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![gravitation limited by a law. The coefficients are naturally separated
into two groups of ten, so that there is no difficulty in choosing
those which are to vanish.
To the uninitiated it may seem surprising that an exact law of
Nature should leave some of the coefficients arbitrary. But we need
to leave something over to be settled when we have specified the
particulars of the problem to which it is proposed to apply the law. A
general law covers an infinite number of special cases. The vanishing
of the ten principal coefficients occurs everywhere in empty space
whether there is one gravitating body or many. The other ten
coefficients vary according to the special case under discussion. This
may remind us that after reaching Einstein’s law of gravitation and
formulating it mathematically, it is still a very long step to reach its
application to even the simplest practical problem. However, by this
time many hundreds of readers must have gone carefully through
the mathematics; so we may rest assured that there has been no
mistake. After this work has been done it becomes possible to verify
that the law agrees with observation. It is found that it agrees with
Newton’s law to a very close approximation so that the main
evidence for Einstein’s law is the same as the evidence for Newton’s
law; but there are three crucial astronomical phenomena in which
the difference is large enough to be observed. In these phenomena
the observations support Einstein’s law against Newton’s.[15]
It is essential to our faith in a theory that its predictions should
accord with observation, unless a reasonable explanation of the
discrepancy is forthcoming; so that it is highly important that
Einstein’s law should have survived these delicate astronomical tests
in which Newton’s law just failed. But our main reason for rejecting
Newton’s law is not its imperfect accuracy as shown by these tests;
it is because it does not contain the kind of information about Nature
that we want to know now that we have an ideal before us which
was not in Newton’s mind at all. We can put it this way. Astronomical
observations show that within certain limits of accuracy both
Einstein’s and Newton’s laws are true. In confirming (approximately)](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-21-2048.jpg)
![other to an interfering field of force. The suggestion that the body
really wanted to go straight but some mysterious agent made it go
crooked is picturesque but unscientific. It makes two properties out
of one; and then we wonder why they are always proportional to
one another—why the gravitational force on different bodies is
proportional to their inertia or mass. The dissection becomes
untenable when we admit that all frames of reference are on the
same footing. The projectile which describes a parabola relative to
an observer on the earth’s surface describes a straight line relative
to the man in the lift. Our teacher will not easily persuade the man
in the lift who sees the apple remaining where he released it, that
the apple really would of its own initiative rush upwards were it not
that an invisible tug exactly counteracts this tendency.[16]
Einstein’s Law of Motion does not recognise this dissection. There
are certain curves which can be defined on a curved surface without
reference to any frame or system of partitions, viz. the geodesics or
shortest routes from one point to another. The geodesics of our
curved space-time supply the natural tracks which particles pursue if
they are undisturbed.
We observe a planet wandering round the sun in an elliptic orbit.
A little consideration will show that if we add a fourth dimension
(time), the continual moving on in the time-dimension draws out the
ellipse into a helix. Why does the planet take this spiral track instead
of going straight? It is because it is following the shortest track; and
in the distorted geometry of the curved region round the sun the
spiral track is shorter than any other between the same points. You
see the great change in our view. The Newtonian scheme says that
the planet tends to move in a straight line, but the sun’s gravity pulls
it away. Einstein says that the planet tends to take the shortest route
and does take it.
That is the general idea, but for the sake of accuracy I must
make one rather trivial correction. The planet takes the longest
route.](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-24-2048.jpg)
![You may remember that points along the track of any material
body (necessarily moving with a speed less than the velocity of light)
are in the absolute past or future of one another; they are not
absolutely “elsewhere”. Hence the length of the track in four
dimensions is made up of time-like relations and must be measured
in time-units. It is in fact the number of seconds recorded by a clock
carried on a body which describes the track.[17] This may be
different from the time recorded by a clock which has taken some
other route between the same terminal points. On p. 39 we
considered two individuals whose tracks had the same terminal
points; one of them remained at home on the earth and the other
travelled at high speed to a distant part of the universe and back.
The first recorded a lapse of 70 years, the second of one year. Notice
that it is the man who follows the undisturbed track of the earth who
records or lives the longest time. The man whose track was violently
dislocated when he reached the limit of his journey and started to
come back again lived only one year. There is no limit to this
reduction; as the speed of the traveller approaches the speed of
light the time recorded diminishes to zero. There is no unique
shortest track; but the longest track is unique. If instead of pursuing
its actual orbit the earth made a wide sweep which required it to
travel with the velocity of light, the earth could get from 1 January
1927 to 1 January 1928 in no time, i.e. no time as recorded by an
observer or clock travelling with it, though it would be reckoned as a
year according to “Astronomer Royal’s time”. The earth does not do
this, because it is a rule of the Trade Union of matter that the
longest possible time must be taken over every job.
Thus in calculating astronomical orbits and in similar problems
two laws are involved. We must first calculate the curved form of
space-time by using Einstein’s law of gravitation, viz. that the ten
principal curvatures are zero. We next calculate how the planet
moves through the curved region by using Einstein’s law of motion,
viz. the law of the longest track. Thus far the procedure is analogous
to calculations made with Newton’s law of gravitation and Newton’s
law of motion. But there is a remarkable addendum which applies](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-25-2048.jpg)
![account given by the policeman is also legitimate; and he
endeavours like a good magistrate to reconcile them both.
Time Geometry. Einstein’s law of gravitation controls a geometrical
quantity curvature in contrast to Newton’s law which controls a
mechanical quantity force. To understand the origin of this
geometrisation of the world in the relativity theory we must go back
a little.
The science which deals with the properties of space is called
geometry. Hitherto geometry has not included time in its scope. But
now space and time are so interlocked that there must be one
science—a somewhat extended geometry—embracing them both.
Three-dimensional space is only a section cut through four-
dimensional space-time, and moreover sections cut in different
directions form the spaces of different observers. We can scarcely
maintain that the study of a section cut in one special direction is the
proper subject-matter of geometry and that the study of slightly
different sections belongs to an altogether different science. Hence
the geometry of the world is now considered to include time as well
as space. Let us follow up the geometry of time.
You will remember that although space and time are mixed up
there is an absolute distinction between a spatial and a temporal
relation of two events. Three events will form a space-triangle if the
three sides correspond to spatial relations—if the three events are
absolutely elsewhere with respect to one another.[18] Three events
will form a time-triangle if the three sides correspond to temporal
relations—if the three events are absolutely before or after one
another. (It is possible also to have mixed triangles with two sides
time-like and one space-like, or vice versa.) A well-known law of the
space-triangle is that any two sides are together greater than the
third side. There is an analogous, but significantly different, law for
the time-triangle, viz. two of the sides (not any two sides) are
together less than the third side. It is difficult to picture such a
triangle but that is the actual fact.](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-31-2048.jpg)
![Einstein’s theory reduces gravitational force to a property of space
ought not to arouse misgiving. In any case the physicist does not
conceive of space as void. Where it is empty of all else there is still
the aether. Those who for some reason dislike the word aether,
scatter mathematical symbols freely through the vacuum, and I
presume that they must conceive some kind of characteristic
background for these symbols. I do not think any one proposes to
build even so relative and elusive a thing as force out of entire
nothingness.
[13] So far as I can tell (without experimental trial) the man who
jumped over a precipice would soon lose all conception of falling;
he would only notice that the surrounding objects were impelled
past him with ever-increasing speed.
[14] It will probably be objected that since the phenomena here
discussed are evidently associated with the existence of a massive
body (the earth), and since Newton makes his tugs occur
symmetrically about that body whereas the apple makes its tugs
occur unsymmetrically (vanishing where the apple is, but strong
at the antipodes), therefore Newton’s frame is clearly to be
preferred. It would be necessary to go deeply into the theory to
explain fully why we do not regard this symmetry as of first
importance; we can only say here that the criterion of symmetry
proves to be insufficient to pick out a unique frame and does not
draw a sharp dividing line between the frames that it would admit
and those it would have us reject. After all we can appreciate that
certain frames are more symmetrical than others without insisting
on calling the symmetrical ones “right” and unsymmetrical ones
“wrong”.
[15] One of the tests—a shift of the spectral lines to the red in the
sun and stars as compared with terrestrial sources—is a test of
Einstein’s theory rather than of his law.
[16] The reader will verify that this is the doctrine the teacher
would have to inculcate if he went as a missionary to the men in
the lift.
[17] It may be objected that you cannot make a clock follow an
arbitrary curved path without disturbing it by impressed forces
(e.g. molecular hammering). But this difficulty is precisely
analogous to the difficulty of measuring the length of a curve with
a rectilinear scale, and is surmounted in the same way. The usual](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-35-2048.jpg)
![theory of “rectification of curves” applies to these time-tracks as
well as to space-curves.
[18] This would be an instantaneous space-triangle. An enduring
triangle is a kind of four-dimensional prism.](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-36-2048.jpg)
![called a demon) is present in the world. An explanation on the lines
of Einstein’s theory must show why something (which we call
principal curvature) is excluded from the world.
In the last chapter the law of gravitation was stated in the form—
the ten principal coefficients of curvature vanish in empty space. I
shall now restate it in a slightly altered form—
The radius of spherical[19] curvature of every three-dimensional
section of the world, cut in any direction at any point of empty
space, is always the same constant length.
Besides the alteration of form there is actually a little difference
of substance between the two enunciations; the second corresponds
to a later and, it is believed, more accurate formula given by Einstein
a year or two after his first theory. The modification is made
necessary by our realisation that space is finite but unbounded (p.
80). The second enunciation would be exactly equivalent to the first
if for “same constant length” we read “infinite length”. Apart from
very speculative estimates we do not know the constant length
referred to, but it must certainly be greater than the distance of the
furthest nebula, say miles. A distinction between so great a
length and infinite length is unnecessary in most of our arguments
and investigations, but it is necessary in the present chapter.
We must try to reach the vivid significance which lies behind the
obscure phraseology of the law. Suppose that you are ordering a
concave mirror for a telescope. In order to obtain what you want
you will have to specify two lengths (1) the aperture, and (2) the
radius of curvature. These lengths both belong to the mirror—both
are necessary to describe the kind of mirror you want to purchase—
but they belong to it in different ways. You may order a mirror of
100 foot radius of curvature and yet receive it by parcel post. In a
certain sense the 100 foot length travels with the mirror, but it does
so in a way outside the cognizance of the postal authorities. The 100
foot length belongs especially to the surface of the mirror, a two-
dimensional continuum; space-time is a four-dimensional continuum,
and you will see from this analogy that there can be lengths](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-38-2048.jpg)
![belonging in this way to a chunk of space-time—lengths having
nothing to do with the largeness or smallness of the chunk, but none
the less part of the specification of the particular sample. Owing to
the two extra dimensions there are many more such lengths
associated with space-time than with the mirror surface. In
particular, there is not only one general radius of spherical curvature,
but a radius corresponding to any direction you like to take. For
brevity I will call this the “directed radius” of the world. Suppose
now that you order a chunk of space-time with a directed radius of
500 trillion miles in one direction and 800 trillion miles in another.
Nature replies “No. We do not stock that. We keep a wide range of
choice as regards other details of specification; but as regards
directed radius we have nothing different in different directions, and
in fact all our goods have the one standard radius, trillion miles.” I
cannot tell you what number to put for because that is still a
secret of the firm.
The fact that this directed radius which, one would think, might
so easily differ from point to point and from direction to direction,
has only one standard value in the world is Einstein’s law of
gravitation. From it we can by rigorous mathematical deduction work
out the motions of planets and predict, for example, the eclipses of
the next thousand years; for, as already explained, the law of
gravitation includes also the law of motion. Newton’s law of
gravitation is an approximate adaptation of it for practical
calculation. Building up from the law all is clear; but what lies
beneath it? Why is there this unexpected standardisation? That is
what we must now inquire into.
Relativity of Length. There is no such thing as absolute length; we
can only express the length of one thing in terms of the length of
something else.[20] And so when we speak of the length of the
directed radius we mean its length compared with the standard
metre scale. Moreover, to make this comparison, the two lengths
must lie alongside. Comparison at a distance is as unthinkable as](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-39-2048.jpg)
![action at a distance; more so, because comparison is a less vague
conception than action. We must either convey the standard metre
to the site of the length we are measuring, or we must use some
device which, we are satisfied, will give the same result as if we
actually moved the metre rod.
Now if we transfer the metre rod to another point of space and
time, does it necessarily remain a metre long? Yes, of course it does;
so long as it is the standard of length it cannot be anything else but
a metre. But does it really remain the metre that it was? I do not
know what you mean by the question; there is nothing by reference
to which we could expose delinquencies of the standard rod, nothing
by reference to which we could conceive the nature of the supposed
delinquencies. Still the standard rod was chosen with considerable
care; its material was selected to fulfil certain conditions—to be
affected as little as possible by casual influences such as
temperature, strain or corrosion, in order that its extension might
depend only on the most essential characteristics of its
surroundings, present and past.[21] We cannot say that it was
chosen to keep the same absolute length since there is no such
thing known; but it was chosen so that it might not be prevented by
casual influences from keeping the same relative length—relative to
what? Relative to some length inalienably associated with the region
in which it is placed. I can conceive of no other answer. An example
of such a length inalienably associated with a region is the directed
radius.
The long and short of it is that when the standard metre takes up
a new position or direction it measures itself against the directed
radius of the world in that region and direction, and takes up an
extension which is a definite fraction of the directed radius. I do not
see what else it could do. We picture the rod a little bewildered in its
new surroundings wondering how large it ought to be—how much of
the unfamiliar territory its boundaries ought to take in. It wants to
do just what it did before. Recollections of the chunk of space that it
formerly filled do not help, because there is nothing of the nature of](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-40-2048.jpg)
![isotropic directed curvature) has a specialisation which requires
explanation; we were then comparing it in our minds with a world
suggested by the pure mathematician which has entirely arbitrary
curvature. But the fact is that a world of arbitrary curvature is a
sheer impossibility. If not the directed radius, then some other
directed length derivable from the metric, is bound to be
homogeneous and isotropic. In applying the ideas of the pure
mathematician we overlooked the fact that he was imagining a world
surveyed from outside with standards foreign to it, whereas we have
to do with a world surveyed from within with standards conformable
to it.
The explanation of the law of gravitation thus lies in the fact that
we are dealing with a world surveyed from within. From this broader
standpoint the foregoing argument can be generalised so that it
applies not only to a survey with metre rods but to a survey by
optical methods, which in practice are generally substituted as
equivalent. When we recollect that surveying apparatus can have no
extension in itself but only in relation to the world, so that a survey
of space is virtually a self-comparison of space, it is perhaps
surprising that such a self-comparison should be able to show up
any heterogeneity at all. It can in fact be proved that the metric of a
two-dimensional or a three-dimensional world surveyed from within
is necessarily uniform. With four or more dimensions heterogeneity
becomes possible, but it is a heterogeneity limited by a law which
imposes some measure of homogeneity.
I believe that this has a close bearing on the rather heterodox
views of Dr. Whitehead on relativity. He breaks away from Einstein
because he will not admit the non-uniformity of space-time involved
in Einstein’s theory. “I deduce that our experience requires and
exhibits a basis of uniformity, and that in the case of nature this
basis exhibits itself as the uniformity of spatio-temporal relations.
This conclusion entirely cuts away the casual heterogeneity of these
relations which is the essential of Einstein’s later theory.”[22] But we
now see that Einstein’s theory asserts a casual heterogeneity of only](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-42-2048.jpg)
![one set of ten coefficients and complete uniformity of the other ten.
It therefore does not leave us without the basis of uniformity of
which Whitehead in his own way perceived the necessity. Moreover,
this uniformity is not the result of a law casually imposed on the
world; it is inseparable from the conception of survey of the world
from within—which is, I think, just the condition that Whitehead
would demand. If the world of space-time had been of two or of
three dimensions Whitehead would have been entirely right; but
then there could have been no Einstein theory of gravitation for him
to criticise. Space-time being four-dimensional, we must conclude
that Whitehead discovered an important truth about uniformity but
misapplied it.
The conclusion that the extension of an object in any direction in
the four-dimensional world is determined by comparison with the
radius of curvature in that direction has one curious consequence.
So long as the direction in the four-dimensional world is space-like,
no difficulty arises. But when we pass over to time-like directions
(within the cone of absolute past or future) the directed radius is an
imaginary length. Unless the object ignores the warning symbol
it has no standard of reference for settling its time extension.
It has no standard duration. An electron decides how large it ought
to be by measuring itself against the radius of the world in its space-
directions. It cannot decide how long it ought to exist because there
is no real radius of the world in its time-direction. Therefore it just
goes on existing indefinitely. This is not intended to be a rigorous
proof of the immortality of the electron—subject always to the
condition imposed throughout these arguments that no agency other
than metric interferes with the extension. But it shows that the
electron behaves in the simple way which we might at least hope to
find.[23]
Predictions from the Law. I suppose that it is at first rather
staggering to find a law supposed to control the movements of stars
and planets turned into a law finicking with the behaviour of](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-43-2048.jpg)
![measuring rods. But there is no prediction made by the law of
gravitation in which the behaviour of measuring appliances does not
play an essential part. A typical prediction from the law is that on a
certain date 384,400,000 metre rods laid end to end would stretch
from the earth to the moon. We may use more circumlocutory
language, but that is what is meant. The fact that in testing the
prediction we shall trust to indirect evidence, not carrying out the
whole operation literally, is not relevant; the prophecy is made in
good faith and not with the intention of taking advantage of our
remissness in checking it.
We have condemned the law of gravitation as a put-up job. You
will want to know how after such a discreditable exposure it can still
claim to predict eclipses and other events which come off.
A famous philosopher has said—
“The stars are not pulled this way and that by mechanical forces;
theirs is a free motion. They go on their way, as the ancients said,
like the blessed gods.”[24]
This sounds particularly foolish even for a philosopher; but I
believe that there is a sense in which it is true.
We have already had three versions of what the earth is trying to
do when it describes its elliptic orbit around the sun.
(1) It is trying to go in a straight line but it is roughly pulled away
by a tug emanating from the sun.
(2) It is taking the longest possible route through the curved
space-time around the sun.
(3) It is accommodating its track so as to avoid causing any
illegal kind of curvature in the empty space around it.
We now add a fourth version.
(4) The earth goes anyhow it likes.
It is not a long step from the third version to the fourth now that
we have seen that the mathematical picture of empty space](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-44-2048.jpg)
![that the unit of length everywhere is a constant fraction of the
directed radius of the world at that point. And as the astronomer
who predicts the future position of the earth does not assume
anything more about what the earth will choose to do than is
expressed in the law so we shall find the same
position of the earth, if we assume nothing more than that the
practical unit of length involved in measurements of the position is a
constant fraction of the directed radius. We do not need to decide
whether the track is to be represented by or , and it
would convey no information as to any observable phenomena if we
knew the representation.
I shall have to emphasise elsewhere that the whole of our
physical knowledge is based on measures and that the physical
world consists, so to speak, of measure-groups resting on a
shadowy background that lies outside the scope of physics.
Therefore in conceiving a world which had existence apart from the
measurements that we make of it, I was trespassing outside the
limits of what we call physical reality. I would not dissent from the
view that a vagary which by its very nature could not be measurable
has no claim to a physical existence. No one knows what is meant
by such a vagary. I said that the earth might go anywhere it chose,
but did not provide a “where” for it to choose; since our conception
of “where” is based on space measurements which were at that
stage excluded. But I do not think I have been illogical. I am urging
that, do what it will, the earth cannot get out of the track laid down
for it by the law of gravitation. In order to show this I must suppose
that the earth has made the attempt and stolen nearer to the sun;
then I show that our measures conspire quietly to locate it back in
its proper orbit. I have to admit in the end that the earth never was
out of its proper orbit;[25] I do not mind that, because meanwhile I
have proved my point. The fact that a predictable path through
space and time is laid down for the earth is not a genuine restriction
on its conduct, but is imposed by the formal scheme in which we
draw up our account of its conduct.](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-48-2048.jpg)
![Non-Empty Space. The law that the directed radius is constant does
not apply to space which is not completely empty. There is no longer
any reason to expect it to hold. The statement that the region is not
empty means that it has other characteristics besides metric, and
the metre rod can then find other lengths besides curvatures to
measure itself against. Referring to the earlier (sufficiently
approximate) expression of the law, the ten principal coefficients of
curvature are zero in empty space but have non-zero values in non-
empty space. It is therefore natural to use these coefficients as a
measure of the fullness of space.
One of the coefficients corresponds to mass (or energy) and in
most practical cases it outweighs the others in importance. The old
definition of mass as “quantity of matter” associates it with a fullness
of space. Three other coefficients make up the momentum—a
directed quantity with three independent components. The
remaining six coefficients of principal curvature make up the stress
or pressure-system. Mass, momentum and stress accordingly
represent the non-emptiness of a region in so far as it is able to
disturb the usual surveying apparatus with which we explore space—
clocks, scales, light-rays, etc. It should be added, however, that this
is a summary description and not a full account of the non-
emptiness, because we have other exploring apparatus—magnets,
electroscopes, etc.—which provide further details. It is usually
considered that when we use these we are exploring not space, but
a field in space. The distinction thus created is a rather artificial one
which is unlikely to be accepted permanently. It would seem that the
results of exploring the world with a measuring scale and a magnetic
compass respectively ought to be welded together into a unified
description, just as we have welded together results of exploration
with a scale and a clock. Some progress has been made towards this
unification. There is, however, a real reason for admitting a partially
separate treatment; the one mode of exploration determines the
symmetrical properties and the other the antisymmetrical properties
of the underlying world-structure.[26]](https://image.slidesharecdn.com/10858-241007031130-d9bb88a8/75/Download-full-ebook-of-a-instant-download-pdf-49-2048.jpg)
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- 4. 14 ©2016 Pearson Education,Inc. Chapter 2 Managing Work Flows and Conducting Job Analysis Managing Human Resources 8th Edition Gomez-Mejia Solutions Manual full chapter at: https://testbankbell.com/product/managing- human-resources-8th-edition-gomez-mejia-solutions-manual/ CHAPTER OVERVIEW Work flow is a highly dynamic process, and a well-designed organizational structure will ensure that work is performed efficiently and produces a high-quality product or service. This chapter discusses the various aspects of organizational design and structure, and the flow of work within the structure to accomplish an organization’s goals and objectives. It is the business strategy selected by management that determines the structure most appropriate for the organization. The term organizational structure refers to relationships among people and groups in an organization. Work flow refers to the way that work is organized within the structure to meet production or service goals. Work flow can be viewed from three different perspectives within the organizational structure: the entire organization, work groups, and individual employees. When business conditions or organizational strategy and objectives change, often the design and structure of the organization will also undergo change in order to adapt, and each of the three elements may be affected. Job analysis is a tool used by organizations to document and describe job content, and measure how much and what types of work are necessary to achieve organizational objectives. Contingent workers and alternative work schedules are also explored in this chapter as a means to create a flexible workforce. Finally, human resource information system (HRIS) applications are discussed, and the management of security and privacy issues is explored. CHALLENGES After reading this chapter, students should be able to deal more effectively with the following challenges: 1. Understand the organizational perspective of work. 2. Understand the group perspective of work. 3. Understand the individual perspective of work. 4. Develop competence in designing jobs and conducting job analysis. 5. Have familiarity with the flexible workforce. 6. Maintain human resource information systems. ANNOTATED OUTLINE CHALLENGE 1 Understand the organizational perspective of work. I. Work: The Organization Perspective
- 5. 15 ©2016 Pearson Education,Inc. The relationship between strategy and organizational structure, the three basic organizational structures, and the uses of work-flow analysis are discussed. A. Strategy and Organizational Structure
- 6. 16 ©2016 Pearson Education,Inc. An organization develops a business strategy by establishing a set of long- term goals. The business strategy selected by management determines the structure and/or restructuring that is most appropriate. Moreover, management selects HR strategies to fit and support its business strategies and organizational structure. B. Designing the Organization Designing an organization involves choosing an organizational structure that will enable the company to most effectively achieve its goals. There are three basic types of organizational structure. Bureaucratic organizations can be described as a pyramid-shaped organization. They consist of hierarchies with many levels of management and are driven by a top-down, or command-and- control, approach in which managers provide considerable direction and have considerable control over others (The classic example is the military). The bureaucratic organization is based on a functional division of labor, where employees are divided into divisions based on their function. Work specialization is another feature of bureaucratic organizations, with employees spending most of their time working individually or at a specialized task. Flat organizations have only a few levels of managers and emphasize a decentralized approach to management, which encourages high employee involvement in business decisions. The purpose of this structure is to create independent small businesses or enterprises that can rapidly respond to customers' needs or changes in the business environment. Flat organizations are useful for organizations that are implementing a management strategy that emphasizes customer satisfaction. Boundaryless organizations enable organizations to form relationships (joint ventures, intellectual property, marketing distribution channels, or financial resources) with customers, suppliers, and/or competitors. Companies often use a boundaryless organizational structure when they (1) collaborate with customers or suppliers to provide better-quality products or services, (2) are entering foreign markets that have entry barriers to foreign competitors,; or (3) need to manage the risk of developing an expensive new technology. Boundaryless organizations share many of the characteristics of flat organizations, with a strong emphasis on teams, which are likely to include employees representing different companies in the joint venture. C. Work-Flow Analysis Managers perform work-flow analysis in order to examine how work creates or adds value to the ongoing processes in a business. Work-flow analysis looks at how work moves from the customer (the demand source) through the organization to the point at which the work leaves the organization as a product or service for the customer (to meet the demand). Work-flow analysis
- 7. 17 ©2016 Pearson Education,Inc. often reveals that some steps or jobs can be combined, simplified, or even eliminated. In other cases, it results in the reorganization of work so that teams rather than individual workers are the source of value creation. D. Business Process Reengineering Business process reengineering (BPR) is another program through which work-flow analysis has helped organizations to make major performance improvements. BPR is different from restructuring in that its focus is not just on eliminating layers of management, but rather a fundamental rethinking and radical redesign of business processes to achieve dramatic improvements in costs, quality, service, and speed. BPR uses work-flow analysis to identify a company’s core processes involved in producing its product or delivering its service to the customer, and organizing its human resources around those core processes to improve organizational performance. Through this analysis, jobs are identified that can be eliminated or recombined to improve company performance. CHALLENGE 2 Understand the group perspective of work. II. Work: The Group Perspective In the flat and boundaryless organizational structures, teamwork is an imperative. Teams are the basic building blocks of both these structures. A team is a small number of people with complementary skills who work toward common goals for which they hold themselves mutually accountable. One type of team that is having a significant impact on U.S. companies today is the self-managed team. A. Self-Managed Teams Self-managed teams (SMTs) are responsible for producing an entire product, a component, or an ongoing service. In most cases, SMT members are cross- trained on the different tasks assigned to the team. Self-managed teams have made some impressive contributions to the bottom line of companies (Shenandoah Life, Xerox, Boeing, and Lucent Technologies) that have used them. Often, these teams are trained in technical, administrative, and interpersonal skills. B. Other Types of Teams Businesses use three other types of teams in addition to SMTs. Unlike SMTs, problem-solving teams do not affect an organization's structure because they exist for only a limited period. They are often used when organizations decide to pursue a quality management effort by making improvements in the quality of a product or service. Special-purpose teams consist of members who span
- 8. 18 ©2016 Pearson Education,Inc. functional or organizational boundaries and whose purpose is to examine complex issues such as introducing new technology, improving the quality of work process, or encouraging cooperation between labor and management in a unionized setting. Virtual teams use interactive computer technologies such as the Internet to work together despite being separated by physical distance. Virtual teams allow organizations to position individuals who might not be otherwise available on teams. CHALLENGE 3 Understand the individual perspective of work. III. Work: The Individual Perspective The third perspective from which the structure and flow of work is examined in this chapter is between the individual employee and the job. Theories of motivation are reviewed along with discussions of job design, job analysis, and job descriptions. Motivating Employees Motivation is that which energizes and sustains human behavior. Motivation theory seeks to explain why employees are more motivated by and satisfied with one type of work than another. To do this, several theories are discussed, including the two-factor theory (which lists the factors that are satisfying and dissatisfying), work adjustment theory (which says that motivation and job satisfaction depend on the fit between the employee's abilities or needs and the job and organizational characteristics), goal-setting theory (which suggests that employees' goals explain motivation and performance), and job characteristic theory (which states that employee motivation depends on job characteristics such as skill variety, task identity, task significance, autonomy, and feedback). CHALLENGE 4 Develop competence in designing jobs and conducting job analysis. IV. Designing Jobs and Conducting Job Analysis All the theories of employee motivation suggest that jobs can be designed to increase motivation and performance. A. Job Design Job design is the process of organizing work into tasks required to perform a specific job. There are three important influences on job design. One is work- flow analysis; the other two are the strategy of the business and the organizational structure that best fits that strategy. The five approaches to job
- 9. 19 ©2016 Pearson Education,Inc. design that are examined are work simplification (versus work elimination), job enlargement, job rotation, job enrichment, and team-based job design. B. Job Analysis A work-flow analysis is followed by a job design and the communication of job expectations to job incumbents. The basis of these things is a job analysis, which requires systematic job data gathering and information organization with respect to the tasks and responsibilities of a particular job. Job analysis is useful for recruitment, selection, performance appraisal, compensation, training, and career development activities. 1. Who performs job analysis? 2. Methods of gathering job information 3. The uses of job analysis 4. The techniques of job analysis a. Task inventory analysis b. Critical incident technique c. Position analysis questionnaire (PAQ) d. Functional job analysis 5. Job analysis and the legal environment 6. Job analysis and organizational flexibility C. Job Descriptions A job description is a portrait of a job. It may be specific (a detailed summary) or general (associated with work-flow strategies that emphasize innovation, flexibility, and loose work planning). Regardless, it is a written document that identifies, defines, and describes a job in terms of its duties, responsibilities, working conditions, and specifications. Job descriptions have four key elements: identification information, job summary, job duties and responsibilities, and job specifications and minimum qualifications. D. Job or Work? In some cases it is more appropriate to focus on the work the individual does rather than the job because some jobs lack clearly defined boundaries. However, the need to assign employees to perform jobs is going to remain an important feature of the work environment. CHALLENGE 5 Have familiarity with the flexible workforce. V. The Flexible Workforce
- 10. 20 ©2016 Pearson Education,Inc. One of the imperatives for many modern organizations is flexibility. Therefore, this section looks at the practice of using contingent workers and examines flexible work schedules. A. Contingent Workers There are two types of workers: contingent (those having a tentative relationship with an employer) and core (those having full-time jobs with an employer). Firms hire contingent workers to help them deal with temporary increases in their workload or to do work that is not part of their core set of capabilities. Contingent workers include temporary employees, part-time employees, outsourced or subcontracted contract workers, and college interns. The jobs held by these workers are diverse, ranging from blue-collar to white-collar executive positions. Outsourcing has increasingly become the wave of the future as more and more companies look to the “virtual corporation” as an organizational model. Consistent with this trend, human resource activities such as payroll, benefits, training, recruiting, and performance evaluation are being outsourced by organizations as well. There are both advantages and disadvantages to outsourcing these activities, and the costs and benefits should be considered before making a decision to outsource or retain a specific activity. B. Flexible Work Schedules Flexible work schedules alter the scheduling of work while leaving intact the job design and the employment relationship. The three most common types of flexible work schedules are flexible work hours, condensed or compressed workweeks, and telecommuting. Employers can use flexible work schedules to provide advantages for both themselves and employees, with employers gaining higher levels of productivity and job satisfaction, and employees feeling that they are trusted by management, which can improve the quality of employee relations. C. The Mobile Workplace Many technology changes have given rise to a mobile work environment. Technology has freed employees to work in different spatial locations, including tea spaces, remote work centers, a home office, or the neighborhood coffee shop. This flexibility allows workers to achieve better work–life balance. CHALLENGE 6 Maintain human resource information systems. VI. Human Resource Information Systems
- 11. 21 ©2016 Pearson Education,Inc. Human resource information systems (HRISs) are systems used to collect, record, store, analyze, and retrieve data concerning an organization's human resources. A. HRIS Applications A computerized HRIS contains hardware and software applications that work together to help managers make HR decisions. HRIS software applications currently available to business include those for employee information, applicant tracking, skills inventory, payroll, time management, and benefits administration. B. HRIS Security and Privacy The HR department must develop policies and guidelines to protect the integrity and security of the HRIS so that private employee information does not fall into the wrong hands. To maintain the security and privacy of HRIS records, companies should control access, develop policies and guidelines that govern the utilization of information, and allow employees to check their records. ANSWERS TO END-OF-CHAPTER DISCUSSION QUESTIONS 1. Are managers likely to question the work commitment of their contingent workers? What might be the consequences for management when the majority of a company's workforce consists of temporary employees and contract workers? Yes, because when there is an economic downturn, contingent workers are the first employees to be discharged. Also, managers might question those workers' commitment because they would suspect that if they can find more permanent positions elsewhere, they will leave. For many this may be true, but for others it may not be. The consequences include a less committed workforce, one that has few loyalties to the company, lower morale, and lower levels of productivity. 2. What are the drawbacks to using flexible work hours from the organization's perspective? Compressed workweeks? Telecommuting? How should the HR department deal with these challenges? The drawbacks of using flexible work hours are that today's greater emphasis on teams requires coordination of work schedules among team members. Concerning compressed workweeks, longer workdays may interfere with job performance. Concerning telecommuting, employers may find themselves with extended obligations under OSHA and other federal laws that cover offsite employees or employees working at home. The HR department can effectively deal with these challenges by identifying each of these drawbacks and developing strategies to
- 12. 21 ©2016 Pearson Education,Inc. address these issues, including creating guidelines and policies regarding work hours, performance expectations, and safety issues. 3. Some management experts do not agree that a virtual team is really a team at all. Based on the definition of a team, what properties of a virtual team satisfy the definition of a team? Do any aspects of a virtual team give rise to doubts over whether it satisfies the definition of a true team? Suppose you needed to organize a virtual team of consultants working in different cities to do an important project for a client. What human resource management practices could you apply that would influence the virtual team members to behave as if they were on a true team such as a self-managed or problem-solving team? The main difference between virtual teams and other teams is that team members interact with each other electronically, rather than face to face. The definition of a team is “a small number of people with complementary skills who work toward common goals for which they hold themselves mutually accountable.” A virtual team has all these properties and satisfies this definition. The one aspect of a virtual team that presents a challenge is the physical proximity of team members. Some practices that could be applied to achieve the same level of accountability that is present with self-managed or problem-solving teams is to integrate team participation into the various programs and policies the organization has in place. For example, including participation in teams as part of the organization’s performance feedback system, such as 360 assessments, would provide firsthand data about individual employee performance on a team. Basing part of employee compensation on team performance is also a method that would influence an employee’s behavior on such a team. 4. A recent trend more and more companies are embracing is to outsource all or most of their human resource management activities. Do you agree or disagree with this trend? What risks is a company taking when it decides to outsource its entire set of human resource management activities? Try to describe a situation where it is most beneficial to retain most of the human resource management activities within a company so that HR is provided by the human resource management department. Students may agree or disagree with this trend. Those who agree will speak of the cost savings gained through outsourcing these activities, which is primarily why companies choose this course of action. The risks associated with outsourcing the entire human resource function, however, are that the firms that perform the outsourced HR functions will not have the knowledge and insight of the organization’s history and culture, which can be very important factors when it comes to decision making. They are also one step removed from the “day to day,” and may not have a firm grasp of changes in employee climate, which affects morale and employee performance. Outsourcing providers also may not be able to offer the presence that is needed in order to gain employee trust, and provide the level of service that companies have come to expect with their in-house HR staffs.
- 13. 22 ©2016 Pearson Education,Inc. An example of a situation where it would be most beneficial to retain the human resource management activities is a company that is going through significant employee relations issues, such as a union organizing drive. A lack of human resource presence would be extremely risky if the company hopes to head off union organizing, as many of the issues are often employee concerns of economics, employment security, and fair working conditions. Outsourced HR providers will most likely be viewed as removed, impersonal, and strictly concerned with the company’s best interests and not necessarily the employees, contributing to a lack of trust by employees and their need for a third party to represent them and their concerns. In contrast, an in-house HR department can serve the role of both company representative and employee advocate, and has a much greater chance of gaining employee trust and confidence, as well as working with frontline supervisors and managers directly to address the issues at hand. 5. In recent years there has been an increase in the number of companies that have wrongly classified an “employee” as a “contract worker,” and consequently were taken to court by workers who believed they were entitled to certain rights and privileges enjoyed by individuals who were given “employee” status. What are some of the rights and privileges that are given to employees and not to contract workers? What advantages do employers gain with contract workers over regular employees? How could a contract worker prove to the courts that he or she is really an employee and was wrongly classified as a contract worker? Some of the rights and benefits given to employees and not to contract workers include higher wages, benefits such as medical plans, 401k plans, stock and savings plans, retirement plans, tuition reimbursement, disability insurance plans, employee assistance programs, vacation time, holiday pay, paid sick time, and various perks such as access to club savings and product discounts. The chief advantage that employers gain with contract workers is that of cost savings. Although some contract workers, especially consultants, may secure a higher rate of pay than a regular employee, the fully loaded cost including fringe benefits is generally less than that of a full-time employee. When the employee is a contract worker through an outside agency, the agency also assumes the cost of workers’ compensation insurance and unemployment insurance for the temporary or contract worker, which results in less liability and cost savings to the employer. Additionally, the employer can employ the contract workers for any period of time and then release them when the company no longer has the need for their services, or is not happy with their work. Employers feel less of an obligation to contract workers than they do to their employees, so they are much less reticent to relieve them from the duties they were hired to perform. Contract workers would have a claim that they are really employees and not contract workers if they have been on a long-term assignment (many companies use the rule of 1,000 hours of service), and, as a result of a ruling in the infamous Microsoft case, become eligible for benefits. ANSWERS TO MY MANAGEMENT LAB DISCUSSION QUESTIONS
- 14. 23 ©2016 Pearson Education,Inc. 6. Are job descriptions really necessary? What would happen if a company decided not to use any job descriptions at all? Job descriptions are necessary and useful for the following HR activities: recruitment, selection, performance appraisals, compensation, training, and career development. If a company decides not to use any job descriptions (documents portraying the jobs) it must determine on what bases it will recruit, select, evaluate performance, compensate, develop training programs, and conduct career development activities. Although it is not wise to eliminate job descriptions altogether, it is possible to use other methods to engage in the HR activities mentioned above. This is mostly successful in organizations with highly educated workers, in an atmosphere of innovation, flexibility, and trust. Often, in such environments, workers will carve their own niches, create their own job descriptions, negotiate their compensation levels in view of market levels and experience, and suggest their own career development. 7. Suggest some ways a manager can make changes in work designs so that employees are able to achieve greater work–life balance. Students will come up with many creative work designs in order for employees to achieve greater work–life balance; however, it is important to discuss the drawbacks to each alternative in addition to the advantages. Students may discuss compressed work weeks, telecommuting, sabbaticals, flex-time, and job-sharing among their alternatives. It is important for them to note that some employees might find these options more stressful than a normal work week. Although not common, changes to routine can sometimes create higher levels of stress than a traditional work schedule; thus, it is of the utmost importance to tailor unique work designs to the needs of the specific employee when possible. 8. Large U.S. companies such as Accenture, AOL, and Dell have outsourced customer service call centers to India. Customers use these call centers for help when they are having difficulty using the services provided by these companies. Many of the outsourced jobs at the call centers were entry-level jobs that had the potential to lead to higher-skilled jobs at those firms. Provide at least three ethical employment issues that managers who use offshore outsource suppliers in India or other low-labor-cost countries should be concerned about. Some students will agree that there are many ethical issues, and some students will see the logic in saving money where you can and see little ethical implications. Managers in the United States have strict labor laws and standards that protect certain employee rights. Other countries have different, often lower standards than the United States. This could cause ethical dilemmas across the firm if outsourced employees are not treated well. Outsourcing can also be detrimental to local economies and take opportunities away from local talent, making it hard to recruit from within the firm.
- 15. 24 ©2016 Pearson Education,Inc. You Manage It! 1: Ethics/Social Responsibility Are Companies Exploiting College Students Who Have Unpaid Internships? Critical Thinking Questions 1. Although it is illegal for profit-based companies to create unpaid internships that require college interns to perform primarily menial tasks, unfortunately this is happening with increasing regularity. What can students do to avoid the experience of having an unpaid internship that consists of mostly menial work with few opportunities to learn new skills? Students’ answers will vary. There are many ways that students may suggest handling this situation. Students may focus on the selection procedure with the understanding that good grades, strong networks, and a focused résumé may lead to better choices for interns. Also, students may share the idea of understanding the job duties and the responsibilities of the internship during the interview or shortly after accepting the position. 2. Does the university have a responsibility to ensure that a student’s unpaid internship will be a legitimate learning experience that earns college credits toward graduation? How can the university ensure that a company provides the unpaid intern a legitimate learning experience while still giving the company the flexibility to deploy the unpaid intern in ways that are useful to the company? Universities certainly have a responsibility to ensure that the student’s internship is a legitimate learning experience. Based on the fact that the internship is unpaid, the university will need to be explicit with expectations for student involvement. The university and its representatives can articulate the elements of the internship position that are most beneficial for the student in achieving course credit for the experience. You Manage It! 2: Emerging Trends Work–Life Balance Is the New Perk Employees Are Seeking Critical Thinking Questions 1. Which types of jobs are best suited for flexibility with regard to hours and office location? Which types of jobs are less likely to afford this type of flexibility? Explain. Although many job types can be made more flexible than standard work arrangements, generally, “professional”-type jobs tend to be more amenable to creating flexibility. These jobs are generally salary-based and contingent on completing jobs/projects on some sort of self-directed schedule or timetable.
- 16. 25 ©2016 Pearson Education,Inc. 2. Earlier in this chapter, you learned that most work in today’s workplace is now being done by teams of employees. In your opinion, does the intensive use of self- managed teams make it easier or more difficult for employees to achieve work–life balance? Explain. Opinions on this question will vary; however, self-managed teams create barriers and opportunities for employees to schedule meetings, prioritize tasks, and complete projects. Although the self-managed team does not subscribe to any organizationally defined meeting or work schedule, it can be difficult to arrange all of the personal preferences of several team members not working a traditional schedule. You Manage It! 3: Technology/Social Media Yahoo CEO Issues a Ban on Telecommuting for Employees Critical Thinking Questions 1. Do you agree or disagree with the CEO’s decision to ban employees from telecommuting at Yahoo? What is the basis of your position? Students’ answers will vary. Many students will probably disagree with the CEO’s decision, citing that individuals are more motivated when they are allowed to work in the space that most satisfies them. However, other students will suggest that it is important for employees to be in the work setting on a daily basis so that creativity and innovation can be part of the conversation. 2. Critics of the decision to restrict telecommuting at Yahoo point to the poor financial and stock market performance of Yahoo in the years prior to this order, and they suggest that the CEO’s motive was to impress investors by displaying more control over Yahoo employees. It is likely that the CEO expected—by mandating that employees be present in the office on a regular basis—that they would have more fortuitous conversations in the corridors of Yahoo that would likely lead to increased levels of innovation and new product development. Can you think of alternative ways that the company could engage employee innovation and creativity without restricting their freedom to work from home? Students will be creative in their response to the question of alternative ways to engage innovation without giving up employee freedom to work at home. Some students may suggest that Yahoo could have leveraged the structure of virtual teams. Innovation and creativity can be a by-product of well-led virtual teams. Students may also point out that competitions for “best practices” often lead to creative ideas and innovation. Using contests and competition can be a useful motivator for innovation. You Manage It! 4: Customer-Driven HR Writing a Job Description
- 17. 26 ©2016 Pearson Education,Inc. Critical Thinking Questions 1. What do you see as the main differences between a specific job description and a general description? The main difference is the level of specificity of tasks. The general description gives a broader picture and allows re-assignment to different jobs or tasks under the same job description. The specific description gives much more detail on the specific tasks that a specific job is expected to perform. It is much more rigid and less flexible than the general description. 2. Suppose several people are employed in the same job as the one for which you are writing a job description. Would it be necessary to write a different job description for each person who works in the same job? It is not necessary to write separate job descriptions for each person. Job descriptions should be written to apply to all people who are in the same job. The description should be specific enough that the employees know what they are to do, but general enough to allow for the minor variations that individuals bring to the job. 3. Carefully follow the format for the “Specific Job Description” provided in Figure 2.6 when writing the job description for the job you selected. Make sure that you include in your job description the following elements: (1) job title and identification information, (2) job summary, (3) job duties and responsibilities, (4) job requirements, and (5) minimum qualifications. Check your work to make sure the style of your job description matches the example in the text as closely as possible. Look for a full and complete job description that matches the style of the example in the textbook. Additional Exercises In-Class or Out-of-Class Group Activities Implicit in this chapter is the view that organizational change is necessary for survival. However, organizational change often places individual employees under considerable stress, particularly the stress resulting from having to learn new skills and job requirements constantly. Is the organization ethically responsible for protecting employees from these stressful changes? Although there may be differing views on this subject, let us suggest one answer that seems the most reasonable. When job loss occurs because of organizational change (or from economic downturn) it seems reasonable that the organization has an ethical responsibility to provide the employee assistance in handling and dealing
- 18. 27 ©2016 Pearson Education,Inc. with this stress. Successful programs would include outplacement services, severance packages to assist the employee during the transitional period, counseling services, and so forth. It is probably unreasonable to expect any organization to protect an individual from the stresses of life. However, it may also be reasonable to assume that the organization may have a stronger ethical responsibility to long- term employees who may be nearing retirement. Many employees and union representatives complain bitterly about the practice of outsourcing work, particularly to foreign countries. Part of the complaint is that companies do this to avoid paying fair wages and providing employee benefits that U.S. workers expect. Is this an ethical issue? Yes, this is an ethical issue. Any time you have a dilemma that pits financial considerations against questions of fair and appropriate treatment of people, you have an ethical question. As with most ethical dilemmas, there is not an easy answer to this one. On one hand you have the financial well-being of the company that is facing worldwide competition, and on the other hand you have the ability of workers to earn a decent wage and to gain reasonable benefits. If a company unilaterally decides to forgo outsourcing of this kind but then is forced out of business because of its competition's lower costs, has an ethical decision been made? Some would say yes; others would say no. Certainly, in a case where a company is very profitable and is not in danger of losing market share due to costs, outsourcing of this kind could easily be seen as unethical. Additionally, loss of U.S. jobs negatively impacts the economy, a topic that has received much attention in recent years. However, most cases are not this clear cut. Use this case as an example to help students understand the difficulties that company executives face in making these decisions. When American Greetings Corporation, the Cleveland greeting card and licensing company, redesigned about 100 jobs in its creative division, it asked workers and managers to reapply for the new jobs. Everyone was guaranteed a position and no one took a pay cut. When the structuring is complete, employees will develop products in teams instead of in assembly-line fashion, and they'll be free to transfer back and forth among teams that make different products instead of working on just one product line, as they have in the past. Give some reasons that you think American Greetings, like many U.S. companies, is restructuring its work to be performed in teams. Would the teams at American Greetings be considered self- managed work teams? Why or why not? Some reasons why American Greetings Corporation, like many other companies, is restructuring is (1) to align the organizational structure with its business strategy, (2) to pursue a TQM effort, (3) to examine complex issues such as the introduction of new technology, and/or (4) to move toward less dependence on supervisors and toward more leadership direction through teams. As companies move to flat and boundaryless organizations, teamwork is an imperative.
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- 20. urgency to saysomething about curvature, he almost automatically said the right thing. I mean that there was only one limitation or law that suggested itself as reasonable, and that law has proved to be right when tested by observation. Some of you may feel that you could never bring your minds to conceive a curvature of space, let alone of space-time; others may feel that, being familiar with the bending of a two-dimensional surface, there is no insuperable difficulty in imagining something similar for three or even four dimensions. I rather think that the former have the best of it, for at least they escape being misled by their preconceptions. I have spoken of a “picture”, but it is a picture that has to be described analytically rather than conceived vividly. Our ordinary conception of curvature is derived from surfaces, i.e. two-dimensional manifolds embedded in a three-dimensional space. The absolute curvature at any point is measured by a single quantity called the radius of spherical curvature. But space-time is a four- dimensional manifold embedded in—well, as many dimensions as it can find new ways to twist about in. Actually a four-dimensional manifold is amazingly ingenious in discovering new kinds of contortion, and its invention is not exhausted until it has been provided with six extra dimensions, making ten dimensions in all. Moreover, twenty distinct measures are required at each point to specify the particular sort and amount of twistiness there. These measures are called coefficients of curvature. Ten of the coefficients stand out more prominently than the other ten. Einstein’s law of gravitation asserts that the ten principal coefficients of curvature are zero in empty space. If there were no curvature, i.e. if all the coefficients were zero, there would be no gravitation. Bodies would move uniformly in straight lines. If curvature were unrestricted, i.e. if all the coefficients had unpredictable values, gravitation would operate arbitrarily and without law. Bodies would move just anyhow. Einstein takes a condition midway between; ten of the coefficients are zero and the other ten are arbitrary. That gives a world containing
- 21. gravitation limited bya law. The coefficients are naturally separated into two groups of ten, so that there is no difficulty in choosing those which are to vanish. To the uninitiated it may seem surprising that an exact law of Nature should leave some of the coefficients arbitrary. But we need to leave something over to be settled when we have specified the particulars of the problem to which it is proposed to apply the law. A general law covers an infinite number of special cases. The vanishing of the ten principal coefficients occurs everywhere in empty space whether there is one gravitating body or many. The other ten coefficients vary according to the special case under discussion. This may remind us that after reaching Einstein’s law of gravitation and formulating it mathematically, it is still a very long step to reach its application to even the simplest practical problem. However, by this time many hundreds of readers must have gone carefully through the mathematics; so we may rest assured that there has been no mistake. After this work has been done it becomes possible to verify that the law agrees with observation. It is found that it agrees with Newton’s law to a very close approximation so that the main evidence for Einstein’s law is the same as the evidence for Newton’s law; but there are three crucial astronomical phenomena in which the difference is large enough to be observed. In these phenomena the observations support Einstein’s law against Newton’s.[15] It is essential to our faith in a theory that its predictions should accord with observation, unless a reasonable explanation of the discrepancy is forthcoming; so that it is highly important that Einstein’s law should have survived these delicate astronomical tests in which Newton’s law just failed. But our main reason for rejecting Newton’s law is not its imperfect accuracy as shown by these tests; it is because it does not contain the kind of information about Nature that we want to know now that we have an ideal before us which was not in Newton’s mind at all. We can put it this way. Astronomical observations show that within certain limits of accuracy both Einstein’s and Newton’s laws are true. In confirming (approximately)
- 22. Newton’s law, weare confirming a statement as to what the appearances would be when referred to one particular space-time frame. No reason is given for attaching any fundamental importance to this frame. In confirming (approximately) Einstein’s law, we are confirming a statement about the absolute properties of the world, true for all space-time frames. For those who are trying to get beneath the appearances Einstein’s statement necessarily supersedes Newton’s; it extracts from the observations a result with physical meaning as opposed to a mathematical curiosity. That Einstein’s law has proved itself the better approximation encourages us in our opinion that the quest of the absolute is the best way to understand the relative appearances; but had the success been less immediate, we could scarcely have turned our back on the quest. I cannot but think that Newton himself would rejoice that after 200 years the “ocean of undiscovered truth” has rolled back another stage. I do not think of him as censorious because we will not blindly apply his formula regardless of the knowledge that has since accumulated and in circumstances that he never had the opportunity of considering. I am not going to describe the three tests here, since they are now well known and will be found in any of the numerous guides to relativity; but I would refer to the action of gravitation on light concerned in one of them. Light-waves in passing a massive body such as the sun are deflected through a small angle. This is additional evidence that the Newtonian picture of gravitation as a tug is inadequate. You cannot deflect waves by tugging at them, and clearly another representation of the agency which deflects them must be found. The Law of Motion. I must now ask you to let your mind revert to the time of your first introduction to mechanics before your natural glimmerings of the truth were sedulously uprooted by your teacher. You were taught the First Law of Motion—
- 23. “Every body continuesin its state of rest or uniform motion in a straight line, except in so far as it may be compelled to change that state by impressed forces.” Probably you had previously supposed that motion was something which would exhaust itself; a bicycle stops of its own accord if you do not impress force to keep it going. The teacher rightly pointed out the resisting forces which tend to stop the bicycle; and he probably quoted the example of a stone skimming over ice to show that when these interfering forces are reduced the motion lasts much longer. But even ice offers some frictional resistance. Why did not the teacher do the thing thoroughly and abolish resisting forces altogether, as he might easily have done by projecting the stone into empty space? Unfortunately in that case its motion is not uniform and rectilinear; the stone describes a parabola. If you raised that objection you would be told that the projectile was compelled to change its state of uniform motion by an invisible force called gravitation. How do we know that this invisible force exists? Why! because if the force did not exist the projectile would move uniformly in a straight line. The teacher is not playing fair. He is determined to have his uniform motion in a straight line, and if we point out to him bodies which do not follow his rule he blandly invents a new force to account for the deviation. We can improve on his enunciation of the First Law of Motion. What he really meant was— “Every body continues in its state of rest or uniform motion in a straight line, except in so far as it doesn’t.” Material frictions and reactions are visible and absolute interferences which can change the motion of a body. I have nothing to say against them. The molecular battering can be recognised by anyone who looks deeply into the phenomenon no matter what his frame of reference. But when there is no such indication of disturbance the whole procedure becomes arbitrary. On no particular grounds the motion is divided into two parts, one of which is attributed to a passive tendency of the body called inertia and the
- 24. other to aninterfering field of force. The suggestion that the body really wanted to go straight but some mysterious agent made it go crooked is picturesque but unscientific. It makes two properties out of one; and then we wonder why they are always proportional to one another—why the gravitational force on different bodies is proportional to their inertia or mass. The dissection becomes untenable when we admit that all frames of reference are on the same footing. The projectile which describes a parabola relative to an observer on the earth’s surface describes a straight line relative to the man in the lift. Our teacher will not easily persuade the man in the lift who sees the apple remaining where he released it, that the apple really would of its own initiative rush upwards were it not that an invisible tug exactly counteracts this tendency.[16] Einstein’s Law of Motion does not recognise this dissection. There are certain curves which can be defined on a curved surface without reference to any frame or system of partitions, viz. the geodesics or shortest routes from one point to another. The geodesics of our curved space-time supply the natural tracks which particles pursue if they are undisturbed. We observe a planet wandering round the sun in an elliptic orbit. A little consideration will show that if we add a fourth dimension (time), the continual moving on in the time-dimension draws out the ellipse into a helix. Why does the planet take this spiral track instead of going straight? It is because it is following the shortest track; and in the distorted geometry of the curved region round the sun the spiral track is shorter than any other between the same points. You see the great change in our view. The Newtonian scheme says that the planet tends to move in a straight line, but the sun’s gravity pulls it away. Einstein says that the planet tends to take the shortest route and does take it. That is the general idea, but for the sake of accuracy I must make one rather trivial correction. The planet takes the longest route.
- 25. You may rememberthat points along the track of any material body (necessarily moving with a speed less than the velocity of light) are in the absolute past or future of one another; they are not absolutely “elsewhere”. Hence the length of the track in four dimensions is made up of time-like relations and must be measured in time-units. It is in fact the number of seconds recorded by a clock carried on a body which describes the track.[17] This may be different from the time recorded by a clock which has taken some other route between the same terminal points. On p. 39 we considered two individuals whose tracks had the same terminal points; one of them remained at home on the earth and the other travelled at high speed to a distant part of the universe and back. The first recorded a lapse of 70 years, the second of one year. Notice that it is the man who follows the undisturbed track of the earth who records or lives the longest time. The man whose track was violently dislocated when he reached the limit of his journey and started to come back again lived only one year. There is no limit to this reduction; as the speed of the traveller approaches the speed of light the time recorded diminishes to zero. There is no unique shortest track; but the longest track is unique. If instead of pursuing its actual orbit the earth made a wide sweep which required it to travel with the velocity of light, the earth could get from 1 January 1927 to 1 January 1928 in no time, i.e. no time as recorded by an observer or clock travelling with it, though it would be reckoned as a year according to “Astronomer Royal’s time”. The earth does not do this, because it is a rule of the Trade Union of matter that the longest possible time must be taken over every job. Thus in calculating astronomical orbits and in similar problems two laws are involved. We must first calculate the curved form of space-time by using Einstein’s law of gravitation, viz. that the ten principal curvatures are zero. We next calculate how the planet moves through the curved region by using Einstein’s law of motion, viz. the law of the longest track. Thus far the procedure is analogous to calculations made with Newton’s law of gravitation and Newton’s law of motion. But there is a remarkable addendum which applies
- 26. only to Einstein’slaws. Einstein’s law of motion can be deduced from his law of gravitation. The prediction of the track of a planet although divided into two stages for convenience rests on a single law. I should like to show you in a general way how it is possible for a law controlling the curvature of empty space to determine the tracks of particles without being supplemented by any other conditions. Fig. 5 Two “particles” in the four-dimensional world are shown in Fig. 5, namely yourself and myself. We are not empty space so there is no limit to the kind of curvature entering into our composition; in fact our unusual sort of curvature is what distinguishes us from empty space. We are, so to speak, ridges in the four-dimensional world where it is gathered into a pucker. The pure mathematician in his unflattering language would describe us as “singularities”. These two non-empty ridges are joined by empty space, which must be free from those kinds of curvature described by the ten principal coefficients. Now it is common experience that if we introduce local puckers into the material of a garment, the remainder has a certain obstinacy and will not lie as smoothly as we might wish. You will
- 27. realise the possibilitythat, given two ridges as in Fig. 5, it may be impossible to join them by an intervening valley without the illegal kind of curvature. That turns out to be the case. Two perfectly straight ridges alone in the world cannot be properly joined by empty space and therefore they cannot occur alone. But if they bend a little towards one another the connecting region can lie smoothly and satisfy the law of curvature. If they bend too much the illegal puckering reappears. The law of gravitation is a fastidious tailor who will not tolerate wrinkles (except of a limited approved type) in the main area of the garment; so that the seams are required to take courses which will not cause wrinkles. You and I have to submit to this and so our tracks curve towards each other. An onlooker will make the comment that here is an illustration of the law that two massive bodies attract each other. We thus arrive at another but equivalent conception of how the earth’s spiral track through the four-dimensional world is arrived at. It is due to the necessity of arranging two ridges (the solar track and the earth’s track) so as not to involve a wrong kind of curvature in the empty part of the world. The sun as the more pronounced ridge takes a nearly straight track; but the earth as a minor ridge on the declivities of the solar ridge has to twist about considerably. Suppose the earth were to defy the tailor and take a straight track. That would make a horrid wrinkle in the garment; and since the wrinkle is inconsistent with the laws of empty space, something must be there—where the wrinkle runs. This “something” need not be matter in the restricted sense. The things which can occupy space so that it is not empty in the sense intended in Einstein’s law, are mass (or its equivalent energy) momentum and stress (pressure or tension). In this case the wrinkle might correspond to stress. That is reasonable enough. If left alone the earth must pursue its proper curved orbit; but if some kind of stress or pressure were inserted between the sun and earth, it might well take another course. In fact if we were to observe one of the planets rushing off in a straight track, Newtonians and Einsteinians alike would infer that there existed a stress causing this behaviour. It is true that causation has
- 28. apparently been turnedtopsy-turvy; according to our theory the stress seems to be caused by the planet taking the wrong track, whereas we usually suppose that the planet takes the wrong track because it is acted on by the stress. But that is a harmless accident common enough in primary physics. The discrimination between cause and effect depends on time’s arrow and can only be settled by reference to entropy. We need not pay much attention to suggestions of causation arising in discussions of primary laws which, as likely as not, are contemplating the world upside down. Although we are here only at the beginning of Einstein’s general theory I must not proceed further into this very technical subject. The rest of this chapter will be devoted to elucidation of more elementary points. Relativity of Acceleration. The argument in this chapter rests on the relativity of acceleration. The apple had an acceleration of 32 feet per second per second relative to the ordinary observer, but zero acceleration relative to the man in the lift. We ascribe to it one acceleration or the other according to the frame we happen to be using, but neither is to be singled out and labelled “true” or absolute acceleration. That led us to reject the Newtonian conception which singled out 32 feet per second per second as the true acceleration and invented a disturbing agent of this particular degree of strength. It will be instructive to consider an objection brought, I think, originally by Lenard. A train is passing through a station at 60 miles an hour. Since velocity is relative, it does not matter whether we say that the train is moving at 60 miles an hour past the station or the station is moving at 60 miles an hour past the train. Now suppose, as sometimes happens in railway accidents, that this motion is brought to a standstill in a few seconds. There has been a change of velocity or acceleration—a term which includes deceleration. If acceleration is relative this may be described indifferently as an acceleration of the train (relative to the station) or an acceleration of
- 29. the station (relativeto the train). Why then does it injure the persons in the train and not those in the station? Much the same point was put to me by one of my audience. “You must find the journey between Cambridge and Edinburgh very tiring. I can understand the fatigue, if you travel to Edinburgh; but why should you get tired if Edinburgh comes to you?” The answer is that the fatigue arises from being shut up in a box and jolted about for nine hours; and it makes no difference whether in the meantime I move to Edinburgh or Edinburgh moves to me. Motion does not tire anybody. With the earth as our vehicle we are travelling at 20 miles a second round the sun; the sun carries us at 12 miles a second through the galactic system; the galactic system bears us at 250 miles a second amid the spiral nebulae; the spiral nebulae.... If motion could tire, we ought to be dead tired. Similarly change of motion or acceleration does not injure anyone, even when it is (according to the Newtonian view) an absolute acceleration. We do not even feel the change of motion as our earth takes the curve round the sun. We feel something when a railway train takes a curve, but what we feel is not the change of motion nor anything which invariably accompanies change of motion; it is something incidental to the curved track of the train but not to the curved track of the earth. The cause of injury in the railway accident is easily traced. Something hit the train; that is to say, the train was bombarded by a swarm of molecules and the bombardment spread all the way along it. The cause is evident— gross, material, absolute—recognised by everyone, no matter what his frame of reference, as occurring in the train not the station. Besides injuring the passengers this cause also produced the relative acceleration of the train and station—an effect which might equally well have been produced by molecular bombardment of the station, though in this case it was not. The critical reader will probably pursue his objection. “Are you not being paradoxical when you say that a molecular bombardment of the train can cause an acceleration of the station—and in fact of
- 30. the earth andthe rest of the universe? To put it mildly, relative acceleration is a relation with two ends to it, and we may at first seem to have an option which end we shall grasp it by; but in this case the causation (molecular bombardment) clearly indicates the right end to take hold of, and you are merely spinning paradoxes when you insist on your liberty to take hold of the other.” If there is an absurdity in taking hold of the wrong end of the relation it has passed into our current speech and thought. Your suggestion is in fact more revolutionary than anything Einstein has ventured to advocate. Let us take the problem of a falling stone. There is a relative acceleration of 32 feet per second per second—of the stone relative to ourselves or of ourselves relative to the stone. Which end of the relation must we choose? The one indicated by molecular bombardment? Well, the stone is not bombarded; it is falling freely in vacuo. But we are bombarded by the molecules of the ground on which we stand. Therefore it is we who have the acceleration; the stone has zero acceleration, as the man in the lift supposed. Your suggestion makes out the frame of the man in the lift to be the only legitimate one; I only went so far as to admit it to an equality with our own customary frame. Your suggestion would accept the testimony of the drunken man who explained that “the paving-stone got up and hit him” and dismiss the policeman’s account of the incident as “merely spinning paradoxes”. What really happened was that the paving-stone had been pursuing the man through space with ever-increasing velocity, shoving the man in front of it so that they kept the same relative position. Then, through an unfortunate wobble of the axis of the man’s body, he failed to increase his speed sufficiently, with the result that the paving-stone overtook him and came in contact with his head. That, please understand, is your suggestion; or rather the suggestion which I have taken the liberty of fathering on you because it is the outcome of a very common feeling of objection to the relativity theory. Einstein’s position is that whilst this is a perfectly legitimate way of looking at the incident the more usual
- 31. account given bythe policeman is also legitimate; and he endeavours like a good magistrate to reconcile them both. Time Geometry. Einstein’s law of gravitation controls a geometrical quantity curvature in contrast to Newton’s law which controls a mechanical quantity force. To understand the origin of this geometrisation of the world in the relativity theory we must go back a little. The science which deals with the properties of space is called geometry. Hitherto geometry has not included time in its scope. But now space and time are so interlocked that there must be one science—a somewhat extended geometry—embracing them both. Three-dimensional space is only a section cut through four- dimensional space-time, and moreover sections cut in different directions form the spaces of different observers. We can scarcely maintain that the study of a section cut in one special direction is the proper subject-matter of geometry and that the study of slightly different sections belongs to an altogether different science. Hence the geometry of the world is now considered to include time as well as space. Let us follow up the geometry of time. You will remember that although space and time are mixed up there is an absolute distinction between a spatial and a temporal relation of two events. Three events will form a space-triangle if the three sides correspond to spatial relations—if the three events are absolutely elsewhere with respect to one another.[18] Three events will form a time-triangle if the three sides correspond to temporal relations—if the three events are absolutely before or after one another. (It is possible also to have mixed triangles with two sides time-like and one space-like, or vice versa.) A well-known law of the space-triangle is that any two sides are together greater than the third side. There is an analogous, but significantly different, law for the time-triangle, viz. two of the sides (not any two sides) are together less than the third side. It is difficult to picture such a triangle but that is the actual fact.
- 32. Let us bequite sure that we grasp the precise meaning of these geometrical propositions. Take first the space-triangle. The proposition refers to the lengths of the sides, and it is well to recall my imaginary discussion with two students as to how lengths are to be measured (p. 23). Happily there is no ambiguity now, because the triangle of three events determines a plane section of the world, and it is only for that mode of section that the triangle is purely spatial. The proposition then expresses that “If you measure with a scale from to and from to the sum of your readings will be greater than the reading obtained by measuring with a scale from to .” For a time-triangle the measurements must be made with an instrument which can measure time, and the proposition then expresses that “If you measure with a clock from to and from to the sum of your readings will be less than the reading obtained by measuring with a clock from to .” In order to measure from an event to an event with a clock you must make an adjustment of the clock analogous to orienting a scale along the line . What is this analogous adjustment? The purpose in either case is to bring both and into the immediate neighbourhood of the scale or clock. For the clock that means that after experiencing the event it must travel with the appropriate velocity needed to reach the locality of just at the moment that happens. Thus the velocity of the clock is prescribed. One further point should be noticed. After measuring with a scale from to you can turn your scale round and measure from to , obtaining the same result. But you cannot turn a clock round, i.e. make it go backwards in time. That is important because it decides which two sides are less than the third side. If you choose the wrong pair the enunciation of the time proposition refers to an impossible kind of measurement and becomes meaningless.
- 33. You remember thetraveller (p. 39) who went off to a distant star and returned absurdly young. He was a clock measuring two sides of a time-triangle. He recorded less time than the stay-at-home observer who was a clock measuring the third side. Need I defend my calling him a clock? We are all of us clocks whose faces tell the passing years. This comparison was simply an example of the geometrical proposition about time-triangles (which in turn is a particular case of Einstein’s law of longest track). The result is quite explicable in the ordinary mechanical way. All the particles in the traveller’s body increase in mass on account of his high velocity according to the law already discussed and verified by experiment. This renders them more sluggish, and the traveller lives more slowly according to terrestrial time-reckoning. However, the fact that the result is reasonable and explicable does not render it the less true as a proposition of time geometry. Our extension of geometry to include time as well as space will not be a simple addition of an extra dimension to Euclidean geometry, because the time propositions, though analogous, are not identical with those which Euclid has given us for space alone. Actually the difference between time geometry and space geometry is not very profound, and the mathematician easily glides over it by a discrete use of the symbol . We still call (rather loosely) the extended geometry Euclidean; or, if it is necessary to emphasise the distinction, we call it hyperbolic geometry. The term non-Euclidean geometry refers to a more profound change, viz. that involved in the curvature of space and time by which we now represent the phenomenon of gravitation. We start with Euclidean geometry of space, and modify it in a comparatively simple manner when the time-dimension is added; but that still leaves gravitation to be reckoned with, and wherever gravitational effects are observable it is an indication that the extended Euclidean geometry is not quite exact, and the true geometry is a non-Euclidean one—appropriate to a curved region as Euclidean geometry is to a flat region.
- 34. Geometry and Mechanics.The point that deserves special attention is that the proposition about time-triangles is a statement as to the behaviour of clocks moving with different velocities. We have usually regarded the behaviour of clocks as coming under the science of mechanics. We found that it was impossible to confine geometry to space alone, and we had to let it expand a little. It has expanded with a vengeance and taken a big slice out of mechanics. There is no stopping it, and bit by bit geometry has now swallowed up the whole of mechanics. It has also made some tentative nibbles at electromagnetism. An ideal shines in front of us, far ahead perhaps but irresistible, that the whole of our knowledge of the physical world may be unified into a single science which will perhaps be expressed in terms of geometrical or quasi-geometrical conceptions. Why not? All the knowledge is derived from measurements made with various instruments. The instruments used in the different fields of inquiry are not fundamentally unlike. There is no reason to regard the partitions of the sciences made in the early stages of human thought as irremovable. But mechanics in becoming geometry remains none the less mechanics. The partition between mechanics and geometry has broken down and the nature of each of them has diffused through the whole. The apparent supremacy of geometry is really due to the fact that it possesses the richer and more adaptable vocabulary; and since after the amalgamation we do not need the double vocabulary the terms employed are generally taken from geometry. But besides the geometrisation of mechanics there has been a mechanisation of geometry. The proposition about the space-triangle quoted above was seen to have grossly material implications about the behaviour of scales which would not be realised by anyone who thinks of it as if it were a proposition of pure mathematics. We must rid our minds of the idea that the word space in science has anything to do with void. As previously explained it has the other meaning of distance, volume, etc., quantities expressing physical measurement just as much as force is a quantity expressing physical measurement. Thus the (rather crude) statement that
- 35. Einstein’s theory reducesgravitational force to a property of space ought not to arouse misgiving. In any case the physicist does not conceive of space as void. Where it is empty of all else there is still the aether. Those who for some reason dislike the word aether, scatter mathematical symbols freely through the vacuum, and I presume that they must conceive some kind of characteristic background for these symbols. I do not think any one proposes to build even so relative and elusive a thing as force out of entire nothingness. [13] So far as I can tell (without experimental trial) the man who jumped over a precipice would soon lose all conception of falling; he would only notice that the surrounding objects were impelled past him with ever-increasing speed. [14] It will probably be objected that since the phenomena here discussed are evidently associated with the existence of a massive body (the earth), and since Newton makes his tugs occur symmetrically about that body whereas the apple makes its tugs occur unsymmetrically (vanishing where the apple is, but strong at the antipodes), therefore Newton’s frame is clearly to be preferred. It would be necessary to go deeply into the theory to explain fully why we do not regard this symmetry as of first importance; we can only say here that the criterion of symmetry proves to be insufficient to pick out a unique frame and does not draw a sharp dividing line between the frames that it would admit and those it would have us reject. After all we can appreciate that certain frames are more symmetrical than others without insisting on calling the symmetrical ones “right” and unsymmetrical ones “wrong”. [15] One of the tests—a shift of the spectral lines to the red in the sun and stars as compared with terrestrial sources—is a test of Einstein’s theory rather than of his law. [16] The reader will verify that this is the doctrine the teacher would have to inculcate if he went as a missionary to the men in the lift. [17] It may be objected that you cannot make a clock follow an arbitrary curved path without disturbing it by impressed forces (e.g. molecular hammering). But this difficulty is precisely analogous to the difficulty of measuring the length of a curve with a rectilinear scale, and is surmounted in the same way. The usual
- 36. theory of “rectificationof curves” applies to these time-tracks as well as to space-curves. [18] This would be an instantaneous space-triangle. An enduring triangle is a kind of four-dimensional prism.
- 37. Chapter VII GRAVITATION—THE EXPLANATION TheLaw of Curvature. Gravitation can be explained. Einstein’s theory is not primarily an explanation of gravitation. When he tells us that the gravitational field corresponds to a curvature of space and time he is giving us a picture. Through a picture we gain the insight necessary to deduce the various observable consequences. There remains, however, a further question whether any reason can be given why the state of things pictured should exist. It is this further inquiry which is meant when we speak of “explaining” gravitation in any far-reaching sense. At first sight the new picture does not leave very much to explain. It shows us an undulating hummocky world, whereas a gravitationless world would be plane and uniform. But surely a level lawn stands more in need of explanation than an undulating field, and a gravitationless world would be more difficult to account for than a world with gravitation. We are hardly called upon to account for a phenomenon which could only be absent if (in the building of the world) express precautions were taken to exclude it. If the curvature were entirely arbitrary this would be the end of the explanation; but there is a law of curvature—Einstein’s law of gravitation—and on this law our further inquiry must be focussed. Explanation is needed for regularity, not for diversity; and our curiosity is roused, not by the diverse values of the ten subsidiary coefficients of curvature which differentiate the world from a flat world, but by the vanishing everywhere of the ten principal coefficients. All explanations of gravitation on Newtonian lines have endeavoured to show why something (which I have disrespectfully
- 38. called a demon)is present in the world. An explanation on the lines of Einstein’s theory must show why something (which we call principal curvature) is excluded from the world. In the last chapter the law of gravitation was stated in the form— the ten principal coefficients of curvature vanish in empty space. I shall now restate it in a slightly altered form— The radius of spherical[19] curvature of every three-dimensional section of the world, cut in any direction at any point of empty space, is always the same constant length. Besides the alteration of form there is actually a little difference of substance between the two enunciations; the second corresponds to a later and, it is believed, more accurate formula given by Einstein a year or two after his first theory. The modification is made necessary by our realisation that space is finite but unbounded (p. 80). The second enunciation would be exactly equivalent to the first if for “same constant length” we read “infinite length”. Apart from very speculative estimates we do not know the constant length referred to, but it must certainly be greater than the distance of the furthest nebula, say miles. A distinction between so great a length and infinite length is unnecessary in most of our arguments and investigations, but it is necessary in the present chapter. We must try to reach the vivid significance which lies behind the obscure phraseology of the law. Suppose that you are ordering a concave mirror for a telescope. In order to obtain what you want you will have to specify two lengths (1) the aperture, and (2) the radius of curvature. These lengths both belong to the mirror—both are necessary to describe the kind of mirror you want to purchase— but they belong to it in different ways. You may order a mirror of 100 foot radius of curvature and yet receive it by parcel post. In a certain sense the 100 foot length travels with the mirror, but it does so in a way outside the cognizance of the postal authorities. The 100 foot length belongs especially to the surface of the mirror, a two- dimensional continuum; space-time is a four-dimensional continuum, and you will see from this analogy that there can be lengths
- 39. belonging in thisway to a chunk of space-time—lengths having nothing to do with the largeness or smallness of the chunk, but none the less part of the specification of the particular sample. Owing to the two extra dimensions there are many more such lengths associated with space-time than with the mirror surface. In particular, there is not only one general radius of spherical curvature, but a radius corresponding to any direction you like to take. For brevity I will call this the “directed radius” of the world. Suppose now that you order a chunk of space-time with a directed radius of 500 trillion miles in one direction and 800 trillion miles in another. Nature replies “No. We do not stock that. We keep a wide range of choice as regards other details of specification; but as regards directed radius we have nothing different in different directions, and in fact all our goods have the one standard radius, trillion miles.” I cannot tell you what number to put for because that is still a secret of the firm. The fact that this directed radius which, one would think, might so easily differ from point to point and from direction to direction, has only one standard value in the world is Einstein’s law of gravitation. From it we can by rigorous mathematical deduction work out the motions of planets and predict, for example, the eclipses of the next thousand years; for, as already explained, the law of gravitation includes also the law of motion. Newton’s law of gravitation is an approximate adaptation of it for practical calculation. Building up from the law all is clear; but what lies beneath it? Why is there this unexpected standardisation? That is what we must now inquire into. Relativity of Length. There is no such thing as absolute length; we can only express the length of one thing in terms of the length of something else.[20] And so when we speak of the length of the directed radius we mean its length compared with the standard metre scale. Moreover, to make this comparison, the two lengths must lie alongside. Comparison at a distance is as unthinkable as
- 40. action at adistance; more so, because comparison is a less vague conception than action. We must either convey the standard metre to the site of the length we are measuring, or we must use some device which, we are satisfied, will give the same result as if we actually moved the metre rod. Now if we transfer the metre rod to another point of space and time, does it necessarily remain a metre long? Yes, of course it does; so long as it is the standard of length it cannot be anything else but a metre. But does it really remain the metre that it was? I do not know what you mean by the question; there is nothing by reference to which we could expose delinquencies of the standard rod, nothing by reference to which we could conceive the nature of the supposed delinquencies. Still the standard rod was chosen with considerable care; its material was selected to fulfil certain conditions—to be affected as little as possible by casual influences such as temperature, strain or corrosion, in order that its extension might depend only on the most essential characteristics of its surroundings, present and past.[21] We cannot say that it was chosen to keep the same absolute length since there is no such thing known; but it was chosen so that it might not be prevented by casual influences from keeping the same relative length—relative to what? Relative to some length inalienably associated with the region in which it is placed. I can conceive of no other answer. An example of such a length inalienably associated with a region is the directed radius. The long and short of it is that when the standard metre takes up a new position or direction it measures itself against the directed radius of the world in that region and direction, and takes up an extension which is a definite fraction of the directed radius. I do not see what else it could do. We picture the rod a little bewildered in its new surroundings wondering how large it ought to be—how much of the unfamiliar territory its boundaries ought to take in. It wants to do just what it did before. Recollections of the chunk of space that it formerly filled do not help, because there is nothing of the nature of
- 41. a landmark. Theone thing it can recognise is a directed length belonging to the region where it finds itself; so it makes itself the same fraction of this directed length as it did before. If the standard metre is always the same fraction of the directed radius, the directed radius is always the same number of metres. Accordingly the directed radius is made out to have the same length for all positions and directions. Hence we have the law of gravitation. When we felt surprise at finding as a law of Nature that the directed radius of curvature was the same for all positions and directions, we did not realise that our unit of length had already made itself a constant fraction of the directed radius. The whole thing is a vicious circle. The law of gravitation is—a put-up job. This explanation introduces no new hypothesis. In saying that a material system of standard specification always occupies a constant fraction of the directed radius of the region where it is, we are simply reiterating Einstein’s law of gravitation—stating it in the inverse form. Leaving aside for the moment the question whether this behaviour of the rod is to be expected or not, the law of gravitation assures us that that is the behaviour. To see the force of the explanation we must, however, realise the relativity of extension. Extension which is not relative to something in the surroundings has no meaning. Imagine yourself alone in the midst of nothingness, and then try to tell me how large you are. The definiteness of extension of the standard rod can only be a definiteness of its ratio to some other extension. But we are speaking now of the extension of a rod placed in empty space, so that every standard of reference has been removed except extensions belonging to and implied by the metric of the region. It follows that one such extension must appear from our measurements to be constant everywhere (homogeneous and isotropic) on account of its constant relation to what we have accepted as the unit of length. We approached the problem from the point of view that the actual world with its ten vanishing coefficients of curvature (or its
- 42. isotropic directed curvature)has a specialisation which requires explanation; we were then comparing it in our minds with a world suggested by the pure mathematician which has entirely arbitrary curvature. But the fact is that a world of arbitrary curvature is a sheer impossibility. If not the directed radius, then some other directed length derivable from the metric, is bound to be homogeneous and isotropic. In applying the ideas of the pure mathematician we overlooked the fact that he was imagining a world surveyed from outside with standards foreign to it, whereas we have to do with a world surveyed from within with standards conformable to it. The explanation of the law of gravitation thus lies in the fact that we are dealing with a world surveyed from within. From this broader standpoint the foregoing argument can be generalised so that it applies not only to a survey with metre rods but to a survey by optical methods, which in practice are generally substituted as equivalent. When we recollect that surveying apparatus can have no extension in itself but only in relation to the world, so that a survey of space is virtually a self-comparison of space, it is perhaps surprising that such a self-comparison should be able to show up any heterogeneity at all. It can in fact be proved that the metric of a two-dimensional or a three-dimensional world surveyed from within is necessarily uniform. With four or more dimensions heterogeneity becomes possible, but it is a heterogeneity limited by a law which imposes some measure of homogeneity. I believe that this has a close bearing on the rather heterodox views of Dr. Whitehead on relativity. He breaks away from Einstein because he will not admit the non-uniformity of space-time involved in Einstein’s theory. “I deduce that our experience requires and exhibits a basis of uniformity, and that in the case of nature this basis exhibits itself as the uniformity of spatio-temporal relations. This conclusion entirely cuts away the casual heterogeneity of these relations which is the essential of Einstein’s later theory.”[22] But we now see that Einstein’s theory asserts a casual heterogeneity of only
- 43. one set often coefficients and complete uniformity of the other ten. It therefore does not leave us without the basis of uniformity of which Whitehead in his own way perceived the necessity. Moreover, this uniformity is not the result of a law casually imposed on the world; it is inseparable from the conception of survey of the world from within—which is, I think, just the condition that Whitehead would demand. If the world of space-time had been of two or of three dimensions Whitehead would have been entirely right; but then there could have been no Einstein theory of gravitation for him to criticise. Space-time being four-dimensional, we must conclude that Whitehead discovered an important truth about uniformity but misapplied it. The conclusion that the extension of an object in any direction in the four-dimensional world is determined by comparison with the radius of curvature in that direction has one curious consequence. So long as the direction in the four-dimensional world is space-like, no difficulty arises. But when we pass over to time-like directions (within the cone of absolute past or future) the directed radius is an imaginary length. Unless the object ignores the warning symbol it has no standard of reference for settling its time extension. It has no standard duration. An electron decides how large it ought to be by measuring itself against the radius of the world in its space- directions. It cannot decide how long it ought to exist because there is no real radius of the world in its time-direction. Therefore it just goes on existing indefinitely. This is not intended to be a rigorous proof of the immortality of the electron—subject always to the condition imposed throughout these arguments that no agency other than metric interferes with the extension. But it shows that the electron behaves in the simple way which we might at least hope to find.[23] Predictions from the Law. I suppose that it is at first rather staggering to find a law supposed to control the movements of stars and planets turned into a law finicking with the behaviour of
- 44. measuring rods. Butthere is no prediction made by the law of gravitation in which the behaviour of measuring appliances does not play an essential part. A typical prediction from the law is that on a certain date 384,400,000 metre rods laid end to end would stretch from the earth to the moon. We may use more circumlocutory language, but that is what is meant. The fact that in testing the prediction we shall trust to indirect evidence, not carrying out the whole operation literally, is not relevant; the prophecy is made in good faith and not with the intention of taking advantage of our remissness in checking it. We have condemned the law of gravitation as a put-up job. You will want to know how after such a discreditable exposure it can still claim to predict eclipses and other events which come off. A famous philosopher has said— “The stars are not pulled this way and that by mechanical forces; theirs is a free motion. They go on their way, as the ancients said, like the blessed gods.”[24] This sounds particularly foolish even for a philosopher; but I believe that there is a sense in which it is true. We have already had three versions of what the earth is trying to do when it describes its elliptic orbit around the sun. (1) It is trying to go in a straight line but it is roughly pulled away by a tug emanating from the sun. (2) It is taking the longest possible route through the curved space-time around the sun. (3) It is accommodating its track so as to avoid causing any illegal kind of curvature in the empty space around it. We now add a fourth version. (4) The earth goes anyhow it likes. It is not a long step from the third version to the fourth now that we have seen that the mathematical picture of empty space
- 45. containing “illegal” curvatureis a sheer impossibility in a world surveyed from within. For if illegal curvature is a sheer impossibility the earth will not have to take any special precautions to avoid causing it, and can do anything it likes. And yet the non-occurrence of this impossible curvature is the law (of gravitation) by which we calculate the track of the earth! The key to the paradox is that we ourselves, our conventions, the kind of thing that attracts our interest, are much more concerned than we realise in any account we give of how the objects of the physical world are behaving. And so an object which, viewed through our frame of conventions, may seem to be behaving in a very special and remarkable way may, viewed according to another set of conventions, be doing nothing to excite particular comment. This will be clearer if we consider a practical illustration, and at the same time defend version (4). You will say that the earth must certainly get into the right position for the eclipse next June (1927); so it cannot be free to go anywhere it pleases. I can put that right. I hold to it that the earth goes anywhere it pleases. The next thing is that we must find out where it has been pleased to go. The important question for us is not where the earth has got to in the inscrutable absolute behind the phenomena, but where we shall locate it in our conventional background of space and time.
- 46. Fig. 6 We musttake measurements of its position, for example, measurements of its distance from the sun. In Fig. 6, shows the ridge in the world which we recognise as the sun; I have drawn the earth’s ridge in duplicate because I imagine it as still undecided which track it will take. If it takes we lay our measuring rods end to end down the ridges and across the valley from to , count up the number, and report the result as the earth’s distance from the sun. The measuring rods, you will remember, adjust their lengths proportionately to the radius of curvature of the world. The curvature along this contour is rather large and the radius of curvature small. The rods therefore are small, and there will be more of them in than the picture would lead you to expect. If the earth chooses to go to the curvature is less sharp; the greater radius of curvature implies greater length of the rods. The number needed to stretch from to will not be so great as the diagram at first suggests; it will not be increased in anything like the proportion of to in the figure. We should not be surprised if the number turned out to be the same in both cases. If so, the surveyor will report the same distance of the earth from the sun whether the track is or . And the Superintendent of the Nautical Almanac who published this same
- 47. distance some yearsin advance will claim that he correctly predicted where the earth would go. And so you see that the earth can play truant to any extent but our measurements will still report it in the place assigned to it by the Nautical Almanac. The predictions of that authority pay no attention to the vagaries of the god-like earth; they are based on what will happen when we come to measure up the path that it has chosen. We shall measure it with rods that adjust themselves to the curvature of the world. The mathematical expression of this fact is the law of gravitation used in the predictions. Perhaps you will object that astronomers do not in practice lay measuring rods end to end through interplanetary space in order to find out where the planets are. Actually the position is deduced from the light rays. But the light as it proceeds has to find out what course to take in order to go “straight”, in much the same way as the metre rod has to find out how far to extend. The metric or curvature is a sign-post for the light as it is a gauge for the rod. The light track is in fact controlled by the curvature in such a way that it is incapable of exposing the sham law of curvature. And so wherever the sun, moon and earth may have got to, the light will not give them away. If the law of curvature predicts an eclipse the light will take such a track that there is an eclipse. The law of gravitation is not a stern ruler controlling the heavenly bodies; it is a kind-hearted accomplice who covers up their delinquencies. I do not recommend you to try to verify from Fig. 6 that the number of rods in (full line) and (dotted line) is the same. There are two dimensions of space-time omitted in the picture besides the extra dimensions in which space-time must be supposed to be bent; moreover it is the spherical, not the cylindrical, curvature which is the gauge for the length. It might be an instructive, though very laborious, task to make this direct verification, but we know beforehand that the measured distance of the earth from the sun must be the same for either track. The law of gravitation, expressed mathematically by , means nothing more nor less than
- 48. that the unitof length everywhere is a constant fraction of the directed radius of the world at that point. And as the astronomer who predicts the future position of the earth does not assume anything more about what the earth will choose to do than is expressed in the law so we shall find the same position of the earth, if we assume nothing more than that the practical unit of length involved in measurements of the position is a constant fraction of the directed radius. We do not need to decide whether the track is to be represented by or , and it would convey no information as to any observable phenomena if we knew the representation. I shall have to emphasise elsewhere that the whole of our physical knowledge is based on measures and that the physical world consists, so to speak, of measure-groups resting on a shadowy background that lies outside the scope of physics. Therefore in conceiving a world which had existence apart from the measurements that we make of it, I was trespassing outside the limits of what we call physical reality. I would not dissent from the view that a vagary which by its very nature could not be measurable has no claim to a physical existence. No one knows what is meant by such a vagary. I said that the earth might go anywhere it chose, but did not provide a “where” for it to choose; since our conception of “where” is based on space measurements which were at that stage excluded. But I do not think I have been illogical. I am urging that, do what it will, the earth cannot get out of the track laid down for it by the law of gravitation. In order to show this I must suppose that the earth has made the attempt and stolen nearer to the sun; then I show that our measures conspire quietly to locate it back in its proper orbit. I have to admit in the end that the earth never was out of its proper orbit;[25] I do not mind that, because meanwhile I have proved my point. The fact that a predictable path through space and time is laid down for the earth is not a genuine restriction on its conduct, but is imposed by the formal scheme in which we draw up our account of its conduct.
- 49. Non-Empty Space. Thelaw that the directed radius is constant does not apply to space which is not completely empty. There is no longer any reason to expect it to hold. The statement that the region is not empty means that it has other characteristics besides metric, and the metre rod can then find other lengths besides curvatures to measure itself against. Referring to the earlier (sufficiently approximate) expression of the law, the ten principal coefficients of curvature are zero in empty space but have non-zero values in non- empty space. It is therefore natural to use these coefficients as a measure of the fullness of space. One of the coefficients corresponds to mass (or energy) and in most practical cases it outweighs the others in importance. The old definition of mass as “quantity of matter” associates it with a fullness of space. Three other coefficients make up the momentum—a directed quantity with three independent components. The remaining six coefficients of principal curvature make up the stress or pressure-system. Mass, momentum and stress accordingly represent the non-emptiness of a region in so far as it is able to disturb the usual surveying apparatus with which we explore space— clocks, scales, light-rays, etc. It should be added, however, that this is a summary description and not a full account of the non- emptiness, because we have other exploring apparatus—magnets, electroscopes, etc.—which provide further details. It is usually considered that when we use these we are exploring not space, but a field in space. The distinction thus created is a rather artificial one which is unlikely to be accepted permanently. It would seem that the results of exploring the world with a measuring scale and a magnetic compass respectively ought to be welded together into a unified description, just as we have welded together results of exploration with a scale and a clock. Some progress has been made towards this unification. There is, however, a real reason for admitting a partially separate treatment; the one mode of exploration determines the symmetrical properties and the other the antisymmetrical properties of the underlying world-structure.[26]