Space-time Coordinate Transformations Research Articles

Geometrical Interpretation of Space Contraction in Two-dimensional Lorentz Transformation

This paper points out that the transformation equations for the spatial and temporal coordinates between two frames of reference in the existing generally accepted version of the Lorentz transformation are deficient, since transformation equations are based on one dimensional motion between inertial frames. Therefore, all possible space-time coordinate transformation equations between moving and stationary frames by prolonging Lorentz transformation in a two-dimensional plane are thoroughly proposed in this article. If denotes the relative velocity between stationary frame (x, y) and moving frame (x', y'), then the transformation equations along X and Y-axis under two-dimensional Lorentz are given, respectively, by the formulas (see full text) . In this work we conclude that length and breadth of a rectangle appears to be shortened to the observer when there is the relative velocity between the rectangle and observer along both X and Y-axis. In particular, we present a concise and carefully reasoned account of a new aspect of Lorentz transformation which decently allows for the determination of space-time coordinates transformation equations in two dimensions of space.

A new paradox and the reconciliation of Lorentz and Galilean transformations

One of the most debated problems in the foundations of the special relativity theory is the role of conventionality. A common belief is that the Lorentz transformation is correct but the Galilean transformation is wrong (only approximately correct in low speed limit). It is another common belief that the Galilean transformation is incompatible with Maxwell equations. However, the “principle of general covariance” in general relativity makes any spacetime coordinate transformation equally valid. This includes the Galilean transformation as well. This renders a new paradox. This new paradox is resolved with the argument that the Galilean transformation is equivalent to the Lorentz transformation. The resolution of this new paradox also provides the most straightforward resolution of an older paradox which is due to Selleri in (Found Phys Lett 10:73–83, 1997). I also present a consistent electrodynamics formulation including Maxwell equations and electromagnetic wave equations under the Galilean transformation, in the exact form for any high speed, rather than in low speed approximation. Electrodynamics in rotating reference frames is rarely addressed in textbooks. The presented formulation of electrodynamics under the Galilean transformation even works well in rotating frames if we replace the constant velocity mathbf {v} with mathbf {v}=varvec{omega }times mathbf {r}. This provides a practical tool for applications of electrodynamics in rotating frames. When electrodynamics is concerned, between two inertial reference frames, both Galilean and Lorentz transformations are equally valid, but the Lorentz transformation is more convenient. In rotating frames, although the Galilean electrodynamics does not seem convenient, it could be the most convenient formulation compared with other transformations, due to the intrinsic complex nature of the problem.

A test of Alzain’s modified inertia model for MOND using galaxy cluster observations

Recently, Alzain proposed a modified inertial formalism of modified Newtonian dynamics (MOND), wherein Galiliean invariance is violated and new Lorentz transformation properties from an inertial to an accelerated frames are posited. Milgrom’s acceleration constant is then found to be invariant under the aegis of these new space-time coordinate transformations. In this model, a modified equation of hydrostatic equilibrium is proposed, which can be applied to relaxed galaxy clusters. However, when we apply this equation to a Chandra sample of 12 clusters, we find that the total dynamical mass obtained using Alzain’s model is much smaller than the baryonic mass for almost all the clusters, thereby implying that this original model is not viable. A variant of this model with a value of $$a_0$$ about two times smaller than that which fits galaxy data, is necessary in order to reconcile the missing mass problem in clusters, without the need for dark matter.

Design of metacontinua in the aeroacoustic spacetime

The effect of background flows on the response of acoustic metamaterials is a key aspect that prevented the full disclosure of their potential in those applications where an aerodynamic velocity field strongly influences the propagation of acoustic disturbances. Indeed, the classic approaches for metamaterial design do not consider the aeroacoustic interaction, and the resulting metamaterials cannot preserve their response when operating in flows. So far, only few authors have addressed the problem, mostly focusing on understanding the phenomenon or identifying corrective techniques with limited usability in practical applications. The present study proposes a general method for the modification of the mechanical properties of acoustic metacontinua to preserve their response in presence of a background flow. The method is based on the application of spacetime coordinate transformations exploiting the spacetime formal invariance of the generalised d’Alembertian. This methodology applies to the equation governing the propagation of acoustic disturbances in a metamaterial having arbitrary constitutive equations independently on the method used for its original design. The approach is validated through numerical simulations, using as a benchmark the problem of the acoustic cloaking of a cylinder impinged by a perturbation generated by an isotropic point source within a flowing medium. Numerical results are obtained for an asymptotic Mach number M_infty le 0.35.

Electrodynamics in Euclidean Space Time Geometries

AbstractIn this article it is proven that Maxwell’s field equations are invariant for a real orthogonal Cartesian space time coordinate transformation if polarization and magnetization are assumed to be possible in empty space. Furthermore, it is shown that this approach allows wave propagation with finite field energy transport. To consider the presence of polarization and magnetization an alternative Poynting vector has been defined for which the divergence gives the correct change in field energy density.

From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that it is equivalent to introducing spacetime-dependent local unitaries —whilst keeping the lattice fixed. By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in (2 + 1)–dimensional curved spacetime. Interestingly, the duality crucially relies on the non linear-independence of the three preferred directions of the honeycomb and triangular lattices: The same construction would fail for the square lattice. At the practical level, this result opens the possibility to simulate field theories on curved manifolds, via the quantum walk on different kinds of lattices.

Modified Newtonian Dynamics (MOND) as a Modification of Newtonian Inertia

We present a modified inertia formulation of Modified Newtonian dynamics (MOND) without retaining Galilean invariance. Assuming that the existence of a universal upper bound, predicted by MOND, to the acceleration produced by a dark halo is equivalent to a violation of the hypothesis of locality (which states that an accelerated observer is pointwise inertial), we demonstrate that Milgrom's law is invariant under a new space-time coordinate transformation. In light of the new coordinate symmetry, we address the deficiency of MOND in resolving the mass discrepancy problem in clusters of galaxies.

Modified constraint algebra in loop quantum gravity and spacetime interpretation

Classically, the constraint algebra of general relativity, which generates gauge transformations, is equivalent to spacetime covariance. In LQG, inverse triad corrections lead to an effective Hamiltonian constraint, which can lead to a modified constraint algebra. We have shown, using example of spherically symmetric spacetimes, that a modified constraint algebra does not correspond to spacetime coordinate transformation. In such a scenario, the notion of black hole horizon, which is based on spacetime notions, also needs to be reconsidered. A possible modification to the classical trapping horizon condition leading to consistent results is suggested. In the case where the constraint algebra is not modified, a spacetime picture is valid and one finds mass threshold for black holes and small corrections to Hawking temperature.

Topological properties of microwave magnetoelectric fields.

Collective excitations of electron spins in a ferromagnetic sample dominated by the magnetic dipole-dipole interaction strongly influence the field structure of microwave radiation. A small quasi-two-dimensional ferrite disk with magnetic-dipolar-mode (MDM) oscillation spectra can behave as a source of specific fields in vacuum, termed magnetoelectric (ME) fields. A coupling between the time-varying electric and magnetic fields in the ME-field structures is different from such a coupling in regular electromagnetic fields. The ME fields are characterized by strong energy confinement at a subwavelength region of microwave radiation, topologically distinctive power-flow vortices, and helicity parameters [E. O. Kamenetskii, R. Joffe, and R. Shavit, Phys. Rev. E 87, 023201 (2013)]. We study topological properties of microwave ME fields by loading a MDM ferrite particle with different dielectric samples. We establish a close connection between the permittivity parameters of dielectric environment and the topology of ME fields. We show that the topology of ME fields is strongly correlated with the Fano-resonance spectra observed at terminals of a microwave structure. We reveal specific thresholds in the Fano-resonance spectra appearing at certain permittivity parameters of dielectric samples. We show that ME fields originated from MDM ferrite disks can be distinguished by topological portraits of the helicity parameters and can have a torsion degree of freedom. Importantly, the ME-field phenomena can be viewed as implementations of space-time coordinate transformations on waves.

四维时空的单边四元数变换

研究新的保持四维间隔不变性的时空坐标变换，即四元数形式单边变换。导出对匀速平动参考系的左右两种单边变换四元数，给出对应的运动学结果。讨论单边变换与Lorentz变换之间的两点重要区别。 New transformations of space-time coordinates preserving the invariance of 4-dimensional interval-unilateral quaternion transformations are studied. The quaternions of left and right unilateral transformations for a reference system of uniform motion with constant velocity are derived. Some results in kinematics of the unilateral transformations are given. Two important differences between the unilateral transformation and Lorentz transformation are discussed.

General relativistic observables of the GRAIL mission

We present a realization of astronomical relativistic reference frames in the Solar System and its application to the Gravity Recovery and Interior Laboratory (GRAIL) mission. We model the necessary space-time coordinate transformations for light-trip time computations and address some practical aspects of the implementation of the resulting model. We develop all the relevant relativistic coordinate transformations that are needed to describe the motion of the GRAIL spacecraft and to compute all observable quantities. We take into account major relativistic effects contributing to the dual one-way range observable, which is derived from one-way signal travel times between the two GRAIL spacecrafts. We develop a general relativistic model for this fundamental observable of GRAIL, accurate to $1\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$. We develop and present a relativistic model for another key observable of this experiment, the dual one-way range rate, accurate to $1\text{ }\text{ }\ensuremath{\mu}\mathrm{m}/\mathrm{s}$. The presented formulation justifies the basic assumptions behind the design of the GRAIL mission. It may also be used to further improve the already impressive results of this lunar gravity recovery experiment after the mission is complete. Finally, we present transformation rules for frequencies and gravitational potentials and their application to GRAIL.

四元数四维间隔不变性和时空坐标变换

利用四元数研究狭义相对论中的时空坐标变换。1) 导出保持四维间隔不变性的时空坐标一般变换的四元数形式，说明四维间隔不变性不能唯一确定Lorentz变换。2) 根据保持时间不变的条件，从一般变换中得到第一类特殊变换，其中包含空间旋转变换。3) 根据保持一个空间坐标不变的条件，从一般变换中得到第二类特殊变换，其中包含正常的Lorentz变换；同时指出某些文献中的四元数形式Lorentz变换式有待商榷之处。4) 从时空坐标一般变换中引入两种不同于传统的空间旋转变换和正常的Lorentz变换的新型变换，一种是离散变换，包含恒等变换，反射变换，换位变换等；另一种是单边变换。 By the use of quaternion, the transformations of space-time coordinates in special relativity are studied. 1) The general transformations in quaternion form are derived, which preserve the invariance of 4-dimensional interval, and it is found that the invariance of the interval can not determine the Lorentz transformation uniquely. 2) Based on the con-dition that preserves the invariance of time, the general transformations reduce to the first kind of special transforma-tions, in which the space rotations are included. 3) Based on the condition that preserves the invariance of a space coor-dinate, the general transformations reduce to the second kind of special transformations, in which the proper Lorentz transformations are included. It is pointed out that the quaternion form of Lorentz transformations in some literatures should be amended. 4) From the general transformations in quaternion form, two types of new transformations are introduced, which are discrete transformations, including identical, reflection and transposition ones, and unilateral transformations. These new transformations are different from the traditional space rotations and the normal Lorentz transformations.

Covariant description of transformation optics in nonlinear media

The technique of transformation optics (TO) is an elegant method for the design of electromagnetic media with tailored optical properties. In this paper, we focus on the formal structure of TO theory. By using a complete covariant formalism, we present a general transformation law that holds for arbitrary materials including bianisotropic, magneto-optical, nonlinear and moving media. Due to the principle of general covariance, the formalism is applicable to arbitrary space-time coordinate transformations and automatically accounts for magneto-electric coupling terms. The formalism is demonstrated for the calculation of the second harmonic wave generation in a twisted TO concentrator.

Light-time computations for the BepiColombo Radio Science Experiment

The Radio Science Experiment is one of the on board experiments of the Mercury ESA mission BepiColombo that will be launched in 2014. The goals of the experiment are to determine the gravity field of Mercury and its rotation state, to determine the orbit of Mercury, to constrain the possible theories of gravitation (for example by determining the post-Newtonian parameters), to provide the spacecraft position for geodesy experiments and to contribute to planetary ephemerides improvement. This is possible thanks to a new technology which allows to reach great accuracies in the observables range and range rate; it is well known that a similar level of accuracy requires studying a suitable model taking into account numerous relativistic effects. In this paper we deal with the modelling of the space-time coordinate transformations needed for the light-time computations and the numerical methods adopted to avoid rounding-off errors in such computations.

Special Theory of Relativity without special assumptions and tachyonic motion

The most general form of transformations of space-time coordinates in Special Theory of Relativity based solely on physical assumptions is described. Only the linearity of space-time transformations and the constancy of the speed of light are used as assumptions. The application to tachyonic motion is indicated.

Gauge gravity in noncommutative de Sitter space and pair creation

From the invariance of the generalized space-time non-commutative commutation relations, local Poincaré and general coordinate transformations are derived. Moreover, a generalized Dirac equation is obtained. Applied to the de Sitter universe, it is shown that the space-time non-commutativity contributes to the particle creation process and induces a Casimir-like effect.

On cloaking for elasticity and physical equations with a transformation invariant form

In this paper, we investigate how the form of the conventional elastodynamic equations changes under curvilinear transformations. The equations get mapped to a more general form in which the density is anisotropic and additional terms appear which couple the stress not only with the strain but also with the velocity, and the momentum gets coupled not only with the velocity but also with the strain. These are a special case of equations which describe the elastodynamic response of composite materials, and which it has been argued should apply to any material which has microstructure below the scale of continuum modelling. If composites could be designed with the required moduli then it could be possible to design elastic cloaking devices where an object is cloaked from elastic waves of a given frequency. To an outside observer it would appear as though the waves were propagating in a homogeneous medium, with the object and surrounding cloaking shell invisible. Other new elastodynamic equations also retain their form under curvilinear transformations. The question is raised as to whether all equations of microstructured continua have a form which is invariant under curvilinear space or space-time coordinate transformations. We show that the non-local bianisotropic electrodynamic equations have this invariance under space-time transformations and that the standard non-local, time-harmonic, electromagnetic equations are invariant under space transformations.

Rare events and scale-invariant dynamics of perturbations in delayed chaotic systems.

We study the dynamics of perturbations in time-delay dynamical systems. Using a suitable space-time coordinate transformation, we find that the time evolution of the linearized perturbations (Lyapunov vector) can be described by the linear Zhang surface growth model [J. Phys. (France) 51, 2129 (1990)]], which is known to describe surface roughening driven by power-law distributed noise. As a consequence, Lyapunov vector dynamics is dominated by rare random events that lead to non-Gaussian fluctuations and multiscaling properties.

Report of the Working Group ‘Relativity for Celestial Mechanics and Astrometry’

Abstract : This is a brief report on the work done by the International Astronomical Union (IAU) working group Relativity for Celestial Mechanics and Astrometry. The first central task of the working group was to recommend some specific form of the metric tensor that is related to the distance of two neighboring points in space-time with coordinates. The metric tensor allows one to derive translational and rotational equations of motion of bodies, to describe the propagation of light, set the rates of atomic clocks, and to model the processes of observation. Meanwhile, it is widely accepted that to adequately describe modern astronomical observations one has to use several relativistic reference systems. The barycentric celestial reference system (BCRS) can be used to model the light propagation from distant celestial objects as well as the motion of bodies within the solar system. The geocentric celestial reference system (GCRS) is physically adequate to describe processes occurring in the vicinity of the Earth (i.e., Earth's rotation, motion of Earth's satellites). The central task of the working group is to specify the metric tensors both in the BCRS and in the GCRS and the corresponding space-time coordinate transformations between these two systems. The IAU resolution A4(1991) contains nine recommendations, the first five of which are relevant to this discussion: (1) space-time cannot be described by a single coordinate system; (2) definition of the origin and orientation of the spatial coordinate grids for the solar system (BCRS) and for the Earth (GCRS); (3) definitions of the TCB (Barycentric Coordinate Time) and TCG (Geocentric Coordinate Time), the time coordinates of the BCRS and GCRS, respectively; (4) definition of Terrestrial Time (TT); and (5) old barycentric dynamical time (TDB) may still be used where discontinuity with previous work is deemed to be undesirable.

Chronostructures of Special Relativity Theory

The class of all theoretically admissiblegeometric structures describing relative rates at whichmoving clocks work (the so called chronostructures) isconsidered. The transformations of space-timecoordinates of events implied by the choice of thechronostructure are determined. It is shown that thechronostructure is invariant with respect to thetransformations which it determines if, and only if, itis correlated with the Lorentz chronostructure. Thischronostructure is the simplest one in the class ofinvariant chronostructures.