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.