CHAPTER XIII
MOTION
49. Analytic Geometry. 49.1 Consider 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.' If be any event-particle, then will denote the -moment which covers . If be 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 moments and intersect in a common level which will be called . Then rects lying in are both -rects and -rects. In particular through in the level pairs of mutually normal rects exist, and every rect through and is a member of one such pair.
49.2 Let be any arbitrarily chosen event-particle, which we will term the origin; and let be 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 as and are each to be taken as one whole. Let denote the matrix containing and , with analogous meanings for and ; and let , and denote respectively the levels containing and , and , and . Let be any other event-particle occupying the point , and let be the -rects through respectively parallel to , , .
In the diagram the third dimension of the moments and namely 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.
49.3 Lengths 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.
49.4 Let the momentary space of be referred to the three rectangular -rects , , as axes of coordinates; and let the momentary space of be 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 -point as axes of co-ordinates.
49.5 Let be any event-particle in the moment , and let occupy the -point which intersects the moment in the event-particle . Let the lapse of time between the moments and be , where is positive when is subsequent to ; and let the coordinates of the -point in the -space be (). Then the coordinates of in the momentary space of and of in the momentary space of are also (). Also the '-coordinates' of in the four-dimensional space of particles are (); this fact for can also be expressed by saying that occupies 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.
49.6 If and be 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 that is identical with and is identical with , where the two rects () must both lie in the level .
Then the matrix normal to this level at will be denoted by ; it contains through one -point , one -point one -rect , and one -rect . Then any event-particle is referred to the axes for the system , and to the axes for the system . Let its -coordinates be () and its -coordinates be (), where , and .
In the diagram, for the sake of simplicity, the particle is 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.
49.7 The formulae for transformation from the a-coordinates to the -coordinates, referred to mutual axes, are obviously of the form where are constants dependent on the two systems and and on the two arbitrarily chosen units of time-lapse in and , but evidently not dependent on the arbitrarily chosen set of rectangular rects and in the level .
The corresponding ()-equations, interchanging and , 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. 50.1 Consider any other time-system . The -point () occupied by () and the -point occupied by the same event-particle lie on a matrix which includes an -line () of which every -point is intersected by . Thus correlates the -point () with the -time and the neighbouring -point on namely (), with the neighbouring -time . In this way makes 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-system correlates a -point () with the velocity () which can be written
Now the time-system indicates 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) of 49.7
Now is any time-system. First identify it with . Then . Hence , and is the velocity of the time-system in the space of [or, more briefly, the 'velocity of in ']. Let this velocity be ; it is evidently along the -axis in the space of , and Again identify the system with . Then ; and hence , and is the velocity of in . Let this velocity be ; it is along the -axis in the space of , and
50.2 We 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 axes and in the matrix and of the axes and . By reference to the figure of subarticle 45.2 it will be seen that, of the four angular regions into which the rects and divide the matrix , two vertically opposite regions include no point-tracks passing through and the remaining two such regions include point-tracks as well as rects through . The standard choice of positive directions for and is 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 for and are 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.
50.3 The principle of kinematic symmetry has two parts, enunciating consequences which flow from the fact that the time-units in two time-systems and are 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 of in and of in ] 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 velocity in , normally transverse to the direction of in , is represented by the velocity () in , where is along the direction of in and ′ is normally transverse to it, then the same magnitude of velocity in , normally transverse to the direction of in , is represented by the velocity () in , where is along the direction of in , and ′ is normally transverse to it.
From the first part of the principle, by (ii) and (iii) of 50.1, we deduce In order to apply the second part of the principle we first identify with ), then from (i) and (ii) of 50.1 Again we identify ( with ), and by interchanging and in the above formulae we find Hence by the second part of the principle
51. Transitivity of Congruence. 51.1 It follows, from (iii) of 49.7, and from (ii) and (iii) of 50.1, and from (i), (ii), (iii) of 50.3, that equations (i) of 50.1 can be written
We can now express and in terms of and an absolute constant by considering deductions from the transitivity of congruence.
51.2 Let be a time-system such that the level contains and , and let these rects be the axes and . Then the matrix contains , , and . Thus we have obtained a set of mutual axes for and ; namely, () and (), where and now play the part that and sustain for and . Thus the velocities of the time-system in and are, by (i) of 51.1, connected by
We have here assumed the congruence of the time-units in and .
Now identify and . Then
Hence from (i) of 51.1
Again identify with . Then
Hence from (i) of this subarticle
From (ii) and (iii) and (i) of 50.3
51.3 Evidently if be any other member of the collinear set of time-systems (, ), then
Hence if be a collinear set of time-systems, and , , , be any four of its members, and hence, since , we obtain where is a constant for the collinear set.
Furthermore, if be a time-system not belonging to but related to and as explained in 51.2,
51.4 Now let , , be any three non-collinear time-systems, and construct a diagram to represent elements in the time-less space of according to the familiar method of geometricians.
The points of the diagram symbolise -points, and the straight lines of the diagram symbolise -lines. Let be any -point and let be the direction in -space of the velocity . Then is the direction in -space of the velocity (positive or negative) of any member of the collinear set ().
Let be the direction in -space of the velocity ; by hypothesis is distinct from . Let be the -line perpendicular to the -plane , and let be a time-system whose velocity in , namely , is along . Let denote the collinear set (), ′ the collinear set (), and ″ the collinear set (). Hence from (vi) of 51.3
Hence from (vii) of 51.3
Hence, since and , it is easy to prove that is the same for any pair of time-systems; in other words, that is an absolute constant.
52. The Three Types of Kinematics. 52.1 There are thus three types of kinematics possible, according as is positive, negative, or infinite. The formally possible type where is zero requires that either or should be zero; by reference to (i) of 49.7 and to (i) of 51.1 this 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.
52.2 First consider the hyperbolic type and put for . The equations of articles 49 and 51 then become
The equations of transformation, namely (ii), can be expressed symmetrically as between and by means of the scheme [where ]
| , | , | , | ||
| , | 0, | 0, | ||
| 0, | 1, | 0, | 0 | |
| 0, | 0, | 1, | 0 | |
| 0, | 0, |
52.3 We notice that
The integral taken throughout the four-dimensional region of the set of event-particles which analyse [cf. 37.3] an event will 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 if be any function of (), it can by (ii) of 52.2 be 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 where is an invariant form for the equations of transformation of 52.2, 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.
52.4 The 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 of is also well known; namely, any velocity which in any time-system is of magnitude is 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 than cannot represent any time-system, and accordingly its physical significance must be entirely different from that of a velocity less than .
52.5 It is easily proved from (ii) of 52.2 that
If the origin and the event-particle , i.e. (), be co-momental and be the time-system whose moment , contains , then by (i)
If and be sequent and on a point-track, and be the time-system whose point is 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-system the -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.
If and be 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.'
52.6 We may conceive a special time-system associated (by some means) with each event-particle (). Thus is 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 time is 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 (), then is 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) of 52.2 can be written
The kinematic symmetry as between and is now apparent in the formulae. The first of equations (ii) can be replaced by
52.7 In considering the elliptic type of kinematics put for . The equations of article 51 are now embodied in the scheme
| , | , | , | ||
| , | 0, | 0, | ||
| 0, | 1, | 0, | 0 | |
| 0, | 0, | 1, | 0 | |
| 0, | 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.
52.8 In the parabolic type of kinematics we put Hence
Then from (ii) of 51.1 and (ii) of 50.3 and (iii) of 50.1
Thus equations (i) of 49.7 give
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, if be 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.
52.9 There 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 velocity with 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 of with 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 taking to 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.