CHAPTER III

DETAILED CONSIDERATION OF VARIOUS

CONCEPTS OF PHYSICS

WE now begin our detailed consideration of the most important concepts

of physics. It is entirely beyond the scope of this essay to attempt

more than an indication of some of the most important matters. Neither

is it to be expected that the parts of this discussion will always have

a very close connection with each other; the purpose of the discussion

is to aid in acquiring the greatest possible self-consciousness of the

whole structure of physics.

THE CONCEPT OF SPACE

A logically satisfying definition of what we understand by the concept

of space is doubtless difficult to give, but we shall not be far from

the mark if we think of it as the aggregate of all those concepts which

have to do with position. Position means position of something. The

position of things is determined by some system of measurement; perhaps

the simplest is that implied in a Cartesian coördinate system with its

three measurements of length. Hence much of the essential discussion of

space has already been given in connection with the concept of length.

We have seen that measurements of length are made with physical

measuring rods applied to some physical object. We cannot measure the

distance between two points in empty space, because if space were empty

there would be nothing to identify the position of the ends of the

measuring rod when we move it from one position to the next. We see,

then, from the point of view of operations that the framework of

Cartesian geometry, often imagined in an ideal mathematical sense, is

really a physical framework, and that what we mean by spatial properties

is nothing but the properties of this framework. When we say that space

is Euclidean, we mean that the physical space of meter sticks is

Euclidean: it is meaningless to ask whether empty space is Euclidean.

Geometry, therefore, in so far as its results are expected to apply to

the external physical world, and in as far as it is not a logical system

built up from postulates, is an experimental science. This view is now

well understood and accepted, but there was a time when it was not

accepted, but vigorously attacked; the change of attitude toward this

question is symptomatic of a change of attitude toward many other

similar questions.

We have already emphasized that the space of astronomy is not a physical

space of meter sticks, but is a space of light waves. We may, therefore,

have different kinds of space, depending on the fundamental operations.

The space of meter sticks we have called "tactual space", and the space

of light beams "optical space". If we ask whether astronomical space is

Euclidean, we mean merely to ask whether those features of optical space

which are within the reach of astronomical measurement are Euclidean.

The only possible attitude with respect to this question, or such

related questions as whether the total volume of space is finite, or

whether space has curvature, is that it is entirely for experiment to

decide, and that we have no right to form any preconceived notion

whatever. It is therefore beyond the scope of this discussion.

It is interesting to notice that the restricted theory of relativity

virtually assumes, although often without making the explicit statement,

that tactual and optical space are the same. This equivalence results

from the properties assumed for light beams. The distance of a mirror

may be found equally well by measuring it with meter sticks, or by

determining the time required by a light signal to travel there and

back. This situation is, however, logically unsatisfying, because it

must be assumed that the operations for measuring time are independently

defined, and we shall see that they are not. It is a consequence of the

assumed equivalence of tactual and optical space that the path of a beam

of light is a straight line, as a straight line is determined by

operations with meter sticks. When we come to astronomical phenomena,

the physical operations with meter sticks can no longer be carried out,

and it is meaningless to ascribe to beams of light on an astronomical

scale the same geometrical properties that we do on a small scale.

THE CONCEPT OF TIME

According to our viewpoint, the concept of time is determined by the

operations by which it is measured. We have to distinguish two sorts of

time; the time of events taking place near each other in space, or local

time, and the time of events taking place at considerably separated

points in space, or extended time. As we now know, the concept of

extended time is inextricably mixed up with that of space. This is not

primarily a statement about nature at all, and might have been made

simply by the observation that the operations by which extended time is

measured involve those for measuring space. Of course historically the

doctrine of relativity was responsible for the critical attitude which

led to an examination of the operations of measuring time, but

relativity was not necessary for a realization of the spatial

implications of time, any more than the discovery of Planck's quantum

unit _h_ was necessary for the invention by Planck of his absolute units

of measurement, although historically he was inspired to make this

invention by discovering _h_, and in his own mind seems to have thought

of the connection as a necessary one.[7]

[Footnote 7: Max Planck, The Theory of Heat Radiation, translated by

Masius, P. Blakiston's Son & Co., 1914 edition, p. 174.]

The physical operations at the basis of the measurement of time have

never been subjected to the critical examination which seems to be

required. One method of measurement, for instance, involves the

properties of light.

A meter stick is set up with mirrors at the two ends, and a light beam

travels back and forth between the two mirrors without absorption. The

time required for a single passage back and forth is defined as the unit

of time, and time is measured simply by counting these intervals. But

such a procedure is unsatisfactory if we are to permit ourselves all

those operations which are demanded by even the simplest postulate of

relativity, for we must be able to move our clock from place to place,

transfer it from one system to another in relative motion, and with it

determine the properties of light beams in the stationary or moving

system. We recognize in principle that the length of the meter stick may

be different when it is in motion, that it may change also during the

acceleration incident to moving it from one place to another, and that

until proved to the contrary the velocity of light may be a function of

velocity or acceleration. The complicated interplay of all these

possibilities leaves us in much doubt as to the physical significance of

such postulates as, for example, that the velocity of light is the same

in the moving system and the stationary system. In order to ascribe any

simple significance to postulates about the velocity of light, it would

seem that we must have an instrument for measuring this velocity, and

therefore for measuring time, which does not itself involve the

properties of light. To do this we might seek to specify the measurement

of time in purely mechanical terms, as for instance in terms of the

vibration of a tuning fork, or the rotation of a flywheel. But here

again we encounter great difficulties, because we recognize that the

dimensions of our mechanical clock may change when it is set in motion,

and that the mass of its parts may also change. We want to use the clock

as a physical instrument in determining the laws of mechanics, which of

course are not determined until we can measure time, and we find that

the laws of mechanics enter into the operation of the clock.

The dilemma which confronts us here is not an impossible one, and is in

fact of the same nature as that which confronted the first physicist who

had to discover simultaneously the approximate laws of mechanics and

geometry with a string which stretched when he pulled it. We must first

guess at what the laws are approximately, then design an experiment so

that, in accordance with this guess, the effect of motion on some

phenomenon is much greater than the expected effect on the clock, then

from measurements with uncorrected clock time find an approximate

expression for the effect of motion on mass or length, with which we

correct the clock, and so on ad infinitum. However, so far as I know,

the possibility of such a procedure has not been analyzed, and until the

analysis is given, our complacency is troubled by a real disquietude,

the intensity of which depends on the natural skepticism of our

temperament.

In practice, the difficulties of such a logical treatment are so great

that the matter has been entirely glossed over. It is convenient to

postulate a clock, of unknown construction, but such that the velocity

of light, when measured in terms of it, has certain properties. Such,

for example, is the point of view in Birkhoff's recent book.[8]

[Footnote 8: G. D. Birkhoff. Relativity and Modern Physics, Harvard

University Press, 1923.]

The difficulty with this method is that the resulting edifice is as

divorced from physical reality as is the logical geometry of postulates.

We cannot be at all sure that the properties of light as measured with

our physical clocks are the same as the theoretical properties. The

difficulty is particularly important and fundamental in the general

theory of relativity; the basis of the whole theory is the infinitesimal

interval _ds_, which is supposed to be given. Once given, the

mathematics follows. But in a physical world, _ds_ is _not_ given, but

must be found by physical operations, and these operations involve

measurements of length and of time with clocks whose construction is not

specified. In any actual physical application the question must be

answered whether the physical instrument used in measuring the temporal

part of _ds_ is really a clock or not. There is at present no criterion

by which this question can be answered. If the vibrating atom is a

clock, then the light of the sun is shifted toward the infra-red, but

how do we know that the atom is a clock (some say yes, others no)? If we

find the displacement physically have we thereby proved that general

relativity is physically true, or have we proved that the atom is a

clock, or have we merely proved that there is a particular kind of

connection between the atom and the rest of nature, leaving the

possibility open that neither is the atom a clock nor general relativity

true? In practice, of course, we shall adopt the solution which is

simplest and most satisfying aesthetically, and doubtless shall say that

the atom is a clock and relativity true. But if we adopt this simple

view, we must also cultivate the abiding consciousness that at some time

in the future troubles may have their origin here.

It seems to me that the logical position of general relativity theory is

merely this: Given any physical system, then it is possible to assign

values to _ds_ such that relations mathematically deduced by the

principle of relativity correspond to relations between measurable

quantities in the physical system; but that the things that we

physically call _ds_ are anything more than approximately connected with

the _ds's_ required to give the mathematical relations, is at present no

more than a pious faith.

To return to the concept of time, we have already stated that there are

two main problems, that of measuring time at a single point of space,

and that of spreading a time system over all space. The second aspect of

the problem is that to which attention has been directed by relativity

theory; the following detailed examination shows how the operations of

relativity for setting and synchronizing clocks at distant places

involve the measurement of space. It is a fundamental postulate that the

adjustment of the clocks is to be accomplished by light signals. The

synchronization of the clocks is now simple enough. We merely demand

that light signals sent from the master clock at intervals of one second

arrive at any distant clock at intervals of one second as measured by

it, and we change the rate of the distant clock until it measures these

intervals as one second. After its rate has been adjusted, the distant

clock is to be so _set_ that when a light signal is despatched from the

master clock at its indicated zero of time the time of arrival recorded

at the distant clock shall be such that the distance of the clock from

the master clock divided by the time of arrival shall give the velocity

of light, assumed already known. This operation involves a measurement

of the distance of the distant clock, so that in spreading the time

coordinates over space the measurement of space is involved by

definition, and the measurement of time is, therefore, not a

self-contained thing. This is the physical basis for the treatment of

space and time as a four-dimensional manifold. Although mathematically

the numbers measuring space and time enter the formulas symmetrically,

nevertheless the physical operations by which these numbers are obtained

are entirely distinct and never fuse, and I believe it can lead only to

confusion to see in the possibility of a four dimensional treatment

anything more than a purely formal matter.

The notion of extended time, therefore, involves the measurement of

space. It is an interesting question whether the notion of local time

also involves the measurement of space. A rigorous answer to this

question involves giving the specifications for the construction of a

clock, which we have seen has not yet been done. It seems to me

probable, however, that the construction of even a single local clock

involves in some way the _measurement_ of space. If, for example, we use

a vibrating tuning fork, we must find how the time of vibration depends

on the amplitude of vibration, and this involves space measurement, or

if we use a rotating flywheel, we have to correct for the change of

moment of inertia due to the change of dimensions when it is set into

motion or brought into a gravitational field, and all this involves

space measurement. However, these considerations are not certain, and

perhaps the question is not important.

There is now the further consideration that actually in practice the

concept of local time is not entirely divorced from that of extended

time, for two bodies cannot occupy the same space at the same time, and

the time of any event is actually measured on an instrument at some

distance, communication being maintained by light or elastic signals.

But experience convinces us that in the limit, as the phenomenon to be

measured gets closer to the clock, there is no measurable difference,

whether communication with the clock is maintained by light, or

acoustical or tactual signals, so that we have come in physical practice

to accept measurement of the time of events in the immediate

neighborhood of the clock (local time) as one of the ultimately simple

things behind which we do not attempt to go.

Local time is, therefore, a concept treated by the physicist even now as

simple and unanalyzable. This concept is what most people have in mind

when they think of time. Time, according to this concept, is something

with the properties of local time; it was something of this kind that

Newton must have meant by his absolute time, and it is the tacit

retention of this sort of concept that is responsible for the difficulty

so often found in grasping the idea of the relativity of simultaneity,

which is of course entirely foreign to our experience of simultaneity in

local time. An examination of the operations involved in extending time

has shown how the concept of extended time is different from that of

simple local time; this difference leads to appreciably different

numerical relations when we are dealing with high velocities or great

distances. Local time is proved by experience not to be a satisfactory

concept for dealing with events separated by great distances in space or

with phenomena involving high velocities. For instance, we must not talk

about the age of a beam of light, although the concept of age is one of

the simplest derivatives of the concept of local time. Neither must we

allow ourselves to think of events taking place in Arcturus _now_ with

all the connotations attached to events taking place _here_ now. It is

difficult to inhibit this habit of thought, but we must learn to do it.

The naïve feeling is very strong that it does _mean_ something to talk

about the entire present state of the universe independent of the

process by which news of the condition of distant parts is determined by

us. I believe that an examination of this feeling will show that it is

psychological in character; what we mean by the totality of the present

is merely the entire present content of our consciousness. This is

apparently a simple direct thing; we do not appreciate until we make

further analysis that our present consciousness of the existence of the

moon or a star is due to light signals, and that therefore the

apparently simple immediate consciousness of events distant in space

involves complicated physical operations.

Similarly, if we continue to use local time, we get into trouble, when

we go to high velocities, with our simple concept of velocity, which may

be defined in terms of a combination of space and time concepts. The

concept of local time thus loses its value and becomes merely a blunted

tool when we try to carry it out of its original range. But the concept

of extended time, with which we have to replace local time, is a

complicated thing, to which we have not yet got ourselves accustomed; it

may perhaps prove to be so complicated as never to be a very useful

intuitive tool of thought.

All these considerations about time have been concerned only with

intervals of such an order of magnitude that they are readily

experienced by any individual. If we have to deal with intervals either

very long or very short, it is obvious that our entire procedure

changes, and consequently the concept changes. In extending the time

concept to eras remote in the past, for example, we try as always, to

choose the new operations so as to piece on continuously with those of

ordinary experience. A precise analysis of the change in the concept of

time when applied to the remote past does not seem to be of great

significance for our present physical purpose, and will not be attempted

here. It is perhaps worth while to point out, however, that all our

other concepts, as well as that of time, must be modified when applied

to the remote past; an example is the concept of truth. It is amusing to

try to discover what is the precise meaning in terms of operations of a

statement like this: "It is true that Darius the Mede arose at 6:30 on

the morning of his thirtieth birthday."

Of more concern for our physical purposes is the modification which the

time concept undergoes when applied to very short intervals. What is the

meaning, for example, in saying that an electron when colliding with a

certain atom is brought to rest in 10^-18 seconds? Here I believe the

situation is very similar to that with regard to short lengths. The

nature of the physical operations changes entirely, and as before, comes

to contain operations of an electrical and optical character. The

immediate significance of 10^-18 is that of a number, which when

substituted into the equations of optics, produces agreement with

observed facts. Thus short intervals of time acquire meaning only in

connection with the equations of electrodynamics, whose validity is

doubtful and which can be tested only in terms of the space and time

coordinates which enter them. Here is the same vicious circle that we

found before. Once again we find that concepts fuse together on the

limit of the experimentally attainable.

This discussion of the concept of time will doubtless be felt by some to

be superficial in that it makes no mention of the _properties_ of the

physical time to which the concept is designed to apply. For instance,

we do not discuss the one dimensional flow of time, or the

irrevocability of the past. Such a discussion, however, is beyond our

present purpose, and would take us deeper than I feel competent to go,

and perhaps beyond the verge of meaning itself. Our discussion here is

from the point of view of operations: we assume the operations to be

given, and do not attempt to ask why precisely these operations were

chosen, or whether others might not be more suitable. Such properties of

time as its irrevocability are implicitly contained in the operations

themselves, and the physical essence of time is buried in that long

physical experience that taught us what operations are adapted to

describing and correlating nature. We may digress, however, to consider

one question. It is quite common to talk about a reversal of the

direction of flow of time. Particularly, for example, in discussing the

equations of mechanics, it is shown that if the direction of flow of

time is reversed, the whole history of the system is retraced. The

statement is sometimes added that such a reversal is actually

impossible, because it is one of the properties of physical time to flow

always forward. If this last statement is subjected to an operational

analysis, I believe that it will be found not to be a statement about

nature at all, but merely a statement about operations. It is

_meaningless_ to talk about time moving backward: by definition,

_forward_ is the direction in which time flows.

THE CAUSALITY CONCEPT

The causality concept is unquestionably one of the most fundamental,

perhaps as fundamental as that of space and time, and therefore at least

equally entitled to a first place in the discussion. But as ordinarily

understood, there are certain spatial and temporal implications in the

causality concept, so that it can best be discussed in this order after

our examination of space and time.

There is an aspect of the causality concept that in many respects is

closely related to the question of "explanation", for to find the causes

of an event usually involves at the same time finding its explanation.

But there are nevertheless sufficient differences to warrant a separate

discussion.

It seems fairly evident that there was originally in the causality

concept an animistic element much like that in the concept of force to

be discussed later. The physical essence of the concept as we now have

it, freed as much as possible from the animistic element, seems to be

somewhat as follows. We assume in the first place an isolated system on

which we can perform unlimited identical experiments, that is, the

system may be started over again from a definite initial condition as

often as desired.[9]

[Footnote 9: We must include in general in the concept of "initial

condition" the past history of the system. In order not to make this

condition so broad as to defeat itself, we have to add the observation

that actually identity of past history is necessary over only a

comparatively short interval of time. Logical precision seems

unattainable here--the physical concepts themselves have not the

necessary precision.]

We assume further that when so started, the system always runs through

exactly the same sequence of events in all its parts. This contains the

assumption that the course of events runs independent of the absolute

time at which they occur--there is no change with time of the properties

of the universe.[10] It is a result of experience that systems with

these properties actually exist. An alternative way of stating our

fundamental hypothesis is that two or more isolated similar systems

started from the same initial condition run through the same future

course of events. Upon the system given in this way, which by itself

runs a definite course of events, we assume that we can superpose from

the outside certain changes, which have no connection with the previous

history of the system, and are completely arbitrary. Now of course in

nature, as we observe it, there is no such thing as an arbitrary change,

without connection with past history, so that strictly our assumption is

a pure fiction. It is here that the animistic element still seems to

persist, although perhaps not necessarily.

[Footnote 10: As so often in physics, we appear to be doing two things

at once here. It is doubtful whether we can give a meaning to "definite

initial condition" apart from the future behavior of the system, so that

we have no real right to infer from uniform future behavior both a

constancy of the laws of nature, independent of time, and a constancy of

initial condition. I very much question whether a thoroughgoing

operational analysis would show that there are really two independent

concepts here, and whether the use of two formally quite different

concepts is anything more than a convenience in expression. It seems to

me that it may be just as meaningless to ask whether the laws of nature

are independent of time as it was to ask with Clifford whether the

absolute scale of magnitude may not be changing as the solar system

travels through space.]

We regard our acts as not determined by the external world, so that

changes produced in the external world by acts of our wills are, to a

certain degree of approximation, arbitrary. The system, then, on which

we are experimenting, is one capable of isolation from us in that we may

regard ourselves as outside the system, and having no connection with

it. The system, furthermore, is capable of isolation from the rest of

the physical universe, in that events taking place outside the system

have no connection with those taking place inside.[11] Experience gives

the justification for assuming that physical isolation of this sort is

possible. Actually, of course, isolation is never complete, but only

partial, up to presumably any desired degree of approximation.

[Footnote 11: Here again, the concept of "isolation" or "connection" is

defined only in terms of the behavior of the system, and it is not clear

whether this is really an operationally independent concept or not.]

The statement that two exactly similar isolated systems, starting from

the same initial conditions (including past history in the general idea

of initial condition) will run through the same future course of events

involves as a corollary that if differences develop in the behavior of

two such apparently similar systems these differences are evidence of

other previous differences. The thesis that this corresponds to

experience may be called the thesis of essential connectivity and is

perhaps the broadest we have: it is the thesis that differences between

the behavior of systems do not occur isolated but are associated with

other differences. It is essentially the same thesis as that already

mentioned in connection with "explanation", namely that it is possible

to correlate any of the phenomena of nature with other phenomena.

If now the connectivity or correlation between phenomena is of a special

kind, we have a causal connection; namely, if whenever we arbitrarily

impress event A on a system we find that event B, always occurs, whereas

if we had not impressed A, B would not have occurred, then we say that A

is the cause of B, and B the effect of A. By suitably choosing the event

A, we may find the effect of any event of which the system is

susceptible.

The relation between A and B is an unsymmetrical one, by the very nature

of the definition, the cause being the arbitrary variable element, and

the effect that which accompanies it. Furthermore, A may obviously be

the cause of more than one event B, and may cause a whole train of

events.

The causal concept analyzed in this way is not simple by any means. We

do not have a simple event A causally connected with a simple event B,

but the whole background of the system in which the events occur is

included in the concept, and is a vital part of it. If the system,

including its past history, were different, the nature of the relation

between A and B might change entirely. The causality concept is

therefore a relative one, in that it involves the whole system in which

the events take place.

In practice we now take an exceedingly pregnant step and seek to extend

the concept, and rid ourselves as much as possible of its relativity. It

is a matter of experience that there are often a great number of systems

in which A is the cause of B. In many cases the causal relation persists

through such a very wide range of systems that we lose sight entirely of

the system, and come to assume that we have an _absolute_ causal

connection between A and B. For instance, when I strike a bell, and hear

the sound, the causal connection persists through such a great number of

different kinds of system that I might think that here is an absolute

causal connection. Such an absolute causal connection would mean that

always under all circumstances, the striking of the bell is accompanied

by a sound. But _all_ conditions means only _all_ those conditions

covered by experiment. Thus in the case of the bell, all our experiments

were made in the presence of the atmosphere. The causal connection

between the striking of the bell and the sound should have been always

recognized in principle as relative to the presence of the atmosphere.

Indeed, later experiments in the absence of the atmosphere show that the

atmosphere does play an essential part. Now as a matter of fact, the

atmosphere is so comparatively easy to remove that we very readily

include the atmosphere in the chain of causal connection. But if the

atmosphere had been impossible to remove, like the old ether of space,

our idea of the causal connections between the striking of the bell and

its sound might have been quite different. In actual physical

applications of the causality concept, the constant background which is

maintained during all the variations by which the causal connection is

established usually has to be inferred from the context.

It is a matter of perhaps universal experience that the event A is

accompanied by not only one event, which is the effect of A by

definition, but A entails a whole causal train of events. It seems to be

a generalization from experience that the causally connected train of

events started by A is a never ending train, provided the system is

large enough. This is perhaps not necessary in the general case, but if

the event A involves imparting external energy to the system, or the

action of external force (momentum change), there can be no question.

That there is a causal train started by A is particularly evident if A

and B are separated in space. Thus in the case of the bell, the impulse

given to the air by the vibration of the bell is propagated through the

air as an elastic wave, which thus constitutes the causal train of

events. The phenomenon of propagation is characteristic of causal

connections of a mechanical character, and is the justification for the

introduction of the time concept in connection with the causality

concept, where it now appears for the first time. It is evident that

when a disturbance is propagated to a distant point, the effect

_follows_ the cause in time, as time is usually measured.

We extend this result, and usually think that the effect _necessarily_

follows the cause. We now examine whether this is a necessary result of

the causality concept. If we are to talk about the time of events at

different places, we must have some way of setting clocks all over

space. If this is done arbitrarily, there is no necessary connection

between the local clock times of a cause and its effect, but

nevertheless the causality concept involves a certain temporal relation

even in this most general case. Suppose that event A takes place at

point 1 and its effect, event B, at point 2. We station a confederate at

2 who sends a light signal (or any other sort of signal) to 1, as soon

as the event B occurs at 2. Then it is a consequence of the nature of

the causality concept that the signal cannot arrive at 1 before event A

occurs. For if it did arrive before A, we should merely omit to perform

A, which by hypothesis is arbitrary, and entirely in our control, and

then our assumption would be violated that the system is such that the

event B occurs only when A also occurs. The same argument shows _a

fortiori_ that if the effect B occurs at the same place as its cause A,

it cannot precede it in time. I cannot see that the nature of the

causality concept imposes any further restriction on the time of B. The

restricted principle of relativity, however, in postulating that no

signal can be propagated faster than a light signal, virtually makes a

further assumption about the temporal connection of causally connected

events, namely, that the event B at 2 cannot occur before the arrival at

2 of a light signal which started from 1 at the instant that A occurred

at 1. For if B did occur earlier, we could use events A and B as a

signaling code, thus violating our hypothesis.

There is thus a closest connection in time, when time is extended over

space as the theory of relativity directs, between cause and effect,

depending on their separation in space; from this arises the relativity

concept of the causal cone, which in the four dimensional manifold of

space-time divides the aggregate of all those events which may be

causally related from the aggregate of those which are separated by such

a small interval in time and such a large interval in space that

communication by light signals and therefore a causal connection is not

possible. Given now two events A and B which are related as cause to

effect in one system of reference, then they must be causally related

also in every other system of reference. For if they were not, we could

by definition of causality suppress the event A in one of the systems in

which the causal relation does not hold, and this, because of the nature

of the concept of event, involves suppressing A in all the systems, thus

violating our hypothesis of a causal connection in the original system.

The concept of event involved in this argument will be examined later. It

appears then, that the fundamental postulate of relativity (that the

form of natural laws is the same in all reference systems) demands that

the temporal order of events causally connected be the same in all

reference systems.

The whole universe at this present moment is often supposed to be

causally connected with all succeeding states. This means that if we

could repeat experience, starting from the same initial conditions, the

future course of events would always be found to be the same. The truth

of this conviction can never be tested by direct experiment, but it is

something at which we arrive by the usual physical process of successive

approximation. It is difficult to formulate precisely what we mean by

"present" state of the universe, and there is every reason to think that

such a formulation is not unique, but the concept contains the necessary

implication that none of the events constituting the "present" can be

causally connected. The events in distant places which constitute the

present must be separated by an interval of time less than time required

by light to travel between the two places.

The conviction, arising from experience, that the future is determined

by the present and correspondingly the present by the past, is often

phrased differently by saying that the present causally determines the

future. This is in a certain sense a generalization of the causality

concept. It is one of the principal jobs of physics to analyze this

complex causal connection into components, representing as far as

possible the future state of the system as the sum of independent trains

of events started by each individual event of the present. How far such

an analysis is possible must be decided by experiment. It is certainly

possible to a very large extent in most cases, but there seems to be no

reason to expect that a complete analysis is possible. So far as the

system is describable in terms of linear differential equations, the

causal trains started by different events propagate themselves in space

and time without interference and with simple addition of effects, and

conversely the present may be analyzed back into the simple sum of

elementary events in the past, but if the equations governing the motion

of the system are not linear, effects are not additive, and such a

causal analysis into elements is not possible. No emphasis is to be laid

here on the _differential_ aspect of the equations: it is quite possible

that finite difference equations may have the same property of

additivity. Although there can be no question that linear equations

enormously preponderate, neither can there be any doubt that some

phenomena cannot be described in terms of linear equations (_e.g._,

ferro-magnetism), so that there seems no reason to think that a causal

analysis is always possible. I believe, however, that the assumption

that such an analysis into small scale elements is possible is tacitly

made in the thought of many physicists. If the analysis is not possible,

we may expect to find results following the cooperation of several

events which cannot be built up from the results of the events occurring

individually.

When a causal analysis is possible, finding the simplest events which

act as the origin of independent causal trains is equivalent to finding

the ultimate elements in a scheme of explanation, so that here we merge

with the concept of explanation, as already mentioned. As was true of

the explanatory sequence, so here there can be no _formal_ end of the

causal sequence, because we can always ask for the cause of the last

member. But it may be physically meaningless to extend the causal

sequence beyond a certain point. We have seen from the point of view of

operations that the causal concept demands the possibility of variation

in the system. It is therefore meaningless to say that A is the cause of

B unless we can experience systems in which A does not occur. Now if in

extending the causal sequence, we eventually arrive at a condition so

broad that physically no further variation can be made, our causal

sequence has to stop.

Corresponding to this property of the causality concept, the causal

sequence may be terminated either formally, by postulate, or naturally,

by the intrinsic physical nature of the elements of the sequence. Thus

if we say that light gets from point to point because it is propagated

by a medium of unalterable properties, which fills all space, which is

always present and can never be eliminated physically, we have by the

postulated properties of the medium brought the possibility of further

inquiry to a close, because to take the next step and ask the cause of

the properties of the ether, demands that we be able to perform

experiments with the ether altered or absent. Such an ending of the

sequence is evidently pure formalism, without physical significance. But

other considerations may give physical significance. Thus if there are

other sorts of experiment that can be explained by assuming a universal

medium of the same properties, the concept proves not only to be useful,

but to have a certain degree of physical significance. An example of an

inevitable termination of the causal sequence is afforded by the

possibility, already mentioned, that the value of the gravitational

constant may be determined by the total quantity of matter in the

universe. Without further qualification, this is an entirely sterile

statement, but if it can be shown that there is a simple numerical

connection, the matter takes on interest, and we may seek further for a

correlation between the numerical relation and other things.

This analysis of the causality concept does not pretend to be complete

and leaves many interesting questions untouched. Perhaps one of the most

interesting of these questions is whether we can separate into cause and

effect two phenomena which _always_ accompany each other, and whether

therefore the classification of phenomena into causally connected groups

is an exhaustive classification. But the discussion is broad enough for

our purpose here; the most important points of view to acquire are that

the causality concept is relative to the whole background of the system

which contains the causally connected events, and that we must assume

the possibility of an unlimited number of identical experiments, so that

the causality concept applies only to sub-groups of events separated out

from the aggregate of all events.

THE CONCEPT OF IDENTITY

One of the most fundamental of all the concepts with which we describe

the external world is that of identity; in fact, thinking would be

almost inconceivable without such a concept. By this concept we bridge

the passage of time; it enables us to say that a particular object in

our present experience is the same as an object of our past experience.

From the point of view of operations, the meaning of identity is

determined by the operations by which we make the judgment that this

object is the _same_ as that one of my past experience. In practice

there are many indirect ways of making this judgment, but I believe the

essence of the situation lies in the possibility of continuous

connection between the object of the present and the past by continuous

observation (either direct or indirect) through all intermediate time.

We must, for example, be able to look continuously at the object, and

state that while we look at it, it remains the same. This involves the

possession by the object of certain characteristics--it must be a

discrete thing, separated from its surroundings by physical

discontinuities which persist. The concept of identifiability applies,

therefore, only to certain classes of physical objects; no one thinks of

trying to identify the wind of to-day with the wind of yesterday. It is

somewhat easier to identify a liquid such as water in its flow in a

stream, because we can make the motion of the water visible by solid

particles suspended in it, but even here it is not easy to prove to a

captious critic that it is really the water and not the suspended

particles of solid that we are identifying. Even solids, when our

measurements are sufficiently refined, seem to lose their discontinuous

edges, as has been suggested in the discussion of the approximate

character of experimental arithmetic, and the identity concept becomes

hazy.

There can be no question that the concept of identity is a tool

perfectly well adapted to deal approximately with nature in the region

of our ordinary experience, but we have to ask a more serious question.

Does not the apparent demand of our thinking apparatus to be furnished

with discrete and identifiable things to think about impose a very

essential restriction on any picture of the physical universe which we

are able to form? We are continually surprising ourselves in the

invention of discrete structure further and further down in the scale of

things, whole sole _raison d'être_ is to be found entirely within

ourselves. Thus Osborne Reynolds[12] has speculated seriously and most

elaborately about an atomic structure in the ether, and we find

Eddington[13] hinting at the existence of structure of an order of

magnitude of 10^-40 cm. On a much larger scale of magnitude we also

think in the same terms, and conceive positive and negative elementary

charges with hard and impenetrable cores, which involves a complete

change in the law of force at points sufficiently close. What physical

assurance have we that an electron in jumping about in an atom preserves

its identifiability in anything like the way that we suppose, or that

the identity concept applies here at all?

[Footnote 12: Osborne Reynolds, The Sub-Mechanics of the Universe, 254

pp., Cambridge University Press, 1903.]

[Footnote 13: A. S. Eddington, Report on Gravitation, Lon. Phys. Soc.,

1918, p. 91.]

In fact, the identity concept seems to lose all meaning in terms of

actual operations on this level of experience.

The mind seems essentially incapable of dealing with continuity as a

property of physical things; it is not even able to talk about

continuity except in negative terms. To each attempted description of

the properties of a truly continuous substance, it can say "No, it is

not that", but cannot imagine experience which corresponds to what it

conceives a really continuous thing ought to feel like. In terms of

operations, continuity has only a sort of negative meaning. Now certain

implications of this inability of the mind can be removed by appropriate

postulates, as, for example, we can postulate the complete annihilation

of a negative by a positive charge, as is now being done in certain

speculations.[14] There is point in doing this, because the annihilation

of two charges has physical meaning. But it is a question whether all

implications of this habit of thought can be removed, and whether any

picture that we can form of nature will not be tinged—sickbed o'er

with the pale cast of thought.

[Footnote 14: For example, J. H. Jeans, Nat. 114, 828-829, 1924.]

The operational view suggests that in this last we are coming perilously

close to a meaningless question, although there is a certain sense in

which there is meaning here. It may turn out as a matter of fact that we

shall not be able to carry our delving into small-scale phenomena deeper

than a certain point, and that nature will appear to be finite downward,

so that we shall bring up against a wall of some kind. But to ask in

such a situation whether we have come to the end because nature is

_really_ finite, or whether we only appear to be at an end because of

some property of our minds, such as inability to deal with continuity,

is, I believe, a meaningless question.

In actual use the identity concept is extended, and identity is used in

other senses than the fundamental one examined above. For instance, we

speak of two observers seeing the same object, or if the object moves or

does something, we may speak of two observers perceiving the same

happening. A happening about which the judgment of sameness is possible

when perceived by different observers (or mathematically expressed when

observed in two reference frames) is what we mean by an event, which is

one of the fundamental concepts of relativity theory. What now is

involved in this concept of event, or what do we mean when we say that

two observers experience the same event? A first crude attempt might say

that the event is the same if it is described in the same way by the two

observers. But this leads us into all the complicated questions of the

meaning of language, which we would gladly avoid, and is furthermore not

true, because the whistle of a locomotive, for example, does not have

the same pitch for two observers moving with different velocities. A

satisfactory analysis of the situation is difficult to give, but I

believe the essence lies in the discrete character of the event, just as

the identity concept when applied only to objects involved discreteness.

The event is bounded on all sides by discontinuities, both in space and

time. Now it seems to be a result of experience that discontinuities

have a certain absolute significance, in that there is a one-to-one

correspondence between the discontinuities observed in any one reference

system and those observed in any other. Corresponding discontinuities in

two reference systems are by definition the same. An event is by

definition the aggregate of all phenomena bounded by certain

discontinuities, and two reference systems are by definition describing

the same event if the discontinuous boundaries of the event are the

same, irrespective of the appearance of the event in the two systems.

The emission of a light signal, for example, is an event according to

this definition, although it may appear as red light in one reference

system and green in another.

We now see that the concept of event is only an approximate concept, as

was also that of identity, and for the same reason, namely, there are no

such things in experience as sharp discontinuities, but as our

measurements become more refined, the edges of supposed discontinuities

become blurred. As we go to smaller scales of magnitude this blurring

becomes more important, until the physical possibility of performing

those operations by which the discontinuities are detected entirely

disappears, and the concept of event acquires, in terms of operations,

an entirely different meaning. We continue to think of the event in the

same way as before in terms of a mental model, but the true operational

significance now depends on the particular phenomenon under

consideration. The concept of event is really not the same sort of thing

when applied to the emission of a quantum of radiation from an atom, or

the emission of gamma radiation from a radioactive disintegration, or

the flashing of a signal from a dark lantern by opening and closing a

shutter. Here as always, when our range of experience is extended, we

must be prepared at some future time to find that, by extending the

ordinary concept of event to small-scale phenomena by the device of the

mental model, we have by implication smuggled into our picture phenomena

which do not exist, so that it will be necessary to revise our thinking,

casting it into terms corresponding to direct experience.

THE CONCEPT OF VELOCITY

The concept of velocity, as ordinarily defined, involves the two

concepts of space and time. The operations by which we measure the

velocity of an object are these: we first observe the time at which the

object is at one position, and then later observe the time at which it

is at a second position, divide the distance between the two positions

by the time interval, and if necessary, when the velocity is variable,

take the limit. As long as we deal with fairly low velocities we do not

have to inquire carefully as to the kind of time we use in these

operations, but when the velocities become high, we do have to take care

to use the local times at the two positions of the body, which means

that we must have a time system spread over space, or, in other words,

the "extended" time system. This velocity concept, defined in this way,

may be used as a tool in describing nature, and it will be found that

nature has certain properties; for example, the velocity of light is 3 x

10^10 cm./sec. Further, no material thing can be given a velocity as

high as this, but as its velocity is made to approach this value,

increments of energy increasing without limit are required.

But now it is very much a question for examination whether the velocity

concept defined in this particular way has been chosen wisely as a tool

for describing natural phenomena. It is quite possible to modify the

velocity concept, that is, to set up other operations which correspond

to our instinctive feeling of what velocity is in terms of immediate

sensation and such that all numerical measures are unmodified at low

velocities.[15] For example, a traveller in an automobile measures his

velocity by observing the clock on his instrument board and the mile

stones which he passes on the road. This operation differs from that of

the definition above in that the time is no longer extended time, but is

the local time of the moving object. The space coordinates used in this

alternative operation at first seem a hybrid sort of thing, but they are

what the observer would actually most naturally use: they are what he

would measure with a tape measure fixed to a point of the road and

allowed to unwind as he proceeds, or what is measured by a vessel at sea

with a log line let out behind.

[Footnote 15: It is an interesting question for the psychologist whether

the velocity concept is not a more primitive thing in order of

apprehension than that of time, and whether the concept of time is not

derived from observing things in motion, or whether indeed there is any

necessary connection at all between velocity and time in terms of

untutored experience.]

Or there is still another most interesting way of defining velocity, in

which the analysis into space and time is not made at all, but velocity

is directly measured by building up the given velocity by physical

addition of a unit velocity selected arbitrarily. This matter is

discussed at some length in my book "_Dimensional Analysis_",[16] but is

of sufficient pertinence here to describe briefly. We may in the first

place construct a concrete standard for velocity, as, for example, by

stretching a string between two pegs on a board with a fixed weight. If

we strike the string, a disturbance travels along the string which we

can follow with the eye, and we define unit velocity as the velocity of

this disturbance. An object has greater than unit velocity if it outruns

the disturbance, and less if it lags behind. We may now duplicate our

standard, making another board with pegs and stretched string, and check

the equality of the two velocities by observing that the two

disturbances run together. We now define two units of velocity as the

velocity of anything which runs with the disturbance of the string of

the second board, when the second board is made to move bodily with such

a velocity that it runs with the disturbance of the first string. The

process may be extended indefinitely, and any velocity measured.

[Footnote 16: Yale University Press, 1922.]

If either of these two alternative definitions of velocity were adopted,

it would be found that the velocity of light is infinite. Further, there

would be no limit to the velocity which can be imparted to material

bodies on giving them unlimited energy, which is what we are prepared to

regard from ordinary experience is natural and simple. The infinite

velocity for light, on the other hand, is most unnatural, particularly

if we favor a medium point of view. We are here faced with a

dilemma--all sorts of phenomena cannot at the same time be treated

simply. If we attach the most fundamental significance to the behavior

of material bodies, we shall do well to adopt one of the alternative

definitions of velocity. If, on the other hand, we regard the phenomena

of light as the most fundamental, we shall endeavor to form our

definition so that the properties of light are simple. This was

precisely the point of view of Einstein; it is characteristic of his

entire scheme of restricted relativity that light is the fundamental

thing, and this influenced him in adopting the first definition of

velocity. Now one can have no quarrel with this desire to make light

fundamental (the wisdom of doing this is to be justified by the

results), and if the properties of light are to be treated

mathematically, one can easily see the desirability of getting rid of

infinite attributes, and so admit the desirability of making the

velocity of light finite. But all this involves another very insidious

assumption which we ourselves have tacitly used in all our preceding

discussion, namely, that the notion of velocity properly pertains to

light at all. Einstein has very definitely adopted this point of view,

and so determined the character of the entire structure of relativity. I

believe, on the contrary, that it is very gravely to be questioned

whether the identification of light with a _thing travelling_, which is

involved in applying the velocity concept, should be made. This

discussion must be postponed, however, until we deal with the properties

of light. The important points for us to notice at present are that the

definition of velocity actually used involves the concept of extended

time, and that it would be possible to define velocity in different

ways, which would give quite a different complexion to phenomena at high

velocities, but which would leave untouched our ordinary experience.

The velocities at which the precise form of definition becomes important

are higher than can be reached in ordinary mechanical experiments. Such

velocities can be attained in terrestrial laboratories only with

electrified particles, as in experiments in high vacua or with

radioactive disintegrations. It is interesting to notice that we very

seldom attempt a direct measurement of velocity in such experiments by

following a discrete particle in its flight and finding the time

required to pass over a measured distance, but the velocities are

measured indirectly, by calculation from the equations of

electrodynamics and in terms of such immediately observed things as

curvature of path. It is true that one or two experiments have attempted

a more direct measure of velocity, but it seems there is room for more

work here.

THE CONCEPTS OF FORCE AND MASS

Another concept of great importance is that of force. Since the usual

analysis finds a connection between force and acceleration, and

acceleration involves velocity, this is a natural place for the

discussion of force. This concept has been subjected to much analysis by

various writers. In origin the concept doubtless arises from the

muscular sensations of resistance experienced from external bodies. This

crude concept may at once be put on a quantitative basis by substituting

a spring balance for our muscles, or instead of the spring balance we

may use any elastic body, and measure the force exerted by it in terms

of its deformation. Of course, the various precautions which must be

taken in carrying out this idea physically are complex; the matter of

precautions against temperature changes, for example, is one of the most

easily understood. The concept of force so defined is limited to static

systems; it is the task of statics to find the relation between the

forces in systems at rest. We next extend the force concept to systems

not in equilibrium, in which there are accelerations, and we must

conceive that at first all our experiments are made in an isolated

laboratory far out in empty space, where there is no gravitational field.

We here encounter a new concept, that of mass, which as it is originally

met is entangled with the force concept, but may later be disentangled

by a process of successive approximations. The details of the various

steps in the process of approximation are very instructive as typical of

all methods in physics, but need not be elaborated here. Suffice it to

say that we are eventually able to give to each rigid material body a

numerical tag characteristic of the body, such that the product of this

number and the acceleration it receives under the action of any given

force applied to it by a spring balance is numerically equal to the

force, the force being defined, except for a correction, in terms of the

deformation of the balance, exactly as it was in the static case. In

particularly, the relation found between mass, force, and acceleration

applies to the spring balance itself by which the force is applied, so

that a correction has to be applied for a diminution of the force

exerted by the balance arising from its own acceleration.

We now extend the scope of our measurements by bringing our laboratory

into the gravitational field of the earth, and immediately our

experience is extended, in that we continually see bodies accelerated

with no spring balance (that is, no force) acting on them. We extend the

concept of force, and say that any body accelerated is acted on by a

force, and the magnitude of this force is defined as that which would

have been necessary to produce in the same body the same acceleration

with a spring balance in empty space. There is physical justification

for this extension in that we find we can remove the acceleration which

a body acquires in a gravitational field by exerting on it with a spring

balance a force of exactly the specified amount in the opposite

direction. This extended idea of force may also be applied to systems in

which there are electrical actions.

We thus see that in extending the notion of force from bodies in rest to

bodies in motion, the character of the concept has changed, because the

operations by which force is measured change--the force acting on a body

is now measured in terms of its acceleration. But in determining the

force from the acceleration, we have to know the mass. This mass has to

be independently measured with the original concept of force; otherwise

we have no basis for such simple statements as that the force of gravity

on a body is proportional to its mass. All this applies to the ordinary

range of experiments with low velocities. If now we extend the range of

measurements, we find phenomena which we had not expected; for example,

there seem to be difficulties in the way of indefinitely increasing the

velocity of a material body, as of a charged atom. We begin to ask

searching questions: is the force of gravity independent of velocity at

high velocities, or is the mass independent of velocity under the same

conditions or independent of the gravitational field, etc.?

In attempting to answer these new questions, we find difficulty with the

concepts in terms of which they are formulated. There are no operations

by which we can find whether force is independent of velocity unless we

first know the mass, or any operations by which a mass can be measured

unless we know a force. The purely mechanical systems with the highest

velocities of which we have any experimental knowledge are the heavenly

bodies. The motion of these is, with the important exception of Mercury,

that predicted by the ordinary laws of mechanics, so that at first it

might appear that we have here confirmation of the laws of mechanics for

bodies with comparatively high velocities. But it must be remembered

that all we can observe of the heavenly bodies is their positions, and

that we cannot perform on these bodies all the operations by which we

can check the laws of mechanics for terrestrial phenomena. If, for

example, mass and the force with which gravity acts on mass were both

equally affected by velocity, the motion of the heavenly bodies would be

exactly the same as that observed now. Hence as we increase the range of

velocity, the concepts of force and mass simultaneously lose their

definiteness, and become partially fused. This is typical of what we

have now come always to expect near the limit of the experimentally

attainable; experience becomes less rich, the choice of physical

operations more restricted, concepts change and become fewer in number.

If we are to retain the same formal number of concepts, we must

introduce arbitrary conventions or definitions. These definitions are to

be determined largely by convenience. In the case of mechanical systems,

this motive of convenience is supplied by considerations from outside

the domain of mechanical phenomena. The highest velocities of practice

are not reached in mechanical, but in electrical systems, in experiments

with vacuum tubes, etc. Considerations of convenience are therefore

dictated from the electrical point of view. These considerations will be

gone into in much more detail later; the conclusion is all that we need

here, which is that it is convenient to assume for the charge of the

electron a constant number, independent of the velocity, and this

involves making its mass variable in a definite way with velocity. Now

if the principle of relativity is accepted, the mass of mechanical

objects must vary with velocity in the same way as the mass of

electrical charges. Since the variability of this latter is fixed,

mechanical mass becomes a definite function of velocity, and the force

is therefore also fixed in any specific physical case.

The fundamental definition of force given above is highly academic,

involving as it does hypothetical experiments in laboratories situated

far out in empty space. Some sort of procedure like this seems to

correspond to more or less explicit statements to be found in the

literature of mechanics. The meaning in terms of actual operations to be

given to such definitions involves complicated inferential reasoning. We

would make much closer connection with the conditions of actual

experiment if in the definition we substituted for the hypothetical

operations in empty space more or less approximately realizable

operations on bodies sliding on level table tops without friction. I

suppose our instinctive feeling for the laws of mechanics is such that

we are convinced that definitions in terms of an interstellar space

laboratory or a level table top are actually the same. But in principle

we must recognize that when the operations are different, the concepts

are different, and if we adopt something equivalent to the table top

definition, as it seems we are physically forced to do, we must leave

open in our thinking the possibility of finding in the present penumbra,

when our accuracy is sufficiently increased, such phenomena perhaps as

directional attributes of mass in a gravitational field.

We have just considered the sort of problem that we encounter on

ordinary scales of magnitude on going from low to high velocities; what

becomes of the concepts of force and mass when we go to a very small

scale? Down to the atomic scale we may at least slur over the new

physical difficulties, for although we cannot of course experiment with

actual atoms, we can nevertheless make measurements of the Brownian[17]

movement of suspensions in liquids settling in a gravitational field,

for example, and the extrapolation to the atom is not a very great one.

The mass of each individual atom is obtained by what is equivalent to a

process of counting, assuming the law of conservation of mass on an

atomic scale. This is justified by all chemical experience. The mass of

the component parts of the atoms, the electrons, may also perhaps be

given a unique significance after we have decided on the laws of the

electrical field, by experiments on acceleration in electrical fields.

The question which interests in principle here is what meaning, if any,

shall be attached to the mass of the elements of the electron.

[Footnote 17: This phenomenon is discussed at length in the book by J.

Perrin, Brownian Movement and Molecular Reality, translated by F. Soddy,

Taylor and Francis, London, 1909.]

It is evident that we here go beyond any possible experience, at least

for the present, and that experience has again become poorer and our

concepts fewer in number. All that we can now demand is that certain

combinations of numbers, some of which represent mechanical mass and

others electrical charge, have proper relations to each other when

integrated throughout the entire body of the electron. Similar questions

confront us when we ask what are the forces which the parts of the

electron exert on each other. We return to this question in considering

the nature of the electrical concepts. In any event, the concepts of

both force and mass are entirely altered in this domain.

It is interesting to note, in passing, that present electrical theory

gives no meaning to the mass of the elements of the electron, since the

total electromagnetic mass of the electron is built up from the _mutual_

terms in the action of the elements--the total mass is not a _linear_

resultant of the action of the elements.

THE CONCEPT OF ENERGY

In examining the concept of energy, we start with purely mechanical

energy. In isolated mechanical systems, in which there are only

conservative forces, the sum of kinetic and potential energy is

constant. The kinetic energy may be defined as ∑ ½ mv^2, formed for

all parts of the body. The potential energy is determined by the

position of the parts of the system, and has physical significance only

with reference to a datum position, that is, only changes of potential

energy have meaning in terms of operations. The total energy ascribed to

the system has therefore an element of arbitrariness in that the datum

position may be chosen at random, and energy acquires meaning only on

tracing the history back to the epoch of the datum position.

The concept of energy may be extended from mechanical systems to all

systems with which we are acquainted; the operations by which meaning is

given to the extended energy concept involve the generalized

conservation principle, or the first law of thermodynamics. The

extension to thermal systems is immediate; the inclusion of optical and

electrical systems in the scheme was a most important physical step,

which of course required careful experimental justification. Because of

its wide range of application, the energy concept has now come to be

regarded as one of the most important in physics; this idea was held by

Ostwald[18] twenty and more years ago, and is now much to the front

because of the connection between mass and energy indicated by the

theory of relativity, and the important rôle assigned to energy levels

in spectrum analysis.

[Footnote 18: W. Ostwald, Die Energie, Barth, Leipzig, 1908.]

What now is the precise nature and significance of the general energy

concept? In the first place the conservation property of energy is one

of the simplest and most obvious of the properties of matter, so that in

this property of energy is seen a reason for ascribing to it certain of

the properties of matter, in particular and most important, that of

localization in space. We must recognize, however, that this idea of a

location in space is injected into the situation entirely by ourselves,

and corresponds to nothing directly given by the operations of

experiment. The idea has had a most important effect, however. Witness,

for instance, the importance ascribed to the discovery by Kelvin of a

function by which the total energy of an electric field can be

represented as distributed through space;[19] this was one of the most

important props of the medium point of view.

[Footnote 19: This function is ⅛π times the scalar product of

electric force and displacement. If Maxwell's definition of displacement

is adopted, the factor ⅛π is replaced by ½, and an accurate analogy

results between the energy stored in the ether and the elastic energy

stored in a bent spring.]

A more critical examination is likely to diminish considerably our

satisfaction with this naive analogy drawn between matter and energy.

With regard to matter, we may still be tolerably satisfied with our

ascription to matter of location in space, but it is quite different

with regard to conservation of matter. In just what sense is matter

conserved? Certainly not in terms of mass, as we at one time thought.

Nevertheless we undeniably have a feeling that there is some sort of

conservation property here, and are driven to formulate it badly in

terms of a hypothetically constant number of protons and electrons. I

have long thought that Newton was groping after some very similar idea

when he so far forgot himself as to define mass as quantity of matter, a

definition perfectly meaningless to a rigorous and unsympathetic

interpretation. On the other hand, whatever meaning may reside in our

idea of conservation of matter, it certainly is not, in at least one

important respect, like the conservation of energy. For the energy of an

isolated mechanical system is a function of the frame of reference in

which it is described; merely by giving velocity to the reference frame

and altering in no way the mechanical system we may change its kinetic,

and so its total, energy by any direct amount. This does not even

remotely resemble ordinary matter. I cannot see that the operations

which are equivalent to the energy concept justify us in saying more

than that energy is a property of a material system; the operations do

not seem to give any unique meaning to a location associated with

energy.

We now ask what significance is to be ascribed to the sort of

conservation that energy does have. We restrict ourselves first to

mechanical systems. The motions of a mechanical system satisfy certain

differential equations of the second order, and the actual motion is to

be found by an integration of the equations. In the integral of a

differential equation certain constants appear which are determined by

the initial conditions, and are therefore the same during all the future

motion of the system; obviously these constants of motion correspond to

conservative properties. This reasoning can of course be at once

extended. Any system, mechanical or not, whose motion is determined by

differential equations, will have certain conservative properties. For

the systems of mechanics energy is one of the conservative functions;

others are momentum and moment of momentum. Energy is particularly

simple, in that it is connected with measurable properties of the system

by a simple formula (∑ ½ mv^2), and is furthermore scalar, which is

also a property of quantity of matter. But to go further and ascribe to

energy other properties of matter, such as localization in space, is

entirely overlooking the essential difference in the character of the

operation by which matter and quantity of energy are measured, that is,

overlooking the essential difference in their physical character.

The possible extension of the energy concept from mechanics to

thermodynamics receives a sufficient physical explanation in terms of

our views of the essentially mechanical character of thermal phenomena.

That the idea can be extended also to simple electrical or magnetic

systems, in which the effect of velocity of propagation is neglected, is

a consequence of the fact that in these systems the equations of motion

remain of the same general mechanical type, it having been shown by

Maxwell that the equations of such systems may be written in the

generalized Lagrangean form. When, however, we extend our formulas to

systems in which the velocity of propagation is important (that is, when

we consider the field equations in their general form) we find that the

Lagrangean equations no longer apply to matter taken by itself, and

energy is no longer conserved in the original sense. A new function

appears, however, which behaves mathematically in the same way that the

energy did before. The equations of motion of the system remain

Lagrangean in form if the mechanical parts of the system are

supplemented by the electric and magnetic fields in space. In this

extended form we have, therefore, a conservative function as before, and

the energy concept may be retained in this enlarged aspect. The physical

operations by which energy is determined are entirely altered, however,

and the physical character of the concept is changed. No more than

before is there justification for localizing energy in space, or

ascribing to it other properties of matter. Yet the materialization of

the energy concept, and the consequent desire that energy be localized

in space, is one of the strongest arguments in many minds for the

existence of a medium.

As far as I can see, therefore, the existence of conservative functions

is involved in the possibility of describing natural phenomena with

differential equations. That further there is a conservative function of

the precise form found in mechanics is a consequence of the particular

form of the equations and the nature of the forces. The question of the

significance of the fact that the forces of nature appear to be

conservative, with respect to this particular function of mechanics, is

of much interest, but it is not our immediate concern now. We are

interested rather to ask under what general conditions we shall have

conservative functions. Quantum theory strongly suggests that when we

pass to phenomena on a small enough scale, we may no longer be able to

employ differential equations in our descriptions, and hence the

previous reason for the existence of constants connected with the motion

disappears. Now there is one obvious remark to be made about this more

general situation. Whenever the future history of a system is so

connected with its present condition that we can retrace our way to the

present from any future configuration, we shall always have conservative

functions. For any future configuration contains certain fixed (or

conservative) features, in that we can reconstruct the unique present

from any future state. There is no reason to expect that the operations

by which we find the fixed features will always be simple, as in the

mechanical case. Now the determination of the future by the present, and

conversely the possibility of reconstructing the present from the future

(or the past from the present), is, we are convinced, a property which

is at least approximately true of phenomena down to a smaller scale of

magnitude than we have yet reached, and so we expect to find these

conservative functions in systems whose ultimate laws of motion are much

more general than any with which we are yet familiar. The particular

form of the conservative function depends on the character of the

system. That there is a scalar conservative function for ordinary

systems depends of course on particular properties of the system, but we

are at least prepared to find that a scalar conservative function does

not necessarily mean a differential equation of the second order.

The potential energy of a system has a particular significance with

respect to this point of view. In an ordinary mechanical system, the

potential energy simply measures the work done by the applied forces in

being displaced from the initial to the final positions; that is, the

potential energy is a measure of the deviation from the initial

position, and so measures a certain feature of the history of the

system. In the more general system, in which we may not have

differential equations, we may look for something analogous to the

potential energy which shall measure the displacement of the system from

its initial configuration. Such a measure is always possible as long as

the past can be reconstructed from the present (or the present from the

future). We recall a remark of Poincaré's[20] to the effect that we of

necessity must always have conservation, because if we have a system in

which conservation apparently fails, we merely have to invent a new form

of potential energy. This remark is obviously not of entire generality,

but applies only to such systems as those considered here, in which the

past may be reconstructed from the present.

[Footnote 20: Henri Poincaré, Wissenschaft und Hypothese, translated

into German by F. and L. Lindemann, Teubner, Leipzig, 1906, Chap. VIII.]

Of late there has been much discussion of the advisability, on the basis

of certain quantum phenomena, of giving up conservation as a principle

applied to the details of the emission and absorption of light,

retaining it only in a statistical sense. It seems to me that the

question here in the minds of physicists was always merely one of

convenience, and that few, if any, doubted the ultimate applicability of

the principle of Poincaré, or thought that we were here concerned with a

system of such great generality that the past could not be reconstructed

from the present. The question was merely whether those variables in

terms of which the potential energy is defined make close enough

connection with other things of immediate experimental significance, or

whether on the whole the retention of a potential energy is not more

trouble than is justified by its convenience, making a treatment from

the statistical point of view preferable. However, this is all now a

matter of more or less past history, because with the recent extension

of the experiments of Compton,[21] we seem to have experimental evidence

for the validity of the conservation law in detail for elementary

quantum processes, with a corresponding simple potential energy.

[Footnote 21: W. Bothe and H. Geiger, 2S. f. Phys. 32, 639-663, 1925. A.

H. Compton, Proc. Nat. Acad. Soc., II, 303-306, 1925.]

Going still deeper, however, there are quantum phenomena which still may

have to be treated by statistical methods, and this may mean giving up

conservation in detail. We have no experimental evidence, for example,

of what the electron is doing while jumping from one quantum orbit to

another. A situation like this merely means that those details which

determine the future in terms of the past may lie so deep in the

structure that at present we have no immediate experimental knowledge of

them, and we may for the present be compelled to give a treatment from a

statistical point of view based on considerations of probability. But I

suppose that no one, except perhaps Norman Campbell,[22] will maintain

that such a situation is more than temporary, or will cease to search

for consequences of these details of structure which may be open to

experimental verification.

[Footnote 22: Norman Campbell, Time and Chance, Phil. Mag. I, 1106-1117,

1926.]

Similarly, we cannot permanently be satisfied with a picture of

radioactive phenomena which represents radioactive disintegration as a

matter of chance.

The general conclusion to which all this discussion leads is that energy

is probably not entitled to the fundamental position that physical

thought is inclined to give it, but that it is a more or less incidental

consequence of more deep-seated properties, and that the character of

these deep-seated properties is subject to only the most general

restrictions, so that very little about the nature of the details can be

inferred from the existence of any energy function.

THE CONCEPTS OF THERMODYNAMICS

We shall not be concerned here with the many technical questions which

are the proper subject of treatises on thermodynamics, but shall attempt

an examination only of some fundamental concepts.

The most fundamental of these, which sets thermodynamics off apart from

the simpler subjects, is probably that of temperature. In origin this

concept was without question physiological, in much the same way as the

mechanical concept of force was physiological. But just as the force

concept was made more precise, so the temperature concept may be more or

less divorced from its crude significance in terms of immediate

sensation and be given a more precise meaning. This precision may be

obtained through the notion of equilibrium states. We have in the first

place the fundamental experimental fact that when a small body is placed

inside a large system, which we recognize by crude means as

comparatively invariable in temperature as time goes on, the small body

very soon acquires a steady condition, that is, it comes to equilibrium

with its surroundings. We now have the further experimental fact that if

the small body A is in equilibrium with its environment, and body B is

also in equilibrium with the same environment, there will be no change

of condition of A and B when they are brought into contact with each

other--that is, A and B are each in equilibrium with the other and also

with the environment and therefore, by definition, at the same

temperature as the environment. The temperature of the environment is

now measured in terms of some of the properties of A and B which crude

experience has shown change with the physiological temperature of A and

B. The physiological notion of temperature is thus made more precise by

being connected with the physical phenomenon of equilibrium.

Now it is at once evident that stated in this way without qualification

we have said things that are not true. It is not true in general that,

when A is in equilibrium with an environment and B is in equilibrium

with the same environment, A will be in equilibrium with B. Suppose, for

example, that the environment is a stream of water and A is a tiny water

wheel moving freely in its bearings, and that B is a similar wheel with

much friction. Then we know that B will become warm, and will not be in

equilibrium with A when brought into contact with it. Or we may choose

for A a mercury thermometer with bulb covered with putty, and for B a

similar thermometer with bulb sheathed in platinum, and we know that the

two thermometers will not register the same temperature in the water

stream. Or still more simply, we may try to read the temperature of the

air in our garden on any bright day with a silvered and with a blackened

bulb thermometer; we know that the two thermometers will read different

temperatures. It is evident, therefore, that we shall have to specify

much more carefully the conditions under which equilibrium holds if we

are to give precise significance to the temperature concept.

It seems fairly evident in the first place that we shall have to rule

out systems in which there is large scale mechanical motion; the simple

notion of temperature does not apply to a system moving with respect to

us. Only when the two thermometers A and B move with the same velocity

as the stream do we have three-fold equilibrium between the stream, A

and B. We may state this in another way by saying that the temperature

of a moving body must be measured on a thermometer stationary with

respect to the body, but this is only a matter of words, and properly

speaking the temperature concept applies only to a certain aspect of the

relation between two bodies mutually at rest. We here entirely neglect

relativity questions such, for example, as the proper way of correcting

for the change of dimensions of moving thermometers.

If now the body whose temperature we are measuring does not move with

the same velocity in all its parts, we may still give a meaning to local

temperature by dividing the body into parts so small that the velocity

of each part is essentially uniform, and measuring the temperature of

each part with a thermometer stationary with respect to it. We are now

confronted with the question of how far to carry the process of

subdivision. Suppose we have a fluid whose motion is completely

turbulent when measured with instruments of the ordinary scale of

magnitude. For such a fluid the fundamental equilibrium proportions hold

between two measuring bodies A and B and the fluid, provided that the

bodies A and B are so large that the motion is completely turbulent on

their scale of magnitude. We may then define the temperature of the

turbulent fluid from the standpoint of these large scale bodies. But we

may also define the temperature from the small scale point of view as

the average of the temperatures recorded by sufficiently small

thermometers, each moving with the velocity of a local bit of the fluid.

These two temperatures will in general be different, and we must more or

less arbitrarily select one which we define as the true temperature. It

would seem that the small scale temperature is the better one to choose,

because there is a certain degree of arbitrariness in specifying the

scale from which the motion shall be judged completely turbulent But on

the other hand, there are difficulties in the small scale definition,

because the turbulence may become more and more fine grained, until we

end with the motion of the molecules themselves, when the operations

certainly fail which give meaning to the temperature concept. In this

case of molecular turbulence, we are driven back to the large scale

definition, which obviously corresponds to ordinary physical practice.

It appears then that the temperature concept is not a clean cut thing,

which can be made to apply to all experience, but that it is more or

less arbitrary, involving the scale of our measuring instruments. In any

special case, the meaning of the temperature concept must be set by

special convention. In practice this does not often make difficulty,

because in the majority of cases there is no large scale motion with

respect to the thermometers.

Consider now the other aspect of the equilibrium problem suggested by

the thermometers with blackened and silvered bulbs in the sunshine. Our

common experience tells us how to deal with this situation effectively

enough for ordinary purposes. We recognize that the possibility of

temperature equilibrium is disturbed by the radiation, and we protect

the bulbs of the thermometers from the sun's radiation by appropriate

shields. But this only minimizes the difficulty. For the shield is

warmed by the sun, and in turn warms to a less degree by its radiation

the bulb of the thermometer within. We must recognize that every body,

no matter what its temperature, is always emitting radiation, so that

the bulb of our thermometer is always in a radiation field. At first

this puts us in a serious quandary as to the whole question of

equilibrium and the meaning of temperature. The situation is saved by

the experimental observation that there is a particular radiation field

which affects all thermometers equally, namely, the field inside an

infinite body all at the same temperature. Logically this looks like the

vicious circle again, for we have not yet defined what we mean by the

same temperature. But actually we avoid the circle here, as in so many

other physical cases, by a process of asymptotic approximation. The

procedure is perhaps something like this: we find that if we experiment

with larger and larger bodies, isolated and at great distances from

other bodies at approximately the same temperature as judged by crude

physiological sensations, two thermometers, identical except that the

bulb of one is blackened and that of the other is silvered, record more

nearly the same temperature as time goes on and as the thermometers are

sunk to greater depths in the body. In actual practice, of course, the

radiational opacity of most bodies is so high that these precautions

against the effects of external radiation can usually be entirely

ignored. At high temperatures, on the other hand, radiation has to be

explicitly dealt with.

The conclusion for us from these considerations is that operationally

the concept of temperature is tied up with that of radiation--the

equilibrium concept of temperature is strictly never exactly applicable;

it is only a limiting sort of concept applicable when the radiation

field is of a special sort, namely, that of a black body.

In spite of the explicit recognition which we have to give radiation in

defining temperature, we usually entirely lose sight of it in thinking

about the mechanism of ordinary physical processes, as for instance when

we picture the temperature of a gas as determined by the kinetic energy

of its molecules. Now I have no doubt that negligence of this sort can

be justified, but the necessary logical analysis is apparently

complicated, and involves a great many different sorts of experiment by

methods of asymptotic approximation, by which we establish the existence

of various sorts of physical constants, such as constants of emission

and absorption and reflection and scattering and fluorescence and

thermal conductivity. We do not need to make the analysis here, but I

believe that some time it would be worth while to attempt it. Such an

analysis will justify the principle so often used: that if a body is in

thermal equilibrium the various processes involved, such as radiation or

thermal conductivity, must when taken separately also be in equilibrium.

Doubtless, if our experience had been confined to higher temperatures,

like that of the sun, this notion of different mechanisms acting

independently would have been more difficult to acquire.

We next consider another fundamental concept of thermodynamics, that of

quantity of heat. We are at first perhaps inclined to think of this as a

comparatively straightforward concept, given immediately in terms of

experience, but an analysis of the operations by which we measure

quantity of heat will show that the situation is really most

complicated. Consider, for example, Joule's experiment in which the

mechanical equivalent of heat was measured by determining the rise of

temperature of the water in a container when stirred by paddles driven

by a falling weight. We do not question that the rise of temperature of

the water has its origin in the mechanical work done on it by the

paddles. But what about the rise of temperature of the container? We

shall doubtless say that part of this rise comes from heat communicated

to it by the warmer water in contact with it, and part from mechanical

work done on it by turbulent impact of the water. But by what operations

shall we measure what part of the energy communicated to the container

is heat and what part mechanical work? We try to give an idealized

answer to this question in terms of Maxwell demons stationed at all

parts of the boundary of the containing vessel with small scale

measuring instruments. To measure the heat entering at any point I can

see nothing else for the Lilliputian observers to do but to determine

the temperature gradient at every point of the boundary from temperature

observations at two different levels, and calculate the heat inflow from

the gradient and the thermal conductivity of the material of the

walls--there seems no way of measuring a flow of heat as such. The

inflow of mechanical energy must be calculated from a detailed knowledge

of the elastic waves and other large scale deformations of the walls.

Here again there is an arbitrary element in our procedure; if our

mechanical measuring instruments are on too gross a scale, we may miss

mechanical energy which we would catch with finer instruments.

This situation which we have just submitted to detailed analysis is, I

believe, typical of the general situation; it is not possible in the

general case to find anything which we can call heat as such. Without

further explicit examination, we can unambiguously speak of a body

losing or gaining heat only when there has been no energy interchange of

any other sort with other bodies. In such a case the heat is measured in

terms of the temperature change of the body. The heat concept is in the

general case a sort of wastebasket concept, defined negatively in terms

of the energy left over when all other forms of energy have been allowed

for.

The essential fact that a quantity of heat can by itself be defined only

in terms of a drop of temperature is somewhat obscured by the usual

method of thermodynamic analysis. In describing a Carnot engine, for

example, it is specified that the engine shall work between a source and

a sink so large that their temperature is not changed by the heat given

out or absorbed by them, so that the impression is natural that heat may

in some way be measured apart from temperature changes. This of course

is not the case; we merely require that the source and sink be so large

that their temperature changes are of a different order of magnitude

from those in the working substance itself, so that with respect to the

working substance, source and sink may be considered to be at constant

temperature.

Assuming now that we are able to measure quantity of heat in those cases

in which the concept has meaning, let us examine the first law of

thermodynamics, which we write in the form:

_dQ_ + _dW_ = _dE_

Here _dQ_ is the heat imparted to a given body by other bodies, _dW_ is

the work of all kinds done on it from outside, and _dE_ is the increase

of internal energy. Now if this equation says what appears at a naïve

first glance, it should say that we find experimentally that the

relation written always holds between the measured quantities _dQ_,

_dW_, and _dE_. We have seen that in the general case it is not possible

to assign a unique operational significance to _dQ_ and _dW_, and

presumably not to their sum. We ignore for the present difficulties of

this kind and confine attention on _dE_; how shall we measure it? I

believe it does not take much examination to convince us that there are

no physical operations for measuring _dE_ as such, and that therefore

the equation expressing the first law must have a different significance

from that which appears on the surface. This is often recognized in the

statement that the essence of the first law is that _dE_ is an exact

differential determined only by the variables which fix the internal

condition of the body, and not a function of the path by which the body

is carried from one condition to another. But what shall we mean by

internal condition, and how shall we be sure that we have found all the

variables required to specify it completely? Internal condition may be a

most complicated thing and require many variables, as shown by a piece

of iron with a complicated magnetic history or by a piece of aluminum

about to undergo recrystallization after overstrain. Here again I

believe there is no physical procedure by which general meaning can be

given to this concept of internal condition. In specific cases we can

state what the variables are which determine internal condition, and the

criterion that we have found the correct internal variables is that _dE_

shall be a complete differential in terms of them. The first law of

thermodynamics properly understood is not at all a statement that energy

is conserved, for the energy concept without conservation is

meaningless. The essence of the first law is contained in the statement

that the energy concept exists (or has meaning in terms of operations).

The first law is often thought to be one of the most general of physics,

but in a paradoxical sense it is the most special of all laws, because

no general meaning can be given to the energy concept, but only specific

meaning in special cases. The first law owes its complete generality to

the fact that no specific case has yet been found of so broad a

character that it cannot be included under one or another special case.

Examination will at once justify this view. Thus we find a great many

systems which are adequately described in terms of two variables,

pressure and temperature, in that a function of _p_ and _t_ can be found

such that its differential equals _dQ_ + _dW_. There are other systems

in which the six components of stress and _t_ completely fix the

internal condition in the sense that they determine a _dE_. In other

systems the specification of a magnetic field may be necessary, or an

electric, or a gravitational field. No case is known which cannot be

handled in terms of the action of external forces of the proper kind,

but there is no general procedure, and the first law owes its generality

to the exhaustive cataloging of special cases.

We may now return to the question left in abeyance above of the

ambiguity in _dQ_ + _dW_. In all the cases in which the specific

variables can be found which define _dE_, _dQ_ and _dW_ also have

meaning. Consider, for example, a gas, the internal condition of which

may be characterized in terms of _t_ and _p_. The mere fact that the

internal condition can be specified in terms of two variables, one a

mechanical variable, shows that the substance is mechanically

homogeneous. Being mechanically homogeneous, we do not have the

possibility of ambiguous values of _dW_ varying with the scale of the

measuring instruments, and in fact we know that dW = p dv. Similarly the

gas being homogeneous and at rest as a whole allows unique values for

_dQ_. Of course this cannot obscure the physical fact that even in such

a gas, when we go to a small enough scale, we find inhomogeneities

arising from the Brownian movement, etc. Practically our statement means

that the inhomogeneities are so fine grained that over a very wide range

of scale of the measuring instruments we find the same definite results.

The same sort of considerations apply to more complicated systems. If

_dE_ is a complete differential in terms of _t_ and six stress

components, this means again that the body is homogeneous, its condition

is determined by temperature and stress, which are the same throughout

the body, and again there is no possible ambiguity from the scale of the

instruments which measure _dW_ and _dQ_. It seems in general, then, that

if the body allows operations by which _dE_ acquires meaning, at the

same time _dQ_ and _dW_ are provided for. In working out this idea in

full detail, some care must be given to the question of order of

differentials. _dQ_, for unit time and unit volume, is strictly equal to

k∇^2 t, where _k_ is thermal conductivity, so that in determining _dQ_

the second derivatives of temperature are involved.

If the body is obviously not homogeneous, it is still a matter of

experience that it can be divided into small pieces, each of which are

by themselves sufficiently homogeneous, and the first law in its usual

form may be applied to each of the pieces.