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In the current study, we conducted a theoretical study to derive the Lorentz transformation between inertial frames of reference moving in two‐dimensional spacetime continuum. The invariance of the space–time interval equation, with use of the derived two‐dimensional transformations, yields the notion of 2 + 2‐dimensional spacetime continuum which consists of two space and two time coordinates. The velocity addition formulas, Lorentz transformations of energy and momentum are then investigated in 2 + 2‐dimensional spacetime frame. Additionally, we investigated the concept of four‐vector in 2 + 2 dimensions and further discussed their transformation based on the matrix equation, which is fully consistent with the Lorentz invariant energy–momentum relation.

We formalize a notion of discrete Lorentz transforms for quantum walks (QW) and quantum cellular automata (QCA), in -dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the clock QW and the clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.

In this work we extend the four-dimensional space time theory of special relativity to six dimensions by adding two extra time coordinates, thereby we propose here six Lorentz transformation equations between inertial frames of reference moving in the three dimensions of space. The Lorentz matrix of order 6 × 6 using new six transformation equations is presented for the case when the coordinate systems have a relative velocity in three dimensions. Also, an elementary derivation of the new transformations in terms of hyperbolic functions including its matrix formulation has been thoroughly discussed. In addition to these, the invariance of differential space-time interval, invariance of d’Alembert operator, transformation of energy and momentum have been theoretically interpreted on the basis of the extended Lorentz transformation equations.

We present a novel derivation of the spacetime metric generated by matter, without invoking Einstein’s field equations. For static sources, the metric arises from a relativistic formulation of D’Alembert’s principle, where the inertial force is treated as a real dynamical entity that exactly compensates gravity. This leads to a conformastatic metric whose geodesic equation—parametrized by proper time—reproduces the relativistic version of Newton’s second law for free fall. To extend the description to moving matter—uniformly or otherwise—we apply a Lorentz transformation to the static metric. The resulting non-static metric accounts for the motion of the sources and, remarkably, matches the weak-field limit of general relativity as obtained from the linearized Einstein equations in the de Donder (or Lorenz) gauge. This approach—at least at Solar System scales, where gravitational fields are weak—is grounded in a new dynamical interpretation of the Equivalence Principle. It demonstrates how gravity can emerge from the relativistic structure of inertia, without postulating or solving Einstein’s equations.

In this article, we present a modified version of the multispring system paradox, and then we introduce a resolution to it by extending the Lorentz transformation for the current and charge densities to the electrically neutral objects, such as springs, in order to demonstrate the nonuniform distribution of the rotating springs in the original paradox. Although the presented resolution seems to interestingly resolve the paradox, it is shown that a simpler version of the said paradox is still valid, to which the introduced resolution is not applicable. In general, it is deduced that the special theory of relativity’s approach to the problem is still vague, and the origin of the paradox cannot be disclosed easily.

We present a theoretical framework for the dynamics of bosonic Bogoliubov quasiparticles. We call it Lorentz quantum mechanics because the dynamics is a continuous complex Lorentz transformation in complex Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the unitary transformation in Hilbert space. In our Lorentz quantum mechanics, three types of state exist: space-like, light-like and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as a matrix form of a Lorentz transformation, and the construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in these mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where bosonic Bogoliubov quasi-particles and their Lorentz quantum dynamics can arise, are presented. They are a one-dimensional fermion gas, Bose-Einstein condensate (or superfluid), and one-dimensional antiferromagnet.

We study the mixing of the Gluino-Glue operator in N=1 Supersymmetric Yang–Mills theory (SYM), both in dimensional regularization and on the lattice. We calculate its renormalization, which is not merely multiplicative, due to the fact that this operator can mix with non-gauge invariant operators of equal or, on the lattice, lower dimension. These operators carry the same quantum numbers under Lorentz transformations and global gauge transformations, and they have the same ghost number. We compute the one-loop quantum correction for the relevant two-point and three-point Green’s functions of the Gluino-Glue operator. This allows us to determine renormalization factors of the operator in the MS¯ scheme, as well as the mixing coefficients for the other operators. To this end our computations are performed using dimensional and lattice regularizations. We employ a standard discretization where gluinos are defined on lattice sites and gluons reside on the links of the lattice; the discretization is based on Wilson’s formulation of non-supersymmetric gauge theories with clover improvement. The number of colors, Nc, the gauge parameter, β, and the clover coefficient, cSW, are left as free parameters.

We constructed gamma operators which are superalgebraic analogs of the Dirac gamma matrices as well as two additional gamma operators which have no analogs in the Dirac theory. We found a new mechanism of the left-right symmetry breaking which is absent in the usual theory of Dirac spinors. We constructed Lorentz invariant gamma operators from operators of creation and annihilation of spinors. These gamma operators are also analogs of the Dirac gamma matrices, however they are not related to Lorentz transformations but generate vector fields as affine spinor connections. We have shown that the theory is equivalent to an extended version of the Pati-Salam theory.

Abstract Albert Einstein's special theory of relativity is encompassed under his general theory of relativity in everything but length contraction and time dilation. In the special theory of relativity, these phenomena are based on relative motion. The general theory of relativity is based on spacetime. This paper corrects that disconnect and shows that spacetime can explain length contraction and time dilation. In this paper, I mathematically step the spacetime interval to the Lorentz transformation. I do this by introducing the spacetime speed triangle which is a geometric representation of spacetime created from the spacetime interval. Not only does the spacetime speed triangle clean up the relationship between the special and general theories but it also brings a connection between spacetime and quantum mechanics. This connection is the spacetime speed triangle is a similar triangle to the energy-momentum relation triangle. This similarity with the matter waves equations brings in other quantum mechanics variables to spacetime.

We use the method of field decomposition, a widely used technique in relativistic magnetohydrodynamics, to study the small velocity approximation (SVA) of the Lorentz transformation in Maxwell equations for slowly moving media. The “deformed” Maxwell equations derived using SVA in the lab frame can be put into the conventional form of Maxwell equations in the medium’s co-moving frame. Our results show that the Lorentz transformation in the SVA of up to O(v/c) (v is the speed of the medium and c is the speed of light in a vacuum) is essential to derive these equations: the time and charge density must also change when transforming to a different frame, even in the SVA, not just the position and current density, as in the Galilean transformation. This marks the essential difference between the Lorentz transformation and the Galilean one. We show that the integral forms of Faraday and Ampere equations for slowly moving surfaces are consistent with Maxwell equations. We also present Faraday equation in the covariant integral form, in which the electromotive force can be defined as a Lorentz scalar that is independent of the observer’s frame. No evidence exists to support an extension or modification of Maxwell equations.