lid, and that he attaches a body to the free end of the rope. The
result of this will be to stretch 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 the end
of Section XVIII. 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.”
XXI.
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 _K′_.[15] Newton saw this objection and attempted to
invalidate it, but without success. But E. Mach recognised 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.
[15] 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.
XXII.
A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
The considerations of Section XX 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 _K′_ which is accelerated relatively to _K_.
But since a gravitational field exists with respect to this new body of
reference _K′_, 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 _K′_ (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
_K′_). 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.[16]
[16] 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.)
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 unlimited 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 laws 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 _K′_ possessing any kind of
motion, then relative to _K′_ there exists a gravitational field which
is variable with respect to space and time.[17] The character of this
field will of course depend on the motion chosen for _K′._ 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.
[17] This follows from a generalisation of the discussion in Section
XX.
XXIII.
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 suppose the
same domain referred to a second body of reference _K′_, which is
rotating uniformly with respect to _K_. In order to fix our ideas, we
shall imagine _K′_ 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 _K′_ 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.[18] 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.
[18] The field disappears at the centre of the disc and increases
proportionally to the distance from the centre as we proceed outwards.
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 _K′_, 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 XII, 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 1, since, according to Section XII, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
measuring-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 π = 3.14 . . ., but a larger number,[19] whereas of course, for
a disc which is at rest with respect to _K_, this operation would yield
π 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 1 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.
[19] 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 _K′_ a gravitational field prevails).
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.
XXIV.
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.[20] The reader will notice that the
situation depicted here corresponds to the one brought about by the
general postulate of relativity (Section XXIII).
[20] 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 Riemann of treating multi-dimensional, non-Euclidean
_continuum_. Thus it is that mathematicians long ago solved the formal
problems to which we are led by the general postulate of relativity.
XXV.
GAUSSIAN CO-ORDINATES
image033
According to Gauss, this combined analytical and geometrical mode of
handling the problem can be arrived at in the following way. We imagine
a system of arbitrary curves (see Fig. 4) drawn on the surface of the
table. These we designate as _u_-curves, and we indicate each of them
by means of a number. The Curves _u_ = 1, _u_ = 2 and _u_ = 3 are drawn
in the diagram. Between the curves _u_ = 1 and _u_ = 2 we must imagine
an infinitely large number to be drawn, all of which correspond to real
numbers lying between 1 and 2. We have then a system of _u_-curves, and
this “infinitely dense” system covers the whole surface of the table.
These _u_-curves must not intersect each other, and through each point
of the surface one and only one curve must pass. Thus a perfectly
definite value of _u_ belongs to every point on the surface of the
marble slab. In like manner we imagine a system of _v_-curves drawn on
the surface. These satisfy the same conditions as the _u_-curves, they
are provided with numbers in a corresponding manner, and they may
likewise be of arbitrary shape. It follows that a value of _u_ and a
value of _v_ belong to every point on the surface of the table. We call
these two numbers the co-ordinates of the surface of the table
(Gaussian co-ordinates). For example, the point _P_ in the diagram has
the Gaussian co-ordinates _u_ = 3, _v_ = 1. Two neighbouring points _P_
and _P′_ on the surface then correspond to the co-ordinates
_P_: _u, v_
_P′_: _u_ + _du, v_ + _dv_,
where _du_ and _dv_ signify very small numbers. In a similar manner we
may indicate the distance (line-interval) between _P_ and _P′_, as
measured with a little rod, by means of the very small number _ds_.
Then according to Gauss we have
_ds_2 = _g_11_du_2 + 2_g_12_du dv_ + _g_22_dv_2,
where _g_11, _g_12, _g_22, are magnitudes which depend in a perfectly
definite way on _u_ and _v_. The magnitudes _g_11, _g_12 and _g_22,
determine the behaviour of the rods relative to the _u_-curves and
_v_-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the _u_-curves and _v_-curves and to attach
numbers to them, in such a manner, that we simply have:
_ds_2 = _du_2 + _dv_2
Under these conditions, the _u_-curves and _v_-curves are straight
lines in the sense of Euclidean geometry, and they are perpendicular to
each other. Here the Gaussian coordinates are simply Cartesian ones. It
is clear that Gauss co-ordinates are nothing more than an association
of two sets of numbers with the points of the surface considered, of
such a nature that numerical values differing very slightly from each
other are associated with neighbouring points “in space.”
So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, _x_1, _x_2, _x_3, _x_4, which are known as “co-ordinates.”
Adjacent points correspond to adjacent values of the coordinates. If a
distance _ds_ is associated with the adjacent points _P_ and _P′_, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:
_ds_2 = _g_11_dx_12 + 2_g_12_dx_1_dx_2 . . . . + _g_44_dx_42,
where the magnitudes _g_11, etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates _x_1 . . _x_4. with the
points of the continuum so that we have simply
_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
In this case relations hold in the four-dimensional continuum which are
analogous to those holding in our three-dimensional measurements.
However, the Gauss treatment for _ds_2 which we have given above is not
always possible. It is only possible when sufficiently small regions of
the continuum under consideration may be regarded as Euclidean
continua. For example, this obviously holds in the case of the marble
slab of the table and local variation of temperature. The temperature
is practically constant for a small part of the slab, and thus the
geometrical behaviour of the rods is _almost_ as it ought to be
according to the rules of Euclidean geometry. Hence the imperfections
of the construction of squares in the previous section do not show
themselves clearly until this construction is extended over a
considerable portion of the surface of the table.
We can sum this up as follows: Gauss invented a method for the
mathematical treatment of continua in general, in which
“size-relations” (“distances” between neighbouring points) are defined.
To every point of a continuum are assigned as many numbers (Gaussian
coordinates) as the continuum has dimensions. This is done in such a
way, that only one meaning can be attached to the assignment, and that
numbers (Gaussian coordinates) which differ by an indefinitely small
amount are assigned to adjacent points. The Gaussian coordinate system
is a logical generalisation of the Cartesian co-ordinate system. It is
also applicable to non-Euclidean continua, but only when, with respect
to the defined “size” or “distance,” small parts of the continuum under
consideration behave more nearly like a Euclidean system, the smaller
the part of the continuum under our notice.
XXVI.
THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED
AS A EUCLIDEAN CONTINUUM
We are now in a position to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in Section XVII. In
accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the
four-dimensional, space-time continuum. We called these “Galileian
co-ordinate systems.” For these systems, the four co-ordinates _x, y,
z, t_, which determine an event or—in other words—a point of the
four-dimensional continuum, are defined physically in a simple manner,
as set forth in detail in the first part of this book. For the
transition from one Galileian system to another, which is moving
uniformly with reference to the first, the equations of the Lorentz
transformation are valid. These last form the basis for the derivation
of deductions from the special theory of relativity, and in themselves
they are nothing more than the expression of the universal validity of
the law of transmission of light for all Galileian systems of
reference.
Minkowski found that the Lorentz transformations satisfy the following
simple conditions. Let us consider two neighbouring events, the
relative position of which in the four-dimensional continuum is given
with respect to a Galileian reference-body _K_ by the space co-ordinate
differences _dx, dy, dz_ and the time-difference _dt_. With reference
to a second Galileian system we shall suppose that the corresponding
differences for these two events are _dx′, dy′, dz′, dt′_. Then these
magnitudes always fulfill the condition.[21]
[21] Cf. Appendixes I and II. The relations which are derived there
for the co-ordinates themselves are valid also for co-ordinate
_differences_, and thus also for co-ordinate differentials
(indefinitely small differences).
_dx_2 + _dy_2 + _dz_2 – _c_2_dt_2 = _dx′_2 + _dy′_2 + _dz′_2 –
_c_2_dt′_2.
The validity of the Lorentz transformation follows from this condition.
We can express this as follows: The magnitude
_ds_2 = _dx_2 + _dy_2 + _dz_2 – _c_2 _dt_2,
which belongs to two adjacent points of the four-dimensional space-time
continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace _x, y, z_,
image034
by _x_1, _x_2, _x_3, _x_4, we also obtain the result that
_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
is independent of the choice of the body of reference. We call the
magnitude _ds_ the “distance” apart of the two events or
four-dimensional points.
Thus, if we choose as time-variable the imaginary variable
image035
instead of the real quantity _t_, we can regard the space-time
contintium—accordance with the special theory of relativity—as a
“Euclidean” four-dimensional continuum, a result which follows from the
considerations of the preceding section.
XXVII.
THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A
EUCLIDEAN CONTINUUM
In the first part of this book we were able to make use of space-time
co-ordinates which allowed of a simple and direct physical
interpretation, and which, according to Section XXVI, can be regarded
as four-dimensional Cartesian co-ordinates. This was possible on the
basis of the law of the constancy of the velocity of light. But
according to Section XXI the general theory of relativity cannot retain
this law. On the contrary, we arrived at the result that according to
this latter theory the velocity of light must always depend on the
co-ordinates when a gravitational field is present. In connection with
a specific illustration in Section XXIII, we found that the presence of
a gravitational field invalidates the definition of the coordinates and
the time, which led us to our objective in the special theory of
relativity.
In view of the resuIts of these considerations we are led to the
conviction that, according to the general principle of relativity, the
space-time continuum cannot be regarded as a Euclidean one, but that
here we have the general case, corresponding to the marble slab with
local variations of temperature, and with which we made acquaintance as
an example of a two-dimensional continuum. Just as it was there
impossible to construct a Cartesian co-ordinate system from equal rods,
so here it is impossible to build up a system (reference-body) from
rigid bodies and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
another, shall indicate position and time directly. Such was the
essence of the difficulty with which we were confronted in Section
XXIII.
But the considerations of Sections XXV and XXVI show us the way to
surmount this difficulty. We refer the four-dimensional space-time
continuum in an arbitrary manner to Gauss co-ordinates. We assign to
every point of the continuum (event) four numbers, _x_1, _x_2, _x_3,
_x_4 (co-ordinates), which have not the least direct physical
significance, but only serve the purpose of numbering the points of the
continuum in a definite but arbitrary manner. This arrangement does not
even need to be of such a kind that we must regard _x_1, _x_2, _x_3, as
“space” co-ordinates and _x_4, as a “time” co-ordinate.
The reader may think that such a description of the world would be
quite inadequate. What does it mean to assign to an event the
particular co-ordinates _x_1, _x_2, _x_3, _x_4, if in themselves these
co-ordinates have no significance? More careful consideration shows,
however, that this anxiety is unfounded. Let us consider, for instance,
a material point with any kind of motion. If this point had only a
momentary existence without duration, then it would to described in
space-time by a single system of values _x_1, _x_2, _x_3, _x_4. Thus
its permanent existence must be characterised by an infinitely large
number of such systems of values, the co-ordinate values of which are
so close together as to give continuity; corresponding to the material
point, we thus have a (uni-dimensional) line in the four-dimensional
continuum. In the same way, any such lines in our continuum correspond
to many points in motion. The only statements having regard to these
points which can claim a physical existence are in reality the
statements about their encounters. In our mathematical treatment, such
an encounter is expressed in the fact that the two lines which
represent the motions of the points in question have a particular
system of co-ordinate values, _x_1, _x_2, _x_3, _x_4, in common. After
mature consideration the reader will doubtless admit that in reality
such encounters constitute the only actual evidence of a time-space
nature with which we meet in physical statements.
When we were describing the motion of a material point relative to a
body of reference, we stated nothing more than the encounters of this
point with particular points of the reference-body. We can also
determine the corresponding values of the time by the observation of
encounters of the body with clocks, in conjunction with the observation
of the encounter of the hands of clocks with particular points on the
dials. It is just the same in the case of space-measurements by means
of measuring-rods, as a little consideration will show.
The following statements hold generally: Every physical description
resolves itself into a number of statements, each of which refers to
the space-time coincidence of two events _A_ and _B_. In terms of
Gaussian co-ordinates, every such statement is expressed by the
agreement of their four co-ordinates _x_1, _x_2, _x_3, _x_4. Thus in
reality, the description of the time-space continuum by means of Gauss
co-ordinates completely replaces the description with the aid of a body
of reference, without suffering from the defects of the latter mode of
description; it is not tied down to the Euclidean character of the
continuum which has to be represented.
XXVIII.
EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
We are now in a position to replace the provisional formulation of the
general principle of relativity given in Section XVIII by an exact
formulation. The form there used, “All bodies of reference _K, K′_,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion,” cannot be maintained, because the use of rigid
reference-bodies, in the sense of the method followed in the special
theory of relativity, is in general not possible in space-time
description. The Gauss co-ordinate system has to take the place of the
body of reference. The following statement corresponds to the
fundamental idea of the general principle of relativity: “_All Gaussian
co-ordinate systems are essentially equivalent for the formulation of
the general laws of nature._”
We can state this general principle of relativity in still another
form, which renders it yet more clearly intelligible than it is when in
the form of the natural extension of the special principle of
relativity. According to the special theory of relativity, the
equations which express the general laws of nature pass over into
equations of the same form when, by making use of the Lorentz
transformation, we replace the space-time variables _x, y, z, t_, of a
(Galileian) reference-body _K_ by the space-time variables _x′, y′, z′,
t′_, of a new reference-body _K′_. According to the general theory of
relativity, on the other hand, by application of _arbitrary
substitutions_ of the Gauss variables _x_1, _x_2, _x_3, _x_4, the
equations must pass over into equations of the same form; for every
transformation (not only the Lorentz transformation) corresponds to the
transition of one Gauss co-ordinate system into another.
If we desire to adhere to our “old-time” three-dimensional view of
things, then we can characterise the development which is being
undergone by the fundamental idea of the general theory of relativity
as follows: The special theory of relativity has reference to Galileian
domains, _i.e._ to those in which no gravitational field exists. In
this connection a Galileian reference-body serves as body of reference,
_i.e._ a rigid body the state of motion of which is so chosen that the
Galileian law of the uniform rectilinear motion of “isolated” material
points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian
domains to _non-Galileian_ reference-bodies also. A gravitational field
of a special kind is then present with respect to these bodies (cf.
Sections XX and XXIII).
In gravitational fields there are no such things as rigid bodies with
Euclidean properties; thus the fictitious rigid body of reference is of
no avail in the general theory of relativity. The motion of clocks is
also influenced by gravitational fields, and in such a way that a
physical definition of time which is made directly with the aid of
clocks has by no means the same degree of plausibility as in the
special theory of relativity.
For this reason non-rigid reference-bodies are used, which are as a
whole not only moving in any way whatsoever, but which also suffer
alterations in form _ad lib._ during their motion. Clocks, for which
the law of motion is of any kind, however irregular, serve for the
definition of time. We have to imagine each of these clocks fixed at a
point on the non-rigid reference-body. These clocks satisfy only the
one condition, that the “readings” which are observed simultaneously on
adjacent clocks (in space) differ from each other by an indefinitely
small amount. This non-rigid reference-body, which might appropriately
be termed a “reference-mollusc”, is in the main equivalent to a
Gaussian four-dimensional co-ordinate system chosen arbitrarily. That
which gives the “mollusc” a certain comprehensibility as compared with
the Gauss co-ordinate system is the (really unjustified) formal
retention of the separate existence of the
space co-ordinates as opposed to the time co-ordinate. Every point on
the mollusc is treated as a space-point, and every material point which
is at rest relatively to it as at rest, so long as the mollusc is
considered as reference-body. The general principle of relativity
requires that all these molluscs can be used as reference-bodies with
equal right and equal success in the formulation of the general laws of
nature; the laws themselves must be quite independent of the choice of
mollusc.
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.
XXIX.
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 _K′_. Then with respect to _K′_ 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 _K′_ simply by mathematical
transformation. We interpret this behaviour as the behaviour of
measuring-rods, clocks and material points under the influence of the
gravitational field _G_. Hereupon we introduce a hypothesis: that the
influence of the gravitational field on measuring-rods, 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 Galileian special case, 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 XV only its energy is
of importance for its effect 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 XXI; 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
deductions from the theory which admit of being tested by observation,
to wit, the curvature of light rays by the gravitational field of the
sun,[22] and a displacement of the spectral lines of light reaching us
from large stars, as compared with the corresponding lines for light
produced in an analogous manner terrestrially (_i.e._ by the same kind
of atom).[23] These two deductions from the theory have both been
confirmed.
[22] First observed by Eddington and others in 1919. (Cf. Appendix
III).
[23] Established by Adams in 1924. (Cf. p. 132)
PART III: CONSIDERATIONS ON THE UNIVERSE AS A WHOLE
XXX.
COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY
Part from the difficulty discussed in Section XXI, there is a second
fundamental difficulty attending classical celestial mechanics, which,
to the best of my knowledge, was first discussed in detail by the
astronomer Seeliger. If we ponder over the question as to how the
universe, considered as a whole, is to be regarded, the first answer
that suggests itself to us is surely this: As regards space (and time)
the universe is infinite. There are stars everywhere, so that the
density of matter, although very variable in detail, is nevertheless on
the average everywhere the same. In other words: However far we might
travel through space, we should find everywhere an attenuated swarm of
fixed stars of approrimately the same kind and density.
This view is not in harmony with the theory of Newton. The latter
theory rather requires that the universe should have a kind of centre
in which the density of the stars is a maximum, and that as we proceed
outwards from this centre the group-density of the stars should
diminish, until finally, at great distances, it is succeeded by an
infinite region of emptiness. The stellar universe ought to be a finite
island in the infinite ocean of space.[24]
[24] _Proof_—According to the theory of Newton, the number of “lines
of force” which come from infinity and terminate in a mass m is
proportional to the mass _m_. If, on the average, the mass density ρ0
is constant throughout the universe, then a sphere of volume _V_ will
enclose the average mass ρ0_V_. Thus the number of lines of force
passing through the surface _F_ of the sphere into its interior is
proportional to ρ0_V_. For unit area of the surface of the sphere the
number of lines of force which enters the sphere is thus proportional
to ρ0_V/F_ or to ρ0_R_. Hence the intensity of the field at the
surface would ultimately become infinite with increasing radius _R_ of
the sphere, which is impossible.
This conception is in itself not very satisfactory. It is still less
satisfactory because it leads to the result that the light emitted by
the stars and also individual stars of the stellar system are
perpetually passing out into infinite space, never to return, and
without ever again coming into interaction with other objects of
nature. Such a finite material universe would be destined to become
gradually but systematically impoverished.
In order to escape this dilemma, Seeliger suggested a modification of
Newton’s law, in which he assumes that for great distances the force of
attraction between two masses diminishes more rapidly than would result
from the inverse square law. In this way it is possible for the mean
density of matter to be constant everywhere, even to infinity, without
infinitely large gravitational fields being produced. We thus free
ourselves from the distasteful conception that the material universe
ought to possess something of the nature of a centre. Of course we
purchase our emancipation from the fundamental difficulties mentioned,
at the cost of a modification and complication of Newton’s law which
has neither empirical nor theoretical foundation. We can imagine
innumerable laws which would serve the same purpose, without our being
able to state a reason why one of them is to be preferred to the
others; for any one of these laws would be founded just as little on
more general theoretical principles as is the law of Newton.
XXXI.
THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE
But speculations on the structure of the universe also move in quite
another direction. The development of non-Euclidean geometry led to the
recognition of the fact, that we can cast doubt on the _infiniteness_
of our space without coming into conflict with the laws of thought or
with experience (Riemann, Helmholtz). These questions have already been
treated in detail and with unsurpassable lucidity by Helmholtz and
Poincaré, whereas I can only touch on them briefly here.
In the first place, we imagine an existence in two dimensional space.
Flat beings with flat implements, and in particular flat rigid
measuring-rods, are free to move in a _plane_. For them nothing exists
outside of this plane: that which they observe to happen to themselves
and to their flat “things” is the all-inclusive reality of their plane.
In particular, the constructions of plane Euclidean geometry can be
carried out by means of the rods _e.g._ the lattice construction,
considered in Section XXIV. In contrast to ours, the universe of 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 π, which is
independent of the diameter of the circle. On their spherical surface
our flat beings would find for this ratio the value
image036
_i.e._ a smaller value than π, 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”
(2π2_R_3). 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_ = 4π_r_2; if it is spherical, then _F_ is always
less than 4π_r_2. 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 moderate degree of certainty, and in
this connection the difficulty mentioned in Section XXX finds its
solution.
XXXII.
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
According to the general theory of relativity, the geometrical
properties of space are not independent, but they are determined by
matter. Thus we can draw conclusions about the geometrical structure of
the universe only if we base our considerations on the state of the
matter as being something that is known. We know from experience that,
for a suitably chosen co-ordinate system, the velocities of the stars
are small as compared with the velocity of transmission of light. We
can thus as a rough approximation arrive at a conclusion as to the
nature of the universe as a whole, if we treat the matter as being at
rest.
We already know from our previous discussion that the behaviour of
measuring-rods and clocks is influenced by gravitational fields, _i.e._
by the distribution of matter. This in itself is sufficient to exclude
the possibility of the exact validity of Euclidean geometry in our
universe. But it is conceivable that our universe differs only slightly
from a Euclidean one, and this notion seems all the more probable,
since calculations show that the metrics of surrounding space is
influenced only to an exceedingly small extent by masses even of the
magnitude of our sun. We might imagine that, as regards geometry, our
universe behaves analogously to a surface which is irregularly curved
in its individual parts, but which nowhere departs appreciably from a
plane: something like the rippled surface of a lake. Such a universe
might fittingly be called a quasi-Euclidean universe. As regards its
space it would be infinite. But calculation shows that in a
quasi-Euclidean universe the average density of matter would
necessarily be _nil_. Thus such a universe could not be inhabited by
matter everywhere; it would present to us that unsatisfactory picture
which we portrayed in Section XXX.
If we are to have in the universe an average density of matter which
differs from zero, however small may be that difference, then the
universe cannot be quasi-Euclidean. On the contrary, the results of
calculation indicate that if matter be distributed uniformly, the
universe would necessarily be spherical (or elliptical). Since in
reality the detailed distribution of matter is not uniform, the real
universe will deviate in individual parts from the spherical, _i.e._
the universe will be quasi-spherical. But it will be necessarily
finite. In fact, the theory supplies us with a simple connection[25]
between the space-expanse of the universe and the average density of
matter in it.
[25] For the radius _R_ of the universe we obtain the equation
image037
The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027;
ρ is the average density of the matter and _k_ is a constant connected
with the Newtonian constant of gravitation.
APPENDICES
APPENDIX I
SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
(SUPPLEMENTARY TO SECTION XI)
For the relative orientation of the co-ordinate systems indicated in
Fig. 2, the _x_-axes of both systems permanently 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 _K′_ 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 _K′_ with
the velocity _c_, the propagation relative to the system _K′_ will be
represented by the analogous formula
_x′_ – _ct′_ = 0 . . . . . (2)
Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation
(_x′_ – _ct′_) = λ(_x_ – _ct_) . . . (3).
is fulfilled in general, where λ 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′_) = (_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 λ and μ
where
image038
and
image039
we obtain the equations
image040
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 _K′_ we have permanently _x′_ = 0, and hence
according to the first of the equations (5)
image041
If we call _v_ the velocity with which the origin of _K′_ is moving
relative to _K_, we then have
image042
The same value _v_ can be obtained from equations (5), if we calculate
the velocity of another point of _K′_ 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 _K′_ 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 _K′_ 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 Δ_x′_ =
1 when measured in the _K′_ system are thus separated in our
instantaneous photograph by the distance
image043
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
image044
From this we conclude that two points on the _x_-axis separated by the
distance 1 (relative to _K_) will be represented on our snapshot by the
distance
image045
But from what has been said, the two snapshots must be identical; hence
Δ_x_ in (7) must be equal to Δ_x′_ in (7_a_), so that we obtain
image046
The equations (6) and (7_b_) 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 XI.
image047
Thus we have obtained the Lorentz transformation for events on the
_x_-axis. It satisfies the condition
_x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (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
image048
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
image049
or, if we square this equation, according to the equation
_x_2 + _y_2 + _z_2 – _c_2_t_2 = 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 _K′_—in accordance with the
corresponding formula
_r′_ = _ct′_
or,
_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_).
In order that equation (10_a_) may be a consequence of equation (10),
we must have
_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2)
(11).
Since equation (8_a_) must hold for points on the _x_-axis, we thus
have σ = 1. It is easily seen that the Lorentz transformation really
satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_)
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 _K′_ be
chosen so that they are spatially parallel to those of _K_. It is also
not essential that the velocity of translation of _K′_ 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_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2
(11_a_).
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 (11_a_) agrees with the
right-hand side.
APPENDIX II
MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)
(SUPPLEMENTARY TO SECTION XVII)
We can characterise the Lorentz transformation still more simply if we
introduce the imaginary
image031
in place of _t_, as time-variable. If, in accordance with this, we
insert
image050
and similarly for the accented system _K′_, 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_12 + _x_22 + _x_32 + _x_42 (12).
That is, by the afore-mentioned choice of “coordinates,” (11_a_) [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_12 + _x_22 + _x_32
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 four-dimensional
“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 present 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°. 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
image051
(_N.B._—One complete revolution corresponds to the angle 2π in the
absolute angular measure customary in physics, and the above expression
given the amount by which the radius sun-planet exceeds this angle
during the interval between one perihelion and the next.) In this
expression _a_ represents the major semi-axis of the ellipse, _e_ its
eccentricity, _c_ the velocity of light, and _T_ the period of
revolution of the planet. Our result may also be stated as follows:
According to the general theory of relativity, the major axis of the
ellipse rotates round the sun in the same sense as the orbital motion
of the planet. Theory requires that this rotation should amount to 43
seconds of arc per century for the planet Mercury, but for the other
Planets of our solar system its magnitude should be so small that it
would necessarily escape detection.[26]
[26] 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.
In point of fact, astronomers have found that the theory of Newton does
not suffice to calculate the observed motion of Mercury with an
exactness corresponding to that of the delicacy of observation
attainable at the present time. After taking account of all the
disturbing influences exerted on Mercury by the remaining planets, it
was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
perihelial movement of the orbit of Mercury remained over, the amount
of which does not differ sensibly from the above mentioned +43 seconds
of arc per century. The uncertainty of the empirical result amounts to
a few seconds only.
(_b_) Deflection of Light by a Gravitational Field
image052
In Section XXII it has been already mentioned that according to the
general theory of relativity, a ray of light will experience a
curvature of its path when passing through a gravitational field, this
curvature being similar to that experienced by the path of a body which
is projected through a gravitational field. As a result of this theory,
we should expect that a ray of light which is passing close to a
heavenly body would be deviated towards the latter. For a ray of light
which passes the sun at a distance of Δ sun-radii from its centre, the
angle of deflection (α) should amount to
image053
It may be added that, according to the theory, half of this deflection
is produced by the Newtonian field of attraction of the sun, and the
other half by the geometrical modification (“curvature”) of space
caused by the sun.
This result admits of an experimental test by means of the photographic
registration of stars during a total eclipse of the sun. The only
reason why we must wait for a total eclipse is because at every other
time the atmosphere is so strongly illuminated by the light from the
sun that the stars situated near the sun’s disc are invisible. The
predicted effect can be seen clearly from the accompanying diagram. If
the sun (_S_) were not present, a star which is practically infinitely
distant would be seen in the direction _D_1, as observed front the
earth. But as a consequence of the deflection of light from the star by
the sun, the star will be seen in the direction _D_2, _i.e._ at a
somewhat greater distance from the centre of the sun than corresponds
to its real position.
In practice, the question is tested in the following way. The stars in
the neighbourhood of the sun are photographed during a solar eclipse.
In addition, a second photograph of the same stars is taken when the
sun is situated at another position in the sky, _i.e._ a few months
earlier or later. As compared with the standard photograph, the
positions of the stars on the eclipse-photograph ought to appear
displaced radially outwards (away from the centre of the sun) by an
amount corresponding to the angle _a_.
We are indebted to the [British] Royal Society and to the Royal
Astronomical Society for the investigation of this important deduction.
Undaunted by the [first world] war and by difficulties of both a
material and a psychological nature aroused by the war, these societies
equipped two expeditions—to Sobral (Brazil), and to the island of
Principe (West Africa)—and sent several of Britain’s most celebrated
astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to
obtain photographs of the solar eclipse of 29th May, 1919. The relative
discrepancies to be expected between the stellar photographs obtained
during the eclipse and the comparison photographs amounted to a few
hundredths of a millimetre only. Thus great accuracy was necessary in
making the adjustments required for the taking of the photographs, and
in their subsequent measurement.
The results of the measurements confirmed the theory in a thoroughly
satisfactory manner. The rectangular components of the observed and of
the calculated deviations of the stars (in seconds of arc) are set
forth in the following table of results:
image054
(_c_) Displacement of Spectral Lines Towards the Red
In Section XXIII it has been shown that in a system _K′_ which is in
rotation with regard to a Galileian system _K_, clocks of identical
construction, and which are considered at rest with respect to the
rotating reference-body, go at rates which are dependent on the
positions of the clocks. We shall now examine this dependence
quantitatively. A clock, which is situated at a distance r from the
centre of the disc, has a velocity relative to _K_ which is given by
_v_ = ω_r_,
where ω represents the angular velocity of rotation of the disc _K′_
with respect to _K_. If _v_0, represents the number of ticks of the
clock per unit time (“rate” of the clock) relative to _K_ when the
clock is at rest, then the “rate” of the clock (_v_) when it is moving
relative to _K_ with a velocity _v_, but at rest with respect to the
disc, will, in accordance with Section XII, be given by
image055
or with sufficient accuracy by
image056
This expression may also be stated in the following form:
image057
If we represent the difference of potential of the centrifugal force
between the position of the clock and the centre of the disc by φ,
_i.e._ the work, considered negatively, which must be performed on the
unit of mass against the centrifugal force in order to transport it
from the position of the clock on the rotating disc to the centre of
the disc, then we have
image058
From this it follows that
image059
In the first place, we see from this expression that two clocks of
identical construction will go at different rates when situated at
different distances from the centre of the disc. This result is also
valid from the standpoint of an observer who is rotating with the disc.
Now, as judged from the disc, the latter is in a gravitational field of
potential φ, hence the result we have obtained will hold quite
generally for gravitational fields. Furthermore, we can regard an atom
which is emitting spectral lines as a clock, so that the following
statement will hold:
_An atom absorbs or emits light of a frequency which is dependent on
the potential of the gravitational field in which it is situated._
The frequency of an atom situated on the surface of a heavenly body
will be somewhat less than the frequency of an atom of the same element
which is situated in free space (or on the surface of a smaller
celestial body).
Now φ = – _K (M/r)_, where _K_ is Newton’s constant of gravitation, and
_M_ is the mass of the heavenly body. Thus a displacement towards the
red ought to take place for spectral lines produced at the surface of
stars as compared with the spectral lines of the same element produced
at the surface of the earth, the amount of this displacement being
image060
For the sun, the displacement towards the red predicted by theory
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 Relativitäts-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.[27]
[27] The displacement 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 XXXII)
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 109 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 remark 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