$$ \frac {2 \overline{AB}}{t'_{A} - t_{A}} = c $$
where _c_ is a universal constant.
We have defined time essentially with a clock at rest in a stationary
system. On account of its adaptability to the stationary system, we call
the time defined in this way as “time of the stationary system.”
§ 2. On the Relativity of Length and Time.
The following reflections are based on the Principle of Relativity and
on the Principle of Constancy of the velocity of light, both of which we
define in the following way:—
1. The laws according to which the nature of physical systems alter are
independent of the manner in which these changes are referred to two
co-ordinate systems which have a uniform translators motion relative to
each other.
2. Every ray of light moves in the “stationary co-ordinate system” with
the same velocity _c_, the velocity being independent of the condition
whether this ray of light is emitted by a body at rest or in motion.[6]
Therefore
velocity = Path of Light/Interval of time,
where, by ‘interval of time’ we mean time as defined in §1.
Let us have a rigid rod at rest; this has a length _l_, when measured by
a measuring rod at rest; we suppose that the axis of the rod is laid
along the X-axis of the system at rest, and then a uniform velocity _v_,
parallel to the axis of X, is imparted to it. Let us now enquire about
the length of the moving rod; this can be obtained by either of these
operations.—
(_a_) The observer provided with the measuring rod moves along with the
rod to be measured, and measures by direct superposition the length of
the rod:—just as if the observer, the measuring rod, and the rod to be
measured were at rest.
(_b_) The observer finds out, by means of clocks placed in a system at
rest (the clocks being synchronous as defined in §1), the points of this
system where the ends of the rod to be measured occur at a particular
time _t_. The distance between these two points, measured by the
previously used measuring rod, this time it being at rest, is a length,
which we may call the “length of the rod.”
According to the Principle of Relativity, the length found out by the
operation _a_), which we may call “the length of the rod in the moving
system” is equal to the length _l_ of the rod in the stationary system.
The length which is found out by the second method, may be called ‘_the
length of the moving rod measured from the stationary system_.’ This
length is to be estimated on the basis of our principle, and _we shall
find it to be different from l_.
In the generally recognised kinematics, we silently assume that the
lengths defined by these two operations are equal, or in other words,
that at an epoch of time _t_, a moving rigid body is geometrically
replaceable by the same body, which can replace it in the condition of
rest.
Relativity of Time.
Let us suppose that the two clocks synchronous with the clocks in the
system at rest are brought to the ends A, and B of a rod, _i.e._, the
time of the clocks correspond to the time of the stationary system at
the points where they happen to arrive; these clocks are therefore
synchronous in the stationary system.
We further imagine that there are two observers at the two watches, and
moving with them, and that these observers apply the criterion for
synchronism to the two clocks. At the time _t__{A}, a ray of light goes
out from A, is reflected from B at the time _t__{B}, and arrives back at
A at time _t′__{A}. Taking into consideration the principle of,
constancy of the velocity of light, we have
_t__{B} - _t__{A} = _r__{AB}/(_c_ - _v_),
and _t′__{A} - _t__{B} = _r__{AB}/(_c_ + _v_),
where _r__{AB} is the length of the moving rod, measured in the
stationary system. Therefore the observers stationed with the watches
will not find the clocks synchronous, though the observer in the
stationary system must declare the clocks to be synchronous. We
therefore see that we can attach no absolute significance to the concept
of synchronism; but two events which are synchronous when viewed from
one system, will not be synchronous when viewed from a system moving
relatively to this system.
§ 3. Theory of Co-ordinate and Time-Transformation from a stationary
system to a system which moves relatively to this with uniform velocity.
Let there be given, in the stationary system two co-ordinate systems,
_i.e._, two series of three mutually perpendicular lines issuing from a
point. Let the X-axes of each coincide with one another, and the Y and
Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be
given to each of the systems, and let the rods and clocks in each be
exactly alike each other.
Let the initial point of one of the systems (_k_) have a constant
velocity in the direction of the X-axis of the other which is stationary
system K, the motion being also communicated to the rods and clocks in
the system (_k_). Any time _t_ of the stationary system K corresponds to
a definite position of the axes of the moving system, which are always
parallel to the axes of the stationary system. By _t_, we always mean
the time in the stationary system.
We suppose that the space is measured by the stationary measuring rod
placed in the stationary system, as well as by the moving measuring rod
placed in the moving system, and we thus obtain the co-ordinates (_x_,
_y_, _z_) for the stationary system, and (ξ, η, ζ) for the moving
system. Let the time _t_ be determined for each point of the stationary
system (which are provided with clocks) by means of the clocks which are
placed in the stationary system, with the help of light-signals as
described in § 1. Let also the time τ of the moving system be determined
for each point of the moving system (in which there are clocks which are
at rest relative to the moving system), by means of the method of light
signals between these points (in which there are clocks) in the manner
described in § 1.
To every value of (_x_, _y_, _z_, _t_) which fully determines the
position and time of an event in the stationary system, there correspond
a system of values (ξ, η, ζ, τ); now the problem is to find out the
system of equations connecting these magnitudes.
Primarily it is clear that on account of the property of homogeneity
which we ascribe to time and space, the equations must be linear.
If we put _x′_ = _x_ - _vt_, then it is clear that at a point relatively
at rest in the system _k_, we have a system of values (_x′_ _y_ _z_)
which are independent of time. Now let us find out τ as a function of
(_x′_, _y_, _z_, _t_). For this purpose we have to express in equations
the fact that τ is not other than the time given by the clocks which are
at rest in the system _k_ which must be made synchronous in the manner
described in § 1.
Let a ray of light be sent at time τ₀ from the origin of the system _k_
along the X-axis towards _x′_ and let it be reflected from that place at
time τ₁ towards the origin of moving co-ordinates and let it arrive
there at time τ₂; then we must have
½ (τ₀ + τ₂) = τ₁
If we now introduce the condition that τ is a function of co-ordinates,
and apply the principle of constancy of the velocity of light in the
stationary system, we have
$$ \frac {1}{2} (\tau (0,0,0,t) + \tau (0,0,0,(t + \frac {x'}{c-v} +
\frac {x'}{c+v}))) $$
$$ = \tau (x',0,0, t + \frac {x'}{c-v}) $$
It is to be noticed that instead of the origin of co-ordinates, we could
select some other point as the exit point for rays of light, and
therefore the above equation holds for all values of (_x′_, _y_, _z_,
_t_,).
A similar conception, being applied to the _y_- and _z_-axis gives us,
when we take into consideration the fact that light when viewed from the
stationary system, is always propagated along those axes with the
velocity √(_c²_ - _v²_), we have the questions
∂τ ∂τ
---- = 0, ---- = 0.
∂y ∂z
From these equations it follows that τ is a linear function of _x′_ and
_t_. From equations (1) we obtain
vx′
τ = a (t - -------- )
c² - v²
where _a_ is an unknown function of _v_.
With the help of these results it is easy to obtain the magnitudes (ξ,
η, ζ) if we express by means of equations the fact that light, when
measured in the moving system is always propagated with the constant
velocity _c_ (as the principle of constancy of light velocity in
conjunction with the principle of relativity requires). For a time τ =
0, if the ray is sent in the direction of increasing ξ, we have
_vx′_
ξ = _c_τ, _i.e._ ξ = _a c_(_t_ - ------------ )
_c²_ - _v²_
Now the ray of light moves relative to the origin of _k_ with a velocity
_c_ - _v_, measured in the stationary system; therefore we have
_x′_
---------- = _t_
_c_ - _v_