Lorentz Matrix Research Articles

Hamilton’s equations in the covariant teleparallel equivalent of general relativity

We present Hamilton's equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism necessary to compute Hamilton's equations in both Weitzenbock gauge and covariant formulation, where for the latter we must introduce new fields: Lorentz matrices and their associated momenta. We also derive explicit relations between the conjugate momenta of the tetrad and the conjugate momenta for the metric that are traditionally defined in GR, which are important to compare both formalisms.

Vertices of Suborbital Graph $F_{u,N}$ under Lorentz Matrix Multiplication

In this study, suborbital graphs, $G_{u,N}$ and $F_{u,N}$ are examined. Modular group $\Gamma$ and its act on $\widehat{\mathbb{Q}}$ are studied. Lorentz matrix that gives the vertices obtained under the classical matrix multiplication in the suborbital graph $F_{u,N}$ is analysed with the Lorentz matrix multiplication. Lorentz matrix written as Möbius transform is normalized and the type of the transform is researched. Moreover, a different element of Modular group $\Gamma$ is scrutinized. The vertices on the path starting with $\infty$ are obtained under this element and the Lorentz matrix multiplication. For this path, it is shown that the vertices obtained in $F_{u,N}$ under the Lorentz matrix multiplication with the Lorentz matrix satisfied the farthest vertex condition for the previous vertex.

Heisenberg Doubles for Snyder-Type Models

A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

Some studies on Lorentz transformation matrix in non-cartesian co-ordinate system

The Lorentz matrices for transformation of co-ordinates in Cartesian system are presented for the cases when the relative velocity between two reference frames is along X , Y and Z axes. The general form of the matrix for transformation of co-ordinates from unprimed to primed frame has been deduced in case of Cartesian co-ordinate system with the help of the above matrices. This matrix has not been transformed to the cases of cylindrical and spherical polar co-ordinates due to the fact that the calculations are cumbersome and lengthy. Hence, considering the relative velocity between two frames along a co-ordinate axis the transformation matrix has been found out for cylindrical and spherical co-ordinates.

Screw Theory in Lorentzian Space

In this paper we present various results about the six dimensional vectors obtained from the tangent operators of spatial motion, called as screws, in Lorentzian space. Each screw has an axis defined by six Plucker coordinates in Lorentzian space. The manipulation of screw coordinate transformations has been simplified by using Lorentz matrix multiplication and dual number algebra. Also, we showed that screw displacement is representation as the exponential of a dual angular velocity matrix by using the dual orthogonal matrices in Lorentzian space.

Using Tangent Boost along a Worldline and Its Associated Matrix in the Lie Algebra of the Lorentz Group

In order to generalize the relativistic notion of boost to the case of non inertial particles and to general relativity, we look closer into the definition of the Lie group of Lorentz matrices and its Lie algebra and we study how this group acts on the Minskowski space. We thus define the notion of tangent boost along a worldline. This very general notion gives a useful tool both in special relativity (for non inertial particles or/and for non rectilinear coordinates) and in general relativity. We also introduce a matrix of the Lie algebra which, together with the tangent boost, gives the whole dynamical description of the considered system (acceleration and Thomas rotation). After studying the properties of Lie algebra matrices and their reduced forms, we show that the Lie group of special Lorentz matrices has four one-parameter subgroups. These tools lead us to introduce the Thomas rotation in a quite general way. At the end of the paper, we present some examples using these tools and we consider the case of an electron rotating on a circular orbit around an atom nucleus. We then discuss the twin paradox and we show that when the one who made a journey into space in a high-speed rocket returns home, he is not only younger than the twin who stayed on Earth but he is also disorientated because his gyroscope has turned with respect to earth referential frame.

Euclidean Complex Relativistic Mechanics: A New Special Relativity Theory

Relativity Theory (RT) was fundamental for the development of Quantum Mechanics (QMs). Special Relativity (SR), as is applied until now, cancels the transitive attribute in parallelism, when three observers are related, because Lorentz Boost (LB) is not closed transformation. In this presentation, considering Linear Spacetime Transformation (LSTT), we demand the maintenance of Minkowski Spacetime Interval (S2). In addition, we demand this LSTT to be closed, so there is no need for axes rotation. The solution is the Vossos Matrix (ΛB) containing real and imaginary numbers. As a result, space becomes complex, but time remains real. Thus, the transitive attribute in parallelism, which is equivalent to the Euclidean Request (ER), is also valid for moving observers. Choosing real spacetime for the unmoved observer (O), all the natural sizes are real, too. Using Vossos Transformation (VT) for moving observers, the four-vectors’ zeroth component (such as energy) is real, in contrast with spatial components that are complex, but their norm is real. It is proved that moving (relative to O) human O' meter length, according to Lorentz Boost (LB). In addition, we find Rotation Matrix Vossos-Lorentz (RBL) that turns natural sizes’ complex components to real. We also prove that Speed of Light in Vacuum (c) is invariant, when complex components are used and VT is closed for three sequential observers. After, we find out the connection between two moving (relative to O) observers: X"= ΛLO"(o) ΛLO(O') X', using Lorentz Matrix (ΛL). We applied this theory, finding relations between natural sizes, that are the same as these extracted by Classic Relativity (CR), when two observers are related (i.e. relativistic Doppler shift is the same). But, the results are different, when more than two observers are related. VT of Electromagnetic Tensor (Fμv), leads to Complex Electromagnetic Fields (CEMFs) for a moving observer. When the unmoved observer O and a moving observer O' are related, O measures the same Real Electromagnetic Fields (REMFs) as those are given, using LB, but O' measure CEMFs with the same formula. Complex Electromagnetic Tensor (F'B) turns to Real Electromagnetic Tensor (F') using the formula . When there are two moving (relative to O) observers O' and O", their real electromagnetic tensors are related, using the form F" = ΛLO"(O) ΛLO(O) F' [ΛLO"(O') ΛLO(O')]T. In addition, we prove that the relation between two moving (relative to O) observers, when they use Real Coordinates (RCs), causes a real rotation between their frames. The turn is opposite to the turn of Thomas and has a little different measure, when the velocities are small. We apply these to the Uniform Circular Motion (UCM) and to the Hydrogen Atom (H), considering that the Proton (p) is the unmoved observer O and the laboratory observer O' has infinitesimal velocity. Using Perturbation Theory (PT) we calculate the position of the fine structure peaks of the atomic spectrum. The result is better not only than this extracted using P. Dirac Theory, but also than that extracted using L. H. Thomas Method.

Rotations in Minkowski Spacetime

With the relation of Olinde Rodrigues-Cartan is obtained an expression for the Lorentz matrix, and it is transformed to a better form for the Newman-Penrose formalism, thus it is possible to realize rotations of the null tetrad of NP.

Didactic derivation of the special theory of relativity from the Klein–Gordon equation

We present a didactic derivation of the special theory of relativity in which Lorentz transformations are ‘discovered’ as symmetry transformations of the Klein–Gordon equation. The interpretation of Lorentz boosts as transformations to moving inertial reference frames is not assumed at the start, but it naturally appears at a later stage. The relative velocity v of two inertial reference frames is defined in terms of the elements of the pertinent Lorentz matrix, and the bound is presented as a simple theorem that follows from the structure of the Lorentz group. The polar decomposition of Lorentz matrices is used to explain noncommutativity and nonassociativity of the relativistic composition (‘addition’) of velocities.

Comment on ‘Lorentz transformations with arbitrary line of motion’

A short comment regarding the derivation of Lorentz transformation proposed by Iyer and Prabhu (2007 Eur. J. Phys. 11 183–90) is given. It is shown that the proposed derivation is similar to that appearing in the standard textbooks of classical mechanics, electrodynamics and the theory of relativity. In fact, those textbooks also provide an elegant form of the Lorentz matrix for the (3+1)-dimensional case, which Iyer and Prabhu claim to be difficult to attain because of its algebraic complexity. We also provide the derivation of the (3+1)-dimensional version of the Lorentz matrix using a method analogous to that proposed by Iyer and Prabhu, and show that the result is completely equivalent to the (3+1)-dimensional version appearing in the textbooks.

H-Unitary and Lorentz Matrices: A Review

Many properties of H-unitary and Lorentz matrices are derived using elementary methods. Complex matrices that are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H are called H-unitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix are called Lorentz. The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H-unitary and Lorentz matrices. Several applications are given, including connected components of Lorentz similarity orbits, products of matrices that are simultaneously positive definite and H-unitary, products of reflections, and stability and robust stability.

Lorentz Transformations from Reflections: Some Applications

We point out, by exhibiting two examples and mentioning a third one, that it is sometimes useful to consider Lorentz transformations as generated from hyperplane or line reflections. One example concerns the construction of boosts linking two given 4-vectors, the other one concerns the Minkowski geometric understanding of V. Moretti's polar decomposition of orthochronous Lorentz matrices.

Thomasrotation: a Lorentz matrix approach

The composition of two pure Lorentz transformations (boosts)parametrized by non-parallel velocities is equivalent to a boostcombined with a pure spatial rotation - the Thomas rotation.Thirty years elapsed from Thomas's 1926 calculation for theprecessional application until an explicit result for theThomas rotation angle appropriate to two finite boostvelocities appeared in the literature. Over the years there havebeen a number of papers that have produced results by variousmethods for the Thomas rotation angle but none by directLorentz matrix methods. This paper repairs that deficiency bypresentation of an explicit Lorentz matrix proof of sufficientconciseness to show that the much-vaunted complexity of thedirect approach has been considerably overstated. Thedemonstration fills a gap at a fundamental level in thedevelopment of the basics of special relativity.

Local filtering operations on two qubits

We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type $\rho'\sim (A\otimes B)\rho(A\otimes B)^{\dagger}$. We consider a real matrix parameterization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filterin operations for concentrating entanglement. Furthermore we give a complete characterization of the mixed states that can be purified arbitrary close to a Bell state. Finally we obtain a new way of calculating the entanglement of formation.

On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L 2 KL 1 where L 1 and L 2 are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.

On holomorphic factorization and meromorphic selectors

We show that, generically, a smooth Lorentzian 2 × 2 matrix valued function A, defined on the unit circle and possessing a so called exterior meromorphic section, can be extented into the unit disk in a way that admits a non-degenerate holomorphic factorization P ∗P , where the adjoint is taken with respect to the constant Lorentz matrix. This has implications for meromorphic approximation and Levi-flat surfaces with generalized disk fibers.

Lorentz matrices: A review

This is an expository review of the Lorentz transformation, which is a change of coordinates used by one “inertial observer” to those used by another one. The transformation can be represented by a four-by-four matrix, the Lorentz matrix or the Minkowski-Lorentz matrix. The most familiar, or “special,” case has thex axis of both observers parallel to their relative velocity. A more general transformation drops this constraint. But then a seeming “paradox” arises when there are three observers, and this has led to a challenge to the self-consistency of the special theory of relativity. It is shown here that this challenge is based on a misunderstanding. The properties of the more general Lorentz transformation are reviewed consistently in terms of the matrix approach, which the author believes is now the easiest approach to understand. The spectral analysis of the Lorentz matrix is also discussed. Several checks are included to “make assurance double sure.”

Relativity in three dimensions

We present a new formulation of relativistic kinematics and field dynamics in an entirely three-dimensional Riemannian space. Here the basic idea is to represent local physical observers by three-dimensional vector fields of bounded length and the dynamics of physical fields by the flows of vector fields, which are one-parameter groups of local smooth transformations of the 3-space. The flow parameter plays the role of local time for each physical observer, and the Lie derivative replaces the time derivative. In the kinematics part we show how the introduction of a relative velocity function in terms of the 3-metric of the space leads immediately to the Lorentz time-dilation formula for two inertially equivalent physical observers. A treatment of equivalence class of inertial observers leads to the definition of generalized Lorentz matrix functions with corresponding generalized properties, which reduce to the usual Lorentz matrices when the underlying space is Euclidean. In the dynamics part we treat the Einstein field equations in Gaussian normal coordinates by reformulating them in a three-dimensional set-up and show how certain solutions, including the de Sitter solution, can be derived from a flat-space 3-metric by subjecting it to a transformation by the flow of a three-dimensional vector field. In this way, we obtain a new solution of the vacuum Einstein field equations which is related to the complex form of (real) Kasner-type solutions. Our formalism can also be applied to other field equations, such as the Maxwell equations, and provide a corresponding set of three-dimensional equations to generate four-dimensional solutions.

The quaternion formalism for Möbius groups in four or fewer dimensions

The Möbius group of R N ∪ {∞} defines N-dimensional inversive geometry. This geometry can serve as an alternative to projective geometry in providing a common foundation for spherical Euclidean and hyperbolic geometry. Accordingly the Möbius group plays an important role in geometry and topology. The modern emphasis on low-dimensional topology makes it timely to discuss a useful quaternion formalism for the Möbius groups in four or fewer dimensions. The present account is self-contained. It begins with the representation of quaternions by 2 x 2 matrices of complex numbers. It discusses 2 x 2 matrices of quaternions and how a suitably normalized subgroup of these matrices, extended by a certain involution related to sense reversal, is 2-1 homomorphic to the Möbius group acting on R 4 ∪ {∞}. It provides details of this action and the relation of this action to various models of the classical geometries. In higher dimensions N ⩾ 5, the best description of the Möbius group is probably by means of ( N + 2) × ( N + 2) Lorentz matrices. In the lower dimensions covered by the quaternion formalism, this alternative Lorentz formalism is a source of interesting homomorphisms. A sampling of these homomorphisms is computed explicitly both for intrinsic interest and for an illustration of the ease with which one can handle the quaternion formalism.

Relativistic-particle interactions—A third world view

A new form for describing interacting relativistic classical particles with invariant world-lines is presented. In contrast to the use of independent particle variables or of «centre of mass» and relative co-ordinates previously considered in the literature, we introduce a Lorentz matrix and an associated conjugate four-dimensional angular momentum as collective variables; these are supplemented with a suitable number of internal variables. The methods of constrained Hamiltonian dynamics are used to ensure that the number of independent degrees of freedom corresponds precisely to that for a system of a given number of particles. Independent particle variables can be defined in the formalism, and interactions included. The eleventh-generator form for relativistic Hamiltonian dynamics is essential to make relativistic invariance, invariant world-lines and interaction compatible with one another.