# Spacetime Deformation-Induced Inertia Effects

Authors: Gagik Ter-Kazarian
Journal: Advances in Mathematical Physics
 (2012)

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## 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-calledmaster 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.


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## Body

## 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, [3–15], 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 [4–6] 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 [16–35], or at galactic and cosmological scales [36–41], 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 [23–26] 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 [29–34], 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, [42–56]. 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 [60–62]. Ever since this question has become a major preoccupation of physicists; see, for example, [63–84]. 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, 60–91]. The experiments by [87–90], 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/m≤10-20. 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 [88–90] 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 [92–94]. 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 [92–95]. 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, 97–101], 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 [105–107]. 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, h=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, [110–113]. 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, T̆x̆V4, spanned by the anholonomic orthonormal frame field, ĕ, as a shorthand for the collection of the 4-tuplet (ĕ0,…,ĕ3), where ĕa=ĕaμ∂̆μ. All magnitudes related to the space, V4, will be denoted by an over “̆”. These then define a dual vector, ϑ̆, of differential forms, ϑ̆=(ϑ̆0⋮ϑ̆3), as a shorthand for the collection of the ϑ̆b=ĕμbdx̆μ, whose values at every point form the dual basis, such that ĕa⌋ϑ̆b=δab, where ⌋ denotes the interior product, namely, this is a C∞-bilinear map ⌋:Ω1→Ω0 where Ωp denotes the C∞-modulo of differential p-forms on V4. In components ĕaμĕμb=δab. On the manifold, V4, the tautological tensor field, id̆, of type (1,1) can be defined which assigns to each tangent space the identity linear transformation. Thus for any point x̆∈V4, and any vector ξ̆∈T̆x̆V4, one has id̆(ξ̆)=ξ̆. In terms of the frame field, the ϑ̆a give the expression for id̆ as id̆=ĕϑ̆=ĕ0⊗ϑ̆0+⋯ĕ3⊗ϑ̆3, in the sense that both sides yield ξ̆ when applied to any tangent vector ξ̆ 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 {ĕa,ϑ̆b} will be used hereinafter for general linear frames. The holonomic metric can be defined in the semi-Riemann space, V4, as(2.1)ğ=ğμνϑ̆μ⊗ϑ̆ν=ğ(ĕμ,ĕν)ϑ̆μ⊗ϑ̆ν,
with components ğμν=ğ(ĕμ,ĕν) in the dual holonomic base {ϑ̆μ≡dx̆μ}. The anholonomic orthonormal frame field, ĕ, relates ğ to the tangent space metric, oab=diag(+---), by oab=ğ(ĕa,ĕb)=ğμνĕaμĕbν, which has the converse ğμν=oabĕμaĕνb because ĕaμĕνa=δνμ.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, ℳ4, in terms of the spacetime structures of V4, as(2.2)ds=eϑ=Ωμνĕν⊗ϑ̆μ=Ωbaĕa⊗ϑ̆b=eρ⊗ϑρ=ea⊗ϑa∈M4,
where Ωμν=πμρπρν is the world deformation tensor, e={ea=eaρeρ} is the frame field, and ϑ={ϑa=eρaϑρ} is the coframe field defined on ℳ4, such that ea⌋ϑb=δab, or in components, eaμeμb=δab; also the procedure can be inverted, eρaeaσ=δρσ. Hence the deformation tensor, Ωba=πcaπbc=Ωμνĕνaĕbμ, yields local tetrad deformations:(2.3)ec=πcaĕa,ϑc=πbcϑ̆b,eϑ=ea⊗ϑa=Ωbaĕa⊗ϑ̆b.
The components of the general spin connection then transform inhomogeneously under a local tetrad deformations (2.3):(2.4)ωbμa=πcaω̆dμcπbd+πca∂μπbc.
This is still a passive transformation, but with inverted factor ordering. The matrices π(x)∶=(πba)(x) are called first deformation matrices, and the matrices γcd(x)=oabπca(x)πdb(x),  second deformation matrices. The matrices πca(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, Λca, 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:(2.5)g=oabπcaπdbϑ̆c⊗ϑ̆d=γcdϑ̆c⊗ϑ̆d=oabϑa⊗ϑb.
In the first case, the contribution of the Christoffel symbols, constructed by the metric γab, reads(2.6)Γbca=12(C̆bca-γaa′γbb′C̆a′cb′-γaa′γcc′C̆a′bc′)+12γaa′(ĕc⌋dγba′-ĕb⌋dγca′-ĕa′⌋dγbc),
with C̆dfe representing the curls of the base members in the semi-Riemann space:(2.7)C̆dfe=ĕf⌋ĕd⌋C̆e=-ϑ̆e([ĕd,ĕf])=ĕdμĕfν(∂̆μĕνe-∂̆νĕμe)=-ĕμe[ĕd(ĕfμ)-ĕf(ĕdμ)],
where C̆a:=dϑ̆a=(1/2)C̆bcaϑ̆b∧ϑ̆c is the anfolonomity 2-form. The deformed metric can be split as follows [96]:(2.8)gμν(π)=Υ2(π)ğμν+γμν(π),
where Υ(π)=πaa, and γμν(π)=[γab-Υ2(π)oab]ĕμaĕνb. In the second case, we may write the commutation table for the anholonomic frame, {ea},(2.9)[ea,eb]=-12Cabcec,
and define the anholonomy objects:(2.10)Cbca=πeaπb-1dπc-1fC̆dfe+2πfaĕgμ(π[b-1g∂μπ[c-1f).
The usual Levi-Civita connection corresponding to the metric (2.8) is related to the original connection by the following relation:(2.11)Γρσμ=Γ̆ρσμ+Πρσμ,
provided that(2.12)Πρσμ=2gμνğν(ρ∇σ)Υ-ğρσgμν∇νΥ+12gμν(∇ργνσ+∇σγρν-∇νγρσ),
where the controvariant deformed metric, gνρ, is defined as the inverse of gμν, such that gμνgνρ=δμρ. Hence, the connection deformation Πρσμ 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∙):=(π•ba)(x∙), satisfies the following peculiar condition: (2.13)π•ca(x∙)∂∙μπ•b-1c(x∙)=ω̆bμa(x̆),
where ω̆bμa(x̆) 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, Γ∙ρσμ, is related to the corresponding spin connectionω∙bμa, through the inverse(2.14)Γ∙ρσμ=e∙aμ∂∙σe∙ρa+e∙aμω∙bσae∙ρb=e∙aμ∂∙σe∙ρa,
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:(2.15)ω′bμa=σcaω∙dμcσbd+σca∂μσbc,
such that(2.16)ωbμ(σ)a∶=ω′bμa=σca∂μσbc
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 through(2.17)Γρσμ=eaμ∂σeρa+eaμωbσ(π)aeρb=σaμ∂σσρa+σaμωbσ(σ)aσρb,
where we have σaμσμb=δab, also the procedure can be inverted, σaμσνa=δνμ, and (2.18)ωbμ(μ)a∶=ωbμa=πcaω̆dμcπdb+πca∂μπcb
is the spin connection. For our convenience, hereinafter the notation {ea(A),ϑ(A)b}(A=π,σ) will be used for general linear frames:(2.19){ea(A),ϑ(A)b}={(ea(π),ea(σ)),(ϑ(π)b,ϑ(σ)b)}≡{(ea,e∙a),(ϑb,ϑ∙b)},
where ea(A)⌋ϑ(A)b=δab, or in components, ea(A)μeμ(A)b=δab; also the procedure can be inverted, eρ(A)aea(A)σ=δρσ, provided that(2.20)ea(A)μ=(ea(A)μ,ea(σ)μ)≡(eaμ,σaμ).
Hence, the affine connection (2.17) can be rewritten in the abbreviated form:(2.21)Γρσ(A)μ=ea(A)μ∂σeρ(A)a+ea(A)μωbσ(A)aeρ(A)b.
Since the first deformation matrices π(x) and σ(x) are arbitrary functions, the inhomogeneously transformed general spin connections ω(π)(x) and ω(σ)(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 Nab(A) and the affine torsion 2-form T(A)a representing a translational misfit (for a comprehensive discussion see [114–118]. These, together with the curvature 2-form Sa(A)b, symbolically can be presented as [119, 120](2.22)(Nab(A),T(A)a,Sa(A)b)~D(A)(gab(A),ϑ(A)a,Γa(A)b),
where for a tensor-valued p-form density of representation type ρ(Lab), the GL(4,R)-covariant exterior derivative reads 𝒟(A):=d+Γa(A)bρ(Lab)∧ and Γa(A)b=Γμa(A)bdxμ is the general nonmetricity connection. This notation will be used instead of ωa(A)b=ωμa(A)bdxμ, such that Γμa(A)b=ea(A)νeσ(A)bΓμνσ+ea(A)ν∂μeν(A)b. In what follows, however, we may still maintain the former notation ωa(A)bto be referred to as corresponding connection constrained by the metricity condition. We may introduce the affine contortion 1-form Kab(A)=-Kab(A) given in terms of the torsion 2-form Ta(A)=ϑb(A)∧Kba(A). In tensor components we have Kμν(A)ρ=2Q(μν)(A)ρ+Qμν(A)ρ, where the torsion tensor Qμν(A)ρ=(1/2),Tμν(A)ρ=Γ[μν](A)ρ given with respect to a holonomic frame, dϑρ(A)=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:(2.23)Rμν(A)ab(ω(A))eρb(A)-Rρμνσ(Γ)eσ(A)a=0,
where(2.24)Rμν(A)ab(ω(A))=∂μω(A)νab-∂νωμ(A)ab+ωμ(A)acωνc(A)b-ων(A)acωμc(A)b,Rρμνσ(Γ)=∂μΓνρσ-∂νΓμρσ-ΓμρλΓνλσ+ΓνρλΓμλσ.
Hence, the relations between the scalar curvatures for an U4 manifold read(2.25)R(A)(ω(A))≡ea(A)μeb(A)νRμν(A)ab(ω(A))=R(g,Γ)≡gρνRρμνμ(Γ).
This means that the Lorentz and diffeomorphism invariant scalar curvature, R, becomes either a function of ea(A)μ 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 ea(A)μ 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 read(2.26)R∘≡Rcd∘(σ)∧ϑ∙c∧ϑ∙d=Rcd∘(π)∧ϑc∧ϑd.
To recover the TSSD-U4 theory, one can choose the EC Lagrangian, LEC, as(2.27)LEC=-12l2R(A)ab∧ηab(A)+12Λη(A)+12Nab(A)∧λab,
where Λ is the cosmological constant, Ra(A)b is the curvature tensor, λab is the Lagrange multiplier, and (1/2)R(A)ab∧ηab(A)=R(A)η(A). The η-basis is consisting in the Hodge dual of exterior products of tetrads by means of the Levi-Civita object: ηabcd∶=ϑ⋆abcd, which yields ηab∶=ϑ⋆ab=(1/2!)ηcdab∧ϑcd and η≔1⋆=(1/4!)ηabcdϑabcd, where we used the abbreviated notations for the wedge product monomials, ϑ(A)μνα⋯=ϑ(A)μ∧ϑ(A)ν∧ϑ(A)α∧⋯, and ⋆ denotes the Hodge dual. The variation of the total action(2.28)S=SEC+Sm(π),
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 ϑ(A)a, 1-form ω(A)ab 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), gives(2.29)(1)12Rca∘(A)∧ϑ(A)c+Ληa(A)=l2θa(A),(2)Θab(A)a′b′∧⋆Ta′b′(A)=l2⋆Σab(π),(3)δLm(π)δΨ=0,δLm(π)δΨ¯=0,
where ℓ is the Planck length,Θa′b′(A)ab(π(x),σ(x))=((∂ω(A)a′b′)/∂ω(π)ab), and ⋆Σab(π)=-⋆Σba(π) is the dual 3-form corresponding to the canonical spin tensor, which is identical with the dynamical spin tensor Sabc(π), namely,(2.30)⋆Σab(π)=Sab(π)μɛμναβϑ(π)ναβ,
provided that,(2.31)⋆T(A)ab∶=12⋆(Qa(A)∧eb(A))=Q(A)c∧ϑ(A)dɛcdab=12Qμν(A)c∧eα(A)dɛabcdϑ(A)μνα,
and that(2.32)Q(A)a=D(A)ϑ(A)a=dϑ(A)a+ωb(A)a∧ϑ(A)b.
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:(2.33)(1)G∘μν+Λgμν=l2θμν(A),(2)∂ων′(A)μ′ρ′∂ων(π)μρ(T(A))Tμ′ρ′(A)ν′=l2Sμρ(π)ν,
where G∘μν≡R∘μν-(1/2)gμνR∘ is Einstein’s tensor, and the modified torsion reads(2.34)Tμρ(A)ν∶=Qμρ(A)ν+δμνQρ(A)-δρνQμ(A).
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:(2.35)(1)G∘μν+Λgμν=l2θμν(σ),(2)Θν′μρ(σ)μ′ρ′ν(T(σ))Tμ′ρ′(σ)ν′=l2Sμρ(π)ν.
We may impose different physical constraints upon the spacetime deformation σ(x), which will be useful for the theory of electromagnetism and charged particles:(2.36)Θν′μρμ′ρ′ν(π(x),σ(x))≡Θν′μρμ′ρ′ν(T(σ))=2φ,σɛμρσν(T-1(σ))ν′μ′ρ′,
with φ as a scalar or pseudoscalar function of relevant variables. Here Θν′μρμ′ρ′ν(T(A))=((∂ων′(A)μ′ρ′)/∂ων(π)μρ(T(A))). Then we obtain(2.37)Tμρ(π)ν=Θν′μρμ′ρ′ν(π(x),σ(x))Tμ′ρ′(σ)ν′=2φ,σɛμρσν,
which recovers the term in the Lagrangian of pseudoscalar-photon interaction theory [41, 97–101], such that the nonmetric part of the Lagrangian can be put in the well-known form of the χ-g framework:(2.38)LI(π)NM=2(-g)1/2AνAμ,ρT(π)νμρ=4(-g)1/2φ,σɛσνμρAνAμ,ρ,(mod⁡div⁡),
where Fμν=Aμ,ν-Aν,μ has the usual meaning for electromagnetism. This is equivalent, up to integration by parts in the action integral (modulo a divergence), to the Lagrangian(2.39)LI(π)NM=(-g)1/2φɛσνμρFσνFμρ.
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.) reads(2.40)χσνμρ=(-g)1/2[12gσμgνρ-12gσρgμν+φɛσνμρ].
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 [105–107].


## 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:(3.1)M2=R(+)1⊕R(-)1.
The ingredient 1D-space RA1 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 is(3.2)g¯=g¯(e¯A,e¯B)ϑ¯A⊗ϑ¯B,
where ϑ¯A=dηA is the infinitesimal displacement. The basis e¯A at the point of interest in M2 consists of two real null vectors:(3.3)g¯(e¯A,e¯B)≡〈e¯A,e¯B〉=o*AB,(o*AB)=(0110).
The norm, id¯≡dη̂, given in this basis reads id¯=e¯ϑ¯=e¯A⊗ϑ¯A, 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 q0∈T1 and a spatial q1∈R1 variables qr(q0,q1)(r=0,1), such that(3.4)M2=R1⊕T1.
The norm, id¯, now can be rewritten in terms of displacement, dqr, as(3.5)id¯=dq̂=e0⊗dq0+e1⊗dq1,
where e0 and e1 are, respectively, the temporal and spatial basis vectors:(3.6)e0=12(e¯(+)+e¯(-)),e1=12(e¯(+)-e¯(-)),g¯(er,es)≡〈er,es〉=ors,(ors)=(100-1).
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 f:M2→M4 is defined to be an immersion—an embedding is a function that is a homeomorphism onto its image:(3.7)q0=12(η(+)+η(-))=t,q1=12(η(+)-η(-))=|x→|.
In fact, we assume that the particle has to be moving simultaneously in the parallel individualM2 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,(3.8)v(±)=dη(±)dq0=12(1±vq),vq=dq1dq0=|v→|=|dx→dt|=const≥0.
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 ofabsolute and relative states of the ingredient spaces RA1. The measure for these states is the very magnitude of the velocity components vA of the particle.Definition 3.1.
The ingredient spaceRA1of the individual MS companion of the particle is said to be in
(3.9)absolute(abs)stateifvA=0,relative(rel⁡)stateifvA≠0.Therefore, the MS can be realized either in thesemiabsolute state (rel, abs), or (abs, rel), or in the total relativestate (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 RA1 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 RA1 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 rateϱ(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 (f⃗) is the immediate cause of a deformation (distortion of the local internal properties) of MS: M2→ℳ̃2.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 Section4 we will determine the function ϱ(η,m,f⃗) 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 f⃗in=-f⃗ (see (2.19)–(5.35)) with respect to the components of inertial force f⃗in in terms of the relativistic force f⃗ 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 e⃗q lies along the net 3-acceleration (e⃗q∥e⃗a),(e⃗a=a⃗net/|a⃗net|) (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 Section2, without going into the subtleties, as applied to the 2D deformation M2→ℳ̃2. To start with, let V2 be 2D semi-Riemann space, which has at each point a tangent space, T̆η̆V2, spanned by the anholonomic orthonormal frame field, ĕ, as a shorthand for the collection of the 2-tuplet (ĕ(+),ĕ(-)), where ĕa=ĕaÃĕÃ, where the holonomic frame is given as ĕÃ=∂̆Ã. 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 “~”—Ã,B̃,…=(±), to denote the holonomic world indices related to either the space V2 or ℳ̃2. All magnitudes referred to the space, V2, will be denoted by an over “̆”. These then define a dual vector, ϑ̆, of differential forms, ϑ̆=(ϑ̆(+)ϑ̆(-)), as a shorthand for the collection of the ϑ̆b=ĕÃbϑ̆Ã, whose values at every point form the dual basis, such that ĕa⌋ϑ̆b=δab. In components ĕaÃĕÃb=δab. On the manifold, V2, the tautological tensor field, id̆, of type (1,1) can be defined which assigns to each tangent space the identity linear transformation. Thus for any point η̆∈V2, and any vector ξ̆∈T̆η̆V2, one has id̆(ξ̆)=ξ̆. In terms of the frame field, the ϑ̆a give the expression for id̆ as id̆=ĕϑ̆=ĕ(+)⊗ϑ̆(+)+ĕ(-)⊗ϑ̆(-), in the sense that both sides yield ξ̆ when applied to any tangent vector ξ̆ 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 {ĕa,ϑ̆b} will be used hereinafter for general linear frames. The holonomic metric can be defined in the semi-Riemann space, V2, as(4.1)ğ=ğÃB̃ϑ̆Ã⊗ϑ̆B̃=ğ(ĕÃ,ĕB̃)ϑ̆Ã⊗ϑ̆B̃,
with components ğÃB̃=ğ(ĕÃ,ĕB̃) in the dual holonomic base {ϑ̆Ã}. The anholonomic orthonormal frame field, ĕ, relates ğ to the tangent space metric, *oab, by *oab=ğ(ĕa,ĕb)=ğÃB̃ĕaÃĕbB̃, which has the converse ğÃB̃=o*abĕÃaĕB̃b because of the relation ĕaÃĕB̃a=δB̃Ã. 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 tensorΩ̃, 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 tensorΩ. Symbolically,(4.2)DC~(D̆,Y̆,Ω)⟶Ω̃.
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 (e¯A;ϑ¯A), at given point (p∈M2), undergoes the distortion transformations, conducted by (D̆,Y̆) and (D,Y), respectively, relating to V2 and ℳ̃2, recast in the form
(4.3)ĕÃ=D̆ÃBe¯B,ϑ̆Ã=Y̆BÃϑ¯B,eÃ=DÃBe¯B,ϑÃ=YBÃϑ¯B.(2) Then, the norm dη̂̃≡id of the infinitesimal displacement dη̃Ã on the general smooth differential 2D-manifold ℳ̃2 can be written in terms of the spacetime structures of V2 and M2:(4.4)id=eϑ=Ω̃ÃB̃ĕB̃⊗ϑ̆Ã=Ωbaĕa⊗ϑ̆b=eC̃⊗ϑC̃=ea⊗ϑa=ΩABe¯B⊗ϑ¯A∈M̃2,
where e={ea=eaC̃eC̃} is the frame field and ϑ={ϑa=eC̃aϑC̃} is the coframe field defined on ℳ̃2, such that ea⌋ϑb=δab. The deformation tensors Ω̃ÃB̃=πÃC̃πC̃B̃ and ΩAB imply(4.5)Ω̃ÃB̃=D̆ÃCΩCDY̆DB̃,ΩAB=YAC̃DC̃B,
provided that (4.6)DC̃A=πC̃B̃D̆B̃A,YBC̃=πÃC̃Y̆BÃ,
such that(4.7)eC̃=πC̃B̃ĕB̃≡∂̃C̃,ϑC̃=πÃC̃ϑ̆Ṽ≡dη̃C̃,η̃C̃∈U∈M̃2.
Hence the anholonomic deformation tensor, Ωba=πcaπbc=Ω̃ÃB̃ĕB̃aĕbÃ, yields local tetrad deformations:(4.8)ec=πcaĕa,ϑc=πbcϑ̆b,eϑ=ea⊗ϑa=Ωbaĕa⊗ϑ̆b.
The matrices π(η̃)∶=(πba)(η̃) are referred to as the first deformation matrices, and the matrices γcd(η̃)=*oabπca(η̃)πdb(η̃),   second deformation matrices. The matrices πca(η̃)∈GL(2,R)forallη̃, 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, Λsr, (r,s=1,0) leave the Minkowski metric invariant. A right multiplication of π(η̃) 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:(4.9)g=o*abπcaπdbϑ̆c⊗ϑ̆d=γcdϑ̆c⊗ϑ̆d=o*abϑa⊗ϑb.
In the first case, the contribution of the Christoffel symbols, constructed by the metric γab, reads(4.10)Γbca=12(C̆bca-γaa′γbb′C̆a′cb′-γaa′γcc′C̆a′bc′)+12γaa′(ĕc⌋dγba′-ĕb⌋dγca′-ĕa′⌋dγbc).
The deformed metric can be split as follows [96]:(4.11)gÃB̃(π)=Υ2(π)ğÃB̃+γÃB̃(π),
where Υ(π)=πaa, and(4.12)γÃB̃(π)=[γab-Υ2(π)o*ab]ĕÃaĕB̃b.
In the second case, we may write the commutation table for the anholonomic frame, {ea},(4.13)[ea,eb]=-12Cabcec,
and define the anholonomy objects:(4.14)Cbca=πeaπ-1bdπ-1cfC̆dfe+2πfaĕgÃ(π-1[bg∂Ãπ-1c]f).
The usual Levi-Civita connection corresponding to the metric (4.11) is related to the original connection by the following relation:(4.15)ΓC̃D̃Ã=Γ̆C̃D̃Ã+ΠC̃D̃Ã,
provided that(4.16)ΠC̃D̃Ã=2gÃB̃ğB̃(C̃∇D̃)Υ-ğC̃D̃gÃB̃∇B̃Υ+12gÃB̃(∇C̃γB̃D̃+∇D̃γC̃B̃-∇B̃γC̃D̃),
where the controvariant deformed metric, gB̃C̃, is defined as the inverse of gÃB̃, such that gÃB̃gB̃C̃=δÃC̃. That is, the connection deformation ΠC̃D̃Ã 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:(4.17)g=gÃB̃ϑÃ⊗ϑB̃=(Ω̃ÃB̃Ω̃C̃D̃)ğB̃D̃ϑ̆Ã⊗ϑ̆C̃=o*abϑa⊗ϑb=(ΩacΩbd)o*cdϑ̆a⊗ϑ̆b=γcdϑ̆c⊗ϑ̆d=(ΩACΩBD)o*CDϑ¯A⊗ϑ¯B.
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 constraints(5.1)DC̃A=D̆B̃A,Y̆BÃ=D̆BÃ,
and, therefore,(5.2)DC~(D,Ω)⟶Ω̃.
The (4.5), by virtue of (4.4) and (5.1), gives(5.3)Ω̃ÃB̃=D̆ÃCΩCDD̆DB̃=πÃB̃,YBC̃=Ω̃ÃC̃D̆BÃ,
where the deformation tensor, Ω̃ÃB̃, yields the partial holonomic frame transformations:(5.4)eC̃=ĕC̃,ϑC̃=Ω̃ÃC̃ϑ̆Ṽ,
or, respectively, the Ωba yields the partial local tetrad deformations:(5.5)ec=ĕc,ϑc=Ωbcϑ̆b,eϑ=ea⊗ϑa=Ωbaĕa⊗ϑ̆b.
Hence, (4.4) defines a diffeomorphism η̃Ã(η):M2→ℳ̃2:(5.6)eÃYBÃ=ΩBAe¯A,
where YBÃ=∂η̃Ã/∂ηB. The conditions of integrability, ∂AYBC̃=∂BYAC̃, and nondegeneracy, det|YBÃ|≠0, immediately define a general form of the flat-deformation tensorΩBA∶=DC̃A∂BΘC̃, where ΘC̃ 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 rateϱ(η,m,f) of instantaneously change of the measure vA of massive (m) test particle under the unbalanced net force (f):(5.7)e(+̃)(ϱ)=D(+̃)B(ϱ)e¯B=e¯(+)-ϱ(η,m,f)v(-)e¯(-),e(-̃)(ϱ)=D(-̃)B(ϱ)e¯B=e¯(-)+ϱ(η,m,f)v(+)e¯(+).
Clearly, these transformations imply a violation of the relation (3.3) (eÃ2(ϱ)≠0) for the null vectors e¯A. Now we can use (4.4) to observe that for dual vectors of differential forms ϑ=(ϑ(+̃)ϑ(-̃)) and ϑ¯=(ϑ¯(+)ϑ¯(-)) we may obtain(5.8)ϑ=(Ω(+)C〈e(+̃),e¯C〉Ω(-)C〈e(+̃),e¯C〉Ω(+)C〈e(-̃),e¯C〉Ω(-)C〈e(-̃),e¯C〉)ϑ¯.
We parameterize the tensor ΩBA in terms of the parameters τ1 and τ2 as(5.9)Ω(+)(+)=Ω(-)(-)=τ1(1+τ2ϱ¯2),Ω(+)(-)=-τ1(1-τ2)ϱv(-),Ω(-)(+)=τ1(1-τ2)ϱv(+),
where ϱ¯2=v2ϱ2,v2=v(+)v(-)=1/2γq2 and γq=(1-vq2)-1/2. Then, the relation (5.8) can be recast in an alternative form:(5.10)ϑ=τ1(1-τ2ϱv(+)τ2ϱv(-)1)ϑ¯.
Suppose that a second observer, who makes measurements using a frame of reference S̃(2) which is held stationary in deformed/distorted space ℳ̃2, uses for the test particle the corresponding spacetime coordinates q̃r̃((q̃0̃,q̃1̃)≡(t̃,q̃)). The (4.4) can be rewritten in terms of spacetime variables as(5.11)id=eϑ≡dq̂̃=ẽ0⊗dt̃+ẽq⊗dq̃,
where ẽ0 and ẽq are, respectively, the temporal and spatial basis vectors:(5.12)ẽ0(ϱ)=12[e(+̃)(ϱ)+e(-̃)(ϱ)],ẽq(ϱ)=12[e(+̃)(ϱ)-e(-̃)(ϱ)].
The transformation equation for the coordinates, according to (5.10), becomes(5.13)ϑ(±̃)=τ1(ϑ¯(±)∓τ2ϱv(±)ϑ¯(∓))=τ1(v(±)∓τ2ϱv2)dt,
which gives the general transformation equations for spatial and temporal coordinates as follows (e⃗q≡e1,q≡q1):(5.14)dt̃=τ1dt,dq̃=τ1[dq(1+τ2ϱvq2)-τ2ϱ2dt]=τ1(dq-τ2ϱ2γq2dt).
Hence, the general metric (4.17) in ℳ̃2 reads
(5.15)g≡ds̃q2=gr̃s̃dq̃r̃⊗dq̃s̃=[(Ω(+)(+))2+Ω(-)(+)Ω(+)(-)]dsq2+Ω(+)(+)(Ω(-)(+)+Ω(+)(-))(dt⊗dt+dq⊗dq)-2Ω(+)(+)(Ω(-)(+)-Ω(+)(-))dt⊗dq,
provided that(5.16)g0̃0̃=(1+ϱvq2)2-ϱ22,g1̃1̃=-(1-ϱvq2)2+ϱ22,g1̃0̃=g0̃1̃=-2ϱ.
The difference of the vector, dq̂∈M2 (3.5), and the vector, dq̂̃∈ℳ̃2 (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 vq≃0, as(5.17)dt̃=τ1dt,dq̃≃τ1(dq-τ2ϱ2dt),
thereby(5.18)Ω(+)(+)=Ω(-)(-)=τ1(1+τ2ϱ¯2),Ω(-)(+)=-Ω(+)(-)=τ1(1-τ2)ϱ¯.
Then (5.17) becomes conventional transformation equations to accelerated (anet≠0) axes if we assume that d(τ2ϱ)/2dt=anet and τ1(vq≃0)=1, where anet is a magnitude of proper net acceleration. In high-velocity limit vq≃1, ϱ¯≃0(dη(-)=v(-)dt≃0,v(+)≃v≃2), we have(5.19)Ω(+)(+)=Ω(-)(-)=τ1,Ω(+)(-)=0,Ω(-)(+)=τ1(1-τ2)2ϱ,
and so (5.14) and (5.15), respectively, give(5.20)dt̃=τ1dt≃τ1dq≃dq̃,(5.21)ds̃q2≃[(1+ϱ2)2-ϱ22]dt̃⊗dt+[-(1-ϱ2)2+ϱ22]dq̃⊗dq̃-22ϱdt̃⊗dq̃≃τ12dsq2=0.
To this end, the inertial effects become zero. Let a⃗net be a local net 3-acceleration of an arbitrary observer with proper linear 3-acceleration a⃗ and proper 3-angular velocity ω⃗ measured in the rest frame:(5.22)a⃗net=du⃗ds=a⃗∧u⃗+ω⃗×u⃗,
where u is the 4-velocity. A magnitude of a⃗net can be computed as the simple invariant of the absolute value |du/ds| as measured in rest frame:(5.23)|a|=|duds|=(dulds,dulds)1/2.
Following [57, 58], let us define an orthonormal frame eâ, carried by an accelerated observer, who moves with proper linear 3-acceleration and a⃗(s) and proper 3-rotation ω⃗(s). Particular frame components are denoted by hats, 0̂,1̂,andsoforth. Let the zeroth leg of the frame e0̂ 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 eî, orthogonal to e0̂, also by (s). The spatial triad eî rotates with proper 3-rotation ω⃗(s). The 4-velocity vector naturally undergoes Fermi-Walker transport along the curve C, which guarantees that e0̂(s) will always be tangent to C determined by xl=xl(s):(5.24)deâds=-Ωeâ,
where the antisymmetric rotation tensor Ω splits into a Fermi-Walker transport part ΩFW and a spatial rotation part ΩSR:(5.25)ΩFWlk=aluk-akul,ΩSRlk=umωnɛmnlk.
The 4-vector of rotation ωl is orthogonal to 4-velocity ul, therefore, in the rest frame it becomes ωl(0,ω⃗), and ɛmnlk is the Levi-Civita tensor with ɛ0123=-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 as(5.26)a⃗abs≡e⃗qd(τ2ϱ)2dsq=e⃗q|de0̂ds|=e⃗q|a|,
where we choose the system S(2) in such a way as the axis e⃗q lies along the net 3-acceleration (e⃗q∥e⃗a),(e⃗a=a⃗net/|a⃗net|). Hereinafter, we may simplify the flat-deformation tensor ΩAB by setting τ2=1, such that (5.9) becomes(5.27)Ω(+)(+)=Ω(-)(-)≡Ω(ϱ¯)=1+ϱ¯2,Ω(+)(-)=Ω(-)(+)=0,
and the general metric (4.17) in ℳ̃2 reads ds̃q2=Ω2(ϱ¯)dsq2. Hence (5.26) gives(5.28)ϱ=2∫0sq|a|dsq′.
Combining (5.14) and (5.26), we obtain the key relation between a so-called inertial acceleration, arisen due to the curvature of MS,(5.29)a⃗in=e⃗aain,ain=d2q̃ds̃q2=-Γr̃s̃1(ϱ)dq̃r̃ds̃qdq̃s̃ds̃q=12(d2η̃(+)ds̃q2-d2η̃(-)ds̃q2),
and a local absolute acceleration as follows:(5.30)Ω2(ϱ¯)γqa⃗in=-a⃗abs,
where Γr̃s̃1(ϱ) are the Christoffel symbols constructed by the metric (5.16). Then (5.30) provides a quantitative means for the inertial forcef⃗(in):(5.31)f⃗(in)=ma⃗in=-mΓr̃s̃1(ϱ)dq̃r̃ds̃qdq̃s̃ds̃q=-ma⃗absΩ2(ϱ¯)γq.
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):(5.32)fl(f0,f⃗)=md2xlds2=Λkl(v⃗)Fk.
Here Fk(0,F⃗) is the force defined in the rest frame of the test particle, and Λkl(v⃗) is the Lorentz transformation matrix (i,j=1,2,3):(5.33)Λji=δij-(γ-1)vivj|v→|2,Λi0=γvi,
where γ=(1-v⃗2)-1/2. So(5.34)|a|=1m|fl|=1m(flfl)1/2=1mγ|f→|,
and hence (5.31), (5.26), and (5.34) give(5.35)f→(in)=-1Ω2(ϱ¯)γqγ[F→+(γ-1)v→(v→⋅F→)|v→|2].
At low velocities vq≃|v⃗|≃0 and tiny accelerations we usually experience, one has Ω(ϱ¯)≃1; therefore (5.35) reduces to the conventional nonrelativistic law of inertia:(5.36)f⃗(in)=-ma⃗abs=-F⃗.
At high velocities vq≃|v⃗|≃1 (Ω(ϱ¯)≃1), if (v⃗·F⃗)≠0, the inertial force (5.35) becomes(5.37)f⃗(in)≃-1γe⃗v(e⃗v⋅F⃗),
and, in agreement with (5.21), it vanishes in the limit of the photon (|v⃗|=1,m=0). Thus, it takes force to disturb an inertia state, that is, to make the absolute acceleration (a⃗abs≠0). 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 [59–66], 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 S̃(2), which is held stationary in the deformed/distorted space ℳ̃2≡V2(ϱ)(ϱ≠0), 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 eâ (5.24), carried by an accelerated observer, with the over “breve” such that(6.1)ĕâ=e¯âμe¯μ=ĕâμĕμ,ϑ̆b̂=e¯μb̂ϑ¯μ=ĕμb̂ϑ̆μ,
with e¯μ=∂μ=∂/(∂x^μ),ĕμ=∂̆μ=∂/∂x̆μ, and ϑ¯μ=dxμ,ϑ̆μ=dx̆. Here, following [58, 64], we introduced a geodesic coordinate system x̆μ—coordinates relative to the accelerated observer (laboratory coordinates)—in the neighborhood of the accelerated path. The coframe members {ϑ̆b̂} are the objects of dual counterpart: ĕâ⌋ϑ̆b̂=δab. We choose the zeroth leg of the frame, ĕ0̂, 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 e¯âμ reads ημνe¯âμe¯b̂ν=oâb̂=diag(+---). The antisymmetric acceleration tensor Φab [64–68, 121–125] is given by(6.2)Φab∶=e¯μb̂de¯âμds=e¯μb̂uλ∇̆λe¯âμ=u⌋Γ̌ab,
provided that Γ̆ab=Γ̆aμbdx̆μ, where Γ̆aμb 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=-ɛijkωk. The invariants constructed out of Φab establish the acceleration scales and lengths. The hypothesis of locality holds for huge proper acceleration lengths |I|-1/2≫1 and |I*|-1/2≫1, where the scalar invariants are given by I=(1/2)ΦabΦab=-a⃗2+ω⃗2 and I*=(1/4)Φab*Φab=-a⃗·ω⃗ (Φab*=ɛabcdΦcd) [64–66, 121–125]. 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 x̆μ, where xμ and x̆μ are connected via x̆0=s and(6.3)xμ=zμ(s)+x̆ie¯îμ(s).
Let q̆r(q̆0,q̆1) be coordinates relative to the accelerated observer in the neighborhood of the accelerated path in MS, with spacetime components implying(6.4)dq̆0=dx̆0,dq̆1=|dx̆→|,ĕ⃗=dx̆→dq̆1=dx̆→|dx̆→|,ĕ→⋅ĕ→=1.
As long as a locality assumption holds, we may describe, with equal justice, the event at xμ (6.3) to be at point q̆r, such that xμ and q̆r, in full generality, are connected via q̆0=s and(6.5)xμ=zqμ(s)+q̆1β¯1̂μ(s),
where the displacement vector from the origin reads dzqμ(s)=β¯0̂μdq̆0, and the components β¯r̂μ can be written in terms of e¯âμ. Actually, from (6.3) and (6.5) we may obtain(6.6)dxμ=dzqμ(s)+dq̆1β¯1̂μ(s)+q̆1dβ¯1̂μ(s)=[β¯0̂μ(1+q̆1φ̆0)+β¯1̂μq̆1φ̆1]dq̆0+β¯1̂μdq̆1≡dzμ(s)+dx̆ie¯îμ(s)+x̆ide¯îμ(s)=[e¯0̂μ(1+x̆iΦi0)+e¯ĵμx̆iΦij)]dx̆0+e¯îμdx̆i,
where dβ¯1̂μ(s) is written in the basis β¯âμ as dβ¯1̂μ=(φ̆0β¯0̂μ+φ̆1β¯1̂μ)dq̆0. Equation (6.6) holds by identifying(6.7)β¯0̂μ(1+q̆1φ̆0)≡e¯0̂μ(1+x̆iΦi0),β¯1̂μq̆1φ̆1≡e¯ĵμx̆iΦij,β¯1̂μdq̆1≡e¯îμdx̆i.
Choosing β¯0̂μ≡e¯0̂μ, we have then(6.8)q̆1φ̆0=x̆iΦi0,β¯1̂μ=e¯îμĕi,q̆1φ̆1=x̆iΦijĕj-1,
with ĕjĕi-1=δij. Consequently, (6.6) yields the standard metric of semi-Riemannian 4D background space V4(0), in noninertial system of the accelerating and rotating observer, computed on the base of hypothesis of locality:(6.9)ğ=ημνdxμ⊗dxν=[(1+a⃗⋅x̆⃗)2+(ω⃗⋅x̆⃗)2-(ω⃗⋅ω⃗)(x̆⃗⋅x̆⃗)]dx̆0⊗dx̆0-2(ω⃗∧x̆⃗)⋅dx̆⃗⊗dx̆0-dx̆⃗⊗dx̆⃗.
This metric was derived by [59] and [63], in agreement with [99] and [62] (see also [64–66]).  We see that the hypothesis of locality leads to the 2D semi-Riemannian MS space: V2(0) with the incomplete metric ğ(ϱ=0):(6.10)ğ=[(1+q̆1φ̆0)2-(q̆1φ̆1)2]dq̆0⊗dq̆0-2(q̆1φ̆1)dq̆1⊗dq̆0-dq̆1⊗dq̆1.
Therefore, our strategy now is to deform the metric (6.10) by carrying out an additional deformation of semi-Riemannian 4D background space V4(0)→ℳ̃4≡V4(ϱ), which, as a corollary, will recover the complete metric g(ϱ≠0) (5.15) of the distorted MS-V2(ϱ). According to (2.3), this means that we should find the first deformation matrices, π(ϱ)∶=(πâb̂)(ϱ), which yield the local tetrad deformations:(6.11)eĉ=πĉâĕâ,ϑĉ=πb̂ĉϑ̆b̂,eϑ=eâ⊗ϑâ=Ωb̂aĕâ⊗ϑ̆b̂,
where Ωb̂â(ϱ)=πĉâ(ϱ)πb̂ĉ(ϱ) is referred to as the anholonomic deformation tensor and that the resulting deformed metric of the space V4(ϱ) can be split as(6.12)gμν(ϱ)=Υ2(ϱ)ğμν+γμν(ϱ),
provided that(6.13)γμν(ϱ)=[γâb̂-Υ2(ϱ)oâb̂]ĕμâĕνb̂,γĉd̂=oâb̂πĉâπd̂b̂,
where Υ(ϱ)=πââ(ϱ) and γâb̂(x̆) are the second deformation matrices. Let the Latin letters r̂,ŝ,…=0,1 be the anholonomic indices referred to the anholonomic frame er̂=er̂s∂s̃, defined on the V2(ϱ), with ∂s̃=∂/∂q̃s̃ as the vectors tangent to the coordinate lines. So, a smooth differential 2D-manifold V2(ϱ) has at each point q̃s a tangent space T̃q̃V2(ϱ), spanned by the frame, {er̂}, and the coframe members ϑr̂=esr̂dq̃s̃, which constitute a basis of the covector space T̃q̃⋆V2(ϱ). All this nomenclature can be given for V2(0) too. Then, we may calculate corresponding vierbein fields ĕrŝ and erŝ from (6.14)ğrs=ĕrr′̂ĕss′̂or′̂s′̂,gr̃s̃=err′̂ess′̂or′̂s′̂,
with ğrs and gr̃s̃ given by (6.10) and (5.16), respectively. Hence(6.15)ĕ00̂=1+a⃗⋅x̆⃗,ĕ01̂=ω⃗∧x̆⃗,ĕ10̂=0,ĕ11̂=1,e00̂=1+ϱvq2,e01̂=ϱ2,e10̂=-ϱ2,e11̂=1-ϱvq2.
Since a distortion of MS may affect only the MS part of the components β¯r̂μ, without relation to the background spacetime part, therefore, a deformation V4(0)→V4(ϱ) is equivalent to a straightforward generalization β¯r̂μ→βr̂μ, where(6.16)βr̂μ=Er̂ŝβ¯ŝμ,Er̂ŝ∶=er̂r′ĕr′ŝ.
Consequently, (6.16) gives a generalization of (6.3) as(6.17)xμ⟶x(ϱ)μ=z(ϱ)μ(s)+x̆ieîμ(s),
provided that, as before, x̆μ denotes the coordinates relative to the accelerated observer in 4D background space V4(ϱ), and according to (6.7), we have(6.18)e0̂μ=β0̂μ,eîμ=β1̂μĕi-1.
A displacement vector from the origin is then dzϱμ(s)=e0̂μdx̆0. Combining (6.16) and (6.18), and inverting erŝ (6.15), we obtain eâμ=πâb̂(ϱ)e¯b̂μ, where(6.19)π0̂0̂(ϱ)≡(1+ϱ22γq2)-1(1-ϱvq2)(1+a⃗⋅x̆⃗),π0̂î(ϱ)≡-(1+ϱ22γq2)-1ϱ2ĕi(1+a⃗⋅x̆⃗),πî0̂(ϱ)≡(1+ϱ22γq2)-1[(ω⃗∧x̆⃗)(1-ϱvq2)-ϱ2]ĕi-1,πîĵ(ϱ)=δijπ(ϱ),π(ϱ)≡(1+ϱ22γq2)-1[(ω⃗∧x̆⃗)ϱ2+1+ϱvq2].
Thus,(6.20)dxϱμ=dzϱμ(s)+dx̆ieîμ+x̆ideîμ(s)=(τb̂dx̆0+πîb̂dx̆i)e¯b̂μ,
where(6.21)τb̂≡π0̂b̂+x̆i(πîâΦab+dπîb̂ds).
Hence, in general, the metric in noninertial frame of arbitrary accelerating and rotating observer in Minkowski spacetime is(6.22)g(ϱ)=ημνdxϱμ⊗dxϱν=Wμν(ϱ)dx̆μ⊗dx̆ν,
which can be conveniently decomposed according to(6.23)W00(ϱ)=π2[(1+a⃗⋅x̆⃗)2+(ω⃗⋅x̆⃗)2-(ω⃗⋅ω⃗)(x̆⃗⋅x̆⃗)]+γ00(ϱ),W0i(ϱ)=-π2(ω⃗∧x̆⃗)i+γ0i(ϱ),Wij(ϱ)=-π2δij+γij(ϱ),
and also(6.24)γ00(ϱ)=π[(1+a⃗⋅x̆⃗)ζ0-(ω⃗∧x̆⃗)⋅ζ⃗]+(ζ0)2-(ζ⃗)2,γ0i(ϱ)=-πζi+τ0̂πî0̂,γij(ϱ)=πî0̂πĵ0̂,ζ0=π(τ0̂-1-a⃗⋅x̆⃗),ζ⃗=π(τ⃗-ω⃗∧x̆⃗).
As we expected, according to (6.22)–(6.24), the matric g(ϱ) is decomposed in the form of (4.11):(6.25)g(ϱ)=π2(ϱ)ğ+γ(ϱ),
where γ(ϱ)=γμν(ϱ)dx̆μ⊗dx̆ν and Υ(ϱ)=πââ(ϱ)=π(ϱ). In general, the geodesic coordinates are admissible as long as(6.26)(1+a⃗⋅x̆⃗+ζ0π)2>(ω⃗∧x̆⃗+ζ⃗π)2.
Equations (6.9) and (6.22) say that the vierbein fields with entries ημνe¯âμe¯b̂ν=oâb̂ and ημνeâμeb̂ν=γâb̂ lead to the relations(6.27)ğ=oâb̂ϑ̆â⊗ϑ̆b̂,g=oâb̂ϑâ⊗ϑb̂=γâb̂ϑ̆â⊗ϑ̆b̂=(ΩâĉΩb̂d̂oĉd̂)ϑ¯â⊗ϑ¯b̂,
and that (6.6) and (6.20) readily give the coframe fields:(6.28)ϑ̆b̂=e¯μb̂dxμ=ĕμb̂dx̆μ,ĕ0b̂=N0b,ĕib̂=Nib,ϑb̂=e¯μb̂dxϱμ=eμb̂dx̆μ=πâb̂ϑ̆â,e0b̂=τb̂,eib̂=πîb̂,
where N00=N≡(1+a⃗·x̆⃗),Ni0=0,N0i=Ni≡(ω⃗·x̆⃗)i,andNij=δij. In the standard (3+1)-decomposition of spacetime, N and Ni are known as lapse function and shift vector, respectively [126, 127]. Hence, we may easily recover the frame field eâ=eâμĕμ=πâb̂ĕb̂ by inverting (6.28):(6.29)e0̂=ππτ0̂-πk̂0̂τk̂ĕ0-τîπτ0̂-πk̂0̂τk̂ĕi,eî=-πî0̂πτ0̂-πk̂0̂τk̂ĕ0+π-1[δij+τjπî0̂πτ0̂-πk̂0̂τk̂]ĕj.
A generalized transport for deformed frame eâ, which includes both the Fermi-Walker transport and distortion of MS, can be written in the following form:(6.30)deâμds=Φ̃abeb̂μ,
where a deformed acceleration tensorΦ̃ab concisely is given by(6.31)Φ̃=(dln⁡πds)+πΦπ-1.
Although the results (6.29)–(6.31) are obtained in the framework of purely classical physics, nevertheless on this base we may straightforwardly put the special-relativistic Dirac equation into a noninertial reference frame by standard method similar to [59]. But we will forbear to write it out here as it is somewhat lengthy and evidently irrelevant to the problem in quest in this paper. It will be interesting topic for another publication.


## 7. Involving the Background Semi-Riemann SpaceV4: Justification for the Introduction of the WPE

We can always choosenatural coordinatesXα(T,X,Y,Z)=(T,X⃗) with respect to the axes of the local free-fall coordinate frame S4(l) in an immediate neighbourhood of any spacetime point (x̆p)∈V4 in question of the background semi-Riemann space, V4, over a differential region taken small enough so that we can neglect the spatial and temporal variations of gravity for the range involved. The values of the metric tensor ğμν and the affine connection Γ̆μνλ at the point (x̆p) are necessarily sufficient information for determination of the natural coordinates Xα(x̆μ) in the small region of the neighbourhood of the selected point [128]. Then the whole scheme outlined in Section 4 will be held in the frame S4(l). The relativistic gravitational force f̆gμ(x̆) exerted on the test particle of the mass (m) is given by(7.1)f̆gμ(x̆)=md2x̆μds̆2=-mΓ̆νλμ(a)dx̆νds̆dx̆λds̆.
The frame S4(l) will be valid if only the gravitational force given in this coordinate frame(7.2)fg(l)α=∂Xα∂x̃μfgμ
could be removed by the inertial force, whereas, as before, the two systems S2 and S4(l) can be chosen in such a way as the axis e⃗q of S(2) lies (e⃗q=e⃗f) along the acting net force f⃗=f⃗(l)+f⃗g(l), where f⃗(l) is the SR value of the unbalanced relativistic force other than gravitational in the frame S4(l), while the time coordinates in the two systems are taken the same, q0=t=X0=T. Then (5.34) now can be replaced by(7.3)12d(τ2ϱ)dsq=1m|f(l)α+fg(l)α|,
and according to (5.31), the general inertial force reads(7.4)f⃗̆(in)=ma⃗in=-ma⃗absΩ2(ϱ¯)γq=-e⃗fΩ2(ϱ¯)γq|f(l)α-m∂Xα∂x̆σΓ̆μνσdx̆μdSdx̆νdS|.
Despite totally different and independent sources of gravitation and inertia, at f(l)α=0, (7.4) establishes the independence of free-fall (vq=0) trajectories of the mass, internal composition, and structure of bodies. This furnishes a justification for the introduction of the WPE. A remarkable feature is that although the inertial force has a nature different than the gravitational force, nevertheless both are due to a distortion of the local inertial properties of, respectively, 2D MS and 4D-background space. The nonvanishing inertial force acting on the photon of energy hν and that of effective mass (hν/c2), after inserting units (h,c) which so far was suppressed, can be obtained from (7.4) (f(l)α=0) as(7.5)f⃗̆(in)=-(hνc2Ω2(ϱ¯))e⃗f|∂Xα∂x̆σΓμνσdx̆μdTdx̆νdT|=-(hνc2Ω2(ϱ¯))e⃗f|(d2t̃dT2)dXαdt̃+(dt̃dT)2∂Xα∂x̆iduidt̃|,
provided that e⃗f=(X⃗/|X⃗|), vq=(e⃗f·u⃗̆)=|u⃗̆|,(γq=γ) where u⃗̆ is the velocity of a photon, (du⃗̆/dt̃) is the acceleration, and, ğμν(dx̆μ/dT)⊗(dx̆ν/dT)=0. Note that Nordtvedt, Will, and others [129–132] were led to provide rigorous underpinnings to the operational significance of various theories, especially in solar system context, developing the parameterized post-Newtonian (PPN) formalism as a theoretical standard for expressing the predictions of relativistic gravitational theories in terms which could be directly related to experimental observations. To obtain some feeling for this, in the PPN approximation we may calculate the inertial force exerted on the photon in a gravitating system of particles that are bound together by their mutual gravitational attraction to order v¯2~GNM¯/r¯ of a small parameter, where v¯,M¯, and r¯ are typically the average values of their velocities, masses, and separations, respectively. To this aim, we may expand the metric tensor to the following order: ğ00=1+g200+g400+⋯,ğij=-δij+g2ij+g4ij+⋯,ği0=g3i0+g5i0+⋯, where gNμν denotes the term of order v¯N. Taking into account the standard expansions of the affine connection [128]: Γ̆μνσ=Γμνσ2+Γμνσ4+⋯ for the components Γ̆00i,Γ̆jki, andΓ̆0i0, and that Γ̆μνσ=Γμνσ3+Γμνσ5+⋯ for the components Γ̆0ji,Γ̆000,andΓ̆ij0, where Γ00i2=Γ0i02=-(1/2)(∂g200/∂x̆i), and so forth; hence to the required accuracy we obtain(7.6)f⃗̆(in)(2)=-(hνc2)e⃗f|(∂Xα∂x̆σ)1(d2x̆σdT2)2|=-(hνc2)(du⃗̆dt̃)2=-(hνγc2)[-2∇⃗ϕ+4u⃗̆(u⃗̆⋅∇⃗ϕ)+O(v¯3)],
where ϕ is the Newton potential, such that g200=2ϕ,g2ij=2δijϕ, and |u⃗̆|=1+2ϕ+O(v¯3).


## 8. RTI in the Background Post-Riemannian Geometry

According to (2.21) and (2.22), if the nonmetricity tensor Nλμν=-𝒟λgμν≡-gμν;λ does not vanish, the general formula for the affine connection written in the spacetime components is (also see [118])(8.1)Γμνρ=Γ∘μνρ+Kμνρ-Nμνρ+12N(μν)ρ,
where the metric alone determines the torsion-free Levi-Civita connection Γ∘μνρ, and Kμνρ∶=2Q(μν)ρ+Qμνρ is the non-Riemann part—the affine contortion tensor. The torsion, Qμνρ=(1/2)Tμνρ=Γ[μν]ρ given with respect to a holonomic frame, dϑρ=0, is a third-rank tensor, antisymmetric in the first two indices, with 24 independent components.


### 8.1. The Principle of Equivalence in the RC Space

The RC manifold,U4, is a particular case of general metric-affine manifold M̃4, restricted by the metricity condition Nab=0, when a nonsymmetric linear connection, Γ, is said to be metric compatible. To avoid any possibility of confusion, here and throughout we again use the first half of Latin alphabet (a,b,c,…=0,1,2,3 rather than (±)) now to denote the anholonomic indices referred to the tangent space, which is endowed with the Lorentzian metric oab∶=diag(+---). The space, U4, also locally has the structure of M4, as has been first pointed out by [133] and developed by [134–137]. In the case of the RC space there also exist orthonormal reference frames which realize an “anholonomic” free-fall elevator. In Hartley’s formulation [137], this reads as follows. For any single pointP∈U4, there exist coordinates{xμ}and an orthonormal frame{ea}in a neighborhood ofPsuch that(8.2)ea=δaμ∂xμΓab=0atP,
where Γab are the connection 1-forms referred to the frame {ea}. Therefore the existence of torsion does not violate the PE. Note that since ∇g=0 holds in U4, the arguments showing that g can be transformed to o at any point P in U4 are the same as in the case of V4, while the treatment of the connection must be different: the antisymmetric part of ω can be eliminated only by a suitable choice for the relative orientation of neighbouring tetrads. Actually, let us choose new local coordinates at P, dxμ→dxa=eμadxμ, related to an inertial frame. Then,(8.3)gab′=eaμebνgμν=oab,Γac′b=eμbeaνecλ(Δνλμ+Kνλμ)≡ecλωaλb.
As it is argued in [138], the metricity condition ensures that this can be done consistently at every point in spacetime. Suppose that we have a tetrad {ea(x)} at the point P and a tetrad {ea(x+dx)} at another point in a neighbourhood of P; then, we can apply a suitable Lorentz rotation to ea(x+dx), so that it becomes parallel to ea(x). Given a vector v at P, it follows that the components vc=v·ec do not change under parallel transport from x to x+dx, provided that the metricity condition holds. Hence, the connection coefficients ωμab(x) at P, defined with respect to this particular tetrad field, vanish: ωμab(P)=0. This property is compatible with gab′=oab, since Lorentz rotation does not influence the value of the metric at a given point. In more general geometries, where the symmetry of the tangent space is higher than the Poincare group, the usual form of the PE is violated and local physics differs from SR.


### 8.2. The Generalized Inertial Force Exerted on the Extended Spinning Body in theU4

We now compute the relativistic inertial force for the motion of the matter, which is distributed over a small region in theU4 space and consists of points with the coordinates xμ, forming an extended body whose motion in the space, U4, is represented by a world tube in spacetime. Suppose that the motion of the body as a whole is represented by an arbitrary timelike world line γ inside the world tube, which consists of points with the coordinates X̃μ(τ), where τ is the proper time on γ. Define(8.4)δxμ=xμ-X̃μ,δx0=0,uμ=dX̃μds.
The Papapetrou equation of motion for the modified momentum (see [118, 139–143]) is(8.5)D∘ΘνDs=-12R∘μσρνuμJσρ-12NμρλKμρλ:ν,
where Kνλμ is the contortion tensor,(8.6)Θν=Pν+1u0Γ∘μρν(uμJρ0+N0μρ)-12u0KμρνNμρ0
is referred to as the modified 4-momentum, Pλ=∫τλ0dΩ is the ordinary 4-momentum, dΩ:=dx4, and the following integrals are defined:(8.7)Mμρ=u0∫τμρdΩ,Mμνρ=-u0∫δxμτνρdΩ,Nμνρ=u0∫sμνρdΩ,Jμρ=∫(δxμτρ0-δxρτμ0+sμρ0)dΩ=1u0(-Mμρ0+Mρμ0+Nμρ0),
where τμρ is the energy-momentum tensor for particles, and sμνρ is the spin density. The quantity Jμρ is equal to ∫(δxμτkl-δxρτμλ+sμρλ)dSλ taken for the volume hypersurface, so it is a tensor, which is called the total spin tensor. The quantity Nμνρ is also a tensor. The relation δx0=0 gives M0νρ=0. It was assumed that the dimensions of the body are small, so integrals with two or more factors δxμ multiplying τνρ and integrals with one or more factors δxμ multiplying sνρλ can be neglected. The Papapetrou equations of motion for the spin (see [118, 139–143]) are(8.8)D∘DsJλν=uνΘλ-uλΘν+KμρλNνμρ+12KμρλNμνρ-KμρνNλμρ-12KμρνNμρλ.
Calculating from (8.5) the particle 4-acceleration is(8.9)1mfgμ(x)=d2xμds̃2=-Γνλμ[uνuλ+1u0Γ∘νρμ(uνJρ0+N0νρ)]+12u0KνρμNνρ0-12R∘νσρμuνJσρ-12NνρλKνρλ:μ.
Thus, the relativistic inertial force, exerted on the extended spinning body moving in the RC space U4, can be found to be(8.10)f⃗(in)(x)=ma⃗in(x)=-ma⃗abs(x)Ω2(ϱ¯)γq=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂xμ×[Γ∘νλμuνuλ+1u0Γ∘νρμ(uνJρ0+N0νρ)-12u0KνρμNνρ0+12R∘νσρμuνJσρ+12NνρλKνρλ:μ]|.
In particular, if the spin density vanishes, sμνρ=0, from the conservation law we get then τμρ=τρμ, Mμρ=Mρμ, Mμνρ=Mμρρ, Nμνρ=0, and (8.11)Jμρ=Lμρ=∫(δxμτρ0-δxρτμ0)dΩ=1u0(-Mμρ0+Mρμ0),
where Lμρ is the angular momentum tensor. The modified 4-momentum (8.6) reduces to(8.12)Θν=Pν+D∘DsLνλuλ.
Equation (8.8) can be recast in the following form:(8.13)D∘DsLλν=uνΘλ-uλΘν,
while (8.5) becomes(8.14)D∘ΘνDs=-12R∘νσρμuμLσρ,
which give the relativistic inertial force exerted on the spinless extended body moving in the RC space U4 as follows:(8.15)f⃗(in)(x)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂xμ[Γ∘νλμuνuλ+1u0Γ∘νρμuνLρ0+12R∘νσρμuνLσρ]|.
If the body is not spatially extended, then it is referred to as a particle. The corresponding condition δxα=0 gives Mμνρ=0,andLμρ=0. Therefore (uλ/u0)Nμν0-Nμνλ=0, which gives Nμνρ=uμJνρ, so Jμν=Sμν=Nμνρuρ, where Sμν is the intrinsic spin tensor. If the body is spatially extended, then the difference Rμρ=Jμρ-Sμρ is the rotational spin tensor. The relativistic inertial force is then(8.16)f⃗(in)(x)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂xμ[Γ∘νλμuνuλ+1u0Γ∘νρμ(uνSρ0+u0¯Sνρ)-12u0KνρμuνSρ0+12R∘νσρμuνSσρ+12uνSρλKνρλ:μ]|.
In case of the Riemann space, V4(Q̆=0), the relativistic inertial force (7.5) exerted on the extended spinning body can be written in terms of the Ricci coefficient of rotation only:(8.17)f⃗̆(in)(x̆)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂x̆μ[Γ̆νλμŭνŭλ+1ŭ0Γ̆νρμ(ŭνJ̆ρ0+N̆0νρ)+12R̆νσρμŭνJ̆σρ]|.
In case of the Weitzenböck space, W4(R∙=0), (7.5) reduces to its teleparallel equivalent:(8.18)f⃗∙(in)(x∙)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂x∙μ[Γ∘νλμu∙νu∙λ+1u∙0Γ∘νλμ(u∙νJ∙ρ0+N∙0νρ)-12u∙0K∙νρμN∙νρ0+12N∙νρλK∙νρλ:μ]|.
All magnitudes related to the teleparallel gravity are denoted by an over “∙”. Finally, the nonvanishing inertial force, f(in)(phot)(x), acting on the photon of energy hν in the U4, can be obtained from (8.16), at f⃗(l)=0, as(8.19)f⃗(in)(phot)(x)=-(hνc2Ω2(ϱ¯))e⃗f|∂Xα∂xμ[Γ∘νλμdxνdTdxλdT+dTdt¯Γ∘νρμ(dxνdTSρ0+dt¯dTSνρ)-dT2dt¯KνρμdxνdTSρ0+12R∘νσρμdxνdTSσρ+12dxνdTSρλKνρλ:μ]|,
where e⃗f=(X⃗/|X⃗|), vq=(e⃗f·u⃗¯)=|u⃗¯|,γq=γ, u⃗¯ is the velocity of the photon in U4, (du⃗¯/dt¯) is the acceleration, and gμν(dxμ/dT)⊗(dxν/dT)=0.


## 8.1. The Principle of Equivalence in the RC Space

The RC manifold,U4, is a particular case of general metric-affine manifold M̃4, restricted by the metricity condition Nab=0, when a nonsymmetric linear connection, Γ, is said to be metric compatible. To avoid any possibility of confusion, here and throughout we again use the first half of Latin alphabet (a,b,c,…=0,1,2,3 rather than (±)) now to denote the anholonomic indices referred to the tangent space, which is endowed with the Lorentzian metric oab∶=diag(+---). The space, U4, also locally has the structure of M4, as has been first pointed out by [133] and developed by [134–137]. In the case of the RC space there also exist orthonormal reference frames which realize an “anholonomic” free-fall elevator. In Hartley’s formulation [137], this reads as follows. For any single pointP∈U4, there exist coordinates{xμ}and an orthonormal frame{ea}in a neighborhood ofPsuch that(8.2)ea=δaμ∂xμΓab=0atP,
where Γab are the connection 1-forms referred to the frame {ea}. Therefore the existence of torsion does not violate the PE. Note that since ∇g=0 holds in U4, the arguments showing that g can be transformed to o at any point P in U4 are the same as in the case of V4, while the treatment of the connection must be different: the antisymmetric part of ω can be eliminated only by a suitable choice for the relative orientation of neighbouring tetrads. Actually, let us choose new local coordinates at P, dxμ→dxa=eμadxμ, related to an inertial frame. Then,(8.3)gab′=eaμebνgμν=oab,Γac′b=eμbeaνecλ(Δνλμ+Kνλμ)≡ecλωaλb.
As it is argued in [138], the metricity condition ensures that this can be done consistently at every point in spacetime. Suppose that we have a tetrad {ea(x)} at the point P and a tetrad {ea(x+dx)} at another point in a neighbourhood of P; then, we can apply a suitable Lorentz rotation to ea(x+dx), so that it becomes parallel to ea(x). Given a vector v at P, it follows that the components vc=v·ec do not change under parallel transport from x to x+dx, provided that the metricity condition holds. Hence, the connection coefficients ωμab(x) at P, defined with respect to this particular tetrad field, vanish: ωμab(P)=0. This property is compatible with gab′=oab, since Lorentz rotation does not influence the value of the metric at a given point. In more general geometries, where the symmetry of the tangent space is higher than the Poincare group, the usual form of the PE is violated and local physics differs from SR.


## 8.2. The Generalized Inertial Force Exerted on the Extended Spinning Body in theU4

We now compute the relativistic inertial force for the motion of the matter, which is distributed over a small region in theU4 space and consists of points with the coordinates xμ, forming an extended body whose motion in the space, U4, is represented by a world tube in spacetime. Suppose that the motion of the body as a whole is represented by an arbitrary timelike world line γ inside the world tube, which consists of points with the coordinates X̃μ(τ), where τ is the proper time on γ. Define(8.4)δxμ=xμ-X̃μ,δx0=0,uμ=dX̃μds.
The Papapetrou equation of motion for the modified momentum (see [118, 139–143]) is(8.5)D∘ΘνDs=-12R∘μσρνuμJσρ-12NμρλKμρλ:ν,
where Kνλμ is the contortion tensor,(8.6)Θν=Pν+1u0Γ∘μρν(uμJρ0+N0μρ)-12u0KμρνNμρ0
is referred to as the modified 4-momentum, Pλ=∫τλ0dΩ is the ordinary 4-momentum, dΩ:=dx4, and the following integrals are defined:(8.7)Mμρ=u0∫τμρdΩ,Mμνρ=-u0∫δxμτνρdΩ,Nμνρ=u0∫sμνρdΩ,Jμρ=∫(δxμτρ0-δxρτμ0+sμρ0)dΩ=1u0(-Mμρ0+Mρμ0+Nμρ0),
where τμρ is the energy-momentum tensor for particles, and sμνρ is the spin density. The quantity Jμρ is equal to ∫(δxμτkl-δxρτμλ+sμρλ)dSλ taken for the volume hypersurface, so it is a tensor, which is called the total spin tensor. The quantity Nμνρ is also a tensor. The relation δx0=0 gives M0νρ=0. It was assumed that the dimensions of the body are small, so integrals with two or more factors δxμ multiplying τνρ and integrals with one or more factors δxμ multiplying sνρλ can be neglected. The Papapetrou equations of motion for the spin (see [118, 139–143]) are(8.8)D∘DsJλν=uνΘλ-uλΘν+KμρλNνμρ+12KμρλNμνρ-KμρνNλμρ-12KμρνNμρλ.
Calculating from (8.5) the particle 4-acceleration is(8.9)1mfgμ(x)=d2xμds̃2=-Γνλμ[uνuλ+1u0Γ∘νρμ(uνJρ0+N0νρ)]+12u0KνρμNνρ0-12R∘νσρμuνJσρ-12NνρλKνρλ:μ.
Thus, the relativistic inertial force, exerted on the extended spinning body moving in the RC space U4, can be found to be(8.10)f⃗(in)(x)=ma⃗in(x)=-ma⃗abs(x)Ω2(ϱ¯)γq=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂xμ×[Γ∘νλμuνuλ+1u0Γ∘νρμ(uνJρ0+N0νρ)-12u0KνρμNνρ0+12R∘νσρμuνJσρ+12NνρλKνρλ:μ]|.
In particular, if the spin density vanishes, sμνρ=0, from the conservation law we get then τμρ=τρμ, Mμρ=Mρμ, Mμνρ=Mμρρ, Nμνρ=0, and (8.11)Jμρ=Lμρ=∫(δxμτρ0-δxρτμ0)dΩ=1u0(-Mμρ0+Mρμ0),
where Lμρ is the angular momentum tensor. The modified 4-momentum (8.6) reduces to(8.12)Θν=Pν+D∘DsLνλuλ.
Equation (8.8) can be recast in the following form:(8.13)D∘DsLλν=uνΘλ-uλΘν,
while (8.5) becomes(8.14)D∘ΘνDs=-12R∘νσρμuμLσρ,
which give the relativistic inertial force exerted on the spinless extended body moving in the RC space U4 as follows:(8.15)f⃗(in)(x)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂xμ[Γ∘νλμuνuλ+1u0Γ∘νρμuνLρ0+12R∘νσρμuνLσρ]|.
If the body is not spatially extended, then it is referred to as a particle. The corresponding condition δxα=0 gives Mμνρ=0,andLμρ=0. Therefore (uλ/u0)Nμν0-Nμνλ=0, which gives Nμνρ=uμJνρ, so Jμν=Sμν=Nμνρuρ, where Sμν is the intrinsic spin tensor. If the body is spatially extended, then the difference Rμρ=Jμρ-Sμρ is the rotational spin tensor. The relativistic inertial force is then(8.16)f⃗(in)(x)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂xμ[Γ∘νλμuνuλ+1u0Γ∘νρμ(uνSρ0+u0¯Sνρ)-12u0KνρμuνSρ0+12R∘νσρμuνSσρ+12uνSρλKνρλ:μ]|.
In case of the Riemann space, V4(Q̆=0), the relativistic inertial force (7.5) exerted on the extended spinning body can be written in terms of the Ricci coefficient of rotation only:(8.17)f⃗̆(in)(x̆)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂x̆μ[Γ̆νλμŭνŭλ+1ŭ0Γ̆νρμ(ŭνJ̆ρ0+N̆0νρ)+12R̆νσρμŭνJ̆σρ]|.
In case of the Weitzenböck space, W4(R∙=0), (7.5) reduces to its teleparallel equivalent:(8.18)f⃗∙(in)(x∙)=-me⃗fΩ2(ϱ¯)γq|1mf(l)α-∂Xα∂x∙μ[Γ∘νλμu∙νu∙λ+1u∙0Γ∘νλμ(u∙νJ∙ρ0+N∙0νρ)-12u∙0K∙νρμN∙νρ0+12N∙νρλK∙νρλ:μ]|.
All magnitudes related to the teleparallel gravity are denoted by an over “∙”. Finally, the nonvanishing inertial force, f(in)(phot)(x), acting on the photon of energy hν in the U4, can be obtained from (8.16), at f⃗(l)=0, as(8.19)f⃗(in)(phot)(x)=-(hνc2Ω2(ϱ¯))e⃗f|∂Xα∂xμ[Γ∘νλμdxνdTdxλdT+dTdt¯Γ∘νρμ(dxνdTSρ0+dt¯dTSνρ)-dT2dt¯KνρμdxνdTSρ0+12R∘νσρμdxνdTSσρ+12dxνdTSρλKνρλ:μ]|,
where e⃗f=(X⃗/|X⃗|), vq=(e⃗f·u⃗¯)=|u⃗¯|,γq=γ, u⃗¯ is the velocity of the photon in U4, (du⃗¯/dt¯) is the acceleration, and gμν(dxμ/dT)⊗(dxν/dT)=0.


## 9. Concluding Remarks

In the framework of TSSD theory, as a preliminary step, we show that by imposing different appropriate physical constraints upon the spacetime deformations, we may recover the term in the Lagrangian of pseudoscalar-photon interaction theory, or we may reproduce the various terms in the Lagrangians of pseudoscalar theories, for example, as intergrand for topological invariant, or pseudoscalar-gluon coupling occurred in QCD in an effort to solve the strong CP problem. We carry out some details of this program to probe the origin and nature of the phenomenon of inertia. We construct the RTI, which treats the inertia as a distortion of local internal properties of hypothetical 2D, so-calledmaster space (MS). The MS is an indispensable companion of individual particle, without relation to the other matter, embedded in the background 4D-spacetime. The RTI allows to compute the inertial force, acting on an arbitrary point-like observer or particle due to its absolute acceleration. In this framework we essentially improve standard metric and other relevant geometrical structures referred to a noninertial frame 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 WPE. We relate the inertia effects to the more general post-Riemannian geometry. We derive a general expression of the relativistic inertial force exerted on the extended spinning body moving in the Rieman-Cartan space.



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*Source: 692030-2012-06-28.xml*