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Elementary Proof of Moretti's Polar Decomposition Theorem for Lorentz Transformations  [PDF]
Helmuth K. Urbantke
Physics , 2002,
Abstract: A proof more elementary than the original one is given for Moretti's theorem that the usual polar decomposition of real matrices when applied to an orthochronous proper Lorentz matrix yields just its standard rotation-boost decomposition. (The complex SL(2,C) analog is well-known.)
Model Reduction Applied to Square to Rectangular Martensitic Transformations Using Proper Orthogonal Decomposition  [PDF]
Linxiang X. Wang,Roderick V. N. Melnik
Physics , 2007,
Abstract: Model reduction using the proper orthogonal decomposition (POD) method is applied to the dynamics of ferroelastic patches to study the first order square to rectangular phase transformations. Governing equations for the system dynamics are constructed by using the Landau-Ginzburg theory and are solved numerically. By using the POD method, a set of empirical orthogonal basis functions is first constructed, then the system is projected onto the subspace spanned by a small set of basis functions determined by the associated singular values. The performance of the low dimensional model is verified by simulating nonlinear thermo-mechanical waves and square to rectangular transformations in a ferroelastic patch. Comparison between numerical results obtained from the original PDE model and the low dimensional one is carried out.
Lorentz transformations with arbitrary line of motion  [PDF]
Chandru Iyer,G. M. Prabhu
Physics , 2008, DOI: 10.1088/0143-0807/28/2/004
Abstract: Sometimes it becomes a matter of natural choice for an observer (A) that he prefers a coordinate system of two-dimensional spatial x-y coordinates from which he observes another observer (B) who is moving at a uniform speed along a line of motion, which is not collinear with As chosen x or y axis. It becomes necessary in such cases to develop Lorentz transformations where the line of motion is not aligned with either the x or the y-axis. In this paper we develop these transformations and show that under such transformations, two orthogonal systems (in their respective frames) appear non-orthogonal to each other. We also illustrate the usefulness of the transformation by applying it to three problems including the rod-slot problem. The derivation has been done before using vector algebra. Such derivations assume that the axes of K and K-prime are parallel. Our method uses matrix algebra and shows that the axes of K and K-prime do not remain parallel, and in fact K and K-prime which are properly orthogonal are observed to be non-orthogonal by K-prime and K respectively. http://www.iop.org/EJ/abstract/0143-0807/28/2/004
Generalized Lorentz Transformations  [PDF]
Virendra Gupta
Physics , 2013,
Abstract: Generalized Lorentz transformations with modified velocity parameter are considered. Lorentz transformations depending on the mass of the observer are suggested.The modified formula for the addition of velocities remarkably preserves the constancy of the velocity of light for all observers. The Doppler red shift is affected and can provide a test of such generalisations.
Lorentz Transformations from Reflections:Some Applications  [PDF]
H. K. Urbantke
Physics , 2002,
Abstract: 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.
Lorentz Transformations  [PDF]
Bernard R. Durney
Physics , 2011,
Abstract: This paper describes a particularly didactic and transparent derivation of basic properties of the Lorentz group. The generators for rotations and boosts along an arbitrary direction, as well as their commutation relations, are written as functions of the unit vectors that define the axis of rotation or the direction of the boost (an approach that can be compared with the one that in electrodynamics, works with the electric and magnetic fields instead of the Maxwell stress tensor). For finite values of the angle of rotation or the boost's velocity, collectively denoted by V, the existence of an exponential expansion for the coordinate transformation's matrix, M (in terms of GV where G is the generator) requires that the matrix's derivative with respect to V, be equal to GM. This condition can only be satisfied if the transformation is additive as it is indeed the case for rotations, but not for velocities. If it is assumed, however, that for boosts such an expansion exists, with V = V(v), v being the velocity, and if the above condition is imposed on the boost's matrix then its expression in terms of hyperbolic cosh(V) and sinh(V} is recovered, and the expression for V(= arc tanh(v)) is determined. A general Lorentz transformation can be written as an exponential containing the sum of a rotation and a boost, which to first order is equal to the product of a boost with a rotation. The calculations of the second and third order terms show that the equations for the generators used in this paper, allow to reliably infer the expressions for the higher order generators, without having recourse to the commutation relations. The transformationmatrices for Weyl spinors are derived for finite values of the rotation and velocity, and field representations, leading to the expression for the angular momentum operator, are studied.
Maps for Lorentz transformations of spin  [PDF]
Thomas F. Jordan,Anil Shaji,E. C. G. Sudarshan
Physics , 2005, DOI: 10.1103/PhysRevA.73.032104
Abstract: Lorentz transformations of spin density matrices for a particle with positive mass and spin 1/2 are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are entangled, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit entangled with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverse maps are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.
Derivation of the Lorentz Transformations  [PDF]
Rostislav Polishchuk
Physics , 2001,
Abstract: The Lorentz Transformations are derived without any linearity assumptions and without assuming that y and z coordinates transform in a Galilean manner. Status of the invariance of the speed of light is reduced from a foundation of the Special Theory of Relativity to just a property which allows to determine a value of the physical constant. While high level of rigour is maintained, this paper should be accessible to a second year university physics student.
On the Lorentz Transformations of Momentum and Energy  [PDF]
M. Toller
Physics , 2003, DOI: 10.1142/S0217732303012027
Abstract: Motivated by ultra-high-energy cosmic ray physics, we discuss all the possible alternatives to the familiar Lorentz transformations of the momentum and the energy of a particle. Starting from natural physical requirements, we exclude all the possibilities, apart from the ones which arise from the usual four-vector transformations by means of a change of coordinates in the mass-shell. This result confirms the remark, given in a preceding paper, that, in a theory without preferred inertial frames, one can always define a linearly transforming energy parameter to which the GZK cutoff argument can be applied. We also discuss the connections between the conservation and the transformation properties of energy-momentum and the relation between energy-momentum and velocity.
Composition of Lorentz Transformations in Terms of Their Generators  [PDF]
B. Coll,F. San Jose Martinez
Physics , 2001, DOI: 10.1023/A:1020018616308
Abstract: Two-forms in Minkowski space-time may be considered as generators of Lorentz transformations. Here, the covariant and general expression for the composition law (Baker-Campbell-Hausdorff formula) of two Lorentz transformations in terms of their generators is obtained. Every subalgebra of the Lorentz algebra of such generators, up to one, may be generated by a sole pair of generators. When the subalgebra is known, the above BCH formula for the two two-forms simplifies. Its simplified expressions for all such subalgebras are also given.
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