Minkowski Metric Research Articles

Lorentz Transformation Under a Discrete Dynamical Time and Continuous Space

The Lorentz transformation of space and time between two reference frames is one of the pillars of the special relativity theory. As a result of the Lorentz transformation, space and time are only relative and are entangled, while the Minkowski metric is Lorentz invariant. For this reason, the Lorentz transformation is one of the major obstructions in the development of physical theories with quantized space and time. Here is described the Lorentz transformation of a physical system with a discrete dynamical time and a continuous space that fulfills Lorentz invariance while approximating the Lorentz transformation at the time continuous limit and the Galilei transformation at the classical limit. Furthermore, the discreteness of time is not mixed with the continuous nature of space, making time distinct from space.

The t-Graphs over Finitely Generated Groups and the Minkowski Metric

In this paper, we introduce t-graphs defined on finitely generated groups. We study some general aspects of the t-graphs on two-generator groups, emphasizing establishing necessary conditions for their connectedness. In particular, we investigate properties of t-graphs defined on finite dihedral groups.

AUTOMATIC CLASSIFICATION OF PAINTINGS BY YEAR OF CREATION

Context. The problem of automatic verification of the legitimacy of the export of works of art is considered. Objective. A method is proposed for automatically determining the age of a painting from a digital photograph using a classification that is performed by an intelligent decision-making system. Method. It is proposed to use the attribute of picture year of creation as the main criterion for making a decision during the customs check of exports legitimacy. Instead of a long and expensive museum examination, photographing works of art in customs conditions and processing photos using a set of descriptors is used. The set of descriptors is proposed, include local binary patterns, their color modification, Haralik’s texture features, the first four moments, Tamura’s texturt features, SIFT descriptor. The data obtained as a result of descriptors action give the values of several dozen private attributes. They form data vectors, which are then concatenated into a generalized object description vector. In the feature space thus created, automatic classification by weighted k-nearest neighbors is performed. The proposed algorithm calculates the distance between objects in a multidimensional space of attribute values and assigns new objects to already formed classes. The criterion for creating classes is the age of the painting from the existing database. As a measure of the objects proximity, it is proposed to use the Euclid and Minkowski metrics. The calculation of weights for the proposed classification algorithm is performed by the Fisher method. Results. The effectiveness of the proposed method was investigated in the course of experiments with an image database containing photos of paintings by world, European and Ukrainian artists. Algorithm configuration parameters that provide high classification accuracy are found. Conclusions. The performed experiments have shown the effectiveness of the selected descriptors for the formation of vector descriptions of images of paintings. The greatest accuracy is provided by descriptor merging, which reveals significant differences in the structural properties of images. This approach to the description of objects in combination with the proposed classification algorithm and the chosen main criterion ensures high accuracy of the obtained solutions. The direction of further research may include the use of convolutional neural networks to improve the accuracy of classification under the condition of a static database.

Z-Score Normalized Features with Maximum Distance Measure Based k-NN Automated Blood Cancer Diagnosis System

Leukemia is a blood-forming cancer disease characterized by the abnormal growth of White Blood Cells. Early detection and treatment of cancer reduce mortality and increase the survival rate. Most of the k-NN approaches used a different combination of statistical and geometrical features of the nucleus and cytoplasm with and without normalization, resulting in an uncertain range of features. Minkowski, Euclidean, City-block, Correlation, and Cosine distance metrics discover the average similarity between features predicts Leukemia with less accuracy. This paper proposes a Z-Score normalized feature set combined with a k-NN maximum distance measure-based automated blood cancer diagnosis system to address this problem. Initially, the features are normalized using the Z-score normalization method, which eliminates the mis-classification outcomes. The Chebyshev distance measure, which finds the maximum similarities between the features when k=1, has the highest accuracy of 97.92%. Furthermore, tuning the parameters by grid-search improves the performance rate by 98.65% .

Prototype based granular neuro-fuzzy system for regression task

Artificial intelligence is often inspired by biological solutions. Prototypes are among these inspirations. Humans often describe complex entities by comparing them to previously known items (prototypes) instead of providing a detailed description. In this study, we apply this approach to neuro-fuzzy systems. Neuro-fuzzy systems operate using fuzzy IF-THEN rules. In our approach, the premises of rules are represented by prototypes. A new item is compared to the prototypes (as wholes) in the rule premises and its similarities to the prototypes in the rules define the firing strengths of the rules. The similarity is assessed based on the Minkowski metric. Prototypes are elaborated by granulation of the input domain with clustering and by applying the principle of justifiable granularity. We tested the proposed method in numerical experiments based on the regression task for benchmark data sets in public data repositories.

Power-law solutions of the f(G) gravity model with electromagnetic and scalar field

In this article, the Gauss-Bonnet model of gravity is investigated, the action of which includes the Gauss-Bonnet invariant, the Maxwell term  and the scalar field. The model is considered in a flat, isotropic and homogeneous Friedman-Robertson-Walker universe in four dimensions at the late stages of the Universe evolution. The Minkowski metric is used. For the researched model, a system of equations of motion and a solution with a scale factor with a power and modified power dependence on time are found. To find a solution, the method of variation was used. Similar solutions are obtained using the Euler-Poisson equation and the method of lowering the order of the derivative, followed by the usage of the Euler-Lagrange equation. The values of , energy density and isotropic pressure are obtained, the graphs of which correspond to modern cosmological data. In the course of the research, it turned out that for a scale factor with a power-law dependence on time within the framework of the researched model, the parameter of the equation of state is equal to the value obtained when solving a model with a modified power-law scale factor. The deceleration parameter is negative, which confirms the realism of the proposed model within the framework of an accelerated expanding universe. When considering the energy conditions, it turned out that the conditions are fulfilled in both models, but the SEC is not fulfilled. A comparison of the values of the energy conditions of the two models is shown.

Einstein–Rosen waves and the Geroch group

Under the action of the Geroch group, the Minkowski metric can be transformed into any vacuum metric with two commuting Killing vectors. In principle, this reduces the problem of deriving vacuum metrics with two commuting Killing vectors to pure algebra. In this article, we use these facts to give a purely algebraic derivation of the Einstein–Rosen metric, which describes a cylindrical gravitational wave. Our derivation has a straightforward extension to gravitational pulse waves.

General fuzzy C-means clustering algorithm using Minkowski metric

As one of the most commonly used clustering methods, fuzzy clustering technique such as the Fuzzy C-means (FCM) has undergone a rapid development. In this paper, a general FCM clustering algorithm based on contraction mapping (cGFCM) is proposed for more general cases of using Minkowski metric (Lp-norm distance) as the similarity measure, and the analytical method for calculating the parameters of the proposed algorithm is given. The core of the proposed cGFCM algorithm lies on constructing a contraction mapping to update the prototypes when an arbitrary Minkowski metric is used to measure the closeness of data points. Subsequently, mainly guided by the Banach contraction mapping principle, the algorithm and implementation approaches are discussed in detail, and the correctness and feasibility of the proposed method are proved. Moreover, the convergence of the proposed algorithm is also discussed. Experimental studies carried out on both synthetic data sets and real-world data sets show that the proposed cGFCM algorithm extends FCM to more general cases without extra time and space costs. Compared with another generalized FCM clustering strategy and other five state-of-the-art clustering methods, the proposed algorithm can not only reach better performance in both clustering accuracy and stability, but reduce the running time several-fold.

Devising an entropy-based approach for identifying patterns in multilingual texts

Even though the plagiarism identification issue remains relevant, modern detection methods are still resource-intensive. This paper reports a more efficient alternative to existing solutions. The devised system for identifying patterns in multilingual texts compares two texts and determines, by using different approaches, whether the second text is a translation of the first or not. This study's approach is based on Renyi entropy. The original text from an English writer's work and five texts in the Russian language were selected for this research. The real and "fake" translations that were chosen included translations by Google Translator and Yandex Translator, an author's book translation, a text from another work by an English writer, and a fake text. The fake text represents a text compiled with the same frequency of keywords as in the authentic text. Upon forming a key series of high-frequency words for the original text, the relevant key series for other texts were identified. Then the entropies for the texts were calculated when they were divided into "sentences" and "paragraphs". A Minkowski metric was used to calculate the proximity of the texts. It underlies the calculations of a Hamming distance, the Cartesian distance, the distance between the centers of masses, the distance between the geometric centers, and the distance between the centers of parametric means. It was found that the proximity of texts is best determined by calculating the relative distances between the centers of parametric means (for "fake" texts ‒ exceeding 3, for translations ‒ less than 1). Calculating the proximity of texts by using the algorithm based on Renyi entropy, reported in this work, makes it possible to save resources and time compared to methods based on neural networks. All the raw data and an example of the entropy calculation on php are publicly available

Off-shell diagrammatics for quantum gravity

This article reports on how diagrammatic identities of Yang–Mills theory translate to diagrammatics for pure gravity. For this, we consider the Einstein–Hilbert action and follow the approach of Capper, Leibbrandt, and Medrano and expand the inverse metric density around the Minkowski metric. By analogy to Yang–Mills theory, cancellation identities are constructed for the graviton as well as the ghost vertices up to the valency of six.

Minkowski space in f(T) gravity

The full set of solutions of $f(T)$ gravity with the Minkowski metric is considered in this note. At the 4-th order in perturbations around the trivial tetrad solution, a new mode is found explicitly. Its presence signals a strong coupling problem that transcends the Minkowski background and also suggests the pathological nature of cosmological solutions.

Casimir forces on deformed fermionic chains

We characterize the Casimir forces for the Dirac vacuum on free-fermionic chains with smoothly varying hopping amplitudes, which correspond to (1+1)D curved spacetimes with a static metric in the continuum limit. The first-order energy potential for an obstacle on that lattice corresponds to the Newtonian potential associated to the metric, while the finite-size corrections are described by a curved extension of the conformal field theory predictions, including a suitable boundary term. We show that, for weak deformations of the Minkowski metric, Casimir forces measured by a local observer at the boundary are metric-independent. We provide numerical evidence for our results on a variety of (1+1)D deformations: Minkowski, Rindler, anti-de Sitter (the so-called rainbow system) and sinusoidal metrics.

Are non-vacuum states much relevant for retrieving shock wave memory of spacetime?

Shock wave gives back reaction to spacetime and the information is stored in the memory of background. Quantum memory effect of localised shock wave on Minkowski metric is investigated here. We find that the Wightman function for massless scalar field, on both sides of the wave location, turns out to be the same for usual Minkowski spacetime. Therefore the observables obtained from this, for any kind of observer, do not show the signature of the shock. Moreover, the vacuum states of field on both sides are equivalent. On the contrary, the correlator for the non-vacuum state does memorise the classical shock wave and hence the effect of it is visible in the observables, even for any frame. We argue that rather than vacuum state, the non-vacuum ones are relevant to retrieve the quantum information of classical memory.

Cauchy–Maxwell equations: A space–time conformal gauge theory for coupled electromagnetism and elasticity

A space–time conformal gauge theory is used to develop a unified continuum model describing myriad electromechanical and magnetomechanical coupling effects observed in solids. Using the pseudo-Riemannian Minkowski metric in a finite-deformation setup and exploiting the Lagrangian’s local conformal symmetry, we derive Cauchy–Maxwell (CM) equations that seamlessly combine, for the first time, Cauchy’s elasto-dynamic equations with Maxwell’s equations for electromagnetism. Maxwell’s equations for vacuum are recoverable from our model, which in itself also constitutes a new derivation of these equations. With deformation gradient and material velocity coupled in the Lagrange density, various pseudo-forces appear in the Euler–Lagrange equations. These forces, not identifiable through classical continuum mechanics, should have significance under specific geometric or loading conditions. As a limited illustration on how the CM equations work, we carry out semi-analytical studies, viz. on an infinite body subject to isochoric deformation and a finite membrane under both tensile and transverse loading, considering piezoelectricity and piezomagnetism. Our results show that under specific loading frequencies and tension, electric and magnetic potentials may increase rapidly in some regions of the membrane. Explorations of this nature via the CM model may have implications in future studies on efficient energy harvesting.

Hadamard renormalization of a two-dimensional Dirac field

The Hadamard renormalization procedure is applied to a free, massive Dirac field $\psi$ on a 2 dimensional Lorentzian spacetime. This yields the state-independent divergent terms in the Hadamard bispinor $G^{(1)}(x, x') = \frac{1}{2} \left\langle \left[ \bar{\psi}(x'), \psi(x) \right] \right\rangle$ as $x$ and $x'$ are brought together along the unique geodesic connecting them. Subtracting these divergent terms within the limit assigns $G^{(1)}(x, x')$, and thus any operator expressed in terms of it, a finite value at the coincident point $x' = x$. In this limit, one obtains a quadratic operator instead of a bispinor. The procedure is thus used to assign finite values to various quadratic operators, including the stress-energy tensor. Results are presented covariantly, in a conformally-flat coordinate chart at purely spatial separations, and in the Minkowski metric. These terms can be directly subtracted from combinations of $G^{(1)}(x, x')$ - themselves obtained, for example, from a numerical simulation - to obtain finite expectation values defined in the continuum.

Origins of Hyperbolicity in Color Perception.

In 1962, H. Yilmaz published a very original paper in which he showed the striking analogy between Lorentz transformations and the effect of illuminant changes on color perception. As a consequence, he argued that a perceived color space endowed with the Minkowski metric is a good approximation to model color vision. The contribution of this paper is twofold: firstly, we provide a mathematical formalization of Yilmaz’s argument about the relationship between Lorentz transformations and the perceptual effect of illuminant changes. Secondly, we show that, within Yilmaz’s model, the color space can be coherently endowed with the Minkowski metric only by imposing the Euclidean metric on the hue-chroma plane. This fact motivates the need of further investigation about both the proper definition and interrelationship among the color coordinates and also the geometry and metrics of perceptual color spaces.

A theory of gravitation

The paper presents a theory of gravitation as a continuum dynamical theory, i.e. the equations of motion are first order in the time derivatives. The theory satisfies the laws of conservation of total energy, momentum and mass in the standard sense, i.e. as applications of Stokes’ theorem. A model in this theory is defined by the specification of an energy density function that accounts for the total mechanical and gravitational energy of the system and which in turn defines an action function. The derivation of the equations of motion is based on Hamilton’s principle of least action with the “same” action function as used for the derivation of the Einstein equation of the theory of general relativity, however, not with respect to variations of the metric but with respect to variations induced by local displacements of space, i.e. variations of the conjugate variable of the momentum and variations of the momentum density. Entropy density is introduced as a variable. This makes it possible to determine the thermodynamic equilibrium conditions for a model. Solutions of the equilibrium conditions are the Schwarzschild and Minkowski metrics on space-time, and the metrics defining the spherical and hyper spherical spaces.

Resolving a spacetime singularity with field transformations

Abstract It is widely believed that classical gravity breaks down and quantum gravity is needed to deal with a singularity. We show that there is a class of spacetime curvature singularities which can be resolved with metric and matter field transformations. As an example, we consider an anisotropic power-law inflation model with scalar and gauge fields in which a space-like curvature singularity exists at the beginning of time. First, we provide a transformation of the metric to the flat geometry, i.e. the Minkowski metric. The transformation removes the curvature singularity located at the origin of time. An essential difference from previous work in the literature is that the origin of time is not sent to past infinity by the transformation but it remains at a finite time in the past. Thus the geometry becomes extensible beyond the singularity. In general, matter fields are still singular in their original form after such a metric transformation. However, we explicitly show that there is a case in which the singular behavior of the matter fields can be completely removed by a redefinition of matter fields. Thus, for the first time, we have resolved a class of initial cosmic singularities and successfully extended the spacetime beyond the singularity in the framework of classical gravity.

On Conformally Flat Cubic (α,β )-Metrics

We prove that in the class of cubic metrics on a manifold M of dimension n ≥ 3, every weakly Einstein metrics is either a Riemannian metric or a locally Minkowski metric. We also prove that the same statement holds for metrics of almost vanishing Ξ-curvature.

Unruh effect universality: emergent conical geometry from density operator

The Unruh effect has been investigated from the point of view of the quantum statistical Zubarev density operator in space with the Minkowski metric. Quantum corrections of the fourth order in acceleration to the energy-momentum tensor of real and complex scalar fields, and Dirac field are calculated. Both massless and massive fields are considered. The method for regularization of discovered infrared divergences for scalar fields is proposed. The calculated corrections make it possible to substantiate the Unruh effect from the point of view of the statistical approach, and to explicitly show its universality for various quantum field theories of massless and massive fields. The obtained results exactly coincide with the ones obtained earlier by calculation of the vacuum aver- age of energy-momentum tensor in a space with a conical singularity. Thus, the duality of two methods for describing an accelerated medium is substantiated. One may also speak about the emergence of geometry with conical singularity from thermodynamics. In particular, the polynomiality of the energy-momentum tensor and the absence of higher-order corrections in acceleration can be explicitly demonstrated.