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Yakov Itin

Publications and source records attributed to Yakov Itin.

At least 19 recordsLinked to original sources
arXiv2024

Acoustic axes conditions revised

The explanation of the basic acoustic properties of crystals requires a recognition of the acoustic axes. To derive the acoustic axes in a given material, one requires both a workable method and the necessary and sufficient criteria for the existence of the acoustic axes in a partial propagation direction. We apply the reduced form of the acoustic tensor to the acoustic axis conditions in the present work. Using this tensor, we obtain in a compact form, allowing for qualitative analysis, the necessary and sufficient criteria for the existence of the acoustic axis. Furthermore, the well-known Khatkevich criteria and their variants are recast in terms of the reduced acoustic tensor. This paper's primary input is an alternate minimal polynomial-based system of acoustic axes conditions. In this approach, we derive an additional characteristic of acoustic axes: the directions in which the minimal polynomial of the third order is reduced to that of the second order. Next, we offer a general solution to the second-order minimum polynomial equation, that utilizes a scalar and a unit vector for defining the acoustic tensor along the acoustic axis. It is shown that the scalar matches the eigenvalue of the reduced acoustic tensor, and the vector corresponds to the polarization into the single eigenvalue direction. We use the minimal polynomial construction to demonstrate the equivalence of different acoustic axis criteria. We demonstrate the applicability of this approach to actual computations of the acoustic axes and their fundamental properties (phase speeds and polarizations) for high symmetry cases, such as isotropic materials and RTHC crystals.

math-ph↗
arXiv2023

Reduced Cristoffel's tensor and acoustic axes: Necessary and sufficient condition

Acoustic axes are spatial directions in media (often crystals) where at least two of the three acoustic waves have the same phase velocity. Identification of such directions for materials with specific elasticity parameters is both theoretically fascinating and practical in acoustic applications. In this paper, we introduce the notion of the reduced acoustic tensor. In contrast to the ordinary Cristoffel's tensor, this tensor is traceless. Then it has only five independent components and fulfills the depressed cubic characteristic equation. We show that the conditions for the existence of an acoustic axes in a given propagation direction are determined solely by the reduced acoustic tensor. The necessary and sufficient conditions takes a simple compact invariant form. We also present the expression for the wave velocities along the acoustic axis. The conditions for the oblate/prolate types of the axis are derived in terms of reduced acoustic tensor.

physics.class-ph↗
arXiv2023

Cauchy relations in linear elasticity: Algebraic and physics aspects

The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure's deviation from the basic physical assumptions of Cauchy's model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke's law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters.

math-ph↗
arXiv2022

Hilbert's energy-momentum tensor extended

A variational derivative of a Lagrangian with regard to the metric tensor is used in classical field models to define Hilbert's energy-momentum tensor for a matter field. In solid-state physics, constitutive relationships between fundamental field variables are a topic that is covered by a broad variety of models. In this context, a constitutive tensor of higher order replaces the of the second-order metric tensor. For the classical field models of gravity and electrodynamics, a similar premetric description with a linear constitutive relation has recently presented. In this paper, we analyze the extension of the Hilbert definition of the energy-momentum tensor to models with general linear constitutive law. Differential forms are required for the covariant treatment of integrals on a differential manifold. The Lagrangian, electromagnetic current, and energy-momentum current must all be represented as twisted 4-forms, 3-forms, and vector-valued 3-forms, respectively. For an arbitrary linear map on forms, we derive a commutative variation identity that allows direct variation procedures without having to deal with the individual components. One can deal with Maxwell-type Lagrangians in any dimension by restricting the linear map to the generalized Hodge dual map (constitutive law). The Hilbert energy-momentum current, which is defined as a variation derivative of the Lagrangian with regard to a coframe field, is derived in differential form. It is demonstrated that the commutative variation identity is closely connected to the explicit form of the energy-momentum current. This construction is applied to a number of field models having a general linear constitutive law.

gr-qc↗
arXiv2021

On the pseudo-Riemann's quartics in Finsler's geometry

An extension of Riemmann's geometry into a direction dependent geometric structure is usually described by Finsler's geometry. Historically, this construction was motivated by the well-known Riemann's quartic length element example. Quite surprisingly, the same quartic expression emerges in solid-state electrodynamics as a basic dispersion relation -- covariant Fresnel equation. Consequently, Riemann's quartic length expression can be interpreted as a mathematical model of a well-established physics phenomena. In this paper, we present various examples of Riemann's quartic that demonstrate that Finsler's geometry is too restrictive even in the case of a positive definite Euclidean signature space. In the case of the spaces endowed with an indefinite (Minkowski) signature, there are much more singular hypersurfaces where the strong axioms of Finsler's geometry are broken down. We propose a weaker definition of Finsler's structure that is required to be satisfied only on open subsets of the tangent bundle. We exhibit the characteristic singular hypersurfaces related to Riemann's quartic and briefly discuss their possible physical interpretation.

math-ph↗
arXiv2020

Decomposition of third-order constitutive tensors

Third-order tensors are widely used as a mathematical tool for modeling physical properties of media in solid state physics. In most cases, they arise as constitutive tensors of proportionality between basic physics quantities. The constitutive tensor can be considered as a complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be seen as a tool for proper identification of natural material, as crystals, and for design the artificial nano-materials with prescribed properties. In this paper, we study the algebraic properties of a generic 3-rd order tensor relative to its invariant decomposition. In a correspondence to different groups acted on the basic vector space, we present the hierarchy of types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a generic 3-rd order tensor, these features are described explicitly. In the case of special tensors of a prescribed symmetry, the decomposition turns out to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of a pair symmetry and the Hall tensor as an example of a pair skew-symmetry.

math-ph↗
arXiv2018

Irreducible matrix resolution of the elasticity tensor for symmetry systems

In linear elasticity, a fourth order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties of a material. Due to Voigt, this tensor is conventionally represented by a $6\times 6$ symmetric matrix. This classical matrix representation does not conform with the irreducible decomposition of the elasticity tensor. In this paper, we construct two alternative matrix representations. The $3\times 7$ matrix representation is in a correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by three $3\times 3$ matrices is suitable for description the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

physics.class-ph↗
arXiv2018

Premetric teleparallel theory of gravity and its local and linear constitutive law

We continue to investigate the premetric teleparallel theory of gravity (TG) with the coframe (tetrad) as gravitational potential. We start from the field equations and a local and linear constitutive law. We create a Tonti diagram of TG in order to disclose the structure of TG. Subsequently we irreducibly decompose the 6th order constitutive tensor under the linear group. Moreover, we construct the most general constitutive tensors from the metric and the totally antisymmetric Levi-Civita symbol, and we demonstrate that they encompass nontrivial axion and skewon type pieces. Using these tools, we derive for TG in the geometric-optics approximation propagating massless spin 0, 1, and 2 waves, including the special case of Einstein's general relativity.

gr-qc↗
arXiv2017

Premetric equivalent of general relativity: Teleparallelism

In general relativity (GR), the metric tensor of spacetime is essential since it represents the gravitational potential. In other gauge theories (such as electromagnetism), the so-called premetric approach succeeds in separating the purely topological field equation from the metric-dependent constitutive law. We show here that GR allows for a premetric formulation, too. For this purpose, we apply the teleparallel approach of gravity, which represents GR as a gauge theory based on the translation group. We formulate the metric-free topological field equation and a general linear constitutive law between the basic field variables. The requirement of local Lorentz invariance turns the model into a full equivalent of GR. Our approach opens a way for a natural extension of GR to diverse geometrical structures of spacetime.

gr-qc↗
arXiv2016

On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics

In 1922, Kottler put forward the program to remove the gravitational potential, the metric of spacetime, from the fundamental equations in physics as far as possible. He successfully applied this idea to Newton's gravitostatics and to Maxwell's electrodynamics, where Kottler recast the field equations in premetric form and specified a metric-dependent constitutive law. We will discuss the basics of the premetric approach and some of its beautiful consequences, like the division of universal constants into two classes. We show that classical electrodynamics can be developed without a metric quite straightforwardly: the Maxwell equations, together with a local and linear response law for electromagnetic media, admit a consistent premetric formulation. Kottler's program succeeds here without provisos. In Kottler's approach to gravity, making the theory relativistic, two premetric quasi-Maxwellian field equations arise, but their field variables, if interpreted in terms of general relativity, do depend on the metric. However, one can hope to bring the Kottler idea to work by using the teleparallelism equivalent of general relativity, where the gravitational potential, the coframe, can be chosen in a premetric way.

gr-qc↗
arXiv2016

Generally Covariant Maxwell Theory for Media with a Local Response: Progress since 2000

In the recent decades, it became more and more popular for engineers, physicists, and mathematicians alike to put the Maxwell equations into a generally covariant form. This is particularly useful for understanding the fundamental structure of electrodynamics (conservation of electric charge and magnetic flux). Moreover, it is ideally suited for applying it to media with local (and mainly linear) response behavior. We try to collect the new knowledge that grew out of this development. We would like to ask the participants of EMTS 2016 to inform us of work that we may have overlooked in our review.

physics.class-ph↗
arXiv2015

Photon propagator in skewon electrodynamics

Electrodynamics with a local and linear constitutive law is used as a framework for models violating Lorentz covariance. The constitutive tensor of such a construction is irreducibly decomposed into three independent pieces. The principal part is the anisotropic generalisation of the standard electrodynamics. The two other parts, axion and skewon, represent non-classical modifications of electrodynamics. We derive the expression for the photon propagator in the Minkowski spacetime endowed with a skewon field. For a relatively small (antisymmetric) skewon field, a modified Coulom law is exhibited.

hep-th↗
arXiv2015

Quadratic invariants of the elasticity tensor

We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.

cond-mat.other↗
arXiv2015

On skewon modification of light cone structure

Electromagnetic media with generic linear response provide a rich class of Lorentz violation models. In the framework of a general covariant metric-free approach, we study electromagnetic wave propagation in these media. We define the notion of an optic tensor and present its unique canonical irreducible decomposition into the principle and skewon parts. The skewon contribution to the Minkowski vacuum is a subject that does not arise in the ordinary models of Lorentz violation based on a modified Lagrangian. We derive the covector parametrization of the skewon optic tensor and discuss its $U(1)$-gauge symmetry. We obtain several compact expressions for the contribution of the principle and skewon optic tensor to the dispersion relation. As an application of the technique proposed here, we consider the case of a generic skewon tensor contributed to a simple metric-type principle part. Our main result: Every solution of the skewon modified Minkowski dispersion relation is necessary spacelike or null. It provides an extreme violation of the Lorentz symmetry. The case of the antisymmetric skewon is studied in detail and some new special cases (electric, magnetic, and degenerate) are discovered. In the case of a skewon represented by a symmetric matrix, we observe a parametric gap that has some similarity to the Higgs model. We worked out a set of specific examples that justify the generic properties of the skewon models and demonstrate the different types of the Lorentz violation phenomena.

hep-th↗
arXiv2014

Finsler-type modification of the Coulomb law

Finsler geometry is a natural generalization of pseudo-Riemannian geometry. It can be motivated e.g. by a modified version of the Ehlers-Pirani-Schild axiomatic approach to space-time theory. Also, some scenarios of quantum gravity suggest a modified dispersion relation which could be phrased in terms of Finsler geometry. On a Finslerian spacetime, the Universality of Free Fall is still satisfied but Local Lorentz Invariance is violated in a way not covered by standard Lorentz Invariance Violation schemes. In this paper we consider a Finslerian modification of Maxwell's equations. The corrections to the Coulomb potential and to the hydrogen energy levels are computed. We find that the Finsler metric corrections yield a splitting of the energy levels. Experimental data provide bounds for the Finsler parameters.

gr-qc↗
arXiv2014

Irreducible decompositions of the elasticity tensor under the linear and orthogonal groups and their physical consequences

We study properties of the fourth rank elasticity tensor C within linear elasticity theory. First C is irreducibly decomposed under the linear group into a "Cauchy piece" S (with 15 independent components) and a "non-Cauchy piece" A (with 6 independent components). Subsequently, we turn to the physically relevant orthogonal group, thereby using the metric. We find the finer decomposition of S into pieces with 9+5+1 and of A into those with 5+1 independent components. Some reducible decompositions, discussed earlier by numerous authors, are shown to be inconsistent. --- Several physical consequences are discussed. The Cauchy relations are shown to correspond to A=0. Longitudinal and transverse sound waves are basically related by S and A, respectively.

cond-mat.other↗
arXiv2014

The Kummer tensor density in electrodynamics and in gravity

Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\cal T}^{ijkl} = - {\cal T}^{jikl} = - {\cal T}^{ijlk}$. Thus, ${\cal K}\sim {\cal T}^3$, see Eq.(46). (i) If $\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If $\cal T$ is identified with the {\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then ${\cal K}\sim R^3$ and, in the special case of general relativity, ${\cal K}$ reduces to the Kummer tensor of Zund (1969). This $\cal K$ is related to the {\it principal null directions} of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose $\cal K$ irreducibly under the 4-dimensional linear group $GL(4,R)$ and, subsequently, under the Lorentz group $SO(1,3)$.

gr-qc↗