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PREFACE
(December, 1916)
The present book is intended, as far as possible, to give an exact
insight into the theory of Relativity to those readers who, from a
general scientific and philosophical point of view, are interested in
the theory, but who are not conversant with the mathematical apparatus
of theoretical physics. The work presumes a standard of education
corresponding to that of a university matriculation examination, and,
despite the shortness of the book, a fair amount of patience and force
of will on the part of the reader. The author has spared himself no
pains in his endeavour to present the main ideas in the simplest and
most intelligible form, and on the whole, in the sequence and
connection in which they actually originated. In the interest of
clearness, it appeared to me inevitable that I should repeat myself
frequently, without paying the slightest attention to the elegance of
the presentation. I adhered scrupulously to the precept of that
brilliant theoretical physicist L. Boltzmann, according to whom
matters of elegance ought to be left to the tailor and to the cobbler.
I make no pretence of having withheld from the reader difficulties
which are inherent to the subject. On the other hand, I have purposely
treated the empirical physical foundations of the theory in a
"step-motherly" fashion, so that readers unfamiliar with physics may
not feel like the wanderer who was unable to see the forest for the
trees. May the book bring some one a few happy hours of suggestive
thought!
December, 1916
A. EINSTEIN
PART I
THE SPECIAL THEORY OF RELATIVITY
PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
In your schooldays most of you who read this book made acquaintance
with the noble building of Euclid's geometry, and you remember --
perhaps with more respect than love -- the magnificent structure, on
the lofty staircase of which you were chased about for uncounted hours
by conscientious teachers. By reason of our past experience, you would
certainly regard everyone with disdain who should pronounce even the
most out-of-the-way proposition of this science to be untrue. But
perhaps this feeling of proud certainty would leave you immediately if
some one were to ask you: "What, then, do you mean by the assertion
that these propositions are true?" Let us proceed to give this
question a little consideration.
Geometry sets out form certain conceptions such as "plane," "point,"
and "straight line," with which we are able to associate more or less
definite ideas, and from certain simple propositions (axioms) which,
in virtue of these ideas, we are inclined to accept as "true." Then,
on the basis of a logical process, the justification of which we feel
ourselves compelled to admit, all remaining propositions are shown to
follow from those axioms, i.e. they are proven. A proposition is then
correct ("true") when it has been derived in the recognised manner
from the axioms. The question of "truth" of the individual geometrical
propositions is thus reduced to one of the "truth" of the axioms. Now
it has long been known that the last question is not only unanswerable
by the methods of geometry, but that it is in itself entirely without
meaning. We cannot ask whether it is true that only one straight line
goes through two points. We can only say that Euclidean geometry deals
with things called "straight lines," to each of which is ascribed the
property of being uniquely determined by two points situated on it.
The concept "true" does not tally with the assertions of pure
geometry, because by the word "true" we are eventually in the habit of
designating always the correspondence with a "real" object; geometry,
however, is not concerned with the relation of the ideas involved in
it to objects of experience, but only with the logical connection of
these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel
constrained to call the propositions of geometry "true." Geometrical
ideas correspond to more or less exact objects in nature, and these
last are undoubtedly the exclusive cause of the genesis of those
ideas. Geometry ought to refrain from such a course, in order to give
to its structure the largest possible logical unity. The practice, for
example, of seeing in a "distance" two marked positions on a
practically rigid body is something which is lodged deeply in our
habit of thought. We are accustomed further to regard three points as
being situated on a straight line, if their apparent positions can be
made to coincide for observation with one eye, under suitable choice
of our place of observation.
If, in pursuance of our habit of thought, we now supplement the
propositions of Euclidean geometry by the single proposition that two
points on a practically rigid body always correspond to the same
distance (line-interval), independently of any changes in position to
which we may subject the body, the propositions of Euclidean geometry
then resolve themselves into propositions on the possible relative
position of practically rigid bodies.* Geometry which has been
supplemented in this way is then to be treated as a branch of physics.
We can now legitimately ask as to the "truth" of geometrical
propositions interpreted in this way, since we are justified in asking
whether these propositions are satisfied for those real things we have
associated with the geometrical ideas. In less exact terms we can
express this by saying that by the "truth" of a geometrical
proposition in this sense we understand its validity for a
construction with rule and compasses.
Of course the conviction of the "truth" of geometrical propositions in
this sense is founded exclusively on rather incomplete experience. For
the present we shall assume the "truth" of the geometrical
propositions, then at a later stage (in the general theory of
relativity) we shall see that this "truth" is limited, and we shall
consider the extent of its limitation.
Notes
*) It follows that a natural object is associated also with a
straight line. Three points A, B and C on a rigid body thus lie in a
straight line when the points A and C being given, B is chosen such
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Even though classical mechanics does not supply us with a sufficiently
broad basis for the theoretical presentation of all physical
phenomena, still we must grant it a considerable measure of " truth,"
since it supplies us with the actual motions of the heavenly bodies
with a delicacy of detail little short of wonderful. The principle of
relativity must therefore apply with great accuracy in the domain of
mechanics. But that a principle of such broad generality should hold
with such exactness in one domain of phenomena, and yet should be
invalid for another, is a priori not very probable.
We now proceed to the second argument, to which, moreover, we shall
return later. If the principle of relativity (in the restricted sense)
does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
which are moving uniformly relative to each other, will not be
equivalent for the description of natural phenomena. In this case we
should be constrained to believe that natural laws are capable of
being formulated in a particularly simple manner, and of course only
on condition that, from amongst all possible Galileian co-ordinate
systems, we should have chosen one (K[0]) of a particular state of
motion as our body of reference. We should then be justified (because
of its merits for the description of natural phenomena) in calling
this system " absolutely at rest," and all other Galileian systems K "
in motion." If, for instance, our embankment were the system K[0] then
our railway carriage would be a system K, relative to which less
simple laws would hold than with respect to K[0]. This diminished
simplicity would be due to the fact that the carriage K would be in
motion (i.e."really")with respect to K[0]. In the general laws of
nature which have been formulated with reference to K, the magnitude
and direction of the velocity of the carriage would necessarily play a
part. We should expect, for instance, that the note emitted by an
organpipe placed with its axis parallel to the direction of travel
would be different from that emitted if the axis of the pipe were
placed perpendicular to this direction.
Now in virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of about
30 kilometres per second. If the principle of relativity were not
valid we should therefore expect that the direction of motion of the
earth at any moment would enter into the laws of nature, and also that
physical systems in their behaviour would be dependent on the
orientation in space with respect to the earth. For owing to the
alteration in direction of the velocity of revolution of the earth in
the course of a year, the earth cannot be at rest relative to the
hypothetical system K[0] throughout the whole year. However, the most
careful observations have never revealed such anisotropic properties
in terrestrial physical space, i.e. a physical non-equivalence of
different directions. This is very powerful argument in favour of the
principle of relativity.
THE THEOREM OF THE
ADDITION OF VELOCITIES
EMPLOYED IN CLASSICAL MECHANICS
Let us suppose our old friend the railway carriage to be travelling
along the rails with a constant velocity v, and that a man traverses
the length of the carriage in the direction of travel with a velocity
w. How quickly or, in other words, with what velocity W does the man
advance relative to the embankment during the process ? The only
possible answer seems to result from the following consideration: If
the man were to stand still for a second, he would advance relative to
the embankment through a distance v equal numerically to the velocity
of the carriage. As a consequence of his walking, however, he
traverses an additional distance w relative to the carriage, and hence
also relative to the embankment, in this second, the distance w being
numerically equal to the velocity with which he is walking. Thus in
total be covers the distance W=v+w relative to the embankment in the
second considered. We shall see later that this result, which
expresses the theorem of the addition of velocities employed in
classical mechanics, cannot be maintained ; in other words, the law
that we have just written down does not hold in reality. For the time
being, however, we shall assume its correctness.
THE APPARENT INCOMPATIBILITY OF THE
LAW OF PROPAGATION OF LIGHT WITH THE
PRINCIPLE OF RELATIVITY
There is hardly a simpler law in physics than that according to which
light is propagated in empty space. Every child at school knows, or
believes he knows, that this propagation takes place in straight lines
with a velocity c= 300,000 km./sec. At all events we know with great
exactness that this velocity is the same for all colours, because if
this were not the case, the minimum of emission would not be observed
simultaneously for different colours during the eclipse of a fixed
star by its dark neighbour. By means of similar considerations based
on observa- tions of double stars, the Dutch astronomer De Sitter was
also able to show that the velocity of propagation of light cannot
depend on the velocity of motion of the body emitting the light. The
assumption that this velocity of propagation is dependent on the
direction "in space" is in itself improbable.
In short, let us assume that the simple law of the constancy of the
velocity of light c (in vacuum) is justifiably believed by the child
at school. Who would imagine that this simple law has plunged the
conscientiously thoughtful physicist into the greatest intellectual
difficulties? Let us consider how these difficulties arise.
Of course we must refer the process of the propagation of light (and
indeed every other process) to a rigid reference-body (co-ordinate
system). As such a system let us again choose our embankment. We shall
imagine the air above it to have been removed. If a ray of light be
sent along the embankment, we see from the above that the tip of the
ray will be transmitted with the velocity c relative to the
embankment. Now let us suppose that our railway carriage is again
travelling along the railway lines with the velocity v, and that its
direction is the same as that of the ray of light, but its velocity of
course much less. Let us inquire about the velocity of propagation of
the ray of light relative to the carriage. It is obvious that we can
here apply the consideration of the previous section, since the ray of
light plays the part of the man walking along relatively to the
carriage. The velocity w of the man relative to the embankment is here
replaced by the velocity of light relative to the embankment. w is the
required velocity of light with respect to the carriage, and we have
w = c-v.
The velocity of propagation ot a ray of light relative to the carriage
thus comes cut smaller than c.
But this result comes into conflict with the principle of relativity
set forth in Section V. For, like every other general law of
nature, the law of the transmission of light in vacuo [in vacuum]
must, according to the principle of relativity, be the same for the
railway carriage as reference-body as when the rails are the body of
reference. But, from our above consideration, this would appear to be
impossible. If every ray of light is propagated relative to the
embankment with the velocity c, then for this reason it would appear
that another law of propagation of light must necessarily hold with
respect to the carriage -- a result contradictory to the principle of
relativity.
In view of this dilemma there appears to be nothing else for it than
to abandon either the principle of relativity or the simple law of the
propagation of light in vacuo. Those of you who have carefully
followed the preceding discussion are almost sure to expect that we
should retain the principle of relativity, which appeals so
convincingly to the intellect because it is so natural and simple. The
law of the propagation of light in vacuo would then have to be
replaced by a more complicated law conformable to the principle of
relativity. The development of theoretical physics shows, however,
that we cannot pursue this course. The epoch-making theoretical
investigations of H. A. Lorentz on the electrodynamical and optical
phenomena connected with moving bodies show that experience in this
domain leads conclusively to a theory of electromagnetic phenomena, of
which the law of the constancy of the velocity of light in vacuo is a
necessary consequence. Prominent theoretical physicists were theref
ore more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were
contradictory to this principle.
At this juncture the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space,
it became evident that in realily there is not the least
incompatibilitiy between the principle of relativity and the law of
propagation of light, and that by systematically holding fast to both
these laws a logically rigid theory could be arrived at. This theory
has been called the special theory of relativity to distinguish it
from the extended theory, with which we shall deal later. In the
following pages we shall present the fundamental ideas of the special
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velocity of light relative to the liquid and the velocity of the
latter relative to the tube are thus known, and we require the
velocity of light relative to the tube.
It is clear that we have the problem of Section 6 again before us. The
tube plays the part of the railway embankment or of the co-ordinate
system K, the liquid plays the part of the carriage or of the
co-ordinate system K1, and finally, the light plays the part of the
Figure 03: file fig03.gif
man walking along the carriage, or of the moving point in the present
section. If we denote the velocity of the light relative to the tube
by W, then this is given by the equation (A) or (B), according as the
Galilei transformation or the Lorentz transformation corresponds to
the facts. Experiment * decides in favour of equation (B) derived
from the theory of relativity, and the agreement is, indeed, very
exact. According to recent and most excellent measurements by Zeeman,
the influence of the velocity of flow v on the propagation of light is
represented by formula (B) to within one per cent.
Nevertheless we must now draw attention to the fact that a theory of
this phenomenon was given by H. A. Lorentz long before the statement
of the theory of relativity. This theory was of a purely
electrodynamical nature, and was obtained by the use of particular
hypotheses as to the electromagnetic structure of matter. This
circumstance, however, does not in the least diminish the
conclusiveness of the experiment as a crucial test in favour of the
theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
which the original theory was based, in no way opposes the theory of
relativity. Rather has the latter been developed trom electrodynamics
as an astoundingly simple combination and generalisation of the
hypotheses, formerly independent of each other, on which
electrodynamics was built.
Notes
*) Fizeau found eq. 10 , where eq. 11
is the index of refraction of the liquid. On the other hand, owing to
the smallness of eq. 12 as compared with I,
we can replace (B) in the first place by eq. 13 , or to the same order
of approximation by
eq. 14 , which agrees with Fizeau's result.
THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
Our train of thought in the foregoing pages can be epitomised in the
following manner. Experience has led to the conviction that, on the
one hand, the principle of relativity holds true and that on the other
hand the velocity of transmission of light in vacuo has to be
considered equal to a constant c. By uniting these two postulates we
obtained the law of transformation for the rectangular co-ordinates x,
y, z and the time t of the events which constitute the processes of
nature. In this connection we did not obtain the Galilei
transformation, but, differing from classical mechanics, the Lorentz
transformation.
The law of transmission of light, the acceptance of which is justified
by our actual knowledge, played an important part in this process of
thought. Once in possession of the Lorentz transformation, however, we
can combine this with the principle of relativity, and sum up the
theory thus:
Every general law of nature must be so constituted that it is
transformed into a law of exactly the same form when, instead of the
space-time variables x, y, z, t of the original coordinate system K,
we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate
system K1. In this connection the relation between the ordinary and
the accented magnitudes is given by the Lorentz transformation. Or in
brief : General laws of nature are co-variant with respect to Lorentz
transformations.
This is a definite mathematical condition that the theory of
relativity demands of a natural law, and in virtue of this, the theory
becomes a valuable heuristic aid in the search for general laws of
nature. If a general law of nature were to be found which did not
satisfy this condition, then at least one of the two fundamental
assumptions of the theory would have been disproved. Let us now
examine what general results the latter theory has hitherto evinced.
GENERAL RESULTS OF THE THEORY
It is clear from our previous considerations that the (special) theory
of relativity has grown out of electrodynamics and optics. In these
fields it has not appreciably altered the predictions of theory, but
it has considerably simplified the theoretical structure, i.e. the
derivation of laws, and -- what is incomparably more important -- it
has considerably reduced the number of independent hypothese forming
the basis of theory. The special theory of relativity has rendered the
Maxwell-Lorentz theory so plausible, that the latter would have been
generally accepted by physicists even if experiment had decided less
unequivocally in its favour.
Classical mechanics required to be modified before it could come into
line with the demands of the special theory of relativity. For the
main part, however, this modification affects only the laws for rapid
motions, in which the velocities of matter v are not very small as
compared with the velocity of light. We have experience of such rapid
motions only in the case of electrons and ions; for other motions the
variations from the laws of classical mechanics are too small to make
themselves evident in practice. We shall not consider the motion of
stars until we come to speak of the general theory of relativity. In
accordance with the theory of relativity the kinetic energy of a
material point of mass m is no longer given by the well-known
expression
eq. 15: file eq15.gif
but by the expression
eq. 16: file eq16.gif
This expression approaches infinity as the velocity v approaches the
velocity of light c. The velocity must therefore always remain less
than c, however great may be the energies used to produce the
acceleration. If we develop the expression for the kinetic energy in
the form of a series, we obtain
eq. 17: file eq17.gif
When eq. 18 is small compared with unity, the third of these terms is
always small in comparison with the second,
which last is alone considered in classical mechanics. The first term
mc^2 does not contain the velocity, and requires no consideration if
we are only dealing with the question as to how the energy of a
point-mass; depends on the velocity. We shall speak of its essential
significance later.
The most important result of a general character to which the special
theory of relativity has led is concerned with the conception of mass.
Before the advent of relativity, physics recognised two conservation
laws of fundamental importance, namely, the law of the canservation of
energy and the law of the conservation of mass these two fundamental
laws appeared to be quite independent of each other. By means of the
theory of relativity they have been united into one law. We shall now
briefly consider how this unification came about, and what meaning is
to be attached to it.
The principle of relativity requires that the law of the concervation
of energy should hold not only with reference to a co-ordinate system
K, but also with respect to every co-ordinate system K1 which is in a
state of uniform motion of translation relative to K, or, briefly,
relative to every " Galileian " system of co-ordinates. In contrast to
classical mechanics; the Lorentz transformation is the deciding factor
in the transition from one such system to another.
By means of comparatively simple considerations we are led to draw the
following conclusion from these premises, in conjunction with the
fundamental equations of the electrodynamics of Maxwell: A body moving
with the velocity v, which absorbs * an amount of energy E[0] in
the form of radiation without suffering an alteration in velocity in
the process, has, as a consequence, its energy increased by an amount
eq. 19: file eq19.gif
In consideration of the expression given above for the kinetic energy
of the body, the required energy of the body comes out to be
eq. 20: file eq20.gif
Thus the body has the same energy as a body of mass
eq.21: file eq21.gif
moving with the velocity v. Hence we can say: If a body takes up an
amount of energy E[0], then its inertial mass increases by an amount
eq. 22: file eq22.gif
the inertial mass of a body is not a constant but varies according to
the change in the energy of the body. The inertial mass of a system of
bodies can even be regarded as a measure of its energy. The law of the
conservation of the mass of a system becomes identical with the law of
the conservation of energy, and is only valid provided that the system
neither takes up nor sends out energy. Writing the expression for the
energy in the form
eq. 23: file eq23.gif
we see that the term mc^2, which has hitherto attracted our attention,
is nothing else than the energy possessed by the body ** before it
absorbed the energy E[0].
A direct comparison of this relation with experiment is not possible
at the present time (1920; see *** Note, p. 48), owing to the fact that
the changes in energy E[0] to which we can Subject a system are not
large enough to make themselves perceptible as a change in the
inertial mass of the system.
eq. 22: file eq22.gif
is too small in comparison with the mass m, which was present before
the alteration of the energy. It is owing to this circumstance that
classical mechanics was able to establish successfully the
conservation of mass as a law of independent validity.
Let me add a final remark of a fundamental nature. The success of the
Faraday-Maxwell interpretation of electromagnetic action at a distance
resulted in physicists becoming convinced that there are no such
things as instantaneous actions at a distance (not involving an
intermediary medium) of the type of Newton's law of gravitation.
According to the theory of relativity, action at a distance with the
velocity of light always takes the place of instantaneous action at a
distance or of action at a distance with an infinite velocity of
transmission. This is connected with the fact that the velocity c
plays a fundamental role in this theory. In Part II we shall see in
what way this result becomes modified in the general theory of
relativity.
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each other, the negative electrical masses constituting the electron
would necessarily be scattered under the influence of their mutual
repulsions, unless there are forces of another kind operating between
them, the nature of which has hitherto remained obscure to us.* If
we now assume that the relative distances between the electrical
masses constituting the electron remain unchanged during the motion of
the electron (rigid connection in the sense of classical mechanics),
we arrive at a law of motion of the electron which does not agree with
experience. Guided by purely formal points of view, H. A. Lorentz was
the first to introduce the hypothesis that the form of the electron
experiences a contraction in the direction of motion in consequence of
that motion. the contracted length being proportional to the
expression
eq. 05: file eq05.gif
This, hypothesis, which is not justifiable by any electrodynamical
facts, supplies us then with that particular law of motion which has
been confirmed with great precision in recent years.
The theory of relativity leads to the same law of motion, without
requiring any special hypothesis whatsoever as to the structure and
the behaviour of the electron. We arrived at a similar conclusion in
Section 13 in connection with the experiment of Fizeau, the result
of which is foretold by the theory of relativity without the necessity
of drawing on hypotheses as to the physical nature of the liquid.
The second class of facts to which we have alluded has reference to
the question whether or not the motion of the earth in space can be
made perceptible in terrestrial experiments. We have already remarked
in Section 5 that all attempts of this nature led to a negative
result. Before the theory of relativity was put forward, it was
difficult to become reconciled to this negative result, for reasons
now to be discussed. The inherited prejudices about time and space did
not allow any doubt to arise as to the prime importance of the
Galileian transformation for changing over from one body of reference
to another. Now assuming that the Maxwell-Lorentz equations hold for a
reference-body K, we then find that they do not hold for a
reference-body K1 moving uniformly with respect to K, if we assume
that the relations of the Galileian transformstion exist between the
co-ordinates of K and K1. It thus appears that, of all Galileian
co-ordinate systems, one (K) corresponding to a particular state of
motion is physically unique. This result was interpreted physically by
regarding K as at rest with respect to a hypothetical æther of space.
On the other hand, all coordinate systems K1 moving relatively to K
were to be regarded as in motion with respect to the æther. To this
motion of K1 against the æther ("æther-drift " relative to K1) were
attributed the more complicated laws which were supposed to hold
relative to K1. Strictly speaking, such an æther-drift ought also to
be assumed relative to the earth, and for a long time the efforts of
physicists were devoted to attempts to detect the existence of an
æther-drift at the earth's surface.
In one of the most notable of these attempts Michelson devised a
method which appears as though it must be decisive. Imagine two
mirrors so arranged on a rigid body that the reflecting surfaces face
each other. A ray of light requires a perfectly definite time T to
pass from one mirror to the other and back again, if the whole system
be at rest with respect to the æther. It is found by calculation,
however, that a slightly different time T1 is required for this
process, if the body, together with the mirrors, be moving relatively
to the æther. And yet another point: it is shown by calculation that
for a given velocity v with reference to the æther, this time T1 is
different when the body is moving perpendicularly to the planes of the
mirrors from that resulting when the motion is parallel to these
planes. Although the estimated difference between these two times is
exceedingly small, Michelson and Morley performed an experiment
involving interference in which this difference should have been
clearly detectable. But the experiment gave a negative result -- a
fact very perplexing to physicists. Lorentz and FitzGerald rescued the
theory from this difficulty by assuming that the motion of the body
relative to the æther produces a contraction of the body in the
direction of motion, the amount of contraction being just sufficient
to compensate for the differeace in time mentioned above. Comparison
with the discussion in Section 11 shows that also from the
standpoint of the theory of relativity this solution of the difficulty
was the right one. But on the basis of the theory of relativity the
method of interpretation is incomparably more satisfactory. According
to this theory there is no such thing as a " specially favoured "
(unique) co-ordinate system to occasion the introduction of the
æther-idea, and hence there can be no æther-drift, nor any experiment
with which to demonstrate it. Here the contraction of moving bodies
follows from the two fundamental principles of the theory, without the
introduction of particular hypotheses ; and as the prime factor
involved in this contraction we find, not the motion in itself, to
which we cannot attach any meaning, but the motion with respect to the
body of reference chosen in the particular case in point. Thus for a
co-ordinate system moving with the earth the mirror system of
Michelson and Morley is not shortened, but it is shortened for a
co-ordinate system which is at rest relatively to the sun.
Notes
*) The general theory of relativity renders it likely that the
electrical masses of an electron are held together by gravitational
forces.
MINKOWSKI'S FOUR-DIMENSIONAL SPACE
The non-mathematician is seized by a mysterious shuddering when he
hears of "four-dimensional" things, by a feeling not unlike that
awakened by thoughts of the occult. And yet there is no more
common-place statement than that the world in which we live is a
four-dimensional space-time continuum.
Space is a three-dimensional continuum. By this we mean that it is
possible to describe the position of a point (at rest) by means of
three numbers (co-ordinales) x, y, z, and that there is an indefinite
number of points in the neighbourhood of this one, the position of
which can be described by co-ordinates such as x[1], y[1], z[1], which
may be as near as we choose to the respective values of the
co-ordinates x, y, z, of the first point. In virtue of the latter
property we speak of a " continuum," and owing to the fact that there
are three co-ordinates we speak of it as being " three-dimensional."
Similarly, the world of physical phenomena which was briefly called "
world " by Minkowski is naturally four dimensional in the space-time
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the following pages, would perhaps have got no farther than its long
clothes. Minkowski's work is doubtless difficult of access to anyone
inexperienced in mathematics, but since it is not necessary to have a
very exact grasp of this work in order to understand the fundamental
ideas of either the special or the general theory of relativity, I
shall leave it here at present, and revert to it only towards the end
of Part 2.
Notes
*) Cf. the somewhat more detailed discussion in Appendix II.
PART II
THE GENERAL THEORY OF RELATIVITY
SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
The basal principle, which was the pivot of all our previous
considerations, was the special principle of relativity, i.e. the
principle of the physical relativity of all uniform motion. Let as
once more analyse its meaning carefully.
It was at all times clear that, from the point of view of the idea it
conveys to us, every motion must be considered only as a relative
motion. Returning to the illustration we have frequently used of the
embankment and the railway carriage, we can express the fact of the
motion here taking place in the following two forms, both of which are
equally justifiable :
(a) The carriage is in motion relative to the embankment,
(b) The embankment is in motion relative to the carriage.
In (a) the embankment, in (b) the carriage, serves as the body of
reference in our statement of the motion taking place. If it is simply
a question of detecting or of describing the motion involved, it is in
principle immaterial to what reference-body we refer the motion. As
already mentioned, this is self-evident, but it must not be confused
with the much more comprehensive statement called "the principle of
relativity," which we have taken as the basis of our investigations.
The principle we have made use of not only maintains that we may
equally well choose the carriage or the embankment as our
reference-body for the description of any event (for this, too, is
self-evident). Our principle rather asserts what follows : If we
formulate the general laws of nature as they are obtained from
experience, by making use of
(a) the embankment as reference-body,
(b) the railway carriage as reference-body,
then these general laws of nature (e.g. the laws of mechanics or the
law of the propagation of light in vacuo) have exactly the same form
in both cases. This can also be expressed as follows : For the
physical description of natural processes, neither of the reference
bodies K, K1 is unique (lit. " specially marked out ") as compared
with the other. Unlike the first, this latter statement need not of
necessity hold a priori; it is not contained in the conceptions of "
motion" and " reference-body " and derivable from them; only
experience can decide as to its correctness or incorrectness.
Up to the present, however, we have by no means maintained the
equivalence of all bodies of reference K in connection with the
formulation of natural laws. Our course was more on the following
Iines. In the first place, we started out from the assumption that
there exists a reference-body K, whose condition of motion is such
that the Galileian law holds with respect to it : A particle left to
itself and sufficiently far removed from all other particles moves
uniformly in a straight line. With reference to K (Galileian
reference-body) the laws of nature were to be as simple as possible.
But in addition to K, all bodies of reference K1 should be given
preference in this sense, and they should be exactly equivalent to K
for the formulation of natural laws, provided that they are in a state
of uniform rectilinear and non-rotary motion with respect to K ; all
these bodies of reference are to be regarded as Galileian
reference-bodies. The validity of the principle of relativity was
assumed only for these reference-bodies, but not for others (e.g.
those possessing motion of a different kind). In this sense we speak
of the special principle of relativity, or special theory of
relativity.
In contrast to this we wish to understand by the "general principle of
relativity" the following statement : All bodies of reference K, K1,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion. But before proceeding farther, it ought to be pointed
out that this formulation must be replaced later by a more abstract
one, for reasons which will become evident at a later stage.
Since the introduction of the special principle of relativity has been
justified, every intellect which strives after generalisation must
feel the temptation to venture the step towards the general principle
of relativity. But a simple and apparently quite reliable
consideration seems to suggest that, for the present at any rate,
there is little hope of success in such an attempt; Let us imagine
ourselves transferred to our old friend the railway carriage, which is
travelling at a uniform rate. As long as it is moving unifromly, the
occupant of the carriage is not sensible of its motion, and it is for
this reason that he can without reluctance interpret the facts of the
case as indicating that the carriage is at rest, but the embankment in
motion. Moreover, according to the special principle of relativity,
this interpretation is quite justified also from a physical point of
view.
If the motion of the carriage is now changed into a non-uniform
motion, as for instance by a powerful application of the brakes, then
the occupant of the carriage experiences a correspondingly powerful
jerk forwards. The retarded motion is manifested in the mechanical
behaviour of bodies relative to the person in the railway carriage.
The mechanical behaviour is different from that of the case previously
considered, and for this reason it would appear to be impossible that
the same mechanical laws hold relatively to the non-uniformly moving
carriage, as hold with reference to the carriage when at rest or in
uniform motion. At all events it is clear that the Galileian law does
not hold with respect to the non-uniformly moving carriage. Because of
this, we feel compelled at the present juncture to grant a kind of
absolute physical reality to non-uniform motion, in opposition to the
general principle of relatvity. But in what follows we shall soon see
that this conclusion cannot be maintained.
THE GRAVITATIONAL FIELD
"If we pick up a stone and then let it go, why does it fall to the
ground ?" The usual answer to this question is: "Because it is
attracted by the earth." Modern physics formulates the answer rather
differently for the following reason. As a result of the more careful
study of electromagnetic phenomena, we have come to regard action at a
distance as a process impossible without the intervention of some
intermediary medium. If, for instance, a magnet attracts a piece of
iron, we cannot be content to regard this as meaning that the magnet
acts directly on the iron through the intermediate empty space, but we
are constrained to imagine -- after the manner of Faraday -- that the
magnet always calls into being something physically real in the space
around it, that something being what we call a "magnetic field." In
its turn this magnetic field operates on the piece of iron, so that
the latter strives to move towards the magnet. We shall not discuss
here the justification for this incidental conception, which is indeed
a somewhat arbitrary one. We shall only mention that with its aid
electromagnetic phenomena can be theoretically represented much more
satisfactorily than without it, and this applies particularly to the
transmission of electromagnetic waves. The effects of gravitation also
are regarded in an analogous manner.
The action of the earth on the stone takes place indirectly. The earth
produces in its surrounding a gravitational field, which acts on the
stone and produces its motion of fall. As we know from experience, the
intensity of the action on a body dimishes according to a quite
definite law, as we proceed farther and farther away from the earth.
From our point of view this means : The law governing the properties
of the gravitational field in space must be a perfectly definite one,
in order correctly to represent the diminution of gravitational action
with the distance from operative bodies. It is something like this:
The body (e.g. the earth) produces a field in its immediate
neighbourhood directly; the intensity and direction of the field at
points farther removed from the body are thence determined by the law
which governs the properties in space of the gravitational fields
themselves.
In contrast to electric and magnetic fields, the gravitational field
exhibits a most remarkable property, which is of fundamental
importance for what follows. Bodies which are moving under the sole
influence of a gravitational field receive an acceleration, which does
not in the least depend either on the material or on the physical
state of the body. For instance, a piece of lead and a piece of wood
fall in exactly the same manner in a gravitational field (in vacuo),
when they start off from rest or with the same initial velocity. This
law, which holds most accurately, can be expressed in a different form
in the light of the following consideration.
According to Newton's law of motion, we have
(Force) = (inertial mass) x (acceleration),
where the "inertial mass" is a characteristic constant of the
accelerated body. If now gravitation is the cause of the acceleration,
we then have
(Force) = (gravitational mass) x (intensity of the gravitational
field),
where the "gravitational mass" is likewise a characteristic constant
for the body. From these two relations follows:
eq. 26: file eq26.gif
If now, as we find from experience, the acceleration is to be
independent of the nature and the condition of the body and always the
t/Relativity.test view on Meta::CPAN
emphasised that this fundamental law can only be valid for bodies of
reference K which possess certain unique states of motion, and which
are in uniform translational motion relative to each other. Relative
to other reference-bodies K the law is not valid. Both in classical
mechanics and in the special theory of relativity we therefore
differentiate between reference-bodies K relative to which the
recognised " laws of nature " can be said to hold, and
reference-bodies K relative to which these laws do not hold.
But no person whose mode of thought is logical can rest satisfied with
this condition of things. He asks : " How does it come that certain
reference-bodies (or their states of motion) are given priority over
other reference-bodies (or their states of motion) ? What is the
reason for this Preference? In order to show clearly what I mean by
this question, I shall make use of a comparison.
I am standing in front of a gas range. Standing alongside of each
other on the range are two pans so much alike that one may be mistaken
for the other. Both are half full of water. I notice that steam is
being emitted continuously from the one pan, but not from the other. I
am surprised at this, even if I have never seen either a gas range or
a pan before. But if I now notice a luminous something of bluish
colour under the first pan but not under the other, I cease to be
astonished, even if I have never before seen a gas flame. For I can
only say that this bluish something will cause the emission of the
steam, or at least possibly it may do so. If, however, I notice the
bluish something in neither case, and if I observe that the one
continuously emits steam whilst the other does not, then I shall
remain astonished and dissatisfied until I have discovered some
circumstance to which I can attribute the different behaviour of the
two pans.
Analogously, I seek in vain for a real something in classical
mechanics (or in the special theory of relativity) to which I can
attribute the different behaviour of bodies considered with respect to
the reference systems K and K1.* Newton saw this objection and
attempted to invalidate it, but without success. But E. Mach recognsed
it most clearly of all, and because of this objection he claimed that
mechanics must be placed on a new basis. It can only be got rid of by
means of a physics which is conformable to the general principle of
relativity, since the equations of such a theory hold for every body
of reference, whatever may be its state of motion.
Notes
*) The objection is of importance more especially when the state of
motion of the reference-body is of such a nature that it does not
require any external agency for its maintenance, e.g. in the case when
the reference-body is rotating uniformly.
A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
The considerations of Section 20 show that the general principle of
relativity puts us in a position to derive properties of the
gravitational field in a purely theoretical manner. Let us suppose,
for instance, that we know the space-time " course " for any natural
process whatsoever, as regards the manner in which it takes place in
the Galileian domain relative to a Galileian body of reference K. By
means of purely theoretical operations (i.e. simply by calculation) we
are then able to find how this known natural process appears, as seen
from a reference-body K1 which is accelerated relatively to K. But
since a gravitational field exists with respect to this new body of
reference K1, our consideration also teaches us how the gravitational
field influences the process studied.
For example, we learn that a body which is in a state of uniform
rectilinear motion with respect to K (in accordance with the law of
Galilei) is executing an accelerated and in general curvilinear motion
with respect to the accelerated reference-body K1 (chest). This
acceleration or curvature corresponds to the influence on the moving
body of the gravitational field prevailing relatively to K. It is
known that a gravitational field influences the movement of bodies in
this way, so that our consideration supplies us with nothing
essentially new.
However, we obtain a new result of fundamental importance when we
carry out the analogous consideration for a ray of light. With respect
to the Galileian reference-body K, such a ray of light is transmitted
rectilinearly with the velocity c. It can easily be shown that the
path of the same ray of light is no longer a straight line when we
consider it with reference to the accelerated chest (reference-body
K1). From this we conclude, that, in general, rays of light are
propagated curvilinearly in gravitational fields. In two respects this
result is of great importance.
In the first place, it can be compared with the reality. Although a
detailed examination of the question shows that the curvature of light
rays required by the general theory of relativity is only exceedingly
small for the gravitational fields at our disposal in practice, its
estimated magnitude for light rays passing the sun at grazing
incidence is nevertheless 1.7 seconds of arc. This ought to manifest
itself in the following way. As seen from the earth, certain fixed
stars appear to be in the neighbourhood of the sun, and are thus
capable of observation during a total eclipse of the sun. At such
times, these stars ought to appear to be displaced outwards from the
sun by an amount indicated above, as compared with their apparent
position in the sky when the sun is situated at another part of the
heavens. The examination of the correctness or otherwise of this
deduction is a problem of the greatest importance, the early solution
of which is to be expected of astronomers.[2]*
In the second place our result shows that, according to the general
theory of relativity, the law of the constancy of the velocity of
light in vacuo, which constitutes one of the two fundamental
assumptions in the special theory of relativity and to which we have
already frequently referred, cannot claim any unlimited validity. A
curvature of rays of light can only take place when the velocity of
propagation of light varies with position. Now we might think that as
a consequence of this, the special theory of relativity and with it
the whole theory of relativity would be laid in the dust. But in
reality this is not the case. We can only conclude that the special
theory of relativity cannot claim an unlinlited domain of validity ;
its results hold only so long as we are able to disregard the
influences of gravitational fields on the phenomena (e.g. of light).
Since it has often been contended by opponents of the theory of
relativity that the special theory of relativity is overthrown by the
general theory of relativity, it is perhaps advisable to make the
facts of the case clearer by means of an appropriate comparison.
Before the development of electrodynamics the laws of electrostatics
were looked upon as the laws of electricity. At the present time we
know that electric fields can be derived correctly from electrostatic
considerations only for the case, which is never strictly realised, in
which the electrical masses are quite at rest relatively to each
other, and to the co-ordinate system. Should we be justified in saying
that for this reason electrostatics is overthrown by the
field-equations of Maxwell in electrodynamics ? Not in the least.
Electrostatics is contained in electrodynamics as a limiting case ;
the laws of the latter lead directly to those of the former for the
case in which the fields are invariable with regard to time. No fairer
destiny could be allotted to any physical theory, than that it should
of itself point out the way to the introduction of a more
comprehensive theory, in which it lives on as a limiting case.
In the example of the transmission of light just dealt with, we have
seen that the general theory of relativity enables us to derive
theoretically the influence of a gravitational field on the course of
natural processes, the Iaws of which are already known when a
gravitational field is absent. But the most attractive problem, to the
solution of which the general theory of relativity supplies the key,
concerns the investigation of the laws satisfied by the gravitational
field itself. Let us consider this for a moment.
We are acquainted with space-time domains which behave (approximately)
in a " Galileian " fashion under suitable choice of reference-body,
i.e. domains in which gravitational fields are absent. If we now refer
such a domain to a reference-body K1 possessing any kind of motion,
then relative to K1 there exists a gravitational field which is
variable with respect to space and time.[3]** The character of this
field will of course depend on the motion chosen for K1. According to
the general theory of relativity, the general law of the gravitational
field must be satisfied for all gravitational fields obtainable in
this way. Even though by no means all gravitationial fields can be
produced in this way, yet we may entertain the hope that the general
law of gravitation will be derivable from such gravitational fields of
a special kind. This hope has been realised in the most beautiful
manner. But between the clear vision of this goal and its actual
realisation it was necessary to surmount a serious difficulty, and as
this lies deep at the root of things, I dare not withhold it from the
reader. We require to extend our ideas of the space-time continuum
still farther.
Notes
*) By means of the star photographs of two expeditions equipped by
a Joint Committee of the Royal and Royal Astronomical Societies, the
existence of the deflection of light demanded by theory was first
confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
III.)
**) This follows from a generalisation of the discussion in
Section 20
BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
Hitherto I have purposely refrained from speaking about the physical
interpretation of space- and time-data in the case of the general
theory of relativity. As a consequence, I am guilty of a certain
slovenliness of treatment, which, as we know from the special theory
of relativity, is far from being unimportant and pardonable. It is now
high time that we remedy this defect; but I would mention at the
outset, that this matter lays no small claims on the patience and on
the power of abstraction of the reader.
We start off again from quite special cases, which we have frequently
used before. Let us consider a space time domain in which no
gravitational field exists relative to a reference-body K whose state
of motion has been suitably chosen. K is then a Galileian
reference-body as regards the domain considered, and the results of
the special theory of relativity hold relative to K. Let us supposse
the same domain referred to a second body of reference K1, which is
rotating uniformly with respect to K. In order to fix our ideas, we
shall imagine K1 to be in the form of a plane circular disc, which
rotates uniformly in its own plane about its centre. An observer who
t/Relativity.test view on Meta::CPAN
circular disc. Thus on our circular disc, or, to make the case more
general, in every gravitational field, a clock will go more quickly or
less quickly, according to the position in which the clock is situated
(at rest). For this reason it is not possible to obtain a reasonable
definition of time with the aid of clocks which are arranged at rest
with respect to the body of reference. A similar difficulty presents
itself when we attempt to apply our earlier definition of simultaneity
in such a case, but I do not wish to go any farther into this
question.
Moreover, at this stage the definition of the space co-ordinates also
presents insurmountable difficulties. If the observer applies his
standard measuring-rod (a rod which is short as compared with the
radius of the disc) tangentially to the edge of the disc, then, as
judged from the Galileian system, the length of this rod will be less
than I, since, according to Section 12, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
measaring-rod will not experience a shortening in length, as judged
from K, if it is applied to the disc in the direction of the radius.
If, then, the observer first measures the circumference of the disc
with his measuring-rod and then the diameter of the disc, on dividing
the one by the other, he will not obtain as quotient the familiar
number p = 3.14 . . ., but a larger number,[4]** whereas of course,
for a disc which is at rest with respect to K, this operation would
yield p exactly. This proves that the propositions of Euclidean
geometry cannot hold exactly on the rotating disc, nor in general in a
gravitational field, at least if we attribute the length I to the rod
in all positions and in every orientation. Hence the idea of a
straight line also loses its meaning. We are therefore not in a
position to define exactly the co-ordinates x, y, z relative to the
disc by means of the method used in discussing the special theory, and
as long as the co- ordinates and times of events have not been
defined, we cannot assign an exact meaning to the natural laws in
which these occur.
Thus all our previous conclusions based on general relativity would
appear to be called in question. In reality we must make a subtle
detour in order to be able to apply the postulate of general
relativity exactly. I shall prepare the reader for this in the
following paragraphs.
Notes
*) The field disappears at the centre of the disc and increases
proportionally to the distance from the centre as we proceed outwards.
**) Throughout this consideration we have to use the Galileian
(non-rotating) system K as reference-body, since we may only assume
the validity of the results of the special theory of relativity
relative to K (relative to K1 a gravitational field prevails).
EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
The surface of a marble table is spread out in front of me. I can get
from any one point on this table to any other point by passing
continuously from one point to a " neighbouring " one, and repeating
this process a (large) number of times, or, in other words, by going
from point to point without executing "jumps." I am sure the reader
will appreciate with sufficient clearness what I mean here by "
neighbouring " and by " jumps " (if he is not too pedantic). We
express this property of the surface by describing the latter as a
continuum.
Let us now imagine that a large number of little rods of equal length
have been made, their lengths being small compared with the dimensions
of the marble slab. When I say they are of equal length, I mean that
one can be laid on any other without the ends overlapping. We next lay
four of these little rods on the marble slab so that they constitute a
quadrilateral figure (a square), the diagonals of which are equally
long. To ensure the equality of the diagonals, we make use of a little
testing-rod. To this square we add similar ones, each of which has one
rod in common with the first. We proceed in like manner with each of
these squares until finally the whole marble slab is laid out with
squares. The arrangement is such, that each side of a square belongs
to two squares and each corner to four squares.
It is a veritable wonder that we can carry out this business without
getting into the greatest difficulties. We only need to think of the
following. If at any moment three squares meet at a corner, then two
sides of the fourth square are already laid, and, as a consequence,
the arrangement of the remaining two sides of the square is already
completely determined. But I am now no longer able to adjust the
quadrilateral so that its diagonals may be equal. If they are equal of
their own accord, then this is an especial favour of the marble slab
and of the little rods, about which I can only be thankfully
surprised. We must experience many such surprises if the construction
is to be successful.
If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a " distance " (line-interval). By
choosing one corner of a square as " origin" I can characterise every
other corner of a square with reference to this origin by means of two
numbers. I only need state how many rods I must pass over when,
starting from the origin, I proceed towards the " right " and then "
upwards," in order to arrive at the corner of the square under
consideration. These two numbers are then the " Cartesian co-ordinates
" of this corner with reference to the " Cartesian co-ordinate system"
which is determined by the arrangement of little rods.
By making use of the following modification of this abstract
experiment, we recognise that there must also be cases in which the
experiment would be unsuccessful. We shall suppose that the rods "
expand " by in amount proportional to the increase of temperature. We
heat the central part of the marble slab, but not the periphery, in
which case two of our little rods can still be brought into
coincidence at every position on the table. But our construction of
squares must necessarily come into disorder during the heating,
because the little rods on the central region of the table expand,
whereas those on the outer part do not.
With reference to our little rods -- defined as unit lengths -- the
marble slab is no longer a Euclidean continuum, and we are also no
longer in the position of defining Cartesian co-ordinates directly
with their aid, since the above construction can no longer be carried
out. But since there are other things which are not influenced in a
similar manner to the little rods (or perhaps not at all) by the
t/Relativity.test view on Meta::CPAN
The great power possessed by the general principle of relativity lies
in the comprehensive limitation which is imposed on the laws of nature
in consequence of what we have seen above.
THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
PRINCIPLE OF RELATIVITY
If the reader has followed all our previous considerations, he will
have no further difficulty in understanding the methods leading to the
solution of the problem of gravitation.
We start off on a consideration of a Galileian domain, i.e. a domain
in which there is no gravitational field relative to the Galileian
reference-body K. The behaviour of measuring-rods and clocks with
reference to K is known from the special theory of relativity,
likewise the behaviour of "isolated" material points; the latter move
uniformly and in straight lines.
Now let us refer this domain to a random Gauss coordinate system or to
a "mollusc" as reference-body K1. Then with respect to K1 there is a
gravitational field G (of a particular kind). We learn the behaviour
of measuring-rods and clocks and also of freely-moving material points
with reference to K1 simply by mathematical transformation. We
interpret this behaviour as the behaviour of measuring-rods, docks and
material points tinder the influence of the gravitational field G.
Hereupon we introduce a hypothesis: that the influence of the
gravitational field on measuringrods, clocks and freely-moving
material points continues to take place according to the same laws,
even in the case where the prevailing gravitational field is not
derivable from the Galfleian special care, simply by means of a
transformation of co-ordinates.
The next step is to investigate the space-time behaviour of the
gravitational field G, which was derived from the Galileian special
case simply by transformation of the coordinates. This behaviour is
formulated in a law, which is always valid, no matter how the
reference-body (mollusc) used in the description may be chosen.
This law is not yet the general law of the gravitational field, since
the gravitational field under consideration is of a special kind. In
order to find out the general law-of-field of gravitation we still
require to obtain a generalisation of the law as found above. This can
be obtained without caprice, however, by taking into consideration the
following demands:
(a) The required generalisation must likewise satisfy the general
postulate of relativity.
(b) If there is any matter in the domain under consideration, only its
inertial mass, and thus according to Section 15 only its energy is
of importance for its etfect in exciting a field.
(c) Gravitational field and matter together must satisfy the law of
the conservation of energy (and of impulse).
Finally, the general principle of relativity permits us to determine
the influence of the gravitational field on the course of all those
processes which take place according to known laws when a
gravitational field is absent i.e. which have already been fitted into
the frame of the special theory of relativity. In this connection we
proceed in principle according to the method which has already been
explained for measuring-rods, clocks and freely moving material
points.
The theory of gravitation derived in this way from the general
postulate of relativity excels not only in its beauty ; nor in
removing the defect attaching to classical mechanics which was brought
to light in Section 21; nor in interpreting the empirical law of
the equality of inertial and gravitational mass ; but it has also
already explained a result of observation in astronomy, against which
classical mechanics is powerless.
If we confine the application of the theory to the case where the
gravitational fields can be regarded as being weak, and in which all
masses move with respect to the coordinate system with velocities
which are small compared with the velocity of light, we then obtain as
a first approximation the Newtonian theory. Thus the latter theory is
obtained here without any particular assumption, whereas Newton had to
introduce the hypothesis that the force of attraction between mutually
attracting material points is inversely proportional to the square of
the distance between them. If we increase the accuracy of the
calculation, deviations from the theory of Newton make their
appearance, practically all of which must nevertheless escape the test
of observation owing to their smallness.
We must draw attention here to one of these deviations. According to
Newton's theory, a planet moves round the sun in an ellipse, which
would permanently maintain its position with respect to the fixed
stars, if we could disregard the motion of the fixed stars themselves
and the action of the other planets under consideration. Thus, if we
correct the observed motion of the planets for these two influences,
and if Newton's theory be strictly correct, we ought to obtain for the
orbit of the planet an ellipse, which is fixed with reference to the
fixed stars. This deduction, which can be tested with great accuracy,
has been confirmed for all the planets save one, with the precision
that is capable of being obtained by the delicacy of observation
attainable at the present time. The sole exception is Mercury, the
planet which lies nearest the sun. Since the time of Leverrier, it has
been known that the ellipse corresponding to the orbit of Mercury,
after it has been corrected for the influences mentioned above, is not
stationary with respect to the fixed stars, but that it rotates
exceedingly slowly in the plane of the orbit and in the sense of the
orbital motion. The value obtained for this rotary movement of the
orbital ellipse was 43 seconds of arc per century, an amount ensured
to be correct to within a few seconds of arc. This effect can be
explained by means of classical mechanics only on the assumption of
hypotheses which have little probability, and which were devised
solely for this purponse.
On the basis of the general theory of relativity, it is found that the
ellipse of every planet round the sun must necessarily rotate in the
manner indicated above ; that for all the planets, with the exception
of Mercury, this rotation is too small to be detected with the
delicacy of observation possible at the present time ; but that in the
case of Mercury it must amount to 43 seconds of arc per century, a
result which is strictly in agreement with observation.
Apart from this one, it has hitherto been possible to make only two
t/Relativity.test view on Meta::CPAN
these beings is two-dimensional; but, like ours, it extends to
infinity. In their universe there is room for an infinite number of
identical squares made up of rods, i.e. its volume (surface) is
infinite. If these beings say their universe is " plane," there is
sense in the statement, because they mean that they can perform the
constructions of plane Euclidean geometry with their rods. In this
connection the individual rods always represent the same distance,
independently of their position.
Let us consider now a second two-dimensional existence, but this time
on a spherical surface instead of on a plane. The flat beings with
their measuring-rods and other objects fit exactly on this surface and
they are unable to leave it. Their whole universe of observation
extends exclusively over the surface of the sphere. Are these beings
able to regard the geometry of their universe as being plane geometry
and their rods withal as the realisation of " distance " ? They cannot
do this. For if they attempt to realise a straight line, they will
obtain a curve, which we " three-dimensional beings " designate as a
great circle, i.e. a self-contained line of definite finite length,
which can be measured up by means of a measuring-rod. Similarly, this
universe has a finite area that can be compared with the area, of a
square constructed with rods. The great charm resulting from this
consideration lies in the recognition of the fact that the universe of
these beings is finite and yet has no limits.
But the spherical-surface beings do not need to go on a world-tour in
order to perceive that they are not living in a Euclidean universe.
They can convince themselves of this on every part of their " world,"
provided they do not use too small a piece of it. Starting from a
point, they draw " straight lines " (arcs of circles as judged in
three dimensional space) of equal length in all directions. They will
call the line joining the free ends of these lines a " circle." For a
plane surface, the ratio of the circumference of a circle to its
diameter, both lengths being measured with the same rod, is, according
to Euclidean geometry of the plane, equal to a constant value p, which
is independent of the diameter of the circle. On their spherical
surface our flat beings would find for this ratio the value
eq. 27: file eq27.gif
i.e. a smaller value than p, the difference being the more
considerable, the greater is the radius of the circle in comparison
with the radius R of the " world-sphere." By means of this relation
the spherical beings can determine the radius of their universe ("
world "), even when only a relatively small part of their worldsphere
is available for their measurements. But if this part is very small
indeed, they will no longer be able to demonstrate that they are on a
spherical " world " and not on a Euclidean plane, for a small part of
a spherical surface differs only slightly from a piece of a plane of
the same size.
Thus if the spherical surface beings are living on a planet of which
the solar system occupies only a negligibly small part of the
spherical universe, they have no means of determining whether they are
living in a finite or in an infinite universe, because the " piece of
universe " to which they have access is in both cases practically
plane, or Euclidean. It follows directly from this discussion, that
for our sphere-beings the circumference of a circle first increases
with the radius until the " circumference of the universe " is
reached, and that it thenceforward gradually decreases to zero for
still further increasing values of the radius. During this process the
area of the circle continues to increase more and more, until finally
it becomes equal to the total area of the whole " world-sphere."
Perhaps the reader will wonder why we have placed our " beings " on a
sphere rather than on another closed surface. But this choice has its
justification in the fact that, of all closed surfaces, the sphere is
unique in possessing the property that all points on it are
equivalent. I admit that the ratio of the circumference c of a circle
to its radius r depends on r, but for a given value of r it is the
same for all points of the " worldsphere "; in other words, the "
world-sphere " is a " surface of constant curvature."
To this two-dimensional sphere-universe there is a three-dimensional
analogy, namely, the three-dimensional spherical space which was
discovered by Riemann. its points are likewise all equivalent. It
possesses a finite volume, which is determined by its "radius"
(2p2R3). Is it possible to imagine a spherical space? To imagine a
space means nothing else than that we imagine an epitome of our "
space " experience, i.e. of experience that we can have in the
movement of " rigid " bodies. In this sense we can imagine a spherical
space.
Suppose we draw lines or stretch strings in all directions from a
point, and mark off from each of these the distance r with a
measuring-rod. All the free end-points of these lengths lie on a
spherical surface. We can specially measure up the area (F) of this
surface by means of a square made up of measuring-rods. If the
universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is
always less than 4pR2. With increasing values of r, F increases from
zero up to a maximum value which is determined by the " world-radius,"
but for still further increasing values of r, the area gradually
diminishes to zero. At first, the straight lines which radiate from
the starting point diverge farther and farther from one another, but
later they approach each other, and finally they run together again at
a "counter-point" to the starting point. Under such conditions they
have traversed the whole spherical space. It is easily seen that the
three-dimensional spherical space is quite analogous to the
two-dimensional spherical surface. It is finite (i.e. of finite
volume), and has no bounds.
It may be mentioned that there is yet another kind of curved space: "
elliptical space." It can be regarded as a curved space in which the
two " counter-points " are identical (indistinguishable from each
other). An elliptical universe can thus be considered to some extent
as a curved universe possessing central symmetry.
It follows from what has been said, that closed spaces without limits
are conceivable. From amongst these, the spherical space (and the
elliptical) excels in its simplicity, since all points on it are
equivalent. As a result of this discussion, a most interesting
question arises for astronomers and physicists, and that is whether
the universe in which we live is infinite, or whether it is finite in
the manner of the spherical universe. Our experience is far from being
sufficient to enable us to answer this question. But the general
theory of relativity permits of our answering it with a moduate degree
of certainty, and in this connection the difficulty mentioned in
Section 30 finds its solution.
t/Relativity.test view on Meta::CPAN
APPENDIX II
MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")
(SUPPLEMENTARY TO SECTION 17)
We can characterise the Lorentz transformation still more simply if we
introduce the imaginary eq. 25 in place of t, as time-variable. If, in
accordance with this, we insert
x[1] = x
x[2] = y
x[3] = z
x[4] = eq. 25
and similarly for the accented system K1, then the condition which is
identically satisfied by the transformation can be expressed thus :
x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
(12).
That is, by the afore-mentioned choice of " coordinates," (11a) [see
the end of Appendix II] is transformed into this equation.
We see from (12) that the imaginary time co-ordinate x[4], enters into
the condition of transformation in exactly the same way as the space
co-ordinates x[1], x[2], x[3]. It is due to this fact that, according
to the theory of relativity, the " time "x[4], enters into natural
laws in the same form as the space co ordinates x[1], x[2], x[3].
A four-dimensional continuum described by the "co-ordinates" x[1],
x[2], x[3], x[4], was called "world" by Minkowski, who also termed a
point-event a " world-point." From a "happening" in three-dimensional
space, physics becomes, as it were, an " existence " in the
four-dimensional " world."
This four-dimensional " world " bears a close similarity to the
three-dimensional " space " of (Euclidean) analytical geometry. If we
introduce into the latter a new Cartesian co-ordinate system (x'[1],
x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are
linear homogeneous functions of x[1], x[2], x[3] which identically
satisfy the equation
x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2
The analogy with (12) is a complete one. We can regard Minkowski's "
world " in a formal manner as a four-dimensional Euclidean space (with
an imaginary time coordinate) ; the Lorentz transformation corresponds
to a " rotation " of the co-ordinate system in the fourdimensional "
world."
APPENDIX III
THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
From a systematic theoretical point of view, we may imagine the
process of evolution of an empirical science to be a continuous
process of induction. Theories are evolved and are expressed in short
compass as statements of a large number of individual observations in
the form of empirical laws, from which the general laws can be
ascertained by comparison. Regarded in this way, the development of a
science bears some resemblance to the compilation of a classified
catalogue. It is, as it were, a purely empirical enterprise.
But this point of view by no means embraces the whole of the actual
process ; for it slurs over the important part played by intuition and
deductive thought in the development of an exact science. As soon as a
science has emerged from its initial stages, theoretical advances are
no longer achieved merely by a process of arrangement. Guided by
empirical data, the investigator rather develops a system of thought
which, in general, is built up logically from a small number of
fundamental assumptions, the so-called axioms. We call such a system
of thought a theory. The theory finds the justification for its
existence in the fact that it correlates a large number of single
observations, and it is just here that the " truth " of the theory
lies.
Corresponding to the same complex of empirical data, there may be
several theories, which differ from one another to a considerable
extent. But as regards the deductions from the theories which are
capable of being tested, the agreement between the theories may be so
complete that it becomes difficult to find any deductions in which the
two theories differ from each other. As an example, a case of general
interest is available in the province of biology, in the Darwinian
theory of the development of species by selection in the struggle for
existence, and in the theory of development which is based on the
hypothesis of the hereditary transmission of acquired characters.
We have another instance of far-reaching agreement between the
deductions from two theories in Newtonian mechanics on the one hand,
and the general theory of relativity on the other. This agreement goes
so far, that up to the preseat we have been able to find only a few
deductions from the general theory of relativity which are capable of
investigation, and to which the physics of pre-relativity days does
not also lead, and this despite the profound difference in the
fundamental assumptions of the two theories. In what follows, we shall
again consider these important deductions, and we shall also discuss
the empirical evidence appertaining to them which has hitherto been
obtained.
(a) Motion of the Perihelion of Mercury
According to Newtonian mechanics and Newton's law of gravitation, a
planet which is revolving round the sun would describe an ellipse
round the latter, or, more correctly, round the common centre of
gravity of the sun and the planet. In such a system, the sun, or the
common centre of gravity, lies in one of the foci of the orbital
ellipse in such a manner that, in the course of a planet-year, the
distance sun-planet grows from a minimum to a maximum, and then
decreases again to a minimum. If instead of Newton's law we insert a
somewhat different law of attraction into the calculation, we find
that, according to this new law, the motion would still take place in
such a manner that the distance sun-planet exhibits periodic
variations; but in this case the angle described by the line joining
sun and planet during such a period (from perihelion--closest
proximity to the sun--to perihelion) would differ from 360^0. The line
of the orbit would not then be a closed one but in the course of time
it would fill up an annular part of the orbital plane, viz. between
the circle of least and the circle of greatest distance of the planet
from the sun.
According also to the general theory of relativity, which differs of
course from the theory of Newton, a small variation from the
Newton-Kepler motion of a planet in its orbit should take place, and
in such away, that the angle described by the radius sun-planet
between one perhelion and the next should exceed that corresponding to
one complete revolution by an amount given by
eq. 41: file eq41.gif
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