Notes 13 and 14.
We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is then at once seen that [=ω] denotes the reciprocal matrix
| ω₁ |
| ω₂ |
| ω₃ |
| ω₄ |
It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]
[ω, s] The vector-product of the four-vector ω and s may be represented by the combination
[ωs] = [=ω]s - ṡω
It is now easy to verify the formula f¹ = A⁻¹fA. Supposing for the sake of simplicity that f represents the vector-product of two four-vectors ω, s, we have
f¹ = [ω¹s¹] = [=ω]¹s¹ - [=s]1ω1]
= [A⁻¹ [=ω]sA - A⁻¹s[=ω]A]
= A⁻¹[=ω]s - s[=ω]A = A⁻¹fA.
Now remembering that generally
f = ρφ + ρ*φ*.
Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular unit planes, there is no difficulty in seeming that
f1 = A⁻¹fA.
Note 15.
The vector product (wf). (P. 36).
This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of frequent occurrence in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.
“We can also form a vectorial combination of a four-vector and a six-vector, giving us a vector of the third type. If the six-vector be of a special type, i.e., a piece of plane, then this vector of the third type denotes the parallelopiped formed of this four-vector and the complement of this piece of plane. In the general case, the product will be the geometric sum of two parallelopipeds, but it can always be represented by a four-vector of the 1st type. For two pieces of 3-space volumes can always be added together by the vectorial addition of their components. So by the addition of two 3-space volumes, we do not obtain a vector of a more general type, but one which can always be represented by a four-vector (loc. cit. p. 759). The state of affairs here is the same as in the ordinary vector calculus, where by the vector-multiplication of a vector of the first, and a vector of the second type (i.e., a polar vector), we obtain a vector of the first type (axial vector). The formal scheme of this multiplication is taken from the three-dimensional case.
Let A = (Ax, Ay, Az) denote a vector of the first type, B = (By z, Bz x, Bx y) denote a vector of the second type. From this last, let us form three special vectors of the first kind, namely—
Bx = (Bx x, Bx y, Bx z) }
By = (By x, By y, By z) } (Bi k = - Bk i, Bi i = 0).
Bz = (Bz x, Bz y, Bz z) }
Since Bj j is zero, Bj is perpendicular to the j-axis. The j-component of the vector-product of A and B is equivalent to the scalar product of A and Bj, i.e.,
(A Bj,) = Ax Bj x + Ay Bj y + Az Bj z.
We see easily that this coincides with the usual rule for the vector-product; e. g., for j = x.
(ABx) = Ay Bx y - Az Bz x.
Correspondingly let us define in the four-dimensional case the product (Pf) of any four-vector P and the six-vector f. The j-component (j = x, y, z, or l) is given by
(Pfj) = Pxfj x + Pyfj y + Pwfj z + Pzfj l
Each one of these components is obtained as the scalar product of P, and the vector fj which is perpendicular to j-axis, and is obtained from f by the rule fj = [(fj x, fj y, fj z, fj l) fj j = 0.]
We can also find out here the geometrical significance of vectors of the third type, when f = φ, i.e., f represents only one plane.
We replace φ by the parallelogram defined by the two four-vectors U, V, and let us pass over to the conjugate plane φ*, which is formed by the perpendicular four-vectors U*, V*. The components of (Pφ) are then equal to the 4 three-rowed under-determinants Dx Dy Dz Dl of the matrix
| Px Py Pz Pl |
| |
| Ux* Uy* Uz* Ul* |
| |
| Vx* Vy* Vz* Vl* |
Leaving aside the first column we obtain
Dx = Py(Uz* Vl* - Ul* Vz*) + Pz(Ul* Vy* - Uy* Vl*)
+ Pl(Uy* Vz* - Uz* Vy*)
= Py φz y* + Pz* φl y + Pl φ*y z.
= Py φx y + Pz φx z + Pl φx l,
which coincides with (Pφx) according to our definition.
Examples of this type of vectors will be found on page 36, Φ = wF, the electrical-rest-force, and ψ = 2wf*, the magnetic-rest-force. The rest-ray Ω = iw[Φψ]* also belong to the same type (page 39). It is easy to show that
Ω = -i | w₁ w₂ w₃ w₄ |
| Φ₁ Φ₂ Φ₃ Φ₄ |
| ψ₁ ψ₂ ψ₃ ψ₄ |
When (Ω₁, Ω₂, Ω₃) = 0, w₄ = i, Ω reduces to the three-dimensional vector
| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |
| |
| ψ₁ ψ₂ ψ₃ |
Since in this case, Φ₁ = w₄ F₁₄ = en (the electric force)
ψ₁ = -iw₄ f₂₃ = mx (the magnetic force)
we have (Ω) = | ex ey ez |
| mx my mz |
[M. N. S.]
Note 16.
The electric-rest force. (Page 37.)
The four-vector φ = wF which is called by Minkowski the electric-rest-force (elektrische Ruh-Kraft) is very closely connected to Lorentz’s Ponderomotive force, or the force acting on a moving charge. If ρ is the density of charge, we have, when ε = 1, μ = 1, i.e., for free space
ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]
= ρ₀/(√(1 - V²/c²)) [dx + 1/c (v₂ h₃ - v₃ h₂)]
Now since ρ₀ = ρ√(1 - V²/c²)
We have ρ₀φ₁ = ρ[dx + 1/c (v₂ h₃ - v₃ h₂)]
N. B.—We have put the components of e equivalent to (dx, dy, dz), and the components of m equivalent to hx hy hz), in accordance with the notation used in Lorentz’s Theory of Electrons.
We have therefore
ρ₀ (φ₁, φ₂, φ₃) = ρ (d + 1/c [v·h]),
i.e., ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron. Compare Lorentz, Theory of Electrons, page 14.
The fourth component φ₄ when multiplied by ρ₀ represents i-times the rate at which work is done by the moving electron, for ρ₀ φ₄ = iρ [vxdx + vydy + vzdz] = vx ρ₀φ₁ + vy ρ₀φ₂ + vz ρ₀φ₃. -√(-1) times the power possessed by the electron therefore represents the fourth component, or the time component of the force-four-vector. This component was first introduced by Poincare in 1906.
The four-vector ψ = iωF* has a similar relation to the force acting on a moving magnetic pole.
[M. N. S.]
Note 17.
Operator “Lor” (§ 12, p. 41).
The operation | ∂/∂x₁ ∂/∂x₂ ∂/∂x₃ ∂/∂x₄ | which plays in four-dimensional mechanics a rôle similar to that of the operator (i∂/∂x, + j∂/∂y, + k∂/∂z = ▽) in three-dimensional geometry has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol □ to denote this operation. In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor. The physical significance of these operations will become clear when along with Minkowski’s method of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with the case of lor S, where S is a six-vector (space-time vector of the 2nd kind).
This being a complicated case, we take the simpler case of lor s,
where s is a four-vector = | s₁, s₂, s₃, s₄ |
and s = | s₁ |
| s₂ |
| s₃ |
| s₄ |
The following geometrical method is taken from Sommerfeld.
Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any shape in the neighbourhood of the space-time point Q, dS denote the three-dimensional bounding surface of ΔΣ, n be the outer normal to dS. Let S be any four-vector, Pn its normal component. Then
Div S = Lim ∫ PndS/ΔΣ.
ΔΣ = 0
Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (dx₁, dx₂, dx₃, dx₄), we have then
Div S = ∂s₁/∂x₁ + ∂s₂/∂x₂ + ∂s₃/∂x₃ + ∂s₄/∂x₄ = lor S.
If f denotes a space-time vector of the second kind, lor f is equivalent to a space-time vector of the first kind. The geometrical significance can be thus brought out. We have seen that the operator ‘lor’ behaves in every respect like a four-vector. The vector-product of a four-vector and a six-vector is again a four-vector. Therefore it is easy to see that lor S will be a four-vector. Let us find the component of this four-vector in any direction s. Let S denote the three-space which passes through the point Q (x₁, x₂, x₃, x₄) and is perpendicular to s, ΔS a very small part of it in the region of Q, dσ is an element of its two-dimensional surface. Let the perpendicular to this surface lying in the space be denoted by n, and let fs n denote the component of f in the plane of (sn) which is evidently conjugate to the plane dσ. Then the s-component of the vector divergence of f because the operator lor multiplies f vectorially.
= Div fs = Lim (∫ fs ndσ)/ΔS.
Δs = 0
Where the integration in dσ is to be extended over the whole surface.
If now s is selected as the x-direction, Δs is then a three-dimensional parallelopiped with the sides dy, dz, dl, then we have
and generally
Div fj = ∂fj x/∂x + ∂fj y/∂y + ∂fj z/∂z + ∂fj l/∂l (where fj, j = 0).
Hence the four-components of the four-vector lor S or Div. f is a four-vector with the components given on page 42.
According to the formulae of space geometry, Dx denotes a parallelopiped laid in the (y-z-l) space, formed out of the vectors (Py Pz Pl), (Uy* Uz* Ul*) (Vy* Vz* Vl*).
Dx is therefore the projection on the y-z-l space of the parallelopiped formed out of these three four-vectors (P, U*, V*), and could as well be denoted by Dyzl. We see directly that the four-vector of the kind represented by (Dx, Dy, Dz, Dl) is perpendicular to the parallelopiped formed by (P U* V*).
Generally we have
(Pf) = PD + P*D*.
∴ The vector of the third type represented by (Pf) is given by the geometrical sum of the two four-vectors of the first type PD and P*D*.
[M. N. S.]
Footnotes
[14]. See notes on § [8] and [10].
[16]. See Note.
[17]. Vide Note.
[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.
[19]. Einzelne stelle der Materie.
[20]. Vide Note.
[26]. Vide note 19.
[27]. Vide note 18.
[28]. Vide note 40.
[29]. Sichel—a German word meaning a crescent or a scythe. The original term is retained as there is no suitable English equivalent.
[30]. Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p. 1.
[31]. H. Minkowski; the passage refers to paper (2) of the present edition.
[32]. Minkowski—Mechanics, appendix, page 65 of paper (2). Planck—Verh. d. D. P. G. Vol. 4, 1906, p. 136.
[33]. Schütz, Gött. Nachr. 1897, p. 110.
[34]. Lienard, L’Eclairage électrique T. 16, 1896, p. 53. Wiechert, Ann. d. Physik, Vol. 4.
[35]. K. Schwarzschild. Gött-Nachr. 1903. H. A. Lorentz, Enzyklopädie der Math. Wissenschaften V. Art 14, p. 199.
- Transcriber’s Notes:
- The book's idiosyncratic spelling, emphasis, punctuation, and symbology especially in mathematical formulas, have been retained.
- Footnotes have been collected at the end of the text, and are linked for ease of reference.