# Translation:On the Theory of Relativity I: Four-dimensional Vector Algebra

On the Theory of Relativity I: Four-dimensional Vector Algebra  (1910)
by Arnold Sommerfeld, translated from German by Wikisource
In German: Zur Relativitätstheorie. I. Vierdimensionale Vektoralgebra, Annalen der Physik, 337 (9), 749-776, Online

On the Theory of Relativity. I.

Four-Dimensional Vector Algebra

by. A. Sommerfeld

In this and some subsequent studies I want to illustrate, how remarkably simplified the electrodynamic concepts and calculations become, when we allow ourselves to be led by the deep space-time understanding of Minkowski. For the friend who suddenly passed away, the following would hardly provide any new insight; yet as elucidation of Minkowski's ideas it may be welcomed by some.

The content of the relativity principle can be formulated according to Minkowski: In the physical equations only space-time vectors are allowed to occur, i.e. quantities that have vector-character in the four-times manifold of space and time, the Minkowskian "world". Therefore when passing from one to a new reference system, its components can be transformed according to the scheme of the coordinate transformation for that manifold ("Lorentz transformation"). Instead of absolute space of the older theory, the absolute world takes its place, i.e. the connection of space and time by the speed of light ${\displaystyle c}$, whose invariability now constitutes the absolute substrate of electrodynamics.

In this first part I confine myself to the algebraic relations of the space-time vectors. A second part "vector analysis" should illustrate the differential properties of the four-dimensional vector-fields. In particular I attached great importance to the geometric description (as far as possible) of the definition and the transformation of the vectors. The required concepts and calculations, that give also a full replacement for the matrix calculus used by Minkowski, are (when admitting imaginary coordinates) direct generalizations of the known three-dimensional vector-procedure.

## § 1. Four- and Six-vectors.

It's well known that in space of three dimensions we have to distinguish between two kinds of vectors, vectors of first kind or polar vectors and vectors of second kind or axial vectors, also denoted as rotors or plan-quantities. A vector of first kind is a straight line provided with a direction-sense, its components are the perpendicular projections upon the coordinate-axes. A vector of second kind is a plane surface provided with a rotation sense, its components are the perpendicular projections of the surface upon the coordinate-planes. The components of the first ones are thus related to one axis for each component, those of the second ones are related to two axes for each component, thus the first has to be written with one and the latter with two indices. In mechanics, force and velocity belong to the first, torque and angular velocity to the second. E. g. the components of an angular velocity were to be written as ${\displaystyle \omega _{yz},\ \omega _{zx},\ \omega _{xy}}$; only after passing to Grassmann's "supplement", i.e. the perpendicular upon the rotating plane, the ordinary notation ${\displaystyle \omega _{x},\ \omega _{y},\ \omega _{z}}$ is justified. Also the magnetic field strength ${\displaystyle {\mathfrak {H}}}$ belongs to the same class of vectors, as it was emphasized by Wiechert. The ordinary notation ${\displaystyle {\mathfrak {H}}_{x}\dots }$ that is also maintained in the following, should consequently be replaced by ${\displaystyle {\mathfrak {H}}_{yz}\dots }$.

In Minkowski[1] space of four dimensions the vectors of first kind have four components, "four-vectors", those of second kind have six components, "six-vectors". There also exist vectors of third kind, that again become a four-component vector.[2]

A four-vector is e.g. a position vector from the origin to a space-time point (x y z l) with components

${\displaystyle x,y,z,l=ict,\ i={\sqrt {-1}}}$

l as well as x, y, z is a length; the expression shall remind of a "way of light". The magnitude of that vector, i.e. the length of the corresponding four-dimensional distance, is:

${\displaystyle {\sqrt {x^{2}+y^{2}+z^{2}+l^{2}}}={\sqrt {x^{2}+y^{2}+z^{2}-c^{2}t^{2}}}}$

Two vectors ${\displaystyle \left(x_{1}y_{1}z_{1}l_{1}\right)}$ and ${\displaystyle \left(x_{2}y_{2}z_{2}l_{2}\right)}$ shall be called perpendicular to each other, if

${\displaystyle x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+l_{1}l_{2}=0}$

The coordinate vector (x y z l) is the type of all four-vectors. A quadruple of quantities only then deserves the name four-vector, when it is transformed by a coordinate change in the same manner as the coordinate vector, i.e. covariant with it. Here, the length of any four-vector is invariant; it is its unique invariant. Especially concerning the coordinate vector, the invariance of its length expresses the principle of the constant propagation of light. By the way, due to the imaginary character of its fourth components, the length of its four-vector can be zero, without that the vector and its components vanish.

A vector of first kind is also the "four-density P", that summarizes the concept of velocity ${\displaystyle {\mathfrak {v}}}$ and density ${\displaystyle \varrho }$ of a moving charge, with the components:

 (1) ${\displaystyle P_{x}=\varrho {\frac {{\mathfrak {v}}_{x}}{c}},\ P_{y}=\varrho {\frac {{\mathfrak {v}}_{y}}{c}},\ P_{z}=\varrho {\frac {{\mathfrak {v}}_{z}}{c}},\ P_{l}=i\varrho ,}$

its length becomes

 (1a) ${\displaystyle \left|P\right|=i\varrho {\sqrt {1-\beta ^{2}}}\ \mathrm {where} \ \beta ^{2}={\frac {1}{c^{2}}}\left({\mathfrak {v}}_{x}^{2}+{\mathfrak {v}}_{y}^{2}+{\mathfrak {v}}_{z}^{2}\right)}$

The vector-character of P is given as follows: We consider the worldline of a charge element de, i.e. the successive space-time locations of it in the xyzl-manifold. The worldline element between to adjacent points 1 and 2

dx, dy, dz, dl

is surely a four-vector covariant with the coordinate-vector. Afterwards we use the important principle of the reference-frame independence of charge, by which the electric mass is fundamentally distinguished before the material mass. This principle is of course in accordance with Maxwell's equations and is mentioned by Einstein as a consequence of it. By that:

${\displaystyle de=\varrho dS=\varrho 'dS'}$

dS = dx dy dz means the three-dimensional space-element, as well as ${\displaystyle dS'}$ is the same in a new "primed" reference system, that emerges from the original by a mere rotation around 0. In Fig. 1, ${\displaystyle dS}$ and ${\displaystyle dS'}$ appear as two arbitrary bisections through the world-line-canal, that corresponds to the limits of de. If we put each of such bisection-pairs through the points 1 and 2, then one cuts out two equal four-dimensional space elements :

${\displaystyle d\Sigma =dS\ dl=dS'dl'}$

where ${\displaystyle dl,dl'}$ are the heights of the two space elements , and at the same time the components of the worldline-element 12 into the axes ${\displaystyle l,l'}$, which as the fourth axes of our unprimed and primed reference system must stand perpendicularly upon ${\displaystyle dS}$ and ${\displaystyle dS'}$.[3] By division of the two preceding equations it is given, that

${\displaystyle {\frac {\varrho }{dl}}={\frac {\varrho '}{dl'}}={\frac {de}{d\Sigma }}}$

is a reference-frame-independent scalar quantity. If we thus multiply the four-vector (dx, dy, dz, dl) by ${\displaystyle i\varrho /dl}$, then it keeps its vector-character; exactly the four-vector P from equation (1) occurs by that.

When passing to the space-time vectors of second kind, we consider in space of four dimensions a plane-section of specific magnitude and location, where we are only concerned with the area and the orientation, but not with the form or the absolute location; the location can be changed by a parallel displacement, the form can be arbitrarily changed by a transformation of same content. Yet this geometric image is still too special; it has only five independent specification-parts, where its magnitude is given by one, and its location by four[4] numbers. But we have to consider, that to a plane in space of four dimensions, also the perpendicular plane (die ensemble of all those lines that are perpendicular to all lines of the first plane) are uniquely[5] specified. In this normal plane we now imagine a second plane-section of specific magnitude. The quintessence of two such plane-sections perpendicular to each other, is the geometric image of a vector of second kind. It depends on six specification-sections, by adding to the five mentioned parts, the magnitude of the plane-section (that is perpendicular to it) as the sixth specification-section. We therefore speak about a six-vector. As its pure right-angled components in a xyzl coordinate space, we denote the sum of the perpendicular projections of the two plane-sections upon one of the six coordinate planes xy, yz, .. zl; the components of a six-vector shall therefore be written by two indices each, that must be mutually different, and their order is of importance.[6]

To the special space-time vector of second kind, the simple plane-section, we can kinematically relate a simple rotation; to the general six-vector, the plane-pair, we relate a general rotation or a helix. Namely we have to note, that for the simplest rotation operation that let unchanged all points of a point in four dimensions, we have to speak not about a rotation axis, but about a rotation plane; that rotation plane it then given by the plane of our plane section, the rotation quantity is given by the magnitude of it. As in three dimensions, a helix[7] is composed of a certain rotation and a translation to the axis of this translation, i.e. a rotation around the infinitely distant axis perpendicular to it, likewise the general rotation in four dimensions emerges from the composition of two simple rotations around two mutually perpendicular planes, in agreement with the constitution of our general six-vector from two perpendicular plane-sections. By this remark the expression "helix" for the general rotation may be justified.

A vector of second kind is e.g. the six-vector f of the electromagnetic field with the components

 (2) ${\displaystyle \left\{{\begin{array}{lrlrlr}f_{yz}=&{\mathfrak {H}}_{x},&f_{zx}=&{\mathfrak {H}}_{y},&f_{xy}=&{\mathfrak {H}}_{z},\\f_{xl}=&-i{\mathfrak {E}}_{x},&f_{yl}=&-i{\mathfrak {E}}_{y},&f_{zl}=&-i{\mathfrak {E}}_{z},\end{array}}\right.}$

The meaning of the six-vector and the relation of its components to two coordinate axis chosen in a certain order, corresponds to the general definition:

 (2a) ${\displaystyle f_{ik}=-f_{ki},\ f_{ii}=0\ (i,k=x,y,z,l)}$

That the electric and magnetic forces ${\displaystyle {\mathfrak {E}}}$ and ${\displaystyle {\mathfrak {H}}}$ are combined by Minkowski to the higher unity of the six-vector f, is characterizing for the relativity theory, in which (depending on the reference system) the same field can appear as purely electric or as electromagnetic. E.g a uniform moving charge shows the co-moving observer only the "electric side" of the six-vector, but to the non-co-moving observer also its "magnetic side".

That the electromagnetic field is indeed to be understood as a six-vector, i.e. that when changing the reference frame its components are transformed accordingly to the type of the four-dimensional plane-section or plane-pair, follows from the fact, that due to Maxwell's equations the field is connected with the four-density P by the divergence condition (see later).

To better become acquainted with the six-vector, we consider the special case of a single plane-section ${\displaystyle \varphi }$ of area 1 and give it the form of a parallelogram that emerges from the origin. The two sides of it may be the four-vectors u, v. Then the projection of that plane-section upon the xy-plane is specified by the projections ${\displaystyle u_{x}u_{y}v_{x}v_{y}}$ of the four-vectors, namely

 (3) ${\displaystyle \varphi _{xy}=u_{x}v_{y}-v_{x}u_{y}}$

Between the thus formed six components of this specific six-vector, of which only five are independent from each other by the counting on p. 753, the identity exists:

 (3a) ${\displaystyle \varphi _{yz}\varphi _{xl}+\varphi _{zx}\varphi _{yl}+\varphi _{xy}\varphi _{zl}=0}$

as it can directly be proven by multiplication of the relevant determinants. Furthermore[8] the area ${\displaystyle \left|\varphi \right|}$ of the surface section must be defined as the square root of the square sum of its six components. In our case we thus have:

 (3b) ${\displaystyle \left|\varphi \right|^{2}=\varphi _{yz}^{2}+\varphi _{zx}^{2}+\varphi _{xy}^{2}+\varphi _{xl}^{2}+\varphi _{yl}^{2}+\varphi _{zl}^{2}=1}$

On the other hand we consider the surface-section ${\displaystyle \varphi ^{*}}$ (perpendicular to ${\displaystyle \varphi }$) of area 1, that we can also denote as the "supplement" of that, and claim that its components are

 (3c) ${\displaystyle \varphi _{yz}^{*}=\varphi _{xl},\ \varphi _{zx}^{*}=\varphi _{yl},\ \varphi _{xy}^{*}=\varphi _{zl},\ \varphi _{xl}^{*}=\varphi _{yz}\dots }$

or in general, when we indicate the indices that are nor illustrated by brackets

 (3d) ${\displaystyle \varphi _{ik}^{*}=\varphi _{(ik)}}$

where in respect to the order of the indices ik(ik) it must be defined, that it must emerge from the order x y z l by an even number of permutations.

For verification we represent ${\displaystyle \varphi ^{*}}$ again by two four-vectors u*v*, which individually are perpendicular to uv. That means (see the above definition of perpendicularity) the existence of the following equations:

${\displaystyle {\begin{array}{ll}u_{x}^{*}u_{x}+u_{y}^{*}u_{y}+u_{z}^{*}u_{z}+u_{l}^{*}u_{l}&=0,\\u_{x}^{*}v_{x}+u_{y}^{*}v_{y}+u_{z}^{*}v_{z}+u_{l}^{*}v_{l}&=0,\\v_{x}^{*}u_{x}+v_{y}^{*}u_{y}+v_{z}^{*}u_{z}+v_{l}^{*}u_{l}&=0,\\v_{x}^{*}v_{x}+v_{y}^{*}v_{y}+v_{z}^{*}v_{z}+v_{l}^{*}v_{l}&=0.\end{array}}}$

By multiplying them by ${\displaystyle v_{l},\ u_{l},\ v_{l},\ u_{l}}$ respectively, and subtracting the second from the first and the fourth from the third, we obtain:

${\displaystyle {\begin{array}{ll}u_{x}^{*}\varphi _{xl}+u_{y}^{*}\varphi _{yl}+u_{z}^{*}\varphi _{zl}&=0,\\v_{x}^{*}\varphi _{xl}+v_{y}^{*}\varphi _{yl}+v_{z}^{*}\varphi _{zl}&=0.\end{array}}}$

Furthermore, if we multiply these two equations by ${\displaystyle v_{z}^{*}u_{z}^{*}}$ or ${\displaystyle v_{y}^{*}u_{y}^{*}}$ respectively, and subtract them from each other, then it follows

${\displaystyle \varphi _{xz}^{*}\varphi _{xl}+\varphi _{yz}^{*}\varphi _{yl}=0}$ or ${\displaystyle \varphi _{xy}^{*}\varphi _{xl}+\varphi _{zy}^{*}\varphi _{zl}=0}$

Due to ${\displaystyle \varphi _{xz}^{*}=-\varphi _{zx}^{*}}$ we can write the proportion:

${\displaystyle \varphi _{yz}^{*}:\varphi _{xy}^{*}:\varphi _{zx}^{*}=\varphi _{xl}:\varphi _{yl}:\varphi _{zl}}$

Accordingly it is given:

${\displaystyle \varphi _{yz}:\varphi _{xy}:\varphi _{zx}=\varphi _{xl}^{*}:\varphi _{yl}^{*}:\varphi _{zl}^{*}}$

and thus, when λ means the factor of proportionality:

${\displaystyle \varphi _{ik}^{*}=\lambda \varphi _{(ik)}}$

However, as ${\displaystyle \varphi ^{*}}$ as well as ${\displaystyle \varphi }$ should have the same area 1, then ${\displaystyle \lambda ^{2}=1}$ in accordance with the claim before (3d).

The identity (3a) can be written, by using the expression of § 3A, also in this way:

 (3a) ${\displaystyle \left\{{\begin{array}{ll}\left(\varphi \varphi ^{*}\right)&=\varphi _{yz}\varphi _{yz}^{*}+\varphi _{xy}\varphi _{xy}^{*}+\dots +\varphi _{zl}\varphi _{zl}^{*}\\&=2\left(\varphi _{yz}\varphi _{xl}+\varphi _{zx}\varphi _{yl}+\varphi _{xy}\varphi _{zl}\right)=0\end{array}}\right.}$

The general six-vector f now emerges from the two special ones ${\displaystyle \varphi }$ and ${\displaystyle \varphi ^{*}}$ in the form

 (4) ${\displaystyle f=\varrho \varphi +\varrho ^{*}\varphi ^{*}}$

${\displaystyle \varrho }$ and ${\displaystyle \varrho ^{*}}$ are the areas of the two mutually perpendicular surface-sections of orientation ${\displaystyle \varphi }$ and ${\displaystyle \varphi ^{*}}$, of which f is composed. To f we form the "dual vector f*", its "supplement" by permutation of the surfaces ${\displaystyle \varrho }$ and ${\displaystyle \varrho ^{*}}$. Namely, f* may be defined in consequence of (4):

 ${\displaystyle f^{*}=\varrho ^{*}\varphi +\varrho \varphi ^{*}}$

From that it follows e.g. for the yz-components of this dual vector

 (4b) ${\displaystyle f_{yz}^{*}=\varrho ^{*}\varphi _{yz}+\varrho \varphi _{yz}^{*}=\varrho ^{*}\varphi _{xl}^{*}+\varrho \varphi _{xl}=f_{xl}}$

in the same way

 (4b) ${\displaystyle f_{zx}^{*}=f_{yl},\ f_{xy}^{*}=f_{zl},\ f_{xl}^{*}=f_{yz},\ f_{yl}^{*}=f_{zx},\ f_{zl}^{*}=f_{xy}}$

and in general

 (4c) ${\displaystyle f_{ik}^{*}=f_{(ik)},\ f_{ki}^{*}=-f_{ik}^{*},\ f_{ii}^{*}=0}$

so that f and f* are, with respect to each other, in the same relation, as the special surface-section ${\displaystyle \varphi }$ with ${\displaystyle \varphi ^{*}}$ perpendicular to it.

Also the quantities ${\displaystyle \left|f\right|^{2}}$ and (ff*) can be expressed by ${\displaystyle \varrho }$ and ${\displaystyle \varrho ^{*}}$ in accordance with (3b, e).

 (5a) ${\displaystyle \left\{{\begin{array}{c}\left|f\right|^{2}=f_{yz}^{2}+f_{zx}^{2}+\dots f_{zl}^{2}\\=\varrho ^{2}\left|\varphi \right|^{2}+\varrho ^{*2}\left|\varphi ^{*}\right|^{2}+2\varrho \varrho *(\varphi \varphi *)=\varrho ^{2}+\varrho ^{*2}\end{array}}\right.}$
 (5b) ${\displaystyle \left\{{\begin{array}{c}\left(ff^{*}\right)=f_{yz}f_{yz}^{*}+f_{xy}f_{xy}^{*}+\cdots +f_{zl}f_{zl}^{*}\\=\varrho ^{2}\left(\varphi \varphi ^{*}\right)^{2}+\varrho ^{*2}\left(\varphi ^{*}\varphi \right)+\varrho \varrho *\left(\left|\varphi \right|^{2}+\left|\varphi ^{*}\right|^{2}\right)=2\varrho \varrho *\end{array}}\right.}$

Both have to be denoted as invariants of the six-vector, as they have, like the surfaces ${\displaystyle \varrho }$ and ${\displaystyle \varrho ^{*}}$, a geometric meaning independent of the location of the coordinate system. We can, for example, denote ${\displaystyle \left|f\right|}$ as (geometric) surface sum, and (ff*) as (double) surface product. In case f means the electromagnetic force, we have by (2) and (5a, b)

 (5c) ${\displaystyle \left|f\right|^{2}={\mathfrak {H}}^{2}-{\mathfrak {E}}^{2}=2L,\ \left(ff^{*}\right)=-2i({\mathfrak {EH}})}$

The expression L means "Lagrangian function of unit volume".

However, besides the components related to the coordinate planes, we can also speak of its components related to the coordinate spaces and coordinate axes.

By the component related to coordinate space, we understand a surface section that lies in this space, which we obtain when we project the two surface sections ${\displaystyle \varrho }$ and ${\displaystyle \varrho ^{*}}$ upon this space, and form the geometric sum of their two projections. E.g. the component of f in relation to the space of x y z, is the three-dimensional vector of second kind:

${\displaystyle \left(f_{yz},\ f_{zx},\ f_{xy}\right)={\mathfrak {H}}}$

By the component related to a coordinate axis[9], we understand a surface-section obtained by splitting off (from the surface sections ${\displaystyle \varrho }$ and ${\displaystyle \varrho ^{*}}$) its components in relation to the space perpendicular with this axis, and the two remaining surface-sections are combined geometrically. If we intersect the resulting surface with the space perpendicular to the axis, then a three-dimensional vector of first kind in this space arises. E.g. the component of f in relation to the axis of l, is the three-dimensional vector of first kind:

 (6) ${\displaystyle f_{l}=\left(f_{lx},\ f_{ly},\ f_{lz}\right)=i{\mathfrak {E}}}$

that also can be interpreted as the special four-vector of the vanishing l-component. In general ${\displaystyle f_{j}}$ shall be the four-vector that is perpendicular to the j-axis:

 (6a) ${\displaystyle f_{j}=\left(f_{jx},\ f_{jy},\ f_{jz},\ f_{jl}\right)}$

The component of the supplement f* related to the j-axis has the corresponding meaning:

 (6b) ${\displaystyle f_{j}^{*}=\left(f_{jx}^{*},\ f_{jy}^{*},\ f_{jz}^{*},\ f_{jl}^{*}\right)}$

and in particular

 (6c) ${\displaystyle f_{l}^{*}=\left(f_{yz},\ f_{zx},\ f_{xy},\ 0\right)={\mathfrak {H}}}$

Besides vectors of first and second kind, also vectors of third kind exist in four-dimensional space, which are again four-vectors given not by a distance, but by a three-dimensional space-section, and which were written with three indices as projections of the space-section upon the four coordinate spaces, e.g.

${\displaystyle {\mathfrak {A}}_{yzl},\ {\mathfrak {A}}_{zlx},\ {\mathfrak {A}}_{lxy},\ {\mathfrak {A}}_{xyz}}$

Geometrically, to such a vector of third kind we can attribute a vector of first kind, its "supplement", namely the line perpendicular to the space section whose length is equal to the size of the space-section. The components of the latter

${\displaystyle {\mathfrak {A}}_{x},\ {\mathfrak {A}}_{y},\ {\mathfrak {A}}_{z},\ {\mathfrak {A}}_{l}}$

are one after the other equal to the components of the vector of third kind. The replaceability of the vector of third kind by its supplement is reaching even further, like the relation of the supplement with the vector of second kind in the case of three dimensions, in so far as in this case the supplement (due to the even dimension number) correctly reproduces the behavior of the vector of third kind, not only regarding coordinate transformations of determinant +1, but also regarding such of determinant -1 (mirroring, inversion). As the summarizing denotation for the vector of first and third kind, the name "four-vector" is recommended.

Since we can geometrically compose any two space sections by vectorial addition of their supplements, we don't obtain a more general vector of third kind through a space-pair as through a single space-section. Indeed the latter has already four mutually independent specification-sections - its size and the three data that are necessary for the definition of its location.

An important example (compare § 3B and §4) for a vector of third kind, we will find in the electrodynamic force (Pf).

## § 2. Component formation into arbitrary directions and planes, and its connection with the Lorentz transformation.

It's characteristic for the vector-concept and its independence of the coordinate system, that we can speak of its components into arbitrary directions (planes as regards the six-vector). By the component of a four-vector ${\displaystyle P}$ into the ${\displaystyle x}$-axis we have to think of the perpendicular projection upon this axis, i.e.

 (7) ${\displaystyle P_{x'}=P_{x}\cos(x'x)+P_{y}\cos(x'y)+P_{z}\cos(x'z)+P_{l}\cos(x'l)}$

If ${\displaystyle x'}$ is a "space-like" axis, then[10] ${\displaystyle \cos(x',x)}$, ${\displaystyle \cos(x',y)}$, ${\displaystyle \cos(x',z)}$ become real, ${\displaystyle \cos(x',l)}$ becomes purely imaginary, so that (see (1) and 7)) ${\displaystyle P_{x'}}$ becomes real. On the other hand, ${\displaystyle P_{l'}}$ is purely imaginary when ${\displaystyle l'}$ is a "time-like" axis.

By the component of a six-vector ${\displaystyle f}$ into an arbitrary plane, which we think as given by two mutually perpendicular directions ${\displaystyle x'y'}$ contained in it, we understand the sum of the perpendicular projections of the surface sections (that forms the vector f) upon this plane; it can be calculated from the two-row determinant of the direction cosine:

 (8) ${\displaystyle \left\{{\begin{array}{lll}f_{x'y'}&=f_{xy}\left|{\cos(x'x)\cos(x'y) \atop \cos(y'x)\cos(y'y)}\right|&+f_{yz}\left|{\cos(x'y)\cos(x'z) \atop \cos(y'y)\cos(y'z)}\right|\\\\&+f_{zx}\left|{\cos(x'z)\cos(x'x) \atop \cos(y'z)\cos(y'x)}\right|&+f_{xl}\left|{\cos(x'x)\cos(x'l) \atop \cos(y'z)\cos(y'l)}\right|\\\\&+f_{yl}\left|{\cos(x'y)\cos(x'l) \atop \cos(y'y)\cos(y'l)}\right|&+f_{zl}\left|{\cos(x'z)\cos(x'l) \atop \cos(y'z)\cos(y'l)}\right|\end{array}}\right.}$

For verification we consider at first the special six-vector ${\displaystyle \varphi }$ (pure surface section of magnitude 1) and illustrate those as in §1 in the form of a parallelogram by two four-vectors u,v. Then the surface-section (projected upon the x'y'-plane, i.e. the sought component ${\displaystyle \varphi _{x'y'}}$, is specified by the projections of the four-vectors ${\displaystyle u_{x'},u_{y'},v_{x'},v_{y'}}$ upon the ${\displaystyle x'}$- and ${\displaystyle y'}$-axis as in equation (3) by:

${\displaystyle \varphi _{x'y'}=u_{x'}v_{y'}-v_{x'}u_{y'}}$

If we substitute into ${\displaystyle u_{x'},\dots }$ its value from (7) and write, according to (3), ${\displaystyle u_{x}v_{y}-u_{y}v_{x}}$ and ${\displaystyle \varphi _{xy}\dots }$ for the sub-determinants, then exactly the equation (8) for the special case ${\displaystyle f=\varphi }$ is given. The same equations also arise for the surface-section ${\displaystyle \varphi ^{*}}$ (perpendicular to ${\displaystyle \varphi }$) and therefore also for the sum ${\displaystyle \varrho \varphi +\varrho ^{*}\varphi ^{*}}$, that means the general six-vector by equation (4). The component rule (8) for the six-vector is also a necessary consequence of that for the four-vector (7).

Regarding the reality-relations, it is similar for the components of the four-vector: ${\displaystyle f_{x'y'}}$ consists of only real, ${\displaystyle f_{x'l'}}$ of only imaginary terms, when the axes ${\displaystyle x',y'}$ are space-like, and ${\displaystyle l'}$ is time-like.

Furthermore we consider the quadruple of four-vectors ${\displaystyle f_{h}}$ derived from the six-vector f, and ask for the expression of the components ${\displaystyle f_{x'}}$ of f into the new ${\displaystyle x'}$-axis (which thus means a surface-section parallel to x', or a four-vector perpendicular to this axis) from the components ${\displaystyle f_{x},f_{y},f_{z},f_{l}}$ into the original axes. Indeed, by stating ${\displaystyle f_{x},f_{y},f_{z},f_{l}}$, also ${\displaystyle f_{x'}}$ must be given. We claim, that the following connection exists:

 (9) ${\displaystyle f_{x'}=f_{x}\cos(x'x)+f_{y}\cos(x'y)+f_{z}\cos(x'z)+f_{l}\cos(x'l)}$

Equation (9) is formally in agreement with (7), but also essentially differs from it, because on the right-hand side of (9) there are four directed four-vectors, on the side of (7) there are four undirected components. Equation (9) is of course to be proved due to the vector-property of f, equation (8). We denote the vector-sum that occurs in (9) by ${\displaystyle \Sigma }$:

${\displaystyle \Sigma =f_{x}\cos(x'x)+f_{y}\cos(x'y)+f_{z}\cos(x'z)+f_{l}\cos(x'l)}$

and form the components into the axis x y z l by equation (6a):

${\displaystyle {\begin{array}{lllll}\Sigma _{x}&=&f_{yx}\cos(x'y)&+f_{zx}\cos(x'z)&+f_{lx}\cos(x'l)\\\Sigma _{y}&=f_{xy}\cos(x'x)&&+f_{zy}\cos(x'z)&+f_{ly}\cos(x'l)\\\Sigma _{z}&=f_{xz}\cos(x'x)+&f_{yz}\cos(x'y)&&+f_{lz}\cos(x'l)\\\Sigma _{t}&=f_{xl}\cos(x'x)+&f_{yl}\cos(x'y)&+f_{zl}\cos(x'z)\end{array}}}$

If we multiply one after the other by ${\displaystyle \cos(j'x)}$, ${\displaystyle \cos(j'y)}$, ${\displaystyle \cos(j'z)}$, ${\displaystyle \cos(j'l)}$ (by ${\displaystyle j'}$ we understand an arbitrary axis at first), and sum up, then by (7) the components of ${\displaystyle \Sigma }$ into this axis emerge, and namely it is given:

for ${\displaystyle j'=x'}$ the sum 0,
for ${\displaystyle j'=y'}$ the expression (8),
for ${\displaystyle j'=z'}$ the same expression by permutation of ${\displaystyle y'}$ with a new direction ${\displaystyle z'}$ (perpendicular to ${\displaystyle x'}$) etc.

The right-hand side of (9) indeed represents the quadruple of that magnitude, denoted by us as ${\displaystyle f_{x'}}$ by equation (6), namely the quadruple

${\displaystyle \left(0\ f_{x'y'}\ f_{x'z'}\ f_{x'l'}\right)}$

For the sake of completeness, also the behavior of a vector of third kind in its projection upon an arbitrary space ${\displaystyle x'y'z'}$ may be mentioned. Its component into space ${\displaystyle x'y'z'}$ would be calculated by a four-parametric expression, in which the three-row sub-determinants of the direction cosine occur as coefficients, yet for which we can put (by the conditions of orthogonality that exist between them) the direction cosine of the common perpendicular ${\displaystyle l'}$. The four-vector of third kind is also projected the same way as the four-vector of first kind forming its supplement.

Now, we prove to ourselves, that the projection- and component-formulas developed here, are identical with the transformation formulas of relativity theory. For this sake we consider, as usual, the special case, that the new reference system of ${\displaystyle x'y'z'l'}$ corresponds in the two middle axes with the original one of x y z l, so that one from the other can obtained by a rotation in the xl-plane (around the yz-plane) (rotation angle ${\displaystyle \varphi }$); namely the ${\displaystyle x'}$-axis may be space-like, thus the ${\displaystyle l'}$-axis is time-like. Then

 (10) ${\displaystyle \left\{{\begin{array}{lrl}\cos(x'x)=&\cos(l'l)&=\cos \varphi \\\cos(x'l)=&-\cos(l'x)&=\sin \varphi \\\cos(y'y)=&\cos(z'z)&=1\\\cos(y'x)=&\dots &=0\end{array}}\right.}$

If we apply the component formula (7) to the coordinate vector (x y z l), then we obtain

 (10a) ${\displaystyle \left\{{\begin{array}{lrl}x'=&x\cos \varphi +l\sin \varphi ,&y'=y\\l'=&-x\sin \varphi +l\cos \varphi ,&z'=z\end{array}}\right.}$

Now ${\displaystyle \cos \varphi >1}$, since by the remark on p. 760 ${\displaystyle \varphi =x_{1}/{\sqrt {x_{1}^{2}+l_{1}^{2}}}}$ and ${\displaystyle l_{1}^{2}<0}$, thus ${\displaystyle \varphi }$, ${\displaystyle \sin \varphi }$ and ${\displaystyle \operatorname {tg} \varphi }$ are purely imaginary, and ${\displaystyle \left|\operatorname {tg} \varphi \right|<1}$. Thus we have, when ${\displaystyle \beta }$ denotes a real fraction:

 (10b) ${\displaystyle \operatorname {tg} \varphi =i\beta ,\ \cos \varphi ={\frac {1}{\sqrt {1-\beta ^{2}}}},\ \sin \varphi ={\frac {i\beta }{\sqrt {1-\beta ^{2}}}}}$

Consequently the equations (10a) become identical with the Lorentz-Einstein transformation and show, that ${\displaystyle v=\beta c}$ means the velocity, by which (in space of xyz) the new system of axis is uniformly displaced in the x-direction against the original one.

At the same time equation (8) shows, as to how the six-vector of the field is transformed (or "projected"). We have by (8) and (10):

${\displaystyle {\begin{array}{ll}f_{x'y'}=f_{xy}\cos \varphi -f_{yl}\sin \varphi ,&f_{z'x'}=f_{zx}\cos \varphi +f_{zl}\sin \varphi ,\\f_{y'z'}=f_{yz},&f_{x'l'}=f_{xl},\\f_{y'l'}=f_{yl}\cos \varphi +f_{xy}\sin \varphi ,&f_{z'l'}=f_{zl}\cos \varphi -f_{zx}\sin \varphi ,\end{array}}}$

or because of (10b) and (2):

 (10c) ${\displaystyle \left\{{\begin{array}{lll}{\mathfrak {H}}_{x'}={\mathfrak {H}}_{x},&{\mathfrak {H}}_{y'}={\frac {{\mathfrak {H}}_{y}+\beta {\mathfrak {E}}_{z}}{\sqrt {1-\beta ^{2}}}},&{\mathfrak {H}}_{z'}={\frac {{\mathfrak {H}}_{z}+\beta {\mathfrak {E}}_{y}}{\sqrt {1-\beta ^{2}}}},\\\\{\mathfrak {E}}_{x'}={\mathfrak {E}}_{x},&{\mathfrak {E}}_{y'}={\frac {{\mathfrak {E}}_{y}+\beta {\mathfrak {H}}_{z}}{\sqrt {1-\beta ^{2}}}},&{\mathfrak {E}}_{z'}={\frac {{\mathfrak {E}}_{z}+\beta {\mathfrak {H}}_{y}}{\sqrt {1-\beta ^{2}}}}.\end{array}}\right.}$

By that also these well-known transformation formulas of the field appear to be the immediate consequence of our geometrical interpretation of the six-vector and its component formation derived from the projection of four-dimensional lines. About Maxwell's equation, from which these formulas were derived by Lorentz and Einstein, we didn't speak at all (except in so far, as we have denoted f as six-vector by referring to those equations). Generally, according to Einstein and Minkowski the relativity principle is to be seen as a superior principle above electrodynamics.

## § 3. Product of four- and six-vectors.

### A. Scalar products.

Two four-vectors P and ${\displaystyle \Phi }$ will be scalar-multiplied by the rule

${\displaystyle (P\Phi )=P_{x}\Phi _{x}+P_{y}\Phi _{y}+P_{z}\Phi _{z}+P_{l}\Phi _{l}}$

E.g., if P means the four-density, ${\displaystyle \Phi }$ the four-potential (combination of scalar and vector potential, see later), then ${\displaystyle (P\Phi )}$ becomes the "electro-kinetic potential" of Schwarzschild. The square of the length of a four-vector is equal to the scalar product by itself, e.g. ${\displaystyle \left|P\right|^{2}=(PP)}$ see (1a).

In case of arbitrary motion of P and ${\displaystyle \Psi }$, the formula taken from ordinary vector calculus

${\displaystyle (P\Phi )=\left|P\right|\left|\Phi \right|\cos(P,\Phi )}$

defines the direction cosine of two arbitrary directions; it was used in this sense already at the beginning of § 2, footnote. The definition of perpendicularity from the beginning of § 1 is a special case of this.

Also two six-vectors f and F can be scalar-multiplied by the rule

${\displaystyle (fF)=f_{xy}F_{xy}+f_{yz}F_{yz}+\dots f_{zl}F_{zl}}$

Into this rule belong the two invariants of the six-vector f, see (5a, b)

${\displaystyle \left|f\right|^{2}=(ff)}$ and ${\displaystyle \left(ff^{*}\right)}$

### B. Vector products.

Two four-vectors u, v, will be vector-multiplied by a rule, which is analogous to the ordinary three-dimensional vector product. Like in that place a vector of second case emerges from two vectors of second kind, here a six-vector results from two four-vectors, namely a special six-vector ${\displaystyle \varphi }$ (simple surface section) with the invariant ${\displaystyle \left(\varphi \varphi ^{*}\right)=0}$. The multiplication rule already used in § 1 reads

${\displaystyle \varphi _{xy}=u_{x}v_{y}-u_{y}v_{x},\dots }$

${\displaystyle \varphi }$ means the parallelogram formed by u and v in terms of size and location.

We can also vector-unify a four-vector and a six-vector, namely to a four-vector of third kind. If the six-vector is in particular a simple surface section, then the vector of third kinds means the parallelepiped directly formed by the four-vector and the supplement of that surface-section. In the general case it will be constructed as the geometric sum of two such parallelepipeds, yet it can always be replaced (see the end of §1) by a four-vector of first kind. This state of facts is the same as in ordinary vector calculus, where a vector of first kind emerges by vector-multiplication of a vector of first and second kind. The formal scheme of that multiplication is taken by us from this three-dimensional case.

${\displaystyle {\mathfrak {A}}=\left({\mathfrak {A}}_{x}{\mathfrak {A}}_{y}{\mathfrak {A}}_{z}\right)}$ be the vector of first kind and ${\displaystyle {\mathfrak {B}}=\left({\mathfrak {B}}_{yz},{\mathfrak {B}}_{zx},{\mathfrak {B}}_{xy}\right)}$ the vector of second kind. From the latter we form three special vectors of first kind, namely

${\displaystyle {\begin{array}{cc}{\mathfrak {B}}_{x}=\left({\mathfrak {B}}_{xx},{\mathfrak {B}}_{xy},{\mathfrak {B}}_{xz}\right),\\{\mathfrak {B}}_{y}=\left({\mathfrak {B}}_{yx},{\mathfrak {B}}_{yy},{\mathfrak {B}}_{yz}\right),&{\mathfrak {B}}_{ik}=-{\mathfrak {B}}_{ki},\ {\mathfrak {B}}_{ii}=0,\\{\mathfrak {B}}_{z}=\left({\mathfrak {B}}_{zx},{\mathfrak {B}}_{zy},{\mathfrak {B}}_{zz}\right),\end{array}}}$

in general ${\displaystyle {\mathfrak {B}}_{j}}$, from which all are perpendicular to the j-axis because of ${\displaystyle {\mathfrak {B}}_{jj}=0}$. Like in §1 equation (6a), they can also be called the components of the surface section ${\displaystyle {\mathfrak {B}}}$ into the j-axis, and originally they denote the surface sections parallel to this axis. As j-component of the vector product of ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$, we now have to define the scalar product of the two vectors of first kind ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}_{j}}$, namely

${\displaystyle \left({\mathfrak {A}}{\mathfrak {B}}_{j}\right)={\mathfrak {A}}_{x}{\mathfrak {B}}_{jx}+{\mathfrak {A}}_{y}{\mathfrak {B}}_{jy}+{\mathfrak {A}}_{z}{\mathfrak {B}}_{jz}}$

This is in agreement with the ordinary rule of the vector product and gives e.g. for ${\displaystyle j=x}$:

${\displaystyle \left({\mathfrak {A}}{\mathfrak {B}}_{x}\right)={\mathfrak {A}}_{y}{\mathfrak {B}}_{xy}-{\mathfrak {A}}_{z}{\mathfrak {B}}_{zx}}$

By that we define in the four-dimensional case the product (PF) from an arbitrary four-vector P and a six-vector f by its four components for j = x, y, z, l, by means of formula:

 (11) ${\displaystyle \left(Pf_{j}\right)=P_{x}f_{jx}+P_{y}f_{jy}+P_{z}f_{jz}+P_{l}f_{jl}}$

Any of this components it thus given by scalar-multiplication of the four-vector P with the four-vector ${\displaystyle f_{j}}$ (perpendicular to the j-axis, defined in (6a), and derived from f). In the next paragraph we consider the electrodynamic force ${\displaystyle {\mathfrak {F}}}$ as an important example of this. In this place, at first the already mentioned geometric meaning of this vector of third kind may be derived, in the special case, where ${\displaystyle f=\varphi }$ represents a simple surface-section.

We replace ${\displaystyle \varphi }$ as a parallelogram by two four-vectors u, v and go over to the dual surface section ${\displaystyle \varphi ^{*}}$ consisting of two four-vectors u*, v* parallel to u, v. Then the components of ${\displaystyle (P\varphi )}$ prove to be equal to the four three-row sub-determinants ${\displaystyle D_{x},D_{y},D_{z},D_{l}}$ of the scheme

 (11a) ${\displaystyle \left|{\begin{array}{cccc}P_{x}&P_{y}&P_{z}&P_{l}\\u_{x}^{*}&u_{y}^{*}&u_{z}^{*}&u_{l}^{*}\\v_{x}^{*}&v_{y}^{*}&v_{z}^{*}&v_{l}^{*}\end{array}}\right|}$

When neglecting the first colon it is namely given:

${\displaystyle {\begin{array}{ll}D_{x}=&P_{y}\left(u_{z}^{*}v_{l}^{*}-u_{l}^{*}v_{z}^{*}\right)+P_{z}\left(u_{l}^{*}v_{y}^{*}-u_{y}^{*}v_{l}^{*}\right)\\&+P_{l}\left(u_{y}^{*}v_{z}^{*}-u_{z}^{*}v_{y}^{*}\right)=P_{y}\varphi _{zl}^{*}+P_{z}\varphi _{ly}^{*}+P_{l}\varphi _{yz}^{*}\end{array}}}$

For that we can write by (5a):

${\displaystyle D_{x}=P_{y}\varphi _{xy}+P_{z}\varphi _{xz}+P_{l}\varphi _{xl}}$

which is in agreement with our definition (11) of ${\displaystyle P\varphi _{xl}}$.

${\displaystyle D_{x}}$ means by the known formulas of space geometry, a parallelepiped situated in the yzl-space formed by the vectors ${\displaystyle \left(P_{y},P_{z},P_{l}\right)}$, ${\displaystyle \left(u_{y}^{*},\ u_{z}^{*},\ u_{l}^{*}\right)}$, ${\displaystyle \left(v_{y}^{*},\ v_{z}^{*},\ v_{l}^{*}\right)}$, i.e. by the projections of the four-vectors P, u*, v* upon the yzl-space; ${\displaystyle D_{x}}$ is at the same time also the projection of the parallelepiped (formed by these four-vectors, situated in four-dimensional space) upon the coordinate space of yzl. By the definition of the end of §1, ${\displaystyle D_{x},\ D_{y},\ D_{z},\ D_{l}}$ indeed represent the components of a vector of third kind. By denoting this components with ${\displaystyle D_{x},\dots }$ instead of the more extensive ${\displaystyle D_{yzl}\dots }$, we already executed the passage from the space section to its supplement (the line perpendicular to the space section). That the quantity ${\displaystyle \left(D_{x},\ D_{y},\ D_{z},\ D_{l}\right)}$ considered as a vector of first kind, is perpendicular to the parallelepiped (formed by the vectors P, u*, v*), can directly be seen. We only have to supplement the above 3 × 4-row determinant scheme by addition of the four components of P or u* or v* to a vanishing four-row determinant, and obtain:

 ${\displaystyle (PD)=\left(u^{*}D\right)=\left(v^{*}D\right)=0}$

If we are dealing with the product of P with a general six-vector ${\displaystyle f=\varrho \varphi +\varrho ^{*}\varphi ^{*}}$, then we shall consider in the same way the two parallelepipeds D and D*, which are formed by the vectors P, u*, v* and P, u, v, respectively. Then ${\displaystyle (Pf)=\varrho D+\varrho ^{*}D^{*}}$ represents the geometric sum of these two space sections, which we can imagine as to be constructed by the supplements (perpendicular to that space section), as the resultant of the two lines ${\displaystyle \varrho D}$ and ${\displaystyle \varrho ^{*}D^{*}}$.

### C. Tensor products.

A vector of second kind gives, completely multiplied by itself, a tensor. We explain the way of constitution at first in the three-dimensional case.

If ${\displaystyle {\mathfrak {B}}}$ is a three-dimensional vector of second kind with components ${\displaystyle {\mathfrak {B}}_{yz},\ {\mathfrak {B}}_{zx},\ {\mathfrak {B}}_{xy}}$, and ${\displaystyle {\mathfrak {B}}_{j},\ {\mathfrak {B}}_{h}}$ are whatever two of the three special vectors derived from it (see p. 765), then it is in general for j, h = x, y, z

${\displaystyle T_{jh}=\left({\mathfrak {B}}_{j}{\mathfrak {B}}_{h}\right)={\mathfrak {B}}_{jx}{\mathfrak {B}}_{hx}+{\mathfrak {B}}_{jy}{\mathfrak {B}}_{hy}+{\mathfrak {B}}_{jz}{\mathfrak {B}}_{hz}}$

From that it directly follows

${\displaystyle T_{jh}=T_{hj},\ T_{jj}\neq 0}$

The thus defined three-dimensional tensor quantity has six components, and differs essentially from our four-dimensional six-vector, for which ${\displaystyle f_{ik}=-f_{ki},\ f_{ii}=0}$ is given. The six components of T can be most conveniently, geometrically interpreted as coefficients of the equation of a surface of second degree.

In the four-dimensional case we accordingly start with a six-vector f, and define a tensor (ff) by all possible scalar products of the special four-vectors ${\displaystyle f_{j},f_{h}}$ (derived by equation (6a) from f) for h, j = x, y, z, l. From that, at first 16 tensor components follow

${\displaystyle \left(f_{j}f_{h}\right)=f_{jx}f_{hx}+f_{jy}f_{hy}+f_{jz}f_{hz}+f_{jl}f_{hl}}$

which, however, are reduced to 10 because of ${\displaystyle \left(f_{j}f_{h}\right)=\left(f_{h}f_{j}\right)}$. The calculation gives, if ${\displaystyle f=({\mathfrak {H}},-i{\mathfrak {E}})}$ especially means the field vector, for j = h:

${\displaystyle {\begin{array}{ll}\left(f_{x}f_{x}\right)=&{\mathfrak {H}}_{y}^{2}+{\mathfrak {H}}_{z}^{2}-{\mathfrak {E}}_{x}^{2}\\\left(f_{y}f_{y}\right)=&{\mathfrak {H}}_{z}^{2}+{\mathfrak {H}}_{x}^{2}-{\mathfrak {E}}_{y}^{2}\\\left(f_{z}f_{z}\right)=&{\mathfrak {H}}_{x}^{2}+{\mathfrak {H}}_{y}^{2}-{\mathfrak {E}}_{z}^{2}\\\left(f_{l}f_{l}\right)=-&{\mathfrak {E}}_{x}^{2}-{\mathfrak {E}}_{y}^{2}-{\mathfrak {E}}_{z}^{2}\end{array}}}$

and for ${\displaystyle j\neq h}$

${\displaystyle {\begin{array}{llr}\left(f_{x}f_{y}\right)=&\left(f_{y}f_{x}\right)=&-\left({\mathfrak {H}}_{x}{\mathfrak {H}}_{y}+{\mathfrak {E}}_{x}{\mathfrak {E}}_{y}\right)\\\left(f_{y}f_{z}\right)=&\left(f_{z}f_{y}\right)=&-\left({\mathfrak {H}}_{y}{\mathfrak {H}}_{z}+{\mathfrak {E}}_{y}{\mathfrak {E}}_{u}\right)\\\left(f_{z}f_{x}\right)=&\left(f_{x}f_{z}\right)=&-\left({\mathfrak {H}}_{z}{\mathfrak {H}}_{x}+{\mathfrak {E}}_{z}{\mathfrak {E}}_{x}\right)\\\left(f_{x}f_{l}\right)=&\left(f_{l}f_{x}\right)=&i\left({\mathfrak {E}}_{y}{\mathfrak {H}}_{z}-{\mathfrak {E}}_{z}{\mathfrak {H}}_{y}\right)\\\left(f_{y}f_{l}\right)=&\left(f_{l}f_{y}\right)=&i\left({\mathfrak {E}}_{z}{\mathfrak {H}}_{x}-{\mathfrak {E}}_{x}{\mathfrak {H}}_{z}\right)\\\left(f_{z}f_{l}\right)=&\left(f_{l}f_{z}\right)=&i\left({\mathfrak {E}}_{x}{\mathfrak {H}}_{y}-{\mathfrak {E}}_{y}{\mathfrak {H}}_{x}\right)\end{array}}}$

Besides we also consider the tensor (f*f*) which is formed by the dual six-vector ${\displaystyle f^{*}=(-i{\mathfrak {E,H}})}$, whose components emerge form the preceding by permutation of ${\displaystyle {\mathfrak {H}}}$ by ${\displaystyle -i{\mathfrak {E}}}$ and ${\displaystyle {\mathfrak {E}}}$ by ${\displaystyle +i{\mathfrak {H}}}$, e.g.

${\displaystyle {\begin{array}{ll}\left(f_{x}^{*}f_{x}^{*}\right)=&-{\mathfrak {E}}_{y}^{2}-{\mathfrak {E}}_{z}^{2}+{\mathfrak {H}}_{x}^{2}\\\left(f_{x}^{*}f_{y}^{*}\right)=&\left(f_{x}^{*}f_{y}^{*}\right)=+\left({\mathfrak {E}}_{x}{\mathfrak {E}}_{y}+{\mathfrak {E}}_{x}{\mathfrak {E}}_{y}\right)\\\left(f_{x}^{*}f_{l}^{*}\right)=&\left(f_{l}^{*}f_{x}^{*}\right)=i\left({\mathfrak {H}}_{y}{\mathfrak {E}}_{z}-{\mathfrak {H}}_{z}{\mathfrak {E}}_{y}\right)\end{array}}}$

and eventually go over to the tensor[11] (composed by both):

 (12) ${\displaystyle T={\frac {1}{2}}\left((f\cdot f)-\left(f^{*}f^{*}\right)\right)}$

with the components

 (12a) ${\displaystyle \left\{{\begin{array}{ll}T_{xx}&={\frac {1}{2}}\left(-{\mathfrak {H}}_{x}^{2}+{\mathfrak {H}}_{y}^{2}+{\mathfrak {H}}_{z}^{2}-{\mathfrak {E}}_{x}^{2}+{\mathfrak {E}}_{y}^{2}+{\mathfrak {E}}_{z}^{2}\right)\\\\T_{yy}&={\frac {1}{2}}\left(-{\mathfrak {H}}_{y}^{2}+{\mathfrak {H}}_{z}^{2}+{\mathfrak {H}}_{x}^{2}-{\mathfrak {E}}_{y}^{2}+{\mathfrak {E}}_{z}^{2}+{\mathfrak {E}}_{x}^{2}\right)\\\\T_{zz}&={\frac {1}{2}}\left(-{\mathfrak {H}}_{z}^{2}+{\mathfrak {H}}_{x}^{2}+{\mathfrak {H}}_{y}^{2}-{\mathfrak {E}}_{z}^{2}+{\mathfrak {E}}_{x}^{2}+{\mathfrak {E}}_{y}^{2}\right)\\\\T_{ll}&={\frac {1}{2}}\left(-{\mathfrak {H}}_{x}^{2}-{\mathfrak {H}}_{y}^{2}-{\mathfrak {H}}_{z}^{2}-{\mathfrak {E}}_{x}^{2}-{\mathfrak {E}}_{y}^{2}-{\mathfrak {E}}_{z}^{2}\right)\end{array}}\right.}$
 (12b) ${\displaystyle \left\{{\begin{array}{ll}T_{xy}=T_{yx}&=-{\mathfrak {H}}_{x}{\mathfrak {H}}_{y}+{\mathfrak {E}}_{x}{\mathfrak {E}}_{y}\\T_{yz}=T_{zx}&=-{\mathfrak {H}}_{y}{\mathfrak {H}}_{z}+{\mathfrak {E}}_{y}{\mathfrak {E}}_{z}\\T_{zx}=T_{xz}&=-{\mathfrak {H}}_{z}{\mathfrak {H}}_{x}+{\mathfrak {E}}_{z}{\mathfrak {E}}_{x}\end{array}}\right.}$
 (12c) ${\displaystyle \left\{{\begin{array}{ll}T_{xl}=T_{lx}&=i\left({\mathfrak {E}}_{y}{\mathfrak {H}}_{z}-{\mathfrak {E}}_{z}{\mathfrak {H}}_{y}\right)\\T_{yl}=T_{ly}&=i\left({\mathfrak {E}}_{z}{\mathfrak {H}}_{x}-{\mathfrak {E}}_{x}{\mathfrak {H}}_{z}\right)\\T_{zl}=T_{lz}&=i\left({\mathfrak {E}}_{x}{\mathfrak {H}}_{y}-{\mathfrak {E}}_{y}{\mathfrak {H}}_{x}\right)\end{array}}\right.}$

As it can be seen, the components ${\displaystyle T_{ik}}$ are identical with the above components ${\displaystyle \left(f_{i}f_{k}\right)}$; the components ${\displaystyle \left(T_{ii}\right)}$ that are different from ${\displaystyle \left(f_{i}f_{i}\right)}$, satisfy the relation ${\displaystyle T_{xx}+T_{yy}+T_{zz}+T_{ll}=0}$. The three components (12c) are, after multiplication by -ic, equal to the components of Poynting's energy flux, the first three components (12a) and the three components (12b) represent Maxwell's stresses, while the fourth component (12a) is the negatively taken energy density.

Also now, it is most convenient to interpret the 16 tensor components as coefficients in the equation of a three-dimensional "space of second degree" located in four-dimensional space.

The product formations discussed here, do all correspond to Minkowski's matrix scheme. Without doubt this offers (in mathematical view) the advantage of greatest generality and simplicity. On the contrary, in the preceding it was tried to emphasize the special geometric meaning of the single products by ordinary vector calculus.

All these products are, by their geometric meaning, independent of the choice of reference frame, namely in the following way: The scalar products (considered in A) are pure invariants of the Lorentz transformation. The vector products (considered in B) were transformed covariantly as a six- or a four-vector; their components in the new reference frame are thus only connected to those of the old reference frame by the formulas of §2.

For the vector product (Pf) we can calculate this directly from equation (9). Namely, if we scalar-multiply the four-vectors located at the left and right sides with the four-vector P, then

${\displaystyle \left(Pf_{x'}\right)=\left(Pf_{x}\right)\cos(x'x)+\left(Pf_{y}\right)\cos(x'y)+\left(Pf_{z}\right)\cos(x'z)+\left(Pf_{l}\right)\cos(x'l)}$

i.e. that formula which, by §2 equation (7), expresses the ${\displaystyle x'}$-component of a four-vector by the old components. (Pf) is thus a true four-vector and ${\displaystyle \left(Pf_{x'}\right)}$ its projection upon the ${\displaystyle x'}$-direction.

At the same time also the new tensor components are calculated from the old tensor components by the simple scheme of the tensor transformation, whose coefficients are quadratically formed from the direction cosine of the new against the old coordinate axes.

## § 4. The electrodynamic force ${\displaystyle {\mathfrak {F}}}$.

The covariant formation (Pf) from the four-density P and the field-vector f (discussed in the previous paragraph under (11)) requires a special interest. Its formal calculation gives:

${\displaystyle {\begin{array}{ll}{\mathfrak {F}}_{x}=\left(Pf_{x}\right)&=\varrho \left({\frac {{\mathfrak {v}}_{x}}{c}}f_{xx}+{\frac {{\mathfrak {v}}_{y}}{c}}f_{xy}+{\frac {{\mathfrak {v}}_{z}}{c}}f_{xz}+if_{xl}\right)\\\\&=\varrho \left({\frac {{\mathfrak {v}}_{y}}{c}}{\mathfrak {H}}_{z}-{\frac {{\mathfrak {v}}_{z}}{c}}{\mathfrak {H}}_{y}+{\mathfrak {E}}_{x}\right)\end{array}}}$

and similarly for the y,z-components; in contrast for the l-component:

${\displaystyle {\begin{array}{ll}{\mathfrak {F}}_{l}=\left(Pf_{l}\right)&=\varrho \left({\frac {{\mathfrak {v}}_{x}}{c}}f_{lx}+{\frac {{\mathfrak {v}}_{y}}{c}}f_{ly}+{\frac {{\mathfrak {v}}_{z}}{c}}f_{lz}+if_{ll}\right)\\\\&={\frac {i\varrho }{c}}\left({\mathfrak {v}}_{x}{\mathfrak {E}}_{x}+{\mathfrak {v}}_{y}{\mathfrak {E}}_{y}+{\mathfrak {v}}_{z}{\mathfrak {E}}_{z}\right)\end{array}}}$

In the expression of the ordinary three-dimensional vector calculation we thus find for j = x, y, z or l:

 (13) ${\displaystyle {\mathfrak {F}}_{j}=\varrho \left({\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {vH}}]\right),\ {\mathfrak {F}}_{l}={\frac {i\varrho }{c}}({\mathfrak {Ev}})}$

${\displaystyle {\mathfrak {F}}_{j}}$ thus represents the j-component of the electrodynamic force (acting on the unit volume), and ${\displaystyle {\mathfrak {F}}_{l}}$ the electrical work (multiplied with i/c) on the moving charge per unit volume and time. The latter shall be denoted energetic, the first dynamic components.

If we decompose ${\displaystyle f=\varrho \varphi +\varrho ^{*}\varphi ^{*}}$ into two simple surface sections, then ${\displaystyle F=\varrho D+\varrho ^{*}D^{*}}$, as discussed in the previous paragraph, is decomposed into two simple space sections, whose suplements are perpendicular to the four-density P. Since the latter (see the beginning of §1) has the direction of the worldline element of the charge (its components were proportional to dx, dy, dz, dl), then also the force ${\displaystyle {\mathfrak {F}}}$ is perpendicular to the worldline of its point of contact. The corresponding requirement is to be extended to an arbitrary force, as long as we exclude dissipative processes like the generation of Joulean heat etc.

The fourth energetic component of ${\displaystyle {\mathfrak {F}}}$ has in no way only a formal meaning, as it might appear at first. Rather (when changing frames) it is located directly at the real dynamic components. If it is (e.g. like in equation (10a)) a reference frame ${\displaystyle x'y'z'l'}$ emerging from the original one by a rotation in the xl-plane, then

 (14) ${\displaystyle {\mathfrak {F}}_{x'}={\mathfrak {F}}_{x}\cos \varphi +{\mathfrak {F}}_{l}\sin \varphi \ \mathrm {and} \ {\mathfrak {F}}_{x}={\mathfrak {F}}_{x'}\cos \varphi -{\mathfrak {F}}_{l'}\sin \varphi }$

The energetic supplementary terms ${\displaystyle {\mathfrak {F}}_{l}\sin \varphi }$ (or ${\displaystyle {\mathfrak {-F}}_{l'}\sin \varphi }$) are small and of order ${\displaystyle 1/c^{2}}$, since ${\displaystyle {\mathfrak {F}}_{l}}$ (or ${\displaystyle {\mathfrak {F}}_{l'}}$) as well as ${\displaystyle \sin \varphi }$ contain the factor ${\displaystyle 1/c}$. Nevertheless this supplementary terms are indispensable for carrying out the principle of relativity, as it will be shown later by the example of the simple Coulomb law.

If the dynamic components ${\displaystyle {\mathfrak {F}}_{x}}$ and ${\displaystyle {\mathfrak {F}}'_{x}}$ for any force ${\displaystyle {\mathfrak {F}}}$ in two relatively moving systems are known, e.g. by sufficiently exact measurements, then we can (by comparison according to the preceding formulas) specify the energetic components ${\displaystyle {\mathfrak {F}}_{l}}$ and ${\displaystyle {\mathfrak {F}}_{l'}}$. Conversely, we can also say: A force (e.g. also gravitation) is only then physically known, i.e. in the sense of the relativity principle specifiable for an arbitrary reference frame, when also its energetic component ${\displaystyle {\mathfrak {F}}_{l}}$ is known.

If we especially choose the primed system as "co-moving", so that the point of contact of force ${\displaystyle {\mathfrak {F}}}$ is at rest in this reference frame, then ${\displaystyle dx'=dy'=dz'=0}$ and also ${\displaystyle {\mathfrak {F}}_{l'}=0}$, since the force on the worldline of its point of contact is perpendicular. Only in this case the energetic component vanishes out of the formulas for the determination of ${\displaystyle {\mathfrak {F}}_{x}\dots {\mathfrak {F}}_{l}}$, thus having the simple form (x-direction = direction of relative motion):

 (14a) ${\displaystyle {\mathfrak {F}}_{x}=\cos \varphi {\mathfrak {F}}_{x'},\ {\mathfrak {F}}_{y}={\mathfrak {F}}_{y'},\ {\mathfrak {F}}_{z}={\mathfrak {F}}_{z'},\ {\mathfrak {F}}_{l}=-\sin \varphi {\mathfrak {F}}_{x'}}$

These are the typical transformation formulas for the passage from the "co-moving" to the "resting system", valid for any vector of first (or third) kind, especially for the electrodynamic force related to unit volume. We assume, that any "specific" force, i.e. calculated per unit volume, also behaves as a vector of first kind.

It is different for the entire force ${\displaystyle {\mathfrak {K}}}$ or ${\displaystyle {\mathfrak {K}}'}$ in the stationary or co-moving system, of which the first, due to its relation to an arbitrary coordinate system, has actually no place in relativity theory. By the specific force is can be thus defined:

 (15) ${\displaystyle {\mathfrak {K}}=\int {\mathfrak {F}}dS,\ {\mathfrak {K}}'=\int {\mathfrak {F}}'dS'}$

if dS = dy dy dz is the three-dimensional volume element in the stationary, ${\displaystyle ds'=dy'dy'dz'}$ is that in the co-moving system. The latter differs from the first, depending on the mutual location of both reference frames. By addition of dS, ${\displaystyle dS'}$, the vector character of the right-hand side will thus be disturbed. The entire force doesn't simply behave as a vector of first kind.

From Fig. 1 it follows

 (15a) ${\displaystyle dS'=dS\cos \varphi ={\frac {dS}{\sqrt {1-\beta ^{2}}}}}$

If, what we don't presuppose, the primed system with its ${\displaystyle l'}$-axis is orientated into the world-line of the contact point of ${\displaystyle dS/dS'={\sqrt {1-\beta ^{2}}}}$, then the ${\displaystyle x'y'z'}$-space and its volume element ${\displaystyle dS'}$ is perpendicular to this worldline, and ${\displaystyle dS'}$ is the perpendicular projection of dS in this space. The projection angle is equal to the angle of the two considered perpendiculars ${\displaystyle l'}$ and l, thus equal to ${\displaystyle \varphi }$. The circumstance, that ${\displaystyle \varphi }$ is imaginary, has the consequence that the projection ${\displaystyle dS'}$ is greater than the projected element dS, opposing to our image in Fig. 1, that operates with a real ${\displaystyle \varphi }$. The ratio ${\displaystyle dS/dS'={\sqrt {1-\beta ^{2}}}}$ is the known Lorentz contraction.

From (15), (15a) and (14a) it directly follows:

${\displaystyle {\begin{array}{l}{\mathfrak {K}}_{x}=\int {\mathfrak {F}}_{x}dS=\cos \varphi \int {\mathfrak {F}}_{x'}dS=\int {\mathfrak {F}}_{x'}dS'={\mathfrak {K}}_{x'}\\\\{\mathfrak {K}}_{y}=\int {\mathfrak {F}}_{y}dS=\int {\mathfrak {F}}_{y'}dS={\frac {1}{\cos \varphi }}\int {\mathfrak {F}}_{y'}dS'{\frac {1}{\cos \varphi }}{\mathfrak {K}}_{y'}\ \mathrm {etc} .\end{array}}}$

and summarizing:

 (14b) ${\displaystyle {\mathfrak {K}}_{x}={\mathfrak {K}}_{x'},\ {\mathfrak {K}}_{y}={\frac {1}{\cos \varphi }}{\mathfrak {K}}_{y'},\ {\mathfrak {K}}_{z}={\frac {1}{\cos \varphi }}{\mathfrak {K}}_{z'},\ {\mathfrak {K}}_{l}=-{\frac {\sin \varphi }{\cos \varphi }}{\mathfrak {K}}_{x'}}$

These formulas (14b), when compared with (14a), prove the preceding explanation, that ${\displaystyle {\mathfrak {K}}}$ doesn't behave like a vector. Namely, if we denote, in formal analogy to our vector notation, by ${\displaystyle \left|{\mathfrak {K}}\right|^{2}}$ and ${\displaystyle \left|{\mathfrak {K}}'\right|^{2}}$ the square sum of the unprimed and primed components (where ${\displaystyle {\mathfrak {K}}'_{l}}$ is zero according to definition equation (15)), then because of (14b), ${\displaystyle \left|{\mathfrak {K}}\right|}$ is not becoming equal to ${\displaystyle \left|{\mathfrak {K}}'\right|}$, but

${\displaystyle \left|{\mathfrak {K}}\right|={\frac {\left|{\mathfrak {K}}'\right|}{\cos \varphi }}=\left|{\mathfrak {K}}'\right|{\sqrt {1-\beta ^{2}}}}$

while from (14a), as it must be the case for a four-vector, it follows of course ${\displaystyle \left|{\mathfrak {F}}\right|=\left|{\mathfrak {F}}'\right|}$.

We also especially consider the case of a normal pressure acting in the primed system, that acts upon a surface ${\displaystyle \sigma '}$ at rest in the primed system, which gives rise to pressure ${\displaystyle p'}$ that coincides with the perpendicular ${\displaystyle n'}$ and which is measured per unit area. We can interpret this case, either as a distribution of infinitely small total-forces ${\displaystyle {\mathfrak {K}}}$ or infinitely great specific forces ${\displaystyle {\mathfrak {F}}}$, according to one or the other of the following two schemes

${\displaystyle {\mathfrak {K}}_{n'}=p'd\sigma '}$ or ${\displaystyle p'={\mathfrak {F}}_{n'}dn'}$

The first standpoint was taken by Einstein[12]. The latter should be preferable because of the vivid vectorial behavior of ${\displaystyle {\mathfrak {F}}}$. From this standpoint we think of ${\displaystyle {\mathfrak {F}}}$ as being distributed over a layer of thickness ${\displaystyle dn'}$ which is adjacent to surface ${\displaystyle \sigma '}$, similarly as we interpret the surface charge as the limit of a spatial distribution.

If we erect upon this three-dimensional layer ${\displaystyle \sigma 'dn'}$ (located in ${\displaystyle x'y'z'}$-space) a four-dimensional cylinder parallel to the ${\displaystyle l'}$-axis, then this encloses the entirety of all worldlines of the layer points. The force ${\displaystyle {\mathfrak {F}}'}$, that should be perpendicular to ${\displaystyle \sigma '}$ in ${\displaystyle x'y'z'}$-space, also coincides with the perpendicular of that four-dimensional cylinder, since it is also perpendicular to the worldline at its contact point. If we lay any inclined intersection as xyz-space through the cylinder, then it intersects (from the cylinder) the image ${\displaystyle \sigma dn}$ of the layer that appears to an observer in the xyzl-system. In Fig. 2, the (by p. 753, Note 2 completely specified) normal plane (laid through the considered point of ${\displaystyle \sigma }$) to the tangential plane of ${\displaystyle \sigma }$, is chosen as drawing plane. This not only contains the normal n of ${\displaystyle \sigma }$ in xyz-space, but also the cylinder-normal ${\displaystyle n'}$ as well as the l-axis, since all three directions are perpendicular to ${\displaystyle \sigma }$. In the ${\displaystyle n'}$-direction falls the direction of vector ${\displaystyle {\mathfrak {F}}}$; by direction and size we have ${\displaystyle {\mathfrak {F}}={\mathfrak {F}}_{n'}}$. For the reference frame xyzl it is decomposed in two components, which both can be seen in the figure: the energetic component ${\displaystyle {\mathfrak {F}}_{l}}$ and the dynamic ${\displaystyle {\mathfrak {F}}_{n}}$.

On the other hand, the component of ${\displaystyle {\mathfrak {F}}}$ perpendicular to the drawing plane in the tangential plane of ${\displaystyle \sigma }$ doesn't exist, i.e. also in the unprimed system (after separation of its energetic component) the force ${\displaystyle {\mathfrak {F}}}$ and its corresponding pressure p normal to its contact-area ${\displaystyle \sigma }$. Regarding the magnitude of p, it directly follows from the figure: If ${\displaystyle \alpha }$ is the angle between n and ${\displaystyle n'}$ (where ${\displaystyle \alpha }$ will lie between 0 and ${\displaystyle \varphi }$ and will assume these most extreme values, when ${\displaystyle n'}$ lies especially perpendicular or parallel to the direction of relative motion of the primed and unprimed system). Then it is obviously given

${\displaystyle {\mathfrak {F}}_{n}={\mathfrak {F}}_{n'}\cos \alpha }$ and simultaneously ${\displaystyle dn'=dn\cos \alpha }$

thus

${\displaystyle {\mathfrak {F}}_{n}dn={\mathfrak {F}}_{n'}dn'}$ or ${\displaystyle p=p'}$

Because the unprimed observer defines its pressure p by the dynamic component ${\displaystyle {\mathfrak {F}}_{n}}$ of ${\displaystyle {\mathfrak {F}}}$, and the layer-thickness dn (seen by him) by ${\displaystyle {\mathfrak {F}}_{n}dn}$ in the same way as the primed by ${\displaystyle {\mathfrak {F}}_{n'}dn'}$. The equality of these two pressures, fundamental for Planck's thermodynamics of moving systems, thus follows here again as purely geometric statement concerning the behavior of four-dimensional vectors and its projection.

When operating with the total-force ${\displaystyle {\mathfrak {K}}}$, such a direct geometric reasoning can hardly be carried out, because ${\displaystyle {\mathfrak {K}}}$ doesn't behave like a vector.

We again come back to the electrodynamic meaning of ${\displaystyle {\mathfrak {F}}=(Pf)}$. While this was, at the beginning of this paragraph, explained by the formal agreement with Lorentz's force-definition, equation (13), it should be directly derived from the relativity principle now.

We consider a unit volume at rest in the primed system, in which the electric charge is distributed by the density ${\displaystyle \varrho '}$. Since the electric field-strength is defined as the moving force upon the charge 1, then the force ${\displaystyle \varrho '{\mathfrak {E}}'}$ is acting on our unit volume. By this the three dynamic components of the specific force ${\displaystyle {\mathfrak {F}}}$ are specified; the energetic component is to be put equal to zero, since the contact-point of ${\displaystyle {\mathfrak {F}}}$ is at rest in the primed system, and its worldline is perpendicular to ${\displaystyle {\mathfrak {F}}}$. The four components of ${\displaystyle {\mathfrak {F}}}$ thus read:

${\displaystyle {\mathfrak {F}}_{x'}=\varrho '{\mathfrak {E}}_{x'},\ {\mathfrak {F}}_{y'}=\varrho '{\mathfrak {E}}_{y'},\ {\mathfrak {F}}_{z'}=\varrho '{\mathfrak {E}}_{z'},\ {\mathfrak {F}}_{l'}=0}$

On the other hand, the vector (Pf) for the ${\displaystyle x'y'z'l'}$-axis gives by equation (11), since ${\displaystyle P_{x'}=P_{y'}=P_{z'}=0}$, ${\displaystyle P_{l'}=i\varrho '}$ (see equation (1)) and ${\displaystyle f_{x'l'}=-i{\mathfrak {E}}_{x'}}$ etc. (see equation (2));

${\displaystyle {\begin{array}{ll}\left(Pf_{x'}\right)=P_{l'}f_{x'l'}=\varrho '{\mathfrak {E}}_{x'},&\left(Pf_{y'}\right)=P_{l'}f_{y'l'}=\varrho '{\mathfrak {E}}_{y'},\\\left(Pf_{z'}\right)=P_{l'}f_{z'l'}=\varrho '{\mathfrak {E}}_{z'},&\left(Pf_{l'}\right)=P_{l'}f_{l'l'}=0.\end{array}}}$

This vector thus agrees for the primed system in all four components with the specific force ${\displaystyle {\mathfrak {F}}}$. Under the single assumption, that the force behaves like a four-vector, ${\displaystyle {\mathfrak {F}}}$ must be given for any reference system by the four-vector (Pf). In relativity theory, Lorentz's force-definition directly follows from the definition of the electric field-strength by the single assumption, that the specific force behaves as a four-vector perpendicular to the worldline of the contact-point. In the original absolute-theory of Lorentz, in contrast, this force-definition had to be introduced as a new fact of experience besides the field-equations. It seems to me a particularly nice achievement of relativity theory, that it makes this fact of experience unnecessary as such, by deriving it from much more general principles.

Following this, a remark of a more critical nature is appended. The relativity principle should be valid only for stationary or quasi-stationary motions of both reference systems against each other. A derivation of Lorentz's force-concept from relativity theory, consequently guarantees it only for the realm of these motions. While in relativity theory this concept was postulated as generally valid, we have (when we consider the relativity principle as its true source) to let it open at all, whether in the expression of ${\displaystyle {\mathfrak {F}}}$ for a real accelerated motion there might occur supplementary terms that might depend on the acceleration of first and higher order. As it seems, all realizable motions fall under the class of quasi-stationary motions, thus such supplementary terms would hardly have a practical interest. Yet in a principle view it seems useful to me, to consider its possibility.

(Received March 17, 1910.)

1. The two papers of Minkowski are: The Fundamental Equations for Electromagnetic Processes in Moving Bodies, Göttinger Nachr. 1908. p. 1 or Mathematische Ann. 68, p. 472. 1910, also separately to be published soon by B.G. Teubner; Space and Time, Physik. Zeitschr. 10. p. 104. 1909, also separately by B.G. Teubner, with a portrait of Minkowski.
2. Corresponding to the number of coordinate axes, -planes and -spaces calculated by the number of combinations of four elements to each 1, 2 or 3, namely:

${\displaystyle {\frac {4}{1}}=4,\ {\frac {4\cdot 3}{1\cdot 2}}=6,\ {\frac {4\cdot 3\cdot 2}{1\cdot 2\cdot 3}}=4}$

3. I think it is allowed to arrange the image and the expression, as if l is real and the perpendicularity is euclidean. Of course, in reality an imaginary l and a non-euclidean perpendicularity is meant, which according to Minkowski can be constructed by a hyperbola or a hyperboloid. It is possible, but not recommended, to reinterpret all that follows accordingly in a non-euclidean way.
4. We can imagine that the location of the plane-section is given by two four-vectors that emerge from two arbitrary points of the plane, and which are arbitrarily located within the plane. Their location is at first specified by three magnitudes, e.g. the relation of its four components. However, any of the two four-vectors can arbitrarily rotated within the plane. By that we can give to each of their three specification-sections an arbitrary value. Thus 2(3-1) = 4 specification-sections for the location of the plane remain only.
5. We think of two lines drawn in the first plane, and we construct the entirety of the lines that are perpendicular to them, where each line fills a linear space. These two spaces intersect each other in a plane containing all common perpendiculars of both starting lines, and which is thus the normal plane of the first plane. The two planes perpendicular to each other, as well as any two planes of general location, have only one point in common. Because any plane in a space of four dimensions is given by two linear equations, and four linear equations uniquely specify a point.
6. Permutation of xy by yx means a projection upon the back of the xy-plane and therefore, as with three dimensions, a sign reversal.
7. To this analogy also Minkowski refers: Space and Time § V.
8. Its square is at first given by the formula of ordinary planimetry:

${\displaystyle \left|u\right|^{2}\left|v\right|^{2}\sin ^{2}(u,v)=\left|u\right|^{2}\left|v\right|^{2}\left(1-\cos ^{2}(u,v)\right)=\left|u\right|^{2}\left|v\right|^{2}-(uv)^{2}}$

compare § 3A, from which equation (3b) directly follows by a suitable combination of the components.

9. This at first somehow complicated term appears indispensable to me for the geometrical interpretation of the subsequent differential vector-operations.
10. If ${\displaystyle x_{1}y_{1}z_{1}l_{1}}$ are the coordinates of an arbitrary point upon the axis of ${\displaystyle x'}$ in the system of x y z l and if we put ${\displaystyle R_{1}^{2}=x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+l_{1}^{2}}$, then by the general definition of § 3 A:

${\displaystyle \cos(x'x)={\frac {x_{1}}{R_{1}}},\ \cos(x'y)={\frac {y_{1}}{R_{1}}},\ \cos(x'z)={\frac {z_{1}}{R_{1}}},\ \cos(x'l)={\frac {l_{1}}{R_{1}}}}$

The axis is space-like if ${\displaystyle R_{1}^{2}>0}$, that is ${\displaystyle x_{1}^{2}+y_{1}^{2}+z_{1}^{2}>\left|l_{1}\right|^{2}}$, and time-like if ${\displaystyle R_{1}^{2}<0}$. In the first case the first three cosines are real and the last is imaginary, in the second case the first three are imaginary and the last is real.

11. According to a friendly suggestion of my colleague M. Laue.
12. A. Einstein, Jahrbuch f. Radioakt. 4. 448.
This work is a translation and has a separate copyright status to the applicable copyright protections of the original content.
Original: This work is in the public domain in the United States because it was published before January 1, 1927. The author died in 1951, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 70 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works. This work is released under the Creative Commons Attribution-ShareAlike 3.0 Unported license, which allows free use, distribution, and creation of derivatives, so long as the license is unchanged and clearly noted, and the original author is attributed.