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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 692030, 41 pages
http://dx.doi.org/10.1155/2012/692030
Research Article

Spacetime Deformation-Induced Inertia Effects

Division of Theoretical Astrophysics, Byurakan Astrophysical Observatory, 378433 Byurakan, Armenia

Received 4 March 2012; Revised 18 April 2012; Accepted 22 April 2012

Academic Editor: Giorgio Kaniadakis

Copyright © 2012 Gagik Ter-Kazarian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We construct a toy model of spacetime deformation-induced inertia effects, in which we prescribe to each and every particle individually a new fundamental constituent of hypothetical 2D, so-called master space (MS), subject to certain rules. The MS, embedded in the background 4D-spacetime, is an indispensable companion to the particle of interest, without relation to every other particle. The MS is not measurable directly, but we argue that a deformation (distortion of local internal properties) of MS is the origin of inertia effects that can be observed by us. With this perspective in sight, we construct the alternative relativistic theory of inertia. We go beyond the hypothesis of locality with special emphasis on distortion of MS, which allows to improve essentially the standard metric and other relevant geometrical structures referred to a noninertial frame in Minkowski spacetime for an arbitrary velocities and characteristic acceleration lengths. Despite the totally different and independent physical sources of gravitation and inertia, this approach furnishes justification for the introduction of the weak principle of equivalence (WPE), that is, the universality of free fall. Consequently, we relate the inertia effects to the more general post-Riemannian geometry.

1. Introduction

Governing the motions of planets, the fundamental phenomena of gravitation and inertia reside at the very beginning of the physics. More than four centuries passed since the famous far-reaching discovery of Galileo (in 1602–1604) that all bodies fall at the same rate [1], which led to an early empirical version of the suggestion that gravitation and inertia may somehow result from a single mechanism. Besides describing these early gravitational experiments, Newton in Principia Mathematica [2] has proposed a comprehensive approach to studying the relation between the gravitational and inertial masses of a body. In Newtonian mechanics, masses are simply placed in absolute space and time, which remain external to them. That is, the internal state of a Newtonian point particle, characterized by its inertial mass, has no immediate connection with the particles’ external state in absolute space and time, characterized by its position and velocity. Ever since, there is an ongoing quest to understand the reason for the universality of the gravitation and inertia, attributing to the WPE, which establishes the independence of free-fall trajectories of the internal composition and structure of bodies. In other words, WPE states that all bodies at the same spacetime point in a given gravitational field will undergo the same acceleration. However, the nature of the relationship of gravity and inertia continues to elude us and, beyond the WPE, there has been little progress in discovering their true relation. Such interesting aspects, which deserve further investigations, unfortunately, have attracted little attention in subsequent developments. Only hypothesis, which in some extent relates inertia and matter, is the Mach principle, see for example, [315], but in the same time it is a subject to many uncertainties. The Mach’s ideas on inertial induction were proposed as the theoretical mechanism for generating the inertial forces felt during acceleration of a reference frame. The ensuing problem of the physical origin of inertial forces led Mach to hypothesize that inertial forces were to be of gravitational origin, occurring only during acceleration relative to the fixed stars. In this model the ratio of inertial to gravitational mass will depend on the average distribution of mass in the universe, in effect making gravitational constant a function of the mass distribution in the universe. The general relativity (GR), which preserves the idea of relativity of all kinds of motion, is built on the so-called strong principle (SPE) that the only influence of gravity is through the metric and can thus (apart from tidal effects) be locally, approximately transformed away by going to an appropriately accelerated reference frame. Despite the advocated success of GR, it is now generally acknowledged, however, that what may loosely be termed Mach principle is not properly incorporated into GR. In particular, the origin of inertia remains essentially the same as in Newtonian physics. Brans thorough analysis [46] has shown that no extra inertia is induced in a body as a result of the presence of other bodies. Various attempts at the resolution of difficulties that are encountered in linking Machs principle with Einsteins theory of gravitation have led to many interesting investigations. For example, by [14] is shown that the GR can be locally embedded in a Ricci-flat 5D manifold such that every solution of GR in 4D can be locally embedded in a Ricci-flat 5D manifold and that the resulting inertial mass of a test particle varies in space time. Anyhow, the difficulty is brought into sharper focus by considering the laws of inertia, including their quantitative aspects. That is, Mach principle and its modifications do not provide a quantitative means for computing the inertial forces. At present, the variety of consequences of the precision experiments from astrophysical observations makes it possible to probe this fundamental issue more deeply by imposing the constraints of various analyses. Currently, the observations performed in the Earth-Moon-Sun system [1635], or at galactic and cosmological scales [3641], probe more deeply both WPE and SPE. The intensive efforts have been made, for example, to clear up whether the rotation state would affect the trajectory of test particle. Shortly after the development of the work by [22], in which is reported that, in weighing gyros, it would be a violation of WPE, the authors of [2326] performed careful weighing experiments on gyros with improved precision but found only null results which are in disagreement with the report of [22]. The interferometric free-fall experiments by [27, 28] again found null results in disagreement with [22]. For rotating bodies, the ultraprecise Gravity Probe B experiment [2934], which measured the frame-dragging effect and geodetic precession on four quartz gyros, has the best accuracy. GP-B serves as a starting point for the measurement of the gyrogravitational factor of particles, whereas the gravitomagnetic field, which is locally equivalent to a Coriolis field and generated by the absolute rotation of a body, has been measured too. This, with its superb accuracy, verifies WPE for unpolarized bodies to an ultimate precision—a four-order improvement on the noninfluence of rotation on the trajectory, and ultraprecision on the rotational equivalence [35]. Moreover, the theoretical models may indicate cosmic polarization rotations which are being looked for and tested in the CMB experiments [40]. To look into the future, measurement of the gyrogravitational ratio of particle would be a further step, see [41] and references therein, towards probing the microscopic origin of gravity. Also, the inertia effects in fact are of vital interest for the phenomenological aspects of the problem of neutrino oscillations; see, for example, [4256]. All these have evoked the study of the inertial effects in an accelerated and rotated frame. In doing this, it is a long-established practice in physics to use the hypothesis of locality for extension of the Lorentz invariance to accelerated observers in Minkowski spacetime [57, 58]. This in effect replaces the accelerated observer by a continuous infinity of hypothetical momentarily comoving inertial observers along its wordline. This assumption, as well as its restricted version, so-called clock hypothesis, which is a hypothesis of locality only concerned about the measurement of time, is reasonable only if the curvature of the wordline could be ignored. As long as all relevant length scales in feasible experiments are very small in relation to the huge acceleration lengths of the tiny accelerations we usually experience, the curvature of the wordline could be ignored and that the differences between observations by accelerated and comoving inertial observers will also be very small. In this line, in 1990, Hehl and Ni proposed a framework to study the relativistic inertial effects of a Dirac particle [59], in agreement with [6062]. Ever since this question has become a major preoccupation of physicists; see, for example, [6384]. Even this works out, still, it seems quite clear that such an approach is a work in progress, which reminds us of a puzzling underlying reality of inertia and that it will have to be extended to describe physics for arbitrary accelerated observers. Beyond the WPE, there is nothing convincing in the basic postulates of physics for the origin and nature of inertia to decide on the issue. Despite our best efforts, all attempts to obtain a true knowledge of the geometry related to the noninertial reference frames of an arbitrary observer seem doomed, unless we find a physical principle the inertia might refer to, and that a working alternative relativistic theory of inertia is formulated. Otherwise one wanders in a darkness. The problem of inertia stood open for nearly four centuries, and the physics of inertia is still an unknown exciting problem to be challenged and allows various attempts. In particular, the inertial forces are not of gravitational origin within GR as it was proposed by Einstein in 1918 [85], because there are many controversies to question the validity of such a description [57, 58, 6091]. The experiments by [8790], for example, tested the key question of anisotropy of inertia stemming from the idea that the matter in our galaxy is not distributed isotropically with respect to the earth, and hence if the inertia is due to gravitational interactions, then the inertial mass of a body will depend on the direction of its acceleration with respect to the direction towards the center of our galaxy. However, these experiments do not found such anisotropy of mass. The most sensitive test is obtained in [88, 89] from a nuclear magnetic resonance experiment with an 𝐿𝑖7 nucleus of spin 𝐼=3/2. The magnetic field was of about 4700 gauss. The south direction in the horizontal plane points within 22 degrees towards the center of our galaxy, and 12 hours later this same direction along the earth’s horizontal plane points 104 degrees away from the galactic center. If the nuclear structure of 𝐿𝑖7 is treated as a single 𝑃3/2 proton in a central nuclear potential, the variation Δ𝑚 of mass with direction, if it exists, was found to satisfy Δ𝑚/𝑚1020. This is by now very strong evidence that there is no anisotropy of mass which is due to the effects of mass in our galaxy. Another experimental test [91] using nuclear-spin-polarized9𝐵𝑒+ ions also gives null result on spatial anisotropy and thus supporting local Lorentz invariance. This null result represents a decrease in the limits set by [8890] on a spatial anisotropy by a factor of about 300. Finally, another theoretical objection is that if the curvature of Riemannian space is associated with gravitational interaction, then it would indicate a universal feature equally suitable for action on all the matter fields at once. The source of the curvature as conjectured in GR is the energy-momentum tensor of matter, which is rather applicable for gravitational fields but not for inertia, since the inertia is dependent solely on the state of motion of individual test particle or coordinate frame of interest. In case of accelerated motion, unlike gravitation, the curvature of spacetime might arise entirely due to the inertial properties of the Lorentz-rotated frame of interest, that is, a “fictitious gravitation” which can be globally removed by appropriate coordinate transformations [57]. This refers to the particle of interest itself, without relation to other systems or matter fields.

On the other hand, a general way to deform the spacetime metric with constant curvature has been explicitly posed by [9294]. The problem was initially solved only for 3D spaces, but consequently it was solved also for spacetimes of any dimension. It was proved that any semi-Riemannian metric can be obtained as a deformation of constant curvature matric, this deformation being parameterized by a 2-form. A novel definition of spacetime metric deformations, parameterized in terms of scalar field matrices, is proposed by [95]. In a recent paper [96], we construct the two-step spacetime deformation (TSSD) theory which generalizes and, in particular cases, fully recovers the results of the conventional theory of spacetime deformation [9295]. All the fundamental gravitational structures in fact—the metric as much as the coframes and connections—acquire the TSSD-induced theoretical interpretation. The TSSD theory manifests its virtue in illustrating how the curvature and torsion, which are properties of a connection of geometry under consideration, come into being. Conceptually and techniquewise this method is versatile and powerful. For example, through a nontrivial choice of explicit form of a world-deformation tensor, which we have at our disposal, in general, we have a way to deform that the spacetime displayed different connections, which may reveal different post-Riemannian spacetime structures as a corollary, whereas motivated by physical considerations, we address the essential features of the theory of teleparallel gravity-TSSD-GR and construct a consistent TSSD-𝑈4 Einstein-Cartan (EC) theory, with a dynamical torsion. Moreover, as a preliminary step, in the present paper we show that by imposing different appropriate physical constraints upon the spacetime deformations, in this framework we may reproduce the term in the well-known Lagrangian of pseudoscalar-photon interaction theory, or terms in the Lagrangians of pseudoscalar theories [41, 97101], or in modification of electrodynamics with an additional external constant vector coupling [102, 103], as well as in case of intergrand for topological invariant [104] or in case of pseudoscalar-gluon coupling occurred in QCD in an effort to solve the strong CP problem [105107]. Next, our purpose is to carry out some details of this program to probe the origin and nature of the phenomenon of inertia. We ascribe the inertia effects to the geometry itself but as having a nature other than gravitation. In doing this, we note that aforementioned examples pose a problem for us that physical space has intrinsic geometrical and inertial properties beyond 4D spacetime derived from the matter contained therein. Therefore, we should conceive of two different spaces: one would be 4D background space-time, and another one should be 2D so-called master space (MS), which, embedded in the 4D background space, is an indispensable individual companion to the particle, without relation to the other matter. That is, the key to our construction procedure is an assignment in which we prescribe to each and every particle individually a new fundamental constituent of hypothetical MS, subject to certain rules. In the contrary to Mach principle, the particle has to live with MS companion as an intrinsic property devoid of any external influence. The geometry of MS is a new physical entity, with degrees of freedom and a dynamics of its own. This together with the idea that the inertia effects arise as a deformation (distortion of local internal properties) of MS is the highlights of the alternative relativistic theory of inertia (RTI), whereas we build up the distortion complex (DC), yielding a distortion of MS, and show how DC restores the world-deformation tensor, which still has to be put in [96] by hand. Within this scheme, the MS was presumably allowed to govern the motion of a particle of interest in the background space. In simple case, for example, of motion of test particle in the free 4D Minkowski space, the suggested heuristic inertia scenario is reduced to the following: unless a particle was acted upon by an unbalanced force, the MS is being flat. This causes a free particle in 4D Minkowski space to tend to stay in motion of uniform speed in a straight line or a particle at rest to tend to stay at rest. As we will see, an alteration of uniform motion of a test particle under the unbalanced net force has as an inevitable consequence of a distortion of MS. This becomes an immediate cause of arising both the universal absolute acceleration of test particle and associated inertial force in 𝑀4 space. This, we might expect, holds on the basis of an intuition founded on past experience limited to low velocities, and these two features were implicit in the ideas of Galileo and Newton as to the nature of inertia. Thereby the major premise is that the centrifugal endeavor of particles to recede from the axis of rotation is directly proportional to the quantity of the absolute circular acceleration, which, for example, is exemplified by the concave water surface in Newton’s famous rotating bucket experiments. In other words, it takes force to disturb an inertia state, that is, to make an absolute acceleration. In this framework, the relative acceleration (in Newton’s terminology) (both magnitude and direction), to the contrary, cannot be the cause of a distortion of MS and, thus, it does not produce the inertia effects. The real inertia effects, therefore, can be an empirical indicator of absolute acceleration. The treatment of deformation/distortion of MS is instructive because it contains the essential quantitative elements for computing the relativistic inertial force acting on an arbitrary observer. On the face of it, the hypothesis of locality might be somewhat worrisome, since it presents strict restrictions, replacing the distorted MS by the flat MS. Therefore, it appears natural to go beyond the hypothesis of locality with spacial emphasis on distortion of MS. This, we might expect, will essentially improve the standard metric, and so forth, referred to a noninertial system of an arbitrary observer in Minkowski spacetime. Consequently, we relate the inertia effects to the more general post-Riemannian geometry. The crucial point is to observe that, in spite of totally different and independent physical sources of gravitation and inertia, the RTI furnishes justification for the introduction of the WPE [108, 109]. However, this investigation is incomplete unless it has conceptual problems for further motivation and justification of introducing the fundamental concept of MS. The way we assigned such a property to the MS is completely ad hoc and there are some obscure aspects of this hypothesis. All these details will be further motivated and justified in subsequent paper. The outline of the rest of the present paper is as follows. In Section 2 we briefly revisit the theory of TSSD and show how it can be useful for the theory of electromagnetism and charged particles. In Section 3, we explain our view of what is the MS and lay a foundation of the RLI. A general deformation/distortion of MS is described in Section 4. In Section 5, we construct the RTI in the background 4D Minkowski space. In Section 6, we go beyond the hypothesis of locality, whereas we compute the improved metric and other relevant geometrical structures in noninertial system of arbitrary accelerating and rotating observer in Minkowski spacetime. The case of semi-Riemann background space 𝑉4 is dealt with in Section 7, whereby we give justification for the introduction of the WPE on the theoretical basis. In Section 8, we relate the RTI to more general post-Riemannian geometry. The concluding remarks are presented in Section 9. We will be brief and often ruthlessly suppress the indices without notice. Unless otherwise stated we take natural units, =𝑐=1.

2. TSSD Revisited: Preliminaries

For the benefit of the reader, this section contains some of the necessary preliminaries on generic of the key ideas behind the TSSD [96], which needs one to know in order to understand the rest of the paper. We adopt then its all ideas and conventions. The interested reader is invited to consult the original paper for further details. It is well known that the notions of space and connections should be separated; see, for example, [110113]. The curvature and torsion are in fact properties of a connection, and many different connections are allowed to exist in the same spacetime. Therefore, when considering several connections with different curvatures and torsions, one takes spacetime simply as a manifold and connections as additional structures. From this view point in a recent paper [96] we have tackled the problem of spacetime deformation. In order to relate local Lorentz symmetry to curved spacetime, there is, however, a need to introduce the soldering tools, which are the linear frames and forms in tangent fiber-bundles to the external curved space, whose components are so-called tetrad (vierbein) fields. To start with, let us consider the semi-Riemann space, 𝑉4, which has at each point a tangent space, ̆𝑇̆𝑥𝑉4, spanned by the anholonomic orthonormal frame field, ̆𝑒, as a shorthand for the collection of the 4-tuplet (̆𝑒0,,̆𝑒3), where ̆𝑒𝑎=̆𝑒𝜇𝑎̆𝜕𝜇. All magnitudes related to the space, 𝑉4, will be denoted by an over “̆”. These then define a dual vector, ̆𝜗, of differential forms, ̆̆𝜗𝜗=0̆𝜗3, as a shorthand for the collection of the ̆𝜗𝑏=̆𝑒𝑏𝜇𝑑̆𝑥𝜇, whose values at every point form the dual basis, such that ̆𝑒𝑎̆𝜗𝑏=𝛿𝑏𝑎, where denotes the interior product, namely, this is a 𝐶-bilinear map Ω1Ω0 where Ω𝑝 denotes the 𝐶-modulo of differential 𝑝-forms on 𝑉4. In components ̆𝑒𝜇𝑎̆𝑒𝑏𝜇=𝛿𝑏𝑎. On the manifold, 𝑉4, the tautological tensor field, 𝑖̆𝑑, of type (1,1) can be defined which assigns to each tangent space the identity linear transformation. Thus for any point ̆𝑥𝑉4, and any vector ̆̆𝑇𝜉̆𝑥𝑉4, one has 𝑖̆̆̆𝜉𝑑(𝜉)=. In terms of the frame field, the ̆𝜗𝑎 give the expression for 𝑖̆𝑑 as 𝑖̆̆𝑑=̆𝑒𝜗=̆𝑒0̆𝜗0+̆𝑒3̆𝜗3, in the sense that both sides yield ̆𝜉 when applied to any tangent vector ̆𝜉 in the domain of definition of the frame field. One can also consider general transformations of the linear group, GL(4,𝑅), taking any base into any other set of four linearly independent fields. The notation {̆𝑒𝑎,̆𝜗𝑏} will be used hereinafter for general linear frames. The holonomic metric can be defined in the semi-Riemann space, 𝑉4, as̆𝑔=̆𝑔𝜇𝜈̆𝜗𝜇̆𝜗𝜈=̆𝑔̆𝑒𝜇,̆𝑒𝜈̆𝜗𝜇̆𝜗𝜈,(2.1) with components ̆𝑔𝜇𝜈=̆𝑔(̆𝑒𝜇,̆𝑒𝜈) in the dual holonomic base {̆𝜗𝜇𝑑̆𝑥𝜇}. The anholonomic orthonormal frame field, ̆𝑒, relates ̆𝑔 to the tangent space metric, 𝑜𝑎𝑏=diag(+), by 𝑜𝑎𝑏=̆𝑔(̆𝑒𝑎,̆𝑒𝑏)=̆𝑔𝜇𝜈̆𝑒𝜇𝑎̆𝑒𝜈𝑏, which has the converse ̆𝑔𝜇𝜈=𝑜𝑎𝑏̆𝑒𝑎𝜇̆𝑒𝑏𝜈 because ̆𝑒𝜇𝑎̆𝑒𝑎𝜈=𝛿𝜇𝜈.

For reasons that will become clear in the sequel, next we write the norm, 𝑑𝑠, of the infinitesimal displacement, 𝑑𝑥𝜇, on the general smooth differential 4D-manifold, 4, in terms of the spacetime structures of 𝑉4, as𝑑𝑠=𝑒𝜗=Ω𝜈𝜇̆𝑒𝜈̆𝜗𝜇=Ω𝑎𝑏̆𝑒𝑎̆𝜗𝑏=𝑒𝜌𝜗𝜌=𝑒𝑎𝜗𝑎4,(2.2) where Ω𝜈𝜇=𝜋𝜌𝜇𝜋𝜈𝜌 is the world deformation tensor, 𝑒={𝑒𝑎=𝑒𝜌𝑎𝑒𝜌} is the frame field, and 𝜗={𝜗𝑎=𝑒𝑎𝜌𝜗𝜌} is the coframe field defined on 4, such that 𝑒𝑎𝜗𝑏=𝛿𝑏𝑎, or in components, 𝑒𝜇𝑎𝑒𝑏𝜇=𝛿𝑏𝑎; also the procedure can be inverted, 𝑒𝑎𝜌𝑒𝜎𝑎=𝛿𝜎𝜌. Hence the deformation tensor, Ω𝑎𝑏=𝜋𝑎𝑐𝜋𝑐𝑏=Ω𝜈𝜇̆𝑒𝑎𝜈̆𝑒𝜇𝑏, yields local tetrad deformations:𝑒𝑐=𝜋𝑎𝑐̆𝑒𝑎,𝜗𝑐=𝜋𝑐𝑏̆𝜗𝑏,𝑒𝜗=𝑒𝑎𝜗𝑎=Ω𝑎𝑏̆𝑒𝑎̆𝜗𝑏.(2.3) The components of the general spin connection then transform inhomogeneously under a local tetrad deformations (2.3):𝜔𝑎𝑏𝜇=𝜋𝑎𝑐̆𝜔𝑐𝑑𝜇𝜋𝑑𝑏+𝜋𝑎𝑐𝜕𝜇𝜋𝑐𝑏.(2.4) This is still a passive transformation, but with inverted factor ordering. The matrices 𝜋(𝑥)=(𝜋𝑎𝑏)(𝑥) are called first deformation matrices, and the matrices 𝛾𝑐𝑑(𝑥)=𝑜𝑎𝑏𝜋𝑎𝑐(𝑥)𝜋𝑏𝑑(𝑥),  second deformation matrices. The matrices 𝜋𝑎𝑐(𝑥)GL(4,𝑅)forall𝑥, in general, give rise to right cosets of the Lorentz group; that is, they are the elements of the quotient group GL(4,𝑅)/SO(3,1), because the Lorentz matrices, Λ𝑎𝑐, leave the Minkowski metric invariant. A right-multiplication of 𝜋(𝑥) by a Lorentz matrix gives an other deformation matrix. If we deform the tetrad according to (2.3), in general, we have two choices to recast metric as follows: either writing the deformation of the metric in the space of tetrads or deforming the tetrad field:𝑔=𝑜𝑎𝑏𝜋𝑎𝑐𝜋𝑏𝑑̆𝜗𝑐̆𝜗𝑑=𝛾𝑐𝑑̆𝜗𝑐̆𝜗𝑑=𝑜𝑎𝑏𝜗𝑎𝜗𝑏.(2.5) In the first case, the contribution of the Christoffel symbols, constructed by the metric 𝛾𝑎𝑏, readsΓ𝑎𝑏𝑐=12̆𝐶𝑎𝑏𝑐𝛾𝑎𝑎𝛾𝑏𝑏̆𝐶𝑏𝑎𝑐𝛾𝑎𝑎𝛾𝑐𝑐̆𝐶𝑐𝑎𝑏+12𝛾𝑎𝑎̆𝑒𝑐𝑑𝛾𝑏𝑎̆𝑒𝑏𝑑𝛾𝑐𝑎̆𝑒𝑎𝑑𝛾𝑏𝑐,(2.6) with ̆𝐶𝑒𝑑𝑓 representing the curls of the base members in the semi-Riemann space:̆𝐶𝑒𝑑𝑓=̆𝑒𝑓̆𝑒𝑑̆𝐶𝑒̆𝜗=𝑒̆𝑒𝑑,̆𝑒𝑓=̆𝑒𝜇𝑑̆𝑒𝜈𝑓̆𝜕𝜇̆𝑒𝑒𝜈̆𝜕𝜈̆𝑒𝑒𝜇=̆𝑒𝑒𝜇̆𝑒𝑑̆𝑒𝜇𝑓̆𝑒𝑓̆𝑒𝜇𝑑,(2.7) where ̆𝐶𝑎̆𝜗=𝑑𝑎̆𝐶=(1/2)𝑎𝑏𝑐̆𝜗𝑏̆𝜗𝑐 is the anfolonomity 2-form. The deformed metric can be split as follows [96]:𝑔𝜇𝜈(𝜋)=Υ2(𝜋)̆𝑔𝜇𝜈+𝛾𝜇𝜈(𝜋),(2.8) where Υ(𝜋)=𝜋𝑎𝑎, and 𝛾𝜇𝜈(𝜋)=[𝛾𝑎𝑏Υ2(𝜋)𝑜𝑎𝑏]̆𝑒𝑎𝜇̆𝑒𝑏𝜈. In the second case, we may write the commutation table for the anholonomic frame, {𝑒𝑎},𝑒𝑎,𝑒𝑏1=2𝐶𝑐𝑎𝑏𝑒𝑐,(2.9) and define the anholonomy objects:𝐶𝑎𝑏𝑐=𝜋𝑎𝑒𝜋𝑏𝑑1𝜋𝑐𝑓1̆𝐶𝑒𝑑𝑓+2𝜋𝑎𝑓̆𝑒𝜇𝑔𝜋1[𝑏𝑔𝜕𝜇𝜋1[𝑐𝑓.(2.10) The usual Levi-Civita connection corresponding to the metric (2.8) is related to the original connection by the following relation:Γ𝜇𝜌𝜎=̆Γ𝜇𝜌𝜎+Π𝜇𝜌𝜎,(2.11) provided thatΠ𝜇𝜌𝜎=2𝑔𝜇𝜈̆𝑔𝜈(𝜌𝜎)Ῠ𝑔𝜌𝜎𝑔𝜇𝜈𝜈1Υ+2𝑔𝜇𝜈𝜌𝛾𝜈𝜎+𝜎𝛾𝜌𝜈𝜈𝛾𝜌𝜎,(2.12) where the controvariant deformed metric, 𝑔𝜈𝜌, is defined as the inverse of 𝑔𝜇𝜈, such that 𝑔𝜇𝜈𝑔𝜈𝜌=𝛿𝜌𝜇. Hence, the connection deformation Π𝜇𝜌𝜎 acts like a force that deviates the test particles from the geodesic motion in the space 𝑉4 (for more details see [96]). Next, we deal with the spacetime deformation 𝜋(𝑥), to be consisted of two ingredient deformations (𝜋(𝑥), 𝜎(𝑥)). Provided, we require that the first deformation matrix, 𝜋(𝑥)=(𝜋𝑎𝑏)(𝑥), satisfies the following peculiar condition: 𝜋𝑎𝑐𝑥𝜕𝜇𝜋𝑏𝑐1𝑥=̆𝜔𝑎𝑏𝜇(̆𝑥),(2.13) where ̆𝜔𝑎𝑏𝜇(̆𝑥) is the spin connection defined in the semi-Riemann space. By virtue of (2.13), the general deformed spin connection vanishes, and a general linear connection, Γ𝜇𝜌𝜎, is related to the corresponding spin connection𝜔𝑎𝑏𝜇, through the inverseΓ𝜇𝜌𝜎=𝑒𝜇𝑎𝜕𝜎𝑒𝑎𝜌+𝑒𝜇𝑎𝜔𝑎𝑏𝜎𝑒𝑏𝜌=𝑒𝜇𝑎𝜕𝜎𝑒𝑎𝜌,(2.14) which is the Weitzenböck connection revealing the Weitzenböck spacetime 𝑊4 of the teleparallel gravity. Thus, 𝜋(𝑥) can be referred to as the Weitzenböck deformation matrix. All magnitudes related to the teleparallel gravity will be denoted by an over “”. The components of the general spin connection then transform inhomogeneously under a local tetrad deformations:𝜔𝑎𝑏𝜇=𝜎𝑎𝑐𝜔𝑐𝑑𝜇𝜎𝑑𝑏+𝜎𝑎𝑐𝜕𝜇𝜎𝑐𝑏,(2.15) such that(𝜎)𝜔𝑎𝑏𝜇=𝜔𝑎𝑏𝜇=𝜎𝑎𝑐𝜕𝜇𝜎𝑐𝑏(2.16) is referred to as the deformation-related frame connection, which represents the deformed properties of the frame only. Then, it follows that the affine connection Γ transforms inhomogeneously throughΓ𝜇𝜌𝜎=𝑒𝜇𝑎𝜕𝜎𝑒𝑎𝜌+𝑒𝜇𝑎(𝜋)𝜔𝑎𝑏𝜎𝑒𝑏𝜌=𝜎𝜇𝑎𝜕𝜎𝜎𝑎𝜌+𝜎𝜇𝑎(𝜎)𝜔𝑎𝑏𝜎𝜎𝑏𝜌,(2.17) where we have 𝜎𝜇𝑎𝜎𝑏𝜇=𝛿𝑏𝑎, also the procedure can be inverted, 𝜎𝜇𝑎𝜎𝑎𝜈=𝛿𝜇𝜈, and (𝜇)𝜔𝑎𝑏𝜇=𝜔𝑎𝑏𝜇=𝜋𝑎𝑐̆𝜔𝑐𝑑𝜇𝜋𝑑𝑏+𝜋𝑎𝑐𝜕𝜇𝜋𝑐𝑏(2.18) is the spin connection. For our convenience, hereinafter the notation {(𝐴)𝑒𝑎,(𝐴)𝜗𝑏}(𝐴=𝜋,𝜎) will be used for general linear frames:(𝐴)𝑒𝑎,(𝐴)𝜗𝑏=(𝜋)𝑒𝑎,(𝜎)𝑒𝑎,(𝜋)𝜗𝑏,(𝜎)𝜗𝑏𝑒𝑎,𝑒𝑎,𝜗𝑏,𝜗𝑏,(2.19) where (𝐴)𝑒𝑎(𝐴)𝜗𝑏=𝛿𝑏𝑎, or in components, (𝐴)𝑒𝑎𝜇(𝐴)𝑒𝜇𝑏=𝛿𝑏𝑎; also the procedure can be inverted, (𝐴)𝑒𝜌𝑎(𝐴)𝑒𝑎𝜎=𝛿𝜎𝜌, provided that(𝐴)𝑒𝑎𝜇=(𝐴)𝑒𝑎𝜇,(𝜎)𝑒𝑎𝜇𝑒𝜇𝑎,𝜎𝜇𝑎.(2.20) Hence, the affine connection (2.17) can be rewritten in the abbreviated form:(𝐴)Γ𝜇𝜌𝜎=(𝐴)𝑒𝑎𝜇𝜕𝜎(𝐴)𝑒𝜌𝑎+(𝐴)𝑒𝑎𝜇(𝐴)𝜔𝑎𝑏𝜎(𝐴)𝑒𝜌𝑏.(2.21) Since the first deformation matrices 𝜋(𝑥) and 𝜎(𝑥) are arbitrary functions, the inhomogeneously transformed general spin connections (𝜋)𝜔(𝑥) and (𝜎)𝜔(𝑥), as well as the affine connection (2.21), are independent of tetrad fields and their derivatives. In what follows, therefore, we will separate the notions of space and connections—the metric-affine formulation of gravity. A metric-affine space (𝑀4,𝑔,Γ) is defined to have a metric and a linear connection that need not be dependent on each other. The lifting of the constraints of metric-compatibility and symmetry yields the new geometrical property of the spacetime, which are the nonmetricity 1-form (𝐴)𝑁𝑎𝑏 and the affine torsion 2-form (𝐴)𝑇𝑎 representing a translational misfit (for a comprehensive discussion see [114118]. These, together with the curvature 2-form (𝐴)𝑆𝑎𝑏, symbolically can be presented as [119, 120](𝐴)𝑁𝑎𝑏,(𝐴)𝑇𝑎,(𝐴)𝑆𝑎𝑏(𝐴)𝒟(𝐴)𝑔𝑎𝑏,(𝐴)𝜗𝑎,(𝐴)Γa𝑏,(2.22) where for a tensor-valued 𝑝-form density of representation type 𝜌(𝐿𝑏𝑎), the 𝐺𝐿(4,𝑅)-covariant exterior derivative reads (𝐴)𝒟=𝑑+(𝐴)Γ𝑎𝑏𝜌(𝐿𝑏𝑎) and (𝐴)Γ𝑎𝑏=(𝐴)Γ𝑏𝜇𝑎𝑑𝑥𝜇 is the general nonmetricity connection. This notation will be used instead of (𝐴)𝜔𝑎𝑏=(𝐴)𝜔𝑏𝜇𝑎𝑑𝑥𝜇, such that (𝐴)Γ𝑏𝜇𝑎=(𝐴)𝑒𝑎𝜈(𝐴)𝑒𝜎𝑏Γ𝜎𝜇𝜈+(𝐴)𝑒𝑎𝜈𝜕𝜇(𝐴)𝑒𝜈𝑏. In what follows, however, we may still maintain the former notation (𝐴)𝜔𝑎𝑏to be referred to as corresponding connection constrained by the metricity condition. We may introduce the affine contortion 1-form (𝐴)𝐾𝑎𝑏=(𝐴)𝐾𝑎𝑏 given in terms of the torsion 2-form (𝐴)𝑇𝑎=(𝐴)𝜗𝑏(𝐴)𝐾𝑎𝑏. In tensor components we have (𝐴)𝐾𝜌𝜇𝜈=2(𝐴)𝑄𝜌(𝜇𝜈)+(𝐴)𝑄𝜌𝜇𝜈, where the torsion tensor (𝐴)𝑄𝜌𝜇𝜈=(1/2),(𝐴)𝑇𝜌𝜇𝜈=(𝐴)Γ𝜌[𝜇𝜈] given with respect to a holonomic frame, 𝑑(𝐴)𝜗𝜌=0, is a third-rank tensor, antisymmetric in the first two indices, with 24 independent components. The TSSD-𝑈4 theory (see [96]) considers curvature and torsion as representing independent degrees of freedom. The RC manifold, 𝑈4, is a particular case of general metric-affine manifold 𝑀4, restricted by the metricity condition, when a nonsymmetric linear connection is said to be metric compatible. Taking the antisymmetrized derivative of the metric condition gives an identity between the curvature of the spin-connection and the curvature of the Christoffel connection:(𝐴)𝑅𝜇𝜈𝑎𝑏(𝐴)𝜔(𝐴)𝑒𝜌𝑏𝑅𝜎𝜌𝜇𝜈(Γ)(𝐴)𝑒𝜎𝑎=0,(2.23) where(𝐴)𝑅𝜇𝜈𝑎𝑏(𝐴)𝜔=𝜕𝜇(𝐴)𝜔𝜈𝑎𝑏𝜕𝜈(𝐴)𝜔𝜇𝑎𝑏+(𝐴)𝜔𝜇𝑎𝑐(𝐴)𝜔𝑏𝜈𝑐(𝐴)𝜔𝜈𝑎𝑐(𝐴)𝜔𝑏𝜇𝑐,𝑅𝜎𝜌𝜇𝜈(Γ)=𝜕𝜇Γ𝜎𝜈𝜌𝜕𝜈Γ𝜎𝜇𝜌Γ𝜆𝜇𝜌Γ𝜎𝜈𝜆+Γ𝜆𝜈𝜌Γ𝜎𝜇𝜆.(2.24) Hence, the relations between the scalar curvatures for an 𝑈4 manifold read(𝐴)𝑅(𝐴)𝜔(𝐴)𝑒𝑎𝜇(𝐴)𝑒𝑏𝜈(𝐴)𝑅𝜇𝜈𝑎𝑏(𝐴)𝜔=𝑅(𝑔,Γ)𝑔𝜌𝜈𝑅𝜇𝜌𝜇𝜈(Γ).(2.25) This means that the Lorentz and diffeomorphism invariant scalar curvature, 𝑅, becomes either a function of (𝐴)𝑒𝑎𝜇 only or a function of 𝑔𝜇𝜈 only. Certainly, it can be seen by noting that the Lorentz gauge transformations can be used to fix the six antisymmetric components of (𝐴)𝑒𝑎𝜇 to vanish. Then in both cases diffeomorphism invariance fixes four more components out of the six 𝑔𝜇𝜈, with the four components 𝑔0𝜇 being non dynamical, obviously, leaving only two dynamical degrees of freedom. This shows that the equivalence of the vierbein and metric formulations holds. According to (2.25), the relations between the Ricci scalars read𝑅(𝜎)𝑅𝑐𝑑𝜗𝑐𝜗𝑑=(𝜋)𝑅𝑐𝑑𝜗𝑐𝜗𝑑.(2.26) To recover the TSSD-𝑈4 theory, one can choose the EC Lagrangian, 𝐿EC, as𝐿EC1=22(𝐴)𝑅𝑎𝑏(𝐴)𝜂𝑎𝑏+12Λ(𝐴)1𝜂+2(𝐴)𝑁𝑎𝑏𝜆𝑎𝑏,(2.27) where Λ is the cosmological constant, (𝐴)𝑅𝑎𝑏 is the curvature tensor, 𝜆𝑎𝑏 is the Lagrange multiplier, and (1/2)(𝐴)𝑅𝑎𝑏(𝐴)𝜂𝑎𝑏=(𝐴)𝑅(𝐴)𝜂. The 𝜂-basis is consisting in the Hodge dual of exterior products of tetrads by means of the Levi-Civita object: 𝜂𝑎𝑏𝑐𝑑=𝜗𝑎𝑏𝑐𝑑, which yields 𝜂𝑎𝑏=𝜗𝑎𝑏=(1/2!)𝜂𝑎𝑏𝑐𝑑𝜗𝑐𝑑 and 𝜂=1=(1/4!)𝜂𝑎𝑏𝑐𝑑𝜗𝑎𝑏𝑐𝑑, where we used the abbreviated notations for the wedge product monomials, (𝐴)𝜗𝜇𝜈𝛼=(𝐴)𝜗𝜇(𝐴)𝜗𝜈(𝐴)𝜗𝛼, and denotes the Hodge dual. The variation of the total action𝑆=𝑆EC+𝑆𝑚(𝜋),(2.28) given by the sum of the gravitational field action, 𝑆EC=𝐿EC, with the Lagrangian (2.27) and the macroscopic matter sources, 𝑆𝑚(𝜋), with respect to the (𝐴)𝜗𝑎, 1-form (𝐴)𝜔𝑎𝑏 and Ψ, which is a 𝑝-form representing a matter field (fundamentally a representation of the SL(4,𝑅) or of some of its subgroups), gives1(1)2(𝐴)𝑅𝑐𝑎(𝐴)𝜗𝑐+Λ(𝐴)𝜂𝑎=2(𝐴)𝜃𝑎,(2)(𝐴)Θ𝑎𝑎𝑏𝑏(𝐴)𝒯𝑎𝑏=2(𝜋)Σ𝑎𝑏,(3)𝛿𝐿𝑚(𝜋)𝛿Ψ=0,𝛿𝐿𝑚(𝜋)𝛿Ψ=0,(2.29) where is the Planck length,(𝐴)Θ𝑎𝑏𝑎𝑏(𝜋(𝑥),𝜎(𝑥))=((𝜕(𝐴)𝜔𝑎𝑏)/𝜕(𝜋)𝜔𝑎𝑏), and (𝜋)Σ𝑎𝑏=(𝜋)Σ𝑏𝑎 is the dual 3-form corresponding to the canonical spin tensor, which is identical with the dynamical spin tensor (𝜋)𝑆𝑎𝑏𝑐, namely,(𝜋)Σ𝑎𝑏=(𝜋)𝑆𝜇𝑎𝑏𝜀𝜇𝜈𝛼𝛽(𝜋)𝜗𝜈𝛼𝛽,(2.30) provided that,(𝐴)𝑇𝑎𝑏1=2(𝐴)𝑄𝑎(𝐴)𝑒𝑏=(𝐴)𝑄𝑐(𝐴)𝜗𝑑𝜀𝑐𝑑𝑎𝑏=12(𝐴)𝑄𝑐𝜇𝜈(𝐴)𝑒𝛼𝑑𝜀𝑎𝑏𝑐𝑑(𝐴)𝜗𝜇𝜈𝛼,(2.31) and that(𝐴)𝑄𝑎=(𝐴)𝐷(𝐴)𝜗𝑎=𝑑(𝐴)𝜗𝑎+(𝐴)𝜔𝑏𝑎(𝐴)𝜗𝑏.(2.32) To obtain some feeling for the tensor language in a holonomic frame then we may recast the first two field equations in (2.29) in the tensorial form:(1)𝐺𝜇𝜈+Λ𝑔𝜇𝜈=2(𝐴)𝜃𝜇𝜈𝜕,(2)(𝐴)𝜔𝜈𝜇𝜌𝜕(𝜋)𝜔𝜈𝜇𝜌(𝐴)𝑇(𝐴)𝑇𝜇𝜌𝜈=2(𝜋)𝑆𝜈𝜇𝜌,(2.33) where 𝐺𝜇𝜈𝑅𝜇𝜈(1/2)𝑔𝜇𝜈𝑅 is Einstein’s tensor, and the modified torsion reads(𝐴)𝑇𝜈𝜇𝜌=(𝐴)𝑄𝜈𝜇𝜌+𝛿𝜈𝜇(𝐴)𝑄𝜌𝛿𝜈𝜌(𝐴)𝑄𝜇.(2.34) Thus, the equations of the standard EC theory can be recovered for 𝐴=𝜋. However, these equations can be equivalently replaced by the set of modified EC equations for 𝐴=𝜎:(1)𝐺𝜇𝜈+Λ𝑔𝜇𝜈=2(𝜎)𝜃𝜇𝜈,(2)(𝜎)Θ𝜈𝜇𝜇𝜌𝜌𝜈(𝜎)𝑇(𝜎)𝑇𝜇𝜌𝜈=2(𝜋)𝑆𝜈𝜇𝜌.(2.35) We may impose different physical constraints upon the spacetime deformation 𝜎(𝑥), which will be useful for the theory of electromagnetism and charged particles:Θ𝜇𝜌𝜈𝜈𝜇𝜌(𝜋(𝑥),𝜎(𝑥))Θ𝜇𝜌𝜈𝜈𝜇𝜌(𝜎)𝑇=2𝜑,𝜎𝜀𝜎𝜈𝜇𝜌(𝜎)𝑇1𝜇𝜌𝜈,(2.36) with 𝜑 as a scalar or pseudoscalar function of relevant variables. Here Θ𝜇𝜌𝜈𝜈𝜇𝜌((𝐴)𝑇)=((𝜕(𝐴)𝜔𝜈𝜇𝜌)/𝜕(𝜋)𝜔𝜈𝜇𝜌((𝐴)𝑇)). Then we obtain(𝜋)𝑇𝜈𝜇𝜌=Θ𝜇𝜌𝜈𝜈𝜇𝜌(𝜋(𝑥),𝜎(𝑥))(𝜎)𝑇𝜇𝜌𝜈=2𝜑,𝜎𝜀𝜎𝜈𝜇𝜌,(2.37) which recovers the term in the Lagrangian of pseudoscalar-photon interaction theory [41, 97101], such that the nonmetric part of the Lagrangian can be put in the well-known form of the 𝜒𝑔 framework:𝐿𝐼(𝜋)𝑁𝑀=2(𝑔)1/2𝐴𝜈𝐴𝜇,𝜌(𝜋)𝒯𝜈𝜇𝜌=4(𝑔)1/2𝜑,𝜎𝜀𝜎𝜈𝜇𝜌𝐴𝜈𝐴𝜇,𝜌,(moddiv),(2.38) where 𝐹𝜇𝜈=𝐴𝜇,𝜈𝐴𝜈,𝜇 has the usual meaning for electromagnetism. This is equivalent, up to integration by parts in the action integral (modulo a divergence), to the Lagrangian𝐿𝐼(𝜋)𝑁𝑀=(𝑔)1/2𝜑𝜀𝜎𝜈𝜇𝜌𝐹𝜎𝜈𝐹𝜇𝜌.(2.39) According to (2.39), the gravitational constitutive tensor 𝜒𝜎𝜈𝜇𝜌=𝜒𝜇𝜌𝜎𝜈=𝜒𝜇𝜌𝜈𝜎 [40] of the gravitational fields (e.g., metric 𝑔𝜇𝜈, (pseudo)scalar field 𝜑, etc.) reads𝜒𝜎𝜈𝜇𝜌=(𝑔)1/212𝑔𝜎𝜇𝑔𝜈𝜌12𝑔𝜎𝜌𝑔𝜇𝜈+𝜑𝜀𝜎𝜈𝜇𝜌.(2.40) The special case 𝜑,𝜎=constant=𝑉𝜎 is considered by [102, 103], for modification of electrodynamics with an additional external constant vector coupling. Imposing other appropriate constraints upon the spacetime deformation 𝜎(𝑥), in the framework of TSSD-𝑈4 theory we may reproduce the various terms in the Lagrangians of pseudoscalar theories, for example, as intergrand for topological invariant [104], or pseudoscalar-gluon coupling occurred in QCD in an effort to solve the strong CP problem [105107].

3. The Hypothetical Flat MS companion: A Toy Model

As a preliminary step we now conceive two different spaces: one would be 4D background Minkowski space, 𝑀4, and another one should be MS embedded in the 𝑀4, which is an indispensable individual companion to the particle, without relation to the other matter. This theory is mathematically somewhat similar to the more recent membrane theory. The flat MS in suggested model is assumed to be 2D Minkowski space, 𝑀2:𝑀2=𝑅1(+)𝑅1().(3.1) The ingredient 1D-space 𝑅1𝐴 is spanned by the coordinates 𝜂𝐴, where we use the naked capital Latin letters 𝐴,𝐵,=(±) to denote the world indices related to 𝑀2. The metric in 𝑀2 is𝑔=𝑔𝑒𝐴,𝑒𝐵𝜗𝐴𝜗𝐵,(3.2) where 𝜗𝐴=𝑑𝜂𝐴 is the infinitesimal displacement. The basis 𝑒𝐴 at the point of interest in 𝑀2 consists of two real null vectors:𝑔𝑒𝐴,𝑒𝐵𝑒𝐴,𝑒𝐵=𝑜𝐴𝐵,𝑜𝐴𝐵=0110.(3.3) The norm, 𝑖𝑑𝑑̂𝜂, given in this basis reads 𝑖𝑑=𝑒𝜗=𝑒𝐴𝜗𝐴, where 𝑖𝑑 is the tautological tensor field of type (1,1), 𝑒 is a shorthand for the collection of the 2-tuplet (𝑒(+),𝑒()), and 𝜗=𝜗(+)𝜗(). We may equivalently use a temporal 𝑞0𝑇1 and a spatial 𝑞1𝑅1 variables 𝑞𝑟(𝑞0,𝑞1)(𝑟=0,1), such that𝑀2=𝑅1𝑇1.(3.4) The norm, 𝑖𝑑, now can be rewritten in terms of displacement, 𝑑𝑞𝑟, as𝑖𝑑=𝑑̂𝑞=𝑒0𝑑𝑞0+𝑒1𝑑𝑞1,(3.5) where 𝑒0 and 𝑒1 are, respectively, the temporal and spatial basis vectors:𝑒0=12𝑒(+)+𝑒(),𝑒1=12𝑒(+)𝑒(),𝑔𝑒𝑟,𝑒𝑠𝑒𝑟,𝑒𝑠=𝑜𝑟𝑠,𝑜𝑟𝑠=.1001(3.6) The MS companion (𝑀2) of this particle is assumed to be smoothly (injective and continuous) embedded in the 𝑀4. Suppose that the position of the particle in the background 𝑀4 space is specified by the coordinates 𝑥𝑙(𝑠)(𝑙=0,1,2,3)(𝑥0=𝑡) with respect to the axes of the inertial system 𝑆(4). Then, a smooth map 𝑓𝑀2𝑀4 is defined to be an immersion—an embedding is a function that is a homeomorphism onto its image:𝑞0=12𝜂(+)+𝜂()=𝑡,𝑞1=12𝜂(+)𝜂()=|||𝑥|||.(3.7) In fact, we assume that the particle has to be moving simultaneously in the parallel individual 𝑀2 space and the ordinary 4D background space (either Minkowskian or Riemannian). Let the nonaccelerated observer uses the inertial coordinate frame 𝑆(2) for the position 𝑞𝑟 of a free test particle in the flat 𝑀2. We may choose the system 𝑆(2) in such a way as the time axis 𝑒0 lies along the time axis of a comoving inertial frame 𝑆4, such that the time coordinates in the two systems are taken the same, 𝑞0=𝑡. For the case at hand,𝑣(±)=𝑑𝜂(±)𝑑𝑞0=121±𝑣𝑞,𝑣𝑞=𝑑𝑞1𝑑𝑞0=|||𝑣|||=|||||𝑑𝑥|||||𝑑𝑡=const0.(3.8) Hence, given the inertial frames 𝑆(4), 𝑆(4), 𝑆(4), in the 𝑀4, in this manner we may define the corresponding inertial frames 𝑆(2), 𝑆(2), 𝑆(2), in the 𝑀2.

Continuing on our quest, we next define the concepts of absolute and relative states of the ingredient spaces 𝑅1𝐴. The measure for these states is the very magnitude of the velocity components 𝑣𝐴 of the particle.

Definition 3.1. The ingredient space𝑅1𝐴of the individual MS companion of the particle is said to be in absolute(abs)stateif𝑣𝐴=0,relative(rel)stateif𝑣𝐴0.(3.9)

Therefore, the MS can be realized either in the semiabsolute state (rel, abs), or (abs, rel), or in the total relative state (rel, rel). It is remarkable that the total-absolute state, (abs, abs), which is equivalent to the unobservable Newtonian absolute two-dimensional spacetime, cannot be realized because of the relation 𝑣(+)+𝑣()=2. An existence of the absolute state of the 𝑅1𝐴 is an immediate cause of the light traveling in empty space 𝑅1 along the 𝑞-axis with a maximal velocity 𝑣𝑞=𝑐 (we reinstate the factor (𝑐)) in the (+)-direction corresponding to the state (𝑣(+),0) (rel, abs), and in the ()-direction corresponding to the state (0,𝑣()) (abs, rel). The absolute state of 𝑅1𝐴 manifests its absolute character in the important for SR fact that the resulting velocity of light in the empty space 𝑅1 is the same in all inertial frames 𝑆(2), 𝑆(2), 𝑆(2),; that is, in empty space light propagates independently of the state of motion of the source—if 𝑣𝐴=0, then 𝑣𝐴=𝑣𝐴==0. Since the 𝑣𝐴 is the very key measure of a deviation from the absolute state, we might expect that this has a substantial effect in an alteration of the particle motion under the unbalanced force. This observation allows us to lay forth the foundation of the fundamental RLI as follows.

Conjecture 1 (RIL conjecture). The nonzero local rate 𝜚(𝜂,𝑚,𝑓) of instantaneously change of a constant velocity 𝑣𝐴 (both magnitude and direction) of a massive (𝑚) test particle under the unbalanced net force (𝑓) is the immediate cause of a deformation (distortion of the local internal properties) of MS: 𝑀22.

We can conclude therefrom that, unless MS is flat, a free particle in 4D background space in motion of uniform speed in a straight line tends to stay in this motion and a particle at rest tends to stay at rest. In this way, the MS companion, therefore, abundantly serves to account for the state of motion of the particle in the 4D background space. The MS companion is not measurable directly, but in going into practical details, in Section 4 we will determine the function 𝜚(𝜂,𝑚,𝑓) and show that a deformation (distortion of local internal properties) of MS is the origin of inertia effects that can be observed by us. Before tempting to build realistic model of accelerated motion and inertial effects, for the benefit of the reader, we briefly turn back to physical discussion of why the MS is two dimensional and not higher. We have first to recall the salient features of MS which admittedly possesses some rather unusual properties; namely, the basis at the point of interest in MS, embedded in the 4D spacetime, would be consisted of the real null vectors, which just allows only two-dimensional constructions (3.3). Next, note that the immediate cause of inertia effects is the nonlinear process of deformation (distortion of local internal properties) of MS, which yields the resulting linear relation 𝑓in𝑓= (see (2.19)–(5.35)) with respect to the components of inertial force 𝑓in in terms of the relativistic force 𝑓 acting on a purely classical particle in 𝑀4. This ultimately requires that MS should only be two dimensional, because to resolve the afore-mentioned relationship of nonlinear and linear processes we may choose the system 𝑆(2) in only allowed way as the time axis 𝑒0 lies along the time axis of a comoving inertial frame 𝑆4, in order that the time coordinates in the two systems are taken the same, 𝑞0=𝑡 and that another axis 𝑒𝑞 lies along the net 3-acceleration (𝑒𝑞𝑒𝑎),(𝑒𝑎=𝑎net/|𝑎net|) (5.26).

4. The General Spacetime Deformation/Distortion Complex

For the self-contained arguments, we now extend just necessary geometrical ideas of the spacetime deformation framework described in Section 2, without going into the subtleties, as applied to the 2D deformation 𝑀22. To start with, let 𝑉2 be 2D semi-Riemann space, which has at each point a tangent space, ̆𝑇̆𝜂𝑉2, spanned by the anholonomic orthonormal frame field, ̆𝑒, as a shorthand for the collection of the 2-tuplet (̆𝑒(+),̆𝑒()), where ̆𝑒𝑎𝐴=̆𝑒𝑎̆𝑒𝐴, where the holonomic frame is given as ̆𝑒𝐴=̆𝜕𝐴. Here, we use the first half of Latin alphabet 𝑎,𝑏,𝑐,=(±) to denote the anholonomic indices to related the tangent space, and the capital Latin letters with an over ---𝐴,𝐵,=(±), to denote the holonomic world indices related to either the space 𝑉2 or 2. All magnitudes referred to the space, 𝑉2, will be denoted by an over ̆. These then define a dual vector, ̆𝜗, of differential forms, ̆𝜗=̆𝜗(+)̆𝜗(), as a shorthand for the collection of the ̆𝜗𝑏=̆𝑒𝑏𝐴̆𝜗𝐴, whose values at every point form the dual basis, such that ̆𝑒𝑎̆𝜗𝑏=𝛿𝑏𝑎. In components 𝐴̆𝑒𝑎̆𝑒𝑏𝐴=𝛿𝑏𝑎. On the manifold, 𝑉2, the tautological tensor field, 𝑖̆𝑑, of type (1,1) can be defined which assigns to each tangent space the identity linear transformation. Thus for any point ̆𝜂𝑉2, and any vector ̆̆𝑇𝜉̆𝜂𝑉2, one has 𝑖̆̆̆𝜉𝑑(𝜉)=. In terms of the frame field, the ̆𝜗𝑎 give the expression for 𝑖̆𝑑 as 𝑖̆̆𝑑=̆𝑒𝜗=̆𝑒(+)̆𝜗(+)+̆𝑒()̆𝜗(), in the sense that both sides yield ̆𝜉 when applied to any tangent vector ̆𝜉 in the domain of definition of the frame field. We may consider general transformations of the linear group, GL(2,𝑅), taking any base into any other set of four linearly independent fields. The notation {̆𝑒𝑎,̆𝜗𝑏} will be used hereinafter for general linear frames. The holonomic metric can be defined in the semi-Riemann space, 𝑉2, as̆𝑔=̆𝑔𝐴𝐵̆𝜗𝐴̆𝜗𝐵=̆𝑔̆𝑒𝐴,̆𝑒𝐵̆𝜗𝐴̆𝜗𝐵,(4.1) with components ̆𝑔𝐴𝐵=̆𝑔(̆𝑒𝐴,̆𝑒𝐵) in the dual holonomic base {̆𝜗𝐴}. The anholonomic orthonormal frame field, ̆𝑒, relates ̆𝑔 to the tangent space metric, 𝑜𝑎𝑏, by 𝑜𝑎𝑏=̆𝑔(̆𝑒𝑎,̆𝑒𝑏)=̆𝑔𝐴𝐵𝐴̆𝑒𝑎𝐵̆𝑒𝑏, which has the converse ̆𝑔𝐴𝐵=𝑜𝑎𝑏̆𝑒𝑎𝐴̆𝑒𝑏𝐵 because of the relation 𝐴̆𝑒𝑎̆𝑒𝑎𝐵𝐴=𝛿𝐵. With this provision, we build up a general distortion-complex, yielding a distortion of the flat space 𝑀2, and show how it recovers the world-deformation tensor Ω, which still has to be put in [96] by hand. The DC members are the invertible distortion matrix 𝐷, the tensor 𝑌, and the flat-deformation tensor Ω. Symbolically,̆̆DC𝐷,𝑌,ΩΩ.(4.2) The principle foundation of a distortion of local internal properties of MS comprises then two steps.

(1) The first is to assume that the linear frame (𝑒𝐴;𝜗𝐴), at given point (𝑝𝑀2), undergoes the distortion transformations, conducted by (̆̆𝐷,𝑌) and (𝐷,𝑌), respectively, relating to 𝑉2 and 2, recast in the form ̆𝑒𝐴=̆𝐷𝐵𝐴𝑒𝐵,̆𝜗𝐴=̆𝑌𝐴𝐵𝜗𝐵,𝑒𝐴=𝐷𝐵𝐴𝑒𝐵𝐴𝐴,𝜗=𝑌𝐵𝜗𝐵.(4.3)

(2) Then, the norm 𝑑̃̂𝜂𝑖𝑑 of the infinitesimal displacement 𝐴𝑑̃𝜂 on the general smooth differential 2D-manifold 2 can be written in terms of the spacetime structures of 𝑉2 and 𝑀2:Ω𝐵𝑖𝑑=𝑒𝜗=𝐴̆𝑒𝐵̆𝜗𝐴=Ω𝑎𝑏̆𝑒𝑎̆𝜗𝑏=𝑒𝐶𝐶𝜗=𝑒𝑎𝜗𝑎=Ω𝐵𝐴𝑒𝐵𝜗𝐴2,(4.4) where 𝑒={𝑒𝑎𝐶=𝑒𝑎𝑒𝐶} is the frame field and 𝜗={𝜗𝑎=𝑒𝑎𝐶𝜗𝐶} is the coframe field defined on 2, such that 𝑒𝑎𝜗𝑏=𝛿𝑏𝑎. The deformation tensors Ω𝐵𝐴𝐶=𝜋𝐴𝜋𝐵𝐶 and Ω𝐵𝐴 implyΩ𝐵𝐴=̆𝐷𝐶𝐴Ω𝐷𝐶̆𝑌𝐵𝐷,Ω𝐵𝐴𝐶=𝑌𝐴𝐷𝐵𝐶,(4.5) provided that 𝐷𝐴𝐶𝐵=𝜋𝐶̆𝐷𝐴𝐵𝐶,𝑌𝐵𝐶=𝜋𝐴̆𝑌𝐴𝐵,(4.6) such that𝑒𝐶𝐵=𝜋𝐶̆𝑒𝐵̃𝜕𝐶𝐶𝐶,𝜗=𝜋𝐴̆𝜗𝑉𝐶𝐶𝑑̃𝜂,̃𝜂𝒰2.(4.7) Hence the anholonomic deformation tensor, Ω𝑎𝑏=𝜋𝑎𝑐𝜋𝑐𝑏=Ω𝐵𝐴̆𝑒𝑎𝐵𝐴̆𝑒𝑏, yields local tetrad deformations:𝑒𝑐=𝜋𝑎𝑐̆𝑒𝑎,𝜗𝑐=𝜋𝑐𝑏̆𝜗𝑏,𝑒𝜗=𝑒𝑎𝜗𝑎=Ω𝑎𝑏̆𝑒𝑎̆𝜗𝑏.(4.8) The matrices 𝜋(̃𝜂)=(𝜋𝑎𝑏)(̃𝜂) are referred to as the first deformation matrices, and the matrices 𝛾𝑐𝑑(̃𝜂)=𝑜𝑎𝑏𝜋𝑎𝑐(̃𝜂)𝜋𝑏𝑑(̃𝜂),   second deformation matrices. The matrices 𝜋𝑎𝑐(̃𝜂)GL(2,𝑅)forall̃𝜂, in general, give rise to right cosets of the Lorentz group; that is, they are the elements of the quotient group GL(2,𝑅)/SO(1,1), because the Lorentz matrices, Λ𝑟𝑠, (𝑟,𝑠=1,0) leave the Minkowski metric invariant. A right multiplication of 𝜋(̃𝜂) by a Lorentz matrix gives another deformation matrix. So, all the fundamental geometrical structures on deformed/distorted MS in fact—the metric as much as the coframes and connections—acquire a deformation/distortion-induced theoretical interpretation. If we deform the tetrad according to (4.8), in general, we have two choices to recast metric as follows: either writing the deformation of the metric in the space of tetrads or deforming the tetrad field:𝑔=𝑜𝑎𝑏𝜋𝑎𝑐𝜋𝑏𝑑̆𝜗𝑐̆𝜗𝑑=𝛾𝑐𝑑̆𝜗𝑐̆𝜗𝑑=𝑜𝑎𝑏𝜗𝑎𝜗𝑏.(4.9) In the first case, the contribution of the Christoffel symbols, constructed by the metric 𝛾𝑎𝑏, readsΓ𝑎𝑏𝑐=12̆𝐶𝑎𝑏𝑐𝛾𝑎𝑎𝛾𝑏𝑏̆𝐶𝑏𝑎𝑐𝛾𝑎𝑎𝛾𝑐𝑐̆𝐶𝑐𝑎𝑏+12𝛾𝑎𝑎̆𝑒𝑐𝑑𝛾𝑏𝑎̆𝑒𝑏𝑑𝛾𝑐𝑎̆𝑒𝑎𝑑𝛾𝑏𝑐.(4.10) The deformed metric can be split as follows [96]:𝑔𝐴𝐵(𝜋)=Υ2(𝜋)̆𝑔𝐴𝐵+𝛾𝐴𝐵(𝜋),(4.11) where Υ(𝜋)=𝜋𝑎𝑎, and𝛾𝐴𝐵𝛾(𝜋)=𝑎𝑏Υ2(𝜋)𝑜𝑎𝑏̆𝑒𝑎𝐴̆𝑒𝑏𝐵.(4.12) In the second case, we may write the commutation table for the anholonomic frame, {𝑒𝑎},𝑒𝑎,𝑒𝑏1=2𝐶𝑐𝑎𝑏𝑒𝑐,(4.13) and define the anholonomy objects:𝐶𝑎𝑏𝑐=𝜋𝑎𝑒𝜋𝑏𝑑1𝜋𝑐𝑓1̆𝐶𝑒𝑑𝑓+2𝜋𝑎𝑓𝐴̆𝑒𝑔𝜋1[𝑏𝑔𝜕𝐴𝜋𝑐1]𝑓.(4.14) The usual Levi-Civita connection corresponding to the metric (4.11) is related to the original connection by the following relation:Γ𝐴𝐶𝐷=̆Γ𝐴𝐶𝐷𝐴+Π𝐶𝐷,(4.15) provided thatΠ𝐴𝐶𝐷𝐴𝐵=2𝑔̆𝑔𝐵(𝐶𝐷)Ῠ𝑔𝐶𝐷𝑔𝐴𝐵𝐵Υ+12𝑔𝐴𝐵𝐶𝛾𝐵𝐷+𝐷𝛾𝐶𝐵𝐵𝛾𝐶𝐷,(4.16) where the controvariant deformed metric, 𝑔𝐵𝐶, is defined as the inverse of 𝑔𝐴𝐵, such that 𝑔𝐴𝐵𝑔𝐵𝐶𝐶=𝛿𝐴. That is, the connection deformation Π𝐴𝐶𝐷 acts like a force that deviates the test particles from the geodesic motion in the space, 𝑉2. Taking into account (4.4), the metric (4.9) can be alternatively written in a general form of the spacetime or frame objects:𝑔=𝑔𝐴𝐵𝜗𝐴𝐵=Ω𝐵𝜗𝐴ΩD𝐶̆𝑔𝐵𝐷̆𝜗𝐴̆𝜗𝐶=𝑜𝑎𝑏𝜗𝑎𝜗𝑏=Ω𝑐𝑎Ω𝑑𝑏𝑜𝑐𝑑̆𝜗𝑎̆𝜗𝑏=𝛾𝑐𝑑̆𝜗𝑐̆𝜗𝑑=Ω𝐶𝐴Ω𝐷𝐵𝑜𝐶𝐷𝜗𝐴𝜗𝐵.(4.17) A significantly more rigorous formulation of the spacetime deformation technique with different applications as we have presented it may be found in [96].

5. Model Building in the 4D Background Minkowski Spacetime

In this section we construct the RTI in particular case when the relativistic test particle accelerated in the Minkowski 4D background flat space, 𝑀4, under an unbalanced net force other than gravitational. Here and henceforth we simplify DC for our use by imposing the constraints𝐷𝐴𝐶=̆𝐷𝐴𝐵,̆𝑌𝐴𝐵=̆𝐷𝐴𝐵,(5.1) and, therefore,DC(𝐷,Ω)Ω.(5.2) The (4.5), by virtue of (4.4) and (5.1), givesΩ𝐵𝐴=̆𝐷𝐶𝐴Ω𝐷𝐶̆𝐷𝐵𝐷𝐵=𝜋𝐴𝐶,𝑌𝐵=Ω𝐶𝐴̆𝐷𝐴𝐵,(5.3) where the deformation tensor, Ω𝐵𝐴, yields the partial holonomic frame transformations:𝑒𝐶=̆𝑒𝐶𝐶=Ω𝐶,𝜗𝐴̆𝜗𝑉,(5.4) or, respectively, the Ω𝑎𝑏 yields the partial local tetrad deformations:𝑒𝑐=̆𝑒𝑐,𝜗𝑐=Ω𝑐𝑏̆𝜗𝑏,𝑒𝜗=𝑒𝑎𝜗𝑎=Ω𝑎𝑏̆𝑒𝑎̆𝜗𝑏.(5.5) Hence, (4.4) defines a diffeomorphism 𝐴̃𝜂(𝜂)𝑀22:𝑒𝐴𝑌𝐴𝐵=Ω𝐴𝐵𝑒𝐴,(5.6) where 𝑌𝐴𝐵𝐴=𝜕̃𝜂/𝜕𝜂𝐵. The conditions of integrability, 𝜕𝐴𝑌𝐶𝐵=𝜕𝐵𝑌𝐶𝐴, and nondegeneracy, 𝐴det|𝑌𝐵|0, immediately define a general form of the flat-deformation tensor Ω𝐴𝐵=𝐷𝐴𝐶𝜕𝐵Θ𝐶, where Θ𝐶 is an arbitrary holonomic function. To make the remainder of our discussion a bit more concrete, it proves necessary to provide, further, a constitutive ansatz of simple, yet tentative, linear distortion transformations, which, according to RLI conjecture, can be written in terms of local rate 𝜚(𝜂,𝑚,𝑓) of instantaneously change of the measure 𝑣𝐴 of massive (𝑚) test particle under the unbalanced net force (𝑓):𝑒(+)(𝜚)=𝐷𝐵+(𝜚)𝑒𝐵=𝑒(+)𝜚(𝜂,𝑚,𝑓)𝑣()𝑒(),𝑒()(𝜚)=𝐷𝐵(𝜚)𝑒𝐵=𝑒()+𝜚(𝜂,𝑚,𝑓)𝑣(+)𝑒(+).(5.7) Clearly, these transformations imply a violation of the relation (3.3) (𝑒2𝐴(𝜚)0) for the null vectors 𝑒𝐴. Now we can use (4.4) to observe that for dual vectors of differential forms 𝜗=𝜗(+)𝜗() and 𝜗=𝜗(+)𝜗() we may obtainΩ𝜗=𝐶(+)𝑒(+),𝑒𝐶Ω𝐶()𝑒(+),𝑒𝐶Ω𝐶(+)𝑒(),𝑒𝐶Ω𝐶()𝑒(),𝑒𝐶𝜗.(5.8) We parameterize the tensor Ω𝐴𝐵 in terms of the parameters 𝜏1 and 𝜏2 asΩ(+)(+)=Ω()()=𝜏11+𝜏2𝜚2,Ω()(+)=𝜏11𝜏2𝜚𝑣(),Ω(+)()=𝜏11𝜏2𝜚𝑣(+),(5.9) where 𝜚2=𝑣2𝜚2,𝑣2=𝑣(+)𝑣()=1/2𝛾2𝑞 and 𝛾𝑞=(1𝑣2𝑞)1/2. Then, the relation (5.8) can be recast in an alternative form:𝜗=𝜏11𝜏2𝜚𝑣(+)𝜏2𝜚𝑣()1𝜗.(5.10) Suppose that a second observer, who makes measurements using a frame of reference 𝑆(2) which is held stationary in deformed/distorted space 2, uses for the test particle the corresponding spacetime coordinates ̃0̃1̃̃𝑞̃𝑟((̃𝑞,̃𝑞)(𝑡,̃𝑞)). The (4.4) can be rewritten in terms of spacetime variables as𝑖𝑑=𝑒𝜗𝑑̃̂𝑞=̃𝑒0̃𝑑𝑡+̃𝑒𝑞𝑑̃𝑞,(5.11) where ̃𝑒0 and ̃𝑒𝑞 are, respectively, the temporal and spatial basis vectors:̃𝑒01(𝜚)=2𝑒(+)(𝜚)+𝑒()(𝜚),̃𝑒𝑞1(𝜚)=2𝑒(+)(𝜚)𝑒().(𝜚)(5.12) The transformation equation for the coordinates, according to (5.10), becomes𝜗(±)=𝜏1𝜗(±)𝜏2𝜚𝑣(±)𝜗()=𝜏1𝑣(±)𝜏2𝜚𝑣2𝑑𝑡,(5.13) which gives the general transformation equations for spatial and temporal coordinates as follows (𝑒𝑞𝑒1,𝑞𝑞1):𝑑̃𝑡=𝜏1𝑑𝑡,𝑑̃𝑞=𝜏1𝜏𝑑𝑞1+2𝜚𝑣𝑞2𝜏2𝜚2𝑑𝑡=𝜏1𝜏𝑑𝑞2𝜚2𝛾2𝑞.𝑑𝑡(5.14) Hence, the general metric (4.17) in 2 reads 𝑔𝑑̃𝑠2𝑞=𝑔̃𝑟=Ω̃𝑠𝑑̃𝑞̃𝑟𝑑̃𝑞̃𝑠(+)(+)2+Ω(+)()Ω()(+)𝑑𝑠2𝑞+Ω(+)(+)Ω(+)()+Ω()(+)(𝑑𝑡𝑑𝑡+𝑑𝑞𝑑𝑞)2Ω(+)(+)Ω(+)()Ω()(+)𝑑𝑡𝑑𝑞,(5.15) provided that𝑔̃0̃0=1+𝜚𝑣𝑞22𝜚22,𝑔̃1̃1=1𝜚𝑣𝑞22+𝜚22,𝑔̃1̃0=𝑔̃0̃1=2𝜚.(5.16) The difference of the vector, 𝑑̂𝑞𝑀2 (3.5), and the vector, 𝑑̃̂𝑞2 (5.11), can be interpreted by the second observer as being due to the deformation/distortion of flat space 𝑀2. However, this difference with equal justice can be interpreted by him as a definite criterion for the absolute character of his own state of acceleration in 𝑀2, rather than to any absolute quality of a deformation/distortion of 𝑀2. To prove this assertion, note that the transformation equations (5.14) give a reasonable change at low velocities 𝑣𝑞0, as𝑑̃𝑡=𝜏1𝑑𝑡,𝑑̃𝑞𝜏1𝜏𝑑𝑞2𝜚2𝑑𝑡,(5.17) therebyΩ(+)(+)=Ω()()=𝜏11+𝜏2𝜚2,Ω(+)()=Ω()(+)=𝜏11𝜏2𝜚.(5.18) Then (5.17) becomes conventional transformation equations to accelerated (𝑎net0) axes if we assume that 𝑑(𝜏2𝜚)/2𝑑𝑡=𝑎net and 𝜏1(𝑣𝑞0)=1, where 𝑎net is a magnitude of proper net acceleration. In high-velocity limit 𝑣𝑞1, 𝜚0(𝑑𝜂()=𝑣()𝑑𝑡0,𝑣(+)𝑣2), we haveΩ(+)(+)=Ω()()=𝜏1,Ω()(+)=0,Ω(+)()=𝜏11𝜏22𝜚,(5.19) and so (5.14) and (5.15), respectively, give𝑑̃𝑡=𝜏1𝑑𝑡𝜏1𝑑𝑞𝑑̃𝑞,(5.20)𝑑̃𝑠2𝑞𝜚1+22𝜚22𝑑̃𝜚𝑡𝑑𝑡+122+𝜚22𝑑̃𝑞𝑑̃𝑞2̃2𝜚𝑑𝑡𝑑̃𝑞𝜏21𝑑𝑠2𝑞=0.(5.21) To this end, the inertial effects become zero. Let 𝑎net be a local net 3-acceleration of an arbitrary observer with proper linear 3-acceleration 𝑎 and proper 3-angular velocity 𝜔 measured in the rest frame:𝑎net=𝑑𝑢𝑑𝑠=𝑎𝑢+𝜔×𝑢,(5.22) where 𝐮 is the 4-velocity. A magnitude of 𝑎net can be computed as the simple invariant of the absolute value |𝑑𝐮/𝑑𝑠| as measured in rest frame:||||𝐚|=𝑑𝐮|||=𝑑𝑠𝑑𝑢𝑙,𝑑𝑠𝑑𝑢𝑙𝑑𝑠1/2.(5.23) Following [57, 58], let us define an orthonormal frame 𝑒̂𝑎, carried by an accelerated observer, who moves with proper linear 3-acceleration and 𝑎(𝑠) and proper 3-rotation 𝜔(𝑠). Particular frame components are denoted by hats, ̂̂0,1,andsoforth. Let the zeroth leg of the frame 𝑒̂0 be 4-velocity 𝐮 of the observer that is tangent to the worldline at a given event 𝑥𝑙(𝑠) and we parameterize the remaining spatial triad frame vectors 𝑒̂𝑖, orthogonal to 𝑒̂0, also by (𝑠). The spatial triad 𝑒̂𝑖 rotates with proper 3-rotation 𝜔(𝑠). The 4-velocity vector naturally undergoes Fermi-Walker transport along the curve 𝐶, which guarantees that 𝑒̂0(𝑠) will always be tangent to 𝐶 determined by 𝑥𝑙=𝑥𝑙(𝑠):𝑑𝑒̂𝑎,𝑑𝑠=Ω𝑒̂𝑎(5.24) where the antisymmetric rotation tensor Ω splits into a Fermi-Walker transport part ΩFW and a spatial rotation part ΩSR:Ω𝑙𝑘FW=𝑎𝑙𝑢𝑘𝑎𝑘𝑢𝑙,Ω𝑙𝑘SR=𝑢𝑚𝜔𝑛𝜀𝑚𝑛𝑙𝑘.(5.25) The 4-vector of rotation 𝜔𝑙 is orthogonal to 4-velocity 𝑢𝑙, therefore, in the rest frame it becomes 𝜔𝑙(0,𝜔), and 𝜀𝑚𝑛𝑙𝑘 is the Levi-Civita tensor with 𝜀0123=1. Then (5.17) immediately indicates that we may introduce the very concept of the local absolute acceleration (in Newton’s terminology) brought about via the Fermi-Walker transported frames as𝑎abs𝑒𝑞𝑑𝜏2𝜚2𝑑𝑠𝑞=𝑒𝑞||||̂0𝑑𝑒||||𝑑𝑠=𝑒𝑞|𝐚|,(5.26) where we choose the system 𝑆(2) in such a way as the axis 𝑒𝑞 lies along the net 3-acceleration (𝑒𝑞𝑒𝑎),(𝑒𝑎=𝑎net/|𝑎net|). Hereinafter, we may simplify the flat-deformation tensor Ω𝐵𝐴 by setting 𝜏2=1, such that (5.9) becomesΩ(+)(+)=Ω()()Ω𝜚=1+𝜚2,Ω()(+)=Ω(+)()=0,(5.27) and the general metric (4.17) in 2 reads 𝑑̃𝑠2𝑞=Ω2(𝜚)𝑑𝑠2𝑞. Hence (5.26) gives𝜚=2𝑠𝑞0|𝐚|𝑑𝑠𝑞.(5.28) Combining (5.14) and (5.26), we obtain the key relation between a so-called inertial acceleration, arisen due to the curvature of MS,𝑎in=𝑒𝑎𝑎in,𝑎in=𝑑2̃𝑞𝑑̃𝑠2𝑞=Γ1̃𝑟̃𝑠(𝜚)𝑑̃𝑞̃𝑟𝑑̃𝑠𝑞𝑑̃𝑞̃𝑠𝑑̃𝑠𝑞=12𝑑2̃𝜂(+)𝑑̃𝑠2𝑞𝑑2̃𝜂()𝑑̃𝑠2𝑞,(5.29) and a local absolute acceleration as follows:Ω2𝜚𝛾𝑞𝑎in=𝑎abs,(5.30) where Γ1̃𝑟̃𝑠(𝜚) are the Christoffel symbols constructed by the metric (5.16). Then (5.30) provides a quantitative means for the inertial force 𝑓(in):𝑓(in)=𝑚𝑎in=𝑚Γ1̃𝑟(̃𝑠𝜚)𝑑̃𝑞̃𝑟𝑑̃𝑠𝑞𝑑̃𝑞̃𝑠𝑑̃𝑠𝑞=𝑚𝑎absΩ2𝜚𝛾𝑞.(5.31) In case of absence of rotation, we may write the local absolute acceleration (5.26) in terms of the relativistic force 𝑓𝑙 acting on a particle with coordinates 𝑥𝑙(𝑠):𝑓𝑙𝑓0,𝑓𝑑=𝑚2𝑥𝑙𝑑𝑠2=Λ𝑙𝑘𝑣𝐹𝑘.(5.32) Here 𝐹𝑘(0,𝐹) is the force defined in the rest frame of the test particle, and Λ𝑙𝑘(𝑣) is the Lorentz transformation matrix (𝑖,𝑗=1,2,3):Λ𝑖𝑗=𝛿𝑖𝑗𝑣(𝛾1)𝑖𝑣𝑗|||𝑣|||2,Λ0𝑖=𝛾𝑣𝑖,(5.33) where 𝑣𝛾=(12)1/2. So1|𝐚|=𝑚||𝑓𝑙||=1𝑚𝑓𝑙𝑓𝑙1/2=1||||𝑚𝛾𝑓||||,(5.34) and hence (5.31), (5.26), and (5.34) give𝑓(in)1=Ω2𝜚𝛾𝑞𝛾𝐹+(𝛾1)𝑣𝑣𝐹|||𝑣|||2.(5.35) At low velocities 𝑣𝑞|𝑣|0 and tiny accelerations we usually experience, one has Ω(𝜚)1; therefore (5.35) reduces to the conventional nonrelativistic law of inertia:𝑓(in)=𝑚𝑎abs=𝐹.(5.36) At high velocities 𝑣𝑞|𝑣|1 (Ω(𝜚)1), if (𝑣𝐹)0, the inertial force (5.35) becomes𝑓(in)1𝛾𝑒𝑣𝑒𝑣𝐹,(5.37) and, in agreement with (5.21), it vanishes in the limit of the photon (|𝑣|=1,𝑚=0). Thus, it takes force to disturb an inertia state, that is, to make the absolute acceleration (𝑎abs0). The absolute acceleration is due to the real deformation/distortion of the space 𝑀2. The relative (𝑑(𝜏2𝜚)/𝑑𝑠𝑞=0) acceleration (in Newton’s terminology) (both magnitude and direction), to the contrary, has nothing to do with the deformation/distortion of the space 𝑀2 and, thus, it cannot produce inertia effects.

6. Beyond the Hypothesis of Locality

The standard geometrical structures, referred to a noninertial coordinate frame of accelerating and rotating observer in Minkowski spacetime, were computed on the base of the hypothesis of locality [5966], which in effect replaces an accelerated observer at each instant with a momentarily comoving inertial observer along its wordline. This assumption represents strict restrictions, because, in other words, it approximately replaces a noninertial frame of reference 𝑆(2), which is held stationary in the deformed/distorted space 2𝑉2(𝜚)(𝜚0), with a continuous infinity set of the inertial frames {𝑆(2),𝑆(2),𝑆(2),} given in the flat 𝑀2(𝜚=0). In this situation the use of the hypothesis of locality is physically unjustifiable. Therefore, it is worthwhile to go beyond the hypothesis of locality with special emphasis on distortion of MS, which, we might expect, will essentially improve the standard results. The notation will be slightly different from the previous section. We denote the orthonormal frame 𝑒̂𝑎 (5.24), carried by an accelerated observer, with the over “breve” such that=̆𝑒̂𝑎𝑒𝜇̂𝑎𝑒𝜇=̆𝑒𝜇̂𝑎̆𝑒𝜇,̆𝜗̂𝑏=𝑒̂𝑏𝜇𝜗𝜇̂𝑏=̆𝑒𝜇̆𝜗𝜇,(6.1) with 𝑒𝜇=𝜕𝜇=𝜕/(𝜕𝑥𝜇),̆𝑒𝜇=̆𝜕𝜇=𝜕/𝜕̆𝑥𝜇, and 𝜗𝜇=𝑑𝑥𝜇,̆𝜗𝜇=𝑑̆𝑥. Here, following [58, 64], we introduced a geodesic coordinate system ̆𝑥𝜇coordinates relative to the accelerated observer (laboratory coordinates)—in the neighborhood of the accelerated path. The coframe members {̆𝜗̂𝑏} are the objects of dual counterpart: ̆𝜗̂𝑏̆𝑒̂𝑎=𝛿𝑏𝑎. We choose the zeroth leg of the frame, ̂0̆𝑒, as before, to be the unit vector 𝐮 that is tangent to the worldline at a given event 𝑥𝜇(𝑠), where (𝑠) is a proper time measured along the accelerated path by the standard (static inertial) observers in the underlying global inertial frame. The condition of orthonormality for the frame field 𝑒𝜇̂𝑎 reads 𝜂𝜇𝜈𝑒𝜇̂𝑎𝑒𝜈̂𝑏̂𝑏=𝑜̂𝑎=diag(+). The antisymmetric acceleration tensor Φ𝑎𝑏 [6468, 121125] is given byΦ𝑏𝑎=𝑒̂𝑏𝜇𝑑𝑒𝜇̂𝑎=𝑑𝑠𝑒̂𝑏𝜇𝑢𝜆̆𝜆𝑒𝜇Γ̂𝑎=𝑢𝑏𝑎,(6.2) provided that ̆Γ𝑏𝑎=̆Γ𝑏𝑎𝜇𝑑̆𝑥𝜇, where ̆Γ𝑏𝑎𝜇 is the metric compatible, torsion-free Levi-Civita connection. According to (5.24) and (5.25), and in analogy with the Faraday tensor, one can identify Φ𝑎𝑏(𝐚,𝜔), with 𝐚(𝑠) as the translational acceleration Φ0𝑖=𝑎𝑖 and 𝜔(𝑠) as the frequency of rotation of the local spatial frame with respect to a nonrotating (Fermi-Walker transported) frame Φ𝑖𝑗=𝜀𝑖𝑗𝑘𝜔𝑘. The invariants constructed out of Φ𝑎𝑏 establish the acceleration scales and lengths. The hypothesis of locality holds for huge proper acceleration lengths |𝐼|1/21 and |𝐼|1/21, where the scalar invariants are given by 𝐼=(1/2)Φ𝑎𝑏Φ𝑎𝑏=𝑎2+𝜔2 and 𝐼=(1/4)Φ𝑎𝑏Φ𝑎𝑏=𝑎𝜔 (Φ𝑎𝑏=𝜀𝑎𝑏𝑐𝑑Φ𝑐𝑑) [6466, 121125]. Suppose that the displacement vector 𝑧𝜇(𝑠) represents the position of the accelerated observer. According to the hypothesis of locality, at any time (𝑠) along the accelerated worldline the hypersurface orthogonal to the worldline is Euclidean space and we usually describe some event on this hypersurface (local coordinate system) at 𝑥𝜇 to be at ̆𝑥𝜇, where 𝑥𝜇 and ̆𝑥𝜇 are connected via ̆𝑥0=𝑠 and𝑥𝜇=𝑧𝜇(𝑠)+̆𝑥𝑖𝑒𝜇̂𝑖(𝑠).(6.3) Let ̆𝑞𝑟(̆𝑞0,̆𝑞1) be coordinates relative to the accelerated observer in the neighborhood of the accelerated path in MS, with spacetime components implying𝑑̆𝑞0=𝑑̆𝑥0,𝑑̆𝑞1=||||𝑑||||,𝑑̆𝑥̆𝑒=̆𝑥𝑑̆𝑞1=𝑑̆𝑥||||𝑑||||,̆𝑥̆𝑒̆𝑒=1.(6.4) As long as a locality assumption holds, we may describe, with equal justice, the event at 𝑥𝜇 (6.3) to be at point ̆𝑞𝑟, such that 𝑥𝜇 and ̆𝑞𝑟, in full generality, are connected via ̆𝑞0=𝑠 and𝑥𝜇=𝑧𝜇𝑞(𝑠)+̆𝑞1𝛽𝜇̂1(𝑠),(6.5) where the displacement vector from the origin reads 𝑑𝑧𝜇𝑞(𝑠)=𝛽𝜇̂0𝑑̆𝑞0, and the components 𝛽𝜇̂𝑟 can be written in terms of 𝑒𝜇̂𝑎. Actually, from (6.3) and (6.5) we may obtain𝑑𝑥