Relativity free of coordinates

Relativity free of coordinates

Serge A. Wagner Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast 141701
July 25, 2019

The concept of a physical space, which actualizes Euclidean geometry, is not confined to the statics of solids but extensible to the phenomena where Newtonian mechanics is valid, defining its concept of time. The laws of propagation of electromagnetic disturbances modify Newtonian formalism for sufficiently fast free motions within each spatial domain of its validity for slow motions and introduce the extended concept of time by uniting those of Newtonian which can exist in different spatial domains of their validity.

A boost direction for a pair of physical spaces is that spatial direction in one of the spaces along which the other space moves. Free motions of point particles make an instrumentation for identifying the boost direction as well as events on a straight line along that direction. The concept of a boost direction secures the physics-based formulation of the basic relativity effects, which eventually results in the relation between two arbitrary spaces in terms of their position vectors and time of a given event.

The obtained transformation rules for the components of the position vector differ from the vector-like relationship known in the literature because the latter actually deals with column vectors made of Cartesian coordinates of true vectors and appears identical to a coordinate transformation referred to as a boost.

Within the physics-based approach, the transformation of coordinates implies specifying coordinate systems in each frame in terms of physical objects/directions. This yields a logically consistent and physically meaningful presentation of the coordinate transformations commonly exploited in the special relativity theory, which makes manifestations of those transformations evident. Specifically, for a Cartesian coordinate system subjected to a boost, the coordinate-free technique of reasoning allows one to evaluate its apparent distortion easily.

I Motivation

The safely correct development and credible presentation of a physical theory implies identifying physical phenomena to which the theory is assuredly applicable, however small such a validity domain would seem to be at the outset.

Historically, the special relativity theory came along in attempts to make Newtonian description of moving macroscopic bodies consistent with optical phenomena observable by means of these bodies. Einstein’s initial defining paperEinstein1905 () on relativity combines the principle of relativity from Newtonian mechanics with an idea that propagation of light does not depend on the motion of its source and can be used to define simultaneous events at different places. This combination turned out to be a physically reasonable way to arrive at Lorentz transformation, reproduced in subsequent monographspost-Einsteinian monographs () and textbookstextbooks ().

The alternative post-Einsteinian presentationFock1964 () of the relativity theory brought Einstein’s attempt to identify the physical foundations of the problem to its actual formal source: the required change of a coordinate system should preserve the form of free motion of point particles while keeping the speed of light as a universal limitation on the speed of any particle. Such a mathematical approach proves to be physically important because it provides a consistent description of the macroscopically perceived (commonly referred to as kinematicHagedorn1964 ()) part of experimental particle physics: Registering devices embedded in the solid wall of an accelerator can embody an inertial reference frame along with its Euclidean geometry; registrable/inferable collisions between particles are events.

Up to the present time, all expositions of the special relativity theory heavily rely on Cartesian coordinates for both formulating premises and developing inferences. Within macroscopic physics, resorting to that handy mathematical technique imposes no restriction on the applicability of the theory, though it comes into apparent conflict with the meaningful practice of exploiting coordinates in physics and engineering. Such a practice implies that the choice of coordinate systems should facilitate the application of basic/general regularities/rules to a particular problem, so that the actual spatial configuration of the problem can determine whether a suitable coordinate system is Cartesian, or orthogonal curvilinear, or even special such as barycentric etc.

In the next section the reader can find a remainder of what physics secures the existence of frames and underlies the validity of Euclidean geometry. Section III presents the concept of a boost direction, which enables the subsequent physics oriented inference of the well known spatiotemporal effects related to the special relativity theory. Section IV exploits these effects to obtain the relation between the time moments and the position vectors of a given event in two frames. This relation entails a logically consistent and physically meaningful presentation of the well known coordinate transformations in Section V.

Ii Physical spaces

ii.1 Statics and foundations of Euclidean geometry

Within macroscopic physics, a physical space can be provisionally viewed as a set of positions of rigidly connected small bodies, where a rigid construction means that the spatial relations between its parts obey the rules of Euclidean geometry. Making use of this mathematical structure in the description of undeformed (or, equivalently, non-accelerating and non-rotating) macroscopic solids is evidently valid, which suggests that the regularities of statics (supplemented with Hooke law to resolve statics’ ambiguities) can underlie Euclidean geometry. This mathematical construct is commonly believed to be applicable to a wider range of spatial scales as well as a larger class of physical phenomena. So other regularities, such as those of electromagnetism and gravity, may take part in maintaining Euclidean geometry for the set of all possible positions of sufficiently small interacting bodies.geometry due to electromagnetism or gravity ()

If one selects from the rigidly connected parts some that make up the coordinate system, then one gets a (necessarily inertialnon-inertial frame ()) reference frame. Usually a way of measuring time is also associated with an inertial reference frame. Since the concept of time is perceived as independent of coordinates and thus exists for a physical space as a whole, in this article the time variable is associated with a physical space, too.

As of now, the group of motions of rigid bodies (described axiomatically) is the only mathematical structure that reminds us of the physical foundation of Euclidean geometry. The group includes spatial translations and rotations . The additive representation of a spatial translation is usually referred to as a spatial vector. One can use rotations to introduce an angle between two vectors etc. When the orthonormal vectors represent the translations along three mutually perpendicular directions, the decomposition


of a displacement (the change of a position vector ) is just what defines Cartesian coordinates.

ii.2 Newtonian dynamics and the applicability extension for Euclidean geometry

The previous section refers those interested in the origin of Euclidean geometry to the physics of solids, which may not be a reasonable starting point of an investigation aimed something in the physics of fields and particles.

Meanwhile, it is Newtonian laws that make an established formalism of mechanics. Within the limitations imposed by microscopic phenomena, Newtonian formalism is believed to be applicable to each sufficiently small part of a macroscopic body, referred to as a point particle.

To calculate how the position vector of each particle is changing over time, a theorist should invoke Newton’s second law


Here and hereinafter the notation is used for the list of expressions where the label runs over all its values.

Within the purely mechanical formalism, the force of action of a particle on a particle cannot be but conservative:

where denotes the nabla operator designed to act on functions of the position vector .

For an isolated system of N particles and given functions , the decomposition (1) turns Eqs. (2) into a closed set of 3N differential equations of second order. Its general solution


involves 6N arbitrary constants, of which one is the time reference shift while seven can be the total energy and the components of the total momentum and the total angular momentum . In other words, inversion of the set of Eqs. (3) can yield 6N-8 constants of motions , possibly specific for each sufficiently small interval of , in addition to the seven universal constants of motion:

Of these functions, only essentially represent the internal/relative motions of the particles while describes the overall intensity of the whole motion, which can be arbitrary normalized by the choice of the unit of the time ; and correspond to the well known global directions related to the whole motion.

The formalism based on the use of position vectors, their decompositions (1) and Eqs. (2) may not be a consistent introduction to foundations since it refers to the regularities/notions left without description in terms of the physics phenomena involved. To build a foundational construction straightforwardly, one should identify some real or possible stationary objects with Euclidean points so that the appropriate constants of motion can approximate the values of the one-point vector field in Eq. (1) , two-point scalar field known as distance (Euclidean length) etc.

The well-known manifestations of quantization preclude implementing such a construction at sufficiently small scales.quantization () As a result, nor Eqs. (2) nor Eqs. (3) appear applicable at those scales. Nevertheless, to construct Euclidean geometry (or, at least, some pregeometry) one could still adopt the motions of the interacting particles as primitive notions described with a set of experimentally identifiable relations between them. The formulation of such relations (which would then make a low-level automatics of mechanics) is far from the goals of this text, but there is hardly any doubt that the set of motions of interacting particles is sufficiently rich to support Euclidean geometry and, therefore, the concept of a space. At this logical level, the relations that underlie Euclidean geometry are not separable from those that eventually give rise to the existence of the constants of motion.

ii.3 Physical spaces and relativity principle

As discussed in the previous section, within the familiar high-level formalism the concept of a physical space manifests itself by means of a position vector , Euclidean nature of which is most likely secured by non-relativistic classical physics. At least one connection between physical spaces and regularities of classical physics reveals itself in terms of a position vector: Eqs. (2) keep their form when one changes a space B for a space A in accordance with the transformation rule


where , and are given parameters. Here and hereinafter the superscript (F) indicates that a quantity is initially defined in a space F. (But as far as the transformation (4) is valid between any pair of spaces, one can actually define a position vector in any space.)

From the early days of the relativity theory, the following generalization of the above statement is regarded as a more or less universal principle, called the principle of relativity: The mutual disposition and the relative translational uniform rectilinear motion of two spaces cannot manifest itself in the description of physical phenomena within one of these spaces. Equivalently, physical laws have the same formulations in different (necessarily non-accelerated non-rotating) spaces.first formulation of the relativity principle ()

In the context of Newtonian mechanics, the meaning of the relativity principle is plain: translational uniform rectilinear motion of a physical system as a whole with respect to some external (reference) bodies does not affect the motion of the internal parts of the system with respect to each other. However, this idea presumes a partition of the physical system that cannot be seamlessly extended to include electromagnetic phenomena since the decomposition of an electromagnetic field into non-interfering components is possible only without electric charges. This apparent gap is accompanied (and aggravated) by the fact that the transformation (4) does not preserve the form of source-free Maxwell’s equations.

Naturally, an attempt to derive the general transformation that keeps the full formalism of Maxwellian electrodynamics would lead a researcher to a complicated problem. So the developers of the special relativity theory cannot but begin with the simple coordinate transformation called Lorentz transformation (which, in accordance with the purpose and logic of this article, is explicitly reproduced as late as in Section V.1.)

Looking into early presentations of the special relativity theory, one can identify the following premise for the formal derivation of Lorentz transformation: the law of motion of a free particle and the laws of propagation of a free electromagnetic field have the same form in all frames.early presentations () To be exact, in a space F the position of a free particle is changing along with the time as


while the positions taken by an electromagnetic wavefront (wave phase-front) at the time from a point source flashed at the position at the time satisfy


The propagation speed of an electromagnetic spherical wavefront is the same in any space. As long as the goals of one’s inference are limited by the derivation of rules equivalent to those of Lorentz transformation, one can confine oneself with the limiting form of Eq. (6) for an infinitely far source position, i.e. the equation


for a plane wavefront which propagates in the direction of the unit vector .

One can view the above statements as some realization of Einstein’s original intention to extend the principle of relativity to the propagation of electromagnetic waves, but without his explicit redefinition of the time variable. However, the next generation of authors has dispensed with both Maxwell equations and Eq. (6) in their introductions to the relativity theory. In the post-Einsteinian derivations of the transformation rules between two frames one finds the principle of relativity replaced by the requirement to preserve the form of Eq. (5) supplemented with the condition


to include ”the motion of a light signal” (in effect, the propagation of the intersection point of an electromagnetic plane wavefront with one of its associated ray paths.)Fock presentation () As a result, the physics that underlies the special relativity theory has been reduced to that of free point particles with the universal limitation on their speed, equal to in any space.

The use of free particles in a reasoning is evidently restricted by the processes of particles’ interaction. But as long as one neglects the extension and the duration of those processes, one can exploit an interaction act as a representation of a basic identifiable entity usually referred to as an event. In other words, each event appears to be real or possible interception of two (or more) free particles. Then one can exploit experimentally identifiable relations between motions of free particles to establish relations between events. If need, the interaction between particles can be assumed so small as not to change their motion. (In general, one need exploiting Eq. (5) for the motion of each particle between the points of interception, where the parameters may change.) This technique along with the basic appliance of the relativity principle is a main tool used in the next section.

Here it is also worth noting that the free motion of particles and the propagation of light rays are not sufficient to construct Euclidean geometry, so the attempts to extend its validity to arbitrarily fast processes could end up with nothing but a new postulate.relativistic construction of geometry () In order to have material carriers of Euclidean geometry, presentations of the relativity theory have no choice but to borrow the notion of space in Newtonian mechanics and then introduce many spaces uniformly moving relative to each other.

When someone applies Newton’s second law (2) to a physical system which consists of weakly interacting (e.g., widely separated) parts, he might think that Newtonian mechanics should involve some means to identify motions in such parts as simultaneous processes. Actually, Eqs. (2) are well known to have originated from the regularities revealed by experiments/observations related to strongly interacting physical bodies, especially gravitationally bound ones, such as Sun and planets. But in such a system, the existence of the time is an inherent property of its (almost periodic or quasi-periodic) motion. If is a characteristic speed of such motion, then for a given timescale it secures synchronization in changing physical quantities over the region of size . However, the special relativity theory implies the speed of light as a characteristic speed and, therefore, considers the region of size . In other words, within the special relativity theory, Newtonian mechanics can secure the synchronization two processes (”clocks”) but only at one spatial point.

Exploiting Eq. (6)/Eq. (7) or the light rays only (without specifying geometrical structures prematurely) one can synchronize events happened to particles at different positions (as far as one neglects the time delay and the position shift due to interaction of a charged particle and electromagnetic field.) Adopting the synchronization properties formulated in Ref. Einstein1905, , S. 894 turns the time into a global variable, similar to a spatial coordinate.

In principle, within sufficiently small timescales, Newtonian mechanics can maintain Euclidean geometry only in the vicinity of each event, which means that the geometry of the subset const, identified in Section II.1 also as a space, might be Riemannian.Fock and geometry () In this text, the global geometry of particles’ positions within each extended space is still assumed Euclidean, since it is appropriate for usual practical applications of the special relativity theory as well as common teaching curricula.

Iii Basic effects of the relativity theory

iii.1 Boost direction

For any pair of spaces A and B, there are the velocity of A with respect to B and, vice versa, the velocity . Since each of these vectors is defined as a spatial object in its own space, there can be no procedure of comparing them directly on the basis of Newtonian mechanics. Nevertheless, since all spaces are supposed to be identical in their essential internal properties, one should accept


due to the symmetry of exchanging A and B. (This should be also considered as a part of establishing universal time unit since one cannot be sure that a non-relativistic standard clocks keeps its rate when set in fast motion.)

Let denote a process that starts with an event and ends with an event , and let a number denote the elapsed time in an arbitrary space F, so that


and .

If in the space A one has for arbitrary events , , , , then in the space B one finds the same equality because the contrary would allow one to judge about the motion of B with respect to A on the basis of internal data in B, in contradiction with the principle of relativity. Further generalization is possible if one exploits Eq. (10) to partition a process into a series of shorter processes over even intervals of time: entails for any natural, rational and, finally, real . It follows that for any space F the relation shows no dependence on the space while for any process the relation depends only on defined by Eq. (9).

Let a point body resting in the space A and a point body resting in the space B meet each other at the event and let another freely moving point body meet and at the events and , respectively. By making use of the laws (5) of free motion, one can get an expression


for the velocity of the body in the space F where

is actually independent of F in accordance with the previous analysis.

Eq. (11) allows one to define a one-parametric family of spaces where is at rest while the bodies and move along the common straight line in opposite directions. Hereinafter this line is referred to as a boost line of a space for a given , and the whole family is referred to as a helicoboost class of spaces, specified by its two members A and B.

If one considers two different values and along with two corresponding bodies and and applies Eq. (11) to the motion of the body in the space , one can find that moves just along the boost line of . Since is an arbitrary member of the helicoboost class and can represent an arbitrary point body (or even a light signal) that moves along the boost line in , the boost lines in different spaces of one helicoboost class prove equivalent in a sense: Any point bodies that stay (moving or resting) on the boost line in one space of a given helicoboost class remain (moving or resting) on the boost line in another space of the same helicoboost class.

If, in addition to the bodies and , one chooses more representatives of the spaces A and B, one gets more parallel boost lines. To avoid specifying particular boost lines in a reasoning, one can invoke the direction of or , which represents a bundle of parallel boost lines in each member G of the helicoboost class. Hereinafter this direction is referred to as a boost direction.

It is possible to define both a boost direction and a helicoboost class of spaces without prior reference to its two members: If there are a space where a set of free point bodies have their velocities parallel or anti-parallel to each other, then there are other spaces where velocities of the these bodies are also parallel or anti-parallel to each other; an equivalence class of such spaces is a helicoboost class; the direction of motion of such free bodies in each space of the helicoboost class is a boost direction.

iii.2 Spatial transverse effect

The motion of light signals along the boost direction in one space of a given helicoboost class is an important limiting case of the motions considered in the previous section. Since the light signals represent the propagation of electromagnetic plane wavefronts (7), one can conclude that they propagate along the boost direction in any space of the helicoboost class. It follows that simultaneous events in a plane perpendicular to the boost direction in one space appear simultaneous in any other space of the same helicoboost class, where they also occupy a plane perpendicular to the boost direction.

Let the locations of the above simultaneous events make a certain instantaneous arrangement over the plane wavefront. To represent it as a stationary geometric configuration, one should consider intersections some bundle of boost lines with a dense series of parallel wavefronts which propagate along the boost direction. Since such wavefronts and the boost lines are observable in any space of the helicoboost class, so is the stationary planar configuration they generate. The possibility of the common geometric configuration shows that the observers in different spaces can come to agreement with each other about the orientation of their spaces around the boost direction or, in other words, to another equivalence relation between two spaces. This relation allows one to partition the helicoboost class into subclasses of identically oriented spaces. In the following, a subclass of this kind is referred to as a boost class.

To present the above equivalence relation in terms of a relative position vector, one can write:


Here and in the rest of the paper, the notation


describes the decompositions of spatial vectors and in their respective spaces.

Since the symbol in Eq. (12) denotes an arbitrary relative position vector, which connects two arbitrary points in a plane perpendicular to , one can view Eq. (12) as a notation for mapping a geometric configuration in the space A to that in the space B. Since the principle of relativity does not allow differences in the geometric properties of these configurations, then




for any spatial vectors in the space A and their counterparts in the space B.

iii.3 Spatial longitudinal effect

Suppose in the space A point bodies P, Q, R, S…are moving with the same velocity while momentarily (detected as) arranged in a straight line along the boost direction, i.e. parallel to . In the space B these point bodies are at rest in a straight line along the boost direction. If in the space A one has , then in the space B one finds the correspondent equality due to the relativity principle. Here the notation refers to a distance between moving bodies G and H momentarily observed in a space A while denotes a distance between stationary bodies G and H in a space B. If one exploits partitioning a line segment in a manner similar to the division (10) of a time interval, one can eventually come to the similar conclusion that entails for any natural, rational and, finally, real . It follows that for any space F of the boost class the relation shows no dependence on the space while for any pair of bodies P and Q stationary in the space B the relation depends only on .

The application of the above result is not bounded by a comparison of the distances between two bodies in the two spaces. In fact, the distance between two moving point bodies in the space A is a distance between two simultaneous events of detecting these bodies in the space A. In the space B, these events are not necessarily simultaneous but happening to the same bodies. So the distance between the bodies is a distance between the events, too. Thus, the relation


is the same for any pair of events in the straight line along the boost direction, provided that they are simultaneous in the space A.

Let simultaneous elementary events in the space A occur over the length in a straight line along the boost direction, i.e. along the direction of motion of the space B. Physically, they together may be an act of detecting a rigid rod embedded into the space A. If a flashlight occurs in the middle of the rod, it takes the same time interval for that light to get to each end of the rod. When instantaneously observed in the space B, the length of the rod appears to be . The arrivals of the above mentioned flashlight at the ends of the moving rod are not simultaneous in the space B: along the direction of the rod’s motion the time difference makes


while the distance between these events is


Here and hereafter .

The equations (17) and (18) take account of the possibility that in different spaces a set of the same events may occupy segments of different sizes along their boost direction. Considering the events involved in the inference and the formulation of Eqs. (17) and (18) one can find that and in accordance with Eq. (16). Then Eq. (18) entails , i.e. the length contraction effect


iii.4 Time dilation effect

To get to another well known effect, one can exploit such thought construct as the light clocklight clock (), where a light pulse propagates back and forth between two mirrors held parallel and apart at the fixed distance in their rest space. In that space, the line of the light propagation is perpendicular to the mirrors and can be referred to as the axis of the clock. The proper duration of the clock’s cycle, i.e. the round-trip time of the light pulse in the clock’s rest space, makes


Let an observer be moving with a speed in the light clock’s rest space and along the axis of the light clock. The observer can detect the contracted length of the light clock and find that the clock’s cycle takes


Here it is worth remarking that the light clock is an auxiliary dedicated thought construct, which can in no way include the actual reflections of an electromagnetic wave from substance of the mirrors but embodies the properly collated sequence of events only. This means that Eq. (20) and Eq. (21) together entail the time dilation effect


which gives the laboratory time interval between two events at a moving point during its given proper time interval .

To show relations between the spatiotemporal effects of the relativity theory in one’s practice of teaching, one may also address the following inference.

Let the light clock of proper length rest in the space A so that the axis of the clock is perpendicular to the direction of motion of the space B, i.e. to the boost direction. In the space B, the axis of the clock is also perpendicular to the boost direction, and the main cycle of the clock corresponds to the light signal path shown in Fig. 1.

Figure 1: The path of the light pulse in the transversally moving light clock.

Here is a length of the clock simultaneously observed in the space B as being in the plane perpendicular to the direction of the clock’s motion. In accordance with Fig. 1 the duration of the clock’s main cycle in the space B should obey the equation

This is consistent with the time dilation effect (22) and the proper duration (20) of the clock’s main cycle because the spatial transversal effect (12) yields when the motion of the light clock and the propagation of the light pulse occur in the same plane.

iii.5 Retardation effect

Eq. (12) suggests that one can generalize Eq. (19) and Eq. (17): If in the space A simultaneous events occur in the transversal spatial slab thick, then in the space B they occur within the slab


and with the time spread


Iv Relation between two spaces

To describe the relation between the spaces A and B in terms of the position vectors and and the time moments and of a given arbitrary event one should first of all identify the place of a common reference event with the origins and of the spaces A and B as well as its time moment with the readings and of their clocks.

Let two elementary events occur in the space A at a given time moment : an event occurs at the origin while occurs at . Due to the time dilation effect, the moment of observing in the space B equals to the reading of a clock fixed at the origin . Due to the time spread effect (24), in the space B the difference in time between and

Therefore, the time moment of observing in the space B


Further, if in the space B the plane is perpendicular to and keeps passing through the origin , then, in accordance with the analysis in Section III.1, in the space A it forms the plane perpendicular to and moving with this velocity. Evidently, the distance from the event to the plane makes . If denotes the position of in the space B, then the length contraction effect (23) helps us conclude that in the space B the distance from the event to the plane is


while Eq. (12) yields


Within the described coordinate-free approach, the relationships (25), (26) and (27) turn out to be transformation rules between spaces. To show the dependence on the spaces A and B explicitly one should only decipher the notation introduced by Eq. (9) and Eq. (13):


In a more compact form, the transformation rules (28)-(30) can also be presented as the mapping




where the symbol unites the meaning of and the meaning of while the symbol unites the meaning of the usual product of two numbers and the meaning of the dot product of two spatial vectors; the symbol denotes the dyadic (outer) product.

Historically, the early attempts to obtain such a mapping did not lead to a correct/unambiguous expression of via because of the failure to distinguish between a column vector and a true vector.early attempts () To keep being mathematically correct the later treatmentCushing1967 () could not avoid resorting to coordinates but appeared limited to the transformation known as a boost (see the final remark in Section V.2.)

V Coordinate transformations in the relativity theory

v.1 Lorentz transformation

As soon as one specifies coordinate systems in the spaces A and B, in terms of relations their unit base vectors and to some physical directions, one can arrive at the relationship


between the column vectors

made of time and spatial coordinates of a given event as observed in the inertial coordinate systems A and B, respectively.

With the choice

for the unit base vectors Eqs. (28)-(30) follow Eq. (33) with where


is the matrix of the original form of Lorentz transformation.

v.2 Boost transformations in the physics literature

Boosts make a well known class of the transformations (33) introduced in graduate level physics courses with an aid of its matrix


whereBoost matrix ()


, the elements of the column vector are the A components of the velocity of B with respect to A.

Due to the identitiesBoost reduction ()



one can say that incorporates as an observable direction.

It is easy to find that textbooks’ authors exploit the boosts only to interpret the Thomas precession contribution to the spin motion of a relativistically moving charged particle.Boosts and Thomas precession () However, despite this seemingly definite connection to observable phenomena, all the texts up till now have favored the presentations that facilitate algebraic operations involving boosts over those which would treat a boost as a physically defined operation/motion or relations between physical objects.

Apparently, such a non-physical cannot lead to any better understanding of the spin precession or anything else because it leaves the physical meaning of itself out of consideration. Its only benefit for physicists seems to be that it has found no reason to consider a boost as the Lorentz transformation “without rotation,” in attempt to generalize the idea of parallel transport, such as the definition (ii) at p. 871 in Ref. Frahm1979, . As a result, to “a boost” mathematicians prefer to apply cautious terms such as “an aligned axis Lorentz transformation” (see p. 236 in Ref. Kennedy2002, .)

The most obvious manifestation of the problem is that no text attempts to formulate a physics based definition of a boost, thereby preventing any futher reasonable use of that concept. The reference formula (36) makes it difficult to get an idea how mutual orientations of several bodies change when set in motion, because (36) alone provides no hints how to choose the axes of the coordinate systems in use. The seemingly key formula (37) appears to be nothing but a relation between two representations (36) and (34) for one relative motion, of which the formal simplicity of (34) may even be misleading:

When asked about the direction of the -axis, someone may correctly infer from (34) that it is the direction of the -axis instantaneously observed in the coordinate system A. But the problem is that a researcher must be able to identify any directions before establishing/verifying (theoretically/experimentally) the relationships (34). To avoid that apparent logical circle, one might take the above description of the direction as a definition and an explicit starting point in a derivation of the transformation rule (34). Needless to say that such an accurate approach can hardly be found in the existing, history-oriented, presentations of the relativity theory.

Now the coordinate-free description for the relation between two arbitrary spaces in Section IV allows one to formulate logically consistent and physically explicit definition of a boost.

v.3 Definition of a boost and derivation of its matrix

Aside from the origins, the above consideration refers to no elements of coordinate systems. To set up Cartesian coordinate systems in the spaces A and B one should specify the direction of their spatial base unit vectors and for . Let




In other words, in view of Eq. (15), makes the same angles with in the space B as does with in the space A. By definition, a boost is the transformation (25) of time along with the transformation of coordinates that satisfies the conditions (38) and (39).

The equivalence relations (38) and (12) along with the property (15) yield the equality of numbers


Eqs. (39) and (26) allows one to rewrite the dot products as


Here the common pithy notation is used: while the repeated index implies the summation over all its values. Then Eq. (40) entails

which is just the spatial part of the transformation (33) with the matrix (35).

In terms of column vectors the above equation can be written as


This form of a boost was obtained from Eq. (37) in Ref. Cushing1967, .

v.4 Boost operation in the laboratory frame

One can address the definition of a boost in the previous section so as to come to physically meaningful conclusions directly. An important example is a simple analytical description for the distortion that a Cartesian coordinate basis exhibits while simultaneously observed in the space where it experiences a boost operation.

Figure 2: The moving vector is instantaneously observed as the vector .

Fig. 2 presents a plane parallel to a spatial vector and the velocity of the space B, both the vectors being sets of simultaneous events in the space A. The boost operation applied to the vector

results in the vector

which is spatial in the frame B and parallel to the above plane, too. Due to the definition of the boost and the length contraction effect, in the frame A the vector is perceived (instantaneously observed) as the vector


for the angle between and as well as between and , and

for the angle between and , one can find that

for the angle between and (see Fig. 2.)

Sometimes it may be reasonable to change the laboratory coordinate system with the aid of a boost so as to reach simpler state of motion, to reduce the form of interaction law etc. Then, in the limit , the angle shows what relative error is not properly compensated in case one fails to transform all relevant quantities appropriately.

Vi Conclusion

A boost direction for a pair of physical spaces is that spatial direction in one of the spaces along which the other space moves. Free motions of point particles make an instrumentation for identifying the boost direction as well as events on a straight line along that direction. The concept of a boost direction secures the formulation of the basic relativity effects in a physics-based manner, which, eventually, results in the relation between two spaces in terms of their position vectors and time moments for a given event.

Within the physics-based approach, addressing the transformation of coordinates implies specifying some coordinate systems in each space in terms of physical objects/directions. This yields a logically consistent and physically meaningful presentation of the coordinate transformations commonly exploited in the special relativity theory, which makes observable effects associated with those transformations evident. In particular, for a Cartesian coordinate system subjected to a boost, the coordinate-free technique of reasoning allows one to evaluate its instantaneously observable apparent distortion easily.


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  • (6) W. Pauli, The Theory of Relativity, (Pergamon Press, 1958)
  • (7) Max Born, Einstein’s Theory of Relativity, (Dover Publications, 1962)
  • (8) See, e.g., chs. I-IV in Ref.  Bergmann1976, , chs. I-II in Ref.  Moeller1955, , ch. 1 in Ref.  Taylor1992, , Sections 11.1-11.4 in Ref.  Jackson1998, , Sections 7.1-7.3 in Ref.  Goldstein2000, .
  • (9) Peter G. Bergmann, Introduction to the Theory of Relativity, (Dover Publications, 1976)
  • (10) C. Møller, The Theory of Relativity, (Oxford University Press, 1955)
  • (11) Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, (W. H. Freeman and Co., 1992)
  • (12) John David Jackson, Classical Electrodynamics, 3rd edition (Wiley, 1998)
  • (13) Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics, 3rd edition (Addison-Wesley, 2000)
  • (14) V. Fock, The Theory of Space Time and Gravitation, 2nd Revised Edition (Pergamon Press, 1964)
  • (15) R. Hagedorn , Relativistic Kinematics. A Guide to Kinematic Problems of High-energy Physics, (W. A. Benjamin, Inc., 1964)
  • (16) Since both electromagnetism and gravity manifest themselves via the concept of force, they allow one to distinguish directions and thereby support projective geometry at least. The next step should involve point particles, i.e. such entities that can strongly interact over a sufficiently short range. Then the application of first Newton’s law to several particles on the same straight line helps one identify stationary particles, which can embody Euclidean points. It enables one to introduce the concept of length in the usual manner. The discussed approach implies that the physical regularities involved can be presented in pre-Euclidean/prenumeric terms to describe experiments/observations more directly. To get an idea about a technique of such description, see Sections V and VI in Ref. prenumeric, .
  • (17) There is no way to introduce a non-inertial reference frame consistent with Euclidean geometry, since the latter exists because of physical regularities that manifest themselves in an inertial reference frame. In other words, one can rotate a real rigid body, but it worsens the physical realization of Euclidean geometry by means of the relationships between parts of this body.
  • (18) Serge A. Wagner, “How to introduce physical quantities physically,” <>
  • (19) For example, if someone would choose to store information about a local direction in the angular momentum of a hydrogen atom, he will inevitably fail both due to the well known unavoidable quantization of that quantity and because of the small lifetime of the corresponding excited state of motion. In accordance to the simple consideration known as Bohr’s theory of a hydrogen atom, it occurs over the spatial scale, called the Bohr radius and recognized as a characteristic atomic scale. In general, the elementary charge , Plank’s constant , electron’s rest mass , proton’s rest mass are well known magnitudes which reveal the quantization of electric charge, angular momentum and mass/energy, respectively. The dimensional and/or provisional theoretical analysis can yield a set of characteristic spatial scales
    where is a dimensionless function of the dimensionless quantities The cases , and yield the well known scales: the classical electron radius (in fact, the characteristic size of proton/neutron), Compton wavelength of electron and the Bohr radius. Here the use of the CGS system of units is assumed, so one can take as a parameter in Maxwell equations, not necessarily the observable speed of light. Though no formal analysis of applicability of Euclidean geometry has yet been published, it is over scales less than Compton wavelength of electron where some authors have found it difficult to provide a consistent concept of a spatial coordinate. See p. 40 in Ref. Bjorken&Drell1964, and p. 2 in Ref. Berestetskii1982, for simple order-of-magnitude considerations and, e.g., Ref. Hegerfeldt1998, for the exemplars of existing theoretical approach to the problem.
  • (20) James D. Bjorken, Sidney D. Drell, Relativistic Quantum Mechanics, (McGraw-Hill, 1964)
  • (21) V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii Quantum Electrodynamics. Course of Theoretical Physics, vol. 4, 2nd edition, (Pergamon Press 1982)
  • (22) G. C. Hegerfeldt, “Instantaneous spreading and Einstein causality in quantum theory,” Ann. d. Phys. 7, 716-725 (1998)
  • (23) The first more or less general formulation of the relativity principle is believed to belong H. Poincaré. See, e.g., p. 111 in Ref. Poincare1905, or p. 107 and p. 300 in Ref. Poincare1913,
  • (24) H. Poincaré, Science and Hypothesis (London and Newcastle-on-Cyne: The Walter Scott Publishing Co., 1905)
  • (25) H. Poincaré, The Foundation of Science (NY: The Science Press New York and Garrison, 1913)
  • (26) The thought that first Newton’s law has the same form in all inertial frames was viewed as trivial and, for this reason, exploited implicitly (until the accurate presentation in Ref.  Fock1964, has appeared.) In contrast, Einstein’s idea that electromagnetic phenomena, such as the propagation of a spherical wave, was apparently perceived as nontrivial. So the statement that a spherical wavefront keeps its form in all frames appeared to be a popular explicit premise for the derivation of Lorentz transformation; see, e.g., p. 100 in Ref.  Silberstein1914, and p. 9 in Ref.  Pauli1958, .
  • (27) This is just the condition in the equations (8.05) and (8.06) at p. 21 in Ref. Fock1964, . Interestingly, Fock starts his consideration with the general wavefront equation, which, in principle, can describe non-spherical wavefronts, e.g., from two interfering point sources. But eventually he has narrowed his inference with no explicit reasoning.
  • (28) See the flat spacetime limit for “the remark” (e) at p. 69 followed by the reasoning at p. 82 in Ref. Ehlers1972, or the description of “the second clock effect” at p. 122 in Ref. Ehlers1973, .
  • (29) J. Ehlers, F. A. E. Pirani, A. Schild, “The Geometry of Free Fall and Light Propagation,” in General relativity, papers honour of J. L. Synge, edited by L. O’Raifeartaigh (Clarendon Press, Oxford, 1972), p. 63–84.
  • (30) J. Ehlers, A. Schild, “Geometry in a Manifold with Projective Structure,” Commun. Math. Phys. 32, 119–146 (1973)
  • (31) This is an impicit reason why FockFock1964 () starts his inference of a boost with a generally nonlinear transformation of the space and time variables and addresses some concepts of Riemannian geometry as elements of a suitable notation.
  • (32) The first description of the light clock is given at Ref. Laue1917, , p. 293.
  • (33) Max von Laue, “Die Nordstrōmsche Gravitationstheorie. (Bericht.),” Jahrbuch der Radioaktivitāt and Elektronik 14, 264-312 (1917)
  • (34) See the footnote 24 at pp. 10 and 11 in Ref. Pauli1958, and the solution to Problem 1-3 at pp. 8 and 9 in Ref. Hagedorn1964, for the examples of such reasoning in educational texts for physicists.
  • (35) James T. Cushing, “Vector Lorentz Transformations,” Am. J. Phys. 35, 858-862 (1967)
  • (36) Cf. Section 7.3 in Ref. Goldstein2000, , Sections 11.7 and 11.8 in Ref. Jackson1998, , §41.4 in Ref. Misner1973, .
  • (37) Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitetion, (Freeman, 1973)
  • (38) See, e.g., Eq. (11.98) at p. 547 in Ref. Jackson1998, or Eq. (24) at p. 872 in Ref. Frahm1979, .
  • (39) Charles P. Frahm, “Representing arbitrary boosts for undergraduates,” Am. J. Phys. 47 (10), 870-872 (1979)
  • (40) See, e.g., Eq. (13) in Ref. Cushing1967, .
  • (41) W. L. Kennedy, “Thomas rotation: a Lorentz matrix approach,” Eur. J. Phys. 23, 235-247 (2002)
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