Chapter IV: Part I

§ 2.
The Limiting Case.
The Fundamental Equations for Äther.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-dynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case ε = 1, μ = 1, σ = 0, they should constitute the laws for ponderable bodies. In this ideal limiting case ε = 1, μ = 1, σ = 0, E will be equal to _e_, and M to _m_. At every space time point (_x_, _y_, _z_, _t_) we shall have the equations[15]

(i) Curl _m_ - (δ_e_/δ_t_) = ρu

(ii) div _e_ = ρ

(iii) Curl _e_ + δ_m_/δ_t_ = 0

(iv) div m = 0

I shall now write (_x₁_ _x₂_ _x₃_ _x₄_) for (_x_, _y_, _z_, _t_) and (ρ₁, ρ₂, ρ₃, ρ₄) for

$$ (\rho u_{x}, \rho u_{y}, \rho u_{z}, i\rho) $$

_i.e._ the components of the convection current ρu, and the electric density multiplied by √ -1

Further I shall write

_f__{2 3}, _f__{3 1}, _f__{1 2}, _f__{1 4}, _f__{2 4}, _f__{3 4}.

for

m_{_x_}, m_{_y_}, m_{_z_}, -ie_{_x_}, -ie_{_y_}, -ie_{_z_}.

_i.e._, the components of m and (-_i.e._) along the three axes; now if we take any two indices (h. k) out of the series

3, 4), _f__{_k h_} = -_f__{_k h_},

Therefore

_f₃₂_ = -_f₂₃_, _f₁₃_ = -_f₃₁_, _f₂₁_ = -_f₁₂_
_f₄₁_ = -_f₁₄_, _f₄₄_ = -_f₂₄_, _f₄₃_ = -_f₃₄_

Then the three equations comprised in (i), and the equation (ii) multiplied by i becomes

$$ \begin{vmatrix} & \frac{\delta f_{1 2}}{\delta x_{2}} & + \frac{\delta f_{1 3}}{\delta x_{3}} & + \frac{\delta f_{1 4}}{\delta x_{4}} & = \rho_{1} \frac{\delta f_{2 1}}{\delta x_{1}} & & + \frac{\delta f_{2 3}}{\delta x_{3}} & \times \frac{\delta f_{2 4}}{\delta x_{4}} & = \rho_{2} \frac{\delta f_{3 1}}{\delta x_{1}} & \times \frac{\delta f_{3 2}}{\delta x_{2}} & & + \frac{\delta f_{3 4}}{\delta x_{4}} & = \rho_{3} \frac{\delta f_{4 1}}{\delta x_{1}} & + \frac{\delta f_{4 2}}{\delta x_{2}} & + \frac{\delta f_{4 3}}{\delta x_{3}} & & = \rho_{4} \end{vmatrix} × $$

On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by (_i_) becomes

$$ \begin{vmatrix} & \frac{\delta f_{3 4}}{\delta x_{2}} & + \frac{\delta f_{4 2}}{\delta x_{3}} & + \frac{\delta f_{2 3}}{\delta x_{4}} & = = \frac{\delta f_{4 3}}{\delta x_{1}} & & + \frac{\delta f_{1 4}}{\delta x_{3}} & + \frac{\delta f_{3 1}}{\delta x_{4}} & = 0 \frac{\delta f_{2 4}}{\delta x_{1}} & + \frac{\delta f_{4 1}}{\delta x_{2}} & & + \frac{\delta f_{1 2}}{\delta x_{4}} & = 0 \frac{\delta f_{3 2}}{\delta x_{1}} & + \frac{\delta f_{1 3}}{\delta x_{2}} & + \frac{\delta f_{2 1}}{\delta x_{3}} & & = - \end{vmatrix} × $$

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices, (1, 2, 3, 4).

§ 3.

It is well-known that by writing the equations i) to iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis, through an amount φ, keeping e, m fixed in space, and introduce new variables _x₁′_ _x₂′_ _x₃′_ _x₄′_ instead of _x₁_ _x₂_ _x₃_ _x₄_ where _x′₁_ = _x₁_ cos φ + _x₂_ sin φ, _x′₂_ = -_x₁_ sin φ + _x₂_ cos φ, _x′₃_ = _x₃_, _x′₄_ = _x₄_, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ and _f′__{1 2}, ... ... _f′__{3 4}, where

_f′₂₃_ = _f₂₃_ cos φ + _f₃₁_ sin φ,
_f′₃₁_ = - _f₂₃_ sin φ + _f₃₁_ cos φ,
_f′₁₂_ = _f₁₂_,
_f′₁₄_ = _f₁₄_ cos φ + _f₂₄_ sin φ,
_f′₂₄_ = - _f₁₄_ sin φ + _f₂₄_ cos φ,
_f′₃₄_ = _f₃₄__{3 4},
_f′__{_k h_} = - _f__{_k h_} (h l k = 1, 2, 3, 4).

then out of the equations (A) would follow a corresponding system of dashed equations (A´) composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning the _suffixes_ (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by _i_ψ a purely imaginary magnitude, and consider the substitution

_x₁′_ = _x₁_,
_x₂′_ = _x₂_,
_x₃′_ = _x₃_ cos _i_ψ + _x₄_ sin _i_ψ, (1)
_x₄′_´ = - _x₃_ sin _i_ψ + _x₄_ cos _i_ψ,

Putting

$$ - i \tan i\psi = \frac{e^{\psi} - e^{-\psi}}{e^{\psi}+e^{-\psi}} = q $$ ,

$$ \psi = \frac{1}{2} \log \frac{1 + q}{1 - q′} $$ (2)

We shall have cos _i_ψ = 1/√(1 - _q²_), sin _i_ψ = _iq_/√(1 - _q²_) where -1 < _q_ < 1, and √(1 - _q²_) is always to be taken with the positive sign.

Let us now write _x′₁_ = _x′_, _x′₂_ = _y′_, _x′₃_ = _z′_, _x′₄_ = _it′_ (3)

then the substitution 1) takes the form

_x′_ = _x_, _y′_ = _y_, _z′_ = (_z_ - _qt_)/√(1 - _q²_), _t′_ =
(-_qz_ + _t_)/√(1 - _q²_), (4)

the coefficients being essentially real.

If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and φ by _i_ψ, we at once perceive that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where

ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos _i_ψ + ρ₄ sin _i_ψ,
ρ′₄ = - ρ₃ sin _i_ψ + ρ₄ cos _i_ψ),

and _f′__{1 2} ... _f′__{3 4}, where

_f′__{4 1} = _f__{4 1} cos _i_ψ + _f__{1 3} sin _i_ψ,
_f′__{1 3} = - _f__{4 1} sin _i_ψ + _f__{1 3} cos _i_ψ,
_f′__{3 4} = _f__{3 4},
_f′__{3 2} = _f__{3 2} cos _i_ψ + _f__{4 2} sin _i_ψ,
_f′__{4 2} = - _f__{3 2} sin _i_ψ + _f__{4 2} cos _i_ψ,
_f′__{1 2} = _f__{1 2}, _f__{_k h_} = - _f′__{_k h_},

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A´), and (B´), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are taken, and _x´_ _y´_ _z´_ _t´_ be taken as a new frame of reference, then we shall have

(5) ρ´ = ρ [(-_qu__{_z_} + 1)/√(1 - _q²_)],
ρ´_u__{_z_}´ = ρ[(_u__{_z_} - _q_)/√(1 - _q²_)],
ρ´_u__{_x_}´ = ρ_u__{_x_},
ρ´_u__{_y_}´ = ρ_u__{_y_}.

(6) _e´__{_x´_} = (_e__{_x_} - _qm__{_y_})/(√(1 - _q²_)),
_m´__{_r´_} = (_qe__{_x_} + _m__{_y_})/(√(1 - _q²_)),
_e´__{_z´_} = _e__{_z_}.

(7) _m´__{_x´_} = (_m__{_x_} - _qe__{_y_})/(√(1 - _q²_)),
_e´__{_y_´} = (_qm__{_x_} + _e__{_y_})/(√(1 - _q²_)),
_m_´_{_z_´} = _m__{_z_}.

Then we have for these newly introduced vectors _u´_, _e´_, _m´_ (with components _u__{_x_}´, _u__{_y_}´, _u__{_z_}´; _e__{_x_}´, _e__{_y_}´, _e__{_z_}´; _m__{_x_}´, _m__{_y_}´, _m__{_z_}´), and the quantity ρ´ a series of equations I´), II´), III´), IV´) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that _e__{_x_} - _qm__{_y_}, _e__{_y_} + _qm__{_x_} are components of the vector _e_ + [_vm_], where _v_ is a vector in the direction of the positive Z-axis, and | _v_ | = _q_, and [_vm_] is the vector product of _v_ and _m_; similarly -_qe__{_x_} + _m__{_y_}, _m__{_x_} + _qe__{_y_} are the components of the vector _m_ - [_ve_].

The equations 6) and 7), as they stand in pairs, can be expressed as.

_e′__{_x′_} + _im′__{_x′_} = (_e__{_x_} + _im__{_x_}) cos _i_ψ +
(_e__{_y_} + _im__{_y_}) sin _i_ψ,

_e′__{_y′_} + _im′__{_y′_} = - (_e__{_x_} + _im__{_x_}) sin _i_ψ +
(_e__{_y_} + _im__{_y_}) cos _i_ψ,

_e′__{_z′_} + _im′__{_z′_} = _e′__{_z_} + _im__{_z_}.

If φ denotes any other real angle, we can form the following combinations:—

(_e′__{_x′_} + _im′__{_x′_}) cos. φ + (_e′__{_y″_} + _im′__{_y′_})
sin φ

= (_e__{_x_} + _im__{_x_}) cos. (φ + _i_ψ) + (_e__{_y_} +
_im__{_y_}) sin (φ + _i_ψ),

= (_e′__{_x′_} + _im′__{_x′_}) sin φ + (_e′__{_y′_} +
_im′__{_y′_}) cos. φ

= - (_e__{_x_} + _im__{_x_}) sin (φ + _i_ψ) + (_e__{_y_} +
_im__{_y_}) cos. (φ + _i_ψ).

§ 4. Special Lorentz Transformation.

The rôle which is played by the Z-axis in the transformation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law:—

Let _v_ be a vector with the components _v__{_x_}, _v__{_y_}, _v__{_z_}, and let | _v_ | = _q_ < 1. By _ṽ_ we shall denote any vector which is perpendicular to _v_, and by _r__{_v_}, _r__{_ṽ_} we shall denote components of _r_ in direction of _ṽ_ and _v_.

Instead of (_x_, _y_, _z_, _t_), new magnetudes (_x′_ _y′_ _z′_ _t′_) will be introduced in the following way. If for the sake of shortness, _r_ is written for the vector with the components (_x_, _y_, _z_) in the first system of reference, _r′_ for the same vector with the components (_x′_ _y′_ _z′_) in the second system of reference, then for the direction of _v_, we have

(10) _r′__{_v_} = (_r__{_v_} - _qt_)/√(1 - _q²_)

and for the perpendicular direction _ṽ_,

(11) _r′__{_ṽ_} = _r__{_ṽ_}

and further (12) _t′_ = (-_qr__{_v_} + _t_)/√(1 - _q²_).

The notations (_r′__{_ṽ_}, _r′__{_v_}) are to be understood in the sense that with the directions _v_, and every direction _ṽ_ perpendicular to _v_ in the system (_x_, _y_, _z_) are always associated the directions with the same direction cosines in the system (_x′_ _y′_ _z′_).

A transformation which is accomplished by means of (10), (11), (12) with the condition 0 < _q_ < 1 will be called a special Lorentz-transformation. We shall call _v_ the vector, the direction of _v_ the axis, and the magnitude of _v_ the moment of this transformation.

If further ρ′ and the vectors _u′_, _e′_, _m′_, in the system (_x′_ _y′_ _z′_) are so defined that,

(13) ρ′ = ρ[(-_qu__{_v_} + 1)/√(1 - _q²_)],
ρ′_u_′_{_v_} = ρ(_u__{_v_} - _q_)/√(1 - _q²_),
ρ′_u__{_ṽ_} = ρ′_u__{_v_},

further

(14) (_e′_ + _im′_)_{_ṽ_} = ((_e_ + _im_) - _i_[_v_, (_e_ +
_im_])']_{_ṽ_})/√(1 - _q²_).

(15) (_e′_ + _im′_)_{_v_} = (_e_ + _im_) - _i_[_u_, (_e_ +
_im_)]_{_v_}.

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

(16) _r__{_v_} = (_r′__{_v_} + _qt′_)/√(1 - _q²_),
_r__{_ṽ_} = _r′__{_ṽ_},
_t_ = (_qr′__{_v_} + _t′_)/√(1 - _q²_),

Now we shall make a very important observation about the vectors _u_ and _u′_. We can again introduce the indices 1, 2, 3, 4, so that we write (_x₁_′, _x₂_′, _x₃_′, _x₄_′) instead of (_x′_, _y′_, _z′_, _it′_) and ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′_u′_{_x′_}, ρ′_u′_{_y′_}, ρ′_u′_{_z′_}, _i_ρ′).

Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant + 1, so that

(17) _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ _i. e._ _x²_ + _y²_ + _z²_ -
_t²_,

is transformed into

_x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_ _i. e._ _x′²_ + _y′²_ + _z′²_ -
_t′²_.

On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃² + ρ₄²) = ρ²(1 - _u__{_x²_}, -_u__{_y²_}, -_u__{_z²_}) = ρ²(1 - _u²_) transformed into ρ²(1 - _u²_) or in other words,

(18) ρ√(1 - _u²_)

is an invariant in a Lorentz-transformation.

If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - _u²_))(_u__{_x_}, _u__{_y_}, _u__{_z_}, _i_) so that ω₁² + ω₂² + ω₃² + ω₄² = -1.

It is apparent that these four values are determined by the vector _u_ and inversely the vector _u_ of magnitude < 1 follows from the 4 values ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -_i_ω₄ real and positive and condition (19) is fulfilled.

The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios of _dx₁_, _dx₂_, _dx₃_, _dx₄_ to

(20) √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_)) = _dt_√(1 - _u²_).

The differentials denoting the displacements of matter occupying the spacetime point (_x₁_, _x₂_, _x₃_, _x₄_) to the adjacent space-time point.

After the Lorentz-transformation is accomplished the velocity of matter in the new system of reference for the same space-time point (_x′_ _y′_ _z′_ _t′_) is the vector _u′_ with the ratios _dx′_/_dt′_, _dy′_/_dt′_, _dz′_/_dt′_, _dl′_/_dt′_, as components.

Now it is quite apparent that the system of values

_x₁_ = ω₁, _x₂_ = ω₂, _x₃_ = ω₃, _x₄_ = ω₄

is transformed into the values

_x₁′_ = ω₁′, _x₂′_ = ω₂′, _x₃′_ = ω₃′, _x₄′_ = ω₄′

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity _u′_ after the transformation as the first system of values has got for _u_ before transformation.

If in particular the vector _v_ of the special Lorentz-transformation be equal to the velocity vector _u_ of matter at the space-time point (_x₁_, _x₂_, _x₃_, _x₄_) then it follows out of (10), (11), (12) that

ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ = _i_

Under these circumstances therefore, the corresponding space-time point has the velocity _v′_ = 0 after the transformation, it is as if we transform to rest. We may call the invariant ρ√(1 - _u²_) the rest-density of Electricity.[16]

§ 5. Space-time Vectors.
Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general _relativity theorem_. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

$$ \begin{vmatrix} x_{1} x_{2} x_{3} x_{4} \end{vmatrix} = \begin{vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & a_{1 4} a_{2 1} & a_{2 2} & a_{2 3} & a_{2 4} a_{3 1} & a_{3 2} & a_{3 3} & a_{3 4} a_{4 1} & a_{4 2} & a_{4 3} & a_{4 4} \end{vmatrix} \begin{vmatrix} x_{1}' x_{2}' x_{3}' x_{4}' \end{vmatrix} $$

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while _a₄₁_, _a₄₂_, _a₄₃_, are purely imaginary, but _a₄₄_ is real and > 0, and _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ transforms into _x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_. The operation shall be called a general Lorentz transformation.

(This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski’s notation, _x₁_ = _a₁₁x₁′_ + _a₁₂x₂′_+ _a₁₃x₃′_+ _a₁₄x₄′_.)

If we put _x₁′_ = _x′_, _x₂′_ = _y′_, _x₃′_ = _z′_, _x₄′_ = _it′_, then immediately there occurs a homogeneous linear transformation of (_x_, _y_, _z_, _t_) to (_x′_, _y′_, _z′_, _t′_) with essentially real co-efficients, whereby the aggregate -_x²_ - _y²_ - _z²_ + _t²_ transforms into -_x′²_ - _y′²_ - _z′²_ + _t′²_, and to every such system of values _x_, _y_, _z_, _t_ with a positive _t_, for which this aggregate > 0, there always corresponds a positive _t’_; this last is quite evident from the continuity of the aggregate _x_, _y_, _z_, _t_.

The last vertical column of co-efficients has to fulfil the condition 22) _a₁₄²_ + _a₂₄²_ + _a₃₄²_ + _a₄₄²_ = 1.

If _a₁₄_ = _a₂₄_ = _a₃₄_ = 0, then _a₄₄_ = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If _a₁₄_, _a₂₄_, _a₃₄_ are not all zero, and if we put _a₁₄_ : _a₂₄_ : _a₃₄_ : _a₄₄_ = _v__{_x_} : _v__{_y_} : _v__{_z_} : _i_

_q_ = √(_v__{_x_}² + _v__{_y_}² +_v__{_z_}²) < 1.

On the other hand, with every set of values of _a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_ which in this way fulfil the condition 22) with real values of _v__{_x_}, _v__{_y_}, _v__{_z_}, we can construct the special Lorentz transformation (16) with (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) can be supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where these are the values of (_x₁′_, _x₂′_, _x₃′_, _x₄′_), obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (_x₁_, _x₂_, _x₃_, _x₄_) in the expression (21).

Besides the time-space vector of the 1st kind (_x₁_, _x₂_, _x₃_, _x₄_) we shall also make use of another space-time vector of the first kind (_y₁_, _y₂_, _y₃_, _y₄_), and let us form the linear combination

(23) _f₂₃_(_x₂__y₃_ - _x₃__y₂_) + _f₃₁_(_x₃__y₁_ - _x₁__y₃_) +
_f₁₂_(_x₁__y₂_
- _x₂__y₁_) + _f₁₄_(_x₁__y₄_ - _x₄__y₁_) + _f₂₄_(_x₂__y₄_ -
_x₄__y₂_) +
_f₃₄_(_x₃__y₄_ - _x₄__y₃_)

with six coefficients _f₂₃_--_f₃₄_. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors.

_x₁_, _x₂_, _x₃_; _y₁_, _y₂_, _y₃_; _f₂₃_, _f₃₁_, _f₁₂_; _f₁₄_, _f₂₄_, _f₃₄_ and the constants _x₄_ and _y₄_, at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject (_x₁_, _x₂_, _x₃_, _x₄_) and (_y₁_, _y₂_, _y₃_, _y₄_) simultaneously to the Lorentz transformation (21), the combination (23) is changed to:

(24) _f₂₃′_(_x₂′__y₃′_ - _x₃′__y₂′_) + _f₃₁_(_x₃′__y₁′_ -
_x₁′__y₃′_) + _f₁₂_
(_x₁′__y₂′_ - _x₂′__y₁′_) + _f₁₄′_(_x₁′__y₄′_) - _x₄′__y₁′_) +
_f₂₄′_(_x₂′__y₄′_
- _x₄′__y₂′_) + _f₃₄′_(_x₃′__y₄′_ - _x₄′__y₃′_),

where the coefficients _f₂₃′_, _f₃₁′_, _f₁₂′_, _f₁₄′_, _f₂₄′_, _f₃₄′_, depend solely on (_f₂₃_ _f₂₄_) and the coefficients _a₁₁_ ... _a₄₄_.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes _f₂₃_, _f₃₁_ ... _f₃₄_, with the condition that when subjected to a Lorentz transformation, it is changed to a new system _f₂₃′_ ... f₃₄, ... which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I)-(iv),—which are the fundamental equations for Äther.

If _x_, _y_, _z_, _it_ (space co-ordinates, and time _it_) is subjected to a Lorentz transformation, and at the same time (_pu__{_x_}, _pu__{_y_}, _pu__{_z_}, _i_ρ) (convection-current, and charge density ρ_i_) is transformed as a space time vector of the 1st kind, further (_m__{_x_}, _m__{_y_}, _m__{_z_}, -_ie__{_x_}, -_ie__{_y_}, -_ie__{_z_}) (magnetic force, and electric induction × (-_i_) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system.

These facts can be more concisely expressed in these words: the system of equations (I and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where (ρ_u_, _i_ρ) is to be transformed as a space time vector of the 1st kind, (_m_ - _ie_) is to be treated as a vector of the 2nd kind, or more significantly,—

(ρ_u_, _i_ρ) is a space time vector of the 1st kind, (_m_ - _ie_)[17] is a space-time vector of the 2nd kind.

I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.

(_i_) _m²_ - _e²_ = _f₂₃²_ + _f₃₁²_ + _f₁₂²_ + _f₁₄²_ + _f₂₄²_ +
_f₂₄²_

_me_ = _i_(_f₂₃__f₁₄_ + _f₃₁__f₂₄_ + _f₁₂__f₃₄_).

A space-time vector of the second kind (_m_ - _ie_), where (_m_ and _e_) are real magnitudes, may be called singular, when the scalar square (_m_ - _ie_)² = 0, _ie_ _m²_ - _e²_ = 0, and at the same time (_m e_) = 0, _ie_ the vector _m_ and _e_ are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every Lorentz-transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product [_me_] coincides with the Z-axis, _i.e._ _m__{_x_} = 0, _e__{_x_} = 0. Then

(_m__{_x_}, -_i e__{_x_})² + (_m__{_y_}, -_i e__{_y_})² ≠ 0.

Therefore (_e__{_y_} + _i m__{_y_})/(_e__{_x_} + _i e__{_x_}) is different from +_i_, and we can therefore define a complex argument (φ + _i_ψ) in such a manner that

tan (φ + _i_ψ)

_e__{_y_} + _i m__{_y_}
= -------------------------
_e__{_x_} + _i m__{_x_}

If then, by referring back to equations (9), we carry out the transformation (1) through the angle ψ and a subsequent rotation round the Z-axis through the angle φ, we perform a Lorentz-transformation at the end of which _m__{_y_} = 0, _e__{_y_} = 0, and therefore _m_ and _e_ shall both coincide with the new Z-axis. Then by means of the invariants _m²_ - _e²_, (_me_) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ 6. Concept of Time.

By the Lorentz transformation, we are allowed to effect certain _changes_ of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.[18] In fact, the following considerations will prove conclusive.

Let a reference system (_x_, _y_, _z_, _t_) for space time points (events) be somehow known. Now if a space point A (_x₀_, _y₀_, _z₀_) the time _t₀_ be compared with a space point P (_x_, _y_, _z_) at the time _t_, and if the difference of time _t_ - _t₀_, (let _t_ > _t₀_) be less than the length A P _i.e._ less than the time required for the propagation of light from A to P, and if _q_ = (_t_ - _t₀_)/(A P) < 1, then by a special Lorentz transformation, in which A P is taken as the axis, and which has the moment _q_, we can introduce a time parameter _t′_, which (see equation 11, 12, § 4) has got the same value _t′_ = _0_ for both space-time points (A, _t₀_), and (P, t). So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time _t₀_ = 0, two different space-points A, B, or three space-points (A, B, C) which are not in the same space-line, and compare therewith a space point P, which is outside the line A B, or the plane A B C, at another time _t_, and let the time difference _t_ - _t₀_ (t > _t₀_) be less than the time which light requires for propagation from the line A B, or the plane (A B C) to P. Let q be the quotient of (_t_ - _t₀_) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C is the axis, and q is the moment, then all the three (or four) events (A, _t₀_), (B, _t₀_), (C, _t₀_) and (P, t) are simultaneous.

If four space-points, which do not lie in one plane, are conceived to be at the same time _t₀_, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.