Research Article | Open Access

Gagik Ter-Kazarian, "Spacetime Deformation-Induced Inertia Effects", *Advances in Mathematical Physics*, vol. 2012, Article ID 692030, 41 pages, 2012. https://doi.org/10.1155/2012/692030

# Spacetime Deformation-Induced Inertia Effects

**Academic Editor:**Giorgio Kaniadakis

#### 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, [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 nucleus of spin . 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 is treated as a single proton in a central nuclear potential, the variation of mass with direction, if it exists, was found to satisfy . 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-polarized 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- and construct a consistent TSSD- 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 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 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, .

#### 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, , which has at each point a tangent space, , spanned by the anholonomic orthonormal frame field, , as a shorthand for the collection of the 4-tuplet , where . All magnitudes related to the space, , will be denoted by an over “”. These then define a dual vector, , of differential forms, as a shorthand for the collection of the , whose values at every point form the dual basis, such that , where denotes the interior product, namely, this is a -bilinear map where denotes the -modulo of differential -forms on . In components . On the manifold, , the tautological tensor field, , of type can be defined which assigns to each tangent space the identity linear transformation. Thus for any point , and any vector , one has . In terms of the frame field, the give the expression for as , in the sense that both sides yield when applied to any tangent vector in the domain of definition of the frame field. One can also consider general transformations of the linear group, , taking any base into any other set of four linearly independent fields. The notation will be used hereinafter for general linear frames. The holonomic metric can be defined in the semi-Riemann space, , as with components in the dual holonomic base . The anholonomic orthonormal frame field, , relates to the tangent space metric, , by , which has the converse because .

For reasons that will become clear in the sequel, next we write the norm, , of the infinitesimal displacement, , on the general smooth differential 4D-manifold, , in terms of the spacetime structures of , as
where is the world *deformation tensor*, is the frame field, and is the coframe field defined on , such that , or in components, ; also the procedure can be inverted, . Hence the deformation tensor, , yields local tetrad deformations:
The components of the general spin connection then transform *inhomogeneously* under a local tetrad deformations (2.3):
This is still a passive transformation, but with inverted factor ordering. The matrices are called *first deformation matrices*, and the matrices , *second deformation matrices*. The matrices , in general, give rise to right cosets of the Lorentz group; that is, they are the elements of the quotient group , because the Lorentz matrices, , leave the Minkowski metric invariant. A right-multiplication of 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:
In the first case, the contribution of the Christoffel symbols, constructed by the metric , reads
with representing the curls of the base members in the semi-Riemann space:
where is the anfolonomity 2-form. The deformed metric can be split as follows [96]:
where , and . In the second case, we may write the commutation table for the anholonomic frame, ,
and define the anholonomy objects:
The usual Levi-Civita connection corresponding to the metric (2.8) is related to the original connection by the following relation:
provided that
where the controvariant deformed metric, , is defined as the inverse of , such that . Hence, the connection deformation acts like a force that deviates the test particles from the geodesic motion in the space (for more details see [96]). Next, we deal with the spacetime deformation , to be consisted of two ingredient deformations (, ). Provided, we require that the first deformation matrix, , satisfies the following peculiar condition:
where 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, , is related to the corresponding spin connection, through the inverse
which is the Weitzenböck connection revealing the Weitzenböck spacetime of the teleparallel gravity. Thus, 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:
such that
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 transforms *inhomogeneously* through
where we have , also the procedure can be inverted, , and
is the spin connection. For our convenience, hereinafter the notation will be used for general linear frames:
where , or in components, ; also the procedure can be inverted, , provided that
Hence, the affine connection (2.17) can be rewritten in the abbreviated form:
Since the first deformation matrices and are arbitrary functions, the *inhomogeneously* transformed general spin connections and , 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 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 and the affine *torsion* 2-form representing a translational misfit (for a comprehensive discussion see [114–118]. These, together with the *curvature* 2-form , symbolically can be presented as [119, 120]
where for a tensor-valued -form density of representation type , the -covariant exterior derivative reads and is the general nonmetricity connection. This notation will be used instead of , such that . In what follows, however, we may still maintain the former notation to be referred to as corresponding connection constrained by the metricity condition. We may introduce the affine *contortion* 1-form given in terms of the torsion 2-form . In tensor components we have , where the torsion tensor given with respect to a holonomic frame, , is a third-rank tensor, antisymmetric in the first two indices, with 24 independent components. The TSSD- theory (see [96]) considers curvature and torsion as representing independent degrees of freedom. The RC manifold, , is a particular case of general metric-affine manifold , 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:
where
Hence, the relations between the scalar curvatures for an manifold read
This means that the Lorentz and diffeomorphism invariant scalar curvature, , becomes either a function of only or a function of only. Certainly, it can be seen by noting that the Lorentz gauge transformations can be used to fix the six antisymmetric components of to vanish. Then in both cases diffeomorphism invariance fixes four more components out of the six , with the four components 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
To recover the TSSD- theory, one can choose the EC Lagrangian, , as
where is the cosmological constant, is the curvature tensor, is the *Lagrange multiplier*, and . The basis is consisting in the Hodge dual of exterior products of tetrads by means of the Levi-Civita object: , which yields and , where we used the abbreviated notations for the wedge product monomials, , and denotes the Hodge dual. The variation of the total action
given by the sum of the gravitational field action, , with the Lagrangian (2.27) and the macroscopic matter sources, , with respect to the , 1-form and , which is a -form representing a matter field (fundamentally a representation of the or of some of its subgroups), gives
where is the Planck length,, and is the dual 3-form corresponding to the canonical spin tensor, which is identical with the dynamical spin tensor , namely,
provided that,
and that
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:
where is Einstein’s tensor, and the *modified torsion* reads
Thus, the equations of the standard EC theory can be recovered for . However, these equations can be equivalently replaced by the set of *modified EC equations* for :
We may impose different physical constraints upon the spacetime deformation , which will be useful for the theory of electromagnetism and charged particles:
with as a scalar or pseudoscalar function of relevant variables. Here . Then we obtain
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 framework:
where has the usual meaning for electromagnetism. This is equivalent, up to integration by parts in the action integral (modulo a divergence), to the Lagrangian
According to (2.39), the gravitational constitutive tensor [40] of the gravitational fields (e.g., metric , (pseudo)scalar field etc.) reads
The special case is considered by [102, 103], for modification of electrodynamics with an additional external constant vector coupling. Imposing other appropriate constraints upon the spacetime deformation , in the framework of TSSD- 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, , and another one should be MS embedded in the , 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, :
The ingredient 1D-space is spanned by the coordinates , where we use the *naked* capital Latin letters to denote the world indices related to . The metric in is
where is the infinitesimal displacement. The basis at the point of interest in consists of two real *null vectors*:
The norm, , given in this basis reads , where is the tautological tensor field of type (1,1), is a shorthand for the collection of the 2-tuplet , and . We may equivalently use a temporal and a spatial variables , such that
The norm, , now can be rewritten in terms of displacement, , as
where and are, respectively, the temporal and spatial basis vectors:
The MS companion () of this particle is assumed to be smoothly (injective and continuous) embedded in the . Suppose that the position of the particle in the background space is specified by the coordinates with respect to the axes of the inertial system . Then, a smooth map is defined to be an immersion—an embedding is a function that is a homeomorphism onto its image:
In fact, we assume that the particle has to be moving simultaneously in the parallel *individual * space and the ordinary 4D background space (either Minkowskian or Riemannian). Let the nonaccelerated observer uses the inertial coordinate frame for the position of a free test particle in the flat . We may choose the system in such a way as the time axis lies along the time axis of a comoving inertial frame , such that the time coordinates in the two systems are taken the same, . For the case at hand,
Hence, given the inertial frames , , , in the , in this manner we may define the corresponding inertial frames , , , in the .

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

*Definition 3.1. *The ingredient spaceof the individual MS companion of the particle is said to be in

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 . An existence of the *absolute* state of the is an immediate cause of the light traveling in empty space along the -axis with a maximal velocity (we reinstate the factor ()) in the -direction corresponding to the state (rel, abs), and in the -direction corresponding to the state (abs, rel). The *absolute* state of manifests its *absolute* character in the important for SR fact that the resulting velocity of light in the empty space is the same in all inertial frames , , ,; that is, in empty space light propagates independently of the state of motion of the source—if , then . Since the 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 of instantaneously change of a constant velocity (both magnitude and direction) of a massive test particle under the unbalanced net force is the immediate cause of a deformation (distortion of the local internal properties) of MS: .

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 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 (see (2.19)–(5.35)) with respect to the components of inertial force in terms of the relativistic force acting on a purely classical particle in . 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 in only allowed way as the time axis lies along the time axis of a comoving inertial frame , in order that the time coordinates in the two systems are taken the same, and that another axis lies along the net 3-acceleration () (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 . To start with, let be 2D semi-Riemann space, which has at each point a tangent space, , spanned by the anholonomic orthonormal frame field, , as a shorthand for the collection of the 2-tuplet , where , where the holonomic frame is given as . Here, we use the first half of Latin alphabet to denote the anholonomic indices to related the tangent space, and the capital Latin letters with an over , to denote the holonomic world indices related to either the space or . All magnitudes referred to the space, , will be denoted by an over . These then define a dual vector, , of differential forms, , as a shorthand for the collection of the , whose values at every point form the dual basis, such that . In components . On the manifold, , the tautological tensor field, , of type (1,1) can be defined which assigns to each tangent space the identity linear transformation. Thus for any point , and any vector , one has . In terms of the frame field, the give the expression for as , in the sense that both sides yield when applied to any tangent vector in the domain of definition of the frame field. We may consider general transformations of the linear group, , taking any base into any other set of four linearly independent fields. The notation will be used hereinafter for general linear frames. The holonomic metric can be defined in the semi-Riemann space, , as
with components in the dual holonomic base . The anholonomic orthonormal frame field, , relates to the tangent space metric, , by , which has the converse because of the relation . With this provision, we build up a general *distortion-complex*, yielding a distortion of the flat space , and show how it recovers the *world-deformation tensor *, which still has to be put in [96] by hand. The DC members are the invertible distortion matrix , the tensor , and the *flat-deformation tensor *. Symbolically,
The principle foundation of a *distortion of local internal properties of MS* comprises then two steps.

The first is to assume that the linear frame , at given point (), undergoes the *distortion* transformations, conducted by and , respectively, relating to and , recast in the form

Then, the norm of the infinitesimal displacement on the general smooth differential 2D-manifold can be written in terms of the spacetime structures of and :
where is the frame field and is the coframe field defined on , such that . The deformation tensors and imply
provided that
such that
Hence the anholonomic deformation tensor, , yields local tetrad deformations:
The matrices are referred to as the *first deformation matrices*, and the matrices , *second deformation matrices*. The matrices , in general, give rise to right cosets of the Lorentz group; that is, they are the elements of the quotient group , because the Lorentz matrices, , leave the Minkowski metric invariant. A right multiplication of 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:
In the first case, the contribution of the Christoffel symbols, constructed by the metric , reads
The deformed metric can be split as follows [96]:
where , and
In the second case, we may write the commutation table for the anholonomic frame, ,
and define the anholonomy objects:
The usual Levi-Civita connection corresponding to the metric (4.11) is related to the original connection by the following relation:
provided that
where the controvariant deformed metric, , is defined as the inverse of , such that . That is, the connection deformation acts like a force that deviates the test particles from the geodesic motion in the space, . Taking into account (4.4), the metric (4.9) can be alternatively written in a general form of the spacetime or frame objects:
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, , under an unbalanced net force other than gravitational. Here and henceforth we simplify DC for our use by imposing the constraints
and, therefore,
The (4.5), by virtue of (4.4) and (5.1), gives
where the deformation tensor, , yields the partial holonomic frame transformations:
or, respectively, the yields the partial local tetrad deformations:
Hence, (4.4) defines a diffeomorphism :
where . The conditions of integrability, , and nondegeneracy, , immediately define a general form of the *flat-deformation tensor *, where 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 * of instantaneously change of the measure of massive test particle under the unbalanced net force :
Clearly, these transformations imply a violation of the relation (3.3) for the null vectors . Now we can use (4.4) to observe that for dual vectors of differential forms and we may obtain
We parameterize the tensor in terms of the parameters and as
where and . Then, the relation (5.8) can be recast in an alternative form:
Suppose that a second observer, who makes measurements using a frame of reference which is held stationary in deformed/distorted space , uses for the test particle the corresponding spacetime coordinates . The (4.4) can be rewritten in terms of spacetime variables as
where and are, respectively, the temporal and spatial basis vectors:
The transformation equation for the coordinates, according to (5.10), becomes
which gives the general transformation equations for spatial and temporal coordinates as follows :
Hence, the general metric (4.17) in reads
provided that
The difference of the vector, (3.5), and the vector, (5.11), can be interpreted by the second observer as being due to the deformation/distortion of flat space . 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 , rather than to any absolute quality of a deformation/distortion of . To prove this assertion, note that the transformation equations (5.14) give a reasonable change at low velocities , as
thereby
Then (5.17) becomes conventional transformation equations to accelerated axes if we assume that and , where is a magnitude of proper net acceleration. In high-velocity limit , ), we have
and so (5.14) and (5.15), respectively, give
To this end, *the inertial effects become zero*. Let be a local net 3-acceleration of an arbitrary observer with proper linear 3-acceleration and proper 3-angular velocity measured in the rest frame:
where is the 4-velocity. A magnitude of can be computed as the simple invariant of the absolute value as measured in rest frame:
Following [57, 58], let us define an orthonormal frame , carried by an accelerated observer, who moves with proper linear 3-acceleration and and proper 3-rotation . Particular frame components are denoted by hats, . Let the zeroth leg of the frame be 4-velocity of the observer that is tangent to the worldline at a given event and we parameterize the remaining spatial triad frame vectors , orthogonal to , also by . The spatial triad rotates with proper 3-rotation . The 4-velocity vector naturally undergoes Fermi-Walker transport along the curve , which guarantees that will always be tangent to determined by :
where the antisymmetric rotation tensor splits into a Fermi-Walker transport part and a spatial rotation part :
The 4-vector of rotation is orthogonal to 4-velocity , therefore, in the rest frame it becomes , and is the Levi-Civita tensor with . 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
where we choose the system in such a way as the axis lies along the net 3-acceleration (). Hereinafter, we may simplify the flat-deformation tensor by setting , such that (5.9) becomes
and the general metric (4.17) in reads . Hence (5.26) gives
Combining (5.14) and (5.26), we obtain the key relation between a so-called *inertial acceleration*, arisen due to the curvature of MS,
and a local *absolute acceleration* as follows:
where are the Christoffel symbols constructed by the metric (5.16). Then (5.30) provides a quantitative means for the *inertial force *:
In case of absence of rotation, we may write the local *absolute acceleration* (5.26) in terms of the relativistic force acting on a particle with coordinates :
Here is the force defined in the rest frame of the test particle, and is the Lorentz transformation matrix :
where . So
and hence (5.31), (5.26), and (5.34) give
At low velocities