Translation:On the Electrodynamics of Minkowski

by Max Abraham

On the Electrodynamics of Minkowski.

Memoir by Max Abraham (Milan)

At the meeting on January 23, 1910.

§ 1. Introduction.

In a preceding paper^[1] I have developed a system of electrodynamics of moving bodies, which – being in accordance with the general principles of the theory of Maxwell and Hertz – embraces the modern theories of E. Cohn, H. A. Lorentz and H. Minkowski. Regarding the special case of Minkowski's theory, an expression of the ponderomotive force resulted, which differs from the expression given by Minkowski himself; I have asserted that this expression satisfies the principle of relativity.

In this present note, this assertion will be confirmed. I will start in § 2 with some theorems related to the four-dimensional vectors, which are essentially already contained in the memoir of Minkowski^[2], that will be applied later; I believed that it was useful to give a four-dimensional vector form to the analysis, which, adapting itself to three dimensional analysis, allows us to quickly get from the four-dimensional variation of space and time to three-dimensional space.

Variables are introduced in § 3, which I will call "four-dimensional tensors." They are a generalization of three dimensional tensors^[3], which characterize, for example, the state of stress of an elastic body. The four-dimensional tensor that should be considered in electrodynamics, contains – in its ten components – the six components of electromagnetic pressure, and the three components of the energy current and the electromagnetic energy density. It will be formed by a four-dimensional tensor, whose components are identical to the values ​​of pressure, the flow of energy, the electromagnetic energy density, which were deduced in my cited work from the general principles of electrodynamics of moving bodies.

As a result, these principles are compatible with the postulate of relativity; the symmetry of the electromagnetic field equations for empty space, which is expressed in the Lorentz transformation, should also be given to the electromagnetic equations for ponderable bodies, by writing them down – either in the form of Minkowski or in the form equivalent to that of Lorentz – without contradicting those principles.

Minkowski has already given to the equations of motion of a material point, a form which is invariant under the Lorentz transformations. However, he thought it necessary to add an additional force to the ponderomotive electromagnetic force, which is incompatible^[4] with my system of electrodynamics. In § 4 I will write the equations of motion so that they satisfy the principle of relativity, without introducing the additional force of Minkowski. However, it must be admitted that the "rest density" of mass is not constant, yet it increases every time, when an electric current generates heat (in Joule) in matter; this hypothesis was already before stated by A. Einstein and M. Planck in relation to the principle of relativity.

But it seems doubtful, if the very concept of space and time developed by Minkowski^[5] is a possible basis of rational mechanics. Indeed, the kinematics of rigid systems, which M. Born^[6] wanted to adapt to the Lorentz group, offers considerable difficulties as shown by G. Herglotz^[7]: the rigid body in the "world" of Minkowski cannot be set into rotation.

§ 2. Four dimensional vectors.

A linear transformation of the four coordinates $x,y,z,z$, which has the invariant

$x^{2}+y^{2}+z^{2}+u^{2}\,$,

is called "Lorentz transformation" according to Minkowski. We will confine ourselves to the following group of orthogonal transformations, i.e. rotations of a space of four dimensions.

A system of four variables which transform as the coordinates $x,y,z,u$, is called a "four-dimensional vector of first kind" $\left(V_{I}^{4}\right)$. When projecting into three-dimensional space $x,y,z$, the three first components of vector $V_{I}^{4}$ constitute a three-dimensional vector $V^{3}$, ${\mathfrak {r}}$, the fourth ($u$) a three-dimensional scalar ($S^{3}$).

A four-dimensional vector of the second kind $\left(V_{II}^{4}\right)$ denotes a system of six magnitudes, which are transformed like the following expressions, formed by the components $x_{1},y_{1},z_{1},u_{1}$ and $x_{2},y_{2},z_{2},u_{2}$ of two $V_{I}^{4}$:

(1)

$\left\{{\begin{array}{ccccc}{\mathfrak {a}}_{x}=\left|{\begin{array}{cc}y_{1}&z_{1}\\y_{2}&z_{2}\end{array}}\right|,&&{\mathfrak {a}}_{y}=\left|{\begin{array}{cc}z_{1}&x_{1}\\z_{2}&x_{2}\end{array}}\right|,&&{\mathfrak {a}}_{z}=\left|{\begin{array}{cc}x_{1}&y_{1}\\x_{2}&y_{2}\end{array}}\right|;\\\\{\mathfrak {b}}_{x}=\left|{\begin{array}{cc}x_{1}&u_{1}\\x_{2}&u_{2}\end{array}}\right|,&&{\mathfrak {b}}_{y}=\left|{\begin{array}{cc}y_{1}&u_{1}\\y_{2}&u_{2}\end{array}}\right|,&&{\mathfrak {b}}_{z}=\left|{\begin{array}{cc}z_{1}&u_{1}\\z_{2}&u_{2}\end{array}}\right|.\end{array}}\right.$

Obviously, when projecting into three-dimensional space, $V_{II}^{4}$ is composed of two $V^{3}$, which, in the symbolism of ordinary vector analysis, we can write:

(1a)	${\mathfrak {a}}=\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right],\ {\mathfrak {b}}={\mathfrak {r_{1}}}u_{2}-{\mathfrak {r}}_{2}u_{1}$

From two $V_{I}^{4}$:

${\mathfrak {r}},\ u$ and ${\mathfrak {r}}_{1},\ u_{1}$

we can compose a four-dimensional scalar ($S^{4}$) as follows:

(2)	$S=xx_{1}+yy_{1}+zz_{1}+uu_{1}={\mathfrak {rr}}_{1}+uu_{1}$

Conversely, from any four-dimensional scalar $\varphi (x,y,z,u)$, we obtain (derived with respect to their coordinates) a $V_{I}^{4}$:

(3)	$X={\frac {\partial \varphi }{\partial x}},\ Y={\frac {\partial \varphi }{\partial y}},\ Z={\frac {\partial \varphi }{\partial z}},\ U={\frac {\partial \varphi }{\partial u}}.$

So the operators

${\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}},\ {\frac {\partial }{\partial u}}$

transform as the components of a $V_{I}^{4}$, and these operators were denoted by Minkowski as the components of the operator "lor".

We can compose a $S^{4}$ from four $V_{I}^{4}$, which determines the space of the parallelepiped of the four vectors:

(4)	$\varphi =\left|{\begin{array}{ccccccc}x&&y&&z&&u\\x_{1}&&y_{1}&&z_{1}&&u_{1}\\x_{2}&&y_{2}&&z_{2}&&u_{2}\\x_{3}&&y_{3}&&z_{3}&&u_{3}\end{array}}\right|$

If we apply scheme (3) to $S^{4}$, we obtain a $V_{I}^{4}$, which is composed of three other $V_{I}^{4},\ {\mathfrak {r}}_{1}u_{1},\ {\mathfrak {r}}_{2}u_{2},\ {\mathfrak {r}}_{3}u_{3}$, whose components are:

(5)

$\left\{{\begin{array}{ccc}X={\frac {\partial \varphi }{\partial x}}=\left|{\begin{array}{ccccc}y_{1}&&z_{1}&&u_{1}\\y_{2}&&z_{2}&&u_{2}\\y_{3}&&z_{3}&&u_{3}\end{array}}\right|,&&Y={\frac {\partial \varphi }{\partial y}}=\left|{\begin{array}{ccccc}z_{1}&&x_{1}&&u_{1}\\z_{2}&&x_{2}&&u_{2}\\z_{3}&&x_{3}&&u_{3}\end{array}}\right|,\\\\Z={\frac {\partial \varphi }{\partial z}}=\left|{\begin{array}{ccccc}x_{1}&&y_{1}&&u_{1}\\x_{2}&&y_{2}&&u_{2}\\x_{3}&&y_{3}&&u_{3}\end{array}}\right|,&&U={\frac {\partial \varphi }{\partial u}}=-\left|{\begin{array}{ccccc}x_{1}&&y_{1}&&z_{1}\\x_{2}&&y_{2}&&z_{2}\\x_{3}&&y_{3}&&z_{3}\end{array}}\right|;\end{array}}\right.$

if we write this in a vectorial way, we have:

(5a)	${\begin{cases}{\mathfrak {R}}=u_{1}\left[{\mathfrak {r}}_{2}{\mathfrak {r}}_{3}\right]+u_{2}\left[{\mathfrak {r}}_{3}{\mathfrak {r}}_{1}\right]+u_{3}\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right],\\U=-{\mathfrak {r}}_{1}\left[{\mathfrak {r}}_{2}{\mathfrak {r}}_{3}\right].\end{cases}}$

Canceling index 3, we write $V_{1}^{4}$ which we obtained:

${\begin{array}{l}{\mathfrak {R}}=u\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right]+\left[{\mathfrak {r}},\ {\mathfrak {r}}_{1}u_{2}-{\mathfrak {r}}_{2}u_{1}\right],\\U=-{\mathfrak {r}}\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right].\end{array}}$

and introduce $V_{II}^{4}\left\{{\mathfrak {a,b}}\right\}$ instead of $V_{I}^{4}\left\{{\mathfrak {r}}_{1},\ u_{1}\right\}$ and $\left\{{\mathfrak {r}}_{2},\ u_{2}\right\}$, composed from them by rule (1a). Then we have:

(6)	${\begin{cases}{\mathfrak {R}}=ua+[{\mathfrak {rb}}],\\U=-{\mathfrak {ra}},\end{cases}}$

that is, a $V_{I}^{4}$ composed of a $V_{I}^{4}$ and a $V_{II}^{4}$.

We obtain another $V_{I}^{4}$, by permutation of $V^{3}{\mathfrak {a}}$ by ${\mathfrak {b}}$ in expressions (6). To demonstrate this, we form the two $S^{4}$:

${\mathfrak {rr}}_{1}+uu_{1}$ and ${\mathfrak {rr}}_{2}+uu_{2}$

Multiplying them respectively with $V_{I}^{4}$:

$-{\mathfrak {r}}_{2},\ -u_{2}$ and $+{\mathfrak {r}}_{1},\ -u_{1}$

and summing, we construct from $V_{I}^{4}$:

${\begin{array}{l}{\mathfrak {R}}'={\mathfrak {r}}_{1}\left({\mathfrak {rr}}_{2}+uu_{2}\right)-{\mathfrak {r}}_{2}\left({\mathfrak {rr}}_{1}+uu_{1}\right),\\U'=u_{1}\left({\mathfrak {rr}}_{2}+uu_{2}\right)-u_{2}\left({\mathfrak {rr}}_{1}+uu_{1}\right)\end{array}}$

that can be written:

${\begin{array}{l}{\mathfrak {R}}'=u\left({\mathfrak {r}}_{1}u_{2}-{\mathfrak {r}}_{2}u_{1}\right)+\left[{\mathfrak {r}}\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right]\right],\\U'=-\left({\mathfrak {r,\ }}{\mathfrak {r}}_{1}u_{2}-{\mathfrak {r}}_{2}u_{1}\right)\end{array}}$

When we introduce $V_{II}^{4}\{a,b\}$ by means of $I_{a}$, then this is resulting in formulas analogous to (6):

(6a)	${\begin{cases}{\mathfrak {R}}'=u{\mathfrak {b}}+[{\mathfrak {ra}}],\\U'=-{\mathfrak {rb}},\end{cases}}$

where ${\mathfrak {a}}$ and ${\mathfrak {b}}$ have changed their place.

In the electrodynamics of Minkowski, four $V^{3}$ take part, i.e, the electric and magnetic excitations ${\mathfrak {D}}$ and ${\mathfrak {B}}$, and two auxiliary vectors ${\mathfrak {E}}$ and ${\mathfrak {H}}$, which form two $V_{II}^{4}$:

${\mathfrak {B}},\ -i{\mathfrak {E}}$ and ${\mathfrak {H}},\ -i{\mathfrak {D}}$

Then we have the $V_{I}^{4}$-"velocity"

${\frac {\mathfrak {q}}{\sqrt {1-{\mathfrak {q}}^{2}}}},\ {\frac {i}{\sqrt {1-{\mathfrak {q}}^{2}}}}$

(${\mathfrak {q}}$ designates the three-dimensional velocity vector related to the speed of light).

If we combine this $V_{I}^{4}$ with $V_{II}^{4}\left\{{\mathfrak {B}},\ -i{\mathfrak {E}}\right\}$ according to scheme (6a), then we obtain the $V_{I}^{4}$

(7)	${\mathfrak {R}}^{e}={\frac {{\mathfrak {E}}+[{\mathfrak {qB}}]}{\sqrt {1-{\mathfrak {q}}^{2}}}},\ U^{e}={\frac {i({\mathfrak {qE}})}{\sqrt {1-{\mathfrak {q}}^{2}}}},$

which were denoted by Minkowski as the "electric rest force". Instead, from the $V_{I}^{4}$-"velocity" and from $V_{II}^{4}[-i{\mathfrak {H}},\ -{\mathfrak {D}}\}$, when combined according to scheme (6), we obtain $V_{I}^{4}$, which is called the "magnetic rest force":

(8)	${\mathfrak {R}}^{m}={\frac {{\mathfrak {H}}-[{\mathfrak {qD}}]}{\sqrt {1-{\mathfrak {q}}^{2}}}},\ U^{m}={\frac {i({\mathfrak {qH}})}{\sqrt {1-{\mathfrak {q}}^{2}}}},$

The vectors "electric rest force and magnetic rest force" are connected with $V^{3}$, which determine the ponderomotive forces on electric and magnetic poles in motion, and which I wrote down in the first paper as ${\mathfrak {E}}'$ and ${\mathfrak {H}}'$:

(9)	${\mathfrak {E}}'={\mathfrak {E}}+[{\mathfrak {qB}}],\ {\mathfrak {H}}'={\mathfrak {H}}-[{\mathfrak {qD}}]$

Evidently we have:

(9a)	${\mathfrak {R}}^{e}=k^{-1}{\mathfrak {E}}',\ U^{e}=ik^{-1}({\mathfrak {qE}}'),$

(9b)	${\mathfrak {R}}^{m}=k^{-1}{\mathfrak {H}}',\ U^{m}=ik^{-1}({\mathfrak {qH}}'),$

setting

(9c)	$k={\sqrt {1-{\mathfrak {q}}^{2}}}$

From the two $V_{I}^{4}$

$\left\{{\mathfrak {R}}^{e},\ U^{e}\right\}$ and $\left\{{\mathfrak {R}}^{m},\ U^{m}\right\}$

which are composed according to formula (1a), we obtain the $V_{II}^{4}$:

${\begin{array}{l}{\mathfrak {a}}=k^{-2}[{\mathfrak {E}}'{\mathfrak {H}}']\\{\mathfrak {b}}=ik^{-2}\left\{{\mathfrak {E}}'({\mathfrak {qH}}')-{\mathfrak {H}}'({\mathfrak {qE}}')\right\}=ik^{-2}\left[{\mathfrak {q}}[{\mathfrak {E}}'{\mathfrak {H}}']\right]\end{array}}$

By insertion of $V^{3}$:

(10a	${\mathfrak {f}}'=[{\mathfrak {E}}'{\mathfrak {H}}']$

the last $V_{II}^{4}$ can be written:

(11)	${\mathfrak {a}}=k^{-2}{\mathfrak {f}}',\ {\mathfrak {b}}=ik^{-2}[{\mathfrak {qf}}']$

By multiplying the $V^{3}{\mathfrak {f}}'$ by the speed of light ($c$), we obtain the "relative radius" vector of my first paper [l. c.), equation IV].

Finally we combine $V_{I}^{4}$ which is represented by (11), with the $V_{I}^{4}$-"velocity" according to scheme (6). $V_{I}^{4}$ is calculated as follows:

${\begin{array}{l}{\mathfrak {R}}=ik^{-3}\left\{{\mathfrak {f}}'+\left[{\mathfrak {q}}[{\mathfrak {qf}}']\right]\right\}\\U=-k^{-3}({\mathfrak {qf}}').\end{array}}$

When multiplied by ($-i$), we arrive at the $V_{I}^{4}$-"rest radius" of Minkowski:

(12)	$\left\{{\begin{array}{l}{\mathfrak {R}}=k^{-1}{\mathfrak {f}}'+k^{-3}{\mathfrak {q}}[{\mathfrak {qf}}']\\U=ik^{-3}({\mathfrak {qf}}').\end{array}}\right.$

§ 3. Four-dimensional tensors .

A "Four-dimensional tensor" ($T^{4}$) is a set of ten variables, which are transformed into the orthogonal Lorentz transformations, such as we transform the squares and products of the coordinates $x,y,z,u$:

$x^{2},\ y^{2},\ z^{2},\ yz,\ zx,\ xy;\qquad xu,\ yu,\ zu;\qquad u^{2}.$

When projecting into the space of three-dimensional ($x,y,z$), then the first six components form a three-dimensional tensor ($T^{3}$), which are transforming as the squares and products of ($x,y,z$); the following three components of $T^{4}$ are forming a $V^{3}$, the tenth a scalar $S^{3}$.

The four components of the operator "lor" transform as components of $V_{I}^{4}$, we can deduce – from a four-dimensional scalar $\varphi$, given as a function of $x,y,z,u$ – a $T^{4}$ which is twice differentiable with respect to $x,y,z,u$:

${\frac {\partial ^{2}\varphi }{\partial x^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial y^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial z^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial y\ \partial z}},\ {\frac {\partial ^{2}\varphi }{\partial z\ \partial x}},\ {\frac {\partial ^{2}\varphi }{\partial x\ \partial y}};\quad {\frac {\partial ^{2}\varphi }{\partial x\ \partial u}},\ {\frac {\partial ^{2}\varphi }{\partial y\ \partial u}},\ {\frac {\partial ^{2}\varphi }{\partial z\ \partial u}};\quad {\frac {\partial ^{2}\varphi }{\partial u^{2}}}.$

A $S^{4}$ is given, being a quadratic homogeneous function of $x,y,z,u$:

(13)	${\begin{cases}\varphi (x,y,z,u)&={\frac {1}{2}}c_{11}x^{2}+{\frac {1}{2}}c_{22}y^{2}+{\frac {1}{2}}c_{33}z^{2}\\\\&+c_{23}yz+c_{31}zx+c_{12}xy\\\\&+c_{14}xu+x_{24}yu+c_{34}zu+{\frac {1}{2}}c_{44}u^{2},\end{cases}}$,

the 10 coefficients:

$c_{11},\ c_{22},\ c_{33},\ c_{23},\ c_{31},\ c_{12};\ c_{14},\ c_{24},\ c_{34};\ c_{44},$

form a four-dimensional tensor.

In the electrodynamics of moving bodies, the equations of the momentum and energy apply:^[8]

(14)

${\begin{cases}&{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}},\\\\&{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}},\\\\&{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}},\\\\\\c{\mathfrak {qK}}+&Q=-{\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}-{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}-{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}}-{\frac {\partial \psi }{\partial t}}.\end{cases}}$

In order to give to these four equations a more symmetrical from, we put:

(15)	$u=ict,\ {\mathfrak {K}}_{u}=i{\mathfrak {qK}}+i{\frac {Q}{c}},\ U_{u}=\psi ;$

(15a)

$X_{u}=-ic{\mathfrak {g}}_{x},\ Y_{u}=-ic{\mathfrak {g}}_{y},\ Z_{u}=-ic{\mathfrak {g}}_{z};$

(15b)

$U_{x}=-{\frac {i}{c}}{\mathfrak {S}}_{x},\ U_{y}={\frac {-i}{c}}{\mathfrak {S}}_{y},\ U_{z}={\frac {-i}{c}}{\mathfrak {S}}_{z}.$

Then they become:

(16)

${\begin{cases}{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}+{\frac {\partial X_{u}}{\partial z}},\\\\{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}+{\frac {\partial Y_{u}}{\partial u}},\\\\{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}+{\frac {\partial Z_{u}}{\partial u}},\\\\{\mathfrak {K}}_{u}={\frac {\partial U_{x}}{\partial x}}+{\frac {\partial U_{y}}{\partial y}}+{\frac {\partial U_{z}}{\partial z}}+{\frac {\partial U_{u}}{\partial u}}.\end{cases}}$

Now, in Minkowski's theory, the system of four variables

${\mathfrak {K}}_{x},\ {\mathfrak {K}}_{y},\ {\mathfrak {K}}_{z},\ {\mathfrak {K}}_{u}$,

where the first three are the components of $V^{3}$ which determines the ponderomotive force per unity of space, must form a $V_{I}^{4}$. The 16 variables $X_{x},\ X_{y},\dots ,U_{z},\ U_{u}$, from which that system is derived using equations (16), must be transformed in such a way that this condition is satisfied.

We determine these 16 variables as follows. They are reduced to 10 ones:

(17)	$\left\{{\begin{array}{ccccccc}X_{x},\ Y_{y},\ Z_{z},&&Y_{z}=Z_{y},&&Z_{x}=X_{z},&&X_{y}=Y_{x};\\X_{u}=U_{x},&&Y_{u}=U_{y},&&Z_{u}=U_{z},&&U_{u},\end{array}}\right.$

which are the components of a $T^{4}$.

Then, from the transformation properties of the components of a $T^{4}$ and the components of operator 'lor', it follows that the four variables, derived from (16), transform as the coordinates $x,y,z,u$ of a point of a four-dimensional space, i.e. as components of a $V_{I}^{4}$, in agreement with the principle of relativity. But this determination chosen by me is not the only one which corresponds to this principle. Indeed, Minkowski himself preferred a different determination, which does not satisfy the symmetry conditions contained in (17). Yet the determination postulated from my system of electrodynamics of moving bodies is demonstrated now.

Our aim is to form a $T^{4}$, whose components correspond to the expressions given in the first paper for the special case of Minkowski's theory. If this is established, it is clear that these expressions satisfy the principle of relativity.

We obtain such a $T^{4}$ by calculating a $S^{4}$ in the form of (13), i.e. a homogeneous quadratic function of $x,y,z,u$, which is invariant under a Lorentz transformations:

(18)	$\varphi (x,y,z,u)=\Phi (x,y,z)-iu({\mathfrak {rf}})+{\frac {1}{2}}u^{2}\psi$

From $S^{3}$, which is a second-order homogeneous function of $x,y,z$:

(18a)

$\Phi (x,y,z)={\frac {1}{2}}X_{x}x^{2}+{\frac {1}{2}}Y_{y}y^{2}+{\frac {1}{2}}Z_{z}z^{2}+Y_{z}yz+Z_{x}zx+X_{y}xy$

we obtain the six pressures of Maxwell; a $V^{3}$ which is now designated by ${\mathfrak {f}}$, gives us at the same time (according to (15ab)) the energy current ${\mathfrak {S}}$ and the electromagnetic momentum density ${\mathfrak {g}}$:

(18b)

${\mathfrak {f}}=c{\mathfrak {g}}={\frac {1}{c}}{\mathfrak {S}};$

finally, the tenth part of $T^{4}$ which is derived from (18), $\psi$, determines the density of electromagnetic energy.

To arrive at a suitable four-dimensional scalar, which is a second-order homogeneous function of coordinates $x,y,z,u$, with bilinear coefficients in the components of the electromagnetic vectors, we form the first one according to scheme (6), the radius vector $\{{\mathfrak {r,u}}\}$ in a space of four dimensions, and $V_{I}^{4}$ from $V_{II}^{4}\{{\mathfrak {a,b}}\}$:

${\mathfrak {R}}=u{\mathfrak {a}}+[{\mathfrak {rb}}],\ U=-({\mathfrak {ra}})$

Similarly, from another $V_{II}^{4}\{{\mathfrak {a',b'}}\}$ and from $V_{I}^{4}\{{\mathfrak {r,a}}\}$ which is comprised of $V_{I}^{4}$:

${\mathfrak {R}}'=u{\mathfrak {a}}'+[{\mathfrak {rb}}'],\ U'=-({\mathfrak {ra}}')$

Now, according to the scheme (2), we obtain the $S^{4}$:

$S={\mathfrak {RR}}'+UU'=u^{2}{\mathfrak {aa}}'+u{\mathfrak {a}}[{\mathfrak {rb}}']+u{\mathfrak {a}}'[{\mathfrak {rb}}]+[{\mathfrak {rb}}][{\mathfrak {rb}}']+({\mathfrak {ra}})({\mathfrak {ra}}')$

that can be written:

(19)	$S=({\mathfrak {ra}})({\mathfrak {ra}}')-({\mathfrak {rb}})({\mathfrak {rb}}')+{\mathfrak {r}}^{2}({\mathfrak {bb}}')+u{\mathfrak {r}}[{\mathfrak {ba}}']+u{\mathfrak {r}}[{\mathfrak {b'a}}]+u^{2}({\mathfrak {aa}}')$

As it follows from (6a), we can permute ${\mathfrak {a}}$ with ${\mathfrak {b}}$ and ${\mathfrak {a'}}$ with ${\mathfrak {b'}}$, and obtain in a corresponding way another $S^{4}$:

(19a)

$S^{*}=({\mathfrak {rb}})({\mathfrak {rb}}')-({\mathfrak {ra}})({\mathfrak {ra}}')+{\mathfrak {r}}^{2}[{\mathfrak {aa}}']+u{\mathfrak {r}}[{\mathfrak {ab}}']+u{\mathfrak {r}}[{\mathfrak {a'b}}]+u^{2}({\mathfrak {bb}}')$

Putting

$4\varphi =S-S^{*}$

it is given:

(20)

$\left\{{\begin{array}{c}2\varphi =({\mathfrak {ra}})({\mathfrak {ra}}')-{\frac {1}{2}}{\mathfrak {r}}^{2}[{\mathfrak {aa}}']-({\mathfrak {rb}})({\mathfrak {rb}}')+{\frac {1}{2}}{\mathfrak {r}}^{2}[{\mathfrak {bb}}']\\\\+u{\mathfrak {r}}[{\mathfrak {b'a}}]+u{\mathfrak {r}}[{\mathfrak {b'a}}]+{\frac {1}{2}}u^{2}\left\{({\mathfrak {aa}}')-({\mathfrak {bb}}')\right\}.\end{array}}\right.$

Now, by identifying the homogeneous second-order function $\varphi$ of $x,y,z,u$ which is invariant under the Lorentz transformation, with $S^{4}$ as given by (18), we find the expressions:

(20a)

$2\Phi =({\mathfrak {ra}})({\mathfrak {ra}}')-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {aa}}')-({\mathfrak {rb}})({\mathfrak {rb}}')+{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {bb}}')$

(20b)

$2{\mathfrak {f}}=i[{\mathfrak {b'a}}]+i[{\mathfrak {b'a}}]$

(20c)

$2\psi =({\mathfrak {aa}}')-({\mathfrak {bb}}')$

We introduce the electrodynamic $V_{II}^{4}$ of Minkowski, by setting

(21)	${\begin{cases}{\mathfrak {a}}={\mathfrak {H}},&{\mathfrak {b}}=-i{\mathfrak {D}};\\{\mathfrak {a}}'={\mathfrak {B}},&{\mathfrak {b}}'=-i{\mathfrak {E}}.\end{cases}}$

By taking into account (18a), the following expressions are resulting now:

(21a)

${\begin{cases}X_{x}x^{2}+Y_{y}y^{2}+Z_{z}z^{2}+2Y_{z}yz+2Z_{x}zx+2X_{y}xy\\=({\mathfrak {rE}})({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {ED}})+({\mathfrak {rH}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {HB}}),\end{cases}}$

(21b)

$2{\mathfrak {f}}=[{\mathfrak {EH}}]+[{\mathfrak {DB}}]$

(21c)

$2\psi ={\mathfrak {ED}}+{\mathfrak {HB}}.$

They give to empty space (where ${\mathfrak {D}}$ and ${\mathfrak {E}}$, ${\mathfrak {H}}$ and ${\mathfrak {B}}$ become identical) the known values ​​of the Maxwell pressure, the current, and the energy density. To ponderable bodies in a resting state, the values ​​(21a) and (21c) of the pressure and energy density are acceptable, yet not the value (21b), because it is

${\mathfrak {D}}=\epsilon {\mathfrak {E}},\ {\mathfrak {B}}=\mu {\mathfrak {H}},$,

then the energy current would be

${\mathfrak {S}}=c{\mathfrak {f}}=\left({\frac {\epsilon \mu +1}{2}}\right)c[{\mathfrak {EH}}],$

which differs from the current given by the Poynting vector

${\mathfrak {S}}=c[{\mathfrak {EH}}]$

by

$\left({\frac {\epsilon \mu +1}{2}}\right)c[{\mathfrak {EH}}]$.

So we must subtract from the invariant $\varphi$ (given by equation (20)), another $S^{4}$, which contains $(\epsilon \mu -1)$ as a factor, and which is equal to zero for empty space.

To obtain such a $S^{4}$, we consider two $V_{I}^{4}$; first the $V_{I}^{4}$-"velocity"

${\mathfrak {r}}_{1}=k^{-1}{\mathfrak {q}},\ u_{1}=ik^{-1},$

then the "rest ray", given by equations (12):

${\mathfrak {R}}=k^{-1}{\mathfrak {f}}'+k^{-3}{\mathfrak {q}}({\mathfrak {qf}}'),\ U=ik^{-3}({\mathfrak {qf}}').$

We introduce the $V^{3}$

(22)	${\mathfrak {W}}=(\epsilon \mu -1)k^{-1}{\mathfrak {R}}=(\epsilon \mu -1)\left\{k^{-2}{\mathfrak {f}}'+k^{-4}{\mathfrak {q}}({\mathfrak {qf}}')\right\}$

with $(\epsilon \mu -1)$ being a $S^{4}$,

${\begin{array}{l}{\mathfrak {r}}_{2}=(\epsilon \mu -1){\mathfrak {R}}=k{\mathfrak {W}},\\u_{2}=(\epsilon \mu -1)U=ik({\mathfrak {qW}}),\end{array}}$

which forms a $V_{I}^{4}$.

Now we compose, according to scheme (2), two $S^{4}$:

${\begin{array}{l}{\mathfrak {rr}}_{1}+uu_{1}=k^{-1}\{({\mathfrak {rq}})+iu\},\\{\mathfrak {rr}}_{2}+uu_{2}=k\{({\mathfrak {rW}})+iu({\mathfrak {qW}})\},\end{array}}$

which are both linear in $x,y,z,u$, and we multiply them. Thus a $S^{4}$ is given, being a homogeneous second-order function of $x,y,z,u$:

(23)	$2\chi =({\mathfrak {rq}})({\mathfrak {rW}})+\left(iu{\mathfrak {r}},\ {\mathfrak {W}}+{\mathfrak {q}}({\mathfrak {qW}})\right)-u^{2}({\mathfrak {qW}}).$

By adding $S^{4}$, $\varphi$ and $\chi$, which are given by (20) and (23), we form the new $S^{4}$

(24)	$f=\varphi +\chi$

and we are using this instead of $\varphi$ as a characteristic invariant, which determines the pressures, the current, and the electromagnetic energy density, by setting:

$f(x,y,z,u)=\Phi (x,y,z)-iu({\mathfrak {rf}})+{\frac {1}{2}}u^{2}\psi .$

Instead of (21a, b, c), thus the formulas follow:

(24a)

${\begin{cases}2\Phi &=X_{x}x^{2}+Y_{y}y^{2}+Z_{z}z^{2}+2Y_{z}yz+2Z_{x}zx+2X_{y}xy\\&=({\mathfrak {rE}})({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {ED}})+({\mathfrak {rH}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {HB}})+({\mathfrak {rq}})({\mathfrak {rW}}),\end{cases}}$

(24b)

$2{\mathfrak {f}}=[{\mathfrak {EH}}]+[{\mathfrak {DB}}]-{\mathfrak {W}}-{\mathfrak {q}}({\mathfrak {qW}})$

(24c)

$2\psi ={\mathfrak {ED}}+{\mathfrak {HB}}-2({\mathfrak {qW}}).$

These values ​​are identical to those values (in the case of Minkowki's theory) derived from my system of electrodynamics of moving bodies in the first paper.

In this theory, we apply the relationship:^[9]

(25)	${\begin{cases}{\mathfrak {D}}=\epsilon {\mathfrak {E}}'-[{\mathfrak {qH}}],&{\mathfrak {E}}={\mathfrak {E}}'-[{\mathfrak {qB}}],\\{\mathfrak {B}}=\mu {\mathfrak {H}}'+[{\mathfrak {qE}}],&{\mathfrak {H}}={\mathfrak {H}}'+[{\mathfrak {qD}}].\end{cases}}$

A calculation, not reproduced here, gives us

$[{\mathfrak {DB}}]-[{\mathfrak {EH}}]=k^{-2}(\epsilon \mu -1)[{\mathfrak {E}}'{\mathfrak {H}}']=k^{-2}(\epsilon \mu -1){\mathfrak {f}}'.$

On the other hand we have, according to (22):

(26)	${\mathfrak {W}}-{\mathfrak {q}}({\mathfrak {qW}})=k^{-2}(\epsilon \mu -1){\mathfrak {f}}'.$

So the following relation holds:

(26e)

${\mathfrak {W}}-{\mathfrak {q}}({\mathfrak {qW}})=[{\mathfrak {DB}}]-[{\mathfrak {EH}}],$

a formula already present in the first paper.^[10]

Equation (24b) therefore can be written

(26b)

${\mathfrak {f}}=[{\mathfrak {EH}}]-{\mathfrak {q}}({\mathfrak {qW}})$

namely

(26c)

${\mathfrak {f}}=[{\mathfrak {DB}}]-{\mathfrak {W}}.$

Evidently from (26b) and (18b), the energy current postulated by the Poynting theorem follows in the case of rest. The values ​​of the energy current and momentum density are consistent with those found in the first paper^[11]. Even the expression (24c) of the energy density was already indicated there^[12].

It remains to prove that the electromagnetic pressures, determined from equation (24a), are those that result from the first paper.

To prove this, we must introduce the "relative pressure", defined by^[13]

${\begin{array}{ccccc}X'_{x}=X_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{x},&&X'_{y}=X_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{x},&&X'_{z}=X_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{x};\\Y'_{x}=Y_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{y},&&Y'_{y}=Y_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{y},&&Y'_{z}=Y_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{y};\\Z'_{x}=Z_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{z},&&Z'_{y}=Z_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{z},&&Z'_{z}=Z_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{z}.\end{array}}$

Taking into account the symmetry conditions:

$Y_{z}=Z_{y},\ Z_{x}=X_{z},\ X_{y}=Y_{x},$

we have

${\begin{array}{c}Y'_{z}-Z'_{y}={\mathfrak {q}}_{z}{\mathfrak {f}}_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{z},\\Z'_{x}-X'_{z}={\mathfrak {q}}_{x}{\mathfrak {f}}_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{x},\\X'_{y}-Y'_{x}={\mathfrak {q}}_{y}{\mathfrak {f}}_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{y},\end{array}}$

relationships that are, as I demonstrated in the first paper, satisfied by the expressions given for relative pressures. Only to prove that, we set the function

$2\Phi '=X'_{x}x^{2}+Y'_{y}y^{2}+Z'_{z}z^{2}+\left(Y'_{z}+Z'_{y}\right)yz+\left(Z'_{x}+X'_{z}\right)zx+\left(X'_{y}+Y'_{x}\right)xy$

equal to

${\begin{array}{ll}2\Phi '&=2\Phi +x^{2}{\mathfrak {q}}_{x}{\mathfrak {f}}_{x}+y^{2}{\mathfrak {q}}_{y}{\mathfrak {f}}_{y}+z^{2}{\mathfrak {q}}_{z}{\mathfrak {f}}_{z}\\&+\left({\mathfrak {q}}_{y}{\mathfrak {f}}_{z}+{\mathfrak {q}}_{z}{\mathfrak {f}}_{y}\right)yz+\left({\mathfrak {q}}_{z}{\mathfrak {f}}_{x}+{\mathfrak {q}}_{x}{\mathfrak {f}}_{z}\right)zx+\left({\mathfrak {q}}_{x}{\mathfrak {f}}_{y}+{\mathfrak {q}}_{y}{\mathfrak {f}}_{x}\right)xy,\end{array}}$

namely

(27)	$2\Phi '=2\Phi +({\mathfrak {rq}})({\mathfrak {rf}})$

and introducing the value (24a) of $2\Phi$, which is an expression identical to the one resulting from the fundamental formulas ($V_{a}$) of the first paper. These finally give

(27a)

$2\Phi '=({\mathfrak {rE}}')({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {E'D}})+({\mathfrak {rH}}')({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {H'B}})$

The identity of the values ​​(27) and (27a) will be demonstrated, by proving that the relationship is satisfied:

(28)

$\left\{{\begin{array}{c}({\mathfrak {rq}})({\mathfrak {rf}})+({\mathfrak {rq}})({\mathfrak {rW}})\\\\=({\mathfrak {r,\ E'-E}})({\mathfrak {rD}})+({\mathfrak {r,\ H'-H}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}\left\{({\mathfrak {E'-E,D}})+({\mathfrak {H'-H,B}})\right\}\end{array}}\right.$

Taking account of (26c) and (25), we can write

(28a)

${\begin{cases}({\mathfrak {rq}})({\mathfrak {r[DB]}})&=\left({\mathfrak {r[qB]}}\right)({\mathfrak {rD}})-\left({\mathfrak {r[qD]}}\right)({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}\left\{\left({\mathfrak {D[qB]}}\right)-{\mathfrak {B[qD]}}\right\}\\&={\mathfrak {r[q,\ B(rD)-D(rB)]+r^{2}(q[DB])}}\end{cases}}$

Now, it is identically

${\mathfrak {[q,\ B(rD)-D(rB)]=-\left[q[r[DB]]\right]=-r(q[DB])+(rq)[DB],}}$

and the second part of equation (28a) gives in fact:

$({\mathfrak {rq}})({\mathfrak {r[DB]}})$

so that the relationship (28) is identically satisfied. So, by formula (27a) which is postulated from our system of electrodynamics, the values ​​of the pressures of Maxwell follow for the special case of the theory of Minkowski, which obeys the principle of relativity in agreement with relation (24a).

§ 4. The equations of motion.

In the mechanics of Minkowski, the so-called "proper time" of a point occurs, i.e. a four-dimensional scalar $(\tau )$, defined by^[14]

(29)	${\frac {dt}{d\tau }}={\frac {1}{\sqrt {1-q^{2}}}}=k^{-1}$

If we differentiate (with respect to $\tau$) the four-dimensional radius vector of the point, and dividing by the speed of light ($c$), then it is resulting in the $V^{4}$-"velocity" of Minkowski:

(30)

${\begin{cases}{\frac {1}{c}}{\frac {dx}{d\tau }}={\frac {1}{c}}{\frac {dx}{dt}}\cdot {\frac {dt}{d\tau }}={\mathfrak {q}}_{x}{\frac {dt}{d\tau }}={\mathfrak {q}}_{x}k^{-1},\\\\{\frac {1}{c}}{\frac {dy}{d\tau }}={\frac {1}{c}}{\frac {dy}{dt}}\cdot {\frac {dt}{d\tau }}={\mathfrak {q}}_{y}{\frac {dt}{d\tau }}={\mathfrak {q}}_{y}k^{-1},\\\\{\frac {1}{c}}{\frac {dz}{d\tau }}={\frac {1}{c}}{\frac {dz}{dt}}\cdot {\frac {dt}{d\tau }}={\mathfrak {q}}_{z}{\frac {dt}{d\tau }}={\mathfrak {q}}_{z}k^{-1},\\\\{\frac {1}{c}}{\frac {du}{d\tau }}={\frac {1}{c}}{\frac {du}{dt}}\cdot {\frac {dt}{d\tau }}=i{\frac {dt}{d\tau }}=ik^{-1},\end{cases}}$

Obviously the four components of the $V^{4}$-"velocity" identically satisfy the equation:

(30a)

$\left({\frac {1}{c}}{\frac {dx}{d\tau }}\right)^{2}+\left({\frac {1}{c}}{\frac {dy}{d\tau }}\right)^{2}+\left({\frac {1}{c}}{\frac {dz}{d\tau }}\right)^{2}+\left({\frac {1}{c}}{\frac {du}{d\tau }}\right)^{2}=-1$

We form now, by the $V_{I}^{4}$-"velocity" and "force" according to the scheme (2), the four-dimensional scalar

(31)	$\Psi ={\mathfrak {K}}_{x}{\frac {dx}{cd\tau }}+{\mathfrak {K}}_{y}{\frac {dy}{cd\tau }}+{\mathfrak {K}}_{z}{\frac {dz}{cd\tau }}+{\mathfrak {K}}_{u}{\frac {du}{cd\tau }}$

Introducing the ponderomotive force of electromagnetic fields, whose components are determined by (16), and taking into account equations (15) and (30), we find:

(31a)

$\Psi =-{\frac {Q}{c}}k^{-1}$

where $Q$ is the Joule-heat, developed in the unity of space and time.

Now, Minkowski gives the equations of motion of an element of matter in the form^[15]

(32)

${\begin{cases}\nu {\frac {d^{2}x}{d\tau ^{2}}}={\mathfrak {K}}_{x}+\Psi {\frac {dx}{cd\tau }},\\\\\nu {\frac {d^{2}y}{d\tau ^{2}}}={\mathfrak {K}}_{y}+\Psi {\frac {dy}{cd\tau }},\\\\\nu {\frac {d^{2}z}{d\tau ^{2}}}={\mathfrak {K}}_{z}+\Psi {\frac {dz}{cd\tau }},\\\\\nu {\frac {d^{2}u}{d\tau ^{2}}}={\mathfrak {K}}_{u}+\Psi {\frac {du}{cd\tau }},\end{cases}}$

where $S^{4}(\nu )$ determines the "rest density" of matter. The identity (30a), from which it follows

${\frac {dx}{d\tau }}{\frac {d^{2}x}{d\tau ^{2}}}+{\frac {dy}{d\tau }}{\frac {d^{2}y}{d\tau ^{2}}}+{\frac {dz}{d\tau }}{\frac {d^{2}z}{d\tau ^{2}}}+{\frac {du}{d\tau }}{\frac {d^{2}u}{d\tau ^{2}}}=0$

is satisfied by equations (32).

Minkowski denotes the ponderomotive force of the electromagnetic field, i.e. the $V^{3}$ whose components are the second members of the first three equations of motion (32), i.e. the $V^{3}$, by:

(32a)

${\mathfrak {K}}+\Psi {\mathfrak {q}}k^{-1}={\mathfrak {K}}-{\frac {{\mathfrak {q}}\cdot Q}{ck^{2}}}$

This vector is not identical to the force determined from the momentum theorem (14), but it differs from that by

$-{\frac {{\mathfrak {q}}\cdot Q}{ck^{2}}}$

Thus, when the Joule-heat emerges in matter, then the mechanics of Minkowski must add this additional force to the ponderomotive force, which is derived from the momentum theorem.

Considering the momentum theorem as being important to electromagnetic mechanics, I prefer to keep this principle of the electrodynamics of moving bodies. We can remove the additional force of Minkowski, by giving the equations of motion, instead of (32), precisely the form suggested by the mechanical law of momentum:

(33)

${\begin{cases}{\frac {d}{d\tau }}\left(\nu {\frac {dx}{d\tau }}\right)={\mathfrak {K}}_{x},\\\\{\frac {d}{d\tau }}\left(\nu {\frac {dx}{d\tau }}\right)={\mathfrak {K}}_{y},\\\\{\frac {d}{d\tau }}\left(\nu {\frac {dz}{d\tau }}\right)={\mathfrak {K}}_{z},\\\\{\frac {d}{d\tau }}\left(\nu {\frac {du}{d\tau }}\right)={\mathfrak {K}}_{u}.\end{cases}}$

Since $\tau$ and $\nu$ are of $S^{4}$, both members of these equations are the components of $V_{I}^{4}$; then these equations agree with the principle of relativity. The identity (30a) is satisfied, if we put

${\frac {d\nu }{d\tau }}\left\{\left({\frac {dx}{d\tau }}\right)^{2}+\left({\frac {dy}{d\tau }}\right)^{2}+\left({\frac {dz}{d\tau }}\right)^{2}+\left({\frac {du}{d\tau }}\right)^{2}\right\}={\mathfrak {K}}_{x}{\frac {dx}{d\tau }}+{\mathfrak {K}}_{y}{\frac {dy}{d\tau }}+{\mathfrak {K}}_{z}{\frac {dz}{d\tau }}+{\mathfrak {K}}_{u}{\frac {du}{d\tau }};$

taking into account (30a), (31), and (31a), we find

${\frac {d\nu }{d\tau }}=-{\frac {\psi }{c}}={\frac {Q}{c^{2}k}}$

namely, according to (29):

(33a)

${\frac {d\nu }{d\tau }}={\frac {Q}{c^{2}}}$

Thus $\nu$, the "rest density" of mass, should be variable, and increases each time when Joule-heat emerges in matter. When we accept this hypothesis, which was first introduced by Einstein and by Planck, then we avoid the additional force of Minkowski.

From the equations of motion (33), which refer to the unit of volume of an extended body, we pass to the equations of motion of a material point, in the same way as it was shown by Minkowski in respect to equations (32).

Milan, January 17, 1910.

Max Abraham.

---

1. M. Abraham, Zur Elektrodynamik bewegter Körper [Rendiconti del Circolo Matematico di Palermo, t. XXVIII (2° sem. 1909), pp. 1-28].
2. H. Minkowski, Die Grundgleichungen far die elektromagnetischen Vorgänge in bewegten Körpern [Nachrichten von der Kgl. Gesellschaft der Wissenschaften zu Göttingen, Jahrgang 1908, pp. 53-111].
3. M. Abraham, Geometrische Grundbegriffe [Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, IV, 2, pp. 3-47].
4. See the discussion between G. Nordström and M. Abraham [Physikalische Zeitschrift, Jahrgang X (1909), pp. 681-687, 737-741].
5. H. Minkowski, Raum und Zeit (Leipzig, Teubner, 1909).
6. M. Born, Die Theorie des starren Elektrons in der Kinematik des Relativitätsprincips [Annalen der Physik, Bd. XXX (1909), pp. 1-56].
7. G. Herglotz, Über den vom Standpunkt des Relativitätsprincips aus als "Starr" zu bezeichnenden Körper [Annalen der Physik, Bd. XXXI (1910), pp. 393-415].
8. M. Abraham, l. c. equations (6) and (7).
9. M. Abraham, l. c., equations (36) and (37).
10. M. Abraham, l. c. equation (40c).
11. M. Abraham, l. c. equations (40), (40a) and (42).
12. M. Abraham, l. c. equation (44a).
13. M. Abraham, l. c. equation (10), where it has to be put ${\mathfrak {w}}=c{\mathfrak {q}},\ {\mathfrak {f}}=c{\mathfrak {g}}$.
14. H. Minkowski, l. c. equation (3), pag. 48.
15. H. Minkowski, l. c. equation (20), pag. 54.