(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

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,

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

and the equations (3) and (4), are

§ 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

with the relation

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

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

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

and from the relation M′ = μm′, we have

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′

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).

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

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,

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,

i.e. for the direction of u,

and for a perpendicular direction ū,

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.

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

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

§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

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.

If we have a second p × q series matrix B,

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

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

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.

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

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

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

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

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

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

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,

is to be replaced by sA in case of a Lorentz transformation

A space-time vector of the 2nd kind^[22] with components f₂₃ ... f₃₄ shall be represented by the alternating matrix

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

Then it follows immediately that the system of six magnitudes (44)

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

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,

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.

(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

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,

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

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

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

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

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

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

i.e.