_v_._c_
τ = φ (_v_). β (_t_ - ----------- ,
c²
ξ = φ (_v_). β (_x_ - _vt_) ,
η = φ (_v_) _y_
ζ = φ (_v_) _z_ ,
where
$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
and
φ (_v_) = _ac_ / √ (_c²_ - _v²_) = _a_ / β
is a function of _v_.
If we make no assumption about the initial position of the moving system
and about the null-point of _t_, then an additive constant is to be
added to the right hand side.
We have now to show, that every ray of light moves in the moving system
with a velocity _c_ (when measured in the moving system), in case, as we
have actually assumed, _c_ is also the velocity in the stationary
system; for we have not as yet adduced any proof in support of the
assumption that the principle of relativity is reconcilable with the
principle of constant light-velocity.
At a time τ = _t_ = 0 let a spherical wave be sent out from the common
origin of the two systems of co-ordinates, and let it spread with a
velocity _c_ in the system K. If (_x_, _y_, _z_), be a point reached by
the wave, we have
_x²_ + _y²_ + _z²_ = _c²__t²_
with the aid of our transformation-equations, let us transform this
equation, and we obtain by a simple calculation,
ξ² + η² + ζ² = _c²_τ².
Therefore the wave is propagated in the moving system with the same
velocity _c_, and as a spherical wave.[7] Therefore we show that the two
principles are mutually reconcilable.
In the transformations we have got an undetermined function φ(_v_), and
we now proceed to find it out.
Let us introduce for this purpose a third co-ordinate system _k′_, which
is set in motion relative to the system _k_, the motion being parallel
to the ξ-axis. Let the velocity of the origin be (-_v_). At the time _t_
= 0, all the initial co-ordinate points coincide, and for _t_ = _x_ =
_y_ = _z_ = 0, the time _t′_ of the system _k′_ = 0. We shall say that
(_x′_ _y′_ _z′_ _t′_) are the co-ordinates measured in the system _k′_,
then by a two-fold application of the transformation-equations, we
obtain
_v_
τ′ = φ(-_v_)β(-_v_){τ + ----- ξ}
_c²_
= φ(_v_)φ(-_v_)t,
_x′_ = φ](_v_)β(_v_)(ξ + _v_τ)
= φ(_v_)φ(-_v_)_x_, etc.
Since the relations between (_x′_, _y′_, _z′_, _t′_), and (_x_, _y_,
_z_, _t_) do not contain time explicitly, therefore K and _k′_ are
relatively at rest.
It appears that the systems K and _k′_ are identical.
∴ φ(_v_)φ(-_v_) = 1.
Let us now turn our attention to the part of the ξ-axis between (ξ = 0,
η = 0, ζ = 0), and (ξ = 0, η = 1, ζ = 0). Let this piece of the _y_-axis
be covered with a rod moving with the velocity _v_ relative to the
system K and perpendicular to its axis;—the ends of the rod having
therefore the co-ordinates
_x₁_ = _vt_, _y₁_ = _l_ / φ(_v_), _z₁_ = 0
_x₂_ = _vt_, _y₂_ = 0, _z₂_ = 0
Therefore the length of the rod measured in the system K is _l_/φ(_v_).
For the system moving with velocity (-_v_), we have on grounds of
symmetry,
_l_ _l_
-------- = ---------
φ(_v_) φ(-_v_)
∴ φ(_v_) = φ(-_v_), ∴ φ(_v_) = 1.
§ 4. The physical significance of the equations obtained concerning
moving rigid bodies and moving clocks.
Let us consider a rigid sphere (_i.e._, one having a spherical figure
when tested in the stationary system) of radius R which is at rest
relative to the system (K), and whose centre coincides with the origin
of K then the equation of the surface of this sphere, which is moving
with a velocity _v_ relative to K, is
ξ² + η² + ζ² = R².
At time _t_ = 0, the equation is expressed by means of (_x_, _y_, _z_,
_t_,) as
$$ \frac {x^2}{(\sqrt {1 - \frac {v_2}{c_2}})^2} + y^2 + z^2 = R^2. $$
A rigid body which has the figure of a sphere when measured in the
moving system, has therefore in the moving condition—when considered
from the stationary system, the figure of a rotational ellipsoid with
semi-axes
$$ R \sqrt {1 - \frac {v^2}{c^2}}, R, R. $$
Therefore the _y_ and _z_ dimensions of the sphere (therefore of any
figure also) do not appear to be modified by the motion, but the _x_
dimension is shortened in the ratio
$$ 1 : \sqrt {1 - \frac {v^2}{c^2}}; $$
the shortening is the larger, the larger is _v_. For _v_ = _c_, all
moving bodies, when considered from a stationary system shrink into
planes. For a velocity larger than the velocity of light, our
propositions become meaningless; in our theory _c_ plays the part of
infinite velocity.
It is clear that similar results hold about stationary bodies in a
stationary system when considered from a uniformly moving system.
Let us now consider that a clock which is lying at rest in the
stationary system gives the time _t_, and lying at rest relative to the
moving system is capable of giving the time τ; suppose it to be placed
at the origin of the moving system _k_, and to be so arranged that it
gives the time τ. How much does the clock gain, when viewed from the
stationary system K? We have,
$$ \tau = \frac {1}{\sqrt {1-\frac {v^2}{c^2}}} (t - \frac {v}{c^2}x),
$$
and _x_ = _vt_,
$$ \therefore \tau - t = (1 - \sqrt {1 - \frac {v^2}{c^2}}) t. $$
Therefore the clock loses by an amount ½(_v²_/_c²_) per second of
motion, to the second order of approximation.
From this, the following peculiar consequence follows. Suppose at two
points A and B of the stationary system two clocks are given which are
synchronous in the sense explained in § 3 when viewed from the
stationary system. Suppose the clock at A to be set in motion in the
line joining it with B, then after the arrival of the clock at B, they
will no longer be found synchronous, but the clock which was set in
motion from A will lag behind the clock which had been all along at B by
an amount ½_t_(_v²_/_c²_), where _t_ is the time required for the
journey.
We see forthwith that the result holds also when the clock moves from A
to B by a polygonal line, and also when A and B coincide.
If we assume that the result obtained for a polygonal line holds also
for a curved line, we obtain the following law. If at A, there be two
synchronous clocks, and if we set in motion one of them with a constant
velocity along a closed curve till it comes back to A, the journey being
completed in _t_-seconds, then after arrival, the last mentioned clock
will be behind the stationary one by ½_t_(_v²_/_c²_) seconds. From this,
we conclude that a clock placed at the equator must be slower by a very
small amount than a similarly constructed clock which is placed at the
pole, all other conditions being identical.
§ 5. Addition-Theorem of Velocities.
Let a point move in the system _k_ (which moves with velocity _v_ along
the _x_-axis of the system K) according to the equation
$$ \xi = w_{\xi} \tau, \eta = w_{\eta} \tau, \zeta = 0, $$
where _w__{ξ} and _w__{η} are constants.
It is required to find out the motion of the point relative to the
system K. If we now introduce the system of equations in § 3 in the
equation of motion of the point, we obtain
$$ x = (\frac {w_{\xi} + v}{1+\frac {vw_{\xi}}{c^2}}) t $$,
$$ y = \frac {(1-\frac {v^2}{c^2})^{\frac {1}{2}} w_{\eta}t} {1+\frac
{vw_{\xi}}{c^2}} $$ ,
z = 0 .
The law of parallelogram of velocities hold up to the first order of
approximation. We can put
$$ U^2 = (\frac {\partial x}{\partial t})^2 + (\frac {\partial
y}{\partial t})^2 $$ ,
$$ w^2 = w_{\xi}^2 + w_{\eta}^2 $$ ,
and
$$ \alpha = tan^{-1} \frac {w}{w_{\xi}} $$
_i.e._, α is put equal to the angle between the velocities _v_, and _w_.
Then we have—
$$ U = \frac {[(v^2 + w^2 + 2 vw \cos \alpha) - (\frac {vw \sin
\alpha}{c})^2]^{\frac {1}{2}}} {1 + \frac {vw \cos \alpha}{c^2}} $$
It should be noticed that _v_ and _w_ enter into the expression for
velocity symmetrically. If _w_ has the direction of the ξ-axis of the
moving system,
$$ U = \frac {v + w}{1 + \frac {vw}{c^2}} $$
From this equation, we see that by combining two velocities, each of
which is smaller than _c_, we obtain a velocity which is always smaller
than _c_. If we put _v_ = _c_ - χ, and _w_ = _c_ - λ, where χ and λ are
each smaller than _c_,[8]
$$ U = c \frac {2c - \chi - \lambda}{2c - \chi - \lambda + \frac {\chi
\lambda}{c^2}} < c $$
It is also clear that the velocity of light _c_ cannot be altered by
adding to it a velocity smaller than _c_. For this case,
$$ U = \frac {c + v}{1 + \frac {cv}{c^2}} = c $$
We have obtained the formula for U for the case when _v_ and _w_ have
the same direction; it can also be obtained by combining two
transformations according to section § 3. If in addition to the systems
K, and k, we introduce the system k´, of which the initial point moves
parallel to the ξ-axis with velocity _w_, then between the magnitudes,
_x_, _y_, _z_, _t_ and the corresponding magnitudes of k´, we obtain a
system of equations, which differ from the equations in § 3, only in the
respect that in place of _v_, we shall have to write,
$$ \frac {v + w}{1 + \frac {vw}{c^2}} $$
We see that such a parallel transformation forms a group.
We have deduced the kinematics corresponding to our two fundamental
principles for the laws necessary for us, and we shall now pass over to
their application in electrodynamics.