Translation:Some General Remarks on the Relativity Principle

by Vladimir Ignatowski

Some general remarks on the relativity principle.

W. v. Ignatowsky (Berlin)

When Einstein introduced the relativity principle some time ago, he simultaneously assumed that the speed of light $c$ shall be a universal constant, i.e. it maintains the same value in all coordinate systems. Also Minkowski started from the invariant $r^{2}-c^{2}t^{2}$ in his investigations, although it is to be concluded from his lecture "Space and Time"^[1], that he attributed to $c$ the meaning of a universal space-time constant rather than that of the speed of light.

Now I've asked myself the question, at which relations or transformation equations one arrives when only the relativity principle is placed at the top of the investigation, and whether the Lorentzian transformation equations are the only ones at all, that satisfy the relativity principle.

In order to answer this question, we again repeat what is given to us by the relativity principle per se.

If we have two coordinate systems $K$ and $K'$, being in translatory motion with respect to each other, then the relativity principle says that both systems can be seen as equally valid, i.e. any of them can be seen as at rest and the other one as in motion. In other words: we cannot determine absolute motion.

However, if $K$ and $K'$ are equally valid, and if we can express in system $K$ any physical quantity $E$ by a function of parameters $a_{1},a_{2},a_{3}\dots$, i.e. by writing

$E=\varphi \left(a_{1},a_{2},a_{3},\dots \right)$,

(1)

then it must be possible that the corresponding quantity $E'$ in system $K'$ can be expressed by the same function $\varphi$ of the corresponding parameters $a_{1}',a_{2}',a_{3}',\dots$, i.e. it will be

$E'=\varphi \left(a_{1}',a_{2}',a_{3}',\dots \right)$.

(2)

If it is assumed that we represent $E'$ by the unprimed parameters, for instance

$E'=f\left(a_{1},a_{2},a_{3},\dots \right)$

(3)

then, since $K$ and $K'$ are equally valid, the equation

$E=f\left(a_{1}',a_{2}',a_{3}',\dots \right)$

(4)

must be correct. Equations (1) to (4) form the mathematical formulation of the relativity principle.

Furthermore, if $q$ denotes the velocity of system $K'$ with respect to $K$ as measured by the latter, and $q'$ the velocity of system $K$ as measured from $K'$, then it evidently must be

$q'=-q$.

(5)

If we now consider a purely kinematic process, i.e., where only $x,y,z$ and $t$ come into consideration, then for instance we can write the following equation

$x'=\varphi (x,y,z,t,q)$

(6)

and similar ones for $y',z'$ and $t'$. Because $x,y,z$ and $t$ are to be considered as parameters by which (among other things) a physical phenomenon can be described, and from (1) to (4) we see that $a_{1}$ in general doesn't have to be equal to $a_{1}'$.

Although the following calculations are very elementary, I only will give the reasoning and the end results in order to save space, and allude for further details to an article of mine which will appear in the Archiv f. Math. u. Phys. soon.

We denote by ${\mathfrak {c}}_{0}$ the unit vector that gives the direction of the motion of $K'$ with respect to $K$, then we lay the $X$- or the $X'$-axis into this direction, and further assume for simplification that the $X'$-axis forms the elongation of the $X$-axis. Since space is to be assumed as being homogeneous and isotropic, it can be shown from that and from reasons of symmetry that $y$ and $z$ can only implicitly occur through $r$ in equation (6), where $r$ is the distance of a point from the $X$-axis. Furthermore it can be shown that $r=r'$, and consequently $x'$ cannot depend on $r$. Therefore we can write instead of (6)

$\left.{\begin{aligned}x'=&\varphi (x,t,q)\\t'=&f(x,t,q)\end{aligned}}\right\}$

(7)

and accordingly because of (3) and (4)

$\left.{\begin{aligned}x=&\varphi (x',t',q')\\t=&f(x',t',q')\end{aligned}}\right\}$.

(8)

If we take the complete differential of (7) and (8), then it is given

$\left.{\begin{aligned}dx'=&pdx+sdt\\dt'=&p_{1}dx+s_{1}dt\end{aligned}}\right\}$

(9)

and

$\left.{\begin{aligned}dx=&p'dx'+s'dt'\\dt=&p_{1}'dx'+s_{1}'dt'\end{aligned}}\right\}$

(10)

where $p,s,p',s'$ etc. denote the corresponding partial derivatives, which we preliminarily must be consider as unknown functions of $x,t,q$ and $x',t',q'$. $D$ shall denote the determinant

$D=\left|{pp_{1} \atop ss_{1}}\right|=ps_{1}-p_{1}s$

(11)

then it follows from (9) and (10)

$\left.{\begin{aligned}p'=&{\frac {s_{1}}{D}};&&&s'=&-{\frac {s}{D}}\\p_{1}'=&-{\frac {p_{1}}{D}};&&&s_{1}'=&{\frac {p}{D}}.\end{aligned}}\right\}$

(12)

Now we take in $K$ and $K'$ two elements $dx$ and $dx'$ of such length, so that they are equal when brought to mutual rest. If we now synchronously measure $dx'$ from $K$ (thus $dt=0$), then we obtain

$dx'=pdx$

(13)

If we synchronously measure $dx$ from $K'$ (thus $dt'=0$), then it follows accordingly

$dx=p'dx'$

(14)

Both systems $K$ and $K'$ are now equally valid, and $dx$ and $dx'$ have the same length when brought to mutual rest. Consequently, the lengths measured from both systems must be equal. Thus

$p=p'$

(15)

From that and (12) it follows

$p^{2}=s_{1}s_{1}'$

(16)

Let us now follow the motion of any substantial point or any phenomenon in space, and denote the corresponding velocity by ${\mathfrak {v}}$ or ${\mathfrak {v}}'$. Then it can be simply demonstrated due to (6), that

${\mathfrak {v}}'={\frac {{\mathfrak {v}}+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {v}}+s{\mathfrak {c}}_{0}}{A}}$,

(17)

where

$A=s_{1}+p_{1}{\mathfrak {c}}_{0}{\mathfrak {v}}$.

(18)

Since ${\mathfrak {v}}$ is totally arbitrary, it is clear that $p,s$ etc. cannot depend on ${\mathfrak {v}}$. Let us assume that the movable point rests with respect to $K'$. Then ${\mathfrak {v}}'=0$ and ${\mathfrak {v}}=q{\mathfrak {c}}_{0}$. From that and from (17) we obtain

$s=-pq$

(19)

Because of the preceding things we obtain by similar considerations

$s_{1}=p$

(20)

so that we can write

$\left.{\begin{aligned}dx'&=pdx-pqdt\\dt'&=p_{1}dx+pdt.\end{aligned}}\right\}$.

(21)

It only remains to determine $p_{1}$ and $p$, because the unprimed quantities can be obtained from (12).

For that purpose we introduce a third coordinate system $K''$, which moves in the same direction ${\mathfrak {c}}_{0}$ with velocity $q_{2}$ measured in $K$. The velocity of $K'$ as measured in $K''$ is $q_{1}$. We denote by ${\overline {p_{1}}},p,q_{1}$ the quantities analogous to $p_{1},p,q$ for couple $K''K'$, and by $p'',p_{1}'',q_{2}$ the ones for couple $KK''$. Then it can easily be demonstrated that the following relation exists:

${\frac {p_{1}}{pq}}={\frac {\overline {p_{1}}}{pq_{1}}}={\frac {p_{1}''}{p''q_{2}}}$

(22)

Since every fraction contains mutually independent quantities here, we can see that it can only be a constant, which we denote by $-n$. Thus we eventually obtain

$\left.{\begin{aligned}dx'&=pdx-pqdt\\dt'&=-pqndx+pdt.\end{aligned}}\right\}$

(23)

Furthermore, it follows from (15) and (12)

$p^{2}={\frac {1}{1-q^{2}n}}$

or

$p={\frac {1}{\sqrt {1-q^{2}n}}}$

(24)

From (24) it follows, that $n$ (which we can denote as a universal space-time constant) is the reciprocal square of a velocity, thus an absolute-positive quantity.

We see that we obtained transformation equations similar to those of Lorentz, except that $n$ is used instead of ${\tfrac {1}{c^{2}}}$. However, the sign is still undetermined, because we could have set the positive sign under the square root in (24) as well.

Now, in order to determine the numerical value and the sign of $n$, we have to look at the experiment. Since we haven't used in the previous derivation any special physical phenomenon, it follows that we can determine $n$ by using an arbitrary phenomenon, and we always must obtain the same value for $n$, since $n$ is indeed a universal constant.

For instance, we can measure the length of a moving meter synchronously. If the measurement shows that it has been contracted, then the negative sign is to be chosen, and $n$ can be calculated from the contraction. However, it's known that the contraction will be so small that we cannot measure it directly.

We now turn to the electrodynamic equations and especially to the case of a uniformly moving point-charge. We know, besides the relativity principle, that the level-surface of the convection potential of the previous point-charge will be a Heaviside-Ellipsoid for the resting observer, whose axis ratio is equal to ${\sqrt {1-q^{2}/c^{2}}}$. Now we must conclude due to the relativity principle, that the level-surface of the potential is spherical for an observer co-moving with the point-charge. However, a sphere will appear to the resting observer as an ellipsoid, with an axis ratio equal to ${\sqrt {1-q^{2}n^{2}}}$. Therefore, ${\sqrt {1-q^{2}/c^{2}}}={\sqrt {1-q^{2}n^{2}}}$. This gives

$n={\frac {1}{c^{2}}}$

(25)

And only from that it follows, that $c$ is constant for all coordinate systems. At the same time we see that the universal space-time constant $n$ is determined by the numerical value of $c$.

Now it is clear that optics lost its special position with respect to the relativity principle by the previous derivation of the transformation equations. By that, the relativity principle itself gains more general importance, because it doesn't depend on a special physical phenomenon any more, but on the universal constant $n$.

Nevertheless we can grant optics or the electrodynamic equations a special position, though not in respect to the relativity principle, but in respect to the other branches of physics, namely in so far as it is possible to determine the constant $n$ from these equations.

On the other hand, when we transform the other physical equations in accordance with the relativity principle, and at that occasion see the occurrence of constant $n$, then we don't have to conclude at all that electrical forces are in play, but we only conclude from the standpoint of the relativity principle, that space and time impress their character upon all physical phenomena by means of constant $n$.

In order to illustrate the meaning of $n$ still further, we use an analogy from optics, namely the relation between image and object. From the pure optical standpoint, object and image are interchangeable. Exactly the same is true when we consider a moving measuring rod that appears to us as being contracted. We can say that space and time project this moving measuring rod to us, so that we can only see the image of it when we assume the resting measuring rod as being an object.

Thus we can completely agree with Minkowski, who says in his lecture "Space and Time"^[2]: "The contraction is not, for example, to be seen as the consequence of resistances in the aether, but as a present from above, as a concomitant of the circumstance of motion", exactly because $n$ is a universal constant.

At the end, I want to mention the velocities possible from the standpoint of the relativity principle.

Consider expression (24) for $p$.

The contraction that we observe at a distance moving with system $K'$ is depending on $p$. Thus it has no meaning to assume that $p$ can become imaginary, i.e., $q$ must always be smaller than $c$. But what is the meaning of $q$? $q$ means the velocity of the coordinate system $K'$, thus it cannot be larger than $c$. In other words: none of the rest coordinate systems can move with superluminal velocity. But we may not see a rest coordinate system only as a mathematical structure, instead we have to think about a material world with its observers and synchronous clocks. Conversely we assume that we can transform any substantial point to rest. By that it follows that a substantial point cannot move with superluminal velocity.

Now the question arises: Is there a velocity, not of substantial points but of phenomena, that are larger than the speed of light, neglecting phase or group velocities? We have to affirm this question.

Without going into details, for which I refer to my last paper in Ann. d. Phys.^[3], I only want to explain this question shortly by an example.

The following can be derived from Lorentz's transformation equations.

$x_{2}-x_{1}$ mean the distance of two fixed points in system $K$ in the direction of ${\mathfrak {c}}_{0}$. This distance is now synchronously measured by the moving observer in $K'$, by which he obtains the distance $l'$. By that, the two synchronous clocks which were placed by the observer in $K'$ at both ends of $l'$ in order to make a synchronous measurement of $x_{2}-x_{1}$, will indicate a time difference $t_{2}-t_{1}$, equal to:

$t_{2}-t_{1}=qn\left(x_{2}-x_{1}\right)$,

(26)

from that if follows

${\frac {x_{2}-x_{1}}{t_{2}-t_{1}}}={\frac {1}{qn}}={\frac {c^{2}}{q}}$

(27)

Now we imagine a rod of length $l'$ in $K'$, and assume that the observers in $K'$ would arrange to elevate the rod at the same time (by aid of their synchronous clocks) perpendicular to ${\mathfrak {c}}_{0}$. However, this won't happen at the same time for the resting observer, namely it will appear to him as being elevated in the moment at which one end of the rod coincides with $x_{1}$, and the other end only then when it coincides with point $x_{2}$, thus after the time $t_{2}-t_{1}$ which is calculated from (26). For the resting observer, the bending of the rod propagates with velocity $V={\tfrac {x_{2}-x_{1}}{t_{2}-t_{1}}}$. From (28) we obtain

$V={\frac {1}{qn}}={\frac {c^{2}}{q}}>c$

(28)

since $q$ is always smaller than $c$, as remarked earlier.

We now consider equation (17), which we can write (because all quantities $p,s,p_{1}$ and $s_{1}$ are known to us) as follows:

${\mathfrak {v}}'={\frac {{\mathfrak {v}}+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {v}}-pq{\mathfrak {c}}_{0}}{p(1-qn{\mathfrak {c}}_{0}{\mathfrak {v}})}}$

(29)

Now we assume that something is moving in the direction of ${\mathfrak {c}}_{0}$ with velocity $v$. Then ${\mathfrak {v}}=v{\mathfrak {c}}_{0}$, and we obtain from (29), when we simultaneously cancel the unit vector ${\mathfrak {c}}_{0}$,

$v'={\frac {v-q}{1-vqn}}$

(30)

From (28) we see, that for us (the resting observer) the bending is moving the quicker, the slower $K'$ is moving with respect to $K$. If we now imagine that we play the role of the moving observer, then the propagation velocity of the bending will appear to us as infinitely large, because the observers elevate the rod at the same time according to their synchronous clocks. The same result can also be obtained from (30). There we replace the value of $v$ by $V$ from (28), thus it is given $v'=\infty$.

Thus we can say, in order to characterize the meaning of $V$ more clearer, that $V$ is the velocity which is required in system $K$ in order to catch up the time in system $K'$.

The existence of velocity

$V={\frac {1}{qn}}={\frac {c^{2}}{q}}>c$

is to be seen as a consequence of the relativity principle, and in particular it is given as the direct consequence of the concept of synchronous clocks and synchronous measurements.

(Received September 23, 1910.)

Discussion.

Sommerfeld: The impossibility of superluminal velocities in connection with process velocities is concluded by Einstein from the circumstance, that (as he drastically says) one can telegraph into the past by superluminal velocities. Let's say that this is related to the signal velocity, not the velocity of an arbitrary process. There are without doubt many processes that are allowed to propagate with superluminal velocity in relativity theory as well. For instance, in anomalous dispersing bodies the phase of light propagates with a velocity that can be faster than light. This is certainly not a contradiction against the relativity principle, because one cannot give a signal by a continuous periodic wave train. Recently Einstein has communicated to me another simple example in which there is superluminal velocity as well, but also in this case it's not about a "signal velocity". Imagine two rulers mutually inclined under a very acute angle, then move one of them with a velocity of 1cm/unit against the other one, then the intersection on the other one propagates with arbitrary large velocity. Isn't it the case that the example of the lecturer has more similarity with this process than with a signal?

Lecturer: Certainly; I didn't call it signal velocity. When we set up two hooks, then (in case we move the rod) the bending will touch the two hooks, and we can measure the velocity of the bending. For the time being I will let it undecided as to whether one can transfer signals by that. I discovered this by my investigations concerning rigid bodies (see Annalen der Physik 33, 607, 1910), in which the rigid body is treated in accordance with Euler's method. There it is given, that the velocity propagates within a rigid body in its own direction with velocity $c^{2}/v$.

Sommerfeld: I believe that the concept of rigid bodies must be modified in so far, that the reactions in its interior cannot propagate with superluminal velocity.

Lecturer: When we assume Born's differential equation, then we come to a propagation velocity larger than that of light. We have a volume element that we consider as being rigid. This volume element at its motion, transformed to rest, conserves its form. This is Born's differential equation. If we keep this condition, then superluminal velocity is given. For the time being we don't know as to how the body will actually move since we can make no statement about the interior forces of the body. Born's differential equations is only one condition which must be satisfied by the motion of rigid bodies.

Born: I want to add only some words to the remark concerning rigid bodies. Within such a body, every action is indeed propagating momentarily for a co-moving observer, and thus with superluminal velocity for a moving observer. It it were possible to show, that it contradicts the relativity principle by "telegraphing into the past" in this way, then the concept of the rigid body must surely be abandoned. The differential expressions by whose disappearance I have defined rigidity would still be useful, since they are a measure of the deformation of the volume element, so they should provide the foundation for an elasticity theory in accordance with the relativity principle. As to how the "propagation with superluminal velocity" mentioned by the lecturer arises, can be made quite illustrative in Minkowski's representation. I imagine a rod of length $l$ (in the following the speaker is drawing at the table) and lay it parallel to the $y$-axis; perpendicular to the $xy$-plane I include time as the third coordinate in order to describe the successive locations of the rod. When the rod is at rest, it will obviously represented by primes lying one above the other in the $t$-direction, and which can be combined to a plane belt parallel to the $yt$-plane. If one now gives the rod a motion in the direction of the $x$-axis, then this results in bending the belt towards the $x$-direction. If the rod should be at rest again at the end, then the belt will eventually be parallel to the $yt$-plane again. In order to show now that the process is propagating with superluminal speed, I take a new coordinate system in motion, that is, I introduce oblique parallel coordinates into our figure in accordance with the Lorentz transformation, at which the $t'$-axis is inclined against the $t$-axis and the $x'y'$-plane against the $xy$-plane. In this system, simultaneous events are represented by planes which are lying obliquely to the old $xy$-plane. If one intersects the bent sheet previously constructed with such an oblique plane, then the intersection is evidently not straight, but has a bending that is moving to the right when one moves the intersecting plane towards above, that is, it is moving to the right with superluminal velocity. The other example mentioned by v. Ignatowsky, concerns the motion denoted by me as hyperbolic motion (the following is again illustrated by a drawing). It will be represented in the $xt$-plane by a bundle of hyperbolas having line $x=\pm ct$ as their asymptote. It satisfies my differential equations of rigidity and one can easily see, that the points having the same velocity are lying on the line passing through the origin. However, these lines are oblique and their inclination is smaller than that of the line $x=\pm ct$. The place where the velocity $q$ is present, thus propagates with superluminal velocity through the body.

Lecturer: I have wanted to add some words about this matter. Einstein arrives at the following formula (Annalen der Physik 23, 381, 1907),

$T=l{\frac {1-Wv/V^{2}}{W-v}}$

and says, that when $W$ is larger than $V$ (Einstein denoted by $V$ which was denoted by $c$ in this lecture), one can always choose $v$ so that $T$ is smaller than zero, thus $T$ becomes negative. That is, one could telegraph into the past. However, if one sets $W$ equal to $V^{2}/v$ ($c^{2}/q$ in this lecture), then the numerator becomes equal to zero here. That is, the velocity becomes infinite for the coordinate system in which the rod is located. Because this velocity $(c^{2}/q)$ is nothing other than the velocity necessary to catch up the time in the moving coordinate system (in which the rod is located) as seen from our viewpoint (in the resting system). Because if we had a clock moving with this velocity as seen from our viewpoint, then it moves with infinitely large velocity for the coordinate system (in which the rod is located) according to the above; this boils down to the circumstance, that everywhere in the system it is simultaneously present. Thus there is an arbitrary amount of synchronous clocks, from which it follows that this velocity is actually the velocity to catch up the time in the moving system. Thus the value $c^{2}/q$ is the limiting velocity, at which the numerator of the previous formula becomes zero, thus $T$ equal to $O$, but not negative. A larger velocity is just not imaginable.

1. This journal 10, 104, 1909.
2. l. c. p. 106.
3. Vol. 33, 607, 1910.