimension) Minkowski manifold to the usual Newtonian spacetime, so that non-relativistic models are described by manifestly covariant Lagrangians. In this paper, we obtain the Fokker–Planck equations from the Euler–Lagrange equations with the extended manifold by using a specific choice of the gauge condition. We work in (1 + 1) spacetime and carry out the analysis for both Abelian and non-Abelian symmetries.

Galilean covariant theories for Bargmann–Wigner fields with arbitrary spin

We construct Lagrangians for non-relativistic massive fields with arbitrary spin. We use a Bargmann–Wigner construction, together with a Galilean covariant approach based on the reduction from an extended (4, 1) Minkowski manifold to the Galilean (3, 1) spacetime. Fierz identities are developed within this framework. By using symmetric spinor fields of rank 2 and rank 3, we can avoid the difficulty arising from the introduction of the minimal electromagnetic interaction in the Bargmann–Wigner wave equations. For fields with spin S, the minimal electromagnetic coupling thereby leads to the gyromagnetic ratio gS = 1/S.

On the electrodynamics of moving bodies at low velocities

We discuss an article by Le Bellac and Lévy-Leblond in which they have identified two Galilean limits of electromagnetism (1973 Nuovo Cimento B 14 217–33). We use their results to point out some confusion in the literature, and in the teaching of special relativity and electromagnetism. For instance, it is not widely recognized that there exist two well-defined non-relativistic limits, so that researchers and teachers are likely to utilize an incoherent mixture of both. Recent works have shed new light on the choice of gauge conditions in classical electromagnetism. We retrieve the results of Le Bellac and Lévy-Leblond first by examining orders of magnitudes and then with a Lorentz-like manifestly covariant approach to Galilean covariance based on a five-dimensional Minkowski manifold. We emphasize the Riemann–Lorenz approach based on the vector and scalar potentials as opposed to the Heaviside–Hertz formulation in terms of electromagnetic fields.

Lorentz-like formulation of Galilean field theories

We review recent articles in which we have constructed nonrelativistic Lagrangian field models by enforcing Galilean covariance on a (4,1) Minkowski manifold and by reducing onto the (3,1) Newtonian space-time. This formalism provides an elegant construction of nonrelativistic Lagrangians, for example, those that describe the electric and magnetic limits of Galilean electromagnetism. We discuss the quantization of scalar and Fermi fields. Scattering amplitudes and cross sections are computed for the Coulomb interaction, as well as electron–electron and electron–positron scattering.PACS Nos.: 03.70.+k, 03.65.Nk, 11.10.Kk, 11.10.Cp

Galilean covariant models of bosons coupled to a Chern–Simons gauge field

We construct non-relativistic models of complex scalar bosons coupled to Chern–Simons gauge fields by using a Galilean covariant formulation based on the embedding of the (d, 1) Newtonian spacetime into a (d +1 , 1) Minkowski manifold with light-cone coordinates. We also examine various generalizations of the Carroll–Field–Jackiw three-dimensional Chern–Simons term for which the usual Lorentz covariance is broken. Models with cubic Chern–Simons term and non-Abelian gauge fields are briefly discussed. Our main result is the application of this covariant formalism to the investigation of the Aharonov– Bohm effect, for which we retrieve the invariant scattering amplitude up to one-loop.

QED Fermi Fields as Operator-Valued Distributions and Anomalies

The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on the importance of causality following the work of Epstein and Glaser. Gauge invariant theories demand the extension of the usual translation operation on OPVD, thereby leading to a generalized $QED$ formulation. At D=2 the solvability of the Schwinger model is totally preserved. At D=4 the paracompactness property of the Euclidean manifold permits using test functions which are decomposition of unity and thereby provides a natural justification and extension of the non perturbative heat kernel method (Fujikawa) for abelian anomalies. On the Minkowski manifold the specific role of causality in the restauration of gauge invariance (and mass generation for $QED_{2}$) is examplified in a simple way.

Galilean covariant Lagrangian models

We construct non-relativistic Lagrangian field models by enforcing Galilean covariance with a (4, 1) Minkowski manifold followed by a projection onto the (3, 1) Newtonian spacetime. We discuss scalar, Fermi and gauge fields, as well as interactions between these fields, preparing the stage for their quantization. We show that the Galilean covariant formalism provides an elegant construction of the Lagrangians which describe the electric and magnetic limits of Galilean electromagnetism. Similarly we obtain non-relativistic limits for the Proca field. Then we study Dirac Lagrangians and retrieve the Levy-Leblond wave equations when the Fermi field interacts with an Abelian gauge field.

Taub numbers at future null infinity: III. The Bondi mass

This work extends the ideas developed in two previous papers by the authors. First- and second-order perturbation solutions of Einstein's equations (in Newman - Penrose form) for the Bondi - Sachs metric are found on a background Minkowski manifold. These solutions allow a tensorial calculation of the Bondi mass using the Taub superpotential.

What is a covariant derivative?

A slight change of emphasis, from geometry towards local symmetry, provides a better framework for the formulation of gauge theories of the gravitational type. General Relativity no longer requires any improvisation, whether the local algebra is based on Poincaré or on De Sitter. Application to the graded super Minkowski manifold, with local algebra based on the super Poincaré algebra, leads to an action principle. The torsion constraints are obtained by straightforward variation of the action . The comparison with component supergravity has not yet been completed, but preliminary indications strongly indicate a close relationship. Local symmetry of the action principle is maintained, the structure is field independent, and the torsion constraints also preserve the local symmetry algebra. The coupling to matter shows that these constraints are necessary for the exclusion of ghosts from the matter sector.

On the Response of a Rindler Particle Detector. III

We investigate the power spectrum of the vacuum noise of a free Dirac field seen by a uniformly accelerated observer in the Minkowski manifold of arbitrary dimension. When the field is massless, it is found to be of Fermi-Dirac type if the dimension of the manifold is even and to be Planckian otherwise. We then consider a simple model of a particle detector of the De Witt monopole type coupled to the Dirac field. When the detector is uniformly accelerated, its response is shown to be expressible in terms of the above power spectrum. These conclusions apply also to a comoving observer (or detector) in the de Sitter manifold if the field is in the conformal vacuum.

On the Response of a Rindler-Particle Detector. II

(Received May 14, 1984) The response function of a uniformly accelerated detector of the De Witt monopole type coupled to a free scalar field in Minkowski manifold of arbitrary dimension is investigated. The principle of detailed balancing is shown to be obeyed. The response is identical to that of the detector at rest in a thermal bath only for a massless field in two or four dimensional spacetime. For a massive field the identity will result provided the line width of the detector is much larger than a value determined by the mass and the acceleration.

On the Response of a Rindler-Particle Detector

The response function of a uniformly accelerated detector of the De Witt monopole type coupled to a free scalar field in Minkowski manifold of arbitrary dimension is investigated. The principle of detailed balancing is shown to be obeyed. The response is identical to that of the detector at rest in a thermal bath only for a massless field in two or four dimensional spacetime. For a massive field the identity will result provided the line width of the detector is much larger than a value determined by the mass and the acceleration.

Factorized wave equations on homogeneous spaces of the Poincaré group

Relativistic quantum-mechanical wave equations are presented for spinor functions defined on the parameter manifold of homogeneous spaces of the Poincare group. These equations are linear in the momentum operators and may be iterated to define certain linear combinations of the Casimir invariants of the Poincare group. The additional variables, over and above those of the Minkowski manifold, are expected to be related to some internal structure which is effectively hidden in the free theory, but which may become observable when interactions are introduced. The minimal coupling to electromagnetism and to other external fields is presented.