# Translation:On the Kinematics of the Rigid Body in the System of the Principle of Relativity

On the Kinematics of the Rigid Body in the System of the Principle of Relativity  (1910)
by Max Born, translated from German by Wikisource
In German: Zur Kinematik des starren Körpers im System des Relativitätsprinzips , Göttinger Nachrichten (2), 161-179, Source

On the kinematics of the rigid body in the system of the principle of relativity.

By

Max Born.

Presented by Mr. W. Voigt in the meeting at 28 May 1910.

Introduction

The definition of rigidity given by me[1] which satisfies the Lorentz-Einstein principle of relativity, has proven to be too narrow,[2] since it was shown that the motion of a body rigid in this sense is in general completely determined by one of its points, save some exceptional cases which were extensively discussed by Herglotz. Therefore this definition in no way can be applied upon ordinary material rigid bodies,[3] even though it suffices for the foundation of the dynamics of the electrons for all observable phenomena. However, one has to view as desirable the introduction of a rigidity definition, which represents the completest possible analogy to the rigid body of ordinary kinematics; because on one side the rigid body is necessary for the formation of mechanics, and in addition it seems to me that an electron theory without this concept is an essential step backwards.

Therefore I have tried to state a new definition of rigidity, which has the largest possible similarity to ordinary rigidity, and I would like to present the definition and these [ 162 ] consequences, which can be easily proceeded without further ado. The properties of the rigid body defined in this way, which I would like to develop here, are the following ones:

1) The rigid body has six degrees of freedom.

2) The definition first given by me corresponds to the case of translation in accordance with the new definition, however, at which the exceptional cases treated by Herglotz don't occur.

3) The velocity field of the points of the body is given by formulas, which are analogues to the corresponding formulas of ordinary kinematics.

4) Neglecting the square of velocity of the points of the bodies against the square of the speed of light, then the motion is identical with the rigid motion of ordinary kinematics.

5) There is an instantaneous axis of rotation, i.e. by a suitable Lorentz transformation one can achieve that all points of a straight line are at rest.

6) All points of the body have a constant "rest distance" with respect to a certain point located upon the instantaneous axis, the center of rotation.

Except the identification of these general properties of the definition, I have computed some special motions, among them the case that a point of the body carries out a uniformly-accelerated motion (hyperbolic motion), and that the body is arbitrarily rotating about the direction of translation. The essential result is, that all points of the body steadily remain upon circular cylinders of constant radius, and their projections upon the direction of motion are carrying out congruent hyperbolic motions.

That the concept of rotation introduced with this definition is useful for some purposes, emerges above all else from the following circumstance. By the aid of a method developed by Minkowski shortly before his death, I have shown[4] that the fundamental equations for electromagnetic processes in moving bodies developed by him, can be derived from the concepts of the theory of electrons. Following Lorentz, the dielectric polarizations and the magnetization of matter are explained by relative motions of electrons against material molecules. The [ 163 ] method is, however, totally different from that of Lorentz; it rests upon the circumstance, that the deviations of motions of the electrons from the motions of the material parts are represented by power series, which are progressing in accordance with a parameter measuring the mean displacement. There it is shown, that the parts of 1. order or the series represent the dielectric polarization, the parts of 2. order the magnetization, and the expressions emerge entirely by themselves as developing coefficients, which represent the magnetic moment, i.e. the torque of the electrons multiplied with the charge. The expressions for the representation of the rotations of the electrons that were obtained in an entirely formal way, are now in agreement with the definition of rotation given here. However, also difficulties for the usefulness of the new definition are standing in the way of this agreement.[5]

## § 1. The rigid body of ordinary mechanics.

While my first definition of rigidity seeks to analogize the property of ordinary rigid bodies, that the distance between any two points of the body is constant at motion, the new definition is similar to another property of ordinary rigid bodies.

The velocity vector ${\displaystyle {\mathfrak {w}}}$ of any of its points ${\displaystyle x,y,z}$ can namely be represented in the following way:[6]

 (1) ${\displaystyle {\mathfrak {w-u=[p,r]}}}$

There, ${\displaystyle {\mathfrak {r}}}$ means the vector with components

${\displaystyle x-a,\ y-b,\ z-c,}$

where ${\displaystyle a,b,c}$ is a certain point of the body; ${\displaystyle {\mathfrak {u}}}$ is the velocity of the point ${\displaystyle a,b,c}$ and ${\displaystyle {\mathfrak {p}}}$ the vector of radial velocity.

From condition (1) one finds the equation of the instantaneous axis of rotation, whose points do all have the same velocity, [ 164 ] by setting ${\displaystyle {\mathfrak {w=u}}}$; thus the vector ${\displaystyle [{\mathfrak {p,r}}]}$ must vanish, i.e. the proportions

 (2) ${\displaystyle x-a:y-b:z-c={\mathfrak {p}}_{x}:{\mathfrak {p}}_{y}:{\mathfrak {p}}_{z}}$

must exist, which represent a straight line. If ${\displaystyle R}$ is the distance of a point ${\displaystyle x,y,z}$ from point ${\displaystyle a,b,c}$, that is

${\displaystyle R^{2}=(x-a)^{2}+(y-b)^{2}+(z-c)^{2},}$

then one finds by differentiation towards time ${\displaystyle t}$:

${\displaystyle R{\frac {dR}{dt}}=({\mathfrak {r,\ w-u}}).}$

Now it follows from equations (1):

 (3) {\displaystyle {\begin{aligned}&({\mathfrak {r,\ w-u}})&=({\mathfrak {r,\ [p,r]}})&&=&0,\\&({\mathfrak {p,\ w-u}})&=({\mathfrak {p,\ [p,r]}})&&=&0\end{aligned}}}

Thus:

 (4) ${\displaystyle R{\frac {dR}{dt}}=0}$.

${\displaystyle {\tfrac {dR}{dt}}}$ vanishes, i.e. the distance of all points with respect to ${\displaystyle a,b,c}$ is constant. These properties characteristic for rigidity will be found again in the new definition.

A remarkable property of ordinary rigid bodies won't be transferred, namely that the distance of any two points ${\displaystyle x',y',z'}$ and ${\displaystyle x'',y'',z''}$ is constant. The latter fact is to be proven in ordinary kinematics as follows. We distinguish all quantities that refer to these two points by one or two primes; then we have the equations:

{\displaystyle {\begin{aligned}{\mathfrak {w'-u}}&={\mathfrak {[p,r']}}\\{\mathfrak {w''-u}}&={\mathfrak {[p,r'']}}\end{aligned}}}

from which it follows by subtraction:

 (5) ${\displaystyle {\mathfrak {w'=w''=[p,r'-r'']}}}$

On the other hand, it is given by differentiation of the relative distance

${\displaystyle \left({\mathfrak {r'-r''}}\right)^{2}=(x'-x'')^{2}+(y'+y'')^{2}+(z'-z'')^{2}}$

towards time ${\displaystyle t}$:

${\displaystyle |{\mathfrak {r'-r''}}|{\frac {d|{\mathfrak {r'-r''}}|}{dt}}=({\mathfrak {r'-r'',\ w'-w''}})}$

[ 165 ] By scalar multiplication with ${\displaystyle {\mathfrak {\left(r'-r''\right)}}}$ it is given from (5):

 (6) ${\displaystyle ({\mathfrak {r'-r'',\ w'-w''}})=({\mathfrak {r'-r'',\ [p,\ r'-r'']}})=0}$

Thus it is

 (7) ${\displaystyle {\frac {d|{\mathfrak {r'-r''}}|}{dt}}=0}$

It won't be possible to transfer this conclusion. However, one will gain a clear insight as to why this property of rigid bodies must cease to apply in the new definition.

## § 2. Rest vectors of first and second kind.

In the following, I will for brevity's sake use the four-dimensional vector analysis introduced by Minkowski,[7] and I therefore allude to the publication of the mentioned author.

An arbitrary reference system of rectangular coordinates ${\displaystyle x,y,z}$ and time ${\displaystyle t}$ should be used; ${\displaystyle c}$ should be the speed of light. We replace

${\displaystyle x,y,z,ict}$

by

${\displaystyle x_{1},x_{2},x_{3},x_{4}}$.

A homogeneous linear substitution

 (8) ${\displaystyle x_{\alpha }=\sum \limits _{\beta =1}^{4}a_{\alpha \beta }x_{\beta }^{'}\qquad (\alpha =1,2,3,4)}$

of determinant ${\displaystyle +1}$, in which all coefficients without index 4 are real, ${\displaystyle \alpha _{14},\alpha _{24},\alpha _{34}}$ as well as ${\displaystyle \alpha _{41},\alpha _{42},\alpha _{43}}$ are purely imaginary (probably zero), eventually ${\displaystyle \alpha _{44}}$ is real again and ${\displaystyle >0}$, and by which

 (8') ${\displaystyle \sum \limits _{\alpha =1}^{4}x_{\alpha }^{2}\ {\textrm {passes\ into}}\ \sum \limits _{\alpha =1}^{4}x_{\alpha }^{'2}}$

is called a Lorentz transformation. The Lorentz transformations contain 6 independent parameter and can be seen as passage from one reference system moving uniformly relative to it, at a simultaneous-timeless rotation of the ${\displaystyle x,y,z}$ axis-cross.

If one neglects this timeless change of space coordinates, one obtains the special Lorentz transformations,[8] which only contain 3 parameters, namely the components of relative velocity ${\displaystyle {\mathfrak {u}}}$ of both reference systems.

A spacetime vector of first kind ${\displaystyle s}$ is a system of 4 quantities [ 166 ] ${\displaystyle s_{1},s_{2},s_{3},s_{4}}$, which transforms at Lorentz transformations as the coordinates ${\displaystyle x_{\alpha }}$. In every reference system, the three first components ${\displaystyle s_{1},s_{2},s_{3}}$ represent a space vector ${\displaystyle {\mathfrak {s}}}$. A spacetime vector of second kind ${\displaystyle f}$ is a system of six quantities

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

which transforms at Lorentz transformations as the 6 determinants, which one can generate from the scheme

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

formed by means of two vectors of first kind.

In every reference system, the six components of ${\displaystyle f}$ represent two space vectors, namely ${\displaystyle f_{23},f_{31},f_{12}}$ may form the space vector ${\displaystyle {\mathfrak {m}}}$, and ${\displaystyle if_{14},if_{24},if_{34}}$ the (real) space vector ${\displaystyle {\mathfrak {n}}}$. Vector ${\displaystyle f^{\ast }}$ which is dual to ${\displaystyle f}$ is the spacetime vector of components ${\displaystyle f_{14},f_{24},f_{34},f_{23},f_{31},f_{12}}$, which is composed of the space vectors ${\displaystyle {\mathfrak {m,n}}}$ in reverse order.

Now we consider the behavior of vectors of first and second kind at special Lorentz transformations with vector ${\displaystyle {\mathfrak {u}}}$, that is, at the passage from reference system ${\displaystyle \Sigma }$ to another ${\displaystyle \Sigma '}$ with parallel spatial coordinate axes, which has the velocity ${\displaystyle {\mathfrak {u}}}$ relative to the first system.

There, a vector of first kind ${\displaystyle s}$ goes over into a vector ${\displaystyle s'}$; its first three components ${\displaystyle s'_{1},s'_{2},s'_{3}}$ represent that space vector ${\displaystyle {\mathfrak {s'}}}$, which is the spacetime vector ${\displaystyle s}$ as it appears to the moving system ${\displaystyle \Sigma '}$. This space vector ${\displaystyle {\mathfrak {s'}}}$ can also be seen as a spacetime vector whose fourth component vanishes in reference system ${\displaystyle \Sigma '}$. This new spacetime vector with components ${\displaystyle s'_{1},s'_{2},s'_{3}}$ measured in ${\displaystyle \Sigma '}$, has the components in ${\displaystyle \Sigma }$

${\displaystyle s_{\alpha }^{0}=s+(u,{\bar {s}})u_{\alpha },\qquad (\alpha =1,2,3,4)}$

i.e. it is identical with the vector

 (9) ${\displaystyle s_{0}=s+(u,{\bar {s}})u,}$

where ${\displaystyle u}$ means the spacetime vector with components:

 (10) {\displaystyle {\begin{aligned}u_{1}={\frac {{\mathfrak {u}}_{x}}{c{\sqrt {1-{\frac {|{\mathfrak {u}}|^{2}}{c^{2}}}}}}},&u_{2}={\frac {{\mathfrak {u}}_{y}}{c{\sqrt {1-{\frac {|{\mathfrak {u}}|^{2}}{c^{2}}}}}}},\\u_{3}={\frac {{\mathfrak {u}}_{z}}{c{\sqrt {1-{\frac {|{\mathfrak {u}}|^{2}}{c^{2}}}}}}},&u_{4}={\frac {i}{\sqrt {1-{\frac {|{\mathfrak {u}}|^{2}}{c^{2}}}}}},\end{aligned}}}

[ 167 ] which satisfy the relation

 (11) ${\displaystyle \Sigma u_{\alpha }^{2}=(u,{\bar {u}})=-1}$.

By its definition, vector ${\displaystyle s_{0}}$ is normal to ${\displaystyle u}$:

 (12) ${\displaystyle (u,{\bar {s}}_{0})=0}$

It is called the rest vector of ${\displaystyle s}$ with respect to velocity ${\displaystyle {\mathfrak {u}}}$, because in that coordinate system ${\displaystyle \Sigma '}$ in which a point moving with velocity ${\displaystyle {\mathfrak {u}}}$ appears to be at rest, the three first components (the space components) of ${\displaystyle s_{0}}$ are identical with the three first components of vector ${\displaystyle s'}$ transformed to ${\displaystyle \Sigma '}$.

Now we consider the vector of second kind ${\displaystyle f}$. At a special Lorentz transformation it goes over into the vector of second kind ${\displaystyle f'}$, which is decomposed in system ${\displaystyle \Sigma '}$ into the two space vectors ${\displaystyle {\mathfrak {m',n'}}}$. These two space vectors can be seen as spacetime vectors whose fourth coordinate vanishes in system ${\displaystyle \Sigma '}$; these spacetime vectors with components ${\displaystyle f'_{23},f'_{31},f'_{12}}$ or ${\displaystyle f'_{14},f'_{24},f'_{34}}$ in system ${\displaystyle \Sigma '}$ shall be denoted by ${\displaystyle \psi }$ and ${\displaystyle \varphi }$. Then one can easily show that the components of ${\displaystyle \varphi }$ are as follows in the original system ${\displaystyle \Sigma }$:

${\displaystyle \varphi _{\alpha }=\sum \limits _{\beta =1}^{4}u_{\alpha }f_{\alpha \beta },\qquad (\alpha =1,2,3,4)}$

i.e. it is

 (13) ${\displaystyle \varphi =-uf}$;

it is also given that

 (14) ${\displaystyle \psi =iuf^{\ast }}$

Obviously both ${\displaystyle \psi }$ and ${\displaystyle \varphi }$ are normal to ${\displaystyle u}$:

 (15) ${\displaystyle (u,{\bar {\varphi }})=0,\ (u,{\bar {\psi }})=0}$

We call ${\displaystyle \psi }$ and ${\displaystyle \varphi }$ the rest vectors belonging to the spacetime vector ${\displaystyle f({\mathfrak {m,n}})}$ with respect to velocity ${\displaystyle {\mathfrak {u}}}$, because in that coordinate system ${\displaystyle \Sigma '}$ in which a point moving with velocity ${\displaystyle {\mathfrak {u}}}$ appears to be at rest, the first three components of ${\displaystyle \psi }$ and ${\displaystyle \varphi }$ are identical with the space vectors ${\displaystyle {\mathfrak {n'}}}$ and ${\displaystyle {\mathfrak {m'}}}$, which compose the transformed vector of second kind ${\displaystyle f'}$.

The verifications of the assertions stated here can be easily given analytically or geometrically. However, they are also [ 168 ] implicitly contained in the considerations of Minkowski[9] concerning the transformation of the electric and magnetic field strength.

## § 3. Definition of the rigid body.

The now developed concepts of rest vectors of first and second kind will now be used in the definition of rigidity.

We denote a certain point of the moving body as the center. This shall be represented in a moment by the spacetime point ${\displaystyle a\left(a_{1},a_{2},a_{3},a_{4}\right)}$, and shall have the velocity defined by the spacetime vector ${\displaystyle u\left(u_{1},u_{2},u_{3},u_{4}\right)}$; thus it is

 (16) ${\displaystyle (u,{\bar {u}})=-1}$

We consider an arbitrary point ${\displaystyle x\left(x_{1},x_{2},x_{3},x_{4}\right)}$ which is located through ${\displaystyle a}$ and normal to ${\displaystyle u}$; thus it is

 (17) ${\displaystyle (u,\ {\overline {x-a}})=0}$.

The velocity vector ${\displaystyle w\left(w_{1},w_{2},w_{3},w_{4}\right)}$ belongs to ${\displaystyle x}$, which satisfies the condition

 (18) ${\displaystyle (u,{\bar {w}})=-1}$

Furthermore ${\displaystyle p}$ shall be a quite arbitrary spacetime vector normal to ${\displaystyle u}$:

 (19) ${\displaystyle (u,{\bar {p}})=0}$

We call it the angular velocity. Its first three components

${\displaystyle p_{1}={\frac {1}{c}}{\mathfrak {p}}_{x},\ p_{2}={\frac {1}{c}}{\mathfrak {p}}_{y},\ p_{3}={\frac {1}{c}}{\mathfrak {p}}_{z}}$

define a space vector ${\displaystyle {\mathfrak {p}}}$, while

 (19') ${\displaystyle p_{4}={\frac {i}{c^{2}}}\left({\mathfrak {u}}_{x}{\mathfrak {p}}_{x}+{\mathfrak {u}}_{y}{\mathfrak {p}}_{y}+{\mathfrak {u}}_{z}{\mathfrak {p}}_{z}\right)={\frac {i}{c^{2}}}({\mathfrak {u,p}})}$

The rest vector of velocity ${\displaystyle w}$ of ${\displaystyle x}$ with respect to ${\displaystyle u}$ is

 (20) ${\displaystyle w^{0}=w+(u,{\bar {w}})u}$;

according to the considerations of the previous paragraph, in a coordinate system ${\displaystyle \Sigma '}$ in which point a rests (i.e. ${\displaystyle {\mathfrak {u}}=0}$), [ 169 ] its first three components are identical with those of the vector of first kind ${\displaystyle w'}$, which emerges from ${\displaystyle w}$ by a special Lorentz transformation ${\displaystyle {\mathfrak {u}}}$. Thus one can denote ${\displaystyle w^{0}}$ also as rest velocity or relative velocity of point ${\displaystyle x}$ with respect to velocity ${\displaystyle u}$ of point ${\displaystyle a}$.

The spacetime vector of second kind

 (21) ${\displaystyle [p,x-a]=Q}$

is denoted by us as the rotation of point ${\displaystyle x}$ about point ${\displaystyle a}$.

We form the two rest vectors belonging to ${\displaystyle Q}$. From the identical relation

${\displaystyle u[p,x-a]=(u,{\bar {p}})(x-a)-(u,{\overline {x-a}})p}$

it follows because of (17) and (19), that one of the two rest vectors is

 (22) ${\displaystyle uQ=0}$.

The other one we call rest rotation and denote it by ${\displaystyle q}$:

 (23) ${\displaystyle iuQ^{\ast }=q}$

According to (15), ${\displaystyle w}$ is normal to ${\displaystyle u}$:

 (24) ${\displaystyle (u,{\bar {q}})=0}$

The first three components of ${\displaystyle q}$,

${\displaystyle q_{1}=-{\frac {{\mathfrak {q}}_{x}}{c}},\ q_{2}=-{\frac {{\mathfrak {q}}_{y}}{c}},\ q_{3}=-{\frac {{\mathfrak {q}}_{z}}{c}}}$

for a space vector, while

 (24') ${\displaystyle q_{4}=-{\frac {i}{c^{2}}}\left({\mathfrak {u}}_{x}{\mathfrak {q}}_{x}+{\mathfrak {u}}_{y}{\mathfrak {q}}_{y}+{\mathfrak {u}}_{z}{\mathfrak {q}}_{z}\right)=-{\frac {i}{c^{2}}}({\mathfrak {u,q}})}$

Formula (22) shows that in the coordinate system ${\displaystyle \Sigma '}$ in which point ${\displaystyle a}$ rests, the vector of second kind ${\displaystyle Q}$ is reduced to a vector of first kind ${\displaystyle q}$, namely it is obvious for ${\displaystyle {\mathfrak {u}}=0}$

 (25) ${\displaystyle {\mathfrak {q=[p,r]}}}$

where ${\displaystyle {\mathfrak {r}}}$ means the space vector with components

${\displaystyle x_{1}-a_{1},\ x_{2}-a_{2},\ x_{3}-a_{3}}$.

One can also express the vector of second kind ${\displaystyle Q}$ by ${\displaystyle q}$. From the identity: [ 170 ]

 (26) ${\displaystyle [u,uQ]+[u,uQ^{\ast }]^{\ast }=(u{\bar {u}})Q}$

it namely follows due to (16) and (22)

 (27) ${\displaystyle [u,q]^{\ast }=-iQ}$.

Now we can define – in agreement with the relativity principle and simultaneously in full analogy to equation (1) – the rigid body as follows:

Definition: The motion of a point system is called rigid, when between the location ${\displaystyle x}$ and the velocity ${\displaystyle w}$ of any point the relation persists that the relative velocity towards the center ${\displaystyle a}$ is equal to the rest rotation:

 (28) ${\displaystyle w^{0}=q}$,

or extensively:

 (28') ${\displaystyle w+(u,{\bar {w}})u=iu[p,x-a)^{\ast }}$.

This definition is covariant with respect to Lorentz transformations. Furthermore it is fully sufficient in connection with conditions (17) and (18), in order to determine the motion as soon as the motion of center ${\displaystyle a}$ and the path of vector ${\displaystyle p}$ are given for all times in agreement with conditions (16) and (19).

Namely, if the motion of ${\displaystyle a}$ is described by giving ${\displaystyle a_{1},a_{2},a_{3},a_{4}}$ as functions of proper time ${\displaystyle \sigma }$; then

${\displaystyle u_{\alpha }={\frac {1}{c}}{\frac {da_{\alpha }}{d\sigma }}\qquad (\alpha =1,2,3,4)}$

are also known functions of ${\displaystyle \sigma }$ which satisfy relation (16).

Also ${\displaystyle p_{\alpha }}$ (of which only three are independent because of (19)) shall be given as functions of ${\displaystyle \sigma }$.

Then ${\displaystyle \sigma }$ can be computed as function of ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$ from equation (17), and then insert its values into (28').

Now when ${\displaystyle \tau }$ means the proper time of point ${\displaystyle x}$, then

${\displaystyle w_{\alpha }={\frac {1}{c}}{\frac {dx_{\alpha }}{d\tau }}.\qquad (\alpha =1,2,3,4)}$

Consequently equations (28'), of which only three are independent because of the orthogonality of both sides to ${\displaystyle u}$, together with (18) are representing a system of equations in the form:

 (29) {\displaystyle \left\{{\begin{aligned}{\frac {dx_{\alpha }}{d\tau }}=\lambda \cdot f_{\alpha }\left(x_{1},x_{2},x_{3},x_{4}\right),&&(\alpha =1,2,3,4)\\\sum \limits _{\alpha =1}^{4}\left({\frac {dx_{\alpha }}{d\tau }}\right)=-c^{2}\end{aligned}}\right.}

[ 171 ] There, ${\displaystyle f_{\alpha }}$ are known functions of their arguments and ${\displaystyle \lambda }$ is a proportionality factor which can be eliminated because of the last equation.

Consequently one has a system of ordinary differential equations for the functions ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$ of ${\displaystyle \tau }$.

The general solution will contain, except a constant corresponding to the arbitrary choice of the origin of ${\displaystyle \tau }$, three parameter ${\displaystyle \xi _{1},\xi _{2},\xi _{3}}$ which individualize the three single moving points.

Thus one gets the motion, expressed in the manner denoted after Lagrange, by equations of the form:

 (30) ${\displaystyle x_{\alpha }=x_{\alpha }\left(\xi _{1},\xi _{2},\xi _{3},\tau \right)}$

as I have used them earlier several times.

## § 4. General properties of rigid bodies.

Now we can derive those properties from the definition of rigid bodies, which are analogous to the ones of the ordinary rigid body discussed in § 1.

First it is clear, that the definition (28') in equation (1) goes over to ordinary kinematics, as soon as terms of order ${\displaystyle {\tfrac {w^{2}}{c^{2}}},\ {\tfrac {wu}{c^{2}}},\ {\tfrac {u^{2}}{c^{2}}}}$ are neglected.

In order to prove the existence of an instantaneous rotation axis, we ask for the spacetime points ${\displaystyle x}$, whose velocity is equal to that of the center. From

${\displaystyle w=u}$

it follows by multiplication with ${\displaystyle u}$ because of (16)

${\displaystyle (u,{\bar {w}})=(u,{\bar {u}})=-1}$,

thus

${\displaystyle w^{0}=w+(u,{\bar {w}})u=w-u=0}$.

Thus it must be ${\displaystyle q=0}$ for the sought point; from (27) it is given ${\displaystyle Q=0}$, or

${\displaystyle [p,\ x-a)=0}$.

However, from that the proportionality of ${\displaystyle p_{\alpha }}$ with ${\displaystyle x_{\alpha }-a_{\alpha }}$ follows.

The sought points are thus located upon the straight worldline

 (31) ${\displaystyle x_{\alpha }-a_{\alpha }=\lambda p_{\alpha }}$,

which obviously is normal to ${\displaystyle u}$. In the coordinate system ${\displaystyle \Sigma '}$, [ 172 ] in which the center ${\displaystyle a}$ is resting, those points form a spatial line which momentarily rests in its entire length. That is the instantaneous rotation axis.

Let ${\displaystyle R}$ be the rest-distance of a point ${\displaystyle x}$ from the center, i.e.

${\displaystyle R^{2}=\sum \limits _{\alpha =1}^{4}\left(x_{\alpha }-a_{\alpha }\right)^{2}}$,

where ${\displaystyle x}$ satisfies condition (17). If the motion is known, thus ${\displaystyle x_{\alpha }}$ are the known functions of ${\displaystyle \tau }$, then one can determine ${\displaystyle \tau }$ as function of proper time ${\displaystyle \sigma }$ of the center from (17), and then insert this value again into ${\displaystyle x_{\alpha }}$. Then ${\displaystyle R}$ becomes a function of ${\displaystyle \sigma }$. If we differentiate this towards ${\displaystyle \sigma }$, it follows:

{\displaystyle {\begin{aligned}R{\frac {dR}{d\sigma }}&=\sum \limits _{\alpha =1}^{4}\left(x_{\alpha }-a_{\alpha }\right)\left(w_{\alpha }{\frac {d\tau }{d\sigma }}-u_{\alpha }\right)\\&=\left(x-a,\ {\overline {w{\frac {d\tau }{d\sigma }}-u}}\right)\end{aligned}}}

Now it follows from (28') by multiplication with ${\displaystyle x-a}$:

${\displaystyle (x-a),\ {\bar {w}})+(u,{\bar {w}})(u,\ {\overline {x-a}})=i(u[p,\ x-a]^{\ast },\ {\overline {x-a}})}$

If one considers (17) and notices that it is identically for two vectors of first kind ${\displaystyle p,q}$

 (32) ${\displaystyle (u[p,q]^{\ast },{\bar {q}})=(u[p,q]^{\ast },{\bar {p}})=0}$,

then one finds:

 (33) ${\displaystyle (x-a),\ {\bar {w}})=0}$

In the same way one finds from (28') by multiplication with ${\displaystyle p}$:

${\displaystyle (p,{\bar {w}})+(u,{\bar {w}})(u,{\bar {p}})=i(u[p,x-a)^{\ast },{\bar {p}})}$

and because of (19) and (32):

 (34) ${\displaystyle (p,{\bar {w}})=0}$

From (33) it is given together with (17)

 (35) ${\displaystyle R{\frac {dR}{d\sigma }}=0}$,

an equation being totally analogous to (4).

From that it follows that the rest-distance of all points from center ${\displaystyle a}$ are independent of the proper time of the latter, thus constant at all.

In this property, the new definition agrees with the ones of the ordinary rigid bodes discussed in § 1. [ 173 ] However, as already said there, that the rest-distance of two points ${\displaystyle x,y}$ doesn't remain constant at motion, is connected with the circumstance that not any arbitrary point of the body can be seen as center, but only the points of the axis. Namely, if ${\displaystyle w,v}$ are the velocities of points ${\displaystyle x,y}$, then the rigidity conditions are in accordance with (28')

 (36) {\displaystyle {\begin{aligned}w+(u,{\bar {w}})u&=iu[p,x-a]^{\ast }\\v+(u,{\bar {v}})u&=iu[p,y-a]^{\ast }\end{aligned}}}

One finds by subtraction:

 (37) ${\displaystyle (w-v)+(u,\ {\overline {w-v}})u=iu[p,\ x-y]^{\ast }}$

If one were allowed to view point ${\displaystyle y}$ as the center of rigid motion, then in accordance with definition (28'), the equation

 (38) ${\displaystyle w+(v,{\bar {w}})v=iu[p,\ x-y]^{\ast }}$

must be valid, which generally contradicts the previous one (37). Thus one cannot choose an arbitrary different point as center, though points of the instantaneous axis. Since for them it is ${\displaystyle v=u}$, the left hand side of (37) becomes

${\displaystyle w-u+(u,\ {\bar {w}})u-(u,{\bar {u}})u=w+(u,{\bar {w}})u}$,

thus identical with the one of (38).

If ${\displaystyle R}$ is the rest-distance of both arbitrary points ${\displaystyle x,y}$, that is

${\displaystyle R^{2}=(x-y,\ {\overline {x-y}})}$,

then it follows by differentiation towards the proper time ${\displaystyle \sigma }$ of the center ${\displaystyle a}$ (as on p. 13):

 (39) ${\displaystyle R{\frac {dR}{d\sigma }}=\left(x-y,\ {\overline {w{\frac {d\tau }{d\sigma }}-v{\frac {d\varkappa }{d\sigma }}}}\right)}$

where ${\displaystyle \tau }$ and ${\displaystyle \varkappa }$ are the proper times of points ${\displaystyle x,y}$.

Even though it follows from (37) by multiplication with ${\displaystyle (x-y)}$ as earlier

${\displaystyle (x-y,\ {\overline {w-v}})=0}$,

one cannot generally conclude the disappearance of the right-hand side of (39).

[ 174 ]

## § 5. Special motions.

If the rotation vector ${\displaystyle {\mathfrak {p}}}$ is identical with zero, from which the identical disappearance of ${\displaystyle p}$ follows because of (19'), then the motion is to be seen as pure translational. In this case, it follows from

${\displaystyle w+(u,{\bar {w}})u=0}$

by multiplication with ${\displaystyle w}$ because of (18)

${\displaystyle (w,{\bar {w}})+(u,{\bar {w}})^{2}=-1+(u,{\bar {w}})^{2}=0}$

thus

${\displaystyle (u,{\bar {w}})=\pm 1}$.

The negative sign has to be chosen here; since it is ${\displaystyle {\mathfrak {u}}=0}$, thus ${\displaystyle u_{1}=u_{2}=u_{3}=0,\ u_{4}=i}$, therefore ${\displaystyle (u,{\bar {w}})}$ goes over into

${\displaystyle iw_{4}={\frac {-1}{\sqrt {1-{\frac {|{\mathfrak {w}}^{2}|}{c^{2}}}}}}}$,

and since the square root is always positive, one has to set

${\displaystyle (u,{\bar {w}})=-1}$.

Thus at translation:

 (40) ${\displaystyle w=u}$.

This equation together with the conditions

 (41) ${\displaystyle (u,{\bar {u}})=-1,\ (w,{\bar {w}})=-1,\ (u,\ {\overline {x-a}})=0}$

is completely equivalent with the regular case of motion of a body to be called rigid according to my first definition of rigidity, as it follows from the results of Herglotz and Noether.[10] Namely one can describe equations (40), (41) as follows:

At translation, the worldlines of the points of the body are the orthogonal trajectories of a bundle of orthogonal spaces.[11]

My application of the first definition of rigidity on the dynamics of the electrons thus retains its full validity, also when the new definition is assumed.

An additional simple special case is the one, that the center [ 175 ] is constantly at rest; then, as already said on p. 18, the rotation reduces itself to the space vector

${\displaystyle {\mathfrak {q=[p,r]}}}$

and the definition equation (28') goes over into

 (42) ${\displaystyle {\frac {\mathfrak {w}}{\sqrt {1-{\frac {|{\mathfrak {w}}|^{2}}{c^{2}}}}}}=[{\mathfrak {p,r}}]}$.

By comparison with (1), § 1, one sees that the motion agrees with the rotation of ordinary kinematics only up to terms of second order.

For instance, if a body rotates about the ${\displaystyle x}$-axis with the constant angular velocity ${\displaystyle {\mathfrak {p}}_{x}}$, then the solution of (42) can be represented in the form

{\displaystyle {\begin{aligned}x&={\textrm {konst}}.\\y&=\varrho \cos \varphi ,\\z&=\varrho \sin \varphi ,\end{aligned}}}

where ${\displaystyle \varrho }$ is a constant and

${\displaystyle \varphi ={\frac {{\mathfrak {p}}_{x}}{\sqrt {1+{\frac {\varrho ^{2}{\mathfrak {p}}_{x}^{2}}{c^{2}}}}}}t}$.

The angular velocity ${\displaystyle {\tfrac {d\varphi }{dt}}}$ measured from the resting system is thus decreasing with increasing distance ${\displaystyle \varrho }$ from the axis.[12] The circumference velocity ${\displaystyle \varrho {\tfrac {d\varphi }{dt}}}$ always remains smaller than ${\displaystyle c}$, so that no restriction of the dimensions of rigid body, as it occurs at accelerated translation, is there at this place.

Another simple example is the following.

The center should carry out a hyperbolic motion (uniformly accelerated translation), and the body should rotate about the direction of translation. Thus it is:

 (43) {\displaystyle {\begin{aligned}a_{1}^{2}+a_{4}^{2}=konst.=\sigma _{1}^{2},&&a_{2}=0,&&a_{3}=0,\\p_{2}=0,\\p_{3}=0.\end{aligned}}}

If we introduce instead of proper time ${\displaystyle \sigma }$ the imaginary quantity

${\displaystyle \sigma _{4}=ic\sigma }$,

then one can represent the motion with the aid of trigonometric [ 176 ] functions of the imaginary argument ${\displaystyle {\tfrac {\sigma _{4}}{\sigma _{1}}}}$ (hyperbolic functions):

 (44) ${\displaystyle a_{1}=\sigma _{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}},\ a_{2}=0,\ a_{3}=0,\ a_{4}=\sigma _{1}\sin {\frac {\sigma _{4}}{\sigma _{1}}}.}$

Then

 (45) ${\displaystyle u_{1}=-i\sin {\frac {\sigma _{4}}{\sigma _{1}}},\ u_{2}=0,\ u_{3}=0,\ u_{4}=i\cos {\frac {\sigma _{4}}{\sigma _{1}}}.}$

We first determine ${\displaystyle \sigma }$ as function of ${\displaystyle x_{\alpha }}$ from equation (17). This reads here:

${\displaystyle x_{1}\sin {\frac {\sigma _{4}}{\sigma _{1}}}-x_{4}\cos {\frac {\sigma _{4}}{\sigma _{1}}}=0,}$

from which it is given

 (46) ${\displaystyle {\frac {x_{4}}{x_{1}}}=\operatorname {tg} {\frac {\sigma _{4}}{\sigma _{1}}}}$

According to (21), the components of ${\displaystyle Q}$ become:

{\displaystyle {\begin{aligned}&\qquad Q_{23}=0,\ Q_{31}=-p_{1}x_{3},\ Q_{12}=p_{1}x_{2},\\&Q_{14}=p_{1}\left(x_{4}-\sigma _{1}\sin {\frac {\sigma _{4}}{\sigma _{1}}}\right)-p_{4}\left(x_{1}-\sigma _{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}\right),\ Q_{24}=-p_{1}x_{2},\\&\qquad \qquad Q_{34}=-p_{4}x_{3}.\end{aligned}}}

From (23) one finds the components of ${\displaystyle q}$:

{\displaystyle {\begin{aligned}q_{1}&=0\\q_{2}&=\left(p_{4}\sin {\frac {\sigma _{4}}{\sigma _{1}}}+p_{1}\sigma _{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}\right)x_{3},\\q_{3}&=-\left(p_{4}\sin {\frac {\sigma _{4}}{\sigma _{1}}}+p_{1}\sigma _{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}\right)x_{2},\\q_{4}&=0\end{aligned}}}

The equations of motion thus read:

 (47a) ${\displaystyle w_{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}+w_{4}\sin {\frac {\sigma _{4}}{\sigma _{1}}}=0}$
 (47b) ${\displaystyle w_{2}=\left(p_{4}\sin {\frac {\sigma _{4}}{\sigma _{1}}}+p_{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}\right)x_{3}}$
 (47c) ${\displaystyle w_{3}=-\left(p_{4}\sin {\frac {\sigma _{4}}{\sigma _{1}}}+p_{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}\right)x_{2}}$

and these completely determine the motion together with

[ 177 ]

 (48) ${\displaystyle w_{1}^{2}=w_{4}^{2}=-1}$.

The agreement of equations (47b) and (47c) with the formulas is obvious, which express the rotation about the ${\displaystyle x}$-axis in ordinary kinematics. From (47a) it follows in connection with (46):

${\displaystyle {\frac {w_{1}}{w_{4}}}=-{\frac {x_{4}}{x_{1}}}}$

or

${\displaystyle x_{1}{\frac {dx_{1}}{d\tau }}+x_{4}{\frac {dx_{4}}{d\tau }}=0.}$

From that it follows:

 (49) ${\displaystyle x_{1}^{2}+x_{4}^{2}={\textrm {konst.}}=\xi _{1}^{2}}$

i.e. the projections of all points of the body upon the direction of translation ${\displaystyle x}$ are carrying out hyperbolic motions.

If we divide (47b) and (47c), it follows:

${\displaystyle {\frac {w_{2}}{w_{3}}}=-{\frac {x_{3}}{x_{2}}}}$

or

${\displaystyle x_{2}{\frac {dx_{2}}{d\tau }}+x_{3}{\frac {dx_{3}}{d\tau }}=0.}$

From that it follows

 (50) ${\displaystyle x_{2}^{2}+x_{3}^{2}={\textrm {konst.}}=\varrho ^{2},}$

i.e. every point of the body at motion remains on a circular cylinder, whose axis is the ${\displaystyle x}$-axis.

Now we set

 (51) {\displaystyle {\begin{aligned}x_{2}&=\varrho \cos \varphi ,\\x_{3}&=\varrho \sin \varphi ,\end{aligned}}}

where ${\displaystyle \varphi }$ is the angle measured in the resting system, around which a point ${\displaystyle x}$ has rotated away from the ${\displaystyle xy}$-plane. Then it is

 (52) {\displaystyle {\begin{aligned}{\frac {dx_{2}}{dx_{4}}}=&&-\varrho \sin \varphi {\frac {d\varphi }{dx_{4}}}=&&-{\frac {d\varphi }{dx_{4}}}x_{3},\\{\frac {dx_{3}}{dx_{4}}}=&&\varrho \cos \varphi {\frac {d\varphi }{dx_{4}}}=&&{\frac {d\varphi }{dx_{4}}}x_{2}.\end{aligned}}}

On the other hand let us consider (47b) and (47c). The equation (19) reads here

${\displaystyle p_{1}\sin {\frac {\sigma _{4}}{\sigma _{1}}}-p_{4}\cos {\frac {\sigma _{4}}{\sigma _{1}}}=0,}$

[ 178 ] from which it follows

${\displaystyle {\frac {p_{4}}{p_{1}}}=\operatorname {tg} {\frac {\sigma _{4}}{\sigma _{1}}}}$.

Consequently the brace in formulas (47b), (47c) becomes:

${\displaystyle p_{4}\sin {\frac {\sigma _{4}}{\sigma _{1}}}+p_{1}\cos {\frac {\sigma _{4}}{\sigma _{1}}}={\frac {p_{1}}{\cos {\frac {\sigma _{4}}{\sigma _{1}}}}}.}$

If we introduce instead of the proper time ${\displaystyle \tau }$ of point ${\displaystyle x}$ the quantity ${\displaystyle \xi _{4}=ic\tau }$, then we can represent the hyperbolic motion (49) as follows:

 (54) ${\displaystyle x_{1}=\xi _{1}\cos {\frac {\xi _{4}}{\xi _{1}}},\ x_{4}=\xi _{1}\sin {\frac {\xi _{4}}{\xi _{1}}},}$

from which it follows

 (55) ${\displaystyle w_{1}=-i\sin {\frac {\xi _{4}}{\xi _{1}}},\ w_{4}=i\cos {\frac {\xi _{4}}{\xi _{1}}}}$

Then it follows from (46) and (54):

${\displaystyle \operatorname {tg} {\frac {\sigma _{4}}{\sigma _{1}}}=\operatorname {tg} {\frac {\xi _{4}}{\xi _{1}}},}$

 (56) ${\displaystyle {\frac {\sigma _{4}}{\sigma _{1}}}={\frac {\xi _{4}}{\xi _{1}}}.}$

If we now divide (47b) and (47c) by ${\displaystyle w_{4}}$ and insert the values (53), (55), (56), then we obtain:

{\displaystyle {\begin{aligned}{\frac {w_{2}}{w_{4}}}&={\frac {p_{1}x_{3}}{i\cos ^{2}{\frac {\xi _{4}}{\xi _{1}}}}},\\{\frac {w_{3}}{w_{4}}}&={\frac {-p_{1}x_{2}}{i\cos ^{2}{\frac {\xi _{4}}{\xi _{1}}}}}.\end{aligned}}}

By introducing ${\displaystyle \cos ^{2}{\frac {\xi _{4}}{\xi _{1}}}=1-\left({\frac {x_{4}}{\xi _{1}}}\right)^{2}}$ (according to (54)) here as well, we obtain

 (57) {\displaystyle {\begin{aligned}{\frac {dx_{2}}{dx_{4}}}&={\frac {-ip_{1}}{1-\left({\frac {x_{4}}{\xi _{1}}}\right)^{2}}}x_{3},\\{\frac {dx_{3}}{dx_{4}}}&={\frac {ip_{1}}{1-\left({\frac {x_{4}}{\xi _{1}}}\right)^{2}}}x_{2}.\end{aligned}}}

[ 179 ] We can compare this with Ansatz (52). One finds

${\displaystyle {\frac {d\varphi }{dx_{4}}}={\frac {ip_{1}}{1-\left({\frac {x_{4}}{\xi _{1}}}\right)^{2}}}.}$

If we now denote the time of the center with ${\displaystyle t}$, so that ${\displaystyle ict=x_{4}}$, then for the angular velocity in the resting system (which is not to be mismatched with the vector ${\displaystyle {\mathfrak {p}}}$), we obtain the equation

 (58) ${\displaystyle {\frac {d\varphi }{dt}}={\frac {-{\mathfrak {p}}_{x}}{1+{\frac {c^{2}}{\xi _{1}^{2}}}t^{2}}}.}$

Since ${\displaystyle {\mathfrak {p}}_{x}}$ is given as function of ${\displaystyle t}$, one obtains ${\displaystyle \varphi }$ by quadrature. For instance, if ${\displaystyle {\mathfrak {p}}_{x}}$ is constant, one finds

 (59) ${\displaystyle \varphi =-{\frac {\xi _{1}{\mathfrak {p}}_{x}}{c}}\operatorname {arctg} {\frac {ct}{\xi _{1}}}.}$

Thus the body carries out ${\displaystyle {\frac {\xi _{1}{\mathfrak {p}}_{x}}{c}}}$ complete rotations, while it approaches from infinity, turns around and leaves into infinity again; at that occasion it only has the full angular velocity ${\displaystyle {\mathfrak {p}}_{x}}$ at ${\displaystyle t=0}$, as seen in the resting coordinate system in accordance with (58), i.e. in the moment at which it turns back and consequently is at rest. It is known that ${\displaystyle b={\tfrac {c^{2}}{\xi _{1}}}}$ is the acceleration of hyperbolic motion.[13] Thus the amount of complete rotations is ${\displaystyle {\tfrac {c{\mathfrak {p}}_{x}}{b}}}$; it is the bigger, the smaller the acceleration.

If one expands (59) with respect to powers of ${\displaystyle b}$, it follows

${\displaystyle \varphi =-{\mathfrak {p}}_{x}t\left(1-{\frac {1}{3}}{\frac {b^{2}t^{2}}{c^{2}}}+\dots \right)}$

Thus as soon as the square of ${\displaystyle bt}$ is to be neglected against the speed of light, the rotation also appears to be uniform from the resting system.

In order to keep the circumference velocity smaller than the speed of light, it must be

 (60) ${\displaystyle |{\mathfrak {p}}_{x}|<{\frac {c}{\varrho }}}$

By that, a new restriction of dimensions of rigid bodies is given, which is analogous to the one for translations given by me earlier.

1. G. Herglotz, Ann. d. Phys. 31, p. 393, 1910. F. Noether, Ann. d. Phys. 31, p. 919, 1910.
2. M. Born, Math. Ann. 68, p. 526, 1910.
3. The definition proposed by Noether, l.c. (Note 2, p. 161), that rigid bodies have 9 degrees of freedom, has the advantage over mine that no point of the body is preferred, but it seems without further ado not to be in agreement with the indicated theory of magnetization nor with the views of Minkowski as regards dynamics.
4. Here, I use the signs ordinarily used in vector analysis; e.g. ${\displaystyle {\mathfrak {AB}}}$ or also (${\displaystyle {\mathfrak {AB}}}$) is the scalar, [${\displaystyle {\mathfrak {AB}}}$] is the vector product of vectors ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$.
5. H. Minkowski, Nachrichten der Kgl. Ges. d. Wiss. zu Göttingen, Math. phys. Kl., p. 54, 1908 (printed in Math. Ann. 68, p. 472, 1910)
6. See Minkowski, (l.c. note 1), § 4.
7. Minkowski, l.c. (Note 1, p. 165).
8. See l.c. (Note 2, p. 161).
9. See G. Herglotz, l.c. (Note 2, p. 161), case A on p. 401.
10. Therefore, Earth would hardly be a "rigid" body in this sense.
11. See M. Born, l.c. (Note 1, p. 161), p. 26.