Eight Lectures on Theoretical Physics/VIII

In the lecture of yesterday we saw, by means of examples, that all continuous reversible processes of nature may be represented as consequences of the principle of least action, and that the whole course of such a process is uniquely determined as soon as we know, besides the actions which are exerted upon the system from without, the kinetic potential $$H$$ as a function of the generalized coordinates and their differential coefficients with respect to time. The determination of this function remains then as a special problem, and we recognize here a rich field for further theories and hypotheses. It is my purpose to discuss with you today an hypothesis which represents a magnificent attempt to establish quite generally the dependency of the kinetic potential $$H$$ upon the velocities, and which is commonly designated as the principle of relativity. The gist of this principle is: it is in no wise possible to detect the motion of a body relative to empty space; in fact, there is absolutely no physical sense in speaking of such a motion. If, therefore, two observers move with uniform but different velocities, then each of the two with exactly the same right may assert that with respect to empty space he is at rest, and there are no physical methods of measurement enabling us to decide in favor of the one or the other. The principle of relativity in its generalized form is a very recent development. The preparatory steps were taken by H. A. Lorentz, it was first generally formulated by A. Einstein, and was developed into a finished mathematical system by H. Minkowski. However, traces of it extend quite far back into the past, and therefore it seems desirable first to say something concerning the history of its development.

The principle of relativity has been recognized in mechanics since the time of Galilee and Newton. It is contained in the form of the simple equations of motion of a material point, since these contain only the acceleration and not the velocity of the point. If, therefore, we refer the motion of the point, first to the coordinates $$x$$, $$y$$, $$z$$, and again to the coordinates $$x'$$, $$y'$$, $$z'$$ of a second system, whose axes are directed parallel to the first and which moves with the velocity $$\nu$$ in the direction of the positive $$x$$-axis:
 * $$\begin{align}&(65){\color{White}.}\qquad&&

x' = x - \nu t,\quad y' = y,\quad z' = z, \end{align}$$ and the form of the equations of motion is not changed in the slightest. Nothing short of the assumption of the general validity of the relativity principle in mechanics can justify the inclusion by physics of the Copernican cosmical system, since through it the independence of all processes upon the earth of the progressive motion of the earth is secured. If one were obliged to take account of this motion, I should have, e. g., to admit that the piece of chalk in my hand possesses an enormous kinetic energy, corresponding to a velocity of something like $$30$$ kilometers per second.

It was without doubt his conviction of the absolute validity of the principle of relativity which guided Heinrich Hertz in the establishment of his fundamental equations for the electrodynamics of moving bodies. The electrodynamics of Hertz is, in fact, wholly built upon the principle of relativity. It recognizes no absolute motion with regard to empty space. It speaks only of motions of material bodies relative to one another. In accordance with the theory of Hertz, all electrodynamic processes occur in material bodies; if these move, then the electrodynamic processes occurring therein move with them. To speak of an independent state of motion of a medium outside of material bodies, such as the ether, has just as little sense in the theory of Hertz as in the modern theory of relativity.



But the theory of Hertz has led to various contradictions with experience. I will refer here to the most important of these. Fizeau brought (1851) into parallelism a bundle of rays originating in a light source $$L$$ by means of a lens and then brought it to a focus by means of a second lens upon a screen $$S$$ (Fig. 2). In the path of the parallel light rays between the two lenses he placed a tube system of such sort that a transparent liquid could be passed through it, and in such manner that in one half (the upper) the light rays would pass in the direction of flow of the liquid while in the other half (the lower), the rays would pass in the opposite direction.

If now a liquid or a gas flow through the tube system with the velocity $$\nu$$, then, in accordance with the theory of Hertz, since light must be a process in the substance, the light waves must be transported with the velocity of the liquid. The velocity of light relative to $$L$$ and $$S$$ is, therefore, in the upper part $$q_{0} + \nu$$, and the lower part $$q_{0} - \nu$$, if $$q_{0}$$ denote the velocity of light relative to the liquid. The difference of these two velocities, $$2\nu$$, should be observable at $$S$$ through corresponding interference of the lower and the upper light rays, and quite independently of the nature of the flowing substance. Experiment did not confirm this conclusion. Moreover, it showed in gases generally no trace of the expected action; i. e., light is propagated in a flowing gas in the same manner as in a gas at rest. On the other hand, in the case of liquids an effect was certainly indicated, but notably smaller in amount than that demanded by the theory of Hertz. Instead of the expected velocity difference $$2\nu$$, the difference $$2\nu(1 - 1/n^{2})$$ only was observed, where $$n$$ is the refractive index of the liquid. The factor $$(1 - 1/n^{2})$$ is called the Fresnel coefficient. There is contained (for $$n = 1$$) in this expression the result obtained in the case of gases.

It follows from the experiment of Fizeau that, as regards electrodynamic processes in a gas, the motion of the gas is practically immaterial. If, therefore, one holds that electrodynamic processes require for their propagation a substantial carrier, a special medium, then it must be concluded that this medium, the ether, remains at rest when the gas moves in an arbitrary manner. This interpretation forms the basis of the electrodynamics of Lorentz, involving an absolutely quiescent ether. In accordance with this theory, electrodynamic phenomena have only indirectly to do with the motion of matter. Primarily all electrodynamical actions are propagated in ether at rest. Matter influences the propagation only in a secondary way, so far as it is the cause of exciting in greater or less degree resonant vibrations in its smallest parts by means of the electrodynamic waves passing through it. Now, since the refractive properties of substances are also influenced through the resonant vibrations of its smallest particles, there results from this theory a definite connection between the refractive index and the coefficient of Fresnel, and this connection is, as calculation shows, exactly that demanded by measurements. So far, therefore, the theory of Lorentz is confirmed through experience, and the principle of relativity is divested of its general significance.

The principle of relativity was immediately confronted by a new difficulty. The theory of a quiescent ether admits the idea of an absolute velocity of a body, namely the velocity relative to the ether. Therefore, in accordance with this theory, of two observers $$A$$ and $$B$$ who are in empty space and who move relatively to each other with the uniform velocity $$\nu$$, it would be at best possible for only one rightly to assert that he is at rest relative to the ether. If we assume, e. g., that at the moment at which the two observers meet an instantaneous optical signal, a flash, is made by each, then an infinitely thin spherical wave spreads out from the place of its origin in all directions through empty space. If, therefore, the observer $$A$$ remain at the center of the sphere, the observer $$B$$ will not remain at the center and, as judged by the observer $$B$$, the light in his own direction of motion must travel (with the velocity $$c - \nu$$) more slowly than in the opposite direction (with the velocity $$c + \nu$$), or than in a perpendicular direction (with the velocity $$\sqrt{c^{2} - \nu^{2}}$$) (cf. Fig. 3). Under suitable conditions the observer $$B$$ should be able to detect and measure this sort of effect.



This elementary consideration led to the celebrated attempt of Michelson to measure the motion of the earth relative to the ether. A parallel beam of rays proceeding from $$L$$ (Fig. 4) falls upon a transparent plane parallel plate $$P$$ inclined at $$45^\circ$$, by which it is in part transmitted and in part reflected. The transmitted and reflected beams are brought into interference by reflection from suitable metallic mirrors $$S_{1}$$ and $$S_{2}$$, which are removed by the same distance $$l$$ from $$P$$. If, now, the earth with the whole apparatus moves in the direction $$PS_{1}$$ with the velocity $$\nu$$, then the time which the light needs in order to go from $$P$$ to $$S_{1}$$ and back is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{l}{c - \nu} + \frac{l}{c + \nu} = \frac{2l}{c} \left(1 + \frac{\nu^{2}}{c^{2}} + \cdots\right). \end{align}$$ On the other hand, the time which the light needs in order to pass from $$P$$ to $$S_{2}$$ and back to $$P$$ is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{l}{\sqrt{c^{2} - \nu^{2}}} + \frac{l}{\sqrt{c^{2} - \nu^{2}}} = \frac{2l}{c} \left(1 + \frac{1}{2} \frac{\nu^{2}}{c^{2}} + \cdots\right). \end{align}$$ If, now, the whole apparatus be turned through a right angle, a noticeable displacement of the interference bands should result, since the time for the passage over the path $$PS_{2}$$ is now longer. No trace was observed of the marked effect to be expected.



Now, how will it be possible to bring into line this result, established by repeated tests with all the facilities of modern experimental art? E. Cohn has attempted to find the necessary compensation in a certain influence of the air in which the rays are propagated. But for anyone who bears in mind the great results of the atomic theory of dispersion and who does not renounce the simple explanation which this theory gives for the dependence of the refractive index upon the color, without introducing something else in its place, the idea that a moving absolutely transparent medium, whose refractive index is absolutely $$= 1$$, shall yet have a notable influence upon the velocity of propagation of light, as the theory of Cohn demands, is not possible of assumption. For this theory distinguishes essentially a transparent medium, whose refractive index is $$= 1$$, from a perfect vacuum. For the former the velocity of propagation of light in the direction of the velocity $$\nu$$ of the medium with relation to an observer at rest is
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

q = c + \frac{\nu^{2}}{c}, \end{align}$$ for a vacuum, on the other hand, $$q = c$$. In the former medium, Cohn's theory of the Michelson experiment predicts no effect, but, on the other hand, the Michelson experiment should give a positive effect in a vacuum.

In opposition to E. Cohn, H. A. Lorentz and FitzGerald ascribe the necessary compensation to a contraction of the whole optical apparatus in the direction of the earth's motion of the order of magnitude $$\nu^{2}/c^{2}$$. This assumption allows better of the introduction again of the principle of relativity, but it can first completely satisfy this principle when it appears, not as a necessary hypothesis made to fit the present special case, but as a consequence of a much more general postulate. We have to thank A. Einstein for the framing of this postulate and H. Minkowski for its further mathematical development.

Above all, the general principle of relativity demands the renunciation of the assumption which led H. A. Lorentz to the framing of his theory of a quiescent ether; the assumption of a substantial carrier of electromagnetic waves. For, when such a carrier is present, one must assume a definite velocity of a ponderable body as definable with respect to it, and this is exactly that which is excluded by the relativity principle. Thus the ether drops out of the theory and with it the possibility of mechanical explanation of electrodynamic processes, i. e., of referring them to motions. The latter difficulty, however, does not signify here so much, since it was already known before, that no mechanical theory founded upon the continuous motions of the ether permits of being completely carried through (cf. previous lecture). In place of the so-called free ether there is now substituted the absolute vacuum, in which electromagnetic energy is independently propagated, like ponderable atoms. I believe it follows as a consequence that no physical properties can be consistently ascribed to the absolute vacuum. The dielectric constant and the magnetic permeability of a vacuum have no absolute meaning, only relative. If an electrodynamic process were to occur in a ponderable medium as in a vacuum, then it would have absolutely no sense to distinguish between field strength and induction. In fact, one can ascribe to the vacuum any arbitrary value of the dielectric constant, as is indicated by the various systems of units. But how is it now with regard to the velocity of propagation of light? This also is not to be regarded as a property of the vacuum, but as a property of electromagnetic energy which is present in the vacuum. Where there is no energy there can exist no velocity of propagation.

With the complete elimination of the ether, the opportunity is now pre\-sent for the framing of the principle of relativity. Obviously, we must, as a simple consideration shows, introduce something radically new. In order that the moving observer $$B$$ mentioned above (Fig. 3) shall not see the light signal given by him travelling more slowly in his own direction of motion (with the velocity $$c - \nu$$) than in the opposite direction (with the velocity $$c + \nu$$), it is necessary that he shall not identify the instant of time at which the light has covered the distance $$c - \nu$$ in the direction of his own motion with the instant of time at which the light has covered the distance $$c + \nu$$ in the opposite direction, but that he regard the latter instant of time as later. In other words: the observer $$B$$ measures time differently from the observer $$A$$. This is a priori quite permissible; for the relativity principle only demands that neither of the two observers shall come into contradiction with himself. However, the possibility is left open that the specifications of time of both observers may be mutually contradictory.

It need scarcely be emphasized that this new conception of the idea of time makes the most serious demands upon the capacity of abstraction and the projective power of the physicist. It surpasses in boldness everything previously suggested in speculative natural phenomena and even in the philosophical theories of knowledge: non-euclidean geometry is child's play in comparison. And, moreover, the principle of relativity, unlike non-euclidean geometry, which only comes seriously into consideration in pure mathematics, undoubtedly possesses a real physical significance. The revolution introduced by this principle into the physical conceptions of the world is only to be compared in extent and depth with that brought about by the introduction of the Copernican system of the universe.

Since it is difficult, on account of our habitual notions concerning the idea of absolute time, to protect ourselves, without special carefully considered rules, against logical mistakes in the necessary processes of thought, we shall adopt the mathematical method of treatment. Let us consider then an electrodynamic process in a pure vacuum; first, from the standpoint of an observer $$A$$; secondly, from the standpoint of an observer $$B$$, who moves relatively to observer $$A$$ with a velocity $$\nu$$ in the direction of the $$x$$-axis. Then, if $$A$$ employ the system of reference $$x$$, $$y$$, $$z$$, $$t$$, and $$B$$ the system of reference $$x'$$, $$y'$$, $$z'$$, $$t'$$, our first problem is to find the relations among the primed and the unprimed quantities. Above all, it is to be noticed that since both systems of reference, the primed and the unprimed, are to be like directed, the equations of transformation between corresponding quantities in the two systems must be so established that it is possible through a transformation of exactly the same kind to pass from the first system to the second, and conversely, from the second back to the first system. It follows immediately from this that the velocity of light $$c'$$ in a vacuum for the observer $$B$$ is exactly the same as for the observer $$A$$. Thus, if $$c'$$ and $$c$$ are different, $$c' > c$$, say, it would follow that: if one passes from one observer $$A$$ to another observer $$B$$ who moves with respect to $$A$$ with uniform velocity, then he would find the velocity of propagation of light for $$B$$ greater than for $$A$$. This conclusion must likewise hold quite in general independently of the direction in which $$B$$ moves with respect to $$A$$, because all directions in space are equivalent for the observer $$A$$. On the same grounds, in passing from $$B$$ to $$A$$, $$c$$ must be greater than $$c'$$, for all directions in space for the observer $$B$$ are now equivalent. Since the two inequalities contradict, therefore $$c'$$ must be equal to $$c$$. Of course this important result may be generalized immediately, so that the totality of the quantities independent of the motion, such as the velocity of light in a vacuum, the constant of gravitation between two bodies at rest, every isolated electric charge, and the entropy of any physical system possess the same values for both observers. On the other hand, this law does not hold for quantities such as energy, volume, temperature, etc. For these quantities depend also upon the velocity, and a body which is at rest for $$A$$ is for $$B$$ a moving body.

We inquire now with regard to the form of the equations of transformation between the unprimed and the primed coordinates. For this purpose let us consider, returning to the previous example, the propagation, as it appears to the two observers $$A$$ and $$B$$, of an instantaneous signal creating an infinitely thin light wave which, at the instant at which the observers meet, begins to spread out from the common origin of coordinates. For the observer $$A$$ the wave travels out as a spherical wave:
 * $$\begin{align}&(66){\color{White}.}\qquad&&

x^{2} + y^{2} + z^{2} - c^{2}t^{2} = 0. \end{align}$$ For the second observer $$B$$ the same wave also travels as a spherical wave with the same velocity:
 * $$\begin{align}&(67){\color{White}.}\qquad&&

x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2} = 0; \end{align}$$ for the first observer has no advantage over the second observer. $$B$$ can exactly with the same right as $$A$$ assert that he is at rest at the center of the spherical wave, and for $$B$$, after unit time, the wave appears as in Fig. 5, while its appearance for the observer $$A$$ after unit time, is represented by Fig. 3.



The equations of transformation must therefore fulfill the condition that the two last equations, which represent the same physical process, are compatible with each other; and furthermore: the passage from the unprimed to the primed quantities must in no wise be distinguished from the reverse passage from the primed to the unprimed quantities. In order to satisfy these conditions, we generalize the equations of transformation $$(65)$$, set up at the beginning of this lecture for the old mechanical principle of relativity, in the following manner:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

x' = \kappa(x - \nu t),\quad y' = \lambda y,\quad z' = \mu z,\quad t' = \nu t + \rho x. \end{align}$$ Here $$\nu$$ denotes, as formerly, the velocity of the observer $$B$$ relative to $$A$$ and the constants $$\kappa$$, $$\lambda$$, $$\mu$$, $$\nu$$, $$\rho$$ are yet to be determined. We must have:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

x = \kappa' (x' - \nu' t'),\quad y = \lambda' y',\quad z = \mu' z',\quad t = \nu' t' + \rho' x'. \end{align}$$ It is now easy to see that $$\lambda$$ and $$\lambda'$$ must both $$= 1$$. For, if, e. g., $$\lambda$$ be greater than $$1$$, then $$\lambda'$$ must also be greater than $$1$$; for the two transformations are equivalent with regard to the $$y$$ axis. In particular, it is impossible that $$\lambda$$ and $$\lambda'$$ depend upon the direction of motion of the other observer. But now, since, in accordance with what precedes, $$\lambda = 1/\lambda'$$, each of the two inequalities contradict and therefore $$\lambda = \lambda' = 1$$; likewise, $$\mu = \mu' = 1$$. The condition for identity of the two spherical waves then demands that the expression $$(66)$$:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

x^{2} + y^{2} + z^{2} - c^{2}t^{2} \end{align}$$ become, through the transformation of coordinates, identical with the expression $$(67)$$:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2}, \end{align}$$ and from this the equations of transformation follow without ambiguity:
 * $$\begin{align}&(68){\color{White}.}\qquad&&

x' = \kappa (x - \nu t),\quad y' = y,\quad z' = z,\quad t' = \kappa \left(t - \frac{\nu}{c^{2}} x\right), \end{align}$$ wherein
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\kappa = \frac{c}{\sqrt{c^{2} - \nu^{2}}}. \end{align}$$ Conversely:
 * $$\begin{align}&(69){\color{White}.}\qquad&&

x = \kappa (x' + \nu t'),\quad y = y',\quad z = z',\quad t = \kappa \left(t' + \frac{\nu}{c^{2}} x'\right). \end{align}$$

These equations permit quite in general of the passage from the system of reference of one observer to that of the other (H. A. Lorentz), and the principle of relativity asserts that all processes in nature occur in accordance with the same laws and with the same constants for both observers (A. Einstein). Mathematically considered, the equations of transformation correspond to a rotation in the four dimensional system of reference $$(x, y, z, ict)$$ through the imaginary angle $$\operatorname{arctg} (i(\nu/c))$$ (H. Minkowski). Accordingly, the principle of relativity simply teaches that there is in the four dimensional system of space and time no special characteristic direction, and any doubts concerning the general validity of the principle are of exactly the same kind as those concerning the existence of the antipodians upon the other side of the earth.

We will first make some applications of the principle of relativity to processes which we have already treated above. That the result of the Michelson experiment is in agreement with the principle of relativity, is immediately evident; for, in accordance with the relativity principle, the influence of a uniform motion of the earth upon processes on the earth can under no conditions be detected.

We consider now the Fizeau experiment with the flowing liquid (see Fig. 2). If the velocity of propagation of light in the liquid at rest be again $$q_{0}$$, then, in accordance with the relativity principle, $$q_{0}$$ is also the velocity of the propagation of light in the flowing liquid for an observer who moves with the liquid, in case we disregard the dispersion of the liquid; for the color of the light is different for the moving observer. If we call this observer $$B$$ and the velocity of the liquid as above, $$\nu$$, we may employ immediately the above formulae in the calculation of the velocity of propagation of light in the flowing liquid, judged by an observer $$A$$ at the screen $$S$$. We have only to put
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{dx'}{dt'} = x' = q_{0}, \end{align}$$ to seek the corresponding value of
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{dx}{dt} = \dot{x}. \end{align}$$ For this obviously gives the velocity sought.

Now it follows directly from the equations of transformation $$(69)$$ that:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{dx}{dt} = \dot{x} = \frac{\dot{x}' + \nu}{1 + \dfrac{\nu \dot{x}'}{c^{2}}}, \end{align}$$ and, therefore, through appropriate substitution, the velocity sought in the upper tube, after neglecting higher powers in $$\nu/c$$ and $$\nu/q_{0}$$, is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\dot{x} = \frac{q_{0} + \nu}{1 + \dfrac{\nu q_{0}}{c^{2}}} = q_{0} + \nu \left(1 - \frac{q_{0}^2}{c^{2}}\right), \end{align}$$ and the corresponding velocity in the lower tube is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

q_{0} - \nu \left(1 - \frac{q_{0}^{2}}{c^{2}}\right). \end{align}$$ The difference of the two velocities is
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

2\nu \left(1 - \frac{q_{0}^{2}}{c^{2}}\right) = 2\nu \left(1 - \frac{1}{n^{2}}\right), \end{align}$$ which is the Fresnel coefficient, in agreement with the measurements of Fizeau.

The significance of the principle of relativity extends, not only to optical and other electrodynamic phenomena, but also to all processes of ordinary mechanics; but the familiar expression ($$\frac{1}{2} mq^{2}$$) for the kinetic energy of a mass point moving with the velocity $$q$$ is incompatible with this principle.

But, on the other hand, since all mechanics as well as the rest of physics is governed by the principle of least action, the significance of the relativity principle extends at bottom only to the particular form which it prescribes for the kinetic potential $$H$$, and this form, though I will not stop to prove it, is characterized by the simple law that the expression
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

H \cdot dt \end{align}$$ for every space element of a physical system is an invariant
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

= H' \cdot dt' \end{align}$$ with respect to the passage from one observer $$A$$ to the other observer $$B$$ or, what is the same thing, the expression $$H/\sqrt{c^{2} - q^{2}}$$ is in this passage an invariant $$= H'/\sqrt{c^{2} - q'^{2}}$$.

Let us now make some applications of this very general law, first to the dynamics of a single mass point in a vacuum, whose state is determined by its velocity $$q$$. Let us call the kinetic potential of the mass point for $$q = 0$$, $$H_{0}$$, and consider now the point at an instant when its velocity is $$q$$. For an observer $$B$$ who moves with the velocity $$q$$ with respect to the observer $$A$$, $$q' = 0$$ at this instant, and therefore $$H' = H_{0}$$. But now since in general:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{H}{\sqrt{c^{2} - q^{2}}} = \frac{H'}{\sqrt{c^{2} - q'^{2}}}, \end{align}$$ we have after substitution:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

H = \sqrt{1 - \frac{q^{2}}{c^{2}}} \cdot H_{0} = \sqrt{1 - \frac{\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}}{c^{2}}} \cdot H_{0}. \end{align}$$ With this value of $$H$$, the Lagrangian equations of motion $$(59)$$ of the previous lecture are applicable.

In accordance with $$(60)$$, the kinetic energy of the mass point amounts to:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

E = \dot{x} \frac{\partial H}{\partial \dot{x}} + \dot{y} \frac{\partial H}{\partial \dot{y}} + \dot{z} \frac{\partial H}{\partial \dot{z}} - H = q \frac{\partial H}{\partial q} - H  = - \frac{H_{0}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}}, \end{align}$$ and the momentum to:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

G = \frac{\partial H}{\partial q} = -\frac{q H_{0}}{c \sqrt{c^{2} - q^{2}}}. \end{align}$$ $$G/q$$ is called the transverse mass $$m_{t}$$, and $$dG/dq$$ the longitudinal mass $$m_{l}$$ of the point; accordingly:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

m_{t} = -\frac{H_{0}}{c \sqrt{c^{2} - q^{2}}}, \quad m_{l} = -\frac{c H_{0}}{(c^{2} - q^{2})^{3/2}}. \end{align}$$ For $$q = 0$$, we have
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

m_{t} = m_{l} = m_{0} = -\frac{H_{0}}{c^{2}}. \end{align}$$ It is apparent, if one replaces in the above expressions the constant $$H_{0}$$ by the constant $$m_{0}$$, that the momentum is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

G = \frac{m_{0}q}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}} \end{align}$$ and the transverse mass:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

m_{t} = \frac{m_{0}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}}, \end{align}$$ and the longitudinal mass:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

m_{l} = \frac{m_{0}}{\left(1 - \dfrac{q^{2}}{c^{2}}\right)^{3/2}}, \end{align}$$ and, finally, that the kinetic energy is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

E = \frac{m_{0} c^{2}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}} = m_{0}c^{2} + \tfrac{1}{2} m_{0}q^{2} + \cdots. \end{align}$$ The familiar value of ordinary mechanics $$\tfrac{1}{2} m_{0}q^{2}$$ appears here therefore only as an approximate value. These equations have been experimentally tested and confirmed through the measurements of A. H. Bucherer and of E. Hupka upon the magnetic deflection of electrons.

A further example of the invariance of $$H \cdot dt$$ will be taken from electrodynamics. Let us consider in any given medium any electromagnetic field. For any volume element $$V$$ of the medium, the law holds that $$V \cdot dt$$ is invariant in the passage from the one to the other observer. It follows from this that $$H/V$$ is invariant; i. e., the kinetic potential of a unit volume or the “space density of kinetic potential” is invariant.

Hence the following relation exists;
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\mathfrak{E} \mathfrak{D} - \mathfrak{H} \mathfrak{B} = \mathfrak{E}' \mathfrak{D}' - \mathfrak{H}' \mathfrak{B}', \end{align}$$ wherein $$\mathfrak{E}$$ and $$\mathfrak{H}$$ denote the field strengths and $$\mathfrak{D}$$ and $$\mathfrak{B}$$ the corresponding inductions. Obviously a corresponding law for the space energy density $$\mathfrak{E} \mathfrak{D} + \mathfrak{H} \mathfrak{B}$$ will not hold.

A third example is selected from thermodynamics. If we take the velocity $$q$$ of a moving body, the volume $$V$$ and the temperature $$T$$ as independent variables, then, as I have shown in the previous lecture, we shall have for the pressure $$p$$ and the entropy $$S$$ the following relations:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{\partial H}{\partial V} = p \quad\text{and}\quad \frac{\partial H}{\partial T} = S. \end{align}$$ Now since $$V/\sqrt{c^{2} - q^{2}}$$ is invariant, and $$S$$ likewise invariant (see above), it follows from the invariance of $$H/\sqrt{c^{2} - q^{2}}$$ that $$p$$ is invariant and also that $$T/\sqrt{c^{2} - q^{2}}$$ is invariant, and hence that:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

p = p' \quad\text{and}\quad \frac{T}{\sqrt{c^{2} - q^{2}}} = \frac{T'}{\sqrt{c^{2} - q'^{2}}}. \end{align}$$

The two observers $$A$$ and $$B$$ would estimate the pressure of a body as the same, but the temperature of the body as different.

A special case of this example is supplied when the body considered furnishes a black body radiation. The black body radiation is the only physical system whose dynamics (for quasi-stationary processes) is known with absolute accuracy. That the black body radiation possesses inertia was first pointed out by F. Hasenöhrl. For black body radiation at rest the energy $$E_{0} = a T^{4}V$$ is given by the Stefan-Boltzmann law, and the entropy $$S_{0} = \int (dE_{0}/T) = \tfrac{4}{3} aT^{3}V$$, and the pressure $$p_{0} = (a/3)T^{4}$$, and, therefore, in accordance with the above relations, the kinetic potential is:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

H_{0} = \frac{a}{3} T^{4} V. \end{align}$$ Let us imagine now a black body radiation moving with the velocity $$q$$ with respect to the observer $$A$$ and introduce an observer $$B$$ who is at rest ($$q = 0$$) with reference to the black body radiation; then:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

\frac{H}{\sqrt{c^{2} - q^{2}}} = \frac{H'}{\sqrt{c^{2} - q'^{2}}} = \frac{H'_{0}}{c}, \end{align}$$ wherein
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

H'_{0} = \frac{a}{3} T'^{4} V'. \end{align}$$ Taking account of the above general relations between $$T'$$ and $$T$$, $$V'$$ and $$V$$, this gives for the moving black body radiation the kinetic potential:
 * $$\begin{align}&{\color{White}.(00)}\qquad&&

H = \frac{a}{3} \frac{T^{4} V}{\left(1 - \dfrac{q^{2}}{c^{2}}\right)^{2}}, \end{align}$$ from which all the remaining thermodynamic quantities: the pressure $$p$$, the energy $$E$$, the momentum $$G$$, the longitudinal and transverse masses $$m_{l}$$ and $$m_{t}$$ of the moving black body radiation are uniquely determined.

Colleagues, ladies and gentlemen, I have arrived at the conclusion of my lectures. I have endeavored to bring before you in bold outline those characteristic advances in the present system of physics which in my opinion are the most important. Another in my place would perhaps have made another and better choice and, at another time, it is quite likely that I myself should have done so. The principle of relativity holds, not only for processes in physics, but also for the physicist himself, in that a fixed system of physics exists in reality only for a given physicist and for a given time. But, as in the theory of relativity, there exist invariants in the system of physics: ideas and laws which retain their meaning for all investigators and for all times, and to discover these invariants is always the real endeavor of physical research. We shall work further in this direction in order to leave behind for our successors where possible—lasting results. For if, while engaged in body and mind in patient and often modest individual endeavor, one thought strengthens and supports us, it is this, that we in physics work, not for the day only and for immediate results, but, so to speak, for eternity.

I thank you heartily for the encouragement which you have given me. I thank you no less for the patience with which you have followed my lectures to the end, and I trust that it may be possible for many among you to furnish in the direction indicated much valuable service to our beloved science.