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Part II: The General Theory of Relativity
18. Special and General Principle of Relativity
19. The Gravitational Field
20. The Equality of Inertial and Gravitational Mass as an Argument
for the General Postulate of Relativity
21. In What Respects are the Foundations of Classical Mechanics
and of the Special Theory of Relativity Unsatisfactory?
22. A Few Inferences from the General Principle of Relativity
23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of
Reference
24. Euclidean and non-Euclidean Continuum
25. Gaussian Co-ordinates
26. The Space-Time Continuum of the Speical Theory of Relativity
Considered as a Euclidean Continuum
27. The Space-Time Continuum of the General Theory of Relativity
is Not a Eculidean Continuum
28. Exact Formulation of the General Principle of Relativity
29. The Solution of the Problem of Gravitation on the Basis of the
General Principle of Relativity
Part III: Considerations on the Universe as a Whole
30. Cosmological Difficulties of Netwon's Theory
31. The Possibility of a "Finite" and yet "Unbounded" Universe
32. The Structure of Space According to the General Theory of
Relativity
Appendices:
01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
03. The Experimental Confirmation of the General Theory of Relativity
04. The Structure of Space According to the General Theory of
Relativity (sup. ch 32)
05. Relativity and the Problem of Space
Note: The fifth Appendix was added by Einstein at the time of the
fifteenth re-printing of this book; and as a result is still under
copyright restrictions so cannot be added without the permission of
the publisher.
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.
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with the specification of the position of the foot of the pole,
supplies us with a complete place specification. On the basis of this
illustration, we are able to see the manner in which a refinement of
the conception of position has been developed.
(a) We imagine the rigid body, to which the place specification is
referred, supplemented in such a manner that the object whose position
we require is reached by. the completed rigid body.
(b) In locating the position of the object, we make use of a number
(here the length of the pole measured with the measuring-rod) instead
of designated points of reference.
(c) We speak of the height of the cloud even when the pole which
reaches the cloud has not been erected. By means of optical
observations of the cloud from different positions on the ground, and
taking into account the properties of the propagation of light, we
determine the length of the pole we should have required in order to
reach the cloud.
From this consideration we see that it will be advantageous if, in the
description of position, it should be possible by means of numerical
measures to make ourselves independent of the existence of marked
positions (possessing names) on the rigid body of reference. In the
physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other and
rigidly attached to a rigid body. Referred to a system of
co-ordinates, the scene of any event will be determined (for the main
part) by the specification of the lengths of the three perpendiculars
or co-ordinates (x, y, z) which can be dropped from the scene of the
event to those three plane surfaces. The lengths of these three
perpendiculars can be determined by a series of manipulations with
rigid measuring-rods performed according to the rules and methods laid
down by Euclidean geometry.
In practice, the rigid surfaces which constitute the system of
co-ordinates are generally not available ; furthermore, the magnitudes
of the co-ordinates are not actually determined by constructions with
rigid rods, but by indirect means. If the results of physics and
astronomy are to maintain their clearness, the physical meaning of
specifications of position must always be sought in accordance with
the above considerations. ***
We thus obtain the following result: Every description of events in
space involves the use of a rigid body to which such events have to be
referred. The resulting relationship takes for granted that the laws
of Euclidean geometry hold for "distances;" the "distance" being
represented physically by means of the convention of two marks on a
rigid body.
Notes
* Here we have assumed that there is nothing left over i.e. that
the measurement gives a whole number. This difficulty is got over by
the use of divided measuring-rods, the introduction of which does not
demand any fundamentally new method.
**A Einstein used "Potsdamer Platz, Berlin" in the original text.
In the authorised translation this was supplemented with "Tranfalgar
Square, London". We have changed this to "Times Square, New York", as
this is the most well known/identifiable location to English speakers
in the present day. [Note by the janitor.]
**B It is not necessary here to investigate further the significance
of the expression "coincidence in space." This conception is
sufficiently obvious to ensure that differences of opinion are
scarcely likely to arise as to its applicability in practice.
*** A refinement and modification of these views does not become
necessary until we come to deal with the general theory of relativity,
treated in the second part of this book.
SPACE AND TIME IN CLASSICAL MECHANICS
The purpose of mechanics is to describe how bodies change their
position in space with "time." I should load my conscience with grave
sins against the sacred spirit of lucidity were I to formulate the
aims of mechanics in this way, without serious reflection and detailed
explanations. Let us proceed to disclose these sins.
It is not clear what is to be understood here by "position" and
"space." I stand at the window of a railway carriage which is
travelling uniformly, and drop a stone on the embankment, without
throwing it. Then, disregarding the influence of the air resistance, I
see the stone descend in a straight line. A pedestrian who observes
the misdeed from the footpath notices that the stone falls to earth in
a parabolic curve. I now ask: Do the "positions" traversed by the
stone lie "in reality" on a straight line or on a parabola? Moreover,
what is meant here by motion "in space" ? From the considerations of
the previous section the answer is self-evident. In the first place we
entirely shun the vague word "space," of which, we must honestly
acknowledge, we cannot form the slightest conception, and we replace
it by "motion relative to a practically rigid body of reference." The
positions relative to the body of reference (railway carriage or
embankment) have already been defined in detail in the preceding
section. If instead of " body of reference " we insert " system of
co-ordinates," which is a useful idea for mathematical description, we
are in a position to say : The stone traverses a straight line
relative to a system of co-ordinates rigidly attached to the carriage,
but relative to a system of co-ordinates rigidly attached to the
ground (embankment) it describes a parabola. With the aid of this
example it is clearly seen that there is no such thing as an
independently existing trajectory (lit. "path-curve"*), but only
a trajectory relative to a particular body of reference.
In order to have a complete description of the motion, we must specify
how the body alters its position with time ; i.e. for every point on
the trajectory it must be stated at what time the body is situated
there. These data must be supplemented by such a definition of time
that, in virtue of this definition, these time-values can be regarded
essentially as magnitudes (results of measurements) capable of
observation. If we take our stand on the ground of classical
mechanics, we can satisfy this requirement for our illustration in the
following manner. We imagine two clocks of identical construction ;
the man at the railway-carriage window is holding one of them, and the
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Notes
*) A simple derivation of the Lorentz transformation is given in
Appendix I.
THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
Place a metre-rod in the x1-axis of K1 in such a manner that one end
(the beginning) coincides with the point x1=0 whilst the other end
(the end of the rod) coincides with the point x1=I. What is the length
of the metre-rod relatively to the system K? In order to learn this,
we need only ask where the beginning of the rod and the end of the rod
lie with respect to K at a particular time t of the system K. By means
of the first equation of the Lorentz transformation the values of
these two points at the time t = 0 can be shown to be
eq. 05a: file eq05a.gif
eq. 05b: file eq05b.gif
the distance between the points being eq. 06 .
But the metre-rod is moving with the velocity v relative to K. It
therefore follows that the length of a rigid metre-rod moving in the
direction of its length with a velocity v is eq. 06 of a metre.
The rigid rod is thus shorter when in motion than when at rest, and
the more quickly it is moving, the shorter is the rod. For the
velocity v=c we should have eq. 06a ,
and for stiII greater velocities the square-root becomes imaginary.
From this we conclude that in the theory of relativity the velocity c
plays the part of a limiting velocity, which can neither be reached
nor exceeded by any real body.
Of course this feature of the velocity c as a limiting velocity also
clearly follows from the equations of the Lorentz transformation, for
these became meaningless if we choose values of v greater than c.
If, on the contrary, we had considered a metre-rod at rest in the
x-axis with respect to K, then we should have found that the length of
the rod as judged from K1 would have been eq. 06 ;
this is quite in accordance with the principle of relativity which
forms the basis of our considerations.
A Priori it is quite clear that we must be able to learn something
about the physical behaviour of measuring-rods and clocks from the
equations of transformation, for the magnitudes z, y, x, t, are
nothing more nor less than the results of measurements obtainable by
means of measuring-rods and clocks. If we had based our considerations
on the Galileian transformation we should not have obtained a
contraction of the rod as a consequence of its motion.
Let us now consider a seconds-clock which is permanently situated at
the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of
this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks :
t = 0
and
eq. 07: file eq07.gif
As judged from K, the clock is moving with the velocity v; as judged
from this reference-body, the time which elapses between two strokes
of the clock is not one second, but
eq. 08: file eq08.gif
seconds, i.e. a somewhat larger time. As a consequence of its motion
the clock goes more slowly than when at rest. Here also the velocity c
plays the part of an unattainable limiting velocity.
THEOREM OF THE ADDITION OF VELOCITIES.
THE EXPERIMENT OF FIZEAU
Now in practice we can move clocks and measuring-rods only with
velocities that are small compared with the velocity of light; hence
we shall hardly be able to compare the results of the previous section
directly with the reality. But, on the other hand, these results must
strike you as being very singular, and for that reason I shall now
draw another conclusion from the theory, one which can easily be
derived from the foregoing considerations, and which has been most
elegantly confirmed by experiment.
In Section 6 we derived the theorem of the addition of velocities
in one direction in the form which also results from the hypotheses of
classical mechanics- This theorem can also be deduced readily horn the
Galilei transformation (Section 11). In place of the man walking
inside the carriage, we introduce a point moving relatively to the
co-ordinate system K1 in accordance with the equation
x1 = wt1
By means of the first and fourth equations of the Galilei
transformation we can express x1 and t1 in terms of x and t, and we
then obtain
x = (v + w)t
This equation expresses nothing else than the law of motion of the
point with reference to the system K (of the man with reference to the
embankment). We denote this velocity by the symbol W, and we then
obtain, as in Section 6,
W=v+w A)
But we can carry out this consideration just as well on the basis of
the theory of relativity. In the equation
x1 = wt1 B)
we must then express x1and t1 in terms of x and t, making use of the
first and fourth equations of the Lorentz transformation. Instead of
the equation (A) we then obtain the equation
eq. 09: file eq09.gif
which corresponds to the theorem of addition for velocities in one
direction according to the theory of relativity. The question now
arises as to which of these two theorems is the better in accord with
experience. On this point we axe enlightened by a most important
experiment which the brilliant physicist Fizeau performed more than
half a century ago, and which has been repeated since then by some of
the best experimental physicists, so that there can be no doubt about
its result. The experiment is concerned with the following question.
Light travels in a motionless liquid with a particular velocity w. How
quickly does it travel in the direction of the arrow in the tube T
(see the accompanying diagram, Fig. 3) when the liquid above
mentioned is flowing through the tube with a velocity v ?
In accordance with the principle of relativity we shall certainly have
to take for granted that the propagation of light always takes place
with the same velocity w with respect to the liquid, whether the
latter is in motion with reference to other bodies or not. The
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
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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
sense. For it is composed of individual events, each of which is
described by four numbers, namely, three space co-ordinates x, y, z,
and a time co-ordinate, the time value t. The" world" is in this sense
also a continuum; for to every event there are as many "neighbouring"
events (realised or at least thinkable) as we care to choose, the
co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely
small amount from those of the event x, y, z, t originally considered.
That we have not been accustomed to regard the world in this sense as
a four-dimensional continuum is due to the fact that in physics,
before the advent of the theory of relativity, time played a different
and more independent role, as compared with the space coordinates. It
is for this reason that we have been in the habit of treating time as
an independent continuum. As a matter of fact, according to classical
mechanics, time is absolute, i.e. it is independent of the position
and the condition of motion of the system of co-ordinates. We see this
expressed in the last equation of the Galileian transformation (t1 =
t)
The four-dimensional mode of consideration of the "world" is natural
on the theory of relativity, since according to this theory time is
robbed of its independence. This is shown by the fourth equation of
the Lorentz transformation:
eq. 24: file eq24.gif
Moreover, according to this equation the time difference Dt1 of two
events with respect to K1 does not in general vanish, even when the
time difference Dt1 of the same events with reference to K vanishes.
Pure " space-distance " of two events with respect to K results in "
time-distance " of the same events with respect to K. But the
discovery of Minkowski, which was of importance for the formal
development of the theory of relativity, does not lie here. It is to
be found rather in the fact of his recognition that the
four-dimensional space-time continuum of the theory of relativity, in
its most essential formal properties, shows a pronounced relationship
to the three-dimensional continuum of Euclidean geometrical
space.* In order to give due prominence to this relationship,
however, we must replace the usual time co-ordinate t by an imaginary
magnitude eq. 25 proportional to it. Under these conditions, the
natural laws satisfying the demands of the (special) theory of
relativity assume mathematical forms, in which the time co-ordinate
plays exactly the same role as the three space co-ordinates. Formally,
these four co-ordinates correspond exactly to the three space
co-ordinates in Euclidean geometry. It must be clear even to the
non-mathematician that, as a consequence of this purely formal
addition to our knowledge, the theory perforce gained clearness in no
mean measure.
These inadequate remarks can give the reader only a vague notion of
the important idea contributed by Minkowski. Without it the general
theory of relativity, of which the fundamental ideas are developed in
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.
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We must note carefully that the possibility of this mode of
interpretation rests on the fundamental property of the gravitational
field of giving all bodies the same acceleration, or, what comes to
the same thing, on the law of the equality of inertial and
gravitational mass. If this natural law did not exist, the man in the
accelerated chest would not be able to interpret the behaviour of the
bodies around him on the supposition of a gravitational field, and he
would not be justified on the grounds of experience in supposing his
reference-body to be " at rest."
Suppose that the man in the chest fixes a rope to the inner side of
the lid, and that he attaches a body to the free end of the rope. The
result of this will be to strech the rope so that it will hang "
vertically " downwards. If we ask for an opinion of the cause of
tension in the rope, the man in the chest will say: "The suspended
body experiences a downward force in the gravitational field, and this
is neutralised by the tension of the rope ; what determines the
magnitude of the tension of the rope is the gravitational mass of the
suspended body." On the other hand, an observer who is poised freely
in space will interpret the condition of things thus : " The rope must
perforce take part in the accelerated motion of the chest, and it
transmits this motion to the body attached to it. The tension of the
rope is just large enough to effect the acceleration of the body. That
which determines the magnitude of the tension of the rope is the
inertial mass of the body." Guided by this example, we see that our
extension of the principle of relativity implies the necessity of the
law of the equality of inertial and gravitational mass. Thus we have
obtained a physical interpretation of this law.
From our consideration of the accelerated chest we see that a general
theory of relativity must yield important results on the laws of
gravitation. In point of fact, the systematic pursuit of the general
idea of relativity has supplied the laws satisfied by the
gravitational field. Before proceeding farther, however, I must warn
the reader against a misconception suggested by these considerations.
A gravitational field exists for the man in the chest, despite the
fact that there was no such field for the co-ordinate system first
chosen. Now we might easily suppose that the existence of a
gravitational field is always only an apparent one. We might also
think that, regardless of the kind of gravitational field which may be
present, we could always choose another reference-body such that no
gravitational field exists with reference to it. This is by no means
true for all gravitational fields, but only for those of quite special
form. It is, for instance, impossible to choose a body of reference
such that, as judged from it, the gravitational field of the earth (in
its entirety) vanishes.
We can now appreciate why that argument is not convincing, which we
brought forward against the general principle of relativity at theend
of Section 18. It is certainly true that the observer in the
railway carriage experiences a jerk forwards as a result of the
application of the brake, and that he recognises, in this the
non-uniformity of motion (retardation) of the carriage. But he is
compelled by nobody to refer this jerk to a " real " acceleration
(retardation) of the carriage. He might also interpret his experience
thus: " My body of reference (the carriage) remains permanently at
rest. With reference to it, however, there exists (during the period
of application of the brakes) a gravitational field which is directed
forwards and which is variable with respect to time. Under the
influence of this field, the embankment together with the earth moves
non-uniformly in such a manner that their original velocity in the
backwards direction is continuously reduced."
IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
We have already stated several times that classical mechanics starts
out from the following law: Material particles sufficiently far
removed from other material particles continue to move uniformly in a
straight line or continue in a state of rest. We have also repeatedly
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
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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
is sitting eccentrically on the disc K1 is sensible of a force which
acts outwards in a radial direction, and which would be interpreted as
an effect of inertia (centrifugal force) by an observer who was at
rest with respect to the original reference-body K. But the observer
on the disc may regard his disc as a reference-body which is " at rest
" ; on the basis of the general principle of relativity he is
justified in doing this. The force acting on himself, and in fact on
all other bodies which are at rest relative to the disc, he regards as
the effect of a gravitational field. Nevertheless, the
space-distribution of this gravitational field is of a kind that would
not be possible on Newton's theory of gravitation.* But since the
observer believes in the general theory of relativity, this does not
disturb him; he is quite in the right when he believes that a general
law of gravitation can be formulated- a law which not only explains
the motion of the stars correctly, but also the field of force
experienced by himself.
The observer performs experiments on his circular disc with clocks and
measuring-rods. In doing so, it is his intention to arrive at exact
definitions for the signification of time- and space-data with
reference to the circular disc K1, these definitions being based on
his observations. What will be his experience in this enterprise ?
To start with, he places one of two identically constructed clocks at
the centre of the circular disc, and the other on the edge of the
disc, so that they are at rest relative to it. We now ask ourselves
whether both clocks go at the same rate from the standpoint of the
non-rotating Galileian reference-body K. As judged from this body, the
clock at the centre of the disc has no velocity, whereas the clock at
the edge of the disc is in motion relative to K in consequence of the
rotation. According to a result obtained in Section 12, it follows
that the latter clock goes at a rate permanently slower than that of
the clock at the centre of the circular disc, i.e. as observed from K.
It is obvious that the same effect would be noted by an observer whom
we will imagine sitting alongside his clock at the centre of the
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
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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
temperature of the table, it is possible quite naturally to maintain
the point of view that the marble slab is a " Euclidean continuum."
This can be done in a satisfactory manner by making a more subtle
stipulation about the measurement or the comparison of lengths.
But if rods of every kind (i.e. of every material) were to behave in
the same way as regards the influence of temperature when they are on
the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behaviour of
our rods in experiments analogous to the one described above, then our
best plan would be to assign the distance one to two points on the
slab, provided that the ends of one of our rods could be made to
coincide with these two points ; for how else should we define the
distance without our proceeding being in the highest measure grossly
arbitrary ? The method of Cartesian coordinates must then be
discarded, and replaced by another which does not assume the validity
of Euclidean geometry for rigid bodies.* The reader will notice
that the situation depicted here corresponds to the one brought about
by the general postitlate of relativity (Section 23).
Notes
*) Mathematicians have been confronted with our problem in the
following form. If we are given a surface (e.g. an ellipsoid) in
Euclidean three-dimensional space, then there exists for this surface
a two-dimensional geometry, just as much as for a plane surface. Gauss
undertook the task of treating this two-dimensional geometry from
first principles, without making use of the fact that the surface
belongs to a Euclidean continuum of three dimensions. If we imagine
constructions to be made with rigid rods in the surface (similar to
that above with the marble slab), we should find that different laws
hold for these from those resulting on the basis of Euclidean plane
geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian co-ordinates in the surface.
Gauss indicated the principles according to which we can treat the
geometrical relationships in the surface, and thus pointed out the way
to the method of Riemman of treating multi-dimensional, non-Euclidean
t/Relativity.test view on Meta::CPAN
APPENDIX I
SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
(SUPPLEMENTARY TO SECTION 11)
For the relative orientation of the co-ordinate systems indicated in
Fig. 2, the x-axes of both systems pernumently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the x-axis. Any such event is
represented with respect to the co-ordinate system K by the abscissa x
and the time t, and with respect to the system K1 by the abscissa x'
and the time t'. We require to find x' and t' when x and t are given.
A light-signal, which is proceeding along the positive axis of x, is
transmitted according to the equation
x = ct
or
x - ct = 0 . . . (1).
Since the same light-signal has to be transmitted relative to K1 with
the velocity c, the propagation relative to the system K1 will be
represented by the analogous formula
x' - ct' = O . . . (2)
Those space-time points (events) which satisfy (x) must also satisfy
(2). Obviously this will be the case when the relation
(x' - ct') = l (x - ct) . . . (3).
is fulfilled in general, where l indicates a constant ; for, according
to (3), the disappearance of (x - ct) involves the disappearance of
(x' - ct').
If we apply quite similar considerations to light rays which are being
transmitted along the negative x-axis, we obtain the condition
(x' + ct') = µ(x + ct) . . . (4).
By adding (or subtracting) equations (3) and (4), and introducing for
convenience the constants a and b in place of the constants l and µ,
where
eq. 29: file eq29.gif
and
eq. 30: file eq30.gif
we obtain the equations
eq. 31: file eq31.gif
We should thus have the solution of our problem, if the constants a
and b were known. These result from the following discussion.
For the origin of K1 we have permanently x' = 0, and hence according
to the first of the equations (5)
eq. 32: file eq32.gif
If we call v the velocity with which the origin of K1 is moving
relative to K, we then have
eq. 33: file eq33.gif
The same value v can be obtained from equations (5), if we calculate
the velocity of another point of K1 relative to K, or the velocity
(directed towards the negative x-axis) of a point of K with respect to
K'. In short, we can designate v as the relative velocity of the two
systems.
Furthermore, the principle of relativity teaches us that, as judged
from K, the length of a unit measuring-rod which is at rest with
reference to K1 must be exactly the same as the length, as judged from
K', of a unit measuring-rod which is at rest relative to K. In order
to see how the points of the x-axis appear as viewed from K, we only
require to take a " snapshot " of K1 from K; this means that we have
to insert a particular value of t (time of K), e.g. t = 0. For this
value of t we then obtain from the first of the equations (5)
x' = ax
Two points of the x'-axis which are separated by the distance Dx' = I
when measured in the K1 system are thus separated in our instantaneous
photograph by the distance
eq. 34: file eq34.gif
But if the snapshot be taken from K'(t' = 0), and if we eliminate t
from the equations (5), taking into account the expression (6), we
obtain
eq. 35: file eq35.gif
From this we conclude that two points on the x-axis separated by the
distance I (relative to K) will be represented on our snapshot by the
distance
eq. 36: file eq36.gif
But from what has been said, the two snapshots must be identical;
hence Dx in (7) must be equal to Dx' in (7a), so that we obtain
eq. 37: file eq37.gif
The equations (6) and (7b) determine the constants a and b. By
inserting the values of these constants in (5), we obtain the first
and the fourth of the equations given in Section 11.
eq. 38: file eq38.gif
Thus we have obtained the Lorentz transformation for events on the
x-axis. It satisfies the condition
x'2 - c^2t'2 = x2 - c^2t2 . . . (8a).
The extension of this result, to include events which take place
outside the x-axis, is obtained by retaining equations (8) and
supplementing them by the relations
eq. 39: file eq39.gif
In this way we satisfy the postulate of the constancy of the velocity
of light in vacuo for rays of light of arbitrary direction, both for
the system K and for the system K'. This may be shown in the following
manner.
We suppose a light-signal sent out from the origin of K at the time t
= 0. It will be propagated according to the equation
eq. 40: file eq40.gif
or, if we square this equation, according to the equation
x2 + y2 + z2 = c^2t2 = 0 . . . (10).
It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place -- as judged from K1 -- in accordance with
the corresponding formula
r' = ct'
or,
x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a).
In order that equation (10a) may be a consequence of equation (10), we
must have
x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11).
Since equation (8a) must hold for points on the x-axis, we thus have s
= I. It is easily seen that the Lorentz transformation really
satisfies equation (11) for s = I; for (11) is a consequence of (8a)
and (9), and hence also of (8) and (9). We have thus derived the
Lorentz transformation.
The Lorentz transformation represented by (8) and (9) still requires
to be generalised. Obviously it is immaterial whether the axes of K1
be chosen so that they are spatially parallel to those of K. It is
also not essential that the velocity of translation of K1 with respect
to K should be in the direction of the x-axis. A simple consideration
shows that we are able to construct the Lorentz transformation in this
general sense from two kinds of transformations, viz. from Lorentz
transformations in the special sense and from purely spatial
transformations. which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing
in other directions.
Mathematically, we can characterise the generalised Lorentz
transformation thus :
It expresses x', y', x', t', in terms of linear homogeneous functions
of x, y, x, t, of such a kind that the relation
x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2 (11a).
is satisficd identically. That is to say: If we substitute their
expressions in x, y, x, t, in place of x', y', x', t', on the
left-hand side, then the left-hand side of (11a) agrees with the
right-hand side.
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
t/Relativity.test view on Meta::CPAN
amounts to about two millionths of the wave-length. A trustworthy
calculation is not possible in the case of the stars, because in
general neither the mass M nor the radius r are known.
It is an open question whether or not this effect exists, and at the
present time (1920) astronomers are working with great zeal towards
the solution. Owing to the smallness of the effect in the case of the
sun, it is difficult to form an opinion as to its existence. Whereas
Grebe and Bachem (Bonn), as a result of their own measurements and
those of Evershed and Schwarzschild on the cyanogen bands, have placed
the existence of the effect almost beyond doubt, while other
investigators, particularly St. John, have been led to the opposite
opinion in consequence of their measurements.
Mean displacements of lines towards the less refrangible end of the
spectrum are certainly revealed by statistical investigations of the
fixed stars ; but up to the present the examination of the available
data does not allow of any definite decision being arrived at, as to
whether or not these displacements are to be referred in reality to
the effect of gravitation. The results of observation have been
collected together, and discussed in detail from the standpoint of the
question which has been engaging our attention here, in a paper by E.
Freundlich entitled "Zur Prüfung der allgemeinen
Relativit¨aut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35,
p. 520: Julius Springer, Berlin).
At all events, a definite decision will be reached during the next few
years. If the displacement of spectral lines towards the red by the
gravitational potential does not exist, then the general theory of
relativity will be untenable. On the other hand, if the cause of the
displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will
furnish us with important information as to the mass of the heavenly
bodies. [5][A]
Notes
*) Especially since the next planet Venus has an orbit that is
almost an exact circle, which makes it more difficult to locate the
perihelion with precision.
The displacentent of spectral lines towards the red end of the
spectrum was definitely established by Adams in 1924, by observations
on the dense companion of Sirius, for which the effect is about thirty
times greater than for the Sun. R.W.L. -- translator
APPENDIX IV
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
(SUPPLEMENTARY TO SECTION 32)
Since the publication of the first edition of this little book, our
knowledge about the structure of space in the large (" cosmological
problem ") has had an important development, which ought to be
mentioned even in a popular presentation of the subject.
My original considerations on the subject were based on two
hypotheses:
(1) There exists an average density of matter in the whole of space
which is everywhere the same and different from zero.
(2) The magnitude (" radius ") of space is independent of time.
Both these hypotheses proved to be consistent, according to the
general theory of relativity, but only after a hypothetical term was
added to the field equations, a term which was not required by the
theory as such nor did it seem natural from a theoretical point of
view (" cosmological term of the field equations ").
Hypothesis (2) appeared unavoidable to me at the time, since I thought
that one would get into bottomless speculations if one departed from
it.
However, already in the 'twenties, the Russian mathematician Friedman
showed that a different hypothesis was natural from a purely
theoretical point of view. He realized that it was possible to
preserve hypothesis (1) without introducing the less natural
cosmological term into the field equations of gravitation, if one was
ready to drop hypothesis (2). Namely, the original field equations
admit a solution in which the " world radius " depends on time
(expanding space). In that sense one can say, according to Friedman,
that the theory demands an expansion of space.
A few years later Hubble showed, by a special investigation of the
extra-galactic nebulae (" milky ways "), that the spectral lines
emitted showed a red shift which increased regularly with the distance
of the nebulae. This can be interpreted in regard to our present
knowledge only in the sense of Doppler's principle, as an expansive
motion of the system of stars in the large -- as required, according
to Friedman, by the field equations of gravitation. Hubble's discovery
can, therefore, be considered to some extent as a confirmation of the
theory.
There does arise, however, a strange difficulty. The interpretation of
the galactic line-shift discovered by Hubble as an expansion (which
can hardly be doubted from a theoretical point of view), leads to an
origin of this expansion which lies " only " about 10^9 years ago,
while physical astronomy makes it appear likely that the development
of individual stars and systems of stars takes considerably longer. It
is in no way known how this incongruity is to be overcome.
I further want to rernark that the theory of expanding space, together
with the empirical data of astronomy, permit no decision to be reached
about the finite or infinite character of (three-dimensional) space,
while the original " static " hypothesis of space yielded the closure
(finiteness) of space.
K = co-ordinate system
x, y = two-dimensional co-ordinates
x, y, z = three-dimensional co-ordinates
x, y, z, t = four-dimensional co-ordinates
t = time
I = distance
v = velocity
F = force
G = gravitational field
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