Part 7
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