CHAPTER XV
MATERIAL OBJECTS
56. Material Objects. 56.1 A material object is essentially a material object of a certain definite sort; namely, we define sorts of material objects, which are sets of objects with certain definite peculiarities, and a material object is such because it is a member of one of these sorts. For example a piece of wood is a material object because it belongs to the class of wooden objects and because this class possesses the requisite peculiarities. Similarly a charge of electricity is a material object for an analogous reason.
The objects which compose a set () form a sort of 'material' objects when (i) the objects of the set are all uniform, (ii) not more than one member of can be located in any volume, (iii) no member of can be located in two volumes of the same moment, (iv) if and be two members of respectively located in non-overlapping volumes in the same moment, then any pair of situations of and respectively are separated events, (v) if be a member of situated in an event , and located in the volume which is a section of , and be any volume which is a portion of , then there is a member of which is located in and is a concurrent component of .
56.2 If be a material object of a certain sort and be a volume in which is located and be a portion of , then the material object of the same sort as which is located in is called an 'extensive component' of .
56.3 It is by means of the properties of material objects that the atomic properties of objects are combined in mathematical calculations with the extensive continuity of events. Apart from material objects mathematical physics as at present developed would be impossible. For example where the physicist sees the electron as an atomic whole, the mathematician sees a distribution of electricity continuous in time and in space and capable of division into component objects which are also analogous distributions.
57. Stationary Events. 57.1 In order to understand the theory of the motion of material objects, it is first necessary to define the concept of a 'stationary' event. Consider some given time-system , and let denote a volume lying in a certain moment of this time-system. Let be a duration of bounded by moments and , and inhered in by ; so that , , are three parallel moments of the time-system , and lies between and . The volume is the locus of a set of event-particles and each of these event-particles lies in one and only one station of the duration . Also each station of either does not intersect or intersects it in one event-particle only. The assemblage of event-particles lying on stations of d which intersect [namely, each event-particle lying on one of these stations] is the complete set of event-particles analysing[8] an event. Such an event is called stationary in the time-system and stretches throughout the duration . It can also be called 'stationary in ,' since defines the time-system . Every event-particle within the event lies on a station of ; and a station of either has all its event-particles lying within the event or none of them. The volume is the section of the event by the moment . Furthermore if ′ be any other moment of the time-system it lying between and , it intersects the event in a volume ′ which is a geometrical replica of the volume . The moments and which bound the duration are the terminal moments of any event which is stationary in . The stations of lying in the event intersect and in terminal volumes and which are geometrical replicas of and ′. A volume, such as ', in which a moment of intersects an event stationary in is called a 'normal cross-section' of the event. A moment of another time-system which intersects the stationary event in a volume , but does not intersect either of the terminal volumes, is said to intersect it in an 'oblique cross-section.' All the oblique cross-sections of a stationary event which are made by moments of the same time-system are geometrical replicas of each other.
57.2 Consider an event stationary in the time-system , and let be another time-system. Let be the measure of the normal cross-sections of and let be the measure of the oblique cross-sections made by moments of . We require the ratio of to . Take (as usual) mutual axes for and , and let the event-particle which is the origin lie in which is the antecedent terminal moment of . Then is at the -time zero, and let (the subsequent terminal moment) be at the -time . Then if () be the -coordinates of the event-particle in which a station (of the set composing the event ) intersects , the -coordinates of the other end (the subsequent end) of in are ().
Furthermore let () be the -coordinates of the antecedent end of , and let () be the -coordinates of the subsequent end of . Then by the usual formulae [cf. subarticle 52.2]
But, by analogous reasoning to that for the elementary case of geometrical parallelograms, the absolute extent of the event can be expressed as and as ). Hence .
57.3 The stations of duration of time-system are portions of points of the time-less space of [the -space].
Thus by prolonging the stations which constitute the stationary event we obtain the assemblage of -points which is the complete assemblage of -points intersecting the cross-sections of , each event-particle in each cross-section lying on one and only one such -point and each of these -points intersecting each cross-section in one event-particle. The assemblage of these -points is a volume of the -space, and the successive instantaneous volumes which are the normal cross-sections of [stationary in ] each occupy this same volume in the -space. Thus the stationary event during the lapse of -time throughout which it endures is happening at the same place in the -space.
But the successive oblique cross-sections of formed by moments of another time-system are instantaneous volumes which successively occupy different volumes in the -space. These instantaneous volumes travel in the -space, sweeping over it with the uniform velocity , namely the velocity due to the time-system in the -space.
57.4 A 'normal slice' of a stationary event is the slice of it cut off between any two normal cross-sections. An 'oblique slice' of a stationary event is the slice of it cut off between any two parallel oblique cross-sections. A normal slice of a stationary event is itself a stationary event in the same time-system.
58. Motion of Objects. 58.1 A material object is 'motionless' within a duration when throughout that duration the material object and its extensive components are all situated in stationary events.
In the case of a motionless material object, Law I for uniform objects can be made more precise, as follows:
If be a material object motionless in the duration and be the stationary event extending throughout in which it is situated, then is situated in any oblique slice of .
The accompanying figures illustrate (i) the kind of slice which is included in this law and (ii) the kind of slice which is excluded.
It immediately follows that—with the nomenclature of the enunciation of the law— is located in every oblique cross-section of .
If be the time-system of the duration in which is motionless and, in some other time-system , ′ be the duration of maximum extent which intersects in an oblique slice, then throughout ′ in the time-less space of a the material object has a uniform motion of translation with the velocity of in .
58.2 This property, possessed by a material object which is motionless in a time-system , of being situated in every oblique slice of its stationary situation is a fundamental physical law of nature. Namely, percipients cogredient with different time-systems can 'recognise' the same material objects. In other words, the character of a material object is not altered by its motion.
58.3 The motion of a material object is 'regular' when if be any volume in which it is located and be any event-particle in , and ′ be any variable volume which contains and is a portion of , and ′ be the extensive component of which is located in ′, then, as ′ is progressively diminished without limit, a time-system can be found such that the errors of calculations, respecting magnitudes exhibited by ′ which assume that ′ is motionless in , tend to the limit zero, provided that the time-lapse of the durations in within which ′ is motionless is also correspondingly diminished without limit.
The above definition of regular motion is a description of the assumptions in the ordinary mathematical treatment of the motion of a material object (not necessarily rigid) which is not moving with a uniform motion of translation. If be the standard time-system to which motions are referred, then the velocity of in is the velocity at the event-particle [i.e. at the -space point at the -time ] of the material object.
59. Extensive Magnitude. 59.1 A theory of extensive magnitude is required to complete the theory of material objects.
Let and ′ be two objects (material or otherwise), then the statement that and ′ possess quantities of a certain kind and that the ratio of the quantity to the quantity ′ has a certain definite numerical value is a reference to some determinate method of comparison of to ′ which is the defining characteristic of that kind of quantity[9].
The quantity of a certain kind possessed by a material object is called 'extensive' when it is a determinate function of the quantities of the same kind possessed by any two of its extensive components which (i) are exhaustive of and (ii) are non-overlapping [i.e. have no extensive component in common].
If the determinate function be that of simple addition [so that, being the quantities possessed respectively by and its two extensive components, ], then the kind of quantity will be called 'absolutely' extensive. When an extensive quantity is not absolutely extensive, it will be called 'semi-extensive.'
59.2 It is usual in philosophical discussions to con- fine the term 'extensive quantity' to what is here defined as 'absolutely extensive quantity,' and to ignore entirely the occurrence of semi-extensive quantities. But in physical science semi-extensive quantities are well known. For example, consider a sphere of radius uniformly charged with electricity throughout its volume. Divide the sphere into two parts, namely a concentric nucleus of radius and a shell of thickness . Then the electromagnetic mass of the whole sphere is not the sum of the electromagnetic masses of these two parts, but is to be calculated by a quadratic law from the charges.
A material object expresses the spatial distribution of a quantity of 'material,' when the quantity is absolutely extensive.
59.3 The volume-density, at a time in the -space of a time-system , of the distribution of any absolutely extensive quantity possessed by a material object is calculated by the ordinary mathematical formula. Consider any event-particle occupying the -point at the -time . Let be the measure of a volume in the -space which contains ; and let ′ be the extensive component of located in , if there be such an extensive component. Let be the measure of the quantity possessed by ′. Then the limit of the ratio of to , as is indefinitely diminished, is the density at at the time of the material [i.e. of the absolutely extensive quantity].
59.4 The above definitions contemplate quantities immediately possessed by the extensive objects as such, for example, charges of electricity and intensities of sense-objects. But there are also quantities which are only mediately possessed by the objects, but are immediately possessed by the events which are their situations. Such quantities may vary with the variation in the situation of the object mediately possessing them.
A mediately possessed quantity may for a certain type of material objects satisfy the characteristic condition for an extensive quantity. In that case it is an extensive quantity mediately possessed by that type of material objects. All variable extensive quantities are of this mediate character. A quantity mediately possessed by a material object at a moment [i.e. at a time ] of a time-system is the limit of the quantity possessed by the successive converging situations of in the successive durations of an abstractive class (of durations in the time-system which converges to .
The volume-density, at a time in the -space of a time-system , of the distribution of any absolutely extensive quantity mediately possessed by a material object is calculated according to the preceding definition for the case of immediately possessed quantities, except that the 'quantity mediately possessed by (or by an extensive component of ) at the time ' must be substituted everywhere for the 'quantity possessed by (or by an extensive component of ).'
59.5 We can compare the volume-densities and of an absolutely extensive quantity for two time-systems and respectively at a given event-particle , assuming, as we may assume, that the motion of the material object possessing (mediately or immediately) the quantity is regular.
Let be the time-system in which the object is stationary at , and let be the volume-density at for the time-system . Let be the moments in , , and respectively which contain . Let be the measure of a small volume in which contains [and therefore the measure of the volume in the timeless -space which this instantaneous volume occupies]. Consider the event () stationary in of which this small volume is a normal cross-section, and bounded by terminal moments ′ and ″ on either side of and both near . Then, by the theory of regular motion, we can take this stationary event () as the situation of an extensive component of , when is small enough and the duration bounded by ′ and ″ is short enough. Let and be the measures of the volumes which are the oblique cross-sections of made by and . Then ultimately , , and are expressions for the measure of the quantity possessed by .
But by equation (1) of 57.2 of this chapter,
Now take the mutual axes for and , and let () and () be the coordinates of in and respectively, and let () and () be the velocities due to it in and respectively. Then by equation (1) of 52.6,
59.6 Now let denote differentiation following the motion () at () and let denote differentiation at the point ().
Then it is easily proved that
Hence from equation (2) of 59.5 above
Again by using the formulae of article 52, we can prove that
From these results we immediately deduce
Now the condition that the total extensive quantity which is the 'charge' of any extensive component never varies when conceived as distributed through the -space is
This is the well-known equation of continuity. Now equation (4) shows that if this equation holds for the space of any time-system, it holds for the spaces of all time-systems.
When the equation of continuity holds, the 'charge' of any extensive component of the material object under consideration never varies. Hence it is a mere matter of words and definition whether the charge is said to be mediately possessed by the object or immediately possessed.
[8]Cf. subarticle 37.3, Chapter X, Part III.
[9]Cf. Principia Mathematica.