Use of Lorentz Transformations for Noninertial Frames of Reference

By analogy with the calculation of the path of a mass point in terms of the integral of the point velocity with respect to time, such that the point has a constant velocity V(t _i) within a time interval dt _i, then changes this velocity stepwise by V(t _i+1), moves with this velocity within a time interval dt _i+1, etc., an accelerated motion of an observer with a clock is represented by alternating states of rest in a sequence of inertial frames of reference and instantaneous jumps from one frame of reference into another. Lorentz transformations are used to calculate the readings of a resting clock observed from a noninertial frame of reference represented in this manner, during the rest of a noninertial observer in a next-in-turn inertial frame of reference belonging to the mentioned sequence, and upon a jump. For the observation from a noninertial frame of reference, the relation of the time interval counted by the resting clock to the time interval counted by the accelerated clock and to the acceleration has been obtained.     

Reciprocal Relativity of Noninertial Frames and the Quaplectic Group

The frame associated with a classical point particle is generally noninertial. The point particle may have a nonzero velocity and force with respect to an absolute inertial rest frame. In time–position–energy–momentum-space {t, q, p, e}, the group of transformations between these frames leaves invariant the symplectic metric and the classical line element ds² = d t². Special relativity transforms between inertial frames for which the rate of change of momentum is negligible and eliminates the absolute rest frame by making velocities relative but still requires the absolute inertial frame. The Lorentz group leaves invariant the symplectic metric and the line elements and . General relativity for particles under only the influence of gravity avoids the issue of noninertial frames as all particles follow geodesics and hence have locally inertial frames. For other forces, the question of the absolute inertial frame remains.) Born conjectured that the line element should be generalized to the pseudo-orthogonal metric . The group leaving this metrics and the symplectic metric invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations also eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is shape reciprocal in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group.     

Revised Robertson's test theory of special relativity: Space-time structure and dynamics

The experimental testing of the Lorentz transformations is based on a family of sets of coordinate transformations that do not comply in general with the principle of equivalence of the inertial frames. The Lorentz and Galilean sets of transformations are the only member sets of the family that satisfy this principle. In the neighborhood of regular points of space-time, all members in the family are assumed to comply with local homogeneity of space-time and isotropy of space in at least one free-falling elevator, to be denoted as Robertson'sab initio rest frame [H. P. Robertson,Rev. Mod. Phys. 21, 378 (1949)].Without any further assumptions, it is shown that Robertson's rest frame becomes a preferred frame for all member sets of the Robertson family except for, again, Galilean and Einstein's relativities. If one now assumes the validity of Maxwell-Lorentz electrodynamics in the preferred frame, a different electrodynamics spontaneously emerges for each set of transformations. The flat space-time of relativity retains its relevance, which permits an obvious generalization, in a Robertson context, of Dirac's theory of the electron and Einstein's gravitation. The family of theories thus obtained constitutes a covering theory of relativistic physics.A technique is developed to move back and forth between Einstein's relativity and the different members of the family of theories. It permits great simplifications in the analysis of relativistic experiments with relevant Robertson's subfamilies. It is shown how to adapt the Clifford algebra version of standard physics for use with the covering theory and, in particular, with the covering Dirac theory.Part of this work was done at the Department of Physics, Utah State University, Logan, Utah 84322.     

Is Minkowski Space-Time Compatible with Quantum Mechanics?

In quantum relativistic Hamiltonian dynamics, the time evolution of interacting particles is described by the Hamiltonian with an interaction-dependent term (potential energy). Boost operators are responsible for (Lorentz) transformations of observables between different moving inertial frames of reference. Relativistic invariance requires that interaction-dependent terms (potential boosts) are present also in the boost operators and therefore Lorentz transformations depend on the interaction acting in the system. This fact is ignored in special relativity, which postulates the universality of Lorentz transformations and their independence of interactions. Taking into account potential boosts in Lorentz transformations allows us to resolve the no-interaction paradox formulated by Currie, Jordan, and Sudarshan [Rev. Mod. Phys. 35, 350 (1963)] and to predict a number of potentially observable effects contradicting special relativity. In particular, we demonstrate that the longitudinal electric field (Coulomb potential) of a moving charge propagates instantaneously. We show that this effect as well as superluminal spreading of localized particle states is in full agreement with causality in all inertial frames of reference. Formulas relating time and position of events in interacting systems reduce to the usual Lorentz transformations only in the classical limit (0) and for weak interactions. Therefore, the concept of Minkowski space-time is just an approximation which should be avoided in rigorous theoretical constructions.     

The introduction of Superluminal Lorentz transformations: A revisitation

We revisit the introduction of the Superluminal Lorentz transformations which carry from bradyonic inertial frames to tachyonic inertial frames, i.e., which transform time-like objects into space-like objects, andvice versa. It has long been known that special relativity can be extended to Superluminal observers only by increasing the number of dimensions of the space-time or—which is in a sense equivalent—by releasing the reality condition (i.e., introducing also imaginary quantities). In the past we always adopted the latter procedure. Here we show the connection between that procedure and the former one. In other words, in order to clarify the physical meaning of the imaginary units entering the classical theory of tachyons, we have temporarily to call into play anauxiliary six-dimensional space-time M(3, 3); however, we are eventually able to go back to the four-dimensional Minkowski space-time. We revisit the introduction of the Superluminal Lorentz transformations also under another aspect. In fact, the generalized Lorentz transformations had been previously written down in a form suited only for the simple case of collinear boosts (e.g., they formed a group just for collinear boosts). We express now the Superluminal Lorentz transformations in a more general form, so that they constitute a group together with the ordinary—orthochronousand antichronous—Lorentz transformations, and reduce to the previous form in the case of collinear boosts. Our approach introduces either real or imaginary quantities, with exclusion of (generic) complex quantities. In the present context, a procedure—in two steps—for interpreting the imaginary quantities is put forth and discussed. In the case of a chain of generalized Lorentz transformations, such a procedure (when necessary) is to be applied only at the end of the chain. Finally, we justify why we call transformations also the Superluminal ones.     

Electrodynamics Under a Possible Alternative to the Lorentz Transformation

A generalization of the classical electrodynamics for systems in absolute motion in presented using a possible alternative to the Lorentz transformation. The main hypothesis assumed in this work are: a) The inertial transformations relate two inertial frames: the privileged frame S and the moving frame S with velocity v with respect to S. b) The transformation of the fields from S to the moving frame S is given by H = a(H – v × D) and E = a(E + v × B), where a is a matrix whose elements depend of the absolute velocity of the system. c) The constitutive relations in the moving frame S are given by D = E, B = H and J = E. It is found that Maxwell's equations, which are transformed to the moving frame, take a new form depending on the absolute velocity of the system. Moreover, differing from classical electrodynamics, it is proven that the electrodynamics proposed explains satisfactorily the Wilson effect.     

Observer-Dependence of Chaos Under Lorentz and Rindler Transformations

The behavior of Lyapunov exponents and dynamical entropies h, whose positivity characterizes chaotic motion, under Lorentz and Rindler transformations is studied. Under Lorentz transformations, and h are changed, but their positivity is preserved for chaotic systems. Under Rindler transformations, and h are changed in such a way that systems, which are chaotic for an accelerated Rindler observer, can be nonchaotic for an inertial Minkowski observer. Therefore, the concept of chaos is observer-dependent.     

Radiation by relativistic dipoles. II

In a continuation of an earlier study, the electromagnetic fields of a point magnetic moment — a magneton — in uniform rectilinear motion, with a given spin precession, are analyzed. It is shown that the same equations can be found through Lorentz transformations from the corresponding expressions in the rest frame. The relationship between the electric and magnetic fieldsE andH radiated by a point magnetic dipole moment and a point electric dipole moment is derived through the use of dual transformations of the electromagnetic field tensor. It is assumed that each moment is in relativistic and otherwise arbitrary motion. In the relativistic case, as in the nonrelativistic case, the switch is accompanied by the replacementsHE, E-H. A covariant formalism is developed for describing the electromagnetic fields in the wave zone. The electromagnetic field tensor associated with the radiation is analyzed.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 73–78, March, 1993.     

“True Transformations Relativity” and Electrodynamics

Different approaches to special relativity (SR) are discussed. The first approach is an invariant approach, which we call the true transformations (TT) relativity. In this approach a physical quantity in the four-dimensional spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant approach is compared with the usual covariant approach, which mainly deals with the basis components of tensors in a specific, i.e., Einstein's coordinatization of the chosen inertial frame of reference. The third approach is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from 3+1 space and time, e.g., the synchronously determined spatial length. This formulation is called the apparent transformations (AT) relativity. It is shown that the principal difference between these approaches arises from the difference in the concept of sameness of a physical quantity for different observers. This difference is investigated considering the spacetime length in the TT relativity and spatial and temporal distances in the AT relativity. It is also found that the usual transformations of the three-vectors (3-vectors) of the electric and magnetic fields E and B are the AT. Furthermore it is proved that the Maxwell equations with the electromagnetic field tensor F^ab and the usual Maxwell equations with E and B are not equivalent, and that the Maxwell equations with E and B do not remain unchanged in form when the Lorentz transformations of the ordinary derivative operators and the AT of E and B are used. The Maxwell equations with F^ab are written in terms of the 4-vectors of the electric E^a and magnetic B^a fields. The covariant Majorana electromagnetic field 4-vector ^a is constructed by means of 4-vectors E^a and B^a and the covariant Majorana formulation of electrodynamics is presented. A Dirac like relativistic wave equation for the free photon is obtained.     

Derivation of inertial forces from the Einstein-de Broglie-Bohm (E.d.B.B.) causal stochastic interpretation of quantum mechanics

The physical origin of inertial forces is shown to be a consequence of the local interaction of Dirac's real covariant ether model⁽¹⁾ with accelerated microobjects, considered as real extended particlelike solitons, piloted by surrounding subluminal real wave fields packets.⁽²⁾ Their explicit form results from the application of local inertial Lorentz transformations to the particles submitted to noninertial velocitydependent accelerations, i.e., constitute a natural extension of Lorentz's interpretation of restricted relativity.⁽³⁾ Indeed Dirac's real physical covariant ether model implies inertial forces if one considers the real accelerated noninertial motions of general relativity, defined within the absolute local inertial frames associated with the observed local isotropy of the 2.7° K background microwave radiation.⁽⁴⁾ Inertia thus appears as a necessary consequence of the real particle motions described by the E.d.B.B. formalism of quantum mechanics.     

Gravitational redshift and the equivalence principle

Two problems have long been confused with each other: the gravitational redshift as discussed by the equivalence principle; and the Doppler shift observed by a detector which moves with constant proper acceleration away from a stationary source. We here distinguish these two problems and give for the first time a solution of the former which is exact within the context of the equivalence principle in a sense discussed in the paper. The equivalence principle leads to transformations between flat spacetimes. These are analyzed, and a generalized Lorentz transformation is proposed which covers transformations from inertial to uniformly accelerated frames of reference.     

Post-Newtonian approximation for an accelerated,rotating frame of reference

The field equations of general relativity are solved to post-Newtonian order for a frame of reference having an arbitrary time-dependent, translational acceleration and an arbitrary time-dependent angular velocity. The derivation is based on a new 3+1 decomposition of the Einstein field equations and geodesic equation of motion. The resulting space-time metric and equation of motion contain gravitational terms, inertial terms, and coupled gravitational-inertial terms. These effects are expressed explicitly in terms of the Newtonian potential and standard post-Newtonian scalar and vector potentials. The physical meaning of the formulas derived is illustrated by application to a system of point-like gravitating masses. These results should be useful for the investigation of general relativistic effects in the analysis of real experimental measurements made with respect to a noninertial frame of reference, such as the surface of the rotating earth or an accelerated spacecraft.     

Kinematical and gravitational analysis of the rocket-borne clock experiment by Vessot and Levine using the revised Robertson's test theory of special relativity

The kinematic aspects of the rocket-borne clock experiment by Vessot and Levine are analyzed with the revised Robertson's test theory of special relativity (Found. Phys. 14, 625 (1984)). Besides the expected time-dilation, it is found that the intermediate steps of this experiment yield in principle Michelson-Morley type information (a relation between longitudinal and transverse length contractions) in the third order of the velocities involved, but no relativity-of-simultaneity related effects.The flat space-time test theory induces a family of spherically symmetric line elements that become the Schwarzschild line element in the relativistic case and also in theabinito rest frame of the theory. These line elements represent the same space-time manifold, but pertain in a one-to-one correspondence to the different flat space-time coordinate transformations of the test theory. The conserved energy is related to the family of local energies in the tangent plane. No deviations from the orthodoxy appear at the pertinent levels of approximation. Hence the unexplained residuals of the Vessot-Levine experiment are not due in obvious ways to kinematic and gravitational frequency shifts caused by deviations of the real coordinate transformations from the Lorentz transformations.This work was started while the author was at Departamento de Fisica, Facultad Experimental de Ciencias, Universidad del Zulia, Maracaibo, Venezuela. It was completed at the Department of Physics, Utah State University, Logan, Utah 83422.     

Covariant Formulation of Electromagnetic 4-Momentum in Terms of 4-Vectors E α and B α

The fundamental difference between the true transformations (TT) and the apparent transformations (AT) is explained. The TT refer to the same quantity, while the AT refer, e.g., to the same measurement in different inertial frames of reference. It is shown that the usual transformations of the three-vectors E and B are - the AT. The covariant electrodynamics with the four-vectors E and B of the electric and magnetic field is constructed. It is also shown that the conventional synchronous definitions of the electromagnetic energy and momentum contain both, the AT of the volume, i.e., the Lorentz contraction, and the AT of E and B, while Rohrlich's expressions contain only the AT of E and B. A manifestly covariant expression for the energy-momentum density tensor and the electromagnetic 4-momentum is constructed using E and B . The 4/3 problem is discussed and it is shown that all previous treatments either contain the AT of the volume, or the AT of E and B, or both of them. In our approach all quantities are four-dimensional spacetime tensors whose transformations are the TT.     

A Lorentz-Poincaré Type Interpretation of the Weak Equivalence Principle

The validity of the Weak Equivalence Principle relative to a local inertial frame is detailed in a scalar-vector gravitation model with Lorentz-Poincaré type interpretation. Given the previously established first Post-Newtonian concordance of dynamics with General Relativity, the principle is to this order compatible with GRT. The gravitationally modified Lorentz transformations, on which the observations in physical coordinates depend, are shown to provide a physical interpretation of parallel transport. A development of ‘geodesic’ deviation in terms of the present model is given as well. PACS subject classifications. 04.20.-q, 04.50.+h     

Relativistic Linear Spacetime Transformations Based on Symmetry

Usually the Lorentz transformations are derived from the conservation of the spacetime interval. We propose here a way of obtaining spacetime transformations between two inertial frames directly from symmetry, the isotropy of the space and principle of relativity. The transformation is uniquely defined except for a constant e, that depends only on the process of synchronization of clocks inside each system. Relativistic velocity addition is obtained, and it is shown that the set of velocities is a bounded symmetric domain. If e=0, Galilean transformations are obtained. If e>0, the speed 1/e and a spacetime interval are conserved. By assuming constancy of the speed of light, we get e=1/c ² and the transformation between the frames becomes the Lorentz transformation. If e<0, a proper speed and a Hilbertian norm are conserved.     

The Proof that Maxwell Equations with the 3D E and B are not Covariant upon the Lorentz Transformations but upon the Standard Transformations: The New Lorentz Invariant Field Equations

In this paper the Lorentz transformations (LT) and the standard transformations (ST) of the usual Maxwell equations (ME) with the three-dimensional (3D) vectors of the electric and magnetic fields, E and B, respectively, are examined using both the geometric algebra and tensor formalisms. Different 4D algebraic objects are used to represent the usual observer dependent and the new observer independent electric and magnetic fields. It is found that the ST of the ME differ from their LT and consequently that the ME with the 3D E and B are not covariant upon the LT but upon the ST. The obtained results do not depend on the character of the 4D algebraic objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors E_Hv and B_Hv and the abstract tensors, the 4-vectors E^a and B^a. All these quantities are defined without reference frames, i.e., as absolute quantities. When some basis has been introduced, they are represented as coordinate-based geometric quantities comprising both components and a basis. It is explicitly shown that this geometric approach agrees with experiments, e.g., the Faraday disk, in all relatively moving inertial frames of reference, which is not the case with the usual approach with the 3D bf E and B and their ST.     

Centrosymmetric lie algebras and boost-dilations

We define the Lie algebrac(n) of centrosymmetric matrices. It generates a noncompact and nonsemisimple local Lie group with the unusual property that expc(n) c(n). The group contains an invariant subgroup of Lorentz boost/ dilation transformations. Forn even, these form a subgroup of the conformal group of the Lorentzian metric with signature (– + – + – +).     

Consequences of the inertial equivalence of energy

The usual macroscopic theory of relativistic mechanics and electromagnetism is formulated so that all assumptions but one are consistent with both special relativity and Newtonian mechanics, the distinguishing assumption being that to any energyE, whatever its form, there corresponds an inertial massE/c ² . The speed of light enters this formulation only as a consequence of the inertial equivalent of energy1/c ² . While, for1/c ² >0 the resulting theory has symmetry under the Poincaré group, including Lorentz transformations, all its physical consequences can be derived and tested in any one inertial frame. In particular, an account is given in one inertial frame for the dynamic causes of relativistic effects for simple accelerated clocks and roads.     

On the Derivation of a High-Velocity Tail from the Boltzmann–Fokker–Planck Equation for Shear Flow

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile U _x (y)=ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f(r,v)=f(V), with Vv–U(r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K()lim ₀ ^–2 (–), where is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker–Planck operator. We have found analytically that for shear rates larger than a certain threshold value a _th0.3520 (where is an average collision frequency and a _th/ is the real root of the cubic equation 64x ³+16x ²+12x–9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f(V;a)|V|^–4–(a) (;a), where tan V _y /V _x and the angular distribution function (;a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition (;a)=(+;a) allows one to obtain the exponent (a) as a function of the shear rate. It diverges when aa _th and tends to a minimum value _min1.252 in the limit a. As a consequence of this power-law decay for a>a _th, all the velocity moments of a degree equal to or larger than 2+(a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~(a), which rotates from ~=–/4,3/4 when aa _th to ~=0, in the limit a.