Part 3
We assume that this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold:—
1. If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B.
2. If the clock at A as well as the clock at B are both synchronous with the clock at C, then the clocks at A and B are synchronous.
Thus with the help of certain physical experiences, we have established what we understand when we speak of clocks at rest at different stations, and synchronous with one another; and thereby we have arrived at a definition of synchronism and time.
In accordance with experience we shall assume that the magnitude
$$ \frac {2 \overline{AB}}{t'_{A} - t_{A}} = c $$
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_
Substituting these values of _t_ in the equation for ξ, we obtain
_c²_ ξ = _a_ ------------- _x′_ _c²_ - _v²_
In an analogous manner, we obtain by considering the ray of light which moves along the _y_-axis,
_vx′_ η = _c_τ = _a c_(_t_ - ------------- ) _c²_ - _v²_
where
_y_ ------------------ = _t_, _x′_ = 0, √ (_c²_ - _v²_)
Therefore
_c_ η = _a_ ------------------ _y_, √ (_c²_ - _v²_)
_c_ ζ = _a_ ----------------- _z_ . √ (_c²_ - _v²_)
If for _x′_, we substitute its value _x_ - _tv_, we obtain
_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.
II.—ELECTRODYNAMICAL PART.
§ 6. Transformation of Maxwell’s equations for Pure Vacuum.
On the nature of the Electromotive Force caused by motion in a magnetic field.
The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that
$$ \frac {1}{c} \frac {\partial}{\partial t} [X, Y, Z] = \begin{vmatrix} \frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac {\partial}{\partial z} L & M & N \end{vmatrix} $$
and
$$ \frac {1}{c} \frac {\partial}{\partial t} [L, M, N] = \begin{vmatrix} \frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac {\partial}{\partial z} X & Y & Z \end{vmatrix} $$ (1)
where [X, Y, Z] are the components of the electric force, L, M, N are the components of the magnetic force.
If we apply the transformations in §3 to these equations, and if we refer the electromagnetic processes to the co-ordinate system moving with velocity _v_, we obtain,
$$ \frac {1}{c} \frac {\partial}{\partial \tau} [X, \beta (Y - \frac {v}{c} N), \beta (Z + \frac {v}{c} M)] = \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L & \beta(M + \frac {v}{c} Z) & \beta(N - \frac {v}{c} Y) \end{vmatrix}
and
$$ \frac {1}{c} \frac {\partial}{\partial \tau} [L, \beta(M + \frac {v}{c} Z), \beta(N - \frac {v}{c} Y)] = - \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X & \beta(Y - \frac {v}{c} N) & \beta(Z + \frac {v}{c} M) \end{vmatrix} $$ ... (2)
where
$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
The principle of Relativity requires that the Maxwell-Hertzian equations for pure vacuum shall hold also for the system k, if they hold for the system K, _i.e._, for the vectors of the electric and magnetic forces
## acting upon electric and magnetic masses in the moving system k, which
are defined by their pondermotive reaction, the same equations hold, ... _i.e._ ...
$$ \frac {1}{c} \frac {\partial}{\partial \tau} (X', Y', Z') = \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L' & M' & N' \end{vmatrix} $$ ,
$$ \frac {1}{c} \frac {\partial}{\partial \tau} (L', M', N') = - \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X' & Y' & Z' \end{vmatrix} $$ ... (3)
Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these systems are equivalent to the Maxwell-Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor ψ(_v_), which depends only on _v_, and is independent of (ξ, η, ζ, τ). Hence the relations,
_v_ _v_ [X′, Y′, Z′] = ψ (_v_) [X, β(Y - ----- N), β(Z + ------ M)], _c_ _c_