CHAPTER XIIIMOTION49. Analytic Geometry.49.1Consider any time-system: we will term the space of this time-system '-space' and its moments '-moments'; also the points and straight lines of-space will be termed '-points' and '-lines,' and rects and levels which lie in-moments will be termed '-rects' and '-levels.' Ifbe any event-particle, thenwill denote the-moment which covers. Ifbe any other time-system, there are no-moments which are also-moments, and no-points which are also-points; but there are-levels which are also-levels and-rects which are also-rects. For the two momentsandintersect in a common level which will be called. Then rects lying inare both-rects and-rects. In particular throughin the levelpairs of mutually normal rects exist, and every rect throughandis a member of one such pair.49.2Letbe any arbitrarily chosen event-particle, which we will term the origin; and letbe the-point occupied by; and let,,be any triad of mutually rectangular-rects in the moment, each containing. In this notation,, etc., do not denote any particular entities, but the symbols such asandare each to be taken as one whole. Letdenote the matrix containingand, with analogous meanings forand; and let,anddenote respectively the levels containingand,and,and. Letbe any other event-particle occupying the point, and letbe the-rects throughrespectively parallel to,,.fig16Fig. 16.In the diagram the third dimension of the momentsandnamely the-dimension, is suppressed, so that these moments are diagrammatically represented as two-dimensional. Point-tracks (in this case-points) are represented by dotted lines. The diagram has the defect of representing matrices, such as, by levels, and is thus liable to lead to unfounded assumptions.49.3Lengths on all rects, whether or no they be-rects, are measurable in terms of one unit length. But time-lapses between-moments—or, what is the same thing, time-lapses along-points—must be measured in a time-unit peculiar to the time-system, since as yet no means of obtaining congruent time-units in different time-systems has been disclosed. We will suppose at present that in each time-system there is a given arbitrarily chosen unit for time-measurement.49.4Let the momentary space ofbe referred to the three rectangular-rects,,as axes of coordinates; and let the momentary space ofbe referred to the three rectangular-rects,,as axes of coordinates; and let the time-less space of a [the-space] be referred to the three rectangular-lines respectively included in the matrices,,as axes of coordinates; and let the four-dimensional space of all particles be referred to the four axes consisting of the three-rects,,, and of the-pointas axes of co-ordinates.49.5Letbe any event-particle in the moment, and letoccupy the-pointwhich intersects the momentin the event-particle. Let the lapse of time between the momentsandbe, whereis positive whenis subsequent to; and let the coordinates of the-pointin the-space be (). Then the coordinates ofin the momentary space ofand ofin the momentary space ofare also (). Also the '-coordinates' ofin the four-dimensional space of particles are (); this fact forcan also be expressed by saying thatoccupies the-point () at the-time.A moment, viewed as a locus of event-particles, is represented by a linear equation in the four coordinates (). But the converse is not true; namely, not every linear equation represents a moment. A pair of linear equations represent a level or a matrix, and three independent linear equations represent a rect or a point-track or a null-track.49.6Ifandbe any two time-systems, two sets of mutually normal axes,, and, can be found as in the previous subarticle. But these two sets can evidently be so adjusted thatis identical withandis identical with, where the two rects () must both lie in the level.fig17Fig. 17.Then the matrix normal to this level atwill be denoted by; it contains throughone-point, one-pointone-rect, and one-rect. Then any event-particle is referred to the axesfor the system, and to the axesfor the system. Let its-coordinates be () and its-coordinates be (), where, and.In the diagram, for the sake of simplicity, the particleis in the matrix; and its coordinates [as in the diagram] in the two systems are () and (), where(with its proper sign) is,(with its proper sign) is,(with its proper sign) is, and(with its proper sign) is.A pair of sets of four axes for a and allied as described in this subarticle are called 'mutual axes' for the two systems.49.7The formulae for transformation from the a-coordinates to the-coordinates, referred to mutual axes, are obviously of the formwhereare constants dependent on the two systemsandand on the two arbitrarily chosen units of time-lapse inand, but evidently not dependent on the arbitrarily chosen set of rectangular rectsandin the level.The corresponding ()-equations, interchangingand, areThe two pairs of ()-equations, (i) and (ii), must be equivalent. The conditions areOnly four out of these five conditions are independent.50. The Principle of Kinematic Symmetry.50.1Consider any other time-system. The-point () occupied by () and the-point occupied by the same event-particle lie on a matrixwhich includes an-line () of which every-point is intersected by. Thuscorrelates the-point () with the-timeand the neighbouring-point onnamely (), with the neighbouring-time. In this waymakes every set of-coordinates of a variable-point to be a function of; namely it correlates an-point () with the velocity (), which can also be writtenAnalogously the same time-systemcorrelates a-point () with the velocity () which can be writtenNow the time-systemindicates a definite transference from an event-particle () to another event-particle () occupying the same-point (), where any mutually normal-coordinates are employed. The former event-particle is that indicated by () and by (), and the latter event-particle by () and by ().Hence from equations (i) of49.7Nowis any time-system. First identify it with. Then. Hence, andis the velocity of the time-systemin the space of[or, more briefly, the 'velocity ofin']. Let this velocity be; it is evidently along the-axis in the space of, andAgain identify the systemwith. Then; and hence, andis the velocity ofin. Let this velocity be; it is along the-axis in the space of, and50.2We will now introduce what we will term the 'Principle of Kinematic Symmetry.'Before enunciating this principle it is necessary to determine a standard method of choosing the positive directions of the axesandin the matrixand of the axesand. By reference to the figure ofsubarticle 45.2it will be seen that, of the four angular regions into which the rectsanddivide the matrix, two vertically opposite regions include no point-tracks passing throughand the remaining two such regions include point-tracks as well as rects through. The standard choice of positive directions forandis such that the two regions bounded one by both positive directions of these axes, and the other by both negative directions, should include only rects passing through.The positive directions forandare settled by the rule that a positive measure of lapse of time should indicate subsequence in the time-order to the moment. This rule is definite because of the ultimate distinction between antecedence and subsequence in time, which has not otherwise been made use of. This standard choice of positive directions along mutual axes for two time-systems will always be adopted.50.3The principle of kinematic symmetry has two parts, enunciating consequences which flow from the fact that the time-units in two time-systemsandare congruent. The first part may be taken as the definition, or necessary and sufficient test, of such congruence.The first part of the principle can be enunciated as the statement that the measures of relative velocities [i.e. the velocity ofinand ofin] are equal and opposite; namelyThe second part is the principle of the symmetry of two time-systems in respect to transverse velocities; namely, if a velocityin, normally transverse to the direction ofin, is represented by the velocity () in, whereis along the direction ofinand′ is normally transverse to it, then the same magnitude of velocityin, normally transverse to the direction ofin, is represented by the velocity () in, whereis along the direction ofin, and′ is normally transverse to it.From the first part of the principle, by (ii) and (iii) of50.1, we deduceIn order to apply the second part of the principle we first identifywith), then from (i) and (ii) of50.1Again we identify (with), and by interchangingandin the above formulae we findHence by the second part of the principle51. Transitivity of Congruence.51.1It follows, from (iii) of49.7, and from (ii) and (iii) of50.1, and from (i), (ii), (iii) of50.3, that equations (i) of50.1can be writtenWe can now expressandin terms ofand an absolute constant by considering deductions from the transitivity of congruence.51.2Letbe a time-system such that the levelcontainsand, and let these rects be the axesand. Then the matrixcontains,,and. Thus we have obtained a set of mutual axes forand; namely, () and (), whereandnow play the part thatandsustain forand. Thus the velocities of the time-systeminandare, by (i) of51.1, connected byWe have here assumed the congruence of the time-units inand.Now identifyand. ThenHence from (i) of51.1Again identifywith. ThenHence from (i) of this subarticleFrom (ii) and (iii) and (i) of50.351.3Evidently ifbe any other member of the collinear set of time-systems (,), thenHence ifbe a collinear set of time-systems, and,,,be any four of its members,and hence, since, we obtainwhereis a constant for the collinear set.Furthermore, ifbe a time-system not belonging tobut related toandas explained in51.2,51.4Now let,,be any three non-collinear time-systems, and construct a diagram to represent elements in the time-less space ofaccording to the familiar method of geometricians.The points of the diagram symbolise-points, and the straight lines of the diagram symbolise-lines. Letbe any-point and letbe the direction in-space of the velocity. Thenis the direction in-space of the velocity (positive or negative) of any member of the collinear set ().fig18Fig. 18.Letbe the direction in-space of the velocity; by hypothesisis distinct from. Letbe the-line perpendicular to the-plane, and letbe a time-system whose velocity in, namely, is along. Letdenote the collinear set (),′ the collinear set (), and″ the collinear set (). Hence from (vi) of51.3Hence from (vii) of51.3Hence, sinceand, it is easy to prove thatis the same for any pair of time-systems; in other words, thatis an absolute constant.52. The Three Types of Kinematics.52.1There are thus three types of kinematics possible, according asis positive, negative, or infinite. The formally possible type whereis zero requires that either orshould be zero; by reference to (i) of49.7and to (i) of51.1this supposition is seen to lead to results in such obvious contradiction to experience as to preclude the necessity for further examination. Let us name the types retained (according to the familiar habit) the 'hyperbolic,' the 'elliptic' and the 'parabolic' types of kinematics.52.2First consider the hyperbolic type and putfor. The equations of articles49and51then becomeThe equations of transformation, namely (ii), can be expressed symmetrically as betweenandby means of the scheme [where],,,,0,0,0,1,0,00,0,1,00,0,52.3We notice thatThe integraltaken throughout the four-dimensional region of the set of event-particles which analyse [cf.37.3] an eventwill be called the 'absolute extent' of. It follows from (i) that the absolute extent of an event is independent of the time-system in which its measure is expressed.Furthermore ifbe any function of (), it can by (ii) of52.2be also expressed as a function of (), and then by (i)or, in more familiar form,where the limits are taken to include some event.We may expect important physical properties to be expressible in terms of such integrals, in particular whereis an invariant form for the equations of transformation of52.2, and when the conditions, which the quantity represented by the integral satisfies, are also invariant in their expression in different time-systems.The formulae of this subarticle hold of each type of kinematics.52.4The hyperbolic type of kinematics has issued in the formulae of the Larmor-Lorentz-Einstein theory of electromagnetic relativity, namely, the theory by which with a certain amount of interpretation the electromagnetic equations are invariant for these transformations.The physical meaning ofis also well known; namely, any velocity which in any time-system is of magnitudeis of the same magnitude in every other time-system. No assumption of the existence of a velocity with this property or of the electromagnetic invariance has entered into the deduction of the kinematic equations of the hyperbolic type. A velocity greater thancannot represent any time-system, and accordingly its physical significance must be entirely different from that of a velocity less than.52.5It is easily proved from (ii) of52.2thatIf the originand the event-particle, i.e. (), be co-momental andbe the time-system whose moment, contains, then by (i)Ifandbe sequent and on a point-track, andbe the time-system whose pointis occupied by, then by (i)Thus there are three ways in which the 'separation' between two event-particles (and) can be estimated; namely, (1) in any assumed time-systemthe-distance between the-points occupied by the event-particles measures-space separation: (2) the lapse of-time between the-moments occupied by the event-particles measures-time separation: and (3) if the event-particles be co-momental,measures the 'proper' space separation and there is no 'proper' time separation; and if the particles be sequent,measures the 'proper' time separation and there is no 'proper' space separation.In the framing of physical laws it is essential to consider what measure of separation is relevant. It is to be noted that there may be time-systems(other than) of special relevance to the phenomena in question. It is not at all obvious that invariance of form in respect to all time-systems is a requisite in the complete expression of such laws; namely, the demand for relativistic equations is only of limited applicability.Ifandbe on a null-trackEvent-particles on the same null-track may be expected to have special physical relations to each other. Call such event-particles 'co-null.'52.6We may conceive a special time-systemassociated (by some means) with each event-particle (). Thusis a function of these four co-ordinates of a particle; or in other words, () are functions of ().A correlation of time-systems to event-particles which is one-many, so that there is one and only one time-system corresponding to each event-particle, is called a 'complete kinematic correlation.' The portion of that correlation which only concerns event-particles at the timeis called a 'kinematic-correlation.' Other portions can be selected by confining the event-particles to certain regions in the-space.If in a certain kinematic correlation the time-system it be correlated to (), thenis called the time-system of () 'proper' to that correlation. The 'proper' time-system of an event-particle always refers to a certain kinematic correlation implicitly understood. Furthermore () is the velocity at () due to the implicitly understood kinematic correlation at the-time.Then,being the proper time-system at (),Then equations (iii) of52.2can be writtenThe kinematic symmetry as betweenandis now apparent in the formulae. The first of equations (ii) can be replaced by52.7In considering the elliptic type of kinematics putfor. The equations ofarticle 51are now embodied in the scheme,,,,0,0,0,1,0,00,0,1,00,0,The fundamental distinction between space and time, i.e. between rects and point-tracks, has failed to find any expression in the formulae for measurement relations. Accordingly with this type of kinematics, it would be natural to suppose that the distinction does not exist and that every rect was a point-track and every point-track a rect. This conception is logically possible but does not appear to correspond to the properties of the external world of events as we know it. Furthermore the electromagnetic equations lose their invariant property.Altogether there appear to be good reasons for putting aside the elliptic type of kinematics as inapplicable to nature.52.8In the parabolic type of kinematics we putHenceThen from (ii) of51.1and (ii) of50.3and (iii) of50.1Thus equations (i) of49.7giveThese are the formulae for the ordinary Newtonian relativity.These formulae are well in accordance with common sense and are in fact the formulae naturally suggested by ordinary experience. To some extent the hyperbolic formulae lead to unexpected results, though, ifbe a velocity not less than that of light, the divergences from the deliverances of common sense take place in respect to phenomena which are not manifest in ordinary experience. But when by refined methods of observation the divergences between the two types of kinematics should be apparent to the senses, experiment has, so far, pronounced in favour of the hyperbolic type. Accordingly it is this type which we consider in the sequel.52.9There is however one objection to the hyperbolic type, as compared to the parabolic type, which is worth considering. In the hyperbolic kinematics there is an absolute velocitywith special properties in nature. The difficulty which is thus occasioned is rather an offence to philosophic instincts than a logical puzzle. But certainly our familiar experience, in some way which it is difficult to formulate in words, leads us to shun the introduction of such absolute physical quantities. This particular difficulty is largely diminished by noting that the existence ofwith its peculiar properties really means that the space-units and time-units are comparable; namely, there is a natural relation between them to be expressed by takingto be unity. Either the time-unit would then be inconveniently small or the space-unit inconveniently large; but this inconvenience does not alter the fact that congruence between time and space is definable. Always when a possible definition of congruence is omitted, such absolute physical quantities occur. The fact that, so far as time and space are concerned, the existence of a congruence theory seems paradoxical is due to absence of any phenomena depending on that theory except in very exceptional circumstances produced by refined observations.
49. Analytic Geometry.49.1Consider any time-system: we will term the space of this time-system '-space' and its moments '-moments'; also the points and straight lines of-space will be termed '-points' and '-lines,' and rects and levels which lie in-moments will be termed '-rects' and '-levels.' Ifbe any event-particle, thenwill denote the-moment which covers. Ifbe any other time-system, there are no-moments which are also-moments, and no-points which are also-points; but there are-levels which are also-levels and-rects which are also-rects. For the two momentsandintersect in a common level which will be called. Then rects lying inare both-rects and-rects. In particular throughin the levelpairs of mutually normal rects exist, and every rect throughandis a member of one such pair.
49.2Letbe any arbitrarily chosen event-particle, which we will term the origin; and letbe the-point occupied by; and let,,be any triad of mutually rectangular-rects in the moment, each containing. In this notation,, etc., do not denote any particular entities, but the symbols such asandare each to be taken as one whole. Letdenote the matrix containingand, with analogous meanings forand; and let,anddenote respectively the levels containingand,and,and. Letbe any other event-particle occupying the point, and letbe the-rects throughrespectively parallel to,,.
fig16Fig. 16.
Fig. 16.
Fig. 16.
In the diagram the third dimension of the momentsandnamely the-dimension, is suppressed, so that these moments are diagrammatically represented as two-dimensional. Point-tracks (in this case-points) are represented by dotted lines. The diagram has the defect of representing matrices, such as, by levels, and is thus liable to lead to unfounded assumptions.
49.3Lengths on all rects, whether or no they be-rects, are measurable in terms of one unit length. But time-lapses between-moments—or, what is the same thing, time-lapses along-points—must be measured in a time-unit peculiar to the time-system, since as yet no means of obtaining congruent time-units in different time-systems has been disclosed. We will suppose at present that in each time-system there is a given arbitrarily chosen unit for time-measurement.
49.4Let the momentary space ofbe referred to the three rectangular-rects,,as axes of coordinates; and let the momentary space ofbe referred to the three rectangular-rects,,as axes of coordinates; and let the time-less space of a [the-space] be referred to the three rectangular-lines respectively included in the matrices,,as axes of coordinates; and let the four-dimensional space of all particles be referred to the four axes consisting of the three-rects,,, and of the-pointas axes of co-ordinates.
49.5Letbe any event-particle in the moment, and letoccupy the-pointwhich intersects the momentin the event-particle. Let the lapse of time between the momentsandbe, whereis positive whenis subsequent to; and let the coordinates of the-pointin the-space be (). Then the coordinates ofin the momentary space ofand ofin the momentary space ofare also (). Also the '-coordinates' ofin the four-dimensional space of particles are (); this fact forcan also be expressed by saying thatoccupies the-point () at the-time.
A moment, viewed as a locus of event-particles, is represented by a linear equation in the four coordinates (). But the converse is not true; namely, not every linear equation represents a moment. A pair of linear equations represent a level or a matrix, and three independent linear equations represent a rect or a point-track or a null-track.
49.6Ifandbe any two time-systems, two sets of mutually normal axes,, and, can be found as in the previous subarticle. But these two sets can evidently be so adjusted thatis identical withandis identical with, where the two rects () must both lie in the level.
fig17Fig. 17.
Fig. 17.
Fig. 17.
Then the matrix normal to this level atwill be denoted by; it contains throughone-point, one-pointone-rect, and one-rect. Then any event-particle is referred to the axesfor the system, and to the axesfor the system. Let its-coordinates be () and its-coordinates be (), where, and.
In the diagram, for the sake of simplicity, the particleis in the matrix; and its coordinates [as in the diagram] in the two systems are () and (), where(with its proper sign) is,(with its proper sign) is,(with its proper sign) is, and(with its proper sign) is.
A pair of sets of four axes for a and allied as described in this subarticle are called 'mutual axes' for the two systems.
49.7The formulae for transformation from the a-coordinates to the-coordinates, referred to mutual axes, are obviously of the formwhereare constants dependent on the two systemsandand on the two arbitrarily chosen units of time-lapse inand, but evidently not dependent on the arbitrarily chosen set of rectangular rectsandin the level.
The corresponding ()-equations, interchangingand, are
The two pairs of ()-equations, (i) and (ii), must be equivalent. The conditions are
Only four out of these five conditions are independent.
50. The Principle of Kinematic Symmetry.50.1Consider any other time-system. The-point () occupied by () and the-point occupied by the same event-particle lie on a matrixwhich includes an-line () of which every-point is intersected by. Thuscorrelates the-point () with the-timeand the neighbouring-point onnamely (), with the neighbouring-time. In this waymakes every set of-coordinates of a variable-point to be a function of; namely it correlates an-point () with the velocity (), which can also be written
Analogously the same time-systemcorrelates a-point () with the velocity () which can be written
Now the time-systemindicates a definite transference from an event-particle () to another event-particle () occupying the same-point (), where any mutually normal-coordinates are employed. The former event-particle is that indicated by () and by (), and the latter event-particle by () and by ().
Hence from equations (i) of49.7
Nowis any time-system. First identify it with. Then. Hence, andis the velocity of the time-systemin the space of[or, more briefly, the 'velocity ofin']. Let this velocity be; it is evidently along the-axis in the space of, andAgain identify the systemwith. Then; and hence, andis the velocity ofin. Let this velocity be; it is along the-axis in the space of, and
50.2We will now introduce what we will term the 'Principle of Kinematic Symmetry.'
Before enunciating this principle it is necessary to determine a standard method of choosing the positive directions of the axesandin the matrixand of the axesand. By reference to the figure ofsubarticle 45.2it will be seen that, of the four angular regions into which the rectsanddivide the matrix, two vertically opposite regions include no point-tracks passing throughand the remaining two such regions include point-tracks as well as rects through. The standard choice of positive directions forandis such that the two regions bounded one by both positive directions of these axes, and the other by both negative directions, should include only rects passing through.
The positive directions forandare settled by the rule that a positive measure of lapse of time should indicate subsequence in the time-order to the moment. This rule is definite because of the ultimate distinction between antecedence and subsequence in time, which has not otherwise been made use of. This standard choice of positive directions along mutual axes for two time-systems will always be adopted.
50.3The principle of kinematic symmetry has two parts, enunciating consequences which flow from the fact that the time-units in two time-systemsandare congruent. The first part may be taken as the definition, or necessary and sufficient test, of such congruence.
The first part of the principle can be enunciated as the statement that the measures of relative velocities [i.e. the velocity ofinand ofin] are equal and opposite; namely
The second part is the principle of the symmetry of two time-systems in respect to transverse velocities; namely, if a velocityin, normally transverse to the direction ofin, is represented by the velocity () in, whereis along the direction ofinand′ is normally transverse to it, then the same magnitude of velocityin, normally transverse to the direction ofin, is represented by the velocity () in, whereis along the direction ofin, and′ is normally transverse to it.
From the first part of the principle, by (ii) and (iii) of50.1, we deduceIn order to apply the second part of the principle we first identifywith), then from (i) and (ii) of50.1Again we identify (with), and by interchangingandin the above formulae we findHence by the second part of the principle
51. Transitivity of Congruence.51.1It follows, from (iii) of49.7, and from (ii) and (iii) of50.1, and from (i), (ii), (iii) of50.3, that equations (i) of50.1can be written
We can now expressandin terms ofand an absolute constant by considering deductions from the transitivity of congruence.
51.2Letbe a time-system such that the levelcontainsand, and let these rects be the axesand. Then the matrixcontains,,and. Thus we have obtained a set of mutual axes forand; namely, () and (), whereandnow play the part thatandsustain forand. Thus the velocities of the time-systeminandare, by (i) of51.1, connected by
We have here assumed the congruence of the time-units inand.
Now identifyand. Then
Hence from (i) of51.1
Again identifywith. Then
Hence from (i) of this subarticle
From (ii) and (iii) and (i) of50.3
51.3Evidently ifbe any other member of the collinear set of time-systems (,), then
Hence ifbe a collinear set of time-systems, and,,,be any four of its members,and hence, since, we obtainwhereis a constant for the collinear set.
Furthermore, ifbe a time-system not belonging tobut related toandas explained in51.2,
51.4Now let,,be any three non-collinear time-systems, and construct a diagram to represent elements in the time-less space ofaccording to the familiar method of geometricians.
The points of the diagram symbolise-points, and the straight lines of the diagram symbolise-lines. Letbe any-point and letbe the direction in-space of the velocity. Thenis the direction in-space of the velocity (positive or negative) of any member of the collinear set ().
fig18Fig. 18.
Fig. 18.
Fig. 18.
Letbe the direction in-space of the velocity; by hypothesisis distinct from. Letbe the-line perpendicular to the-plane, and letbe a time-system whose velocity in, namely, is along. Letdenote the collinear set (),′ the collinear set (), and″ the collinear set (). Hence from (vi) of51.3
Hence from (vii) of51.3
Hence, sinceand, it is easy to prove thatis the same for any pair of time-systems; in other words, thatis an absolute constant.
52. The Three Types of Kinematics.52.1There are thus three types of kinematics possible, according asis positive, negative, or infinite. The formally possible type whereis zero requires that either orshould be zero; by reference to (i) of49.7and to (i) of51.1this supposition is seen to lead to results in such obvious contradiction to experience as to preclude the necessity for further examination. Let us name the types retained (according to the familiar habit) the 'hyperbolic,' the 'elliptic' and the 'parabolic' types of kinematics.
52.2First consider the hyperbolic type and putfor. The equations of articles49and51then become
The equations of transformation, namely (ii), can be expressed symmetrically as betweenandby means of the scheme [where]
52.3We notice that
The integraltaken throughout the four-dimensional region of the set of event-particles which analyse [cf.37.3] an eventwill be called the 'absolute extent' of. It follows from (i) that the absolute extent of an event is independent of the time-system in which its measure is expressed.
Furthermore ifbe any function of (), it can by (ii) of52.2be also expressed as a function of (), and then by (i)or, in more familiar form,where the limits are taken to include some event.
We may expect important physical properties to be expressible in terms of such integrals, in particular whereis an invariant form for the equations of transformation of52.2, and when the conditions, which the quantity represented by the integral satisfies, are also invariant in their expression in different time-systems.
The formulae of this subarticle hold of each type of kinematics.
52.4The hyperbolic type of kinematics has issued in the formulae of the Larmor-Lorentz-Einstein theory of electromagnetic relativity, namely, the theory by which with a certain amount of interpretation the electromagnetic equations are invariant for these transformations.
The physical meaning ofis also well known; namely, any velocity which in any time-system is of magnitudeis of the same magnitude in every other time-system. No assumption of the existence of a velocity with this property or of the electromagnetic invariance has entered into the deduction of the kinematic equations of the hyperbolic type. A velocity greater thancannot represent any time-system, and accordingly its physical significance must be entirely different from that of a velocity less than.
52.5It is easily proved from (ii) of52.2that
If the originand the event-particle, i.e. (), be co-momental andbe the time-system whose moment, contains, then by (i)
Ifandbe sequent and on a point-track, andbe the time-system whose pointis occupied by, then by (i)
Thus there are three ways in which the 'separation' between two event-particles (and) can be estimated; namely, (1) in any assumed time-systemthe-distance between the-points occupied by the event-particles measures-space separation: (2) the lapse of-time between the-moments occupied by the event-particles measures-time separation: and (3) if the event-particles be co-momental,measures the 'proper' space separation and there is no 'proper' time separation; and if the particles be sequent,measures the 'proper' time separation and there is no 'proper' space separation.
In the framing of physical laws it is essential to consider what measure of separation is relevant. It is to be noted that there may be time-systems(other than) of special relevance to the phenomena in question. It is not at all obvious that invariance of form in respect to all time-systems is a requisite in the complete expression of such laws; namely, the demand for relativistic equations is only of limited applicability.
Ifandbe on a null-track
Event-particles on the same null-track may be expected to have special physical relations to each other. Call such event-particles 'co-null.'
52.6We may conceive a special time-systemassociated (by some means) with each event-particle (). Thusis a function of these four co-ordinates of a particle; or in other words, () are functions of ().
A correlation of time-systems to event-particles which is one-many, so that there is one and only one time-system corresponding to each event-particle, is called a 'complete kinematic correlation.' The portion of that correlation which only concerns event-particles at the timeis called a 'kinematic-correlation.' Other portions can be selected by confining the event-particles to certain regions in the-space.
If in a certain kinematic correlation the time-system it be correlated to (), thenis called the time-system of () 'proper' to that correlation. The 'proper' time-system of an event-particle always refers to a certain kinematic correlation implicitly understood. Furthermore () is the velocity at () due to the implicitly understood kinematic correlation at the-time.
Then,being the proper time-system at (),
Then equations (iii) of52.2can be written
The kinematic symmetry as betweenandis now apparent in the formulae. The first of equations (ii) can be replaced by
52.7In considering the elliptic type of kinematics putfor. The equations ofarticle 51are now embodied in the scheme
The fundamental distinction between space and time, i.e. between rects and point-tracks, has failed to find any expression in the formulae for measurement relations. Accordingly with this type of kinematics, it would be natural to suppose that the distinction does not exist and that every rect was a point-track and every point-track a rect. This conception is logically possible but does not appear to correspond to the properties of the external world of events as we know it. Furthermore the electromagnetic equations lose their invariant property.
Altogether there appear to be good reasons for putting aside the elliptic type of kinematics as inapplicable to nature.
52.8In the parabolic type of kinematics we putHence
Then from (ii) of51.1and (ii) of50.3and (iii) of50.1
Thus equations (i) of49.7give
These are the formulae for the ordinary Newtonian relativity.
These formulae are well in accordance with common sense and are in fact the formulae naturally suggested by ordinary experience. To some extent the hyperbolic formulae lead to unexpected results, though, ifbe a velocity not less than that of light, the divergences from the deliverances of common sense take place in respect to phenomena which are not manifest in ordinary experience. But when by refined methods of observation the divergences between the two types of kinematics should be apparent to the senses, experiment has, so far, pronounced in favour of the hyperbolic type. Accordingly it is this type which we consider in the sequel.
52.9There is however one objection to the hyperbolic type, as compared to the parabolic type, which is worth considering. In the hyperbolic kinematics there is an absolute velocitywith special properties in nature. The difficulty which is thus occasioned is rather an offence to philosophic instincts than a logical puzzle. But certainly our familiar experience, in some way which it is difficult to formulate in words, leads us to shun the introduction of such absolute physical quantities. This particular difficulty is largely diminished by noting that the existence ofwith its peculiar properties really means that the space-units and time-units are comparable; namely, there is a natural relation between them to be expressed by takingto be unity. Either the time-unit would then be inconveniently small or the space-unit inconveniently large; but this inconvenience does not alter the fact that congruence between time and space is definable. Always when a possible definition of congruence is omitted, such absolute physical quantities occur. The fact that, so far as time and space are concerned, the existence of a congruence theory seems paradoxical is due to absence of any phenomena depending on that theory except in very exceptional circumstances produced by refined observations.