LECTURE II 

CONCLUSIONS FROM THE RELATIVITY THEORY 

A. INTRODUCTION 

The theory of relativity of Einstein and his collaborators 
has profoundly revolutionized our conceptions of nature. 
Time and space have ceased to be entities and have become 
mere forms of conception. The length of a body and the 
time on it and the mass have ceased to be fixed properties 
and have become dependent on the conditions of obser- 
vation. The law of conservation of matter thus had to be 
abandoned and mass became a manifestation of energy. 
The law of gravitation has been recast, and the force of 
gravitation has become an effect of inertial motion, like 
centrifugal force. The ether has been abandoned, and the 
field of force of Faraday and Maxwell has become the 
fundamental conception of physics. The laws of mechanics ^ 
have been changed, and time and space have been bound' 
together in the four-dimensional world space, the dimen- 
sions of which are neither space nor time, but a symmetrical 
combination of both. 

With such profound changes in the laws and conceptions 
of nature, it is startling to see that all the numerical results 
of calculations have remained the same. With a very few 
exceptions, the differences between the results of the old 
and of the new conceptions are so small that they usually 
cannot be observed even by the most accurate scientific 
investigation, and in the few instances where the differences 
have been measured, as in the disturbances of Mercury's 
orbit, the bending of the beam of light in the gravitational 
field, etc., they are close to the limits of observation. 

12 


CONCLUSIONS FROM RELATIVITY THEORY 13 

We have seen that the length of a body and the time on it 
change with the relative velocity of the observer. The 
highest velocities which we can produce (outside of ionic 
velocities) are the velocity of the rifle bullet (1000 meters 
per second) , the velocity of expansion of high-pressure steam 
into a vacuum (2000 meters per second), and the velocity of 
propagation of the detonation in high explosives (6000 meters 
per second). At these velocities the change of length and 
time is one part in 180,000 millions, 22,000 millions and 
5000 millions respectively. The highest cosmic velocity 
probably is that of a comet passing the sun at grazing 
distance, 200 kilometers per second. The shortening of the 
length even then would be only one in four millions. 

The bending of a beam of light in the gravitational field 
of the sun is only a fraction of a thousandth of a degree. 

The overrunning of the perihelium of the planet Mercury 
is only about 20 miles out of more than a hundred million 
miles. 

Therefore the principal value of the relativity theory 
thus far consists in the better conception of nature and its 
laws which it affords. Some of the most interesting illustra- 
tions of this will be discussed in the following pages. 

B. THE ETHER AND THE FIELD OF FORCE 

Newton's corpuscular theory of light explained radiation 
as a bombardment by minute particles projected at 
extremely high velocities, in much the same way as the 
alpha and the beta rays are explained today. This 
corpuscular theory was disproven by the phenomenon of 
interference, in the following manner: If the corpuscular 
theory is right, then two equal beams of light, when super- 
imposed, must always combine to a beam of twice the 
intensity. Experience, however, shows that two equal 
beams of light when superimposed, may give a beam of 
double intensity, or may extinguish each other and give 
darkness, or may give anything between these two 


14 RELATIVITY AND SPACE 

extremes. This can be explained only by assuming light 
to be a wave, like an alternating current. Depending on 
their phase relation, the combination of two waves (as two 
beams of light or two alternating currents) may be anything 
between their sum and their difference. Thus the two 
alternating currents consumed by two incandescent lamps 
add, being in phase; the two alternating currents consumed 
respectively by an inductance and by a capacity subtract, 
giving a resultant equal to their difference; that is, if they 
are equal, they extinguish each other. The phenomenon of 
interference thus leads to the wave theory of light. 

If light is a wave motion, there must be something to 
move, and this hypothetical carrier of the light wave has 
been called the ether. Here our troubles begin. The 
phenomenon of polarization shows that light is a transverse 
wave; that is, the ether atoms move at right angles to the 
light beam, and not in its direction as. is the case with 
sound waves. In such transverse motion a vibrating ether 
atom neither approaches nor recedes from the next ether 
atom, and the only way in which in the propagation of the 
light wave the vibratory motion of each ether atom can be 
transmitted to the next one is by forces acting between the 
ether atoms so as to hold them together in their relative 
positions. Bodies in which the atoms are held together 
in their relative positions are solid bodies. That is, trans- 
verse waves can exist only in solid bodies. As the velocity 
of light is extremely high, the forces between the ether 
atoms which transmit the vibrations must be very great. 
That is, the ether is a solid body of very high rigidity, 
infinitely more rigid than steel. 

At the same time, the ether must be of extremely high 
tenuity, since all the cosmic bodies move through it at high 
velocities without meeting any friction. In the revolution 
of the earth around the sun either the ether stands still and 
the earth moves through the ether, at 20 miles per second, 
or the earth carries a mass of ether with it (''ether drift"). 
In the first case there should be friction between the mass 


\ 

\ 

\ 

\ 
\ 
\ 
\ 

0 o 


CONCL USIONS FROM RELA TIVITY THEOR Y 15 

of the earth and the ether; in the last case there should be 
friction between the ether carried along with the earth and 
the stationary ether. But in either case the frictional 
energy would come from the earth, would slow down the 
speed of the earth and show astronomically as a change of 
the orbit of the earth, and no 
such evidence of ether friction 
is observed. g' 

Which of the two alter- 
native possibilities- — a sta- 
tionary ether or an ether 
moving with the earth — is 
true can be determined ex- 
perimentally. Suppose, in 
Fig. 1, i^ is a railway train 
moving at speed v, and I p^^ ^ 

shoot a rifle bullet through 

the train, in the direction at right angles to the track, at 
velocity c. The bullet enters the train at the point A 
of the track and leaves it at the point B of the track. But 
while the bullet passes from A to 5 the train has moved 
forward and the point B' of the train has come to the 
point B of the track. Thus with regard to the train- — that 
is, for an observer in the train- — the bullet moves from A to 
B' and thus appears to have come not from 0 but from 0' , 
from a direction farther forward by angle a = OAO', where 
tan a = v/c. 

Now, instead of the train consider the earth; instead of 
the bullet, a beam of light from some fixed star. If, then, 
the ether stands still, the beam of light from a fixed star, 
carried by the ether, would go in a straight line OABC, and 
from the moving earth we would see the fixed star, not 
where it really is, at 0, but deflected in the direction of the 
earth's motion, toward 0', and a half a year later, when the 
earth in its orbit around the sun is moving in the opposite 
direction, we should see the star deflected in the opposite 
direction. During the annual revolution of the earth 


16 RELATIVITY AND SPACE 

around the sun all the fixed stars thus would describe small 
circles. This is the case, and this phenomenon, called 
'^ aberration," proves that the ether stands still and is not 
carried along by the cosmic bodies. 

If the ether stands still and the earth is moving through 
it, then by the Newtonian mechanics the velocity of light 
relative to the earth — that is, as observed here on earth — 
should in the direction of the earth's motion be 20 miles less, 
in the opposite direction 20 miles more, than the veloc- 
ity with regard to the stationary ether. If, however, the 
ether moves with the earth, then obviously the velocity of 
light on earth should be the same in all directions. The 
latter is the case, and thus it is proved that the ether 
moves with the earth and does not stand still. This is 
exactly the opposite conclusion from that given by the 
aberration. 

Thus the conception of the ether is one of those untenable 
hypotheses which have been made in the attempt to 
explain some difficulty. The more it is studied and con- 
clusions drawn from it, the more contradictions we get, and 
the more unreasonable and untenable it becomes. It has 
been merely conservatism or lack of courage which has 
kept us from openly abandoning the ether hypothesis. 
The belief in an ether is in contradiction to the relativity 
theory, since this theory shows that there is no absolute 
position nor motion, but that all positions and motions are 
relative and equivalent. If, however, an ether existed, 
then the position at rest with regard to the ether, and the 
motion relative to the ether, would be absolute and different 
from other positions and motions, and the assumption of an 
ether thus leads to the conclusion of the existence of abso- 
lute motion and position and so contradicts the relativity 
theory. 

Thus the hypothesis of the ether has been finally dis- 
proven and abandoned. There is no such thing as the 
ether, and light and the wireless waves are not wave motions 
of the ether. 


CONCLUSIONS FROM RELATIVITY THEORY 17 

What, then, is the fallacy in the wave theory of light 
which has led to the erroneous conception of an ether? 

The phenomenon of interference proves that light is a 
wave, a periodic phenomenon, just like an alternating 
current. Thus the wave theory of light and radiation 
stands today as unshaken as ever. However, when this 
theory was established, the only waves with which people 
were familiar were the waves in water and the sound waves, 
and both are wave motions. As the only known waves 
were wave motions, it was natural that the light wave also 
was considered as a wave motion. This led to the question 
of what moves in the light wave, and thus to the hypothesis 
of the ether, with all its contradictory and illogical attri- 
butes. But there is no more reason to assume the light 
wave to be a wave motion than there is to assume the 
alternating-current wave to be a motion of matter. We 
know that nothing material is moving in the alternating- 
current or voltage wave, and if the wave theory of light 
had been propounded after the world had become familiar 
with electric waves^ — that is, with waves or periodic phenom- 
ena which are not wave motions of matter^ — the error of 
considering the light wave as a wave motion would never 
have been made and the ether theory would never have been 
propounded. 

Hence the logical error which led to the ether theory is 
the assumption that a w^ave must necessarily be a wave 
motion. A wave may be a wave motion of matter, as the 
water wave and sound wave, or it may not be a wave 
motion. Electrical engineering has dealt with alternating- 
current and voltage waves, calculated their phenomena and 
applied them industrially, but has never considered that 
anything material moves in the alternating-current wave 
and has never felt the need of an ether as the hypothetical 
carrier of the electric wave. When Maxwell and Hertz 
proved the identity of the electromagnetic wave and the 
light wave, the natural conclusion was that light is an 
electromagnetic wave, that the ether was unnecessary also 

2 


M 


Fig. 2. 


18 RELATIVITY AND SPACE 

in optics, and, as it was illogical, to abandon it. But, 
curiously enough, we then began to talk about electric 
waves in the ether, about ether telegraphy, etc.- — instead of 
abandoning it, that is, we dragged the conception of the 

ether into electrical en- 
\ gineering, where it never 

had been found necessary 

before. 

What, then, is the 

mechanism of the light 
wave and the electromagnetic wave? 

Suppose we have a permanent bar magnet M (Fig. 2) and 
bring a piece of iron / near it. It is attracted, or moved; 
that is, a force is exerted on it. We bring a piece of copper 
near the magnet, and nothing happens. We say the space 
surrounding the magnet is a magnetic field. A field, or 
field of force, we define as "a condition in space exerting a 
force on a body susceptible to this field." Thus, a piece of 
iron being magnetizable — that is, susceptible to a magnetic 
field^ — ^will be acted upon; a piece of copper, not being 
magnetizable, shows no action. A field is completely 
defined and characterized at any point by its intensity and 
its direction, and in Faraday's pictorial representation of 
the field by the lines of force, the direction of the lines of 
force represents the direction of the field, and the density 
of the lines of force represents the intensity of the field. 
To produce a field of force requires energy, and this 
energy is stored in the space we call the field. Thus we can 
go further and define the field as ''a condition of energy 
storage in space exerting a force on a body susceptible to 
this energy.'' 

The space surrounding a magnet is a magnetic field. 
If we electrify a piece of sealing wax by rubbing it, it 
surrounds itself by a dielectric or electrostatic field, 
and bodies susceptible to electrostatic forces- — such as light 
pieces of paper — are attracted. The earth is surrounded 
by a gravitational field, the lines of gravitational force 


CONCLUSIONS FROM RELATIVITY THEORY 19 

issuing radially from the earth. If a stone falls to the 
earth, it is due to the. stone being in the gravitational 
field of the earth and being acted upon by it. 

This illustrates the difference between the conception 
of the field held by Faraday and Maxwell, in explaining 
force action, and the Newtonian theory of action at a 
distance. To Newton the earth is attracted by the sun 
and therefore revolves around it, because the force of 
gravitation acts across the distance between sun and 
earth in a manner proportional to the mass and inversely 
proportional to the square of the distance. To us the 
earth revolves around the sun because it is in the gravi- 
tational field of the sun and this field exerts a force on the 
earth. The force is proportional to the mass of the 
earth and to the intensity of the gravitational field — 
that is, the density of the lines of gravitational force. As 
the lines of force issue radially from the sun, their density 
decreases with the square of the distance. 

Both conceptions, that of action at a distance and that of 
the field of force, thus give the same result and in some 
respects are merely different ways of looking at the same 
thing. But the first, the action at a distance, is logically 
repugnant to our ideas, as we cannot conceive how a body 
can act across empty space at a place where it is not and 
with which nothing connects it. 

However, there is something more than mere logical 
preference in favor of the conception of the field. We may 
illustrate this on the magnetic field (Fig. 2). The old con- 
ception, before Faraday, was that the poles of the mag- 
net M act across the distance on the magnet poles induced 
in the iron I. Accepting this action at a distance, we 
should expect that as long as the magnet M and the iron / 
remain the same and in the same relative position the force 
should be the same, no matter what happens elsewhere in 
the space which we called the magnetic field. This, how- 
ever, is not the case, but the conditions existing anywhere 
in the field, outside of M and /, may affect and greatly 


20 RELATIVITY AND SPACE 

modify the action of M on I. This is difficult to explain 
in a simple manner by the theory of the action at a dis- 
tance, but very simple and obvious by the field theory, as 
anything done anywhere in the space outside of M and I 
which changes the field at / also must change the force 
exerted on /. Thus a piece of iron A (Fig. 3) interposed 
between M and / increases the force by concentrating the 
field on /. A copper disk C inserted between I and M 
(Fig. 4) as long as it is at rest has no effect, because copper 
is not susceptible to the magnetic field. If, however, the 
copper disk C is revolved, the force on / decreases with 
increasing speed of C and finally virtually vanishes, 
because electric currents induced in C screen off the field 
from I. Pieces of iron, like B and C in Fig. 5, may reverse 
the force exerted by M on / from an attraction to a repul- 
sion by reversing the magnetic field at I. 

Thus the theory of the field of force has proven simpler 
and more workable than the conception of the action 
at a distance, and for this reason it has been generally 
accepted. 

Suppose now, in Fig. 2, instead of a periuanent magnet 
M, we have a bundle of soft iron wires with a coil of insul- 
ated copper wire around it and send a constant direct cur- 
rent through the latter. We then have an electromagnet, 
and the space surrounding M is a magnetic field, character- 
ized at every point by an intensity and a direction. If 
now we increase the current, the magnetic field increases; 
if we decrease the current, the field decreases ; if we reverse 
the current, the field reverses; if we send an alternating 
current through the coil, the magnetic field alternates — 
that is, is a periodic phenomenon or a wave, an alternating 
magnetic field wave. 

Similarly, by connecting an insulated conductor to a 
source of voltage we produce surrounding it an electro- 
static or dielectric field — a constant field if the voltage is 
constant, an alternating dielectric field — that is, a periodic 
or wave phenomenon^ — ^if we use an alternating voltage. 


CONCLUSIONS FROM RELATIVITY THEORY 21 

Magnetic and dielectric fields are usually combined, since 
where there is a current producing a magnetic field there 
is a voltage producing a dielectric field. Thus the space 
surrounding a conductor carrying an electric current is an 
electromagnetic field — that is, a combination of a magnetic 
field, concentric with the conductor, and a dielectric field, 
radial to the conductor. 

If the current and voltage ^ 

are constant, the electro- <^^^^ 

magnetic field is constant 


M 


or stationary relative to the 
conductor, just as the ^°' ^' 

gravitational field of the earth is stationary with regard 
to the earth. If the current and voltage alternate, the 
electromagnetic field alternates — ^that is, is a periodic field 
or an electromagnetic wave. 

Maxwell then has deduced mathematically, and Hertz 
demonstrated experimentally, that the alternating electro- 
magnetic field — that is, the electromagnetic wave — has the 
same speed of propagation as the light wave, and has shown 
that the electromagnetic wave and the (polarized) light 
wave are identical in all their properties. Hence light is 
an electromagnetic wave — that is, an alternating electro- 
magnetic field of extremely high frequency. 

Electrophysics has been successfully developed to its 
present high state, and has dealt with alternating currents, 

voltages and electromag- 
netic fields, without ever 
requiring or considering a 
medium such as the ether. 
Whatever may be the 
mechanism of the electro- 
magnetic wave, it certainly 
is not a mere transverse wave motion of matter, and 
the light, being shown to be a high-frequency electro- 
magnetic wave, cannot be considered any more as a 
wave motion of the ether. The ether thus vanishes. 


M 


Fig. 4. 


22 RELATIVITY AND SPACE 

following the phlogiston and other antiquated physical 
conceptions. 

The conception of the field of force, or, as we should more 
correctly say, the field of energy, thus takes the place of 
the conception of action at a distance and of the ether. 
The beam of light and the electromagnetic wave (like that 
of the radio communication station or that surrounding a 
power transmission line) are therefore periodic alternations 
of the electromagnetic energy field in space, and the differ- 
ences are merely those due to the differences of frequency. 
Thus the electromagnetic field of the 60-cycle transmission 
line has a wave length of 3 X lO^V^O cm. = 5000 km. Its 
extent is limited to the space between the conductors and 
their immediate surroundings, being therefore extremely 
small compared with the wave length, and under these 
conditions the part of the electromagnetic energy which is 
radiated into space is extremely small. It is so small that 
it may be neglected and that it may be said that all the energy 
supplied by the source of power which is consumed in produc- 
ing the electromagnetic field is returned to the supply circuit 
at the disappearance of the field. In radio communication 
wave lengths of 15,000 to 200 meters and less^ — that is, fre- 
quencies of 20,000 to 1,500,000 cycles and more — are used, 
and the circuit is arranged so as to give the electromagnetic 
field the greatest possible extent, it being the field which 

carries the message. 
Then a large part, or 
even the major part, of 
the energy of the electro- 
magnetic field is radiated. 
At the frequency of the 
light wave, about 600 
millions of millions of cycles, the wave length, about 50 
micro cm., is an insignificant part of the extent of the 
field- — ^that is, of the distance to which the beam travels — 
and therefore virtually all the energy of the field is radiated, 
none returned to the radiator. 


Fig. 5. 


CONCLUSIONS FROM RELATIVITY THEORY 23 

As the electromagnetic field represents energy storage in 
space, it cannot extend through space instantaneously, but 
must propagate through space at a finite velocity, the rate 
at which the power radiated by the source of the field can 
fill up the space with the field energy. The field energy is 
proportional to the energy radiation of the source of the 
field (transmission line, radio antenna, incandescent body) 
and to the electromagnetic constants of space (permeability, 
or specific inductance, and permittivity, or specific capac- 
ity), and the velocity of propagation of the electromagnetic 
field — that is, the velocity of light — ^thus is: 

1 

c = ~7E=^ = 3 X IQio cm., 


where L is the inductance, C the capacity per unit space. 

As has been seen, the velocity of light has nothing to do 
with any rigidity and elasticity constants of matter, but is 
merely a function of the electromagnetic field constants of 
space. 

Lack of familiarity with the conception of the energy field 
in space, and familiarity with the conception of matter as 
the (hypothetical) carrier of energy, may lead to the 
question: What is the carrier of the field energy in space? 
Would not the ether be needed as the hypothetical carrier 
of the field energy? 

All that we know of the world is derived from the percep- 
tions of our senses. They are the only real facts; all things 
else are conclusions from them. All sense perceptions are 
exclusively energy effects. That is, energy is the only real 
existing entity, the primary conception, which exists for us 
because our senses respond to it. All other conceptions are 
secondary, conclusions from the energy perceptions of our 
senses. Thus space and time and motion and matter are 
secondary conceptions with which our mind clothes the 
events of nature — that is, the hypothetical cause of our 
sense perceptions. Obviously, then, by carrying the 
explanation of light and electromagnetic waves back to the 


24 RELATIVITY AND SPACE 

energy field^ — that is, to energy storage in space^ — ^we have 
carried it back as far as possible, to the fundamental or 
primary conceptions of the human mind, the perceptions of 
the senses, which give us the entity energy and the form 
under which the human mind conceives it, that of space and 
time. 

C. THE FOUR-DIMENSIONAL TIME-SPACE OF 
MINKOWSKI 

The relativity theory shows that length is not a constant 
property of a body but depends on the conditions under 
which it is observed. This does not mean that a body, like 
the railway train of our previous instance, has at some time 
one length, U, and at another time another length, Zi, 
but it means that at the same time the railway train has 
different lengths to different observers. It has the length 
lo to one observer — for instance, the observer in the railway 
train, who is at rest with regard to it^ — and at the same time 
a different (and shorter) length, U, to another observer — for 
instance, the observer standing near the track and watching 
the train passing by^ — and it would have still another length, 
^2, to a third observer having a different relative speed with 
regard to the train. The same applies to the time. That 
is, the beat of the second-pendulum in the train has the 
duration to to the observer in the train, and the same beat of 
the same second-pendulum in the train has a different (and 
longer) duration, ti, to an observer on the track; and so on. 

Thus the length of an object depends on the velocity of 
its relative motion to the observer, and as velocity is length 
divided by time, this makes the length of an object depen- 
dent on the time. Inversely, as the time depends on the 
velocity of the relative motion, the time depends on length. 
Thus length^ — ^that is, space dimension — and time become 
dependent upon each other. 

We always have known that this world of ours is in reality 
four-dimensional — that is, every point event in the world is 
given by four numerical values, data, coordinates or 


CONCLUSIONS FROM RELATIVITY THEORY 


25 


dimensions, whatever we may call them, three dimensions 
in space and one dimension in time. But because in the 
physics before Einstein space and time were always 
independent of each other, we never realized this or found 
any object or advantage in considering the world as four- 
dimensional, but always considered the point events as 
three-dimensional in space and one-dimensional in time, 
treating time and space as separate and incompatible 
entities. The relativity theory, by interrelating space and 
time, thus changes our entire world conception. 

The dependence of length and time on the relative veloc- 
ity and thus on each other is an inevitable conclusion from 
the relativity theory — that is, from the two assumptions. 
(1) That all motion is relative, the motion of the railway 
train relative to the track being the same as the motion 
of the track relative to the train, and (2) that the laws of 
nature, and thus the velocity of light, are the same 
everywhere. 

Consider, in Fig. 6 ; our illustration of a railway train R, 
moving with the velocity v, for instance, at 60 miles per 


•^W///y/y/y/////////////'//vy/////////'//^/^^////^>>^ 


Fig. 6. 

hour, relative to the track B. Let us denote the distance 
relative to the train — that is, measured in the train^ — ^by 
x', and the time in the train by t'. The distance measured 
along the track may be denoted by x and the time on the 
track by t. For simplicity we may count distance and 
time, in the train and on the track, from the same zero 
value — that is, assume x = 0, t = 0, x' = 0, t' = 0, 


26 RELATIVITY AND SPACE 

(This obviously makes no essential difference, but merely 
eliminates unnecessary constant terms in the equations of 
transformation from train to track and inversely.) 

x' and t\ the coordinates with regard to the train, thus 
are moving at velocity v relative to the coordinates x and t 
with regard to the track, and by the conventional or 
Newtonian mechanics, we would have: 

t' = t, 

that is, the time is the same on the track and in the train, 
and 

X = x'-\- vt' , 

that is, the distance along the track x of a point of the train 
increases during the time t by vt' , that is, with the velocity 
V. These equations do not apply any more in the relativity 
theory as they would give different velocities of light rela- 
tive to the train and relative to the track. To find the 
equations which apply, we start with the most general 
relations between x, t and x' , t' , that is:^ 

x' = ax — bt, 
t' = pt — qx, 

and then determine the constants a, b, p, q by the three 
conditions which must be fulfilled. 

1. The relative velocity of the train coordinates x', t' 
with regard to the track x, t is v. 

2. By the relativity theory, the relative motion of the 
track with regard to the train is the same as the relative 
motion of the train with regard to the track; that is, x', t' are 
related to x, t by the same equations as x, t are related to 
x', t'. 

3. The velocity of light c on the track, in the x, t coordi- 
nates, is the same as in the train, in the x' , t' coordinates. 

These three conditions give four equations between the 
four constants a, b, p, q, and thereby determine these 
constants and give, as the relations between the coordinates 

* The relation must be linear, as it is univalent. 


CONCLUSIONS FROM RELATIVITY THEORY 27 

of events (that is, a material point and a time moment) 
relative to the moving train, x' and t', and the coordinates 
relative to the track, x, t, the equations:^ 

1 In the most general expressions the train coordinates x', t' are related 
to the track coordinates x, t by the coordinate transformation equations : 

x' = ax — bt\ 

t' = pt - qx j ^"^ 

(These equations must be linear, as one point of the train can correspond 
to one point of the track only, and inversely.) 

1. Since x'i' has relative to xi the velocity f, it is, for a;' = 0:ax — bt = 0, 
and since x/t = v, it follows : 

b/a = V 


0 = av j ' 
Thus: 

x' = ax — avt . . 

t' = pt - qx j ^'^^ 

2. From the conditions of relativity it follows from equation (c): 
X — ax' + avt' 


t = pt' + qx' ' ^^^ 

where the reversal of the sign is obvious as the track relative to the train 
moves in the opposite direction to the relative motion of the train to the 
track : 

Substituting (c) into (d) gives: 

x(a^ — avq — 1) + avt{p — a) = 0 
tip"^ — avq — 1) — qx{p — a) = 0 
and as these must be identities, the coefficients of x and t must individually 
vanish; that is: 

p = a ] 

^ «" - 1 (e) 

av ] 
Thus, substituting into (c) 

x' = a{x — vt) 1 

t = at — X \ 

av J 

3. From the constancy of the velocity of light it follows, that ,if : 
then it must be: > _ u (?) 

Substituting (gr) into (/) and dividing, t and t' cancel, and an equation in 
(d) remains, from which follows: 

1 


and by substitution into (6) and (e), the values of a, b, p and q are arrived at. 


28 RELATIVITY AND SPACE 

, X — Vt x' + Vt' 

X' = . X = 


^FI " ¥~- 


(1) 


i' = ^= or i = ^= (2) 


From these equations (1) and (2) it follows: 

1. One and the same point x' of the train, at two different 
times, U and U ^ appears as two different points of the 
track : 

_x' -V vU _x' ^ vU 

This is obvious and merely means that during the time 
interval from U to U the point x' of the train has moved 
from the point Xi to the point Xi of the track. 

2. Two events occurring in the train at one and the 
same time t' — that is, simultaneously — ^at two different 
points Xi and x^ of the train, are not simultaneous as seen 
from the track, but occur at two different times: 

I' + ^x/ (' + "L ^,' 

i\ — — , and ti = 


/■ 


and inversely. 

Thus, if X'^> Xx and the two events at X\ and X'l occurred 
not simultaneously, but the event at Xi later than that 
at X2', but by a time difference less than that between U 
and ii, then, seen from the track, the second event would be 
the later, the first one the earlier, while seen from the train 
the second event would be the earlier and the first one the 
later. 

In other words, simultaneousness in time and being 
earlier or later in time are only relative, and two events may 


CONCL USIONS FROM RELA TIVITY THEOR Y 29 

be simultaneous to one observer but not simultaneous to 
another observer because of a different relative motion; 
or one event may be earlier than another one to one observer 
and later to another observer. 

3. The distance, at a given time, between two points Pi 
and F2 of the train, in train coordinates — ^that is, as seen 
from the train^ — is U = X2' — Xi ; in track coordinates — 
that is, as seen from the track^ — ^the same distance is L = 
X2 — Xi. However, by (1) : 


x-i — Xx 


Xi — Xi 


i-% 


or (3) 


That is, a length L' in the train appears from the track 

shorter by the factor - h (the more, the faster the 

speed), and if the train were to move at the velocity of 
light, V = c, the length L' in the train would from the track 
appear as L = 0, that is, would vanish, while at a velocity 
greater than that of light the length L would become 
imaginary^ — ^that is, no velocity greater than that of light 
can exist. 

4. The time difference between two events occurring 
at a point P in the train, by the time as observed by an 
observer in the train — that is, in train coordinates — is 
T' = to' — ti ; but seen from the track — that is, for an 
observer watching the clock in the train while standing 
on the track, or in track coordinates^ — the time difference 
between the same events is T = to — ti. However, by (2) : 

t -I -->-'-A' 

ll — ti — I 

L _ v^ 

~2 


30 


RELATIVITY AND SPACE 


or: 


T = 


r 


(4) 


That is, to the observer from the track, comparing the 
clock in the train (T') with a clock on the track (T), the 
clock in the train appears slow; that is, the time in the train 
has slowed down by the factor . 


1 - 


c- 


The straight-line motion of a point (as, for instance, the 
front of the railway train) can conveniently be represented 
graphically by plotting the distance x as abscissa and the 
time t as ordinate. A motion at constant speed then 
gives a straight line for path curve, as shown by Pq. .Pi in 
Fig. 7, where for convenience we chose t = 0 for x = 0. The 

X 

velocity then is given by Vo = j = tan PiPqT. An 

extended body like our railway train would at time t = 0 
be represented by a length PoPoo, and at any other time 


ti by P1P2 parallel to PqPoq, and the motion of the train then 
is represented by the area between the lines PoFi — the 
path curve of the front of the train — and P00P2 — the path 


CONCLUSIONS FROM RELATIVITY THEORY 


31 


curve of the back of the train. The horizontal Une P1P2 
then gives the distance Xi — Xi occupied by the train at 
the time i\\ that is, the length of the train is L = X\— X2. 
The vertical line P1P3 gives the time t^ — ti required by the 
train to pass a given point Xi at velocity Vq] that is, the 
duration of the passage of the train is T = ts — U. 

Now, instead of plotting the path curves of the train as 
in Fig. 7, with x and t as coordinates, let us plot them in 
the coordinates x' , t' (1) and (2), as the train motion would 
appear to an observer having the velocity v relative to the 
first observer. 

The equations relating x, t, to x', t', given by equations (1) 
and (2), are very similar to those representing a rotation of 


Fig. 8. 


the coordinate axes by an angle tan co = v/c. If it were such 
a simple rotation, the new axes X' and T" would then form 
with the axes X and T of Fig. 7 the angle w, as shown in 
Fig. 8. For the new coordinate axes X' and T' — that is, 
for the observer at relative velocity v- — the length of the 
train would be the width of the path curve parallel to X' — ■ 


82 


RELATIVITY AND SPACE 


that is, would be P1P2, instead of P1P2 — or the length of 
the train would be shorter, and the duration of the passage 
of the train over a given point of the track would be P1P3 
instead of PiP^ — that is, the time would be longer. 

To the second observer P1P2' is the train length, while 
to the first observer P1P2 is the train length and P1P2' not 
the train length but a combination of length and time. 
Inversely, to the second observer P1P2 — which is the train 
length to the first observer — is not the train length but is 
a combination of length and time. Analogously, to the 
first observer P1P3 is the time of the train passage, while 


r 7' 


Fig. 9. 

P1P3' is not the time but a combination of time and length. 
Inversely, to the second observer P1P3' is the time and 
P1P3 a combination of time and length. 

Consider, however, two point events in the train, Xi'h' 
and x^/W — that is, an occurrence at point Xi and time i/ 
and an occurrence at point x^ and time t^. Then, from 
the track, the same two point events are given by Xi, h and 

Consider now these point events Pi and Pi represented 
graphically, with the distance as abscissa and the time as 
ordinate, as is done in Fig. 9, for both observers, at relative 


CONCLUSIONS FROM RELATIVITY THEORY 33 

velocity v with each other — that is, for coordinate axes X, 
T and X' , T' turned against each other by angle w, where 
tan CO = v/c. 

The distance X2 — Xi (that is, the distance between the 
two points as seen from the track) differs from x-i — Xi 
(that is, the distance between the same two points as seen 
from the train), and the time (2 — U differs from t^ — // simi- 
larly, as would be the case if x, t were one set of coordinates 
and x', t' a second set of coordinates, rotated with respect 
to the first one by angle tan w = v/c] but the distance 
between the two points Pi and P2 obviously would be the 
same, whatever change of coordinates we apply; that is, it 
would be: 

s^ = (X2 - x^y + {u -hy = (xo' - xi'Y + {k - uY =s'^ 

In the relation between the train coordinates x', t' and 
the track coordinates x, t, as given by equations (1) and (2), 
this, however, is not the case. That is, the relation between 
X, t and x', t', as given by equations (1) and (2)- — ^that is, 
the difference of the viewpoints of the two observers — is 
not a simple rotation by angle co as we have assumed above, 
but it is: 

{X2 - xi)2 - c\U -hy = (x./ - Xi'Y - c\t,' - h'Y, (5) 

as easily seen from equations (1) and (2). 

The appearance of the factor c^ in equation (5) is merely 
due to the choice of the units of x and t. 

The disadvantage, leading to complexity of equations 
(1) and (2), is that time and distance are given in different 
units, and as both equations involve both factors, they 
naturally would be different when given in feet and seconds 
from their form when given in miles and seconds, or in 
feet and minutes, etc., just as we would get differences and 
complications, in mere space relations, if, for instance, we 
expressed the two horizontal distances in miles and the 
vertical distance in feet. 


34 RELATIVITY AND SPACE 

The first requirement to simplify conditions is therefore 
to express time and distance in the same units. That is, 
if the distance is given in miles, express the time not in 
seconds, but also in miles, namely, by the distance traveled 
by light during the time, using the distance traveled by 
light in one second as the unit of time. Or, if it is desired 
to keep the second as unit of time, express the distance not 
in feet or miles, but in time measure — that is, the time 
required by the light to go the unit of distance. In other 
words, use the distance measure for time or the time meas- 
ure for distance. This idea is not new. Astronomers have 
for long time, though for other reasons, used a time meas- 
ure for large distances, the ''light-year," that is, the 
distance traveled by light in one year.^ 

As in the world of events we have three space coordinates 
and one time coordinate, it is simpler to express the time in 
space measure — that is, to express it not in seconds (or 
minutes, years, etc.), but in miles, or centimeters, or what- 
ever unit is used in the distance measurements. That is, 
substitute 

w = ct (6) 

where c is the velocity of light. 

1 The value of the use of time measure for the distance, or the distance 
measure for the time, may be very great wherever time and distance enter 
the same equations, and it is therefore useful in electrical engineering, for 
instance, when dealing with transmission line phenomena. Thus in my 
paper on the "General Equations of the Electric Circuit" {A.I.E.E. 
Transactions, 1907, also "Transient Phenomena," Section IV) the equa- 
tions contain exponential and trigonometric functions of time t and distance 
I, of the form cos {qt ± kl), etc. By choosing time measure for the distance 
(as more convenient in this case, since the time is the dominant term) : 
X = al, where a = s/hC is the reciprocal of the velocity of light, the equa- 
tions simplify to the form cos q{t ± X). Introducing now the local time 
^ = t ± \, the complex expression of the two variables I and t simplifies 
into an expression of a single variable only, the "local" time t?; that is, the 
time counted at every point from the moment as stai-ting point where the 
wave front reaches this point, in other words, the local time on the moving 
wave. 


CONCLUSIONS FROM RELATIVITY THEORY 35 

The transformation equations between train and track 
then become: 


X' = 


w = 


V 

X w 

c 

or: 

X + ~IV 

c 

V 
IV X 

c 

w' + x' 
c 

V-:^ 

^ xF? 

(7) 


(8) 


where x and x' are the respective distances and w and w' the 
respective times in distance measure. 
Then we get the relation: 

*S^ = (0:2 — Xi)^ — {w-i — WiY = 

{x,' - x,y - {w,' - w.'Y (9) 

This, however, is not the distance between two points 
with coordinates Xx, Wi and Xi, w^, because the expression of 
the distance contains the plus sign. 

Now, suppose we use as time measure not the distance 
w = ct, but the imaginary value of this distance, as explained 
later. That is, use as time coordinate: 

u = jet', (10) 

hence, represent the time by the imaginary value of the 
distance traveled by the light during the time. 

Then the transformation equations (1) and (2) between 
train and track become: 

X + JU X — J u 

x' = —r==l or: X = ~. (11) 


^/ 


1 C2 V^ C' 

u — j-x u -\- J X 


36 RELATIVITY AND SPACE 

These are the transformation equations of a rotation of the 
coordinates from x, u to x' , u' , by an angle tan w = j v/c, 
and it is then: 

^2 = {X2 - x,y ^ iu2 - u,y ^ {x^' - xy'y + {u,' - wy 

(13) 

That is: Expressing the time by the imaginary distance 
unit u = jet, the relation between the events as seen from 
the observer in the train and the same events as seen from 
an observer on the track (or in any other relative motion) 
is a rotation of the coordinate system x, u by the imaginary 
angle jco, given by tan co = v/c, and all the expressions 
are symmetrical in x and in u; that is, there is no difference 
between the distance and the time coordinates. 

To the observer in the train distance and time are sepa- 
rate coordinates of the phenomenon occurring in the 
train — that is, a phenomenon regarding which the observer 
is at rest; but to any observer in relative motion to the 
phenomenon which he observes, what appear to him as 
distance and as time are not the same distance and time as 
to the observer at rest, but are compounds of distance and 
time. Now, physics and engineering deal with motion, 
and when investigating motion we obviously cannot 
be at rest for every motion; and therefore what we call 
distance and time are not absolute and intrinsically different 
quantities, but are combinations of the two symmetrical 
coordinates x and u. 

It is similar to the relation, in mere space, between 
horizontal and vertical directions. At a given place on 
earth horizontal and vertical directions are intrinsically 
different. But, comparing two different places on earth, 
the horizontal and vertical directions at one place are not 
the same as those at the other place, but differ by a rotation 
of coordinates and are related to each other by the same 
equations as x, u and x' , u' . 

In the preceding we have for simplicity considered one 
space direction x only. This, with the time coordinate 


CONCLUSIONS FROM RELATIVITY THEORY 37 

u = jet, gives us two coordinates, x and u, and thus permits 
graphical illustration. In the events of the general world 
we have three space coordinates, x, y, z, and the time 
coordinate t, and from the relativity theory it thus follows : 

Space, as represented by three dimensions, x, y, z, and 
time, as represented by one dimension, t, are not separate 
and intrinsically different, but the world and its events are a 
four-dimensional system, and all point events are repre- 
sented by four symmetrical coordinates: x, y, z, u. 

In the special case concerning an event stationary with 
regard to the observer, x, y and z are the three space coordi- 
nates of the Newtonian mechanics, and u = jet is the time 
coordinate. For every event, however, in relative motion 
to the observer, x, y, z and u are four symmetrical coordi- 
nates, none having a preference or difference from the other, 
each involving the space and the time conceptions of 
Newtonian mechanics. 

The expression of an event in coordinates x, y, z and u 
differs from the expression of the same event by another 
observer in relative motion with regard to the first, and 
therefore represented by coordinates x', y' , z' and u' , by 
a rotation of the coordinate system x, y, z, u against the 
coordinate system x/, y', z', u', in the four-dimensional 
manifold, by an angle tan w = j v/e, where v is the relative 
velocity. 

The distance between two point events Pi and P^ in the 
four-dimensional manifold remains the same whatever 
coordinate system may be used^ — that is, is independent of 
the relative velocity of the observer. 

S'- = (x, - x,y + (i/2 - yiY + (22 - z,y + {U2 - u^y = 

{x,' - x,'Y + {y,' - y,'Y + {z,' - z,'Y + {u,' - u,'Y. 

Thus, if we consider x, y, z as space and t as time distance, 
relative motion v changes the space and time distance, 
changes the length and duration, but the total distance S 
in the four-dimensional manifold remains unchanged. 

This four-dimensional manifold is a Euclidean space. 


38 RELATIVITY AND SPACE 

The equations (6) to (9) appear simpler than those of 
Minkowskian space, (10) to (13), as they do not contain the 
imaginary unit. But the distance S — equation (9)^ — is 
not the expression of the EucUdean space, and the effect 
of relative velocity is not a mere rotation of the coordinate 
system, and thus the point events do not give the same 
simplicity of expression as in the Minkowskian space. 

Now what does this mean, rotation by an imaginary 
angle? It sounds unreal and meaningless. But it is no 
more and no less so than rotation by a negative angle. 
Physically, rotation by a negative angle means rotation in 
opposite direction, and rotation by an imaginary angle 
then means rotation in quadrature direction — that is, in 
the direction of right angle to the positive and the negative 
direction. 

Intrinsically, only the absolute integer number has a 
meaning^ — 4 horses, 4 dollars, 4 miles. Already the frac- 
tion has no intrinsic meaning; }i horse, for instance, 
is meaningless. It acquires a meaning only by defining it 
as denoting a relation : }i dollar. So the negative number 
intrinsically is unreal and meaningless: —4 horses has 
no meaning. But we attribute to it a meaning by conven- 
tion, as representing the opposite direction from the positive 
number. Thus —4 degrees means 4 degrees below zero 
temperature, when +4 means 4 degrees above zero tempera- 
ture, and in this relation both are equally real. But just 
as the negative number means the opposite direction, so 
the imaginary number means the quadrature direction, and 
5j miles north of New York is just as reasonable as —10 
miles north. The latter means 10 miles in the opposite 
direction from the northern direction, that is, south, and 
the former 5 miles in the quadrature direction from the 
northern direction, that is, west (or east). Thus the 
statements: Yonkers is +15 miles, Staten Island —10 
miles, Jersey City +3j miles, Brooklyn — 3j miles north of 
New York, are equally real and rational. When deahng 
with individuals, as when dealing with horses, neither the 


CONCLUSIONS FROM RELATIVITY THEORY 39 

fraction nor the negative nor the imaginary number has 
any meaning. When deaUng with divisible quantities 
the fraction receives a meaning. When deahng with 
directional quantities of one dimension, as time, tempera- 
ture, etc., the negative number acquires a meaning as 
denoting the opposite direction to the positive. When 
dealing with two-dimensional functions, as geographical 
location, vector representation of alternating currents, 
etc., the imaginary number also acquires a meaning, as 
denoting the quadrature direction^ — that is, the direction 
at right angles to the positive and the negative. 

The only difference between the conception of the nega- 
tive and the conception of the imaginary number is that 
we have been introduced to the negative number in school 
and use it in everyday life and thus have become familiar 
with it, while this is not the case yet with the imaginary 
number. But inherently the imaginary number is no more 
and no less unreal than the negative number. 

Thus, if by a rotation by angle -fco we mean a counter- 
clockwise rotation, a rotation by — co would be a clockwise 
rotation, like that shown in Fig. 8, and a rotation by angle 
jo) would be a rotation at right angles; that is (in Fig. 8), 
out of plane of the paper, for instance a rotation around the 
T axis. 

If, as is done in Fig. 8, we represent the relation between 
the viewpoints of the two observers at relative velocity v 
to each other, by a rotation of the coordinates x, t into x,' f 
by angle w in clockwise direction (where tan co = v/c), 
then we get a shortening of the length, from PiP-z to P1P2 , 
and a slowing down of the time, from P1P3 to P\Pz , as 
required by the equations (1) and (2) of the relativity theory. 
But with increasing v, and thus increasing angle w, the 
length as given by the equations (1) and (2) continuously 
decreases and becomes zero for v = c, while in the clockwise 
rotation of Fig. 8 the length P1P2 decreases, reaches a 
minimum and then increases again. Thus Fig. 8 does not 
physically represent the rotation given by the equations 


40 RELATIVITY AND SPACE 

(1) and (2). However, if we assume as representing the 
relation (1) and (2) a rotation by angle jw — ^that is, a 
rotation at right angles, out of the plane of the paper, for 
instance around the T axis — ^then with increasing w the 
length of the train- — that is, the spoor or projection of 
P1P2 on the new plane — indefinitely decreases and finally 
becomes zero, just as required by the equations (1) and (2). 

However, the quadrature rotation, which represents the 
relation between x, t and x', t', is not a rotation around the T 
axis, as a rotation around the T axis carries us from the 
X axis toward the YZ plane, while the quadrature rotation 
jui carries us outside of the space coordinates x, y, z into a 
direction at right angles to XYZ — that is, a fourth dimen- 
sion of the world space of Minkowski^ — and therefore cannot 
graphically be represented any more in the three-dimen- 
sional space manifold. 

In this four-dimensional manifold of Minkowski, this 
world or time space, which includes symmetrically the 
space and the time, with x^ y, z, u as coordinates, we cannot 
say that x, y, z are space coordinates and u the time coordi- 
nate, but all four dimensions are given in the same units, 
centimeters or miles, or, if we wish to use the time unit as 
measure, seconds; but all four dimensions are symmetrical, 
and each contains the space and time conceptions. Thus 
there is no more reason to consider x, ?/, z as space coordinates 
and u as time coordinate than there is to consider x and 
u as space and y and z as time coordinates, etc. 

Only in the special case of an observer at rest with regard 
to the phenomenon does x, y, z become identical with the 
space coordinates, and u becomes jc^, where i is the time of 
the Newtonian mechanics. 

But as soon as the observer is in motion relative to the 
phenomenon his viewpoint is that of a system x', y' , z' , u' , 
rotated out of the Newtonian space and time, and the dis- 
tinction between space and time coordinates then vanishes. 

Owing to the extremely limited range of possible veloci- 
ties, we cannot get far outside of the Newtonian time space. 


CONCL US IONS FROM RELA TIVITY THEOR Y 41 

In other words, of the four-dimensional world space of 
Minkowski, only a very narrow range is accessible to us, 
that near to the Newtonian space within rotation of a small 
fraction of a degree. The viewpoint from a comet passing 
the sun at grazing distance at 200 km. per second would 
differ from the Newtonian only by a rotation into the general 
Minkowski space by 0.04 degree. 

However, within the three-dimensional timeless space 
of Newton the conditions are similar. We can move in the 
two horizontal directions x and y to unlimited distance; 
but in the third dimension, the vertical z, we are limited to a 
very few miles, so that in the Newtonian space we are 
practically limited to the two horizontal dimensions, just 
as in the general world space we are limited to the range 
near the Newtonian time space. 

D. MASS AND ENERGY 

If a body moves with the velocity v relative to the 
observer, from the relativity theory it follows that the 
length (in the direction of motion) on the body is shortened 

and the time lengthened by the factor . M ^, where c = 

velocity of light in vacuum. 


T = 


r 


i-K 


For V = c — that is, a body moving with the velocity of 
light— by (3) and (4), L = 0 and T = oo. That is, on 
a body moving with the velocity of light the length vanishes, 
becomes zero, and the time stops. 

For v>c — ^that is, velocities greater than the velocity of 
light^ — length and time become imaginary. That is, such 
velocities cannot exist. The velocity of light thus is the 


42 RELATIVITY AND SPACE 

greatest possible physically existing velocity, and no greater 
relative velocity can exist or is conceivable. 

This conclusion appears at first unreasonable. 

Suppose we have a body moving with a velocity 90 per 
cent that of light, Vi = 0.9c, and a second body moving 
with the same velocity, but in opposite direction, V2 = 0.9c. 
The relative velocity between these two bodies thus would 
be V = Vi -{- V2 = 1.8c, or greater than the velocity of 
light, we would think. However, this is not so, and the 
error which we have made is in adding the velocities Vi 
and V2 to get the resultant velocity. This is the law of the 
Newtonian or pre-Einsteinian mechanics, but does not 
apply any more in the relativity theory, since velocity is 
distance — that is, length — divided by time, and as the 
length varies with the velocity, the velocity Vi for a station- 
ary observer, is not Vi any more for an observer moving with 
the velocity i'2. 

Thus velocities do not add algebraically, even when in the 
same direction. 

Suppose a body moves with the velocity Vi relative to an 
observer (for instance, a railway train relative to the 
observer on the track), and a second body moves in the 
same direction relative to the first body with the velocity 
V2 (for instance, I walk forward in the train with the veloc- 
ity ^2) . What is the resultant velocity^ — ^that is, the relative 
velocity of the second body with regard to the observer 
(for instance, my velocity relative to the observer on the 
track) ? 

A point Xiti relative to the train has relative to the track, 
by (1) and (2), the coordinates: 

Xi + Vih 


V^ 


4 1 ^1 

^1 + -o Xl 


'= ■ V 


^. 


CONCL US IONS FROM RELA TIVITY THEOR Y 43 

If now this point moves relative to the train with the 
velocity V2, it is Xi = v^h, and substituting this in the 
preceding equations and dividing gives: 

„ = 5 = 'llAjH (14) 

as the resultant velocity v of the two velocities V\ and Vi. 

Equation (14) thus is the law of addition of velocities by 
the relativity theory. 

If then vx = 0.9c and V2 = 0.9c, v = 1.8c/1.81 = 0.9945c; 
that is, two velocities each 90 per cent of the velocity of light 
add to a resultant velocity 99.45 per cent of the velocity 
of light. 

From (14) it follows that as long as Vi and V2 are less than 
the velocity of light c, no matter how close they approach 
it, their sum v also is less than c. 

If one of the velocities equals the velocity of light c, 
then, substituting in (14), we get: 

Vi + c ^ 


V = 


1+E! 
c 


That is, adding — or subtracting — ^any velocity ^i to or from 
the velocity of light c still gives the same velocity c. This 
explains why in the previous instance, if a train moves at 
the velocity v and the light along the track at the velocity c, 
the velocity of light relative to the train is the combination 
of c and v, which is again c. 

The velocity of light c thus has the characteristic that 
any velocity (whether less than, equal to or even greater 
than c) may be added to it or any velocity less than 
c may be subtracted from it without changing it; that is, 
it has the characteristic of the mathematical conception 
of infinity : 00 . 

Therefore there can be no velocity greater than c, since 
whatever velocity may be added to c still leaves it 
unchanged at c. 


44 RELATIVITY AND SPACE 

The kinetic energy of a mass m, moving at velocity v, is, 
in the Newtonian mechanics given by : 

2" 

In the relativity theory the total kinetic energy of a mass 
m moving at the relative velocity v is given by 

JllO = 


E = 


4^-% 


and thus becomes infinite, for y = c, the velocity of light. 
This energy, for y = 0, or the mass at rest, becomes: 

Eoo = wc^, 

which may be considered as the ''kinetic energy of mass," 
while m is a constant, similar to permeability or specific 
capacity. 

The kinetic energy required to give a mass m the relative 
velocity v then is given by: 

hi = — , — mc^. 

This expanded into a series gives: 

rp _ mv'^ . 3 my^ . _ mv"^ fi i 3 y^ , | 

^-"^ + 8^+ • • • ""2"r + 8c^+ • • •} 

The second term already is negligible for all velocities 
except those comparable with the velocity of light. The 
first term is the kinetic energy of the Newtonian mechanics. 

Mass therefore appears as a form of energy, kinetic 
energy, and the "energy equivalent of mass, "or the "kinetic 
energy of mass," is £"00 = mc''-. 

This is an enormous energy, almost beyond conception. 

One kilogram of coal, when burned, is equivalent to about 
3,400,000 kgm. (kilogram-meters) or about 10 kw.-hr. 
(kilowatt-hours) . 


CONCLUSIONS FROM RELATIVITY THEORY 45 

The earth revolves in its orbit around the sun at about 20 
miles per second; that is, about thirty times as fast as the 
fastest rifle bullet. Its kinetic energy, }^mv-, therefore is 
enormous; 1 kg. weight on the earth has, owing to this high 
velocity, the kinetic energy of 50,000,000 kgm. or about 
150 kw.-hr. ; that is, fifteen times as much energy as would 
be given as heat by the combustion of the same weight of 
coal. (Therefore, if the earth were stopped by a collision 
and its kinetic energy converted into heat, its temperature 
would be raised by about 500,000 deg. C.) 

The kinetic energy of 1 kg. weight of matter, Eq = mc^, 
however, is about 9000 millions of millions of kilogram- 
meters — or 25 thousand million kilowatt-hours — thousands 
of million times larger than the energy of coal. 

Estimating the total energy consumed during the year on 
earth for heat, light, power, etc., as about 15 millions of 
millions of kilowatt-hours, 600 kg., or less than two-thirds 
of a ton of dirt, if it could be disintegrated into energy, 
mc^, would supply all the heat, light and energy demand of 
the whole earth for a year. 

Or, the energy equivalent mc- of one pound of dirt would 
run all the factories, mills, railroads, etc., and light all 
the cities and villages of the United States for a month. 
It would supply the fuel for the biggest transatlantic liner 
for 300 trips from America to Europe and back. And if this 
energy of one pound of dirt could be let loose instanta- 
neously, it would be equal in destructive power to over 
a million tons of dynamite.