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 - δe/δt = C,

(ii) div e = lρ.

(iii) Curl E + δM/δt = 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 = μm, 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, iρ) 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, ip) 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′ = εE′, M’ = μm′, C′ = σE′

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 + [uM], and M - [uE] 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 + [uM]),

and from the relation M′ = μm′, 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 + [uM])_u,

C_ū = σ (E + [uM])_u/√(1 - u²).

In consideration of the manner in which σ enters into these relations, it will be convenient to call the vector C - ρu 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 = ρu; 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).

(IIIa″) Curl (H - [uE]) = J + dD/dt + u div D

- curl [uD].

(I″) div D = ρ

(IV″) curl E = - dB/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

(εqXXXIV″, p. 327) D - E = (ε - 1) {E + [uB]}

(εqXXXIII′, p. 223) J = σ(E + [uB])

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, = uρ (the current being zero for σ = 0,

(29) Curl [H - (u, E)] = C + dD/dt - curl [uD],

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 - [uD] = H - [uD] (M - [uE] = 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 - [uE]. 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 + [uB])_u = (D - E)_u

and for a perpendicular direction ū,

(ε - 1) [E + (uB)]_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]) = dE/dt + u div. E + J

- Curl [E - (u. M)] = dM/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 .......... a₁_q |

| |

| a_{p 1} .......... a_{p q} |

B = | b₁₁ .......... b₁_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 .......... c₁_r |

| |

| c_pr .......... 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 f₁₃ f₁₄ |

| f₂₁ 0 f₂₃ f₂₄ |

| f₃₁ f₃₂ 0 f₃₄ |

| f₄₁ f₄₂ f₄₃ 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 f₄₂ f₂₃ |

| f₄₃ 0 f₁₄ f₃₁ |

| f₂₄ f₄₁ 0 f₁₂ |

| f₃₂ f₁₃ f₂₁ 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^½f we shall denote the expression

Det^½f

= f₃₂ f₁₄ f₁₃ f₂₄ + f₂₁ f₃₄·

4⁰. A linear transformation

x_h = α_h1 x₁′ + α_h2 x₂′ + α_h3 x₃′ + α_h4 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_1k α_1k + α_2h α_2k + α_3h α_3k + α_4h α_4k 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 sA 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 f₁₂ f₁₃ f₁₄ |

|f₂₁ 0 f₂₃ f₂₄ |

|f₃₁ f₃₂ 0 f₃₄ |

|f₄₁ f₄₂ f₄₃ 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 (Āf*A) (A⁻¹fA) = A⁻¹f*fA = Det^½ function. A⁻¹A = Det^½f 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 (wA) (Āṡ) = 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′ = wA, f into f′ = A⁻¹ f A,—therefore w′ f′ becomes = (wA 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, wf] + [w, wf*]* = (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,

ωf = | if₄₁, if₄₂, if₄₃, 0 |;

ωf* = | if₃₂, if₁₃, if₂₁, 0 |

[ω · ωf] = 0, 0, 0, f₄₁, f₄₂, f₄₃;

[ω · ωf*]* = 0, 0, 0, f₃₂, f₁₃, f₂₁.

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 -iω₄ > 0.

By F and f shall be understood the space time vectors of the second kind M - iE, 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 -ωf, e, m, u, and in particular -ωf is normal to ω. The relation (C) can be written as

{C} ωf = εωF.

The expression (ωf) gives four components, but the fourth can be derived from the first three.

Let us now form the time-space vector 1st kind, ψ - iωf*, whose components are

ψ₁ = -i(ω₂ f₃₄ + ω₃ f₄₂ + ω₄ f₂₃)

ψ₂ = -i(ω₁ f₄₃ + ω₃ f₄₄ + ω₄ f₃₁)

ψ₃ = -i(ω₁ f₂₄ + ω₂ f₄₁ + ω₄ f₁₂)

ψ₄ = -i(ω₁ f₃₂ + ω₂ f₁₃ + ω₃ f₂₁)

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(um)/√(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* = μψf*.

We can use the relations (C) and (D) to calculate F and f from Φ and ψ we have

ωF = -Φ, ωF* = -iμψ, ωf = -εΦ, ωf* = -iψ.

and applying the relation (45) and (46), we have

F = [ω. Φ] + iμ[ω. ψ]* 55)

f = ε[ω. Φ] + i[ω. ψ]* 56)

i.e.

F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + iμ [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 = iω[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_xΩ₁ + ω_yΩ₂ + ω_zΩ₃) 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 = -(ω₁s₁ + ω₂s₂ + ω₃s₃ + ω₄s₄) = (- | u | C_u + ρ)/√(1 - u²) = ρ′.

This expression gives us the rest-density of electricity (see §8 and §4).

Then 61) = s + (ωṡ)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 - ρu, 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 + (ωṡ)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 fF of two such matrices, for by a transformation A, it is replaced by (A⁻¹fA·A⁻¹FA) = A⁻¹fFA,

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_hΩ_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_1k/∂x₁ + ∂S_2k/∂x₂ + ∂S_3k/∂x₃ + ∂S_4h/∂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′_1k/∂x₁′ + ∂S′_2k/∂x₂′

+ ∂S′_3k/∂x₃′ + ∂S′_4k/∂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_1k + ∂/∂x₂ a_2k + ∂/∂x₃ a_3k + ∂/∂x₄ a_4k.

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. Ās = 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,-iE)

(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, iρ)

(ρ = electrical space-density, C - ρu = 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} ωf = εωF

{D} ωF* = μωf*

{E} s + (ωṡ), w = - σωF.

ω ῶ = -1, and ωF, ωf, ωF*, ωf*, s + (ωs)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) fF =| 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 + f₃₁F₃₁ + 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 -iT_x |

| X_y Y_y Z_y -iT_y |

| X_z Y_z Z_z -iT_z |

| -iX_t -iY_t -iZ_t T_t |

Now X_x = ½[m_xM_x - m_yM_y - m_zM_z + e_xE_x - e_yE_y - e_zE_z]

so

(75) X_y = m_xM_y + e_yE_x, Y_x = m_yM_x + e_xE_y etc.

X_t = e_yM_z - e_zM_y, T_x = m_xE_y - m_yE_z, etc.

T_t = ½[m_xM_x + m_yM_y + m_zM_z + e_xE_x + e_yE_y + e_zE_z]

L_t = ½[m_xM_x + m_yM_y + m_zM_z - e_xE_x - e_yE_y - e_zE_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) fF = S - L, F*f* = -S - L,

where by L, we mean L-times the unit matrix, i.e. the matrix with elements

| Le_hk |, (e_hh = 1, e_hk = 0, h ≠ k h, k = 1, 2, 3, 4).

Since here SL = LS, we deduce that,

F*f*fF = (-S - L)(S - L) = -SS + L²,

and find, since f*f = Det^½f, F*F = Det^½F, we arrive at the interesting

conclusion

(79) SS = L² - Det^½f Det^½F

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_h1 S_1k + S_h2 S_2k + S_h3 S_3k + S_h4 S_4k = 0,

h, k being unequal indices in the series 1, 2, 3, 4, and

(81) S_h1 S_1h + S_h2 S_2h + S_h3 S_3h + S{h4} S_4h = L² -

Det^½f Det^½F,

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 = μm on the other hand.

The expression on the right-hand side of (81), which equals

[½ (m M - eE)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) Ff = 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_x² = Y_y² = Z_z² = T_t², = (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^1/4S.

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 + fF, and it follows that

lor S = lor L + lor fF.

The symbol ‘lor’ denotes a differential process which in lor fF, operates on the one hand upon the components of f, on the other hand also upon the components of F. Accordingly lor fF 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 fF, in which the diffentiations operate upon the components of F alone. From 78) we obtain

fF = -F*f* - 2L;

hence the second part of lor fF = -(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 f₃₁ - ∂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 = -sF + 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 = -sF + 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_yM_z - s_zM_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_zM_x - s_xM_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_xM_y - s_yM₄

- 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_xE_x + s_yE_y + s_zE_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.