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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