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.
PART II. ELECTRO-MAGNETIC PHENOMENA. § 7. Fundamental Equations for bodies at rest.
After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limiting case ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us—when proper fundamental data are given—to obtain the following quantities at every place and time, and therefore at every space-time point as functions of (_x_, _y_, _z_, _t_):—the vector of the electric force E, the magnetic induction M, the electrical induction _e_, the magnetic force _m_, the electrical space-density ρ, the electric current s (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector _v_, the velocity of matter.
The relations in question can be divided into two classes.
Firstly—those equations, which,—when _v_, the velocity of matter is given as a function of (_x_, _y_, _z_, _t_),—lead us to a knowledge of other magnitude as functions of _x_, _y_, _z_, _t_—I shall call this first class of equations the fundamental equations—
Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector _u_ as functions of (_x_, _y_, _z_, _t_).
For the case of bodies at rest, _i.e._ when _u_ (_x_, _y_, _z_, _t_) = 0 the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are;—
(1) The Differential Equations:—which contain no constant referring to matter:—
(_i_) Curl _m_ - δ_e_/δ_t_ = C, (_ii_) div _e_ = lρ. (_iii_) Curl E + δM/δ_t_ = 0, (_iv_) Div M = 0.
(2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves _i.e._ for isotopic bodies;—they are comprised in the equations
(V) _e_ = ε E, M = μ_m_, C = σE.
where ε = dielectric constant, μ = magnetic permeability, σ = the conductivity of matter, all given as function of _x_, _y_, _z_, _t_; _s_ is here the conduction current.
By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,
_x₁_ = _x_, _x₂_ = _y_, _x₃_ = _z_, _x₄_ = _it_,
and write _s₁_, _s₂_, _s₃_, _s₄_ for C_{_x_}, C_{_y_}, C_{_z_} (√-1)ρ.
Further _f₂₃_, _f₃₁_, _f₁₂_, _f₁₄_, _f₂₄_, _f₃₄_
for _m__{_x_}, _m__{_y_}, _m__{_z_}, -_i_(_e__{_x_}, _e__{_y_}, _e__{_z_}),
and F₂₃, F₃₁, F₁₂, F₁₄, F₂₄, F₃₄
for M_{_x_}, M_{_y_}, M_{_z_}, -_i_(E_{_x_}, E_{_y_}, E_{_z_})
lastly we shall have the relation _f__{k h} = - _f__{_h k_}, _F__{_k h_} = -_F__{_h k_}, (the letter _f_, F shall denote the field, _s_ the (_i.e._ current).
Then the fundamental Equations can be written as
(A) ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = s₁
∂_f₂₁_/∂_x₁_ + + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = s₂
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + + ∂_f₃₄_/∂_x₄_ = s₃
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = s₄
and the equations (3) and (4), are
∂F₃₄/∂_x₂_ + ∂F₄₂/∂_x₃_ + ∂F₂₃/∂_x₄_ = 0
∂F₄₃/∂_x₁_ + + ∂F₁₄/∂_x₃_ + ∂F₃₁∂_x₄_ = 0
∂F₂₄/∂_x₁_ + ∂F₄₁/∂_x₂_ + + ∂F₁₂/∂_x₄_ = 0
∂F₃₂/∂_x₁_ + ∂F₁₃/∂_x₂_ + ∂F₂₁/∂_x₃_ = 0
§ 8. The Fundamental Equations.
We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.
The first Axion shall be,—
When a detached region[19] of matter is at rest at any moment, therefore the vector _u_ is zero, for a system (_x_, _y_, _z_, _t_)—the neighbourhood may be supposed to be in motion in any possible manner, then for the space-time point _x_, _y_, _z_, _t_, the same relations (A) (B) (V) which hold in the case when all matter is at rest, shall also hold between ρ, the vectors C, _e_, _m_, _M_, _E_ and their differentials with respect to _x_, _y_, _z_, _t_. The second axiom shall be:—
Every velocity of matter is < 1, smaller than the velocity of propagation of light.[20]
The fundamental equations are of such a kind that when (_x_, _y_, _z_, _it_) are subjected to a Lorentz transformation and thereby (_m_ - _ie_) and (_M_ - _iE_) are transformed into space-time vectors of the second kind, (C, _i_ρ) as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.
Shortly I can signify the third axiom as:—
(_m_, -_ie_), and (_M_, -_iE_) are space-time vectors of the second kind, (C, _i_p) is a space-time vector of the first kind.
This axiom I call the Principle of Relativity.
In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.
According to the second axiom, the magnitude of the velocity vector | _u_ | is < 1 at any space-time point. In consequence, we can always write, instead of the vector _u_, the following set of four allied quantities
ω₁ = u_{_x_}/√(1 - _u²_), ω₂ = u_{_y_}/√(1 - u²), ω₃ = u_{_z_}/√(1 - u²), ω₄ = _i_/√(1 - u²)
with the relation
(27) ω₁² + ω₂² + ω₃² + ω₄² = - |
From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space-time vector of the 1st kind.
Let us now fix our attention on a certain point (_x_, _y_, _z_) of matter at a certain time (_t_). If at this space-time point _u_ = 0, then we have at once for this point the equations (_A_), (_B_) (_V_) of § 7. If _u_ ≠ 0, then there exists according to 16), in case | _u_ | < 1, a special Lorentz-transformation, whose vector _v_ is equal to this vector _u_ (_x_, _y_, _z_, _t_), and we pass on to a new system of reference (_x′_ _y′_ _z′_ _t′_) in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = _i_, therefore the new velocity vector ω′ = 0, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point (_x′_ _y′_ _z′_ _t_) involves the newly introduced magnitude (_u′_ ρ′, C′, _e′_, _m′_, _E′_, _M′_) and their differential quotients with respect to (_x′_, _y′_, _z′_, _t′_) in the same manner as the original equations for the point (_x_, _y_, _z_, _t_). But according to the first axiom, when _u′_ = 0, these equations must be exactly equivalent to
(1) the differential equations (_A′_), (_B′_), which are obtained from the equations (_A_), (_B_) by simply dashing the symbols in (_A_) and (_B_).
(2) and the equations
(V′) _e′_ = ε_E′_, _M’_ = μ_m′_, _C′_ = σ_E′_
where ε, μ, σ are the dielectric constant, magnetic permeability, and conductivity for the system (_x′_ _y′_ _z′_ _t′_) _i.e._ in the space-time point (_x_ _y_, _z_ _t_) of matter.
Now let us return, by means of the reciprocal Lorentz-transformation to the original variables (_x_, _y_, _z_, _t_), and the magnitudes (_u_, ρ, C, _e_, _m_, _E_, _M_) and the equations, which we then obtain from the last mentioned, will be the fundamental equations sought by us for the moving bodies.
Now from § 4, and § 6, it is to be seen that the equations _A_), as well as the equations _B_) are covariant for a Lorentz-transformation, _i.e._ the equations, which we obtain backwards from _A′_) _B′_), must be exactly of the same form as the equations _A_) and _B_), as we take them for bodies at rest. We have therefore as the first result:—
The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in ρ and the vectors C, _e_, _m_, E, M, are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have
I) curl _m_ - ∂_e_/∂_t_ = C₁,
II) div _e_ = ρ
III) curl E + ∂M/∂_t_ = 0
IV) div M = 0
The velocity of matter occurs only in the auxiliary equations which characterise the influence of matter on the basis of their characteristic constants ε, μ, σ. Let us now transform these auxiliary equations V′) into the original co-ordinates (_x_, _y_, _z_, and _t_.)
According to formula 15) in § 4, the component of _e′_ in the direction of the vector _u_ is the same as that of (_e_ + [_u_ _m_]), the component of _m′_ is the same as that of _m_ - [_u_ _e_], but for the perpendicular direction _ū_, the components of _e′_, _m′_ are the same as those of (_e_ + [_u_ _m_]) and (_m_ - [_u_ _e_], multiplied by 1/√(1 - _u²_). On the other hand E′ and M′ shall stand to E + [_u_M], and M - [_u_E] in the same relation as _e′_ and _m′_ to _e_ + [_um_], and _m_ - (_ue_). From the relation _e′_ = εE′, the following equations follow
(C) _e_ + [_um_] = ε(E + [_u_M]),
and from the relation M′ = μ_m′_, we have
(D) M - [_u_ E] = μ(_m_ - [_ue_]),
For the components in the directions perpendicular to _u_, and to each other, the equations are to be multiplied by √(1 - _u²_).
Then the following equations follow from the transformation? equations (12), (10), (11) in § 4, when we replace q, _r__{_v_}, _r__{_ṽ_}, _t_, _r′__{_v_}, _r′__{_ṽ_}, _t’_ by |_u_|, C_{_u_}, C_{_ū_}, ρ, C′_{_u_}, C′_{_ū_}, ρ′
ρ′ = (-|_u_| C_{_u_} + ρ)/√(1 - _u²_), C’_{_u_} = (C_{_u_} - |_u_|ρ)/√(1 - _u²_), C′_{_ū_} = C_{_ū_},
E) (C_{_u_} - |_u_|ρ)/√(1 - _u²_) = σ(E + [_u_M])_{_u_},
C_{_ū_} = σ (E + [_u_M])_{_u_}/√(1 - _u²_).
In consideration of the manner in which σ enters into these relations, it will be convenient to call the vector C - ρ_u_ with the components C_{_u_} - ρ|_u_| in the direction of _u_, and C′_{_ū_} in the directions _ū_ perpendicular to _u_ the “Convection current.” This last vanishes for σ = 0.
We remark that for ε = 1, μ = 1 the equations _e′_ = E′, _m′_ = M′ immediately lead to the equations _e_ = E, _m_ = M by means of a reciprocal Lorentz-transformation with -_u_ as vector; and for σ = 0, the equation C′ = 0 leads to C = ρ_u_; that the fundamental equations of Äther discussed in § 2 becomes in fact the limitting case of the equations obtained here with ε = 1, μ = 1, σ = 0.
§ 9. The Fundamental Equations in Lorentz’s Theory.
Let us now see how far the fundamental equations assumed by Lorentz correspond to the Relativity postulate, as defined in §8. In the article on Electron-theory (Ency., Math., Wiss., Bd. V. 2, Art 14) Lorentz has given the fundamental equations for any possible, even magnetised bodies (see there page 209, Eqn XXX′, formula (14) on page 78 of the same (part).
(III_a″_) Curl (H - [_u_E]) = J + _d_D/_dt_ + _u_ div D - curl [_u_D].
(I″) div D = ρ
(IV″) curl E = - _d_B/_dt_, Div B = 0 (V′)
Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) μ = 1, B = H, and in addition to that takes account of the occurrence of the di-electric constant ε, and conductivity σ according to equations
(ε_q_XXXIV″, p. 327) D - E = (ε - 1) {E + [_u_B]}
(ε_q_XXXIII′, p. 223) J = σ(E + [_u_B])
Lorentz’s E, D, H are here denoted by E, M, _e_, _m_ while J denotes the conduction current.
The three last equations which have been just cited here coincide with eqn (II), (III), (IV), the first equation would be, if J is identified with C, = _u_ρ (the current being zero for σ = 0,
(29) Curl [H - (_u_, E)] = C + _d_D/_dt_ - curl [_u_D],
and thus comes out to be in a different form than (1) here. Therefore for magnetised bodies, Lorentz’s equations do not correspond to the Relativity Principle.
On the other hand, the form corresponding to the relativity principle, for the condition of non-magnetisation is to be taken out of (D) in §8, with μ = 1, not as B = H, as Lorentz takes, but as (30) B - [_u_D] = H - [_u_D] (M - [_u_E] = _m_ - [_ue_]. Now by putting H = B, the differential equation (29) is transformed into the same form as eqn (1) here when _m_ - [_ue_] = M - [_u_E]. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for moving non-magnetised bodies at last agree with the relativity postulate.
If we make use of (30) for non-magnetic bodies, and put accordingly H = B + [_u_, (D - E)], then in consequence of (C) in §8,
(ε - 1) (E + [_u_, B]) = D - E + [_u_. [_u_, D - E]],
_i.e._ for the direction of _u_,
(ε - 1) (E + [_u_B])_{_u_} = (D - E)_{_u_}
and for a perpendicular direction ū,
(ε - 1) [E + (_u_B)]_{_u_} = (1 - _u²_) (D - E)_{_u_}
_i.e._ it coincides with Lorentz’s assumption, if we neglect _u²_ in comparison to 1.
Also to the same order of approximation, Lorentz’s form for J corresponds to the conditions imposed by the relativity principle [comp. (E) § 8]—that the components of J_{_u_}, J_{_ū_} are equal to the components of σ (E + [_u_ B]) multiplied by √(1 - _u²_) or 1 / √(1 - _u²_) respectively.
§10. Fundamental Equations of E. Cohn.
E. Cohn assumes the following fundamental equations.
(31) Curl (M + [_u_ E]) = _d_E/_dt_ + u div. E + J
- Curl [E - (_u_. M)] = _d_M/_dt_ + u div. M.
(32) J = σ E, = ε E - [_u_ M], M = μ (_m_ + [_u_ E.])
where E M are the electric and magnetic field intensities (forces), E, M are the electric and magnetic polarisation (induction). The equations also permit the existence of true magnetism; if we do not take into account this consideration, div. M. is to be put = 0.
An objection to this system of equations, is that according to these, for ε = 1, μ = 1, the vectors force and induction do not coincide. If in the equations, we conceive E and M and not E - (U. M), and M + [U E] as electric and magnetic forces, and with a glance to this we substitute for E, M, E, M, div. E, the symbols _e_, M, E + [U M], _m_ - [_u_ _e_], ρ, then the differential equations transform to our equations, and the conditions (32) transform into
J = σ(E + [_u_ M]) _e_ + [_u_, (_m_ - [_u_ _e_])] = ε(E + [_u_ M]) M - [_u_, (E + _u_ M)] = μ(_m_ - [_u_ _e_])
then in fact the equations of Cohn become the same as those required by the relativity principle, if errors of the order _u²_ are neglected in comparison to 1.
It may be mentioned here that the equations of Hertz become the same as those of Cohn, if the auxiliary conditions are
(33) E = εE, M = μM, J = σE.
§11. Typical Representations of the Fundamental Equations.
In the statement of the fundamental equations, our leading idea had been that they should retain a covariance of form, when subjected to a group of Lorentz-transformations. Now we have to deal with ponderomotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations, satisfying the conditions of covariance. In order to arrive at such forms, I shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a Lorentz-transformation. Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below.
A system of magnitudes _a__{_h_ _k_} formed into the matrix
| _a₁₁_...................._a__{1 _q_} | | | | | | | | _a__{_p_ 1}..........._a__{_p_ _q_} |
arranged in _p_ horizontal rows, and _q_ vertical columns is called a _p_ × _q_ series-matrix, and will be denoted by the letter A.
If all the quantities _a__{_h_ _k_} are multiplied by C, the resulting matrix will be denoted by CA.
If the roles of the horizontal rows and vertical columns be intercharged, we obtain a _q_ × _p_ series matrix, which will be known as the transposed matrix of A, and will be denoted by Ā.
Ā = | _a₁₁_ ...................... _a__{_p_ 1} | | | | _a__{1 _q_} ............ _a__{_p_ _q_} |
If we have a second _p_ × _q_ series matrix B,
B = | _b₁₁_ ......................... _b₁__{_q_} | | | | _b__{_p_ 1} ............. b_{_p_ _q_} |
then A + B shall denote the _p_ × _q_ series matrix whose members are _a__{_h_ _k_} + _b__{_h_ _k_}.
2⁰ If we have two matrices
A = | _a₁₁_ ..................... _a__{1 _q_} | | | | _a__{_p_ 1} ........... _a__{_p_ _q_} |
B = | _b__{1 1} .............. _b__{1 _r_} | | | | _b__{_q_ 1} .......... _b__{_p_ _r_} |
where the number of horizontal rows of B, is equal to the number of vertical columns of A, then by AB, the product of the matrices A and B, will be denoted the matrix
C = | _c₁₁_ ...................... _c__{1 _r_} | | | | _c__{_p_ _r_} ........... _c__{_p_ _p_} |
where _c__{_h_ _k_} = _a__{_h_ 1} _b₁__{_k_} + _a__{_h_ 2} _b__{2 _h_} + ... _a__{_k_ _s_} _b__{_s_ _k_} + ... + _a__{_k_ _q_} _b__{_q_ _h_}
these elements being formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law (AB)S = A(BS) holds, where S is a third matrix which has got as many horizontal rows as B (or AB) has got vertical columns.
For the transposed matrix of C = BA, we have Ċ = ḂĀ
3⁰. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows.
As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 × 4 series) with the elements.
(34) | e₁₁ e₁₂ e₁₃ e₁₄ | = | 1 0 0 0 | | e₂₁ e₂₂ e₂₃ e₂₄ | | 0 1 0 0 | | e₃₁ e₃₂ e₃₃ e₃₄ | | 0 0 1 0 | | e₄₁ e₄₂ e₄₃ e₄₄ | | 0 0 0 1 |
For a 4 × 4 series-matrix, Det A shall denote the determinant formed of the 4 × 4 elements of the matrix. If det A ≠ 0, then corresponding to A there is a reciprocal matrix, which we may denote by A⁻¹ so that A⁻¹A = 1.
A matrix
_f_ = | 0 _f₁₂_ _f_₁₃ _f₁₄_ | | _f_₂₁ 0 _f₂₃_ _f₂₄_ | | _f₃₁_ _f_₃₂ 0 _f₃₄_ | | _f_₄₁ _f_₄₂ _f_₄₃ 0 |
in which the elements fulfil the relation _f__{_h_ _k_} = -_f__{_h_ _k_}, is called an alternating matrix. These relations say that the transposed matrix _ḟ_ = -_f_. Then by _f_^{*} will be the _dual_, alternating matrix
(35)
_f_^{*} = | 0 _f₃₄_ _f_₄₂ _f₂₃_ | | _f_₄₃ 0 _f₁₄_ _f₃₁_ | | _f₂₄_ _f_₄₁ 0 _f₁₂_ | | _f_₃₂ _f_₁₃ _f_₂₁ 0 |
Then (36) _f_* _f_ = _f₃₄_ _f₂₂_ + _f₄₂_ _f₃₁_ + _f₃₂_ _f₂₄_
_i.e._ We shall have a 4 × 4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).
The determinant of _f_ is therefore the square of the combination, by Det^{½}_f_ we shall denote the expression
Det^{½}_f_ = _f₃₂_ _f₁₄_ _f₁₃_ _f₂₄_ + _f₂₁_ _f₃₄_·
4⁰. A linear transformation
_x__{_h_} = α_{_h_1} _x₁′_ + α_{_h_2} _x₂_′ + α_{_h_3} _x₃′_ + α_{_h_4} _x₄′_ (_h_ = 1,2,3,
which is accomplished by the matrix
A = | α₁₁, α₁₂, α₁₃, α₁₄ | | | | α₂₁, α₂₂, α₂₃, α₂₄ | | | | α₃₁, α₃₂, α₃₃, α₃₄ | | | | α₄₁, α₄₂, α₄₃, α₄₄ |
will be denoted as the transformation A.
By the transformation A, the expression
_x²₁_ + _x²₂_ + _x²₃_ + _x²₄_ is changed into the quadratic for _m_ ∑ α_{_hk_} _x__{_h_}′ _x__{_k_}′,
where α_{_hk_} = α_{1_k_} α_{1_k_} + α_{2_h_} α_{2_k_} + α_{3_h_} α_{3_k_} + α_{4_h_} α_{4_k_} are the members of a 4 × 4 series matrix which is the product of Ā A, the transposed matrix of A into A. If by the transformation, the expression is changed to
_x′₁²_ + _x₂′_^2 + _x₃′_^2 + _x′₄²_,
we must have Ā A = 1.
A has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of A) it follows out of (39) that (Det A)² = 1, or Det A = ± 1.
From the condition (39) we obtain
A⁻¹ = Ā,
_i.e._ the reciprocal matrix of A is equivalent to the transposed matrix of A.
For A as Lorentz transformation, we have further Det A = +1, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and _a₄₄_ > 0.
5⁰. A space time vector of the first kind[21] which s represented by the 1 × 4 series matrix,
(41) _s_ = |_s₁_ _s₂_ _s₃_ _s₄_|
is to be replaced by _s_A in case of a Lorentz transformation
A. _i.e._ _s′_ = | _s₁′_ _s₂′_ _s₃′_ _s₄′_| = |_s₁_ _s₂_ _s₃_ _s₄_| A;
A space-time vector of the 2nd kind[22] with components _f₂₃_ ... _f₃₄_ shall be represented by the alternating matrix
(42) _f_ = | 0 _f_₁₂ _f₁₃_ _f₁₄_ |
|_f₂₁_ 0 _f_₂₃ _f₂₄_ |
|_f_₃₁ _f₃₂_ 0 _f₃₄_ |
|_f_₄₁ _f_₄₂ _f_₄₃ 0 |
and is to be replaced by A⁻¹ _f_ A in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression (37), we have the identity Det^{½} (Ā _f_ A) = Det A. Det^{½} _f_. Therefore Det^{½} _f_ becomes an invariant in the case of a Lorentz transformation [see eq. (26) See. § 5].
Looking back to (36), we have for the dual matrix (Ā_f_*A) (A⁻¹_f_A) = A⁻¹_f_*_f_A = Det^{½} function. A⁻¹A = Det^{½}_f_ from which it is to be seen that the dual matrix _f_* behaves exactly like the primary matrix _f_, and is therefore a space time vector of the II kind; _f_* is therefore known as the dual space-time vector of _f_ with components (_f₁₄_, _f₂₄_, _f₃₄_,), (_f₂₃_}, _f₃₁_, _f₁₂_).
6. If _w_ and _s_ are two space-time rectors of the 1st kind then by _w_ _ṡ_ (as well as by _s_ _ẇ_) will be understood the combination (43) _w₁_ _s₁_ + _w₂_ _s₂_ + _w₃_ _s₃_ + _w₄_ _s₄_.
In case of a Lorentz transformation A, since (_w_A) (Ā_ṡ_) = _w_ _s_, this expression is invariant.—If _w_ _ṡ_ = 0, then _w_ and _s_ are perpendicular to each other.
Two space-time rectors of the first kind (_w_, _s_) gives us a 2 × 4 series matrix
| _w₁_ _w₂_ _w₃_ _w₄_ | | _s₁_ _s₂_ _s₃_ _s₄_ |
Then it follows immediately that the system of six magnitudes (44)
_w₂_ _s₃_ - _w₃_ _s₂_, _w₃_ _s₁_ - _w₁_ _s₃_, _w₁_ _s₂_ - _w₂_ _s₁_, _w₁_ _s₄_ - _w₄_ _s₁_, _w₂_ _s₄_ - _w₄_ _s₂_, _w₃_ _s₄_ - _w₄_ _s₃_,
behaves in case of a Lorentz-transformation as a space-time vector of the II kind. The vector of the second kind with the components (44) are denoted by [_w_, _s_]. We see easily that Det^{½} [_w_, _s_] = 0. The dual vector of [_w_, _s_] shall be written as [_w_, _s_].
If _ẇ_ is a space-time vector of the 1st kind, _f_ of the second kind, _w_ _f_ signifies a 1 × 4 series matrix. In case of a Lorentz-transformation A, _w_ is changed into _w′_ = _w_A, _f_ into _f′_ = A⁻¹ _f_ A,—therefore _w′_ _f′_ becomes = (_w_A A⁻¹ _f_ A) = _w_ _f_ A _i.e._ _w_ _f_ is transformed as a space-time vector of the 1st kind.[23] We can verify, when _w_ is a space-time vector of the 1st kind, _f_ of the 2nd kind, the important identity
(45) [_w_, _w__f_] + [_w_, _w__f_*]* = (_w_] _ẇ_)_f_.
The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.
For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_,
ω_f_ = | _i__f_₄₁, _i__f_₄₂, _i__f_₄₃, 0 |; ω_f_* = | _i__f_₃₂, _i__f_₁₃, _i__f_₂₁, 0 |
[ω · ω_f_] = 0, 0, 0, _f_₄₁, _f_₄₂, _f_₄₃; [ω · ω_f_*]* = 0, 0, 0, _f_₃₂, _f_₁₃, _f_₂₁.
The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in (ω₁, ω₂, ω₃, ω₄).
After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants ε μ, σ will be introduced.
Instead of the space vector _u_, the velocity of matter, we shall introduce the space-time vector of the first kind ω with the components.
ω₁ = _u__{_x_}/√(1 - _u²_), ω₂ = _u__{_y_}/√(1 - _u²_), ω₃ = _u__{_z_}/√(1 - _u²_), ω₄ = _i_/√(1 - _u²_).
(40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 and -_i_ω₄ > 0.
By F and _f_ shall be understood the space time vectors of the second kind M - _i_E, _m_ - _ie_.
In Φ = ωF, we have a space time vector of the first kind with components
Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄
Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄
Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄
Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃
The first three quantities (φ₁, φ₂, φ₃) are the components of the space-vector (E + [_u_, M])/√(1 - _u²_),
and further (φ₄ = _i_[_u_ E]/√(1 - _u²_).
Because F is an alternating matrix,
(49) ωΦ = ω₁ φ₁ + ω₂ Φ₂ + ω₃ Φ₃ + ω₄ Φ₄ = 0.
_i.e._ Φ is perpendicular to the vector ω; we can also write Φ₄ = _i_[ω_{x} Φ₁ + ω_{y} Φ₂ + ω_{z} Φ₃].
I shall call the space-time vector Φ of the first kind as the _Electric Rest Force_.[24]
Relations analogous to those holding between -ωF, E, M, U, hold amongst -ω_f_, _e_, _m_, _u_, and in particular -ω_f_ is normal to ω. The relation (C) can be written as
{C} ω_f_ = εωF.
The expression (ω_f_) gives four components, but the fourth can be derived from the first three.
Let us now form the time-space vector 1st kind, ψ - _i_ω_f_*, whose components are
ψ₁ = -_i_(ω₂ _f₃₄_ + ω₃ _f_₄₂ + ω₄ _f₂₃_) ψ₂ = -_i_(ω₁ _f_₄₃ + ω₃ _f_₄₄ + ω₄ _f₃₁_) ψ₃ = -_i_(ω₁ _f₂₄_ + ω₂ _f_₄₁ + ω₄ _f₁₂_) ψ₄ = -_i_(ω₁ _f_₃₂ + ω₂ _f_₁₃ + ω₃ _f_₂₁)
Of these, the first three ψ₁, ψ₂, ψ₃, are the _x_, _y_, _z_ components of the space-vector 51) (m - (_ue_))/√(1 - _u²_) and further (52) ψ₄ = _i_(_u_m)/√(1 - _u²_).
Among these there is the relation
(53) ωψ = ω₁ ψ₁ + ω₂ ψ₂ + ω₃ ψ₃ + ω₄ ψ₄ = 0
which can also be written as ψ₄ = _i_ (_u__{_x_} ψ₁ + _u__{_y_} ψ₂ + _u__{_z_} ψ₃).
The vector ψ is perpendicular to ω; we can call it the _Magnetic rest-force_.
Relations analogous to these hold among the quantities ωF*, M, E, _u_ and Relation (D) can be replaced by the formula
{ D } -ωF* = μψ_f_*.
We can use the relations (C) and (D) to calculate F and _f_ from Φ and ψ we have
ωF = -Φ, ωF* = -_i_μψ, ω_f_ = -εΦ, ω_f_* = -_i_ψ.
and applying the relation (45) and (46), we have
F = [ω. Φ] + _i_μ[ω. ψ]* 55) _f_ = ε[ω. Φ] + _i_[ω. ψ]* 56)
_i.e._
F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + _i_μ [ω₃ Ψ₄ - ω₄ ψ₃], etc. _f₁₂_ = ε(ω₁ Φ₂ - ω₂ φ₁) + _i_ [ω₃ ψ₄ - ω₄ ψ₃]., etc.
Let us now consider the space-time vector of the second kind [Φ ψ], with the components
[ Φ₂ ψ₃ - Φ₃ ψ₂, Φ₃ ψ₁ - Φ₁ ψ₃, Φ₁ ψ₂ - Φ₂ ψ₁ ] [ Φ₁ ψ₄ - Φ₄ ψ₁, Φ₂ ψ₄ - Φ₄ ψ₂, Φ₃ ψ₄ - Φ₄ ψ₃ ]
Then the corresponding space-time vector of the first kind ω[Φ, ψ] vanishes identically owing to equations 9) and 53)
for ω[Φ.ψ] = -(ωψ)Φ + (ωΦ)ψ
Let us now take the vector of the 1st kind
(57) Ω = _i_ω[Φψ]*
with the components
Ω₁ = -_i_ | ω₂ ω₃ ω₄ | | Φ₂ Φ₃ Φ₄ | | ψ₂ ψ₃ ψ₄ |, etc.
Then by applying rule (45), we have
(58) [Φ.ψ] = _i_[ωΩ]*
_i.e._ Φ₁ψ₂ - Φ₂ψ₁ = _i_(ω₃Ω₄ - ω₄Ω₃) etc.
The vector Ω fulfils the relation
(ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0,
(which we can write as Ω₄ = _i_(ω_{x}Ω₁ + ω_{y}Ω₂ + ω_{z}Ω₃) and Ω is also normal to ω. In case ω = 0, we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and
[Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ | |ψ₁ ψ₂ ψ₃ |.
I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.
As for the relation E), which introduces the conductivity σ we have -ωS = -(ω₁_s₁_ + ω₂_s₂_ + ω₃_s₃_ + ω₄_s₄_) = (- | _u_ | C_{_u_} + ρ)/√(1 - _u²_) = ρ′.
This expression gives us the rest-density of electricity (see §8 and §4).
Then 61) = _s_ + (ω_ṡ_)ω represents a space-time vector of the 1st kind, which since ωω = -1, is normal to ω, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the (_x_-_y_-_z_) co-ordinates of the space-vector, then the component in the direction of _u_ is
C_{_u_} - (| _u_ | ρ′)/√(1 - _u²_) = (_c__{_u_} - | _u_ |ρ)/√(1 - _u²_) = J_{_u_}/(1 - _u²_)
and the component in a perpendicular direction is C_{_u_} = J_{_ū_}.
This space-vector is connected with the space-vector J = C - ρ_u_, which we denoted in §8 as the conduction-current.
Now by comparing with Φ = -ωF, the relation (E) can be brought into the form
{E} _s_ + (ω_ṡ_)ω = - σωF,
This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to ω.
Lastly, we shall transform the differential equations (A) and (B) into a typical form.
§12. The Differential Operator Lor.
A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S_{_kh_} | | S₂₁ S₂₂ S₂₃ S₂₄ | | S₃₁ S₃₂ S₃₃ S₃₄ | | S₄₁ S₄₂ S₄₃ S₄₄ |
with the condition that in case of a Lorentz transformation it is to be replaced by ĀSA, may be called a space-time matrix of the II kind. We have examples of this in:—
1) the alternating matrix _f_, which corresponds to the space-time vector of the II kind,—
2) the product _f_F of two such matrices, for by a transformation A, it is replaced by (A⁻¹_f_A·A⁻¹FA) = A⁻¹_f_FA,
3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time vectors of the 1st kind, the 4 × 4 matrix with the element S_{_hk_} = ω_{_h_}Ω_{_k_},
lastly in a multiple L of the unit matrix of 4 × 4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.
We shall have to do constantly with functions of the space-time point (_x_, _y_, _z_, _it_), and we may with advantage
employ the 1 × 4 series matrix, formed of differential symbols,—
| ∂/∂_x_, ∂/∂_y_, ∂/∂_z_, ∂/_i_∂_t_,| or (63) | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ |
For this matrix I shall use the shortened from “lor.”[25]
Then if S is, as in (62), a space-time matrix of the II kind, by lor S′ will be understood the 1 × 4 series matrix
| K₁ K₂ K₃ K₄ |
where K_{_k_} = ∂S_{1_k_}/∂_x₁_ + ∂S_{2_k_}/∂_x₂_ + ∂S_{3_k_}/∂_x₃_ + ∂S_{4_h_}/∂_x₄_.
When by a Lorentz transformation A, a new reference system (_x′₁_ _x′₂_ _x′₃_ _x₄_) is introduced, we can use the operator
lor′ = | ∂/∂_x₁′_ ∂/∂_x₂′_ ∂/∂_x₃′_ ∂/∂_x₄′_ |
Then S is transformed to S′= Ā S A = | S′_{_hk_} |, so by lor 'S′ is meant the 1 × 4 series matrix, whose element are
K’_{_k_} = ∂S′_{1_k_}/∂_x₁′_ + ∂S′_{2_k_}/∂_x₂′_ + ∂S′_{3_k_}/∂_x₃′_ + ∂S′_{4_k_}/∂_x₄′_.
Now for the differentiation of any function of (_x_ _y_ _z_ _t_) we have the rule ∂/∂_x__{_k_}′ = ∂/∂_x₁_ ∂_x₁_/∂_x__{_k_}′ + ∂/∂_x₂_ ∂_x₂_/∂_x__{_k_}′ + ∂/∂_x₃_ ∂_x₃_/∂_x__{_k_}′ + ∂/∂_x₄_ ∂_x₄_/∂_x__{_k_}′ = ∂/∂_x₁_ _a__{1_k_} + ∂/∂_x₂_ _a__{2_k_} + ∂/∂_x₃_ _a__{3_k_} + ∂/∂_x₄_ _a__{4_k_}.
so that, we have symbolically lor′ = lor A.
Therefore it follows that
lor ′S′ = lor (A A⁻¹ SA) = (lor S)A.
_i.e._, lor S behaves like a space-time vector of the first kind.
If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements
| ∂L/∂_x₁_ ∂L/∂_x₂_ ∂L/∂_x₃_ ∂L/∂_x₄_ |
If _s_ is a space-time vector of the 1st kind, then
lor _ṡ_ = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_.
In case of a Lorentz transformation A, we have
lor ′_ṡ′_ = lor A. Ā_s_ = lor _s_.
_i.e._, lor _s_ is an invariant in a Lorentz-transformation.
In all these operations the operator lor plays the part of a space-time vector of the first kind.
If _f_ represents a space-time vector of the second kind,—lor _f_ denotes a space-time vector of the first kind with the components
∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_, ∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_, ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_, ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_
So the system of differential equations (A) can be expressed in the concise form
{A} lor f = -_s_,
and the system (B) can be expressed in the form
{B} log F* = 0.
Referring back to the definition (67) for log _ṡ_, we find that the combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish identically, when _f_ and F* are alternating matrices. Accordingly it follows out of {A}, that
(68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) = 0,
while the relation
(69) lor (lor F*) = 0,
signifies that of the four equations in {B}, only three represent independent conditions.
I shall now collect the results.
Let ω denote the space-time vector of the first kind
(_u_/√(1 - _u²_}), _i_/√(1 - _u²_))
(_u_ = velocity of matter),
F the space-time vector of the second kind (M,-_i_E)
(M = magnetic induction, E = Electric force,
_f_ the space-time vector of the second kind (_m_,-_ie_)
(_m_ = magnetic force, _e_ = Electric Induction.
_s_ the space-time vector of the first kind (C, _i_ρ)
(ρ = electrical space-density, C - ρ_u_ = conductivity current,
ε = dielectric constant, μ = magnetic permeability,
σ = conductivity,
then the fundamental equations for electromagnetic processes in moving bodies are[26]
{A} lor _f_ = -_s_
{B} log F* = 0
{C} ω_f_ = εωF
{D} ωF* = μω_f_*
{E} _s_ + (ω_ṡ_), _w_ = - σωF.
ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time vectors of the first kind are all normal to ω, and for the system {B}, we have
lor (lor F*) = 0.
Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the motion of matter as well as the vector _u_ as a function of _x_, _y_, _z_, _t_, when proper fundamental data are given.
§ 13. The Product of the Field-vectors _f_ F.
Finally let us enquire about the laws which lead to the determination of the vector ω as a function of (_x_, _y_, _z_, _t_.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices
_f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ | | _f₂₁_ 0 _f₂₃_ _f₂₄_ | | _f₃₁_ _f₃₂_ 0 _f₃₄_ | | _f₄₁_ _f₄₂_ _f₄₃_ 0 |
F = | 0 F₁₂ F₁₃ F₁₄ | | F₂₁ 0 F₂₃ F₂₄ | | F₃₁ F₃₂ 0 F₃₄ | | F₄₁ F₄₂ F₄₃ 0 |
are of much importance. Let us write,
(70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ - L S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ - L S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ - L |
Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.
Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by
(72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄ + _f₂₄_ F₂₄ + _f₃₄_ F₃₄)
Then we shall have
(73) S₁₁ = ½(_f₂₃_ F₂₃ + _f₃₄_ F₃₄ + _f₄₂_ F₄₂ - _f₁₂_ F₁₂ - _f₁₃_ F₁₃ _f₁₄_ F₁₄)
S₁₂ = _f₁₃_ F₃₂ + _f₁₄_ F₄₂ etc....
In order to express in a real form, we write
(74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
= | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
| X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
| X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
| -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} |
Now X_{_x_} = ½[_m__{_x_}M_{_x_} - _m__{_y_}M_{_y_} - _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]
so
(75) X_{_y_} = _m__{_x_}M_{_y_} + _e__{_y_}E_{_x_}, Y_{_x_} = _m__{_y_}M_{_x_} + _e__{_x_}E_{_y_} etc.
X_{_t_} = _e__{_y_}M_{_z_} - _e__{_z_}M_{_y_}, T_{_x_} = _m__{_x_}E_{_y_} - _m__{_y_}E_{_z_}, etc.
T_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} + _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} + _e__{_y_}E_{_y_} + _e__{_z_}E_{_z_}]
L_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} + _m__{_z_}M_{_z_} - _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]
These quantities[27] are all real. In the theory for bodies at rest, the combinations (X_{_x_}, X_{_y_}, X_{_z_}, Y_{_z_}, Y_{_y_}, Y_{_z_}, Z_{_x_}, Z_{_y_}, Z_{_z_}) are known as “Maxwell’s Stresses,” T_{_x_}, T_{_y_}, T_{_z_} are known as the Poynting’s Vector, T_{_t_} as the electromagnetic energy-density, and L as the Langrangian function.
On the other hand, by multiplying the alternating matrices of _f_* and F*, we obtain
(77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |
| -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |
| -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |
| -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |
and hence, we can put
(78) _f_F = S - L, F*_f_* = -S - L,
where by L, we mean L-times the unit matrix, _i.e._ the matrix with elements
| L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_ = 1, 2, 3, 4).
Since here SL = LS, we deduce that,
F*_f_*_f_F = (-S - L)(S - L) = -SS + L²,
and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the interesting
conclusion
(79) SS = L² - Det^{½}_f_ Det^{½}F
_i.e._ the product of the matrix S into itself can be expressed as the multiple of a unit matrix—a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations
(80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} + S_{_h_4} S_{4_k_} = 0,
_h_, _k_ being unequal indices in the series 1, 2, 3, 4, and
(81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} + S{_h_4} S_{4_h_} = L² - Det^{½}_f_ Det^{½}F,
for _h_ = 1, 2, 3, 4.
Now if instead of F, and _f_ in the combinations (72) and (73), we introduce the electrical rest-force Φ, the magnetic rest-force ψ, and the rest-ray Ω [(55), (56) and (57)], we can pass over to the expressions,—
(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
(83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_} + ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_} + μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ ω_{_h_} Ω_{_k_} (_h₁_ _k_ = 1, 2, 3, 4).
Here we have
Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²
_e__{_hh_} = 1, _e__{_hk_} = 0 (_h_ ≠ _k_).
The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the right side of (83) as well as S_{_k_ _h_} represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and _e_ = εE, M = μ_m_ on the other hand.
The expression on the right-hand side of (81), which equals
[½ (_m_ M - _e_E)²] + (_em_) (EM),
is >= 0, because (_em_ = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back to 79), we can denote the positive square root of this expression as Det^{1/4} S.
Since _ḟ_ = -_f_, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of S, the following relations from (78),
(84) F_f_ = Ṡ - L, _f_* F* = -Ṡ - L,
Then is
Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} |
an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,
(85) S - Ṡ = - (εμ - 1) [ω, Ω],
from which we deduce that [see (57), (58)].
(86) ω (S - Ṡ)* = 0,
(87) ω (S - Ṡ) = (εμ - 1) Ω
When the matter is at rest at a space-time point, ω = 0, then the equation 86) denotes the existence of the following equations
Z_{_y_} = Y_{_z_}, X_{_z_} = Z_{_x_}, Y_{_x_} = X_{_y_},
and from 83),
T_{_x_} = Ω₁, T_{_y_} = Ω₂, T_{_z_} = Ω₃
X_{_t_} = εμΩ₁, Y_{_t_} = εμΩ₂, Z_{_t_} = εμΩ₃
Now by means of a rotation of the space co-ordinate system round the null-point, we can make,
Z_{_y_} = Y_{_z_} = 0, X_{_z_} = Z_{_x_} = 0, X_{_x_} = X_{_y_} = 0,
According to 71), we have
(88) X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} = 0,
and according to 83), T_{_t_} > 0. In special cases, where ω vanishes it follows from 81) that
X_{_x_}² = Y_{_y_}² = Z_{_z_}² = T_{_t_}², = (Det^{1/4} S)²,
and if T, and one of the three magnitudes X_{_x_}, Y_{_y_}, Z_{_z_} are = ±Det^{1/4} S, the two others = -Det^{1/4} S. If Ω does not vanish let Ω ≠ 0, then we have in particular from 80)
T_{_z_} X_{_t_} = 0, T_{_z_} Y_{_t_} = 0, Z_{_z_} T_{_z_} + T_{_z_} T_{_t_} = 0,
and if Ω₁ = 0, Ω₂ = 0, Z_{_z_} = -T_{_t_} It follows from (81), (see also 83) that
X_{_x_} = -Y_{_y_} = ±Det^{1/4} S,
and -Z_{_z_} = T_{_t_} = √(Det^{½} S + εμΩ₃²) > Det^{1/4}S.
The space-time vector of the first kind
(89) K = lor S,
is of very great importance for which we now want to demonstrate a very important transformation
According to 78), S = L + _f_F, and it follows that
lor S = lor L + lor _f_F.
The symbol ‘lor’ denotes a differential process which in lor _f_F, operates on the one hand upon the components of _f_, on the other hand also upon the components of F. Accordingly lor _f_F can be expressed as the sum of two parts. The first part is the product of the matrices (lor _f_) F, lor _f_ being regarded as a 1 × 4 series matrix. The second part is that part of lor _f_F, in which the diffentiations operate upon the components of F alone. From 78) we obtain
_f_F = -F*_f_* - 2L;
hence the second part of lor _f_F = -(lor F*)_f_* + the part of -2 lor L, in which the differentiations operate upon the components of F alone. We thus obtain
lor S = (lor _f_)F - (lor F*)_f_* + N,
where N is the vector with the components
N_{_h_} = ½(∂_f₂₃_/∂_x__{_h_} F₂₃ + ∂_f₃₁_/∂_x__{_h_} F₃₁ + ∂_f₁₂_/∂_x__{_h_} F₁₂ + ∂_f₁₄_/∂_x__{_h_} F₁₄ + ∂_f₂₄_/∂_x__{_h_} F₂₄ + ∂_f₃₄_/∂_x__{_h_} F₃₄ - ∂F₂₃/∂_x__{_h_} _f₂₃_ - ∂F₃₁/∂_x__{_h_} _f_₃₁ - ∂F₁₂/∂_x__{_h_} _f₁₂_ - ∂F₁₄/∂_x__{_h_} _f₁₄_ - ∂F₂₄/∂_x__{_h_} _f₂₄_ - ∂F₃₄/∂_x__{_h_} _f₃₄_),
(_h_ = 1, 2, 3, 4)
By using the fundamental relations A) and B), 90) is transformed into the fundamental relation
(91) lor S = -_s_F + N.
In the limitting case ε = 1, μ = 1, _f_ = F, N vanishes identically.
Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of N,—
(92) N_{_h_} = - ½ Φ[=Φ]∂ε/∂_x__{_h_} - ½ ψ[=ψ]∂μ/∂_x__{_h_} + (εμ - 1)(Ω₁ ∂ω₁/∂_x__{_h_} + Ω₂ ∂ω₂/∂_x__{_h_} + Ω₃ ∂ω₃/∂_x__{_h_} + Ω₄ ∂ω₄/∂_x__{_h_})
for _h_ = 1, 2, 3, 4.
Now if we make use of (59), and denote the space-vector which has Ω₁, Ω₂, Ω₃ as the _x_, _y_, _z_ components by the symbol W, then the third component of 92) can be expressed in the form
(93) (εμ - 1)/√(1 - _u²_) (W ∂_u_/∂_x__{_h_}),
The round bracket denoting the scalar product of the vectors within it.
§ 14. The Ponderomotive Force.[28]
Let us now write out the relation K = lor S = -_s_F + N in a more practical form; we have the four equations
(94) K₁ = ∂X_{_x_}/∂_x_ + ∂X_{_y_}/∂_y_ + ∂X_{_y_}/∂_z_ - ∂X_{_t_}/∂_t_ = ρE_{_x_} + _s__{_y_}M_{_z_} - _s__{_z_}M_{_x_}
- ½ Φ[=Φ] ∂ε/∂_x_ - ½ ψ[=ψ]∂μ/∂_x_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_x_),
(95) K₂ = ∂Y_{_x_}/∂_x_ + ∂Y_{_y_}/∂_y_ + ∂Y_{_z_}/∂_z_ - ∂Y_{_t_}/∂_t_ = ρE_{_y_} + _s__{_z_}M_{_x_} - _s__{_x_}M_{_y_}
- ½ Φ[=Φ]∂ε/∂_y_ - ½ ψ[=ψ]∂μ/∂_y_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_y_),
(96) K₃ = ∂Z_{_x_}/∂_x_ + ∂Z_{_y_}/∂_y_ + ∂Z_{_z_}/∂_z_ - ∂Z_{_t_}/∂_t_ = ρE₂ + _s__{_x_}M_{_y_} - _s__{_y_}M₄
- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂_z_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_z_),
(97) (1/_i_)K₄ = ∂T_{_y_}/∂_x_ - ∂T_{_y_}/∂_y_ - ∂T_{_z_}/∂_z_ - ∂T_{_t_}/∂_t_ = _s__{_x_}E_{_x_} + _s__{_y_}E_{_y_} + _s__{_z_}E_{_z_}
- ½ Φ[=Φ]∂ε/∂_t_ - ½ ψ[=ψ]∂μ/∂_t_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_t_).
It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point _x_, _y_, _z_, _t_, it has got, _x_, _y_, _z_ components as the first three components of the space-time vector
K + (ωK)ω,
This vector is perpendicular to ω; the law of Energy finds its expression in the fourth relation.
The establishment of this opinion is reserved for a separate tract.
In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK = 0, and we obtain the ordinary equations in the theory of electrons.
Footnote 9:
_Vide_ Note 1.
Footnote 10:
Note 2.
Footnote 11:
_Vide_ Note 3.
Footnote 12:
_Vide_ Note 4.
Footnote 13:
Note 5.
Footnote 14:
See notes on § 8 and 10.
Footnote 15:
See note 9.
Footnote 16:
See Note.
Footnote 17:
Vide Note.
Footnote 18:
Just as beings which are confined within a narrow region surrounding a point on a spherical surface, may fall into the error that a sphere is a geometric figure in which one diameter is particularly distinguished from the rest.
Footnote 19:
Einzelne stelle der Materie.
Footnote 20:
Vide Note.
Footnote 21:
_Vide_ note 13.
Footnote 22:
_Vide_ note 14.
Footnote 23:
_Vide_ note 15.
Footnote 24:
_Vide_ note 16.
Footnote 25:
_Vide_ note 17.
Footnote 26:
_Vide_ note 19.
Footnote 27:
_Vide_ note 18.
Footnote 28:
Vide note 40.
APPENDIX Mechanics and the Relativity-Postulate.
It would be very unsatisfactory, if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.
Now many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basis of the new Electro-dynamics.
In order to decide this let us fix our attention upon a special Lorentz transformation represented by (10), (11), (12), with a vector _v_ in any direction and of any magnitude _q_ < 1 but different from zero. For a moment we shall not suppose any special relation to hold between the unit of length and the unit of time, so that instead of _t_, _t′_, _q_, we shall write _ct_, _ct′_, and _q_/_c_, where _c_ represents a certain positive constant, and _q_ is < _c_. The above mentioned equations are transformed into
_r′__{_ṽ_} = _r__{_ṽ_}, _r′__{_v_} = _c_(_r__{_v_} - _qt_)/√(_c²_ - _q²_), _t′_ = (_qr__{_v_} + _c²__t_)/_c_√(_c²_ - _q²_)
They denote, as we remember, that _r_ is the space-vector (_x_, _y_, _z_), _r′_ is the space-vector (_x′_ _y′_ _z′_)
If in these equations, keeping _v_ constant we approach the limit _c_ = ∞, then we obtain from these
_r′__{_ṽ_} = _r__{_ṽ_}, _r′__{_v_} = _r__{_v_} - _qt_, _t′_ = _t_.
The new equations would now denote the transformation of a spatial co-ordinate system (_x_, _y_, _z_) to another spatial co-ordinate system (_x′_ _y′_ _z′_) with parallel axes, the null point of the second system moving with constant velocity in a straight line, while the time parameter remains unchanged. We can, therefore, say that classical mechanics postulates a covariance of Physical laws for the group of homogeneous linear transformations of the expression
-_x²_ - _y²_ - _z²_ + _c²_ (1)
when _c_ = ∞.
Now it is rather confusing to find that in one branch of Physics, we shall find a covariance of the laws for the transformation of expression (1) with a finite value of _c_, in another part for _c_ = ∞.
It is evident that according to Newtonian Mechanics, this covariance holds for _c_ = ∞ and not for _c_ = velocity of light.
May we not then regard those traditional covariances for _c_ = ∞ only as an approximation consistent with experience, the actual covariance of natural laws holding for a certain finite value of _c_.
I may here point out that by if instead of the Newtonian Relativity-Postulate with _c_ = ∞, we assume a relativity-postulate with a finite _c_, then the axiomatic construction of Mechanics appears to gain considerably in perfection.
The ratio of the time unit to the length unit is chosen in a manner so as to make the velocity of light equivalent to unity.
While now I want to introduce geometrical figures in the manifold of the variables (_x_, _y_, _z_, _t_), it may be convenient to leave (_y_, _z_) out of account, and to treat _x_ and _t_ as any possible pair of co-ordinates in a plane, referred to oblique axes.
A space time null point 0 (_x_, _y_, _z_, _t_ = 0, 0, 0, 0) will be kept fixed in a Lorentz transformation.
The figure -_x²_ - _y²_ - _z²_ + _t²_ = 1, _t_ > 0 ... (2)
which represents a hyper boloidal shell, contains the space-time points A (_x_, _y_, _z_, _t_ = 0, 0, 0, 1), and all points A′ which after a Lorentz-transformation enter into the newly introduced system of reference as (_x′_, _y′_, _z′_, _t′_ = 0, 0, 0, 1).
The direction of a radius vector 0A′ drawn from 0 to the point A′ of (2), and the directions of the tangents to (2) at A′ are to be called normal to each other.
Let us now follow a definite position of matter in its course through all time _t_. The totality of the space-time points (_x_, _y_, _z_, _t_) which correspond to the positions at different times _t_, shall be called a space-time line.
The task of determining the motion of matter is comprised in the following problem:—It is required to establish for every space-time point the direction of the space-time line passing through it.
To transform a space-time point P (_x_, _y_, _z_, _t_) to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference (_x′_, _y′_, _z′_, _t′_), in which the _t′_ axis has the direction 0A′, 0A′ indicating the direction of the space-time line passing through P. The space _t′_ = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P.
To the increment _dt_ of the time of P corresponds the increment
_d_τ = √(_dt²_ - _dx²_ - _dy²_) - _dz²_ = _dt_√(1 - _u²_)
of the newly introduced time parameter _t′_. The value of the integral
∫ _dτ_ = ∫ √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_))
when calculated upon the space-time line from a fixed initial point P₀ to the variable point P, (both being on the space-time line), is known as the ‘Proper-time’ of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.)
If we take a body R₀ which has got extension in space at time _t₀_, then the region comprising all the space-time line passing through R₀ and _t₀_ shall be called a space-time filament.
If we have an analytical expression θ(_x_ _y_, _z_, _t_) so that θ(_x_, _y_ _z_ _t_) = 0 is intersected by every space time line of the filament at one point,—whereby
-(∂Θ/∂_x_)², -(∂Θ/∂_y_)², -(∂Θ/∂_z_)², -(∂Θ/∂_t_)² > 0, ∂Θ/∂_t_ > 0.
then the totality of the intersecting points will be called a cross section of the filament.
At any point P of such across-section, we can introduce by means of a Lorentz transformation a system of reference (_x′_, _y_, _z′_ _t_), so that according to this
∂Θ/∂_x′_ = 0, ∂Θ/∂_y′_ = 0, ∂Θ/∂_z′_ = 0, ∂Θ/∂_t′_ > 0.
The direction of the uniquely determined _t′_—axis in question here is known as the upper normal of the cross-section at the point P and the value of _d_J = ∫∫∫ _dx′ dy′ dz′_ for the surrounding points of P on the cross-section is known as the elementary contents (Inhalts-element) of the cross-section. In this sense R₀ is to be regarded as the cross-section normal to the _t_ axis of the filament at the point _t_ = _t₀_, and the volume of the body R₀ is to be regarded as the contents of the cross-section.
If we allow R₀ to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a case, a space-time line will be thought of as a principal line and by the term ‘Proper-time’ of the filament will be understood the ‘Proper-time’ which is laid along this principal line; under the term normal cross-section of the filament, we shall understand the cross-section upon the space which is normal to the principal line through P.
We shall now formulate the principle of conservation of mass.
To every space R at a time _t_, belongs a positive quantity—the mass at R at the time _t_. If R converges to a point (_x_, _y_, _z_, _t_), then the quotient of this mass, and the volume of R approaches a limit μ(_x_, _y_, _z_, _t_), which is known as the mass-density at the space-time point (_x_, _y_, _z_, _t_).
The principle of conservation of mass says—that for an infinitely thin space-time filament, the product μ_d_J, where μ = mass-density at the point (_x_, _y_, _z_, _t_) of the filament (_i.e._, the principal line of the filament), _d_J = contents of the cross-section normal to the _t_ axis, and passing through (_x_, _y_, _z_, _t_), is constant along the whole filament.
Now the contents _d_J_{n} of the normal cross-section of the filament which is laid through (_x_, _y_, _z_, _t_) is
(4) _d_J_{n} = (1/√(1 - _u²_))_d_J = -_i_ω₄ _d_J = (_dt_/_d_τ)_d_J.
and the function
ν = μ/-_i_ω₄ = μ√(1 - _u²_)) = μ(∂τ/∂_t_. (5)
may be defined as the rest-mass density at the position (_x_ _y_ _z_ _t_). Then the principle of conservation of mass can be formulated in this manner:—
_For an infinitely thin space-time filament, the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament._
In any space-time filament, let us consider two cross-sections Q° and Q′, which have only the points on the boundary common to each other; let the space-time lines inside the filament have a larger value of _t_ on Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be called a space-time _sichel_,[29] Q′ is the lower boundary, and Q′ is the upper boundary of the _sichel_.
If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of the _sichel_, there corresponds an exit point of the same by the upper boundary, whereby for both, the product νdJ_{n} taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals ∫ν_dJ__{n} (the first being extended over the upper, the second upon the lower boundary) vanishes. According to a well-known theorem of Integral Calculus the difference is equivalent to
∫∫∫∫ lor ν[=ω] _dx dy dz dt_,
the integration being extended over the whole range of the _sichel_, and (comp. (67), § 12)
lor ν[=ω] = (∂νω₁/∂_x₁_) + (∂νω₂/∂_x₂_) + (∂νω₃/∂_x₃_) + (∂νω₄/∂_x₄_).
If the _sichel_ reduces to a point, then the differential equation
lor ν[=ω] = 0, (6)
which is the condition of continuity
(∂μ_u__{_x_}/∂_x_) + (∂μ_u__{_y_}/∂_y_) + (∂μ_u__{_z_}/∂_z_) + (∂μ/∂_t_) = 0.
Further let us form the integral
N = ∫ ∫∫∫ ν _dx dy dz dt_ (7)
extending over the whole range of the space-time _sichel_. We shall decompose the _sichel_ into elementary space-time filaments, and every one of these filaments in small elements _d_τ of its proper-time, which are however large compared to the linear dimensions of the normal cross-section; let us assume that the mass of such a filament ν_d_J_{_n_} = _dm_ and write τ⁰, τ^l for the ‘Proper-time’ of the upper and lower boundary of the _sichel_.
Then the integral (7) can be denoted by
∫∫ ν_d_J_{_n_} _d_τ = ∫ (τ′-τ⁰) _dm_.
taken over all the elements of the sichel.
Now let us conceive of the space-time lines inside a space-time _sichel_ as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sichel in the following manner. The entire curves are to be varied in any possible manner inside the _sichel_, while the end points on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they always move forward normal to the curves. The whole process may be analytically represented by means of a parameter λ, and to the value λ = 0, shall correspond the actual curves inside the _sichel_. Such a process may be called a virtual displacement in the sichel.
Let the point (_x_, _y_, _z_, _t_) in the sichel λ = 0 have the values _x_ + δ_x_, _y_ + δ_y_, _z_ + δ_z_, _t_ + δ_t_, when the parameter has the value λ; these magnitudes are then functions of (_x_, _y_, _z_, _t_, λ). Let us now conceive of an infinitely thin space-time filament at the point (_x_ _y_ _z_ _t_) with the normal section of contents _d_J_{_n_} and if _d_J_{_n_} + δ_d_J_{_n_} be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass—(ν + _d_ν being the rest-mass-density at the varied position),
(8) (ν + δν) (_d_J_{_n_} + δ_d_J_{_n_}) = ν_d_J_{_n_} = _dm_.
In consequence of this condition, the integral (7) taken over the whole range of the _sichel_, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as the _mass action_ of the virtual displacement.
If we now introduce the method of writing with indices, we shall have
(9) _d_(_x__{_h_} + δ_x__{_h_}) = _dx__{_h_} + ∑_{_k_} ∂δ_x__{_h_}/∂_x__{_k_} + ∂δ_x__{_h_}/∂λ _d_λ
_k_ = 1, 2, 3, 4 _h_ = 1, 2, 3, 4
Now on the basis of the remarks already made, it is clear that the value of N + δN, when the value of the parameter is λ, will be:—
(10) N + δN = ∫∫∫∫ ((ν_d_(τ + δτ))/_d_τ)_dx_ _dy_ _dz_ _dt_,
the integration extending over the whole sichel _d_(τ + δτ) where _d_(τ + δτ) denotes the magnitude, which is deduced from
√(-(_dx₁_ + _d_δ_x₁_)² - (_dx₂_ + _d_δ_x₂_)² - (_dx₃_ + _d_δ_x₃_)² - (_dx₄_ + _d_δ_x₄_)²)
by means of (9) and
_dx₁_ = ω₁ _d_τ, _dx₂_ = ω₂ _d_τ, _dx₃_ = ω₃ _d_τ, _dx₄_ = ω₄ _d_τ, _d_λ = 0
therefore:—
(11) (_d_(τ + δτ))/_d_τ = √( -∑(ω_{_h_} + ∑(∂δ_x__{_h_}/∂_x__{_k_})ω_{_k_})²)
_k_ = 1, 2, 3, 4. _h_ = 1, 2, 3, 4.
We shall now subject the value of the differential quotient
(12) ((_d_(N + δN))/_d_λ) (λ = 0)
to a transformation. Since each δ_x__{_h_} as a function of (_x_, _y_, _z_, _t_) vanishes for the zero-value of the parameter λ, so in general _d_δ_x__{_k_}/(∂_x__{_h_} = 0, for λ = 0.
Let us now put (∂δ_x__{_h_}/∂λ) = ξ_{_h_} (_h_ = 1, 2, 3, 4) (13)
λ = 0
then on the basis of (10) and (11), we have the expression (12):—
= -∫∫∫∫ ∑ ω_{_h_}((∂ξ_{_h_}/∂_x₁_)ω₁ + (∂ξ_{_h_}/∂_x₂_)ω₂ +(∂ξ_{_h_}/∂_x₃_)ω₃ + (∂ξ_{_h_}/∂_x₄_)ω₄) _dx dy dz dt_
for the system (_x₁_ _x₂_ _x₃_ _x₄_) on the boundary of the _sichel_, (δ_x₁_ δ_x₂_ δ_x₃_ δ_x₄_) shall vanish for every value of λ and therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial integration, the integral is transformed into the form
∫∫∫∫ ∑ ξ_{_h_}(∂νω_{_h_}ω₁/∂_x₁_ + ∂νω_{_h_}ω₂/∂_x₂_ + ∂νω_{_h_}ω₃/∂_x₃_ + ∂νω_{_h_}ω₄/∂_x₄_) _dx dy dz dt_
the expression within the bracket may be written as
= ω_{_h_} ∑ ∂νω_{_k_}/∂_x__{_k_} + ν∑ω_{_k_}∂ω_{_h_}/∂_x__{_k_}.
The first sum vanishes in consequence of the continuity equation (_b_). The second may be written as
(∂ω_{_h_}/∂_x₁_)(_dx₁_/_d_τ) + (∂ω_{_h_}/∂_x₂_)(_dx₂_/_d_τ) + (∂ω_{_h_}/∂_x₃_)(_dx₃_/_d_τ) + (∂ω_{_h_}/∂_x₄_)(_dx₄_/_d_τ)
= _d_ω_{_h_}/_d_τ = (_d_/_d_τ)(_dx__{_h_}/_d_τ)
whereby (_d_/_d_τ) is meant the differential quotient in the direction of the space-time line at any position. For the differential quotient (12), we obtain the final expression
(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)
_dx dy dz dt_.
For a virtual displacement in the _sichel_ we have postulated the condition that the points supposed to be substantial shall advance normally to the curves giving their actual motion, which is λ = 0; this condition denotes that the ξ_{_h_} is to satisfy the condition
_w₁_ξ₁ + _w₂_ξ₂ + _w₃_ξ₃ + _w₄_ξ₄ = 0. (15)
Let us now turn our attention to the Maxwellian tensions in the electrodynamics of stationary bodies, and let us consider the results in § 12 and 13; then we find that Hamilton’s Principle can be reconciled to the relativity postulate for continuously extended elastic media.
At every space-time point (as in § 13), let a space time matrix of the 2nd kind be known
(16) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
| S₂₁ S₂₂ S₂₃ S₂₄ | = | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
| S₃₁ S₃₂ S₃₃ S₃₄ | = | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
| S₄₁ S₄₂ S₄₃ S₄₄ | = | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} |
where X_{_n_} Y_{_x_} .....X_{_z_}, T_{_t_} are real magnitudes.
For a virtual displacement in a space-time sichel (with the previously applied designation) the value of the integral
(17) W + δW = ∫∫∫∫ (∑S_{_h k_} (∂(_x__{_k_} + δ_x__{_k_}))/∂_x__{_h_} _dx dy dz dt_
extended over the whole range of the _sichel_, may be called the tensional work of the virtual displacement.
The sum which comes forth here, written in real magnitudes, is
X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} + X_{_x_} (∂δ_x_)/∂_x_ + X_{_y_} (∂δ_x_)/∂_y_ + ... Z_{_z_} (∂δ_z_)/∂_z_
- X_{_t_} (∂δ_x_/∂_t_ - ... + T_{_x_} (∂δ_t_)/∂_x_ + ... T_{_t_} (∂δ_t_)/∂_t_
we can now postulate the following _minimum principle in mechanics_.
_If any space-time Sichel be bounded, then for each virtual displacement in the Sichel, the sum of the mass-works, and tension works shall always be an extremum for that process of the space-time line in the Sichel which actually occurs._
The meaning is, that for each virtual displacement,
([_d_(·δN + δW)]/_d_λ)_{λ = 0} = 0 (18)
By applying the methods of the Calculus of Variations, the following four differential equations at once follow from this minimal principle by means of the transformation (14), and the condition (15).
(19) ν ∂_w__{_h_}/∂τ = K_{_h_} + χ_w__{_h_} (_h_ = 1, 2, 3, 4)
whence K_{_h_} = ∂S_{1 _h_}/∂_x₁_ + ∂S_{2 _h_}/∂_x₂_ + ∂S_{3 _h_}/∂_x₃_ + ∂S_{4 _h_}/∂_x₄_, (20)
are components of the space-time vector 1st kind K = lor S, and X is a factor, which is to be determined from the relation _w__ẇ_ = - 1. By multiplying (19) by _w__{_h_}, and summing the four, we obtain X = K_ẇ_, and therefore clearly K + (K_ẇ_)_w_ will be a space-time vector of the 1st kind which is normal to _w_. Let us write out the components of this vector as
X, Y, Z, ·_i_T
Then we arrive at the following equation for the motion of matter,
(21) ν _d_/_d_τ (_dx_/_d_τ) = X, ν _d_/_d_τ (_dy_/_d_τ) = Y, ν _d_/_d_τ (_dz_/_d_τ) = Z,
ν _d_/_d_τ (_dx_/_d_τ) = T, and we have also
(_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² > (_dt_/_d_τ)² = -1,
and X _dx_/_d_τ + Y _dy_/_d_τ + Z _dz_/_d_τ = T _dt_/_d_τ.
On the basis of this condition, the fourth of equations (21) is to be regarded as a direct consequence of the first three.
From (21), we can deduce the law for the motion of a material point, _i.e._, the law for the career of an infinitely thin space-time filament.
Let _x_, _y_, _z_, _t_, denote a point on a principal line chosen in any manner within the filament. We shall form the equations (21) for the points of the normal cross section of the filament through _x_, _y_, _z_, _t_, and integrate them, multiplying by the elementary contents of the cross section over the whole space of the normal section. If the integrals of the right side be R_{_x_} R_{_y_} R_{_z_} R_{_t_} and if _m_ be the constant mass of the filament, we obtain
(22) _m_ _d_/_d_τ _dx_/_d_τ = R_{_x_}, _m_ _d_/_d_τ _dy_/_d_τ = R_{_y_}, _m_ _d_/_d_τ _dz_/_d_τ = R_{_z_}, _m_ _d_/_d_τ _dt_/_d_τ = R_{_t_}
R is now a space-time vector of the 1st kind with the components (R_{_x_} R_{_y_} R_{_z_} R_{_t_}) which is normal to the space-time vector of the 1st kind _w_,—the velocity of the material point with the components
_dx_/_d_τ, _dy_/_d_τ, _dz_/_d_τ, _i_ _dt_/_d_τ.
We may call this vector R _the moving force of the material point_.
If instead of integrating over the normal section, we integrate the equations over that cross section of the filament which is normal to the _t_ axis, and passes through (_x_, _y_, _z_, _t_), then [See (4)] the equations (22) are obtained, but
are now multiplied by _d_τ/_dt_; in particular, the last equation comes out in the form,
_m_ _d_/_dt_ (_dt_/_d_τ) = _w__{_x_} R_{_x_} _d_τ/_dt_ + _w__{_y_} R_{_y_} _d_τ/_dt_ + _w__{_z_} R_{_z_} _d_τ/_dt_.
The right side is to be looked upon _as the amount of work done per unit of time_ at the material point. In this equation, we obtain the energy-law for the motion of the material point and the expression
_m_ (_dt_/_d_τ - 1) = _m_ [1/√(1 - _w²_) - 1] = _m_ (½ |_w₁²_ + 3/8 |_w₁⁴_ + )
may be called the kinetic energy of the material point.
Since _dt_ is always greater than _d_τ we may call the quotient (_dt_ - _d_τ)/_d_τ as the “Gain” (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed;—The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.
The set of four equations (22) again shows the symmetry in (_x_, _y_, _z_, _t_), which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. On the ground of this demand for symmetry, the triplet consisting of the first three equations are to be constructed after the model of the fourth; remembering this circumstance, we are justified in saying,—
“If the relativity-postulate be placed at the head of mechanics, then the whole set of laws of motion follows from the law of energy.”
I cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation.
If B*(_x_*, _y_*, _z_*, _t_*) be a solid (fester) space-time point, then the region of all those space-time points B (_x_, _y_, _z_, _t_), for which
(23) (_x_ - _x_*)² + (_y_ - _y_*)² + (_z_ - _z_*)² = (_t_ - _t_*)²
_t_ - _t_* >= 0
may be called a “Ray-figure” (Strahl-gebilde) of the space time point B*.
A space-time line taken in any manner can be cut by this figure only at one particular point; this easily follows from the convexity of the figure on the one hand, and on the other hand from the fact that all directions of the space-time lines are only directions from B* towards to the concave side of the figure. Then B* may be called the light-point of B.
If in (23), the point (_x_ _y_ _z_ _t_) be supposed to be fixed, the point (_x_* _y_* _z_* _t_*) be supposed to be variable, then the relation (23) would represent the locus of all the space-time points B*, which are light-points of B.
Let us conceive that a material point F of mass _m_ may, owing to the presence of another material point F*, experience a moving force according to the following law. Let us picture to ourselves the space-time filaments of F and F* along with the principal lines of the filaments. Let BC be an infinitely small element of the principal line of F; further let B* be the light point of B, C* be the light point of C on the principal line of F*; so that OA′ is the radius vector of the hyperboloidal fundamental figure (23) parallel to B*C*, finally D* is the point of intersection of line B*C* with the space normal to itself and passing through B. The moving force of the mass-point F in the space-time point B is now the space-time vector of the first kind which is normal to BC, and which is composed of the vectors
(24) _mm_*(OA′/B*D*)³ BD* in the direction of BD*, and another vector of suitable value in direction of B*C*.
Now by (OA′/B*D*) is to be understood the ratio of the two vectors in question. It is clear that this proposition at once shows the covariant character with respect to a Lorentz-group.
Let us now ask how the space-time filament of F behaves when the material point F* has a uniform translatory motion, _i.e._, the principal line of the filament of F* is a line. Let us take the space time null-point in this, and by means of a Lorentz-transformation, we can take this axis as the t-axis. Let _x_, _y_, _z_, _t_, denote the point B, let τ* denote the proper time of B*, reckoned from O. Our proposition leads to the equations
(25) _d²__x_/_d_τ² = - _m_*_x_/(_t_ - τ*)², _d²__y_/_d_τ² = - _m_*_y_/(_t_ - τ*)³
_d²__z_/_d_τ² = -_m_*_z_/(_t_ - τ*)³, (26) _d²__t_/_d_τ² = -_m_*/(_t_ - τ*)² _d_(_t_ - τ*)/_dt_
where (27) _x²_ + _y²_ + _z²_ = (_t_ - τ*)²
and (28) (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² = (_dt_/_d_τ)² - 1.
In consideration of (27), the three equations (25) are of the same form as the equations for the motion of a material point subjected to attraction from a fixed centre according to the Newtonian Law, only that instead of the time _t_, the proper time τ of the material point occurs. The fourth equation (26) gives then the connection between proper time and the time for the material point.
Now for different values of τ′, the orbit of the space-point (_x_ _y_ _z_) is an ellipse with the semi-major axis _a_ and the eccentricity _e_. Let E denote the eccentric anomaly, Τ the increment of the proper time for a complete description of the orbit, finally _n_Τ = 2π, so that from a properly chosen initial point τ, we have the Kepler-equation
(29) _n_τ = E - _e_ sin E.
If we now change the unit of time, and denote the velocity of light by _c_, then from (28), we obtain
(30) (_dt_/_d_τ)² - 1 = (_m_*/_ac²_) (1 + _e_ cos E)/(1 - _e_ cos E)
Now neglecting _c⁻⁴_ with regard to 1, it follows that
_ndt_ = _nd_τ [ 1 + ½ _m_*/_ac²_ (1 + _e_ cos E)/(1 - _e_ cos E) ]
from which, by applying (29),
(31) _nt_ + const = (1 + ½ _m_*/_ac²_) _n_τ + _m_*/_ac²_ Sin E.
the factor _m_*/_ac²_ is here the square of the ratio of a certain average velocity of F in its orbit to the velocity of light. If now _m_* denote the mass of the sun, _a_ the semi major axis of the earth’s orbit, then this factor amounts to 10⁻⁸.
The law of mass attraction which has been just described and which is formulated in accordance with the relativity postulate would signify that gravitation is propagated with the velocity of light. In view of the fact that the periodic terms in (31) are very small, it is not possible to decide out of astronomical observations between such a law (with the modified mechanics proposed above) and the Newtonian law of attraction with Newtonian mechanics.
Footnote 29:
Sichel—a German word meaning a crescent or a scythe. The original term is retained as there is no suitable English equivalent.
