$$ - i \tan i\psi = \frac{e^{\psi} - e^{-\psi}}{e^{\psi}+e^{-\psi}} = q
$$ ,
$$ \psi = \frac{1}{2} \log \frac{1 + q}{1 - q′} $$ (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.