Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies

by Hermann Minkowski

INTRODUCTION.

At the present time, different opinions are being held about the fundamental equations of Electrodynamics for moving bodies. The Hertzian^[1] forms must be given up, for it has appeared that they are contrary to many experimental results.

In 1895 H. A. Lorentz^[2] published his theory of optical and electrical phenomena in moving bodies; this theory was based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been ushered into existence. In his theory^[3], Lorentz proceeds from certain equations, which must hold at every point of "Æther"; then by forming the average values over "physically infinitely small" regions, which however contain large numbers of electrons, the equations for electro-magnetic processes in moving bodies can be successfully built up.

In particular, Lorentz's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous "Æther"; it shows that this fact is connected with the covariance of the original equation, at certain simultaneous transformations of the space and time co-ordinates; these transformations have obtained from H. Poincaré^[4] the name of Lorentz-transformations. The covariance of these fundamental equations, when subjected to the Lorentz-transformation, is a purely mathematical fact; I will call this the Theorem of Relativity; this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light.

Now without recognizing any hypothesis about the connection between "Æther" and matter, we can expect these mathematically evident theorems to have their consequences so far extended — that thereby even those laws of ponderable media which are yet unknown may anyhow possess this covariance when subjected to a Lorentz-transformation; by saying this, we do not indeed express an opinion, but rather a conviction, — and this conviction I may be permitted to call the Postulate of Relativity. The position of affairs here is almost the same as when the Principle of Conservation of Energy was postulated in cases, where the corresponding forms of energy were unknown.

Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite connection may be styled the Principle of Relativity.

These differentiations seem to me to be useful for enabling us to characterise the present day position of the electro-dynamics for moving bodies.

H. A. Lorentz has found out the Relativity theorem and has created the Relativity postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.

A. Einstein^[5] has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept which is forced upon us by observation of natural phenomena.

The Principle of Relativity has not yet been formulated for electro-dynamics of moving bodies in the sense characterized by me. In the present essay, while formulating this principle, I shall obtain the fundamental equations for moving bodies in a sense which is uniquely determined by this principle. But it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle.

We would at first expect that the fundamental equations which are assumed by Lorentz for moving bodies would correspond to the Relativity Principle. But it will be shown that this is not the case for the general equations which Lorentz has for any possible, and also for magnetic bodies; but this is approximately the case (if we neglect the square of the velocity of matter in comparison to the velocity of light) for those equations which Lorentz hereafter infers for non-magnetic bodies. But this latter accordance with the relativity principle is due to the fact that the condition of non-magnetisation has been formulated in a way not corresponding to the relativity principle; therefore the accordance is due to the fortuitous compensation of two contradictions to the relativity postulate. But meanwhile enunciation of the Principle in a rigid manner does not signify any contradiction to the hypotheses of Lorentz's molecular theory, but it shall become clear that the assumption of the contraction of the electron in Lorentz's theory must be introduced at an earlier stage than Lorentz has actually done.

In an appendix, I have gone into discussion of the position of Classical Mechanics with respect to the relativity postulate. Any easily perceivable modification of mechanics for satisfying the requirements of the Relativity theory would hardly afford any noticeable difference in observable processes; but would lead to very surprising consequences. By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone.

§ 1. NOTATIONS.

Let a rectangular system (x, y, z, t,) of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the unit of length that the velocity of light in space becomes unity.

Although I would prefer not to change the notations used by Lorentz, it appears important to me to use a different selection of symbols, for thereby certain homogeneity will appear from the very beginning. I shall denote the vector

electric force by ${\mathfrak {E}}$, the magnetic induction by ${\mathfrak {M}}$, the electric induction by ${\mathfrak {e}}$ and the magnetic force by ${\mathfrak {m}}$,

so that ${\mathfrak {E,M,e,m}}$ are used instead of Lorentz's ${\mathfrak {E,B,D,H}}$ respectively.

I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, i.e., instead of operating with t, I shall operate with it, where i denotes ${\sqrt {-1}}$. If now instead of (x, y, z, it), I use the method of writing with indices, certain essential circumstances will come into evidence; on this will be based a general use of the suffixes (1, 2, 3, 4). The advantage of this method will be, as I expressly emphasize here, that we shall have to handle symbols which have a purely real appearance; we can however at any moment pass to real equations if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities.

An individual system of values of x, y, z t, i. e., of $x_{1},\ x_{2},\ x_{3},\ x_{4}$ shall be called a space-time point.

Further let ${\mathfrak {w}}$ denote the velocity vector of matter, $\epsilon$ the dielectric constant, $\mu$, the magnetic permeability, $\sigma$ the conductivity of matter, while $\varrho$ denotes the density of electricity in space, and ${\mathfrak {s}}$ the vector of "Electric Current" which we shall come across in §7 and §8.

PART I. Consideration of the Limiting Case Æther.

§ 2. The Fundamental Equations for Æther.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electrodynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case $\epsilon =1,\ \mu =1,\ \sigma =0$, they should constitute the laws for ponderable bodies. In this ideal limiting case $\epsilon =1,\ \mu =1,\ \sigma =0$, we shall have ${\mathfrak {E}}={\mathfrak {e}},{\mathfrak {M}}={\mathfrak {m}}$. At every space time point x, y, z, t we shall have the equations:

${\begin{array}{lcrl}(I)&\qquad &curl\ {\mathfrak {m}}-{\frac {\partial e}{\partial t}}&=\varrho {\mathfrak {w,}}\\\\(II)&&div\ {\mathfrak {e}}&={\mathfrak {\varrho ,}}\\\\(III)&&curl\ {\mathfrak {e}}+{\frac {\partial {\mathfrak {m}}}{\partial t}}&=0,\\\\(IV)&&div\ {\mathfrak {m}}&=0.\end{array}}$

I shall now write $x_{1},\ x_{2},\ x_{3},\ x_{4}$ for x, y, z, it $\left(i={\sqrt {-1}}\right)$ and

$\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}$

for

$\varrho {\mathfrak {w}}_{x},\ \varrho {\mathfrak {w}}_{y},\ \varrho {\mathfrak {w}}_{z},\ i\varrho$

i.e. the components of the convection current $\varrho {\mathfrak {w}}$, and the electric density multiplied by ${\sqrt {-1}}$.

Further I shall write

$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$

for

${\mathfrak {m}}_{x},\ {\mathfrak {m}}_{y},\ {\mathfrak {m}}_{z},\ -i{\mathfrak {e}}_{x},\ -i{\mathfrak {e}}_{y},\ -i{\mathfrak {e}}_{z}$,

i.e., the components of ${\mathfrak {m}}$ and $-i{\mathfrak {e}}$ along the three axes; now if we take any two indices h, k out of the series

$f_{kh}=-f_{hk}$,

therefore

$f_{32}=-f_{23},\ f_{13}=-f_{31},\ f_{21}=-f_{12}$, $f_{41}=-f_{14},\ f_{42}=-f_{24},\ f_{43}=-f_{34}$,

Then the three equations comprised in (I), and the equation (II) multiplied by i becomes

(A)

${\begin{array}{ccccccccc}&&{\frac {\partial f_{12}}{\partial x_{2}}}&+&{\frac {\partial f_{13}}{\partial x_{3}}}&+&{\frac {\partial f_{14}}{\partial x_{4}}}&=&\varrho _{1},\\\\{\frac {\partial f_{21}}{\partial x_{1}}}&&&+&{\frac {\partial f_{23}}{\partial x_{3}}}&+&{\frac {\partial f_{24}}{\partial x_{4}}}&=&\varrho _{2},\\\\{\frac {\partial f_{31}}{\partial x_{1}}}&+&{\frac {\partial f_{32}}{\partial x_{2}}}&&&+&{\frac {\partial f_{34}}{\partial x_{4}}}&=&\varrho _{3},\\\\{\frac {\partial f_{41}}{\partial x_{1}}}&+&{\frac {\partial f_{42}}{\partial x_{2}}}&+&{\frac {\partial f_{43}}{\partial x_{3}}}&&&=&\varrho _{4}.\end{array}}$

On the other hand, the three equations comprised in (III) multiplied by -i, and equation (IV) multiplied by -1, become

(B)

${\begin{array}{ccccccccc}&&{\frac {\partial f_{34}}{\partial x_{2}}}&+&{\frac {\partial f_{42}}{\partial x_{3}}}&+&{\frac {\partial f_{23}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial f_{43}}{\partial x_{1}}}&&&+&{\frac {\partial f_{14}}{\partial x_{3}}}&+&{\frac {\partial f_{31}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial f_{24}}{\partial x_{1}}}&+&{\frac {\partial f_{41}}{\partial x_{2}}}&&&+&{\frac {\partial f_{12}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial f_{32}}{\partial x_{1}}}&+&{\frac {\partial f_{13}}{\partial x_{2}}}&+&{\frac {\partial f_{21}}{\partial x_{3}}}&&&=&0.\end{array}}$

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices (1,2,3,4).

§ 3. The Theorem of Relativity of Lorentz.

It is well-known that by writing the equations I) to IV) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis. through an amount $\varphi$, keeping ${\mathfrak {e,m,w}}$ fixed in space, and introduce new variables $x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$ instead of $x_{1},\ x_{2},\ x_{3},\ x_{4}$, where

$x'_{1}=x_{1}\cos \varphi +x_{2}\sin \varphi ,\ x'_{2}=-x_{1}\sin \varphi +x_{2}\cos \varphi ,\ x'_{3}=x_{3},\ x'_{4}=x_{4}$,

and introduce magnitudes

$\varrho '_{1},\ \varrho '_{2},\ \varrho '_{3},\ \varrho '_{4}$,

where

$\varrho '_{1}=\varrho _{1}\cos \varphi +\varrho _{2}\sin \varphi ,\ \varrho '_{2}=-\varrho _{1}\sin \varphi +\varrho _{2}\cos \varphi ,\ \varrho '_{3}=\varrho _{3},\ \varrho '_{4}=\varrho _{4}$,

and $f'_{12},\dots f'_{34}$, where

$f'_{23}=f_{23}\cos \varphi +f_{31}\sin \varphi ,\ f'_{31}=-f_{23}\sin \varphi +f_{31}\cos \varphi ,\ f'_{12}=f_{12}$, $f'_{14}=f_{14}\cos \varphi +f_{24}\sin \varphi ,\ f'_{24}=-f_{14}\sin \varphi +f_{24}\cos \varphi ,\ f'_{34}=f_{34}$, $f'_{kh}=-f'_{hk}\qquad (h,k=1,2,3,4)$,

then out of the equations (A) would follow a corresponding system of dashed equations (A') composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning the suffixes (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by $i\psi$ a purely imaginary magnitude, and consider the substitution

(1)	${\begin{array}{ccc}&x'_{1}=x_{1},\ x'_{2}=x_{2},\\x'_{3}=x_{3}\cos \ i\psi +x_{4}\sin \ i\psi ,&&x'_{4}=-x_{3}\sin \ i\psi +x_{4}\cos \ i\psi \end{array}}$

Putting

(2)	$-i\ tg\ i\psi ={\frac {e-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q,\ \psi ={\frac {1}{2}}\log \ nat\ {\frac {1+q}{1-q}}$

We shall have

$\cos \ i\psi ={\frac {1}{\sqrt {1-q^{2}}}},\ \sin \ i\psi ={\frac {iq}{\sqrt {1-q^{2}}}}$,

where $-1<q<1$, and ${\sqrt {1-q^{2}}}$ is always to be taken with the positive sign.

Let us now write

(3)	$x'_{1}=x',\ x'_{2}=y',\ x'_{3}=z',\ x'_{4}=it'$,

then the substitution 1) takes the form

(4)	$x'=x,\ y'=y,\ z'={\frac {z-qt}{\sqrt {1-q^{2}}}},\ t'={\frac {-qz+t}{\sqrt {1-q^{2}}}}$

the coefficients being essentially real.

If now in the above-mentioned rotation round the z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and $\varphi$ by $i\psi$, we at once perceive that simultaneously, new magnitudes $\varrho '_{1},\ \varrho '_{2},\ \varrho '_{3},\ \varrho '_{4}$, where

${\begin{array}{ccc}&\varrho '_{1}=\varrho _{1},\ \varrho '_{2}=\varrho _{2},\\\varrho '_{3}=x_{3}\cos \ i\psi +\varrho _{4}\sin \ i\psi ,&&\varrho '_{4}=-\varrho _{3}\sin \ i\psi +\varrho _{4}\cos \ i\psi \end{array}}$

and $f'_{12},\dots f'_{34}$, where

$f'_{41}=f_{41}\cos \ i\psi +f_{13}\sin \ i\psi ,\ f'_{13}=-f_{41}\sin \ i\psi +f_{13}\cos \ i\psi ,\ f'_{34}=f_{34}$, $f'_{32}=f_{32}\cos \ i\psi +f_{42}\sin \ i\psi ,\ f'_{42}=-f_{32}\sin \ i\psi +f_{42}\cos \ i\psi ,\ f'_{12}=f_{12}$, $f'_{kh}=-f'_{hk}\qquad (h,k=1,2,3,4)$,

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A'), and (B'), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are taken, and x', y', z', t' be taken as a new frame of reference, then we shall have

(5)

$\varrho '=\varrho \left({\frac {-q{\mathfrak {w}}_{z}+1}{\sqrt {1-q^{2}}}}\right),\ \varrho '{\mathfrak {w}}'_{z'}=\varrho \left({\frac {{\mathfrak {w}}_{z}-q}{\sqrt {1-q^{2}}}}\right)$, $\varrho '{\mathfrak {w}}'_{x'}=\varrho {\mathfrak {w}}_{x},\ \varrho '{\mathfrak {w}}'_{y'}=\varrho {\mathfrak {w}}_{y}$,

furthermore

(6)	$e'_{x'}={\frac {{\mathfrak {e}}_{x}-q{\mathfrak {m}}_{y}}{\sqrt {1-q^{2}}}},\ {\mathfrak {m}}'_{y'}={\frac {-q{\mathfrak {e}}_{x}+{\mathfrak {m}}_{y}}{\sqrt {1-q^{2}}}},\ {\mathfrak {e}}'_{z'}={\mathfrak {e}}_{z}$

and

(7)	${\mathfrak {m}}'_{x'}={\frac {{\mathfrak {m}}_{x}+q{\mathfrak {e}}_{y}}{\sqrt {1-q^{2}}}},\ {\mathfrak {e}}'_{y'}={\frac {q{\mathfrak {m}}_{x}+{\mathfrak {e}}_{y}}{\sqrt {1-q^{2}}}},\ {\mathfrak {m}}'_{z'}={\mathfrak {m}}_{z}$[6]

Then we have for these newly introduced vectors ${\mathfrak {w',e',m'}}$ with components ${\mathfrak {w}}'_{x},{\mathfrak {w}}'_{y},{\mathfrak {w}}'_{z};{\mathfrak {e}}'_{x},{\mathfrak {e}}'_{y},{\mathfrak {e}}'_{z}$; ${\mathfrak {m}}'_{x},{\mathfrak {m}}'_{y},{\mathfrak {m}}'_{z}$ and the quantity $\varrho '$ a series of equations I'), II'), III'), IV) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that e${\mathfrak {e}}_{x}-q{\mathfrak {m}}_{y},\ {\mathfrak {e}}_{y}+q{\mathfrak {m}}_{x},\ {\mathfrak {e}}_{z}$ are components of the vector ${\mathfrak {e}}+[{\mathfrak {vm}}]$, where ${\mathfrak {v}}$ is a vector in the direction of the positive z-axis, and $\left|{\mathfrak {v}}\right|=q$, and $[{\mathfrak {vm}}]$ is the vector product of ${\mathfrak {v}}$ and ${\mathfrak {m}}$; similarly ${\mathfrak {m}}_{x}+q{\mathfrak {e}}_{y},\ {\mathfrak {m}}_{y}-q{\mathfrak {e}}_{x},\ {\mathfrak {m}}_{z}$ are the components of the vector ${\mathfrak {m}}-[{\mathfrak {ve}}]$.

The equations 6) and 7), as they stand in pairs, can be expressed as.

${\mathfrak {e}}'_{x'}+i{\mathfrak {m}}'_{x'}=({\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x})\cos \ i\psi +({\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y})\sin \ i\psi$, ${\mathfrak {e}}'_{y'}+i{\mathfrak {m}}'_{y'}=-({\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x})\sin \ i\psi +({\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y})\cos \ i\psi$, ${\mathfrak {e}}'_{z'}+i{\mathfrak {m}}'_{z'}={\mathfrak {e}}_{z}+i{\mathfrak {m}}_{z}$

If $\varphi$ denotes any other real angle, we can form the following combinations : —

(8)	$({\mathfrak {e'}}_{x'}+i{\mathfrak {m}}'_{x'})\cos \ \varphi +({\mathfrak {e'}}_{y'}+i{\mathfrak {m}}'_{y'})\sin \ \psi$ $=({\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x})\cos \ (\varphi +i\psi )+({\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y})\sin \ (\varphi +i\psi )$,

(9)	$-({\mathfrak {e'}}_{x'}+i{\mathfrak {m}}'_{x'})\sin \ \varphi +({\mathfrak {e'}}_{y'}+i{\mathfrak {m}}'_{y'})\cos \ \varphi$ $=-({\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x})\sin \ (\varphi +i\psi )+({\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y})\cos \ (\varphi +i\psi )$

§ 4. Special Lorentz-Transformation.

The role which is played by the z-axis in the transformation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law: —

Let ${\mathfrak {v}}$ be a vector with the components ${\mathfrak {v}}_{x},\ {\mathfrak {v}}_{y},\ {\mathfrak {v}}_{z}$, and let $\left|{\mathfrak {v}}\right|=q<1$. By ${\mathfrak {\bar {v}}}$ we shall denote any vector which is perpendicular to ${\mathfrak {v}}$, and by ${\mathfrak {r_{v}}}$, ${\mathfrak {r_{\bar {v}}}}$ we shall denote components of ${\mathfrak {r}}$ in direction of ${\mathfrak {\bar {v}}}$ and $\left|{\mathfrak {v}}\right|$.

Instead of x, y, z, t, new magnetudes x,' y,' z,' t' will be introduced in the following way. If for the sake of shortness, ${\mathfrak {r}}$ is written for the vector with the components x, y, z in the first system of reference, ${\mathfrak {r}}'$ for the same vector with the components x', y', z' in the second system of reference, then for the direction of ${\mathfrak {v}}$ we have

(10)	${\mathfrak {r'_{v}}}={\frac {r_{v}-qt}{\sqrt {1-q^{2}}}}$,

and for every perpendicular direction ${\mathfrak {\bar {v}}}$

(11)	${\mathfrak {r'_{\bar {v}}}}={\mathfrak {r_{\bar {v}}}}$,

and further

(12)	$t'={\frac {-q{\mathfrak {r_{v}}}+t}{\sqrt {1-q^{2}}}}$

The notations ${\mathfrak {r'_{v}}}$ and ${\mathfrak {r'_{\bar {v}}}}$ are to be understood in the sense that with the directions ${\mathfrak {v}}$, and every direction ${\mathfrak {v}}$ perpendicular to ${\mathfrak {\bar {v}}}$ in the system x, y, z are always associated the directions with the same direction cosines in the system x', y', z' ,

A transformation which is accomplished by means of (10), (11), (12) with the condition $0<q<1$ will be called a special Lorentz-transformation. We shall call ${\mathfrak {v}}$ the vector, the direction of ${\mathfrak {v}}$ the axis, and the magnitude of ${\mathfrak {v}}$ the moment of this transformation.

If further $\varrho '$ and the vectors ${\mathfrak {w}}',\ {\mathfrak {e}}',\ {\mathfrak {m}}'$, in the system x', y', z' are so defined that,

(13)	$\varrho '={\frac {\varrho (-q{\mathfrak {w_{v}}}+1)}{\sqrt {1-q^{2}}}}$,

(14)	$\varrho '{\mathfrak {w'_{v}}}={\frac {\varrho {\mathfrak {w_{v}}}-\varrho q}{\sqrt {1-q^{2}}}},\ \varrho '{\mathfrak {w'_{\bar {v}}}}=\varrho {\mathfrak {w_{\bar {v}}}}$,

further^[7]

(15)

${\begin{array}{c}({\mathfrak {e}}'+i{\mathfrak {m}}')_{\mathfrak {\bar {v}}}={\frac {({\mathfrak {e}}+i{\mathfrak {m}}-i[{\mathfrak {w}},\ {\mathfrak {e}}+i{\mathfrak {m}}])_{\bar {v}}}{\sqrt {1-q^{2}}}}\\({\mathfrak {e}}'+i{\mathfrak {m}}')_{\mathfrak {v}}=({\mathfrak {e}}+i{\mathfrak {m}}-i[{\mathfrak {w}},\ {\mathfrak {e}}+i{\mathfrak {m}}])_{\mathfrak {v}}\end{array}}$,

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

(16)	${\mathfrak {r_{v}}}={\frac {{\mathfrak {r'_{v}}}+qt'}{\sqrt {1-q^{2}}}},\ {\mathfrak {r_{\bar {v}}}}={\mathfrak {r'_{\bar {v}}}},\ t={\frac {q{\mathfrak {r'_{v}}}+t'}{\sqrt {1-q^{2}}}}$.

Now we shall make a very important observation about the vectors ${\mathfrak {w}}$ and ${\mathfrak {w}}'$. We can again introduce the indices 1, 2, 3, 4, so that we write $x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$ instead of x,' y,' z,' it' , and $\varrho '_{1},\ \varrho '_{2},\ \varrho '_{3},\ \varrho '_{4}$ instead of $\varrho '{\mathfrak {w}}'_{x'}\ \varrho '{\mathfrak {w}}'_{y'}\ \varrho '{\mathfrak {w}}'_{z'}\ i\varrho '$. Like the rotation round the z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant +1, so that

(17)	$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2},$ d. i. $x^{2}+y^{2}+z^{2}-t^{2}$

is transformed into

$x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2},$ d. i. $x'^{2}+y'^{2}+z'^{2}-t'^{2}.$

On the basis of the equations (13), (14), we shall have

$-(\varrho _{1}^{2}+\varrho _{2}^{2}+\varrho _{3}^{2}+\varrho _{4}^{2})=\varrho ^{2}(1-{\mathfrak {w}}_{x}^{2}-{\mathfrak {w}}_{y}^{2}-{\mathfrak {w}}_{z}^{2})=\varrho ^{2}(1-{\mathfrak {w}}^{2})$

transformed into $\varrho '(1-{\mathfrak {w}}'^{2})$ or in other words,

(18)	$\varrho {\sqrt {1-{\mathfrak {w}}^{2}}}$,

is an invariant in a Lorentz-transformation.

If we divide $\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}$ by this magnitude, we obtain the four values

$w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}$,

so that

(19)	$w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}=-1$.

It is apparent that these four values, are determined by the vector ${\mathfrak {w}}$ and inversely the vector ${\mathfrak {w}}$ of magnitude $<1$ follows from the 4 values $w_{1},\ w_{2},\ w_{3},\ w_{4}$, where $w_{1},\ w_{2},\ w_{3}$ are real, $-iw_{4}$ real and positive and condition (19) is fulfilled.

The meaning of $w_{1},\ w_{2},\ w_{3},\ w_{4}$ here is, that they are the ratios of $dx_{1},\ dx_{2},\ dx_{3},\ dx_{4}$ to

(20)	${\sqrt {-(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2})}}=dt{\sqrt {1-{\mathfrak {w}}^{2}}}$

The differentials denoting the displacements of matter occupying the spacetime point $x_{1},\ x_{2},\ x_{3},\ x_{4}$ to the adjacent space-time point.

After the Lorentz-transfornation is accomplished the velocity of matter in the new system of reference for the same space-time point x', y', z', t' is the vector ${\mathfrak {w}}'$ with the ratios ${\frac {dx'}{dt'}},\ {\frac {dy'}{dt'}},\ {\frac {dz'}{dt'}}$ as components.

Now it is quite apparent that the system of values

$x_{1}=w_{1},\ x_{2}=w_{2},\ x_{3}=w_{3},\ x_{4}=w_{4}$

is transformed into the values

$x'_{1}=w'_{1},\ x'_{2}=w'_{2},\ x'_{3}=w'_{3},\ x'_{4}=w'_{4}$

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity ${\mathfrak {w}}'$ after the transformation as the first system of values has got for ${\mathfrak {w}}$ before transformation.

If in particular the vector ${\mathfrak {v}}$ of the special Lorentz-transformation be equal to the velocity vector ${\mathfrak {w}}$ of matter at the space-time point $x_{1},\ x_{2},\ x_{3},\ x_{4}$, then it follows out of (10), (11), (12) that

$w'_{1}=0,\ w'_{2}=0,\ w'_{3}=0,\ w'_{4}=i$,

Under these circumstances therefore, the corresponding space-time point has the velocity ${\mathfrak {w}}'=0$ after the transformation, it is as if we transform to rest. We may call the invariant $\varrho {\sqrt {1-{\mathfrak {w}}^{2}}}$ as the rest-density of Electricity.

§ 5. Space-time Vectors. Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

(21)

${\begin{array}{c}x_{1}=\alpha _{11}x'_{1}+\alpha _{12}x'_{2}+\alpha _{13}x'_{3}+\alpha _{14}x'_{4},\\\\x_{2}=\alpha _{21}x'_{1}+\alpha _{22}x'_{2}+\alpha _{23}x'_{3}+\alpha _{24}x'_{4},\\\\x_{3}=\alpha _{31}x'_{1}+\alpha _{32}x'_{2}+\alpha _{33}x'_{3}+\alpha _{34}x'_{4},\\\\x_{4}=\alpha _{41}x'_{1}+\alpha _{42}x'_{2}+\alpha _{43}x'_{3}+\alpha _{44}x'_{4}\end{array}}$

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while $\alpha _{14},\ \alpha _{24},\ \alpha _{34}$, are purely imaginary, but $\alpha _{44}$ is real and $>0$, and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ transforms into $x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2}$. The operation shall be called a general Lorentz transformation.

If we put $x'_{1}=x',\ x'_{2}=y',\ x'_{3}=z',\ x'_{4}=it'$ then immediately there occurs a homogeneous linear transformation of x, y, z, t in x', y', z', t' with essentially real co-efficients, whereby the aggregrate $-x^{2}-y^{2}-z^{2}+t^{2}$ transforms into $-x'^{2}-y'^{2}-z'^{2}+t'^{2}$, and to every such system of values x, y, z, t with a positive t, for which this aggregate $>0$, there always corresponds a positive t'; this last is quite evident from the continuity of the aggregate x, y, z, t.

The last vertical column of co-efficients has to fulfill, the condition

(22)	$\alpha _{14}^{2}+\alpha _{24}^{2}+\alpha _{34}^{2}+\alpha _{44}^{2}=1$

If $\alpha _{14}=0,\ \alpha _{24}=0,\ \alpha _{34}=0$ then $\alpha _{44}=1$, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If $\alpha _{14},\ \alpha _{24},\ \alpha _{34}$ are not all zero, and if we put

$\alpha _{14}:\alpha _{24}:\alpha _{34}:\alpha _{44}={\mathfrak {v}}_{x}:{\mathfrak {v}}_{y}:{\mathfrak {v}}_{z}:i$,

$q={\sqrt {{\mathfrak {v}}_{x}^{2}+{\mathfrak {v}}_{y}^{2}+{\mathfrak {v}}_{z}^{2}}}<1$.

On the other hand, with every set of value of $\alpha _{14},\ \alpha _{24},\ \alpha _{34},\ \alpha _{44}$ which in this way fulfill the condition 22) with real values of ${\mathfrak {v}}_{x}+{\mathfrak {v}}_{y}+{\mathfrak {v}}_{z}$, we can construct the special Lorentz-transformation (16) with $\alpha _{14},\ \alpha _{24},\ \alpha _{34},\ \alpha _{44}$ as the last vertical column, — and then every Lorentz-transformation with the same last vertical column $\alpha _{14},\ \alpha _{24},\ \alpha _{34},\ \alpha _{44}$ supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a group.

Under a space-time vector of the 1st kind shall be understood a system of four magnitudes $\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}$ with the condition that in case of a Lorentz-transformation it is to be replaced by the set $\varrho '_{1},\ \varrho '_{2},\ \varrho '_{3},\ \varrho '_{4}$, where these are the values $x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$ obtained by substituting $\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}$ for $x_{1},\ x_{2},\ x_{3},\ x_{4}$ in the expression (21).

Besides the time-space vector of the 1st kind $x_{1},\ x_{2},\ x_{3},\ x_{4}$ we shall also make use of another spacetime vector of the first kind $u_{1},\ u_{2},\ u_{3},\ u_{4}$, and let us form the linear combination

(23)	${\begin{array}{c}f_{23}(x_{2}u_{3}-x_{3}u_{2})+f_{31}(x_{3}u_{1}-x_{1}u_{3})+f_{12}(x_{1}u_{2}-x_{2}u_{1})\\\\+f_{14}(x_{1}u_{4}-x_{4}u_{1})+f_{24}(x_{2}u_{4}-x_{4}u_{2})+f_{34}(x_{3}u_{4}-x_{4}u_{3})\end{array}}$

with six coefficients $f_{23},\dots f_{34}$. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors

$x_{1},\ x_{2},\ x_{3};u_{1},\ u_{2},\ u_{3};f_{23},\ f_{31},\ f_{12};f_{14},\ f_{24},\ f_{34}$

and the constants $x_{4}$ and $u_{4}$ at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject $x_{1},\ x_{2},\ x_{3},\ x_{4}$ and $u_{1},\ u_{2},\ u_{3},\ u_{4}$ simultaneously to the Lorentz transformation (21), the combination (23) is changed to.

(24)	${\begin{array}{c}f'_{23}(x'_{2}u'_{3}-x'_{3}u'_{2})+f'_{31}(x'_{3}u'_{1}-x'_{1}u'_{3})+f'_{12}(x'_{1}u'_{2}-x'_{2}u'_{1})\\\\+f'_{14}(x'_{1}u'_{4}-x'_{4}u'_{1})+f'_{24}(x'_{2}u'_{4}-x'_{4}u'_{2})+f'_{34}(x'_{3}u'_{4}-x'_{4}u'_{3})\end{array}}$

where the coefficients $f'_{23},\dots f'_{34}$ depend solely on $f_{23},\dots f_{34}$ and the coefficients $\alpha _{11},\ \alpha _{12},\dots \alpha _{44}$.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes $f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$ with the condition that when subjected to a Lorentz transformation, it is changed to a new system $f'_{23},\ f'_{31},\ f'_{12},\ f'_{14},\ f'_{24},\ f'_{34}$ which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I) — (IV), — which are the fundamental equations for Æther.

If x, y, z, it (space co-ordinates, and time it) is subjected to a Lorentz transformation, and at the same time $\varrho {\mathfrak {w}}_{x},\ \varrho {\mathfrak {w}}_{y},\ \varrho {\mathfrak {w}}_{z},\ i\varrho$ (convection-current, and charge density × i) is transformed as a space time vector of the 1st kind, further ${\mathfrak {m}}_{x},\ {\mathfrak {m}}_{y},\ {\mathfrak {m}}_{z},\ -i{\mathfrak {e}}_{x}\ -i{\mathfrak {e}}_{y},\ -i{\mathfrak {e}}_{z}$ (magnetic force, and electric induction × i) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced info the system.

These facts can be more concisely expressed in these words: the system of equations (I, and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where $\varrho {\mathfrak {w}},\ i\varrho$ is to be transformed as a space time vector of the 1st kind, ${\mathfrak {m}},\ -i{\mathfrak {e}}$ is to be treated as a vector of the 2nd kind, or more significantly, —

$\varrho {\mathfrak {w}},\ i\varrho$ is a space time vector of the 1st kind, ${\mathfrak {m}},\ -i{\mathfrak {e}}$ is a space-time vector of the 2nd kind.

I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.

(25)	${\mathfrak {m}}^{2}-{\mathfrak {e}}^{2}=f_{23}^{2}+f_{31}^{2}+f_{12}^{2}+f_{14}^{2}+f_{24}^{2}+f_{34}^{2}$,

(26)	${\mathfrak {me}}=i(f_{23}f_{14}+f_{31}f_{24}+f_{12}f_{34})$,

A space-time vector of the second kind ${\mathfrak {m}},\ -i{\mathfrak {e}}$, where ${\mathfrak {m}}$ and ${\mathfrak {e}}$ are real magnitudes, may be called singular, when the scalar square $({\mathfrak {m}}-i{\mathfrak {e}})^{2}=0$, i.e. ${\mathfrak {m}}^{2}-{\mathfrak {e}}^{2}=0$, and at the same time $({\mathfrak {me}})=0$, i.e. the vector ${\mathfrak {m}}$ and ${\mathfrak {e}}$ are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every Lorentz-transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product $[{\mathfrak {me}}]$ coincides with the z-axis, i.e. ${\mathfrak {m}}_{z}=0,{\mathfrak {e}}_{z}=0$. Then

${\mathfrak {m}}_{x}-i{\mathfrak {e}}_{x})^{2}+({\mathfrak {m}}_{y}-i{\mathfrak {e}}_{y})^{2}\neq 0$

Therefore ${\frac {{\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y}}{{\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x}}}$ is different from $\pm i$, and we can therefore define a complex argument $\varphi +i\psi$ in such a manner that

$tg(\varphi +i\psi )={\frac {{\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y}}{{\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x}}}$

If then, by referring back to equations (9), we carry out the transformation (1) through the angle $\psi$ and a subsequent rotation round the z-axis through the angle $\varphi$, we perform a Lorentz-transformation at the end of which ${\mathfrak {m}}_{y}=0,\ {\mathfrak {e}}_{y}=0$, and therefore ${\mathfrak {m}}$ and ${\mathfrak {e}}$ shall both coincide with the new x-axis. Then by means of the invariants ${\mathfrak {m}}^{2}+{\mathfrak {e}}^{2}$ and $({\mathfrak {me}})$ the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ 6. Concept of Time.

By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.^[8] In fact, the following considerations will prove conclusive.

Let a reference system x, y, z, t for space time points (events) be somehow known. Now if a space point $A(x_{0},\ y_{0},\ z_{0})$ at the time $t_{0}=0$ be compared with a space point P(x, y, z) at the time t and if the difference of time $t-t_{0}$, (let $t>t_{0}$) be less than the length AP, i.e. less than the time required for the propagation of light from A to P, and if $q={\frac {t-t_{0}}{AP}}<1$, then by a special Lorentz transformation, in which AP is taken as the axis, and which has the moment q, we can introduce a time parameter t which (see equation 11, 12, § 4) has got the same value $t'=0$ for both space-time points $A,t_{0}$, and P, t. So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time $t_{0}=0$, two different space-points A, B, or three space-points A, B, C which are not in the same space-line, and compare therewith a space point P, which is outside the line AB, or the plane ABC at another time t, and let the time difference $t-t_{0}$ be less than the time which light requires for propagation from the line AB, or the plane ABC to P. Let q be the ratio of $t-t_{0}$ by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on AB, or from P on the plane ABC is the axis, and q is the moment, then all the three (or four) events $A,\ t_{0};\ B,\ t_{0};\ (C,\ t_{0})$ and P, t are simultaneous.

If four space-points, which do not lie in one plane are conceived to be at the same time to, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.

PART II. ELECTRO-MAGNETIC PHENOMENA.

§ 7. Fundamental Equations for bodies at rest.

After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case $\epsilon =1,\ \mu =1,\ \sigma =1$, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us — when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every spacetime point as functions of x, y, z, t: — the vector of the electric force ${\mathfrak {E}}$, the magnetic induction ${\mathfrak {M}}$, the electrical induction ${\mathfrak {e}}$, the magnetic force ${\mathfrak {m}}$, the electrical space-density $\varrho$, the electric current ${\mathfrak {s}}$ (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector ${\mathfrak {w}}$, the velocity of matter.

The relations in question can be divided into two classes.

Firstly — those equations, which, — when ${\mathfrak {w}}$, the velocity of matter is given as a function of x, y, z, t, — lead us to a knowledge of other magnitude as functions of x, y, z, t — I shall call this first class of equations the fundamental equations —

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector ${\mathfrak {w}}$ as functions of x, y, z, t.

For the case of bodies at rest, i.e. when ${\mathfrak {w}}(x,\ y,\ z,\ t)=0$, the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are ; —

(1) The Differential Equations: — which contain no constant referring to matter: —

${\begin{array}{rcrl}(I)&\qquad &curl\ {\mathfrak {m}}-{\frac {\partial e}{\partial t}}&={\mathfrak {s}},\\\\(II)&&div\ {\mathfrak {e}}&=\varrho ,\\\\(III)&&curl\ {\mathfrak {E}}+{\frac {\partial {\mathfrak {M}}}{\partial t}}&=0,\\\\(IV)&&div\ {\mathfrak {M}}&=0\end{array}}$;

(2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves, i.e. for isotopic bodies; — they are comprised in the equations

(V)	${\mathfrak {e}}=\epsilon {\mathfrak {E}},\ {\mathfrak {M}}=\mu {\mathfrak {m}},\ {\mathfrak {s}}=\sigma {\mathfrak {E}}$,

where $\epsilon$ = dielectric constant, $\mu$ = magnetic permeability, $\sigma$ = the conductivity of matter, all given as function of x, y, z, t. ${\mathfrak {s}}$ is here the conduction current.

By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,

$x_{1}=x,\ x_{2}=y,\ x_{3}=z,\ x_{4}=it$

and write $s_{1},\ s_{2},\ s_{3},\ s_{4}$ for ${\mathfrak {s}}_{x},\ {\mathfrak {s}}_{y},\ {\mathfrak {s}}_{z},\ i\varrho$,

further $f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$

for ${\mathfrak {m}}_{x},\ {\mathfrak {m}}_{y},\ {\mathfrak {m}}_{z},\ -i{\mathfrak {e}}_{x},\ -i{\mathfrak {e}}_{y},\ -i{\mathfrak {e}}_{z}$,

and $F_{23},\ F_{31},\ F_{12},\ F_{14},\ F_{24},\ F_{34}$

for ${\mathfrak {M}}_{x},\ {\mathfrak {M}}_{y},\ {\mathfrak {M}}_{z},\ -i{\mathfrak {E}}_{x},\ i{\mathfrak {E}}_{y},\ i{\mathfrak {E}}_{z}$;

lastly we shall have the relation $f_{kh}=-f_{hk},\ F_{kh}=-F_{hk}$, (the letter f, F shall denote the field, s the (i.e. current).

Then the fundamental Equations can be written as

(A)

${\begin{array}{ccccccccc}&&{\frac {\partial f_{12}}{\partial x_{2}}}&+&{\frac {\partial f_{13}}{\partial x_{3}}}&+&{\frac {\partial f_{14}}{\partial x_{4}}}&=&s_{1},\\\\{\frac {\partial f_{21}}{\partial x_{1}}}&&&+&{\frac {\partial f_{23}}{\partial x_{3}}}&+&{\frac {\partial f_{24}}{\partial x_{4}}}&=&s_{2},\\\\{\frac {\partial f_{31}}{\partial x_{1}}}&+&{\frac {\partial f_{32}}{\partial x_{2}}}&&&+&{\frac {\partial f_{34}}{\partial x_{4}}}&=&s_{3},\\\\{\frac {\partial f_{41}}{\partial x_{1}}}&+&{\frac {\partial f_{42}}{\partial x_{2}}}&+&{\frac {\partial f_{43}}{\partial x_{3}}}&&&=&s_{4}.\end{array}}$

and the equations (III) and (IV), are

(B)

${\begin{array}{ccccccccc}&&{\frac {\partial F_{34}}{\partial x_{2}}}&+&{\frac {\partial F_{42}}{\partial x_{3}}}&+&{\frac {\partial F_{23}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial F_{43}}{\partial x_{1}}}&&&+&{\frac {\partial F_{14}}{\partial x_{3}}}&+&{\frac {\partial F_{31}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial F_{24}}{\partial x_{1}}}&+&{\frac {\partial F_{41}}{\partial x_{2}}}&&&+&{\frac {\partial F_{12}}{\partial x_{4}}}&=&0,\\\\{\frac {\partial F_{32}}{\partial x_{1}}}&+&{\frac {\partial F_{13}}{\partial x_{2}}}&+&{\frac {\partial F_{21}}{\partial x_{3}}}&&&=&0.\end{array}}$

§ 8. The Fundamental Equations for Moving Bodies.

We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.

The first Axion shall be, —

When a detached region of matter is at rest at any moment, therefore the vector ${\mathfrak {w}}$ is zero, for a system x, y, z, t — the neighbourhood may be supposed to be in motion in any possible manner, then for the spacetime point x, y, z, t the same relations (A) (B) (V) which hold in the case when all matter is at rest, snail also hold between $\varrho$, the vectors ${\mathfrak {s,e,m,E,M}}$ and their differentials with respect to x, y, z, t.

The second axiom shall be : —

Every velocity of matter is < 1, smaller than the velocity of propagation of light.

The third axiom shall be : —

The fundamental equations are of such a kind that when x, y, z, it are subjected to a Lorentz transformation and thereby ${\mathfrak {m}},\ -i{\mathfrak {e}}$ and ${\mathfrak {M}},\ -i{\mathfrak {E}}$ are transformed into space-time vectors of the second kind, ${\mathfrak {s}},\ i\varrho$ as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.

Shortly I can signify the third axiom as ; —

${\mathfrak {m}},\ -i{\mathfrak {e}}$ and ${\mathfrak {M}},\ -i{\mathfrak {E}}$ are space-time vectors of the second kind, ${\mathfrak {s}},\ i\varrho$ is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.

In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.

According to the second axiom, the magnitude of the velocity vector $\left|{\mathfrak {w}}\right|$ is < 1 at any space-time point. In consequence, we can always write, instead of the vector ${\mathfrak {w}}$, the following set of four allied quantities

$w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}$,

with the relation

(27)	$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=-1$

From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space-time vector of the 1st kind, and we want to call it space-time vector velocity.

Let us now fix our attention on a certain point x, y, z of matter at a certain time t. If at this space-time point ${\mathfrak {w}}=0$, then we have at once for this point the equations (A), (B) (V) of § 7. If ${\mathfrak {w}}\neq 0$, then there exists according to 16), in case $\left|{\mathfrak {w}}\right|<1$, a special Lorentz-transformation, whose vector ${\mathfrak {v}}$ is equal to this vector ${\mathfrak {w}}(x,\ y,\ z,\ t)$ and we pass on to a new system of reference x', y', z', t' in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values

(28)	$w'_{0}=0,\ w'_{2}=0,\ w'_{3}=0,\ w'_{4}=i$,

therefore the new velocity vector ${\mathfrak {w}}'=0$, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point x, y, z, t involves the newly introduced magnitude ${\mathfrak {w}}',\varrho ',{\mathfrak {s',e',m',E',M'}}$ and their differential quotients with respect to x', y', z, t' in the same manner as the original equations for the point x, y, z, t. But according to the first axiom, when ${\mathfrak {w}}'=0$ these equations must be exactly equivalent to

(1) the differential equations (A'), (B'), which are obtained from the equations (A), (B) by simply dashing the symbols in (A) and (B).

(2) and the equations

(V')	${\mathfrak {e}}'=\epsilon {\mathfrak {E}}',\ {\mathfrak {M}}'=\mu {\mathfrak {m}}',\ {\mathfrak {s}}'=\sigma {\mathfrak {E}}'$,

where $\epsilon ,\ \mu ,\ \sigma$ are the dielectric constant, magnetic permeability, and conductivity for the system x', y', z', t', i.e. in the space-time point x, y, z, t of matter.

Now let us return, by means of the reciprocal Lorentz-transformation to the original variables x, y, z, t, and the magnitudes ${\mathfrak {w}},\varrho ,{\mathfrak {s,e,m,E,M}}$ and the equations, which we then obtain from the last mentioned, will be the fundamentil equations sought by us for the moving bodies.

Now from § 4, and § 6, it is to be seen that the equations A), as well as the equations B) are covariant for a Lorentz-transformation, i.e. the equations, which we obtain backwards from A') B'), must be exactly of the same form as the equations A) and B), as we take them for bodies at rest. We have therefore as the first result: —

The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in $\varrho$ and the vectors ${\mathfrak {s,\ e,\ m,\ E,\ M}}$ are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have

(I)	${\begin{array}{rcrl}(I)&\qquad &curl\ {\mathfrak {m}}-{\frac {\partial e}{\partial t}}&={\mathfrak {s}},\\\\(II)&&div\ {\mathfrak {e}}&=\varrho ,\\\\(III)&&curl\ {\mathfrak {E}}+{\frac {\partial {\mathfrak {M}}}{\partial t}}&=0,\\\\(IV)&&div\ {\mathfrak {M}}&=0\end{array}}$,

The velocity of matter occurs only in the auxilliary equations which characterise the influence of matter on the basis of their characteristic constants $\epsilon ,\ \mu ,\ \sigma$. Let us now transform these auxilliary equations x, y, z into the original co-ordinates x, y, z, and t.)

According to formula 15) in § 4, the component of ${\mathfrak {e}}'$ in the direction of the vector ${\mathfrak {w}}$ is the same us that of ${\mathfrak {e}}+[{\mathfrak {wm}}]$, the component of ${\mathfrak {m}}'$ is the same as that of ${\mathfrak {m}}-[{\mathfrak {we}}]$, but for the perpendicular direction ${\mathfrak {\bar {w}}}$, the components of ${\mathfrak {e}}'$ and ${\mathfrak {m}}'$ are the same as those of ${\mathfrak {e}}+[{\mathfrak {wm}}]$ and ${\mathfrak {m}}-[{\mathfrak {we}}]$, multiplied by ${\frac {1}{\sqrt {1-{\mathfrak {w}}^{2}}}}$. On the other hand ${\mathfrak {E}}'$ and ${\mathfrak {M}}'$ shall stand to ${\mathfrak {E}}+[{\mathfrak {wM}}]$, and ${\mathfrak {M}}-[{\mathfrak {wE}}]$ in the same relation us ${\mathfrak {e}}'$ and ${\mathfrak {m}}'$ to ${\mathfrak {e}}+[{\mathfrak {wm}}]$ and ${\mathfrak {m}}+[{\mathfrak {we}}]$. From the relation ${\mathfrak {e}}'=\epsilon {\mathfrak {E}}'$, the following equations follow

(C)	${\mathfrak {e}}+[{\mathfrak {wm}}]=\epsilon ({\mathfrak {E}}+[{\mathfrak {wM}}])$.

and from the relation ${\mathfrak {M}}'=\mu {\mathfrak {m}}'$ we have

(D)	${\mathfrak {M}}-[{\mathfrak {wE}}]=\mu ({\mathfrak {m}}-[{\mathfrak {we}}])$

For the components in the directions perpendicular to ${\mathfrak {w}}$, and to each other, the equations are to be multiplied by ${\sqrt {1-{\mathfrak {w}}^{2}}}$.

Then the following equations follow from the transfermation equations (12), 10), (11) in § 4, when we replace $q,\ {\mathfrak {r_{v},\ r_{\bar {v}}}},t,{\mathfrak {r'_{v},\ r'_{\bar {v}}}},t'$ by $\left|{\mathfrak {w}}\right|,\ {\mathfrak {s_{w},s_{\bar {w}}}},\varrho ,{\mathfrak {s'_{w},s'_{\bar {w}}}},\varrho '$.

$\varrho '={\frac {-\left|{\mathfrak {w}}\right|{\mathfrak {s_{w}}}+\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}},\ s'_{w}={\frac {s_{w}-\left|{\mathfrak {w}}\right|\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}},\ {\mathfrak {s'_{\bar {w}}}}={\mathfrak {s}}_{\bar {w}}$,

(E)

${\begin{array}{c}{\frac {{\mathfrak {s_{w}}}-\left|{\mathfrak {w}}\right|\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}}=\sigma ({\mathfrak {E}}+[{\mathfrak {wM}}])_{\mathfrak {w}},\\\\{\mathfrak {s_{\bar {w}}}}={\frac {\sigma ({\mathfrak {E}}+[{\mathfrak {wM}}])_{\mathfrak {\bar {w}}}}{\sqrt {1-{\mathfrak {w}}^{2}}}}\end{array}}$

In consideration of the manner in which $\sigma$ enters into these relations, it will be convenient to call the vector ${\mathfrak {s}}-\varrho {\mathfrak {w}}$ with the components ${\mathfrak {s_{w}}}-\varrho {\mathfrak {\left|w\right|}}$ in the direction of ${\mathfrak {w}}$ and ${\mathfrak {s_{\bar {w}}}}$ in the directions ${\mathfrak {w}}$ perpendicular to ${\mathfrak {\bar {w}}}$ the Convection current. This last vanishes for $\sigma =0$.

We remark that for $\epsilon =1,\ \mu =1$ the equations ${\mathfrak {e'=E',\ m'=M'}}$ immediately lead to the equations ${\mathfrak {e=E,\ m=M}}$ by means of a reciprocal Lorentz-transformation with $-{\mathfrak {w}}$ as vector; and for $\sigma =0$, the equation ${\mathfrak {s}}'=0$ leads to ${\mathfrak {s}}=\varrho {\mathfrak {w}}$, that the "fundamental equations of Æther" discussed in § 2 becomes in fact the limitting case of the equations obtained here with $\epsilon =1,\ \mu =1,\ \sigma =0$.

§ 9. The Fundamental Equations in Lorentz's Theory.

Let us now see how far the fundamental equations assumed by Lorentz correspond to the Relativity postulate, as defined in §8. In the article on Electron-theory (Ency, Math., Wiss., Bd. V. 2, Art 14) Lorentz has given the fundamental equations for any possible, even magnetised bodies (see there page 209, Eq. XXX', formula (14) on page 78 of the same (part).

${\begin{array}{rcrl}(IIIa'')&\qquad &curl\ ({\mathfrak {H}}-[{\mathfrak {wE}}]&={\mathfrak {F}}+{\frac {\partial {\mathfrak {D}}}{\partial t}}+{\mathfrak {w}}\ div\ {\mathfrak {D}}-curl[{\mathfrak {wD}}],\\\\(I'')&&div\ {\mathfrak {D}}&=\varrho ,\\\\(IV'')&&curl\ {\mathfrak {E}}&=-{\frac {\partial {\mathfrak {B}}}{\partial t}},\\\\(V'')&&div\ {\mathfrak {B}}&=0.\end{array}}$

Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) $\mu =1$, ${\mathfrak {B}}={\mathfrak {H}}$, and in addition to that takes account of the occurrence of the di-electric constant $\epsilon$, and conductivity $\sigma$ according to equations

(Eq. XXXIV"', p. 227) (Eq. XXXIII", p. 223)

${\begin{array}{rl}{\mathfrak {D}}-{\mathfrak {E}}&=\left(\epsilon -1\right)\left({\mathfrak {E}}+[{\mathfrak {wB}}]\right)\\\\{\mathfrak {F}}&=\sigma ({\mathfrak {E}}+[{\mathfrak {wB}}])\end{array}}$

Lorentz's ${\mathfrak {E,B,D,H}}$ are here denoted by ${\mathfrak {E,M,e,m}}$ while ${\mathfrak {F}}$ denotes the conduction current.

The three last equations which have been just cited here coincide with eq. (II), (III), (IV), the first equation would be, if ${\mathfrak {F}}$ is identified with ${\mathfrak {s}}-{\mathfrak {w}}\sigma$ (the current being zero for $\sigma =0$),

(29)	$curl\ ({\mathfrak {H}}-[{\mathfrak {wE}}])={\mathfrak {s}}+{\frac {\partial {\mathfrak {D}}}{\partial t}}-curl[{\mathfrak {wD}}]$

and thus comes out to in in a different form than (1) here. Therefore for magnetised bodies, Lorentz's equations do not correspond to the Relativity Principle.

On the other hand, the form corresponding to the relativity principle, for the condition of non-magnetisation is to be taken out of (D) in §8, with $\mu =1$, not as ${\mathfrak {B}}={\mathfrak {H}}$, as Lorentz takes, but as

(30)	${\mathfrak {B}}-[{\mathfrak {wE}}]={\mathfrak {H}}-[{\mathfrak {wD}}]$ (hier ${\mathfrak {M}}-[{\mathfrak {wE}}]={\mathfrak {m}}-[{\mathfrak {we}}]$)

Now by putting ${\mathfrak {H}}={\mathfrak {B}}$, the differential equation (29) is transformed into the same form as eq. (1) here when ${\mathfrak {m}}-[{\mathfrak {we}}]={\mathfrak {M}}-[{\mathfrak {wE}}]$. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for moving non-magnetised bodies at last agree with the relativity postulate.

If we make use of (30) for non-magnetic bodies, and put accordingly ${\mathfrak {H}}={\mathfrak {B}}+[{\mathfrak {w}},\ {\mathfrak {D}}-{\mathfrak {E}}]$, then in consequence of (C) in §8,

$(\epsilon -1)({\mathfrak {E}}+({\mathfrak {wB}}])={\mathfrak {D}}-{\mathfrak {E}}+({\mathfrak {w}}[{\mathfrak {w}},\ {\mathfrak {D}}-{\mathfrak {E}}])$

,

i.e. for the direction of ${\mathfrak {w}}$

$(\epsilon -1)({\mathfrak {E}}+({\mathfrak {wB}}])_{\mathfrak {w}}=({\mathfrak {D}}-{\mathfrak {E}})_{\mathfrak {w}}$,

and for a perpendicular direction ${\mathfrak {\bar {w}}}$,

$(\epsilon -1)({\mathfrak {E}}+({\mathfrak {wB}}])_{\mathfrak {\bar {w}}}=(1-{\mathfrak {w}}^{2})({\mathfrak {D}}-{\mathfrak {E}})_{\mathfrak {\bar {w}}}$,

i.e. it coincides with Lorentz's assumption, if we neglect ${\mathfrak {w}}^{2}$ in comparison to 1.

Also to the same order of approximation, Lorentz's form for ${\mathfrak {F}}$ corresponds to the conditions imposed by the relativity principle [comp. (E) § 8] — that the components of ${\mathfrak {F_{w}}}$, ${\mathfrak {F_{\bar {w}}}}$ are equal to the components of $\sigma ({\mathfrak {E}}+({\mathfrak {wB}}])$ multiplied by ${\sqrt {1-{\mathfrak {w}}^{2}}}$ or ${\frac {1}{\sqrt {1-{\mathfrak {w}}^{2}}}}$ respectively.

§ 10. Fundamental Equations of E. Cohn.

E. Cohn^[9] assumes the following fundamental equations

(31)	${\begin{array}{c}curl\ (M+[{\mathfrak {wE}}])={\frac {\partial {\mathfrak {E}}}{\partial t}}+{\mathfrak {w}}\ div\ {\mathfrak {E}}+{\mathfrak {F}},\\\\-curl\ ({\mathfrak {E}}-[{\mathfrak {wM}}])={\frac {\partial {\mathfrak {M}}}{\partial t}}+{\mathfrak {w}}\ div\ {\mathfrak {M}},\end{array}}$

(32)	${\mathfrak {F}}=\sigma E,\ {\mathfrak {E}}=\epsilon E-[{\mathfrak {w}}M],\ {\mathfrak {M}}=\mu M+[{\mathfrak {w}}E]$,

where E, M are the electric and magnetic field intensities (forces), ${\mathfrak {E,M}}$ are the electric and magnetic polarisation (induction). The equations also permit the existence of true magnetism; if we do not take into account this consideration, $div\ {\mathfrak {M}}$ is to be put = 0.

An objection to this system of equations, is that according to these, for $\epsilon =1,\ \mu =1$, the vectors force and induction do not coincide. If in the equations, we conceive E and M and not $E-[{\mathfrak {wM}}]$ and $M+[{\mathfrak {wE}}]$ as electric and magnetic forces, and with a glance to this we substitute for ${\mathfrak {E,M}},E,M,div\ {\mathfrak {E}}$ the symbols ${\mathfrak {e,M,\ E}}+[{\mathfrak {wM}}],\ {\mathfrak {m}}-[{\mathfrak {we}}],\ \varrho$, then the differential equations transform to our equations, and the conditions (32) transform into

${\begin{array}{rl}{\mathfrak {F}}&=\sigma ({\mathfrak {E}}+[{\mathfrak {wM}}]),\\\\{\mathfrak {e}}+[{\mathfrak {w,m}}-[{\mathfrak {we}}]]&=\epsilon ({\mathfrak {E}}+[{\mathfrak {wM}}]),\\\\{\mathfrak {M}}-[{\mathfrak {w,\ E}}+[{\mathfrak {wM}}]]&=\mu ({\mathfrak {m}}-[{\mathfrak {we}}])\end{array}}$;

then in fact the equations of Cohn become the same as those required by the relativity principle, if errors of the order ${\mathfrak {w}}^{2}$ are neglected in comparison to 1.

It may be mentioned here that the equations of Hertz become the same as those of Cohn, if the auxilliary conditions are

(33)	${\mathfrak {E}}=\epsilon E,\ {\mathfrak {M}}=\mu M,\ {\mathfrak {F}}=\sigma E$;

§ 11. Typical Representations of the fundamental equations.

In the statement of the fundamental equations, our leading idea had been that they should retain a covariance of form, when subjected to a group of Lorentz-transformations. Now we have to deal with ponderomotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations, satisfying the conditions of covariance. In order to arrive at such forms, I shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a Lorentz-transformation. Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below.

1°. A system of magnitudes $a_{hk}$, formed into the matrix

$\left|{\begin{array}{ccc}a_{11},&\dots &a_{1q}\\\vdots &&\vdots \\a_{p1},&\dots &a_{pq}\end{array}}\right|$

arranged in p horizontal rows, and q vertical columns is called a $p\times q$ series-matrix,^[10] and will be denoted by the letter A.

If all the quantities $a_{hk}$ are multiplied by c, the resulting matrix will be denoted by $cA$.

If the roles of the horizontal rows and vertical columns be intercharged, we obtain a $q\times p$ series matrix, which will be known as the transposed matrix of A, and will be denoted by A.

${\bar {A}}=\left|{\begin{array}{ccc}a_{11},&\dots &a_{q1}\\\vdots &&\vdots \\a_{1p},&\dots &a_{pq}\end{array}}\right|$.

If we have a second $p\times q$ series matrix B.

$B=\left|{\begin{array}{ccc}b_{11},&\dots &b_{1q}\\\vdots &&\vdots \\b_{p1},&\dots &b_{pq}\end{array}}\right|$,

then A+B shall denote the $p\times q$ series matrix whose members are $a_{hk}+b_{hk}$.

2° If we have two matrices

$A=\left|{\begin{array}{ccc}a_{11},&\dots &a_{1q}\\\vdots &&\vdots \\a_{p1},&\dots &a_{pq}\end{array}}\right|,\ B=\left|{\begin{array}{ccc}b_{11},&\dots &b_{1r}\\\vdots &&\vdots \\b_{p1},&\dots &b_{qr}\end{array}}\right|$

where the number of horizontal rows of B, is equal to the number of vertical columns of A, then by AB, the product of the matrices A and B, will be denoted the matrix

$C=\left|{\begin{array}{ccc}c_{11},&\dots &c_{1r}\\\vdots &&\vdots \\c_{p1},&\dots &c_{pr}\end{array}}\right|$

where

$c_{hk}=a_{h1}b_{1k}+a_{h2}b_{2k}+\dots +a_{hq}b_{qk}\quad \left({h=1,2,\dots p \atop k=1,2,\dots r}\right)$

these elements being formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law $(AB)S=A(BS)$ holds, where S is a third matrix which has got as many horizontal rows as B (or AB) has got vertical columns.

For the transposed matrix of $C=AB$, we have ${\bar {C}}={\bar {B}}{\bar {A}}$.

3°. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows.

As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 ✕ 4 series) with the elements.

(34)	$\left|{\begin{array}{cccc}e_{11},&e_{12},&e_{13},&e_{14}\\e_{21},&e_{22},&e_{23},&e_{24}\\e_{31},&e_{32},&e_{33},&e_{34}\\e_{41},&e_{42},&e_{43},&e_{44}\end{array}}\right|=\left|{\begin{array}{cccc}1,&0,&0,&0\\0,&1,&0,&0\\0,&0,&1,&0\\0,&0,&0,&1\end{array}}\right|$

For a 4✕4 series-matrix, Det A shall denote the determinant formed of the 4✕4 elements of the matrix. If $DetA\neq 0$, then corresponding to A there is a reciprocal matrix, which we may denote by $A^{-1}$ so that $A^{-1}A=1$

A matrix

$f=\left|{\begin{array}{cccc}0,&f_{12},&f_{13},&f_{14}\\f_{21},&0,&f_{23},&f_{24}\\f_{31},&f_{32},&0,&f_{34}\\f_{41},&f_{42},&f_{43},&0\end{array}}\right|$,

in which the elements fulfill the relation $f_{kh}=-f_{hk}$, is called an alternating matrix. These relations say that the transposed matrix ${\bar {f}}=-f$. Then by $f^{*}$ will be the dual, alternating matrix

(35)	$f^{*}=\left|{\begin{array}{cccc}0,&f_{34},&f_{42},&f_{23}\\f_{43},&0,&f_{14},&f_{31}\\f_{24},&f_{41},&0,&f_{12}\\f_{32},&f_{13},&f_{21},&0\end{array}}\right|$,

Then

(36)	$f^{*}f=f_{32}f_{14}+f_{13}f_{24}+f_{21}f_{34}$,

i.e. We shall have a 4✕4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).

The determinant of f is therefore the square of the combination, by $Det^{\frac {1}{2}}f$ we shall denote the expression

(37)	$Det^{\frac {1}{2}}f=f_{32}f_{14}+f_{13}f_{24}+f_{21}f_{34}$

4°. A linear transformation

(38)	$x_{h}=\alpha _{h1}x'_{1}+\alpha _{h2}x'_{2}+\alpha _{h3}x'_{3}+\alpha _{h4}x'_{4}\qquad (h=1,2,3,4)$

which is accomplished by the matrix

${\mathsf {A}}=\left|{\begin{array}{cccc}\alpha _{11},&\alpha _{12},&\alpha _{13},&\alpha _{14}\\\alpha _{21},&\alpha _{22},&\alpha _{23},&\alpha _{24}\\\alpha _{31},&\alpha _{32},&\alpha _{33},&\alpha _{34}\\\alpha _{41},&\alpha _{42},&\alpha _{43},&\alpha _{44}\end{array}}\right|$,

will be denoted as the transformation ${\mathsf {A}}$

By the transformation ${\mathsf {A}}$, the expression

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$

is changed into the quadratic form

$\Sigma a_{hk}x'_{h}x'_{k}\qquad (h,k=1,2,3,4)$

where

$a_{hk}=\alpha _{1h}\alpha _{1k}+\alpha _{2h}\alpha _{2k}+\alpha _{3h}\alpha _{3k}+\alpha _{4h}\alpha _{4k}$

are the members of a 4✕4 series matrix which is the product of ${\mathsf {{\bar {A}}A}}$, the transposed matrix of ${\mathsf {A}}$ into ${\mathsf {A}}$. If by the transformation, the expression is changed to

$x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2}$

we must have

(39)	${\mathsf {{\bar {A}}A}}=1$

${\mathsf {A}}$ has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of ${\mathsf {A}}$ it follows out of (39) that $(Det{\mathsf {A}})^{2}=1,Det{\mathsf {A}}=\pm 1$.

From the condition (39) we obtain

(40)	${\mathsf {A}}^{-1}={\mathsf {\overline {A}}}$

i.e. the reciprocal matrix of ${\mathsf {A}}$ is equivalent to the transposed matrix of ${\mathsf {A}}$.

For ${\mathsf {A}}$ as Lorentz transformation, we have further $Det{\mathsf {A}}=+1$, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and $\alpha _{44}>0$.

5°. A space time vector of the first kind which is represented by the 1✕4 series matrix,

(41)	$s=|s_{1},\ s_{2},\ s_{3},\ s_{4}|$

is to be replaced by $s{\mathsf {A}}$ in case of a Lorentz transformation

A space-time vector of the 2nd kind with components $f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$ shall be represented by the alternating matrix

(42)	$f=\left|{\begin{array}{cccc}0,&f_{12},&f_{13},&f_{14}\\f_{21},&0,&f_{23},&f_{24}\\f_{31},&f_{32},&0,&f_{34}\\f_{41},&f_{42},&f_{43},&0\end{array}}\right|$

and is to be replaced by ${\mathsf {\overline {A}}}f{\mathsf {A}}={\mathsf {A}}^{-1}f{\mathsf {A}}$ in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression (37), we have the identity $Det^{\frac {1}{2}}({\mathsf {\overline {A}}}f{\mathsf {A}})=Det\ {\mathsf {A}}\ Det^{\frac {1}{2}}f$. Therefore $Det^{\frac {1}{2}}f$ becomes an invariant in the case of a Lorentz transformation [see eq. (26) Sec. § 5].

Looking back to (36), we have for the dual matrix

$({\mathsf {A}}^{-1}f^{*}{\mathsf {A}})({\mathsf {A}}^{-1}f{\mathsf {A}})={\mathsf {A}}^{-1}f^{*}f{\mathsf {A}}=Det^{\frac {1}{2}}f.{\mathsf {A}}^{-1}{\mathsf {A}}=Det^{\frac {1}{2}}f$,

from which it is to be seen that the dual matrix $f^{*}$ behaves exactly like the primary matrix f, and is therefore a space time vector of the II kind; $f^{*}$ is therefore known as the dual space-time vector of f with components $f_{14},\ f_{24},\ f_{34},\ f_{23},\ f_{31},\ f_{12}$.

6°.If w and s are two space-time vectors of the 1st kind then by $w{\bar {s}}$ (as well as by $s{\bar {w}})$) will be understood the combination

(43)	$w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+w_{4}s_{4}$

In case of a Lorentz transformation ${\mathsf {A}}$, since $(w{\mathsf {A}})({\mathsf {\bar {A}}}{\bar {s}})=w{\bar {s}}$ this expression is invariant. — If $w{\bar {s}}=0$, then w and s are perpendicular to each other.

Two space-time rectors of the first kind w, s gives us a 2✕4 series matrix

$\left|{\begin{array}{cccc}w_{1},&w_{2},&w_{3},&w_{4}\\s_{1},&s_{2},&s_{3},&s_{4}\end{array}}\right|$

Then it follows immediately that the system of six magnitudes

(44)	$w_{2}s_{3}-w_{3}s_{2},\ w_{3}s_{1}-w_{1}s_{3},\ w_{1}s_{2}-w_{2}s_{1},\ w_{1}s_{4}-w_{4}s_{1},\ w_{2}s_{4}-w_{4}s_{2},\ w_{3}s_{4}-w_{4}s_{3}$

behaves in case of a Lorentz-transformation as a space-time vector of the II. kind. The vector of the second kind with the components (44) are denoted by [w,s]. We see easily that $Det^{\frac {1}{2}}[w,s]=0$. The dual vector of [w,s] shall be written as [w,s]*.

If w is a space-time vector of the 1st kind, f of the second kind, wf signifies a 1✕4 series matrix. In case of a Lorentz-transformation ${\mathsf {A}}$, w is changed into $w'=w{\mathsf {A}}$, f into $f'={\mathsf {A}}^{-1}f{\mathsf {A}}$, therefore $w'f'=w{\mathsf {A}}\ {\mathsf {A}}^{-1}f{\mathsf {A}}=(wf){\mathsf {A}}$, i.e., wf is transformed as a space-time vector of the 1st kind. We can verify, when w is a space-time vector of the 1st kind, f of the 2nd kind, the important identity

(45)	$[w,wf]+[w,wf^{*}]^{*}=(w{\bar {w}})f$.

The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.

For example, for $w_{1}=0,\ w_{2}=0,\ w_{3}=0,\ w_{4}=i$

$wf=\left|if_{41},\ if_{42},\ if_{43},\ 0\right|;\ wf^{*}=\left|if_{32},\ if_{13},\ if_{21},\ 0\right|$; $[w,wf]=0,0,0,f_{41},\ f_{42},\ f_{43};\ [w,wf^{*}]=0,0,0,\ f_{32},\ f_{13},\ f_{21}$;

The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in $w_{1},\ w_{2},\ w_{3},\ w_{4}$.

---

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants $\epsilon ,\ \mu ,\ \sigma$ will be introduced.

Instead of the space vector ${\mathfrak {w}}$, the velocity of matter, we shall introduce the space-time vector of the first kind w with the components.

$w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}$

where

(46)	$w{\overline {w}}=w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}=-1$

and $-iw_{4}>0$..

By F and f shall be understood the space time vectors of the second kind ${\mathfrak {M}},\ -i{\mathfrak {E}}$, ${\mathfrak {m}},\ -i{\mathfrak {e}}$.

In $\Phi =-wF$, we have a space time vector of the first kind with components

${\begin{array}{ccccccccc}\Phi _{1}&=&&&w_{2}F_{12}&+&w_{3}F_{13}&+&w_{4}F_{14},\\\Phi _{2}&=&w_{1}F_{21}&&&+&w_{3}F_{23}&+&w_{4}F_{24},\\\Phi _{3}&=&w_{1}F_{31}&+&w_{2}F_{32}&&&+&w_{4}F_{34},\\\Phi _{4}&=&w_{1}F_{41}&+&w_{2}F_{42}&+&w_{3}F_{43}.\end{array}}$.

The first three quantities $\Phi _{1},\ \Phi _{2},\ \Phi _{3}$ are the components of the space-vector

(47)	${\frac {{\mathfrak {E}}+[{\mathfrak {wM}}]}{\sqrt {1-{\mathfrak {w}}^{2}}}}$,

and further

(48)	$\Phi _{4}={\frac {i[{\mathfrak {wE}}]}{\sqrt {1-{\mathfrak {w}}^{2}}}}$,

Because F is an alternating matrix,

(49)	$w{\overline {\Phi }}=w_{1}\Phi _{1}+w_{2}\Phi _{2}+w_{3}\Phi _{3}+w_{4}\Phi _{4}=0$,

i.e. $\Phi$ is perpendicular to the vector to w; we can also write

(50)	$\Phi _{4}=i({\mathfrak {w}}_{x}\Phi _{1}+{\mathfrak {w}}_{y}\Phi _{2}+{\mathfrak {w}}_{z}\Phi _{3})$,

I shall call the space-time vector $\Phi$ of the first kind as the Electric Rest Force.

Relations analogous to those holding between $-wF,\ {\mathfrak {E,\ M,\ w}}$, hold amongst $-wf,\ {\mathfrak {e,\ m,\ w}}$, and in particular -wf is normal to w. The relation (C) can be written as

{C}	$wf=\epsilon wF$

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

Let us now form the time-space vector 1st kind $\Psi =iwf^{*}$, whose components are

${\begin{array}{cccccccccc}\Psi _{1}&=&-i(&&&w_{2}f_{34}&+&w_{3}f_{42}&+&w_{4}f_{23}),\\\Psi _{2}&=&-i(&w_{1}f_{43}&&&+&w_{3}f_{14}&+&w_{4}f_{31}),\\\Psi _{3}&=&-i(&w_{1}f_{24}&+&w_{2}f_{41}&&&+&w_{4}f_{12}),\\\Psi _{4}&=&-i(&w_{1}f_{32}&+&w_{2}f_{13}&+&w_{3}f_{21}&&).\end{array}}$

Of these, the first three $\Psi _{1},\ \Psi _{2},\ \Psi _{3}$ are the x-, y-, z-components of the space-vector

(51)	${\frac {{\mathfrak {m}}-[{\mathfrak {we}}]}{\sqrt {1-{\mathfrak {w}}^{2}}}}$,

and further

(52)	$\Psi _{4}={\frac {i[{\mathfrak {wm}}]}{\sqrt {1-{\mathfrak {w}}^{2}}}}$;

Among these there is the relation

(53)	$w{\overline {\Psi }}=w_{1}\Psi _{1}+w_{2}\Psi _{2}+w_{3}\Psi _{3}+w_{4}\Psi _{4}=0$,

which can also be written as

(54)	$\Psi _{4}=i({\mathfrak {w}}_{x}\Psi _{1}+{\mathfrak {w}}_{y}\Psi _{2}+{\mathfrak {w}}_{z}\Psi _{3})$

The vector $\Psi$ is perpendicular to w; we can call it the Magnetic rest-force.

Relations analogous to these hold among the quantities $iwF^{*},{\mathfrak {M,E,w}}$ and Relation (D) can be replaced by the formula

{D}	$wF^{*}=\mu wf^{*}$

We can use the relations (C) and (D) to calculate F and f from $\Phi$ and $\Psi$, we have

$wF=-\Phi ,\ wF^{*}=-i\mu \Psi ,\ wf=-\epsilon \Phi ,\ wf^{*}=-i\Psi$

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

(55)	$F=[w,\Phi ]+i\mu [w,\Psi ]^{*}$,

(56)	$f=\epsilon [w,\Phi ]+i[w,\Psi ]^{*}$,

i.e.

$F_{12}=(w_{1}\Phi _{2}-w_{2}\Phi _{1})+i\mu (w_{3}\Psi _{4}-w_{4}\Psi _{3})$, etc. $f_{12}=\epsilon (w_{1}\Phi _{2}-w_{2}\Phi _{1})+i(w_{3}\Psi _{4}-w_{4}\Psi _{3})$, etc.

Let us now consider the space-time vector of the second kind $[\Phi \Psi ]$