Section 2 of 12 - Table of Contents
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
man on the footpath the other. Each of the observers determines the
position on his own reference-body occupied by the stone at each tick
of the clock he is holding in his hand. In this connection we have not
taken account of the inaccuracy involved by the finiteness of the
velocity of propagation of light. With this and with a second
difficulty prevailing here we shall have to deal in detail later.
Notes
*) That is, a curve along which the body moves.
THE GALILEIAN SYSTEM OF CO-ORDINATES
As is well known, the fundamental law of the mechanics of
Galilei-Newton, which is known as the law of inertia, can be stated
thus: A body removed sufficiently far from other bodies continues in a
state of rest or of uniform motion in a straight line. This law not
only says something about the motion of the bodies, but it also
indicates the reference-bodies or systems of coordinates, permissible
in mechanics, which can be used in mechanical description. The visible
fixed stars are bodies for which the law of inertia certainly holds to
a high degree of approximation. Now if we use a system of co-ordinates
which is rigidly attached to the earth, then, relative to this system,
every fixed star describes a circle of immense radius in the course of
an astronomical day, a result which is opposed to the statement of the
law of inertia. So that if we adhere to this law we must refer these
motions only to systems of coordinates relative to which the fixed
stars do not move in a circle. A system of co-ordinates of which the
state of motion is such that the law of inertia holds relative to it
is called a " Galileian system of co-ordinates." The laws of the
mechanics of Galflei-Newton can be regarded as valid only for a
Galileian system of co-ordinates.
THE PRINCIPLE OF RELATIVITY
(IN THE RESTRICTED SENSE)
In order to attain the greatest possible clearness, let us return to
our example of the railway carriage supposed to be travelling
uniformly. We call its motion a uniform translation ("uniform" because
it is of constant velocity and direction, " translation " because
although the carriage changes its position relative to the embankment
yet it does not rotate in so doing). Let us imagine a raven flying
through the air in such a manner that its motion, as observed from the
embankment, is uniform and in a straight line. If we were to observe
the flying raven from the moving railway carriage. we should find that
the motion of the raven would be one of different velocity and
direction, but that it would still be uniform and in a straight line.
Expressed in an abstract manner we may say : If a mass m is moving
uniformly in a straight line with respect to a co-ordinate system K,
then it will also be moving uniformly and in a straight line relative
to a second co-ordinate system K1 provided that the latter is
executing a uniform translatory motion with respect to K. In
accordance with the discussion contained in the preceding section, it
follows that:
If K is a Galileian co-ordinate system. then every other co-ordinate
system K' is a Galileian one, when, in relation to K, it is in a
condition of uniform motion of translation. Relative to K1 the
mechanical laws of Galilei-Newton hold good exactly as they do with
respect to K.
We advance a step farther in our generalisation when we express the
tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course
with respect to K1 according to exactly the same general laws as with
respect to K. This statement is called the principle of relativity (in
the restricted sense).
As long as one was convinced that all natural phenomena were capable
of representation with the help of classical mechanics, there was no
need to doubt the validity of this principle of relativity. But in
view of the more recent development of electrodynamics and optics it
became more and more evident that classical mechanics affords an
insufficient foundation for the physical description of all natural
phenomena. At this juncture the question of the validity of the
principle of relativity became ripe for discussion, and it did not
appear impossible that the answer to this question might be in the
negative.
Nevertheless, there are two general facts which at the outset speak
very much in favour of the validity of the principle of relativity.
Even though classical mechanics does not supply us with a sufficiently
broad basis for the theoretical presentation of all physical
phenomena, still we must grant it a considerable measure of " truth,"
since it supplies us with the actual motions of the heavenly bodies
with a delicacy of detail little short of wonderful. The principle of
relativity must therefore apply with great accuracy in the domain of
mechanics. But that a principle of such broad generality should hold
with such exactness in one domain of phenomena, and yet should be
invalid for another, is a priori not very probable.
We now proceed to the second argument, to which, moreover, we shall
return later. If the principle of relativity (in the restricted sense)
does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
which are moving uniformly relative to each other, will not be
equivalent for the description of natural phenomena. In this case we
should be constrained to believe that natural laws are capable of
being formulated in a particularly simple manner, and of course only
on condition that, from amongst all possible Galileian co-ordinate
systems, we should have chosen one (K[0]) of a particular state of
motion as our body of reference. We should then be justified (because
of its merits for the description of natural phenomena) in calling
this system " absolutely at rest," and all other Galileian systems K "
in motion." If, for instance, our embankment were the system K[0] then
our railway carriage would be a system K, relative to which less
simple laws would hold than with respect to K[0]. This diminished
simplicity would be due to the fact that the carriage K would be in
motion (i.e."really")with respect to K[0]. In the general laws of
nature which have been formulated with reference to K, the magnitude
and direction of the velocity of the carriage would necessarily play a
part. We should expect, for instance, that the note emitted by an
organpipe placed with its axis parallel to the direction of travel
would be different from that emitted if the axis of the pipe were
placed perpendicular to this direction.
Now in virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of about
30 kilometres per second. If the principle of relativity were not
valid we should therefore expect that the direction of motion of the
earth at any moment would enter into the laws of nature, and also that
physical systems in their behaviour would be dependent on the
orientation in space with respect to the earth. For owing to the
alteration in direction of the velocity of revolution of the earth in
the course of a year, the earth cannot be at rest relative to the
hypothetical system K[0] throughout the whole year. However, the most
careful observations have never revealed such anisotropic properties
in terrestrial physical space, i.e. a physical non-equivalence of
different directions. This is very powerful argument in favour of the
principle of relativity.
THE THEOREM OF THE
ADDITION OF VELOCITIES
EMPLOYED IN CLASSICAL MECHANICS
Let us suppose our old friend the railway carriage to be travelling
along the rails with a constant velocity v, and that a man traverses
the length of the carriage in the direction of travel with a velocity
w. How quickly or, in other words, with what velocity W does the man
advance relative to the embankment during the process ? The only
possible answer seems to result from the following consideration: If
the man were to stand still for a second, he would advance relative to
the embankment through a distance v equal numerically to the velocity
of the carriage. As a consequence of his walking, however, he
traverses an additional distance w relative to the carriage, and hence
also relative to the embankment, in this second, the distance w being
numerically equal to the velocity with which he is walking. Thus in
total be covers the distance W=v+w relative to the embankment in the
second considered. We shall see later that this result, which
expresses the theorem of the addition of velocities employed in
classical mechanics, cannot be maintained ; in other words, the law
that we have just written down does not hold in reality. For the time
being, however, we shall assume its correctness.
THE APPARENT INCOMPATIBILITY OF THE
LAW OF PROPAGATION OF LIGHT WITH THE
PRINCIPLE OF RELATIVITY
There is hardly a simpler law in physics than that according to which
light is propagated in empty space. Every child at school knows, or
believes he knows, that this propagation takes place in straight lines
with a velocity c= 300,000 km./sec. At all events we know with great
exactness that this velocity is the same for all colours, because if
this were not the case, the minimum of emission would not be observed
simultaneously for different colours during the eclipse of a fixed
star by its dark neighbour. By means of similar considerations based
on observa- tions of double stars, the Dutch astronomer De Sitter was
also able to show that the velocity of propagation of light cannot
depend on the velocity of motion of the body emitting the light. The
assumption that this velocity of propagation is dependent on the
direction "in space" is in itself improbable.
In short, let us assume that the simple law of the constancy of the
velocity of light c (in vacuum) is justifiably believed by the child
at school. Who would imagine that this simple law has plunged the
conscientiously thoughtful physicist into the greatest intellectual
difficulties? Let us consider how these difficulties arise.
Of course we must refer the process of the propagation of light (and
indeed every other process) to a rigid reference-body (co-ordinate
system). As such a system let us again choose our embankment. We shall
imagine the air above it to have been removed. If a ray of light be
sent along the embankment, we see from the above that the tip of the
ray will be transmitted with the velocity c relative to the
embankment. Now let us suppose that our railway carriage is again
travelling along the railway lines with the velocity v, and that its
direction is the same as that of the ray of light, but its velocity of
course much less. Let us inquire about the velocity of propagation of
the ray of light relative to the carriage. It is obvious that we can
here apply the consideration of the previous section, since the ray of
light plays the part of the man walking along relatively to the
carriage. The velocity w of the man relative to the embankment is here
replaced by the velocity of light relative to the embankment. w is the
required velocity of light with respect to the carriage, and we have
w = c-v.
The velocity of propagation ot a ray of light relative to the carriage
thus comes cut smaller than c.
But this result comes into conflict with the principle of relativity
set forth in Section V. For, like every other general law of
nature, the law of the transmission of light in vacuo [in vacuum]
must, according to the principle of relativity, be the same for the
railway carriage as reference-body as when the rails are the body of
reference. But, from our above consideration, this would appear to be
impossible. If every ray of light is propagated relative to the
embankment with the velocity c, then for this reason it would appear
that another law of propagation of light must necessarily hold with
respect to the carriage -- a result contradictory to the principle of
relativity.
In view of this dilemma there appears to be nothing else for it than
to abandon either the principle of relativity or the simple law of the
propagation of light in vacuo. Those of you who have carefully
followed the preceding discussion are almost sure to expect that we
should retain the principle of relativity, which appeals so
convincingly to the intellect because it is so natural and simple. The
law of the propagation of light in vacuo would then have to be
replaced by a more complicated law conformable to the principle of
relativity. The development of theoretical physics shows, however,
that we cannot pursue this course. The epoch-making theoretical
investigations of H. A. Lorentz on the electrodynamical and optical
phenomena connected with moving bodies show that experience in this
domain leads conclusively to a theory of electromagnetic phenomena, of
which the law of the constancy of the velocity of light in vacuo is a
necessary consequence. Prominent theoretical physicists were theref
ore more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were
contradictory to this principle.
At this juncture the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space,
it became evident that in realily there is not the least
incompatibilitiy between the principle of relativity and the law of
propagation of light, and that by systematically holding fast to both
these laws a logically rigid theory could be arrived at. This theory
has been called the special theory of relativity to distinguish it
from the extended theory, with which we shall deal later. In the
following pages we shall present the fundamental ideas of the special
theory of relativity.
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