International Journal of Theoretical and Mathematical Physics 2019, 9(3): 81-96 DOI: 10.5923/j.ijtmp.20190903.03

Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

Maciej Rybicki

Sas-Zubrzyckiego 8/27 Kraków, Poland

Abstract A thought experiment concerning specific case of length contraction demonstrates inconsistency of the Special Theory of Relativity (STR). When two extended objects in the mutual relative motion occupy the same area in spacetime (e.g. elevator moving in the elevator shaft; here considered in 2D representation), then Lorentz transformation appears generally unable to handle this situation in a self-consistent way. Namely, transformation from either elevator or shaft rest frames to the observer’s rest frame, non-collinear with the direction of relative motion between the said objects, leads to a contradiction. This happens because the elevator and shaft are subject to the Lorentz length contraction along two different directions in plane, due to different directions of their motion in the observer’s frame. In result, they no longer match each other in this frame. A direct cause of this contradiction is the resulting from the principle of relativity concept of relative motion, regarded𝑥𝑥𝑥𝑥 as the source of “relativistic” effects. The revealed inconsistency, to which the name “elevator-shaft paradox” is assigned, makes necessary replacing STR by a different theory based on the postulate of preferred frame of reference and the related concepts of absolute rest and absolute motion. The relevant Preferred Frame Theory (PFT) is discussed, with the particular focus on partial experimental equivalence between PFT and STR. The proposed theory predicts an invariant hierarchy of energy, which opens up new prospects for the sought theory of quantum gravity. Keywords Length contraction, Lorentz transformation, Preferred frame of reference, Spacetime, Special relativity, Tangherlini transformation

contrast to that, our goal is to examine Special Relativity in 1. Introduction application to the “real” two-dimensional case, i.e. the one in which the action takes place so to say simultaneously in two All the textbooks on Special Relativity are full of rockets, spatial dimensions in plane. Anticipating the final result twins, clocks, trains, platforms, etc. This may suggest that of that examination (which is the fundamental revision of STR has been thoroughly verified as a consistent description STR, followed by an indication𝑥𝑥𝑥𝑥 of its due replacement), let us of the real three-dimensional world with time taken as the precede the main body of this paper by a brief glance at the fourth dimension. Meanwhile, the more careful insight origin of Special Relativity and its potential preferred frame shows that nearly all the considered cases exploit a single alternative. spatial dimension, namely the one parallel to the relative STR arose as a solution to the problem of incompatibility motion between two inertial frames, conventionally between the Maxwell’s electromagnetic equations and the identified with -axis. Even when two spatial dimensions Newtonian mechanics. Unlike the Newton’s laws of motion, are taken into account (as e.g. in the Sagnac effect), the the Maxwell’s equations were found to vary under the respective transformations𝑥𝑥 between selected frames are Galilean transformation. This suggested existence of a always considered in separation, i.e. as connecting certain ubiquitous medium in which the electromagnetic consecutive inertial frames, all linked to the single (curved) waves would propagate just as sound in the air. The relevant course of motion. This general observation applies both to hypothesis of the motionless luminiferous ether (alternative the algebraical description of theory, in particular to Lorentz to the dragged or partly dragged ether hypotheses) meant the transformation, and to respective geometrical representation split of physical reality into the two separate domains: expressed by Minkowski or Loedel spacetime diagrams. In mechanics - ruled by the Galilean principle of relativity, and electromagnetism - complying with the idea of the preferred * Corresponding author: frame of reference. However, in a series of experiments [email protected] (Maciej Rybicki) conducted by Michelson and Morley no motion of Earth Published online at http://journal.sapub.org/ijtmp Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing through ether had been detected. This unexpected result led This work is licensed under the Creative Commons Attribution International to several new concepts including length contraction and License (CC BY). http://creativecommons.org/licenses/by/4.0/ time dilation (“local time”) (FitzGerald [1], Lorentz [2],

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Larmor [3]), subsequently developed by Lorentz in his STR inconsistency as a theory applied to more than one of theory of electron, also known as Lorentz ether theory (LET) spatial dimensions. Instead, in Part II we outline a theory [4]. The failures in detecting absolute motion had been alternative to STR (the second option, mentioned above), elevated by Poincare [5] to the rank of universal law of mathematically consistent, compatible with all known nature, called by him the principle of relativity, which paved experiments and predicting new subtle effects so far the way to Special Relativity. undiscovered. In this pivotal moment of physics development, the basic principles once again hang in the balance. The postulated unity of nature (and consequently of physics) required new 2. Part I model of transformation, under which all phenomena would An evidence of the geometrical inconsistency of Special behave uniformly. Logically, the main alternative was: Relativity either extension of the Galilean principle of relativity to electrodynamics and other domains, or its opposition, a 1. Description of the elevator-shaft thought-experiment theory related to the notions of absolute space and absolute and its preliminary analysis motion, with the Galilean relativity “emerging” as an In the following reasoning we shall strictly adhere to the approximation in the limit of small velocities. Each of these STR rules implemented to 3-dimensional spacetime , , options involved different type of transformation and (all the time = 0). Assume the 2D “elevator” moving different conversion of Newtonian mechanics to a new uniformly, say, upward, with the relativistic velocity𝑐𝑐𝑐𝑐 𝑥𝑥 adequate form. However, due to specific “state of excitement” 𝑦𝑦relative to the 2D𝑧𝑧 vertical elevator shaft, hereinafter referred featured by the abundance of competitive hypotheses, only to as “rails”. Let be the rest frame for the elevator, and𝐕𝐕 the first of mentioned two possibilities had been developed the rest frame for′ the rails. To get a better insight, assume to the stage of complete theory, namely STR. Once it already that in the elevator𝒦𝒦 is square shaped. The spacing happened, the “natural selection” did its work eliminating between𝒦𝒦 the′ rails is assumed to be equal to the width of the second option. elevator,𝒦𝒦 while the rails length is treated as infinite. In the As shown in Einstein’s seminal paper [6], Lorentz moving elevator is contracted by the Lorentz factor 2 2 1 2 𝒦𝒦 transformation did not need any physical “content” to appear ( ) = (1 ) from square to the horizontal except the two fundamental postulates: the principle of rectangle (Fig. 1). − ⁄ 𝛾𝛾 𝑉𝑉 − 𝑉𝑉 ⁄𝑐𝑐 relativity and the constancy (isotropy) of the velocity of light. Due to the radical simplicity of these postulates, STR evinced itself as a “spacetime physics” [7], which means it basically refers to geometry. This became clearly visible owing to Minkowski geometrical interpretation of Special Relativity [8]. The relevant Minkowski spacetime (or space) provides a fusion of the Euclidean three-dimensional space with time interpreted as the fourth dimension of space (with opposite sign) into a single inseparable four-dimensional continuum otherwise called the Lorentzian manifold. An invariant element of Minkowski space is the “spacetime interval” - the Pythagorean-like construct connecting two timelike (i.e. causally connectible) “events” in spacetime, in analogy to distance in Euclidean space obtainable from the Pythagorean theorem. The pseudo-Euclidean geometry of the (flat) “hyperbolic” Minkowski space determines all Figure 1. Elevator moving upwards in becomes contracted by the kinematical properties of any physical object connected with Lorentz factor ( ), from the square to rectangle. The dashed line depicts 𝓚𝓚 its motion relative to a freely chosen inertial observer. The the square shape of𝑽𝑽 elevator in its rest frame . Both in frames and elevator tightly 𝜸𝜸adheres to the rails ′ ′ fact that STR can be interpreted as a purely geometrical 𝓚𝓚 𝓚𝓚 𝓚𝓚 model is widely considered to ensure “by definition” its Assume that rails are divided into the square “cages” internal coherence. It could be therefore surprising that identified with sequential floors. The relationship between careful examination of a certain specific class of cases the height of elevator and the height of any particular cage reveals fallacy hidden behind the symmetry. As a result, would then correspond to the relationship between the pole Special Relativity, along with Minkowski space, turns out to and the barn in the pole-in-barn paradox. Namely, in frame be an inadequate description of nature. Although quoting the moving upwards elevator is shorter in height Einstein against himself may seem a bit unfair; yet his compared with any sequential cage passed by, whereas in ingenious remark perfectly matches this case: “As far as the 𝒦𝒦frame each moving downwards cage is shorter in height laws of mathematics refer to reality, they are not certain, and than the elevator.′ As is known, Special Relativity provides as far as they are certain, they do not refer to reality” [9]. consistent𝒦𝒦 explanation to this apparent paradox, based on the Part I includes a thought experiment demonstrating the relativity of simultaneity. However, we aim to consider here

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1 a different problem. To formulate it, let us assume that + ( ) = = + − (1) elevator and rails are observed from the train moving relative 1+ 𝛾𝛾1+𝑢𝑢 𝐕𝐕⊥ ′′ ′′ ′′ 𝐮𝐮 𝐕𝐕2∥ 2 𝑢𝑢 𝑉𝑉 𝑢𝑢 𝑉𝑉 to with the relativistic velocity in positive direction. 𝐕𝐕 𝐕𝐕∥ ⊕ 𝐕𝐕⊥ ∥ ∥ Let be the rest frame for the train observer (Fig. 2). where and are, respectively,𝑐𝑐 the parallel𝑐𝑐 ( ) and perpendicular′′ ( )′′ components of , and and ′′ are, 𝒦𝒦 ′′ 𝐮𝐮 𝑥𝑥 ∥ ⊥ respectively,𝐕𝐕 𝐕𝐕and′′ components of′′ . Since in𝑥𝑥 the 𝒦𝒦 ∥ ⊥ elevator travels𝑦𝑦 in direction, so𝐕𝐕 respective𝐕𝐕 components𝐕𝐕 are: = 0 and𝑥𝑥 𝑦𝑦 . Consequently 𝐕𝐕Eq. (1) reduces𝒦𝒦 to: 𝑦𝑦 1 ∥ ⊥ = + (2) 𝐕𝐕 𝐕𝐕 ≡ 𝐕𝐕 ( ) ′′ − It follows that the elevator moves𝑢𝑢 ⊥in in the direction 𝐕𝐕 𝐮𝐮 𝛾𝛾 𝐕𝐕 1 determined by the components: = and′′ = ( ) , ′′ 𝒦𝒦 ′′ − with the value of being: ∥ ⊥ 𝑢𝑢 ⊥ ′′ 𝐕𝐕 𝐮𝐮 𝐕𝐕 𝛾𝛾 𝐕𝐕 2 2 𝐕𝐕 = + ( ) (3) ′′ In any case different𝑉𝑉 from�𝑢𝑢 �=𝑉𝑉⁄0𝛾𝛾, and/or𝑢𝑢 � = 0, the elevator is contracted in ′′ by the Lorentz′′ factor ∥ ⊥ 1 2 𝐕𝐕 𝐕𝐕 = 1 2 2 in ′′the direction diagonal to Figure 2. The rails at rest in frame , elevator (at rest in ) moving ⁄ 𝒦𝒦 ′′ − uniformly upwards, and the train-frame moving uniformly to′ the right; both′′ of its horizontal and vertical sides. Hence, all four 𝛾𝛾�𝑉𝑉 � � − 𝑉𝑉 ⁄𝑐𝑐 � all three as observed from the rail-frame𝓚𝓚 ′′ 𝓚𝓚 sides are tilted towards the direction transverse to vector . 𝓚𝓚 In result, the elevator reshapes in from the square to the′′ As far as the train ( ) moves𝓚𝓚 to the right relative to rhomboid, with all its sides diagonal′′ in plane (Fig.𝐕𝐕 4). vertical rails ( ), the rails′′ move to the left relative to 𝒦𝒦 with the velocity (or𝒦𝒦 , provided we revert the sense of′′ 𝑥𝑥′′𝑦𝑦′′ -axes). Same𝒦𝒦 as in , the vector is perpendicular𝒦𝒦 to the′′ rails in −. 𝐮𝐮Consequently,𝐮𝐮 the rails described as “vertical”𝑥𝑥𝑥𝑥 in ′′(parallel𝒦𝒦 to -axis) are,𝐮𝐮 regardless of their motion, also vertical𝒦𝒦 in (parallel to -axis). Compared with respective𝒦𝒦 spacing in′′ 𝑦𝑦, in the′′ rails get closer to each other due to the length𝒦𝒦 contraction′′ by𝑦𝑦 the Lorentz factor 2 2 1 2 ( ) = (1 ) (Fig.𝒦𝒦 3). 𝒦𝒦 − ⁄ 𝛾𝛾 𝑢𝑢 − 𝑢𝑢 ⁄𝑐𝑐

Figure 4. Elevator observed in frame . Due to the length contraction, the elevator reshapes from the square to ′′rhomboid. Both horizontal and vertical sides are skewed in 𝓚𝓚 ′′ It turns out that, in 𝓚𝓚 , the rhomboidal shape of elevator does not match the vertical′′ direction of the rails. Unlike the pole-in-barn paradox 𝒦𝒦in which time variables play crucial role (in rebutting this paradox), the revealed incompatibility cannot be solved by applying the relativity of simultaneity. Likewise, but this time in agreement with expectation, the Figure 3. The rails contracted in frame . The dashed lines show the respective spacing in frame ′′ mismatching does not depend on whether the coordinate 𝓚𝓚 systems and are collinear or not. This prediction In turn, the elevator𝓚𝓚 travels in with the velocity remains in conflict′′ with the respective compatibility in along the diagonal direction in ′′ plane, from bottom′′ frames 𝒦𝒦 and 𝒦𝒦 , which eventually indicates an right to top left. Both the exact value𝒦𝒦 and the direction𝐕𝐕 of inconsistency of STR′ (Fig. 5). vector follow from the STR𝑥𝑥 ′′velocity𝑦𝑦′′ addition in the 𝒦𝒦 𝒦𝒦 general configuration:′′ 𝐕𝐕

84 Maciej Rybicki: Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

odds with the perfect fit in frames and , hence, as previously, indicates an inconsistency of STR. ′ Concluding, the reasoning contained𝒦𝒦 in this𝒦𝒦 section, if thoroughly considered, provides a wholly adequate evidence of the inconsistency of Special Relativity. Nevertheless, the two counter-arguments have been presented to the author in response to the “working” version of the present paper; the one in private correspondence [10] and the other one [11] aimed at justifying the rejection of the submitted manuscript. Regardless of the fundamental disagreement with these arguments, the author is truly grateful to both prominent proponents of Einstein’s theory for their stimulating contribution in improving this paper. All the more so that typical reaction in this case is no reaction. In short, these

two counter-arguments are, respectively, the following: 1) Figure 5. Square elevator contracted along the diagonal direction and vertical rails contracted along the horizontal direction do not match each Elevator cannot match the rails in one frame and not match in other in the frame . This contradicts respective fit in frames and the other one. Lorentz transformation certainly cannot lead ′′ ′ to such nonsensical result. In particular, a single spacetime 𝓚𝓚 𝓚𝓚 𝓚𝓚 Let us consider now a simplified version of the above point in one frame cannot split into the two separate points in experiment, reduced to the basically one-dimensional objects another frame; 2) The composition of two non-collinear embedded in the 3D spacetime. Let only one rail be left (we Lorentz boosts does not give a pure Lorentz boost along the shall call it “pipe”), as well as only one vertical side of the resultant velocity, but Lorentz boost and rotation. It is this elevator (we shall call it “rod”). Beside this difference (in rotation that matches the contracted elevator to the rails in fact coming down to assumption that elevator and shaft are observer’s frame. both infinitely thin), the whole previous arrangement The following two sections are devoted to the analysis of remains unchanged (Fig. 6). these arguments. We shall consider them separately, although it is clear that they interconnect with each other to a certain degree. In particular, the second argument might be regarded as validating the first one. 2. Argument from the “singleness” of four-position of adjacent points The argument proceeds as follows. In accordance with the previous arrangement, consider the Lorentz transformation from to of the spacetime coordinates of elevator four vertices (which′′ suffices to transform the whole object). In frame𝒦𝒦 these𝒦𝒦 vertices are constantly in contact with the rails. Hence, in any particular instant determined in , each vertex comes𝒦𝒦 into contact with a certain point on the rail. In terms of Minkowski geometry, this means that both adjacent𝒦𝒦 points (vertex and rail-point) constitute a single point (event) in spacetime, represented by a single position 4-vector (four-position), specified by the single set of space and time Figure 6. The rod sliding in the vertical pipe at rest in , observed in coordinates (despite considering the 2D space, we shall stick frame . Due to the diagonal direction of contraction in the rod deviates from′′ vertical direction, hence can no longer slide inside𝒦𝒦 the′′ pipe to the established nomenclature). Let us call the so-defined 𝓚𝓚 𝓚𝓚 spacetime point the “composite point”. Each “composite In frame , considering respective shapes, the pipe point” in , represented by respective four-position, after moves perpendicularly′′ to vector , while the rod moves being transformed to makes also a single “composite diagonally to𝒦𝒦 vector . Due to the length contraction along point” represented𝒦𝒦 by the′′ new four-position, which can be vector the rod skews′′ in −𝐮𝐮towards the transverse written as: , , 𝒦𝒦 , , . The resultant shape of direction ′′to this vector.𝐕𝐕 This is′′ an obvious geometrical elevator obtained by transforming′′ ′′ ′′the four-positions of all consequence𝐕𝐕 of length contraction𝒦𝒦 , considering e.g. how vertices from𝑐𝑐𝑐𝑐 𝑥𝑥 𝑦𝑦to ⟶ 𝑐𝑐𝑡𝑡 must𝑥𝑥 therefore𝑦𝑦 comply with the does behave the diagonal of the rectangle contracted along shape (direction and spacing)′′ of the rails. Consequently, the its sides. Hence, in result of contraction in , the direction elevator contracted𝒦𝒦 in 𝒦𝒦 must adhere to the rails with both of rod differs from the direction of pipe that′′ itself remains of its side-edges, just as ′′it does in . There is only “floor” vertical. Consequently, according to observer𝒦𝒦 , the rod and “ceiling” that are 𝒦𝒦skewed at an equal angle in result of and pipe do not match each other, so that the rod′′ can no length contraction. This distortion is𝒦𝒦 a direct consequence of longer slide inside the pipe. This ridiculous prediction𝒦𝒦 is at the relativity of simultaneity, manifesting itself as the

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difference in determining the simultaneity of distanced match each other in the elevator frame and in the rails frame, events in and along the direction. Similarly, they must also match each other in any other frame. The transformation from′′ to , specified′′ as , , property of contacting, connected with the demand of the , 𝒦𝒦 , 𝒦𝒦 must′ also preserve′′ 𝑥𝑥𝑥𝑥 the singleness′ of′ singleness of four-positions, cannot be frame-depended. respective′ ′′ “composite′′ ′′ 𝒦𝒦points”. 𝒦𝒦 𝑐𝑐𝑡𝑡 𝑥𝑥 Indeed, it must be not, and really, it is not! However, this is 𝑦𝑦 Let⟶ us𝑡𝑡 recap𝑥𝑥 : the𝑦𝑦 elevator indeed reshapes in from the an argument from nature, not from theory. Nature is always square to rhomboid due to the diagonal direction′′ of length internally coherent, or self-consistent, but its description contraction in plane; nevertheless, its vertical𝒦𝒦 (in (theory) does not automatically possess this vital property. and in ) side-edges remain vertical in . In that way, What we really claim in our rebuttal is not denying the 𝑥𝑥′′𝑦𝑦′′ 𝒦𝒦 the singleness′ of the four-positions of respective′′ “composite “conservation of the singleness of four-position”, but 𝒦𝒦 𝒦𝒦 points” is preserved. denying the ability of Lorentz transformation to fulfil both The above argument seems to be conclusive by the demands together, i.e. to predict the correct length strength of pure logic rather than by the rules specific to contraction and to preserve the integrity of “composite Special Relativity. Beyond any doubt, the property of points” due to the “singleness” of four-positions. The right “contacting” cannot be frame-dependent. A single set of theory should certainly satisfy both demands, namely spacetime coordinates cannot split into the two separated preserve the singleness of four-positions in transformations sets of coordinates (one for the vertex and one for the and predict the correct length contraction. However, as crazy rail-point) in result of the Lorentz (or any other) as it sounds, STR fails in this regard. transformation from one frame to another. This is a “mathematically trivial fact”, while its negation is “really 3. Argument from the Thomas-Wigner rotation absurd” [10]. A specific prediction connected with STR states that In reply to this argument, we shall also use logic rather composition of two (or more) non-collinear Lorentz boosts than physics. Let us call, just for our present purpose, the does not give a pure Lorentz boost along the final velocity above-mentioned property of Lorentz transformation the as observed in the observer’s (initial) frame, but a “conservation of the singleness of four-position”. The composition of Lorentz boost and rotation, known as the respective demand is indeed trivial and obvious. The point is, however, that along with this one, another demand Thomas-Wigner rotation (TWR). In the case of orbital should also be made and fulfilled. Namely, the right motion, we deal with the sequence of (infinitesimal) TWRs, transformation must give (predict) the correct length resulting in a periodic change of the spatial orientation of contraction. Note that unlike the “ordinary” point, i.e. the orbiting object, known as Thomas precession (TP) [12]. one fixed to a single object, e.g. elevator or rail as taken This effect explains the otherwise unexplainable separately, each “composite point” here considered does not observation in quantum physics, namely relativistic trace out a single trajectory (worldline) in spacetime. This is correction to the spin-orbit interaction. A typical example is because any particular “composite point”, although defined the precession of spin of electron orbiting around the by a single position 4-vector, is however represented by two nucleus. Within STR, Thomas precession is a direct different velocity 4-vectors. In frame these vectors consequence of the relativity of simultaneity associated with are different both as to the value and the direction.′′ This fact each participating TWR. As a result, TP is absent in cases determines the way that “composite object𝒦𝒦 ” (i.e. the one when direction of spin is perpendicular to the plane of formed in by the set of four “composite points”) is rotation, i.e. parallel to -axis (see e.g. comprehensive contracted in . According to STR, contraction always explanation by Taylor & Wheeler [7]). takes place 𝒦𝒦along′′ the direction of relative motion, hence According to the above𝑦𝑦 description as applied to the along respective𝒦𝒦 relative velocity vector. In the here elevator case, the composition of two Lorentz boosts: from considered case, this means that rails are transformed from to , and from to , is thought to give the Lorentz to along vector ; instead, the elevator is boost′ between and ,′′ complemented by TWR of the transformed′′ from to along vector . These two elevator𝒦𝒦 𝒦𝒦 together ′ with𝒦𝒦 frame𝒦𝒦′′ . Specifically, this would different𝒦𝒦 𝒦𝒦 directions of′ motion′′𝐮𝐮 in one reference ′′frame ( mean that Lorentz𝒦𝒦 boost𝒦𝒦 along velocity′ is followed by plane), applied to the𝒦𝒦 Lorentz𝒦𝒦 transformation𝐕𝐕 of any single′′ ′′ the rotation of elevator through𝒦𝒦 a definite angle,′′ namely just “composite point” determine two different directions 𝑥𝑥along𝑦𝑦 the one needed to “verticalize” the contracted𝐕𝐕 elevator, which the said objects are contracted. As a result, we obtain thereby matching it to the rails. Consequently, despite an apparent split of particular “composite points” followed diagonal direction of the elevator contraction in by the resultant decoherence of shapes (despite their plane, no incompatibility between elevator and rails is compatibility in and , elevator and rails no longer expected (Fig. 7). 𝑥𝑥′′𝑦𝑦′′ match each other in )′. An alternative to this, from Nevertheless, it is not a difficult task to disprove the somewhere else nonsensical𝒦𝒦 𝒦𝒦′′ result, is preserving (in one argument based on the Thomas-Wigner rotation. The point way or another) the singleness𝒦𝒦 of four-positions to satisfy is not that TWR and TP are unreal in any sense. By all the demand of logic, which is however achieved at the means they are real, both as an inherent prediction of STR expense of breaking the rules of length contraction. and as the really observed effect (precession of spin). Let us recap this section. Provided that elevator and rails However, the crucial point is that TWR does not apply to

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the elevator case. A fundamental difference between the as well as to all remaining inertial frames. It would be case of orbiting body, or generally, a body changing therefore illegitimate to assume that transformation between direction of its motion (due to the forces acting on it) and these very frames should involve the Thomas-Wigner the here considered motion of elevator, is that motion in the rotation. Hence, what we really deal with in the first case is non-inertial, while it is doubtlessly inertial in elevator-shaft paradox is not the composition of two the present (elevator) case. Namely, in opposition to a body consecutive Lorentz boosts resulting in the Thomas-Wigner moving along the curvilinear path (smooth curve or rotation (of elevator), but the pair of two concurrent Lorentz polygonal chain), thereby changing (constantly or boosts: from to and from to , applying occasionally) inertial frames, the elevator moves respectively to the rails and′′ to the elevator,′ both these′′ objects rectilinearly, hence occupies all the time the same inertial contributing to𝒦𝒦 a one 𝒦𝒦“composite object”,𝒦𝒦 i.e. 𝒦𝒦to the elevator frame . Consequently, the Lorentz boost between the “framed” by the rails. elevator frame′ and the observer’s frame should In order to avoid confusion, we should bear in mind an not be𝒦𝒦 considered′ as a composition of Lorentz′′ boosts elementary distinction between the two notions: inertial between and𝒦𝒦 , and and in 𝒦𝒦the sense frame (of reference) - physical entity of the properties corresponding′ to TWR. It is pretty′′ clear that defined by the laws of motion, and coordinate system - Thomas-Wigner𝒦𝒦 rotation𝒦𝒦 pertains𝒦𝒦 solely𝒦𝒦 and exclusively to geometrical construct imposing a definite order on given the composition of mutually non-collinear inertial frames (at inertial frame. According to this distinction, a single inertial least three of them, including observer’s frame) frame is represented by infinitely many coordinate systems, participating in motion, i.e. to the frames contributing to the differing by locations of the origin and orientation of axes. curvilinear trajectory of a given body in spacetime. With no The rails and elevator are contracted in one and the same doubt, the frame , despite its specific role in the inertial frame along two different directions, elevator-shaft paradox, does not participate in the elevator corresponding to different′′ directions of respective Lorentz motion. The elevator𝒦𝒦 is not even for a moment at rest with boosts. Consequently,𝒦𝒦 the frame is represented here by respect to the rails, that is to frame . the two coordinate systems, each of′′ them collinear with one of the mentioned directions. 𝒦𝒦 𝒦𝒦 4. A “symmetrical” version of the elevator-shaft thought experiment Let us consider yet another one version of the thought experiment, which will provide us with an “argument from symmetry”. Let the velocity of rails in be (as previously), and the velocity of elevator measured′′ in be specified as = ( ) . Then, according to𝒦𝒦 Eq. (2),𝐮𝐮 the horizontal and vertical components of the velocity vector𝒦𝒦 𝑢𝑢 are = 𝐕𝐕( 𝐮𝐮) 𝛾𝛾and = ( ); hence are equal in′′ value.′′ For these ′′components,′′ vector′′ and thereby the𝐕𝐕 ∥ ⊥ path𝐕𝐕 traced− out𝐮𝐮 𝑥𝑥by elevator𝐕𝐕 in 𝐮𝐮 𝑦𝑦is inclined′′ at the angle 135° in plane (or 45°, while′′ looking𝐕𝐕 in negative direction).′′ Consequently,′′ the 𝒦𝒦length contraction′′ of the (square 𝑥𝑥at 𝑦𝑦rest) elevator takes place in the𝑥𝑥 exactly Figure 7. Thomas-Wigner rotation as the putative solution to the symmetrical direction to its diagonals, namely parallelly to elevator-shaft paradox. Rotation of frame “verticalizes” the side-edges one of square diagonals and perpendicularly to the other one. of contracted elevator (dashed line), matching′ them to the rails (solid line) 𝓚𝓚 In result, the square reshapes to the rhomb, with all four sides To better realize the problem, consider the motion of a equally tilted with respect to direction, hence equally body along the curvilinear path, e.g. the orbital motion. The deviated from and directions.′′ Now, the only real trajectory of orbiting body between two given points laid unsymmetrical element′′ in ′′the 𝐕𝐕whole composition are the on the circumference (say, distanced of less than half of it) is rails. We may however𝑥𝑥 complement𝑦𝑦 our (anyway) imaginary neither physically equivalent to the “shortcut” charted along experimental setup with the horizontal rails moving upwards the chord connecting these points, nor to the “shortcut” in direction with velocity ( ) . Despite this comprehended as a “jump” between observer’s rest frame rearrangement, direction of the elevator′′ motion in and the final inertial frame, both tangent to the orbit in said remains𝑦𝑦′′ untouched, being described𝐮𝐮 by𝑦𝑦 the same, equal in′′ points. In contrast to this, the elevator trajectory is itself a value velocity components. Consequently, we deal with two𝒦𝒦 “shortcut”, i.e. makes the straight line in spacetime, no alternative scenarios as to the “factual” interpretation of the matter what (inertial) frame it is observed from. In other horizontal and vertical components of the diagonal words, the straight diagonal path of the elevator motion in trajectory of elevator. According to the first one, the plane constitutes real trajectory in , and not the elevator moves upwards with respect to the vertical rails imaginary one. The frames (at rest to the′′ elevator) and moving to the left; according to the second one, the elevator 𝑥𝑥′′𝑦𝑦 ′′(at rest to the train) are, at′ least from 𝒦𝒦the viewpoint of moves to the left with respect to the horizontal rails moving the′′ principle of relativity, physically𝒦𝒦 identical to each other, upwards (Fig. 8). 𝒦𝒦 International Journal of Theoretical and Mathematical Physics 2019, 9(3): 81-96 87

In particular, in Sec.10 we’ll demonstrate how the introduced theory (and respective transformation) deals with the presented elevator case in various configurations. For now, let us only mention that the difference between the Lorentz transformation and its competitive equivalent consists (in this regard) in the distinction between relative and absolute motion, each one assumed as the source of “relativistic” effects.

3. Part II Alternative to Special Relativity: theory based on the assumption of preferred frame

1. Transformation regarding the postulate of preferred frame Figure 8. Elevator moving in along the direction inclined at the angle ° in plane. This motion′′ can be alternatively described as 1) The demonstrated inconsistency of Special Relativity directed upwards′′ relative′′ to vertical𝓚𝓚 rails moving to the left; or 2) directed to entails the need of introducing an alternative theory, 𝟏𝟏𝟏𝟏𝟏𝟏 𝒙𝒙 𝒚𝒚 the left relative to horizontal rails moving upwards. The contracted elevator mathematically consistent and compatible with all known does not match the rails in regardless of the “factual” scenario ′′ experiments. In the following sections we shall present and The above case evidently𝓚𝓚 shows that neither of directions discuss the Preferred Frame Theory (PFT) in this regard. connected with the shape of elevator can be treated as In contrast to STR founded on the principle of relativity privileged. Although the “argument from symmetry” adds (first postulate), PFT is founded on assumption that the nothing really substantial to that already has been said, yet so-called “relativistic effects” are in fact “absolute”, ergo this example is particularly expressive. detectable (directly or indirectly) in any inertial frame, 5. Conclusion to Part I including the rest frame of the observed object. Let us take a glance at the elevator-shaft thought Consequently, these effects are thought to be caused by the experiment describing it in a slightly different way, so to say, absolute motion, defined as the motion with respect to a in the “reverse order”, i.e. starting from the observer’s frame. certain physically preferred frame (absolute space). In a Assume that in the 2-dimensional observer’s frame, the two close link to that assumption (and in opposition to the second extended objects: the rectangular elevator and the parallel STR postulate) the velocity of light is assumed to be constant rails (2-dimensional elevator shaft) travel with the uniform (isotropic) in the preferred frame only, while in any other rectilinear motion, each one along different path; both inertial frame it is thought to depend on the observer’s forming the acute angle. With reference to the mentioned absolute velocity and the direction of emission, due to the objects, the respective directions of motion are defined as: real distortion of clocks and measuring rods at rest in the perpendicular to the rails and diagonal to the elevator, e.g. observer’s frame. If consequently applied, these assumptions parallel to the diagonal of rectangle. All the time, these do not lead to the Lorentz transformation (LT), but to a objects remain in a close mutual spatial relationship, namely different transformation formulated only some half-century elevator moves along (inside) the rails, while both in the rails later by Tangherlini [13], hereinafter called the Tangherlini and elevator rest frames the vertical side-edges of elevator transformation (TT): adhere to the rails. However, in the observer’s frame, each of = = + these objects is subject to the length contraction along ′ = = ′ ′ direction of its motion in this frame. In result, the rails get 𝑥𝑥 𝑥𝑥𝛾𝛾 − 𝑣𝑣𝑣𝑣𝑣𝑣 𝑥𝑥 𝑥𝑥 ⁄𝛾𝛾 𝑣𝑣𝑡𝑡 𝛾𝛾 ′ = = ′ closer to each other without changing their direction, while 𝑦𝑦 𝑦𝑦 𝑦𝑦 𝑦𝑦 the elevator reshapes from rectangle to rhomboid, therefore ′ = = ′ (4) 𝑧𝑧 𝑧𝑧 𝑧𝑧 𝑧𝑧 changing direction of all its sides. In consequence, elevator where Lorentz factor′ = (1 2 2′) 1 2 is the function and rails do not match each other in the observer’s frame. 𝑡𝑡 𝑡𝑡⁄𝛾𝛾 𝑡𝑡 𝑡𝑡 𝛾𝛾 of the absolute velocity . In contrast to− symmetry⁄ featuring Considering the perfect fit between elevator and rails in both LT, the primed and𝛾𝛾 unprimed− 𝑣𝑣 ⁄ 𝑐𝑐 coordinates are not their rest frames, this means a contradiction indicating an exchangeable in TT; unprimed𝑣𝑣 coordinates refer to the inconsistency of STR. preferred frame , while the primed ones refer to - the In order to find out a way from the above puzzle so as to frame in absolute motion. TT is usually regarded as′ the overcome this purely negative result, one has to take into special case of a 𝒦𝒦wider class of “equivalent” transformations𝒦𝒦 account the crucial (though persistently ignored) fact, including Lorentz transformation, based on the namely that Lorentz transformation is not the only Reichenbach’s synchronization parameter , related to the transformation predicting the length contraction and time possible difference between the one-way and two-way dilation. We shall examine this question in details in Part II. velocities of light (Reichenbach [14], Seller𝜀𝜀i [15], [16]). In

88 Maciej Rybicki: Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

general, Reichenbach’s parameter admits free value of measured in velocity of the preferred frame relates such that 0 < < 1, wherein = 1 2 corresponds with to as: ′ 𝜀𝜀 𝒦𝒦 𝒦𝒦 the one-way velocity of light equal to , in agreement with 2 𝜀𝜀 𝜀𝜀 ⁄ 𝑣𝑣 = = (5) the STR second postulate. In fact, as far as we can judge, the 𝑣𝑣𝑣𝑣𝑣𝑣 0 < < 1 = 1 2 𝑐𝑐 ′ case (opposing ) does not refer to a We call and the “corresponding”𝑣𝑣 𝑡𝑡⁄𝛾𝛾 𝑣𝑣𝛾𝛾 velocities. hypothetical set of “in-between” transformations, but applies 3. Velocity of light′ according to PFT specifically𝜀𝜀 to TT, whereas 𝜀𝜀the “freedom”⁄ of coefficient 𝑣𝑣 𝑣𝑣 corresponds with the differences in values of absolute The velocity of light undergoes basically the same rules as velocity of different frames, combined with different angles𝜀𝜀 any other absolute velocity. In the preferred frame , the of emission in each particular frame (see Sec.3 for details). light signal travels with the velocity c regardless of the In some respects, TT reminds the Galilean transformation motion of the source and the direction of emission𝒦𝒦 (is as opposed to the Lorentz transformation. Namely, unlike in isotropic). Provided Galilean transformation, the velocities LT, the time coordinate does not depend on space of light traveling along -axis in frame would be , coordinates; hence simultaneity is defined as absolute. On in positive direction (i.e. ′direction of motion′ of relative + 𝑥𝑥 𝒦𝒦 ′ 𝑐𝑐 − 𝑣𝑣 the other hand, TT is not symmetrical due to the principle of to ) and in negative direction. However, in 𝒦𝒦 relativity (i.e. does not comply neither with Lorentz nor accordance with TT, length contraction and time dilation 𝒦𝒦 𝑐𝑐 𝑣𝑣 Galilean symmetry), which distinguishes it from both affecting standards of length and time in contribute to respective velocities, yielding: ′ Galilean and Lorentz transformations. It forms instead a sort 2 𝒦𝒦2 of axial symmetry with the preferred frame acting as “axis of 1 = ( ) 2 = ( + ) (6) symmetry”. From the viewpoint of TT, the principle of where stands′ for the velocity′ of light in the preferred 𝑐𝑐 𝑐𝑐 − 𝑣𝑣 𝛾𝛾 𝑐𝑐 𝑐𝑐 𝑣𝑣 𝛾𝛾 relativity is a pure Galilean property connected with Galilean frame. For the so-defined velocities, one can prove: transformation, “emerging” at the limit 0 . 𝑐𝑐 0.5 < 1 2 < (7) Understandably enough, Tangherlini transformation did not In the general case, ′i.e. for arbitrary′ angle of receive broad attention so far. According to the prevailing𝑣𝑣 → 𝑐𝑐 𝑐𝑐 ≤ 𝑐𝑐 𝑐𝑐 ≤ 𝑐𝑐 ∞ opinion (e.g. [17]), TT is wholly equivalent in experiments to propagation of the ray in relation to positive direction, as 𝜑𝜑 the Lorentz transformation. Meanwhile, it is admittedly observed in , the velocity of light is:′ ′ cos 𝑥𝑥 equivalent to it in a wide class of cases, but not totally [18a, = 2 (8) 𝒦𝒦 1 cos 2 18b]. We shall develop this point in Ss. 4 and 5. ′ 𝑐𝑐−𝑣𝑣 𝜑𝜑 𝑣𝑣 The above form of Tangherlini transformation does not The light front determined𝑐𝑐 −�𝑐𝑐� by this𝜑𝜑 equation forms the exhaust all possibilities resulting from PFT; hence in this prolate spheroid (reducible to sphere in ), with the source regard it is more “specific” than Lorentz transformation and of emission placed in the “frontal” focus. In this regard, even the Lorentz boost. While in Eq. (4) the non-primed difference between STR and PFT reminds𝒦𝒦 the one between coordinates are the preferred ones, in the general case, both the circular orbits (as they were thought to be up to Galileo) frames may differ from the preferred frame; consequently, and Kepler’s ellipses (Fig. 9). they may remain in a free relationship as to their individual absolute velocities (including symmetrical case with equal absolute velocities) and to the direction of respective absolute velocity vectors. This all makes the general form of TT more complicated, nevertheless reducible to simpler cases according to the principle of correspondence [19]. Anyhow, for our present purposes the above form (TT) is fully sufficient and adequate. 2. “Corresponding” velocities between two frames vs. single relative velocity Unlike in LT, the velocity present in TT is not a unique “relative” velocity between the frames and . Instead, 𝑣𝑣 it stands solely for the velocity of measured′ in the Figure 9. Velocity of light in the absolutely moving inertial frame. The preferred frame , i.e. the absolute𝒦𝒦 ′ velocity𝒦𝒦 of . light front forms the prolate spheroid (ellipse in 2D depiction), with the Likewise, the Lorentz factor relates to 𝒦𝒦, being the function′ source of emission located in the “frontal” focus of absolute velocit𝒦𝒦y . Because the effects of length𝒦𝒦 ( ) contraction and time dilation are asymmetrical𝒦𝒦 with respect Because cos = cos + 180 , so the time of a round to and (they primarily𝑣𝑣 affect objects at rest in trip along the straight path 2 in frame∘ equals: 𝜑𝜑 −2 𝜑𝜑 ′ 2 ′ only), so mutual′ velocities of and cannot be equal.′ 2 2 1 cos 𝐿𝐿 1 cos𝒦𝒦 𝒦𝒦 𝒦𝒦 𝒦𝒦 = 𝑣𝑣 + 𝑣𝑣 The measured in velocity of is ′determined, apart ′ ( ) ′ ( ) 𝐿𝐿 � −�𝑐𝑐�cos 𝜑𝜑� 𝐿𝐿 � −+�𝑐𝑐�cos 𝜑𝜑� ′ from the absolute velocity′ of 𝒦𝒦 , also𝒦𝒦 by “distortion” of 2 2 𝑐𝑐−𝑣𝑣 𝜑𝜑 2 𝑐𝑐 𝑣𝑣 𝜑𝜑 2 measuring rods and𝒦𝒦 clocks at ′rest𝒦𝒦 in this frame. The 𝑇𝑇 2 1 cos 2 cos 2 = 𝑣𝑣 = 𝑣𝑣 = (9) 𝒦𝒦 ′ 2 2 cos 2 ′ 2 2 cos 2 ′ respective effects accumulate, in result of which the 𝐿𝐿 � −�𝑐𝑐� 𝜑𝜑�𝑐𝑐 𝐿𝐿 �𝑐𝑐− 𝑐𝑐 𝜑𝜑� 𝐿𝐿 𝑐𝑐 −𝑣𝑣 𝜑𝜑 𝑐𝑐 −𝑣𝑣 𝜑𝜑 𝑐𝑐 International Journal of Theoretical and Mathematical Physics 2019, 9(3): 81-96 89

Hence the averaged velocity on distance 2 is . indications between the moments of passing the front and Generally, the averaged velocity of light traveling along′ any rear ends of be . Assuming that velocity of in is closed path demarcated in any inertial frame equals𝐿𝐿 . 𝑐𝑐A , the velocity′ of clock relative to would′ be, in mathematical basis for this property is a theorem according accordance with𝐿𝐿 LT:𝑇𝑇 = , and according′ to𝒦𝒦 TT: 𝒦𝒦 = 2 to which an integral over free (smooth) closed curve𝑐𝑐 in 𝑣𝑣 . In turn, the travel time′ of the clock along𝒦𝒦 distance ′, as relation to an arbitrary straight line equals zero. determined in is,𝑣𝑣 according𝑣𝑣 to LT: = , 𝑣𝑣′and 4. STR vs. PFT: empirical equivalence in the area of 𝑣𝑣according𝛾𝛾 to TT: ′ = . Hence STR predicts:′ 𝐿𝐿 𝒦𝒦 𝑇𝑇 𝑇𝑇𝑇𝑇 kinematics ′ = = (10) 𝑇𝑇 𝑇𝑇⁄𝛾𝛾 a) Transformations of distance and time according to LT Instead, according′ to PFT′ ′ we have: and TT 𝐿𝐿 𝑣𝑣 𝑇𝑇 𝑣𝑣𝑣𝑣𝑣𝑣 = = 2 = (11) In order to describe the motion of an object in kinematical It follows that′ despite′ ′ the difference in predictions as to terms, one needs reliable measures of space and time, as 𝐿𝐿 𝑣𝑣 𝑇𝑇 𝑣𝑣𝛾𝛾 𝑇𝑇⁄𝛾𝛾 𝑣𝑣𝑣𝑣𝑣𝑣 the velocity of clock and the time of its travel along the referred to the observer’s frame. This demand does not measuring rod (“dependent” quantities in ) the effective present any difficulties in the case of Galilean transformation result (“independent” quantity ) is identical,′ which means (hence in the range of small velocities in the real world) in empirical equivalence of STR and′ PFT in this𝒦𝒦 case. which the mentioned measures are invariant. Instead, for the Case 2. Consider the motion𝐿𝐿 in of the measuring rod velocities comparable with it gets more complicated since at rest in , whose length as measured′ in equals . Let the definition of “measurement” depends on the choice of the rod, oriented in line with -axes,𝒦𝒦 pass a single clock at transformation. In a given reference𝑐𝑐 frame there are available, rest in 𝒦𝒦. We ask for the time′ indicated𝒦𝒦 by this 𝐿𝐿clock as obtained directly, only the readings of a single clock at rest between passing′ it by the front𝑥𝑥𝑥𝑥 and′ rear ends of the rod. As and the distance measured by stationary measuring rod. We for the point𝒦𝒦 -object formerly considered,𝑇𝑇 the velocity of rod call these quantities “independent”. The remaining quantities in is, according to LT: = , and according to 2 have to be obtained indirectly and depend on transformation; TT: ′ = . The length of the rod′ in is, according to hence we call them “dependent”. In other words, “dependent” LT:𝒦𝒦 ′ = , and according to𝑣𝑣 TT:𝑣𝑣 =′ . Hence, in means that respective quantity varies due to the choice of accord𝑣𝑣 ′ with𝑣𝑣𝛾𝛾 STR we have: 𝒦𝒦′ transformation (here alternatively LT and TT), while 𝐿𝐿 𝐿𝐿⁄𝛾𝛾 𝐿𝐿 𝐿𝐿𝐿𝐿 = = (12) “independent” means the opposite. Instead, PFT predicts:′ ′ ′ Let be the observer’s rest frame and the observed 𝑇𝑇 𝐿𝐿 ⁄𝑣𝑣 𝐿𝐿⁄𝑣𝑣𝑣𝑣 frame. According′ to PFT/TT (and to our formerly introduced = = 2 = (13) notation𝒦𝒦), is absolutely moving and 𝒦𝒦 determines As before, in spite′ of′ differences′ in predictions as to the absolute rest. ′Instead, according to STR/LT, both frames are 𝑇𝑇 𝐿𝐿 ⁄𝑣𝑣 𝐿𝐿𝐿𝐿⁄𝑣𝑣𝛾𝛾 𝐿𝐿⁄𝑣𝑣𝑣𝑣 length and velocity taken separately (“dependent” quantities entirely equivalent𝒦𝒦 in agreement with the𝒦𝒦 principle of in ), the effective result (“independent” quantity ) is relativity. In this subsection we shall test the uniform motion identical,′ which, as previously, means empirical equivalence′ using transformation of time and of length (distance) from of both𝒦𝒦 transformations in this case. 𝑇𝑇 to alternatively according to LT and TT, as well as defining ′ the velocity due to each transformation. In that b) Equivalence of synchronization conventions 𝒦𝒦 𝒦𝒦 way we shall infer “dependent” values straight from In this subsection we shall take virtual measurement of the transformation, treating respective values in as known velocity of a point-object using the readings of two distanced data. clocks at rest in . This task requires defining simultaneity 𝒦𝒦 In the case of point-object at rest in , defining its in the observer’s frame,′ which in turn involves the necessity uniform motion in means juxtaposing two “dependent” of choice of a definite𝒦𝒦 synchronizing convention related to 𝒦𝒦 quantities (velocity ′and time) with one “independent” given transformation. In particular, we shall test whether the (distance), measured𝒦𝒦 in . Instead, in the case of extended velocity defined according to STR, combined with object (measuring rod) at′ rest in , the “dependent” respective synchronization, is equivalent to the velocity quantities are its length and𝒦𝒦 velocity as referred to , while defined according to PFT, also combined with respective the “independent” quantity is time, comprehended𝒦𝒦 ′ as the synchronization. time-difference between two indications of a single𝒦𝒦 clock at Consider the motion in of a point-object (particle) at rest in . Hence, the criterion of equivalence consists rest in , along distance ′(measuring rod at rest in ) in satisfying′ the following condition: two “dependent” with two clocks 1 and 𝒦𝒦2′ on its both ends. As before,′ quantities𝒦𝒦 should give (respectively: in product and in velocity𝒦𝒦 of relative to𝐿𝐿 is . We synchronize these𝒦𝒦 quotient) a definite “independent” quantity, the same for clocks by the light′ 𝐶𝐶 ray, having𝐶𝐶 regard to the velocity of light each transformation. Below we shall test LT and TT defined alternatively𝒦𝒦 by LT 𝒦𝒦and TT.𝑣𝑣 Likewise, alternatively, according to these two arrangements. we determine velocity of the particle traveling along distance Case 1. Consider the motion of a point-object (clock) at from 2 to 1. An ultimate criterion of equivalence and rest in , on distance marked out in frame . Let the self′ -consistency of both descriptions is the indication of moving clock measure the′ proper time during its′ motion clock𝐿𝐿 𝐶𝐶 in the𝐶𝐶 moment of passing it by the particle. 𝒦𝒦 𝐿𝐿 𝒦𝒦 1 along . Let this time, defined as the difference of clock ′ 𝐶𝐶 𝐿𝐿 90 Maciej Rybicki: Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

Synchronization according to STR means the Einstein c) Empirical equivalence of the velocity addition formulae convention (EC) based on the STR light postulate. Hence, Let the frame move relative to frame with the while sending the ray from the clock 1 indicating time 1 velocity . Let the ′particle at rest in another frame ′ to the clock 2 distanced of , we should set it in the move with respect𝒦𝒦 to with the velocity𝒦𝒦 in the′′ ′ 𝐶𝐶 𝑡𝑡 moment of reaching it by the ray for the time: direction𝑣𝑣 parallel to the motion′ of relative to 𝒦𝒦 . 𝐶𝐶 𝐿𝐿 2 = 1 + (14) According to the Galilean𝒦𝒦 transformation,′ the 𝑉𝑉resultant 𝒦𝒦 𝒦𝒦 Still according to LT,′ the ′velocity′ of a particle along the velocity of this particle in would be: 𝑡𝑡 𝑡𝑡 𝐿𝐿 ⁄𝑐𝑐 distance from to is ; hence, the (denoted by ) = + (18) 2 1 3 𝒦𝒦 indication ′of clock 1 in the moment of passing it by the′ Instead, the respective velocity derived from LT is: 𝐿𝐿 𝐶𝐶 𝐶𝐶 𝑣𝑣 𝑡𝑡 𝜎𝜎 𝑣𝑣 𝑉𝑉 particle is: + 𝐶𝐶 = (19) = + + = + ( + ) 1+ 2 3 1 1 (15) 𝑣𝑣 𝑉𝑉 ′ ′ ′ ′ ′ ′ Instead, according to TT, the light synchronization in the Let us find an analogous 𝑣𝑣formula𝑣𝑣⁄𝑐𝑐 for PFT/TT. Since we 𝑡𝑡 𝑡𝑡 𝐿𝐿 ⁄𝑐𝑐 𝐿𝐿 ⁄𝑣𝑣 𝑡𝑡 𝐿𝐿 𝑣𝑣 𝑐𝑐 ⁄𝑣𝑣𝑣𝑣 𝜎𝜎 direction from 1 to 2 means that (due to Eq. (6)) one has consider the result of summation as referred to that is to set the clock 2 in the moment of reaching it by the ray assumed to be the preferred frame, so velocities and , as for: 𝐶𝐶 𝐶𝐶 absolute velocities of and respectively,𝒦𝒦 will not 𝐶𝐶 ′ ′′ 𝑣𝑣 𝜎𝜎 = + ( ) 2 (16) change in the PFT scenario. However, the question of 2 1 velocity looks different.𝒦𝒦 This velocity𝒦𝒦 refers to frame ′ ′ ′ Because, according to TT, the velocity of particle that according to STR is equivalent (physically identical)′ 𝑡𝑡 𝑡𝑡 𝐿𝐿 ⁄ 𝑐𝑐 − 𝑣𝑣 𝛾𝛾2 traveling along , from 2 to 1 is , so the clock 1 with , 𝑉𝑉whereas according to PFT differs from . 𝒦𝒦 will indicate in the′ moment of passing it by the particle the From the fact that and remain unchanged in PFT time: 𝐿𝐿 𝐶𝐶 𝐶𝐶 𝑣𝑣𝛾𝛾 𝐶𝐶 (compared𝒦𝒦 to respective values in STR) it follows𝒦𝒦 that 2 2 3 = 1 + ( ) + velocity of the particle𝜎𝜎 relative𝑣𝑣 to , as referred to , ′ ′ ′(1 2 2) ′ (1 2 2) equals: ′ 𝑡𝑡 = 𝑡𝑡 + 𝐿𝐿 ⁄ 𝑐𝑐 − 𝑣𝑣 𝛾𝛾 +𝐿𝐿 ⁄𝑣𝑣𝛾𝛾 𝒦𝒦 𝒦𝒦 1 ′ ( ) ′ = (20) ′ 𝐿𝐿 − 𝑣𝑣 ⁄𝑐𝑐 𝐿𝐿 − 𝑣𝑣 ⁄𝑐𝑐 𝑡𝑡 ( 3 2 + 2 + 3 2) Because the measuring rods𝒦𝒦 are contracted in and the = + 𝑐𝑐 − 𝑣𝑣 𝑣𝑣 𝑉𝑉 𝜎𝜎 − 𝑣𝑣 1 ′ 2 clocks are retarded by the Lorentz factor (respectively:′ ′ = ( ), = ( )), hence in frame the 𝒦𝒦velocity of 𝐿𝐿 𝑣𝑣 −2 𝑣𝑣 ⁄𝑐𝑐 𝑐𝑐 −( +𝑣𝑣 )− 𝑣𝑣 ⁄𝑐𝑐 𝑣𝑣 ⁄𝑐𝑐 = 𝑡𝑡1 + = 1 + (17) the′ particle is:′ ′ ′ 2 𝑣𝑣𝑣𝑣′ − 𝑣𝑣 𝑣𝑣 𝑣𝑣 𝐿𝐿 �𝑐𝑐−𝑣𝑣 ⁄𝑐𝑐� 𝐿𝐿 𝑣𝑣 𝑐𝑐 𝑙𝑙 𝑙𝑙⁄𝛾𝛾 𝑡𝑡 𝑡𝑡𝛾𝛾 𝒦𝒦 ′ ′ = 2 (21) This𝑡𝑡 result𝑣𝑣𝑣𝑣 −is𝑣𝑣 identical𝑡𝑡 with𝑣𝑣𝑣𝑣 that obtained in Eq. (15), ( ) which signifies empirical equivalence between STR and PFT ′ (We use symbol (𝑉𝑉)𝒦𝒦 instead𝛾𝛾 𝑣𝑣 𝑉𝑉of𝒦𝒦 simply , since, in the in this case. In general, we may take for granted (and thus following equations, we shall use the Lorentz factor also as a 𝑣𝑣 formulate as a rule) that relative simultaneity combined with function of other than𝛾𝛾 velocities, measured𝛾𝛾 in ). constant velocity of light due to STR always gives identical Eventually, the PFT velocity addition formula applied to results to these obtained from absolute simultaneity summation in the preferred𝑣𝑣 frame has the form: 𝒦𝒦 combined with inconstant velocity of light due to PFT. The = + 2 (22) so-defined equivalence covers large experimental area, 𝒦𝒦( ) ′ which provides an explanation of why STR has not been Let us compare𝜎𝜎 this𝑣𝑣 formula𝑉𝑉𝒦𝒦 ⁄𝛾𝛾 𝑣𝑣with the STR velocity questioned in experiments (or disproved by thought addition (Eq. (19)). Our goal is to test whether velocity , experiments) for more than a hundred years. defined according to LT, is empirically equivalent to However, for the reasons of geometrical inconsistency of velocity , defined according to TT. Let us convert Eq.𝑉𝑉 STR demonstrated in Part I; and also considering departure (19) to the form:′ 𝒦𝒦 from empirical equivalence in dynamics (discussed in Part II, 𝑉𝑉 1 2 2 = + = + (23) 1+ 2 (1+ 2) 2 Sec. 6), synchronization according to Einstein convention is −𝑣𝑣 ⁄𝑐𝑐 𝑉𝑉 ( ) applicable in the preferred frame only. In any other inertial Now, the𝜎𝜎 STR𝑣𝑣 velocity𝑉𝑉 � 𝑣𝑣 𝑣𝑣addition⁄𝑐𝑐 � 𝑣𝑣 reminds𝑣𝑣𝑣𝑣 ⁄respective𝑐𝑐 𝛾𝛾 𝑣𝑣 PFT frame, synchronization utilizing velocities of light defined formula (i.e. Eq. (22)); moreover, both formulae will totally according to Eq. (8) will confirm (will be identical with) the coincide if we define velocity as: synchronization obtained from EC in the preferred frame. ′ = 𝒦𝒦 (24) Concluding: synchronization performed according to EC in (1+𝑉𝑉 2) 𝑉𝑉 the preferred frame determines absolute simultaneity. As is ′ Hence, the question of𝒦𝒦 equivalence𝑣𝑣𝑣𝑣⁄𝑐𝑐 between the formulae known, Einstein convention considered as applicable in any 𝑉𝑉 (19) and (22) reduces to another question, namely whether inertial frame is inseparable from the postulate of relative velocities and are distinguishable in kinematical simultaneity; consequently, synchronization according to experiments. To answer′ this question, let us consider the 𝒦𝒦 PFT is inseparable from the postulate of absolute following example.𝑉𝑉 𝑉𝑉 Let the “added” velocity (defined by simultaneity. STR as , and by PFT as ) be the velocity of a rod at rest in of the length , measured′ in . Let this rod pass the 𝒦𝒦 ′′ 𝑉𝑉 𝑉𝑉 𝒦𝒦 𝐿𝐿 𝒦𝒦 International Journal of Theoretical and Mathematical Physics 2019, 9(3): 81-96 91

clock at rest in . As it was in Case 2, the criterion of where ( ) denotes the absolute Lorentz factor for the empirical equivalence′ between LT and TT are the readings observer’s𝑂𝑂 frame , ( 1) - absolute Lorentz factor for (equal or not) of the𝒦𝒦 clock in the moment of passing it by the 𝛾𝛾 ′′ “intermediate” frame ,𝑉𝑉 1 - velocity of “intermediate” rear end of the rod. To fulfil the criterion of equivalence, frame relative𝒦𝒦 to 𝛾𝛾′ , as measured in the preferred respective readings should be identical despite different (due frame .′ Velocity 1 𝒦𝒦corresponds′′ 𝑉𝑉 to velocity in Eq. (22). to LT and TT) values of the rod length and its velocity, both Instead,𝒦𝒦 2 denotes velocity𝒦𝒦 of the particle in “intermediate” inserted to the equation for time. We assume that clock, frame 𝒦𝒦 , namely: 𝑉𝑉 𝑣𝑣 when passed by the front end of the rod, in both scenarios 𝑉𝑉 ′ = ( ) 2 (32) indicates identical time 0 = 0. According to LT, the length 𝒦𝒦 2 1 ( 1) of rod in is = . Instead, according to TT, the ( ) corresponding to velocity𝑉𝑉 𝜎𝜎 − 𝑉𝑉in Eq.𝛾𝛾 𝑉𝑉 (22). length of rod′ results′ from𝑡𝑡 the mutual relation of observer’s 𝑉𝑉 5. STR vs. PFT: identical predictions′ concerning frame and𝒦𝒦 the rod𝐿𝐿 rest𝐿𝐿 ⁄frame𝛾𝛾 (expressed by the quotient of 𝑉𝑉𝒦𝒦 momentum respective Lorentz factors), both referred to the preferred frame : Assume that an object (particle) at rest in falls apart into two parts (here called “forward” and “backward”′ = (25) 𝒦𝒦 ( ) ( ) particles, denoted respectively and )𝒦𝒦 receding in ′ 𝑣𝑣 𝜎𝜎 opposite directions along -axis so that net momentum in The requirement 𝐿𝐿of identical𝐿𝐿𝛾𝛾 ⁄𝛾𝛾 clock indications is tantamount to the requirement of equality between respective amounts to zero. Let ′“forward”ℱ meansℬ the direction times. Hence, we have to settle if: compatible′ with the motion𝑥𝑥 of relative to , while ? “backward”𝒦𝒦 - the opposite direction.′ Let the rest mass of = ( ) (26) each particle be 0 . In accordance𝒦𝒦 with STR,𝒦𝒦 the ( ) (𝐿𝐿𝛾𝛾) 𝑣𝑣 𝐿𝐿 = 1 ′ momentum of each particle in (for ) is: Because: 𝛾𝛾 𝑉𝑉 𝑉𝑉 𝛾𝛾 𝜎𝜎 𝑉𝑉𝒦𝒦 𝑚𝑚 = = ′ (1 2)1 2 (33) = (1 + 2) (27) 0 ( ) 𝒦𝒦0 𝑐𝑐 ⁄ where is the relative𝑉𝑉 velocity between each of the so, Eq. (23) takes the form:′ 𝑝𝑝 𝑚𝑚 𝑉𝑉𝛾𝛾 𝑚𝑚 𝑉𝑉⁄ − 𝑉𝑉 𝑉𝑉 𝑉𝑉𝒦𝒦 𝑣𝑣𝑣𝑣⁄𝑐𝑐 considered particles and frame . Instead, in accordance ( ) 𝑉𝑉 2 = (28) with PFT, the respective momentum′ is: ( ) (1+ ) ( ) 𝐿𝐿 𝐿𝐿𝛾𝛾 𝑣𝑣 𝒦𝒦 2 = 0( ) (diff ) (34) Simplifying and rearranging𝛾𝛾 𝑉𝑉 𝑉𝑉𝒦𝒦′ 𝑣𝑣𝑣𝑣 ⁄gives:𝑐𝑐 𝛾𝛾 𝜎𝜎 𝑉𝑉𝒦𝒦′ ′ 2 The term 0( ) denotes𝐵𝐵 𝐴𝐴the mass 𝐴𝐴of a given particle ( ) 1+ 1 𝑝𝑝 𝑚𝑚 𝛾𝛾 ⁄𝛾𝛾 𝑣𝑣 𝛾𝛾 = (29) as measured in , where = (1 2) 1 2 is the 𝛾𝛾 𝑣𝑣 � ( 𝑣𝑣)𝑣𝑣⁄𝑐𝑐 � ( ) 𝐵𝐵 𝐴𝐴 Lorentz factor𝑚𝑚 for𝛾𝛾 ⁄ 𝛾𝛾the′ observer’s frame −, ⁄and = After rewriting Lorentz factors𝛾𝛾 𝜎𝜎 , we obtain𝛾𝛾 𝑉𝑉 (for = 1): 2 1 2 𝐴𝐴 (1 ) the Lorentz𝒦𝒦 factor for𝛾𝛾 the particle,− 𝑣𝑣 ′ both related 𝐵𝐵 2 2 2 to the preferred− ⁄ frame . Instead, 𝒦𝒦 denotes𝛾𝛾 the 1 + (1 + ) (𝑐𝑐 + ) (diff ) 1 (1 + ) = − 𝜎𝜎 1 2 1 + 1 2 measured in relative velocity between each of 𝑣𝑣 𝑉𝑉 𝑣𝑣𝑣𝑣 − 𝑣𝑣 𝑉𝑉 𝒦𝒦 𝑣𝑣 � − � � 𝑣𝑣𝑣𝑣 � (alternatively considered) particles: or and frame √ − 𝑣𝑣 1+( )2𝑣𝑣𝑣𝑣2 2 (1 2)(1 2) − 𝑣𝑣1 (precisely: the 𝒦𝒦absolute value of the vector difference′ = = = (30) 1 2 1 2 𝑣𝑣𝑣𝑣 −𝑣𝑣 −𝑉𝑉 −𝑣𝑣 −𝑉𝑉 ( ) between velocity of given particle andℱ velocityℬ of , each𝒦𝒦 | | which confirms� empirical−𝑣𝑣 equivalence� − 𝑣𝑣between 𝛾𝛾Eq.𝑉𝑉 (19) and case measured in ; hence (diff ) = ). Because′ both 𝒦𝒦 Eq. (22). the mass and velocity of and depend on the direction 𝒦𝒦 𝑣𝑣 𝛔𝛔 − 𝐯𝐯 Eq. (22) is however not the general PFT formula for the of their motion in , hence and (diff ) are defined separately for each particle,′ ℱ henceforthℬ respectively denoted: velocity addition since (first of all) it applies exclusively to 𝐵𝐵 summation in the preferred system . This in particular ( ) , ( ) and ( 𝒦𝒦) , ( ) . Instead,𝛾𝛾 Lorentz𝑣𝑣 factor remains unchanged in both cases. means that final velocity (here denoted ) is identical for ℱ ℬ ℱ ℬ 𝐴𝐴 STR and PFT. Meanwhile, in a more𝒦𝒦 general case of 𝛾𝛾 Our 𝛾𝛾goal is to 𝑣𝑣check 𝑣𝑣whether the momenta of and 𝛾𝛾 summation in positive direction, performed𝜎𝜎 in a frame equal each other, and whether they equal the momentum different from the preferred one (here denoted ′′), the final defined according to STR. Before entering to analysis,ℱ let’sℬ velocities derived𝑥𝑥 alternatively from LT and TT might be make an obvious simplification of Eq. (34) reducing its different. Apart from regarding the absolute Lorentz𝒦𝒦 factor “explanatory” form to: ′ for the “intermediate” frame ( in Eq. (22)) we must also = 0 (diff ) (35) ′′ take into account the absolute Lorentz factor for , i.e. the ′ Taking advantage of the equivalence𝐴𝐴 𝐵𝐵 of addition formulae one in which summation is performed.𝒦𝒦 In other words, Eq. 𝑝𝑝 𝑚𝑚 𝛾𝛾 𝛾𝛾 𝑣𝑣 (22) refers to the particular case in which the Lorentz𝒦𝒦 factor according to STR (19) and PFT (22), we define the particles for the observer equals 1, while in the general case it may velocity in as: ± differ from 1. The generalized “collinear” velocity addition ( , ) = (36) 𝒦𝒦 1± formula reads: 𝑣𝑣 𝑉𝑉 with positive signs for and ne𝑣𝑣gative𝑣𝑣 for . Hence, = 2 + 2 (31) 𝜎𝜎 ℱ ℬ ( ) 1 2 ( 1) respective absolute Lorentz factors are: ℱ ℬ 𝜎𝜎 𝛾𝛾 𝑂𝑂 �𝑉𝑉 𝑉𝑉 ⁄𝛾𝛾 𝑉𝑉 �

92 Maciej Rybicki: Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

+ 2 We call this effect “energy-momentum disparity” - a = 1 (37) ( ) 1+ phenomenon expected to take place in any absolutely 𝑣𝑣 𝑉𝑉 moving frame. The “energy-momentum disparity” has and 𝛾𝛾 ℱ � − � 𝑣𝑣𝑣𝑣� (among others) the following properties. Independently of 2 = 1 (38) the absolute velocity of inertial observer, the momenta of ( ) 1 𝑣𝑣−𝑉𝑉 moving particles are the same as they are in STR. Besides, ℬ After substituting to𝛾𝛾 Eq. (35�) we− obtain� −𝑣𝑣𝑣𝑣� the momentum 1 for equal momenta of particles moving in opposite directions, for particle : ′ their averaged energy is for all angles identical with that + 𝑝𝑝 defined by STR. In this specific sense, we may call the ℱ 0 1 + = = = averaged energy isotropic. 1 0 ( ) ( ) 𝑣𝑣 𝑉𝑉 ′ 𝑚𝑚 � + 2− 𝑣𝑣� 7. PFT: energy of massive body 𝑝𝑝 𝑚𝑚 𝛾𝛾𝐴𝐴𝛾𝛾 ℱ 𝑣𝑣 ℱ 1 𝑣𝑣𝑣𝑣 (1 2) 1 + Let us replace the “working” symbols , and 𝑣𝑣 𝑉𝑉 ( ) ( ) �� −2 � � � − 𝑣𝑣 0 (1 ) 𝑣𝑣𝑣𝑣 used in previous sections by the “general” ℱones:ℬ and 𝐴𝐴, = = where will denote the absolute Lorentz𝛾𝛾 factor𝛾𝛾 for the𝛾𝛾 +𝑚𝑚 𝑉𝑉2 − 𝑣𝑣 𝛾𝛾 Γ 1 (1 + )2(1 2) observed object and - the absolute Lorentz factor for 1 + observer.𝛾𝛾 According to PFT, in an arbitrary, absolutely 𝑣𝑣 𝑉𝑉 �� − � 2 � � 𝑣𝑣𝑣𝑣 − 𝑣𝑣 moving inertial frame (hereΓ represented by ) total energy 0 1 𝑣𝑣𝑣𝑣 0 = = = 0 ( ) (39) of the observed object is given by formula: ′ (1 2)(1 2)2 1 2 𝑚𝑚 𝑉𝑉� −𝑣𝑣 � 𝑚𝑚 𝑉𝑉 0 2 𝒦𝒦 𝑉𝑉 = (45) Similarly,� we−𝑉𝑉 derive−𝑣𝑣 the momentum√ −𝑉𝑉 𝑚𝑚2 for𝑉𝑉𝛾𝛾 particle : ′ 𝑚𝑚 𝛾𝛾 ′ For the object at rest𝐸𝐸 in Γ 𝑐𝑐 one has = 1 ; 0 1 𝑝𝑝 ℬ 2 = = = consequently, the above equation ′reduces to = 0 , 2 0 ( ) ( ) 𝑣𝑣 − 𝑉𝑉 ′ 𝑚𝑚 � 2 � which coincides with STR formula𝒦𝒦 for the body𝛾𝛾⁄ Γat rest. 𝐴𝐴 ℬ ℬ − 𝑣𝑣𝑣𝑣(1 2) 𝑝𝑝 𝑚𝑚 𝛾𝛾 𝛾𝛾 𝑣𝑣 1 1 Instead, in the preferred frame (i.e. for =𝐸𝐸1) 𝑚𝑚for 𝑐𝑐an 𝑣𝑣 − 𝑉𝑉 arbitrary value of : (0 < < ) Eq. (45) reduces to: �� −2� � � − 𝑣𝑣 0 (1 ) − 𝑣𝑣𝑣𝑣 𝒦𝒦 2 Γ = = = 0 (46) 2 𝑣𝑣 𝑣𝑣 𝑐𝑐 𝑚𝑚 𝑉𝑉 − 𝑣𝑣 2 2 which coincides with the STR formula for the moving body. 1 (1 ) (1 ) 𝐸𝐸 𝑚𝑚 𝛾𝛾𝑐𝑐 1 In turn, in the case of an object at rest in the preferred frame 𝑣𝑣 − 𝑉𝑉 �� − � �2� − 𝑣𝑣𝑣𝑣 − 𝑣𝑣 , observed in the moving frame (i.e., for = 1 , 0−(𝑣𝑣1𝑣𝑣 ) = = 0 ( ) (40) (1+ 2 2 2 2)(1 2) > 1) Eq. (45) reduces to: ′ 𝑚𝑚 𝑉𝑉 −𝑣𝑣 𝒦𝒦 2 𝒦𝒦 𝛾𝛾 𝑉𝑉 0 This means� 𝑣𝑣 equality𝑉𝑉 −𝑣𝑣 −𝑉𝑉 of momenta−𝑣𝑣 𝑚𝑚 of 𝑉𝑉the𝛾𝛾 particles and , Γ = (47) as well as equality of respective momenta with the ′ 𝑚𝑚 𝑐𝑐 momentum defined according to STR. ℱ ℬ meaning that in this case 𝐸𝐸total energyΓ is smaller than rest energy - the result only apparently paradoxical. 6. PFT: disparity between energy and momentum in the Let us express energy in terms of energy-momentum absolutely moving frames (asymmetry of energy disparity and, generally, in terms of energy-momentum distribution) relation. The respective STR equation is: Before rewriting Eq. (31), we have reduced it to Eq. (35); 2 2 2 now however, we’ll turn back to the original form since we = ( ) + ( 0 ) (48) need separate expressions for mass in frame . The mass According to PFT, in the general case related to an 𝐸𝐸 � 𝑝𝑝𝑝𝑝 𝑚𝑚 𝑐𝑐 of particle is: ′ arbitrary absolutely moving observer’s frame , energy 2 𝒦𝒦 ′ ( ) 0 1 depends on the magnitude of energy-momentum disparity ℱ 1 = 0 = (41) + 2 that, in turn, depends on the direction of motion𝒦𝒦 of the 𝛾𝛾 ℱ 𝑚𝑚1 � −𝑣𝑣 ′ 1+ particle in . The respective formula reads: 𝑚𝑚 𝑚𝑚 𝛾𝛾𝐴𝐴 𝑣𝑣 𝑉𝑉 � −� 𝑣𝑣𝑣𝑣 � ′ 2 Instead, the mass of particle is: 2 2 2 =𝒦𝒦 ( ) + ( 0 ) 1 cos (49) ( ) 1 2 0 ′ ′ 𝑣𝑣 2 = 0 ℬ= 2 (42) 𝛾𝛾 ℬ 𝑚𝑚1 � −𝑣𝑣 where 𝐸𝐸 is the� velocity𝑝𝑝 𝑐𝑐 of𝑚𝑚 𝑐𝑐 measured� − � 𝑐𝑐in� preferred𝜑𝜑� frame ′ 1 𝛾𝛾𝐴𝐴 𝑣𝑣−𝑉𝑉 , and - the measured in ′ angle between the velocity 𝑚𝑚 𝑚𝑚 � −� � It follows that 1 2 . Provided−𝑣𝑣𝑣𝑣 the mass-energy vector 𝑣𝑣of the motion of 𝒦𝒦relative′ to and the velocity equivalence, this implies′ different′ total energies of particles vector𝒦𝒦 for𝜑𝜑 the observed particle𝒦𝒦 (hence, ′the case = 0 , 𝑚𝑚 ≠ 𝑚𝑚 and , despite equality of their momenta. Let us write this cos = 1 corresponds to𝒦𝒦 the “backward”𝒦𝒦 direction of the∘ result: particle motion; in the particular case of a particle at𝜑𝜑 rest in ℱ ℬ ( ) = ( ) (43) , coinciding𝜑𝜑 with Eq. (47)). It is clear that “orthogonal” case = 90 , cos = 0 is the only one for which the STR (ℱ) (ℬ ) (44) 𝑝𝑝 𝑝𝑝 energy𝒦𝒦 -momentum∘ relation holds in the absolutely moving 𝜑𝜑 𝜑𝜑 𝐸𝐸 ℱ ≠ 𝐸𝐸 ℬ frame.

International Journal of Theoretical and Mathematical Physics 2019, 9(3): 81-96 93

8. PFT: energy and momentum of photon the theoretically more general but observationally less According to STR, energy of photon is given by the helpful terms, the preferred frame is the one described by the formula: FLRW metric. Eventually, one could identify the preferred frame with a hypothetic ubiquitous field endowed with = = (50) fundamental properties, possibly connected with gravity. Instead, the photon momentum defined by STR is: 𝐸𝐸 ℎ𝑓𝑓 ℎ𝑐𝑐⁄𝜆𝜆 As is known from the relevant Doppler effect (known as “dipole pattern”), the velocity of Earth (taken together with = = (51) ℎ ℎ𝑓𝑓 Solar System) with respect to isotropic CMB amounts to ca. ( - Planck constant, 𝑝𝑝 - frequency,𝜆𝜆 𝑐𝑐 - speed of light, 370 km/s in a certain definite direction, which gives the - photon wavelength) Let the source of emission be at rest in value of absolute Lorentz factor ~1 + 10 7. Apart from the ℎabsolutely moving frame𝑓𝑓 . According𝑐𝑐 to PFT, the𝜆𝜆 this main source of absolute motion we should− take into velocity of light in depends on′ absolute velocity of account the orbital motion of Earth𝛾𝛾 with the relative velocity and on direction of emission,′ in𝒦𝒦 accordance with Eq. (8). For′ ±30 km/s, yielding the difference in absolute Lorentz factors the source of emission𝒦𝒦 at rest in the frequency observed𝒦𝒦 ~10 9, and the rotational motion of Earth, on the average in is identical in all directions ′(is isotropic), which is a much −less than 1km/s, yielding respective difference 12 consequence′ of the homogeneity𝒦𝒦 of time in any inertial ∆𝛾𝛾~10 . These values, combined with each other, as well frame.𝒦𝒦 More specifically, for any particular absolute velocity as with −different orientation in space of experimental devices of combined with any arbitrary frequency of the source ∆(e.g.𝛾𝛾 linacs), should be compared with the “basic” Lorentz ( 0) and′ the angle of emission ( ), one has: factors gained by the particles accelerated in experiments. 𝒦𝒦 They depend on the type of matter particle and on the power ( ) 𝑓𝑓 ′ = ( ) =𝜑𝜑0 = . (52) and size of accelerator; yet we may assume that extreme 𝑐𝑐(𝜑𝜑 ) values vary between ~103 (for protons) and ~105 (for ′ 𝜑𝜑 where denotes 𝜆𝜆the𝜑𝜑 angle𝑓𝑓 of emission,𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 and 0 - frequency electrons). This shows difficulties, or even impossibility, in of the source. This explains the negative results of settling between STR𝛾𝛾 and PFT in “usual” experiments𝛾𝛾 Michelson𝜑𝜑 -Morley and Kennedy-Thorndike𝑓𝑓 experiments, concerning energy. Fortunately, at least in the part despite the predicted by PFT inconstant (anisotropic) concerning the question of validity of STR, this goal can be velocity of light and anisotropic wavelength. Rearranging achieved by pure logic, as shown in Part I. However, as it and implementing the Planck constant gives: many times happened in the past, even very faint effects may not surpass the capability of experiments specially aimed for = = 0 (53) ( ) ( ) that purpose. ′ ℎ𝑐𝑐 𝑐𝑐 ′ ′ 𝜆𝜆 𝑐𝑐 It follows that, for𝐸𝐸 a given𝜑𝜑 frequencyℎ𝑓𝑓 𝜑𝜑 0 of the source, the 10. Back to the elevator: description according to PFT observed energy of photon depends on the direction of The case discussed in Part I may have various options emission, being inversely proportional𝑓𝑓 to the wavelength when described according to PFT, due to different location (and to velocity), while the observed frequency remains of rails, elevator and observer with respect to the preferred constant, regardless of the direction of mission. This frame. However, there is no need to consider them in large differentiates PFT from the respective STR description, number; few characteristic examples will do just fine. More according to which “inversely proportional to wavelength” is important are the general rules concerning length contraction, tantamount with “proportional to frequency”. The latter different from those appropriate for STR. The main holds (exactly) in PFT in the preferred frame only. From Eq. difference is the fact that the primal source of length (53) one derives the momentum of photon: contraction and time dilation is the absolute (hence not relative) motion. Specifically the PFT rules are: 1) length = 2 ( ) = 2 ( ) = (54) ( ) ′ 𝐸𝐸 ′ 𝑐𝑐 ′ ℎ𝑓𝑓 contraction of an object in absolute motion is a real physical 𝜑𝜑 ′ 𝜑𝜑 identical with 𝑝𝑝that predicted𝑐𝑐 𝑐𝑐 byℎ𝑓𝑓 STR.𝑐𝑐 𝜑𝜑 𝑐𝑐 It𝑐𝑐 turns out𝑐𝑐 that, unlike effect; it occurs always in line of this motion; 2) Inertial, energy, momentum does not depend on the direction of absolutely moving observer (inertial frame) equipped with emission (is isotropic). Concluding: As in the case with measuring rods undergoes the length contraction on the same massive particles, the photon is subject to the footing as the observed object; 3) Contraction of the “energy-momentum disparity” and the relevant “asymmetry observer’s measuring rods superimposes on contraction of of energy distribution”. In general, PFT is not exactly the observed object yielding the final effect in the observer’s equivalent to STR in the predictions regarding energy. reference frame. In particular, contraction of the measuring rod at rest in observer’s frame results in an adequate 9. Difference between the STR and PFT predictions as to elongation of the observed object in the direction of the energy “in practice” absolute motion of observer. (The same pertains to clocks It is a matter of reasonable presumption (at least until and time dilation.) Below, we shall adhere to these rules. better arguments for or against are provided) that physically Let us first consider the case analogous to the one preferred frame coincides with the cosmologically preferred discussed in Part I; however now will be defined as the frame, i.e. the one described by the exactly isotropic preferred frame. The vertical rails are at rest in , the radiation of the cosmic microwave background (CMB). In elevator, at rest in , moves in 𝒦𝒦 with velocity , and ′ 𝒦𝒦 𝒦𝒦 𝒦𝒦 𝐕𝐕 94 Maciej Rybicki: Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

frame moves to the right perpendicularly to the rails to be a contradiction, similar to that revealed in Part I. But it with velocity′′ , exactly as shown in Fig. 2. While the is not; looks can be deceiving. The elevator is indeed elevator𝒦𝒦 matches the rails in and in , the question is if contracted in in the diagonal direction; however, this it does the same𝐮𝐮 in , i.e. whether the′ observations in only means that if this elevator (i.e. the one moving upwards , and are mutually′′ 𝒦𝒦 compatible𝒦𝒦 . (We could, of in ) would 𝒦𝒦move upwards (or stay at rest) in frame , course,′ consider′′ also the𝒦𝒦 cases in which the elevator does not then its′′ shape in would be rhomboidal, not rectangular. 𝒦𝒦match𝒦𝒦 the rails𝒦𝒦 in all three frames; however, for obvious Observer𝒦𝒦 should take this into account once received 𝒦𝒦an reasons, the “well operating” elevator provides us with the order from the 𝒦𝒦observer for making an elevator more conclusive examples.) well-operating𝒦𝒦 in ! The elongated′′ rhomboidal elevator Because the elevator moves in in vertical direction, undergoes in the length′′ contraction𝒦𝒦 in line of its diagonal so contraction takes place in this very direction, i.e. motion in this frame,𝒦𝒦 as a result of which it takes the shape of perpendicularly to vector . In consequence,𝒦𝒦 contraction in a rectangle in𝒦𝒦 (granted that its rhomboidal shape was becomes preserved in . Instead, contraction of the properly matched ′′in ). At the same time, distance between frame superimposes 𝐮𝐮on ′′that primal contraction. Any the rails becomes𝒦𝒦 contracted in in the horizontal direction. 𝒦𝒦horizontal′′ (parallel to )𝒦𝒦 measuring rod at rest in Consequently, if elevator𝒦𝒦 and rails fit together in they becomes𝒦𝒦 contracted by the Lorentz factor ( ). This, of′′ also fit together in frame 𝒦𝒦. In general, compatibility′′ course, is not perceived 𝐮𝐮directly in , but just in result𝒦𝒦 of between respective observations follows from the 𝒦𝒦fact that, that the elevator, together with rails, ′′undergoes𝛾𝛾 𝑢𝑢 in an according to PFT, contraction𝒦𝒦 is a real physical effect adequate elongation in the horizontal𝒦𝒦 direction. Finally,′′ resulting from the absolute motion (Fig. 12). despite the difference in proportions in the frames compared𝒦𝒦 among themselves, elevator and rails match each other in frame just like they do in frames and (Fig. 10). ′′ ′ 𝒦𝒦 𝒦𝒦 𝒦𝒦

Figure 11. Elevator (at rest in ), rails (at rest in ) and frame moving to the left; all three observed′ in frame ′′ 𝓚𝓚 ′′ 𝓚𝓚 𝓚𝓚 𝓚𝓚

Figure 10. The rails (at rest in ), elevator (at rest in ), both observed in . In frame elevator and rails match each other just′ like they do in . Interpreting′′ the′′ length contraction𝓚𝓚 in terms of real𝓚𝓚 effect caused by absolute𝓚𝓚 motion ensures𝓚𝓚 compatibility between observations performed in 𝓚𝓚 and ′′ 𝓚𝓚 Consider𝓚𝓚 now another example. Let an ultimate observer’s frame be the preferred frame . Let , now being the rest frame for vertical rails, move to the right′′ with velocity . Let be the rest frame for𝒦𝒦 the elevator𝒦𝒦 moving upwards in ′ with velocity . (Notice that frame is now𝐮𝐮 different𝒦𝒦′′ from in′′ previous example.) From′ the viewpoin𝒦𝒦 t of , frame′ 𝐕𝐕 moves to the left with𝒦𝒦 velocity 2 ( ) (Fig. 11). ′′ 𝒦𝒦 𝒦𝒦 𝒦𝒦 As𝑢𝑢 observed in , the elevator (generally, frame ) 𝐮𝐮moves𝛾𝛾 in the diagonal direction, from bottom left to top right,′ 𝒦𝒦 𝒦𝒦 Figure 12. The rails (at rest in ), elevator (at rest in ), both observed with a certain resultant velocity , related to and . in frame . Elevator reshapes in ′′ from rhomboid to rectangle;′ in result, The motion with respect to is the absolute motion; hence′′ elevator and rails match each 𝓚𝓚other in just like they𝓚𝓚 do in . The elevator undergoes the (real) length𝐕𝐕 contraction𝐮𝐮 along 𝐕𝐕the dashed line𝓚𝓚 depicts the shapes of elevator𝓚𝓚 and of rails as if they were′′ at rest diagonal direction parallel to𝒦𝒦 vector . Seemingly, it looks in 𝓚𝓚 𝓚𝓚 𝓚𝓚 𝐕𝐕 International Journal of Theoretical and Mathematical Physics 2019, 9(3): 81-96 95

Let us stay at the above arrangement. Consider now 4. Conclusions observations made by observer traveling, say, inside the elevator. The elevator is contracted′ in from the rhomboid The evidence of inconsistency taken in the thought to rectangle; however, observer𝒦𝒦 does not observe this experiment (Part I) leaves no doubt that Special Theory of Relativity should be replaced by a different theory. Likewise, effect directly, still perceiving elevator′ 𝒦𝒦 as rhomboidal. This the arguments presented in Part II, despite their elementary fact, obvious for STR, is obvious for𝒦𝒦 PFT either, considering form, speak in favour of the Preferred Frame Theory as the that length contraction (as well as time dilation) affects all right alternative. The elevator-shaft paradox shows that this objects at rest in a given system, conventionally described as is only the length contraction (i.e. the effect manifesting measuring rods and clocks. itself at the velocities comparable with the speed of light) How then the rhomboidal shape of elevator agrees with that enables settling an old controversy as to the absolute vs. vertical rails in and ? The measuring rods at rest in relative nature of motion, dated from Newton and Leibniz. are contracted in by′′ the Lorentz factor ( ), in the In terms of logic, Special Relativity originated from two diagonal′ direction𝒦𝒦 parallel𝒦𝒦 to . Hence, all objects observed in𝒦𝒦 are elongated in𝒦𝒦 this very direction (compared𝛾𝛾 𝑉𝑉 with basic postulates translated into algebraic form of Lorentz respective′ lengths measured in𝐕𝐕 ), including the objects at transformation. Over time, STR gained however advanced rest𝒦𝒦 in . Consequently, the rails, (already contracted in mathematical shape, to mention “hyperbolic” Minkowski in the horizontal′ direction), become𝒦𝒦 elongated in in the geometry, matrix algebra and group theory. Consequently, it direction𝒦𝒦 parallel to the diagonal vector . Imagine′ 𝒦𝒦a should be possible to disprove STR using more advanced rectangle formed by vertical rails at rest in intersected𝒦𝒦 tools than the ones used in this paper. It is likely that, in by two horizontal (in ) lines, e.g. the rectangular𝐕𝐕 ′′ cage accordance with the spirit of Einstein’s remark [9], STR is tightly framing the elevator.′′ This rectangle𝒦𝒦 becomes self-consistent as a mathematical structure applicable to the elongated in to the 𝒦𝒦shape of rhomboid in the direction of toy model of physical reality reduced to a single one spatial vector . Of course,′ this pertains not only to this particular dimension, but not adequate as a description of the real rectangle, but𝒦𝒦 to all others and, in consequence, to the entire three-dimensional world embedded in time. In connection rails that𝐕𝐕 due to this elongation become skewed in frame , with that claim, it has to be examined whether the in the direction parallel to the skewed vertical sides o′f non-collinear Lorentz transformations form a group as they elevator. Hence, elevator and rails match each other in 𝒦𝒦 , do in the collinear case. just as they do in and (Fig. 13). ′ The failure of STR breaks the present consent in physics ′′ 𝒦𝒦 as to the basic correctness of both relativity theory and 𝒦𝒦 𝒦𝒦 quantum mechanics. This can be (and really is, as this author could find out) considered disastrous, considering that Einstein’s theory makes a foundation of modern physics. However, this consent does not include the main unsolved problem in modern physics which is the lacking theory of quantum gravity, in fact tantamount with the split of physics as a whole. Revealing the STR inconsistency should be therefore perceived as a challenge and chance in opening up the new prospects for physics at the fundamental level. As vividly pointed out by Magueijo [20]) the quantum-gravity effects, whatever they are, should be absolute, ergo independent of the choice of reference frame. This disagrees with STR, according to which velocity (and consequently total energy) is always relative, i.e. depends on the choice of Figure 13. The rails (at rest in ), elevator (at rest in ), both observed reference frame. In contrast to that, PFT predicts that in frame . Elevator and rails′′ subject to the elongation′ in diagonal particular energies are basically related to the absolute direction, parallel′ to the direction𝓚𝓚 of the elevator motion𝓚𝓚 in . In result, velocities. Although specific values of energy still depend in 𝓚𝓚 rhomboidal elevator matches the tilted rails, which indicates the PFT on the absolute velocity of observer, yet, in any compatibility of observations made in with observations𝓚𝓚 made in and ′ particular reference frame they form an invariant hierarchy. ′′ 𝓚𝓚 𝓚𝓚 Consequently, energy perceived in a given frame as extreme Let𝓚𝓚 us conclude: The above representative examples show (in the sense of expected quantum effects) is as well extreme that the elevator-shaft paradox can be resolved in a in any other frame. This property of PFT provides dramatical self-consistent way if, and only if, contraction is interpreted change to all present models aimed at searching for the as an absolute effect, caused by the absolute motion, in spontaneous Lorentz violation in Planck scale; thereby may accordance with the Preferred Frame Theory and respective appear crucial for solving the problem of quantum gravity. Tangherlini transformation.

96 Maciej Rybicki: Failure of Special Relativity as a Theory Applied to More Than One of Spatial Dimensions

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