Further Considerations on the Physical Interpretation of Lorentz Transformations

L. Jánossy

§ 1. In the preceding article[^1] I discussed the question of the physical interpretation of the Lorentz transformations. I presented arguments indicating that the interpretation of the Lorentz transformations given by Einstein, on the one hand, and by Lorentz and Fitzgerald, on the other, must be reconsidered; I gave a number of arguments in favor of an interpretation close to, but not identical with, the old Lorentz–Fitzgerald interpretation.

After the publication of the above-mentioned article I had the opportunity to discuss this question with many physicists. Although these discussions did not lead to essentially new results, many details were clarified in them, and I shall attempt to formulate my considerations anew, making use of the results of these discussions.

In the present article I shall not consider all the problems discussed in the previous work, but shall examine in greater detail certain selected aspects of the problem. In particular, I shall consider much more fully than before the questions of the dynamics of a rigid body (or of a closed physical system), and shall dwell especially on what happens when such a system is accelerated.

I. Experimental Facts

§ 2. The theory of relativity proceeded from the negative results obtained in a number of experiments. Later the theory predicted a series of positive effects, which were in part confirmed experimentally. For clarity of argument we shall briefly discuss these experimental results:

1) The Michelson–Morley experiment and a number of analogous experiments.

2) Experiments proving the “slowing down of the rate of clocks.” Two experiments were carried out in which this effect appeared:

a) It was found that the lifetime \(\tau'\) of a fast \(\mu\)-meson is considerably longer than the lifetime of a \(\mu\)-meson at rest. In the experiment the following relations were qualitatively proved:

\[ \tau' \sim \tau \cdot \frac{P}{mc^2}, \qquad 1 < \frac{P}{mc^2} < 50, \tag{1} \]

where \(\tau = 2.2 \cdot 10^{-6}\) sec is the half-life of mesons at rest, which is measured directly. The time \(\tau'\) is measured by determining the fraction of \(\mu\)-mesons that disappears from the beam as a result of decay. Substituting into relation (1) the relativistic expression for the momentum, we obtain, instead of expression (1):

\[ \tau' \simeq \frac{\tau}{\sqrt{1-\left(\frac{v}{c}\right)^2}} . \tag{2} \]

b) The wavelength of the light emitted by a rapidly moving atom proves to be greater than the corresponding wavelength emitted by an atom at rest.^2 The difference

\[ \Delta\lambda=\lambda' - \lambda, \]

where \(\lambda'\) is the wavelength emitted by the fast atom, and \(\lambda\) is the corresponding wavelength of the atom at rest. It was found that

\[ \Delta\lambda = a \frac{v^2}{c^2} a \simeq 0.5 . \tag{3} \]

If one assumes that \(a=0.5\), then formula (3) is compatible with the relativistic relation

\[ \nu'=\nu \sqrt{1-\frac{v^2}{c^2}}, \tag{4} \]

where \(\nu'\) is the frequency of the light emitted by an atom moving with velocity \(v\). Relation (4) can be interpreted as follows: if the atom is regarded as a clock, then this clock slows its rate when the atom is given a greater velocity.

3) Change of mass with velocity. By determining the deflection of an electron in combined electric and magnetic fields, one can establish that mass changes with velocity. By means of well-known measurements it has been possible to prove that the mass of electrons and protons increases with velocity.

The results of the measurements agree better with the expression given by Lorentz than with the older expression of Abraham.

In another article^3 we dealt with an exact evaluation of the experimental data. The accuracy of the best measurements is about \(1\%\).

4) Mass defect. Atomic nuclei constructed from a definite number of nucleons have an inertial (and also gravitational) mass \(M\), which is less than the sum of the masses of the particles \(m_i\) forming the nucleus. According to the theory one should expect

\[ M=\sum m_i-\frac{U}{c^2}, \tag{5} \]

where \(U\) is the binding energy.

Measurements confirm the theory; however, according to Flugge’s analysis^4 it appears desirable to obtain more experimental data which would confirm relation (5) than are available at the present time.

§ 3. Thus, we have summarized the most important experimental confirmations of the theory of relativity. In our opinion, a very useful task will be accomplished if these fundamental experiments are repeated, and if their number is also increased, and also if the results obtained are subjected to critical analysis.

In interpreting the experiments considered in § 2, it is usually asserted that these experiments prove effects caused by the motion of an object relative to an observer.

In fact, these experimental results prove not this, but the following: if these results are considered without prejudice, then they point to the effects of accelerated motion relative to some physical system.

1) Let us consider the Michelson–Morley experiment; here the experimental result consists simply in the fact that the fringes visible in the interferometer are not displaced when the interferometer is rotated. This rotation is accelerated motion, i.e. we are studying the influence of accelerated motion on the instrument. More precisely, accelerated motion gives no residual effects which

could have been observed after the acceleration had ended. Moreover, the Michelson—Morley experiment was repeated at different times of the year in order to find out whether the accelerated motion of the Earth in its orbit gives rise to any accumulating effects that would affect the instrument.

The same is true also for the Trouton—Noble experiment and other well-known experiments of the same type. In all these experiments it was established that a certain accelerated motion of the system does not produce noticeable residual effects in the apparatus (rotation of a capacitor, etc.).

2) Experiments on determining the lifetime of the meson prove that a meson moving very rapidly with respect to the Earth (or the solar or galactic system) has a lifetime \(\tau' > \tau\), where \(\tau\) is the lifetime of a meson moving with relatively small velocity with respect to the Earth. In fact, mesons are usually born with large velocities, so that the experiments apparently show that, when mesons are slowed down, we shorten their lifetime. Thus, the experiment gives information about the influence of acceleration on the mechanism of decay of \(\mu\)-mesons.

The same remarks may be made concerning experiments on the transverse Doppler effect. Atoms possessing known properties are accelerated; as a result of the accelerated motion, the frequency of the light emitted by them changes. Moreover, although such experiments have not in fact been carried out, there is no doubt that if the motion of the atoms forming canal rays is slowed and they are collected in a discharge tube, then afterward they will emit the same frequencies that they emitted before the acceleration. Thus, it is very probable that the slowing of fast atoms leads to an increase in the frequency emitted by them, with the initial frequency being reached as soon as they are slowed down to velocities small in comparison with the speed of light.

3) Considering experiments on the variation of mass with velocity, we shall find that if electrons or protons are accelerated to very great velocities, then their interaction with electric and magnetic fields differs from the interaction of slow electrons or protons. Again, fast electrons are fast with respect to the Earth.

§ 4. Finally, in order to interpret the experimental facts, we do not so much need a theory that would predict how one and the same system looks from the point of view of different observers*). For the interpretation of experimental results it is necessary to have a theory that would investigate how a physical system changes under acceleration, during which the state of its translational motion changes. In particular, the theory must predict what happens in a system accelerated to velocities close to the speed of light.

II.

1. The Lorentz—FitzGerald Point of View

§ 5. The negative result of the Michelson—Morley experiments and of a number of analogous experiments was interpreted by Lorentz and FitzGerald in the following way.

They assumed the existence of an ether that was in a state of absolute rest. From their point of view, the ether was the carrier of electromagnetic waves. It was supposed that light waves propagate in the ether isotropically in all directions with velocity \(c\). The negative result of experiments of the Michelson—Morley type was explained by the well-known contraction hypothesis. It was believed that the Lorentz—FitzGerald deformation

*) See, for example, Ioffe’s remark\(^5\).

completely compensates the effect which, without this compensation, would be observed as a consequence of the ether wind. The following types of deformation, if they exist, exactly compensate all possible effects of translational motion relative to the ether.

Fig. 1. Scheme of the Lorentz deformation.

Let us consider a material system consisting of a rod of length \(a\) and two clocks at each end of the rod; these clocks are controlled by one and the same mechanism, which is also connected with the rod (Fig. 1). If we accelerate the latter system from a state of rest into a state with translational velocity \(v\), parallel to the rod, then, by assumption, the following deformations must take place:

1) the rod contracts and reaches the length

\[ a' = a \sqrt{1 - \frac{v^2}{c^2}}; \tag{6a} \]

2) the clocks slow their rate in the ratio

\[ 1 : \sqrt{1 - \frac{v^2}{c^2}}; \tag{6б} \]

3) the clocks undergo a phase shift \(\Delta t\) relative to one another

\[ \Delta t = - \frac{av}{\sqrt{c^2 - v^2}}; \tag{6в} \]

4) a cylinder of length \(a\), rotating about an axis parallel to the direction of acceleration, undergoes a torsional deformation about the axis, the angle of torsion being

\[ \Delta \varphi = - \frac{a v \omega}{c^2}. \tag{6г} \]

In connection with the Lorentz–FitzGerald deformations we shall note that deformation 1) is necessary for explaining the Michelson–Morley experiments; deformation 2) is necessary for explaining the transverse Doppler effect and the decay of \(\mu\)-mesons; deformations 3) and 4) do not lead to effects which up to now could have been experimentally tested. However, the latter effects should be expected on general grounds if effects (1) and (2) actually occur.

The shift in the readings of clocks and the torsional deformation of a rotating cylinder play an important role in the discussion of thought experiments (see, for example, Cohn\(^6\), especially p. 1408).

Deformations 3) and 4) are of considerable physical interest, because they compensate certain effects which, if the corresponding deformation did not arise, could be expected from thought experiments. But, as far as I can see, there exist no experimentally observed effects in which these deformations would play an actually important role.

§ 6. To admit the Lorentz deformations as an ad hoc assumption merely in order to explain the negative result of the Michelson–Morley experiment would, of course, be most unsatisfactory. Four rather strange deformations, together with the assumption of a somewhat mystical ether, seemed to Einstein and to most physicists completely unacceptable, and therefore the Lorentz–FitzGerald point of view was wholly rejected in favor of Einstein’s point of view.

Considering this problem retrospectively, I have come to the conclusion that the discussion of these two possible points of view should be resumed anew in the light of the abundant amount of new material that we now possess. Having carefully examined this problem, I believe, on the one hand, that on closer inspection Einstein’s point of view is logically and philosophically far less satisfactory than is commonly thought, and, on the other hand, that the Lorentz–FitzGerald point of view can be modified in such a way that it will give a far more satisfactory picture than is usually considered possible.

In the present article I shall set out this modified Lorentz–FitzGerald point of view in detail and shall try to show its internal consistency. As for objections concerning Einstein’s point of view, here I shall confine myself to a few brief remarks.

2. Einstein’s Point of View

§ 7. In this point of view it is stated that the conception of the ether is unsatisfactory from a philosophical point of view; it is argued that, owing to the large number of failures in attempts to detect the ether wind, it is necessary to construct a theory without an ether, which cannot be observed. Proceeding from this, it is assumed that inertial systems \(K_1, K_2\), etc., which move translationally with respect to one another, are completely equivalent; the laws of nature have one and the same form in any of them. In a consistent formulation of this idea we must transform coordinates and times not according to the Galilean transformations, but according to the Lorentz transformations.

Thus, whether two events are simultaneous or not under the Lorentz transformations depends, to a certain extent, on the reference system and, consequently, on the observer. Moreover, the time order of two events, from Einstein’s point of view, may turn out to be different when observed from two different reference systems. There is an essential limitation on the latter assertion: if two events occur sufficiently close to one another, or if the interval of time between them is sufficiently large, then their time order is one and the same in all (admissible) inertial systems.

Einstein pointed out, and here we agree with him, that a change in the time order of two events \(A\) and \(B\) in passing from one reference system to another could lead to difficulties only in the case where there existed a causal connection between these events. Einstein’s hypothesis on the equivalence of inertial systems connected by Lorentz transformations can be maintained only if there are no signals in nature propagating with a speed greater than the speed of light. This condition was explicitly expressed by Einstein.

Einstein’s arguments are often reversed and it is declared as though the theory of relativity “proves” that there is no action propagating with a speed greater than the speed of light. This reversal of the arguments is a mystification of the state of affairs. If signals propagating with a speed exceeding the speed of light are discovered, then Einstein’s point of view will be rejected. We note this here in order to emphasize (this was accepted by Einstein) that the hypothesis of the impossibility of the propagation of signals with a speed greater than the speed of light is in fact not proved by the theory of relativity, but is an essential hypothesis on which the theory of relativity is based, and a hypothesis that cannot be positively proved.

§ 8. Einstein proposed his interpretation of the Lorentz transformations instead of the Lorentz–FitzGerald interpretation for purely philosophical consid-

... experiments, an experimentum crucis for these two points of view does not exist, since within the framework of classical theory they are mathematically equivalent. If such an experiment were found, it would, if possible, be in favor of the Lorentz–FitzGerald point of view (against Einstein’s point of view); an experimentum crucis in favor of Einstein’s point of view (against the Lorentz–FitzGerald point of view) is impossible.

“If a universal law of nature is discovered that does not satisfy this condition (namely, covariance with respect to Lorentz transformations), then at least one of the two basic premises of the theory will be refuted” (see 7, p. 29).

In fact, future experiments may perhaps prove that signals exist which propagate with a speed greater than the speed of light, but no experiment can prove that such signals do not exist. Similarly, some experiments in the future may perhaps give definite indications of the absolute motion of the Earth, but it is perfectly clear that no experiment can rule out the possibility that some physicist, after some time, will succeed in determining this motion. Thus this part of the theory of relativity always stands in need of defense.

III. DYNAMICS OF A PHYSICAL SYSTEM UNDERGOING ACCELERATION

§ 9. In the present section, as well as in Sections IV and V, we shall consider a physical system from the point of view of only one frame of reference, and we shall examine the behavior of this system under acceleration. We shall denote our frame of reference by \(K_0\). It is assumed that this frame of reference is inertial, at rest, for example, with respect to the solar system, although it need not be exactly such a system. In § 39 we shall consider the philosophical and methodological significance of the fact that our considerations are based only on the examination of a single frame \(K_0\). Without entering into the details of these questions, we shall at present simply postulate that \(K_0\) is a frame moving with a small velocity, say, with respect to the solar system, whose coordinates and clocks are adjusted in such a way that Maxwell’s equations hold in it. Thus we have assumed that light in the system \(K_0\) propagates isotropically with velocity \(c\). Once this assumption has been made, we can measure distances and check clocks by means of light signals.

§ 10. The electromagnetic field. Thus, we have assumed that in the system \(K_0\) Maxwell’s equations hold. Although, for the time being, we are considering phenomena in only one frame of reference, we can nevertheless make use of the property of Maxwell’s equations that they are invariant with respect to Lorentz transformations. For if we subject the coordinates and times, together with the field strength and the current and charge density, to Lorentz transformations and refer these transformed quantities again to the original system \(K_0\), then we obtain a new distribution of the electromagnetic field. This transformed distribution satisfies Maxwell’s equations if the original field satisfied them.

Consider, for example, the case of a charge at rest. The field of the charge is the well-known Coulomb field. If we now subject the charge and its field, as a single system, to Lorentz transformations, we obtain a distribution that represents a charge moving with constant velocity and its field. The field of a moving charge is the well-known Lorentz field, compressed in the longitudinal direction. Thus we have obtained the stationary field of a moving charge. However, by means of this method it is impossible to determine,

what happened to the charge when it was accelerated from a state of rest to a state with velocity \(v\). To investigate the latter question we must have a solution of Maxwell’s equations for a charge distribution which was initially at rest, was then gradually accelerated, and finally moves with velocity \(v\). Such a distribution is given by the formula

\[ \rho(\mathbf r,t)=\rho_2\left(\mathbf r-\int^t \mathbf v(t')\,dt',\,t\right), \tag{7} \]

\[ \mathbf v(t)= \begin{cases} 0 & \text{for } t<0,\\ \mathbf v & \text{for } t>t_1>0 \end{cases} \tag{8} \]

and

\[ \rho_2(\mathbf r,t)=0 \quad \text{for } |\mathbf r-\mathbf r_1|>\varepsilon \text{ for all values of } t . \tag{9} \]

(\(\mathbf r_1\) denotes the center of the charge cloud.) Thus, \(\rho(\mathbf r,t)\) represents a charge distribution which was initially at rest, began to move at the time \(t=0\), and reached the velocity \(v\) no later than the time \(t_1>0\). We shall assume that the distribution \(\rho_2\) depends explicitly on the time \(t\), so that during acceleration any deformation of the charge may occur. We have postulated condition (9) in order to guarantee that the charge distribution is always located inside a sphere of radius \(\varepsilon\) (but does not necessarily fill it completely). Further, the distribution behaves as a rigid one both before the acceleration is switched on and after it is switched off; however, the outlines of the distribution after the time \(t_1\) may differ from the outlines before the time \(t\).

§ 11. One can obtain a solution of Maxwell’s equations for the charge distribution (7)—(9), but we must make further assumptions about the current density.

The solution of Maxwell’s equations for the case of an arbitrarily accelerated charge distribution was given explicitly by Heitler\(^8\) for the limiting case \(\varepsilon\to 0\), i.e., Heitler determined only that part of the field which does not depend on the structure of the charge distribution. For distributions with \(\varepsilon\ne 0\), to Heitler’s solution one must add solutions corresponding to definite electric and magnetic multipoles, which move together with the charge.

The expression describing the field of an arbitrarily moving charge has a simple physical meaning, which may be described qualitatively as follows.

§ 12. For times \(t\gg t_1\), i.e., after the acceleration, we must consider four zones of the field surrounding the charge. These zones are denoted \(I, II, III, IV\) in Fig. 2; they may be described as follows:

I. A zone with radius \(|\mathbf r-\mathbf r_1-\mathbf v t|<R_I\sim \varepsilon\), immediately surrounding the charge. The field here depends essentially on the structure of the charge and cannot be determined solely with the aid of Maxwell’s equations.

II. The zone following zone I; this is the compressed Lorentz field, which can be determined with sufficient accuracy by means of the formulas given by Lorentz. The outer boundary of this region propagates with the speed of light around the charge, beginning from the moment when the charge reached the velocity \(v\); thus, the radius of the zone is \(c(t-t_1)\) for \(t>t_1\).

III. The zone surrounding zone II; this zone contains spherical waves emitted during the process of acceleration. The field in this receding zone is not stationary.

IV. The zone \(|\mathbf r|>ct\); the effects of acceleration have not yet reached this last zone, and therefore in it we find the field of a resting charge.

Considering the entire picture, we may conclude the following. When a charge distribution (occupying only a small region) is accelerated to

the velocity \(\mathbf{v}\), the field of this distribution undergoes a Lorentz transformation. An exception to this can occur only in the immediate vicinity of this distribution (zone \(I\)), where the field that arises depends essentially on the structure of the charge. Moreover, the transformed field is established not instantaneously, but only to the extent that zone \(II\) propagates into the part of space that interests us. We can express this result

Fig. 2. Location of Lorentz-contracted regions around a moving charge.

in another way, saying that when the charge is accelerated we disturb its field, and when we allow the disturbed field to relax, a compressed Lorentz field is established.

Neglecting zone \(I\) for the time being, we may assert that the Lorentz contraction of the charge field follows directly from Maxwell’s equations without any additional hypotheses about the structure of the charge.

§ 13. Zone \(I\), where the field depends on the structure of the charge distribution, is inessential if we consider a system of interacting charges located at distances exceeding their diameters. However, in such a system, in addition to electromagnetic forces, non-electromagnetic forces must also act in order for the system to be stable; we shall consider this problem below.

The field in the immediate vicinity of the charge distribution (zone \(I\)) plays an essential role if we wish to determine the self-action force of the charge. The well-known calculations of Lorentz give the following result. If the charge is accelerated, then the reaction force of the charge’s self-action on itself has a component opposite to the direction of the acceleration. Thus, if we place the charge distribution, say, in an external electrostatic field, then part of the action of the external field will be compensated by the reaction of the self-action of the charge’s field. In this way, this reaction decreases the magnitude of the acceleration produced by the external field. The exact magnitude of this reaction depends on the assumption about the properties of the function \(\rho(\mathbf{r}, t)\), which specifies the distribution of the charge during the process of acceleration. Assuming, for example,

\[ \rho(\mathbf{r}, t)=\rho_1\left(\mathbf{r}-\int_0^t \mathbf{v}(t')\,dt'\right), \]

i.e., acceleration without deformation, we shall obtain the Abraham electron, for which the dependence of mass on velocity is given by Abraham’s formula. If, following Lorentz, we assume that the distribution undergoes a contraction in length in the ratio \(1 : \sqrt{1-\frac{v^{2}}{c^{2}}}\), then we shall obtain the dependence of mass on velocity given by Lorentz. Considering higher approximations, we shall also obtain the contribution of higher derivatives, i.e., radiation corrections.

Thus we see that from Maxwell’s equations it follows, at least qualitatively, that the mass of the electron depends on its velocity, if mass is taken as the ratio of the total force to the acceleration produced.

We can calculate a quantitative expression for this dependence only in the case where we know the structure of the electron. Proceeding in the reverse order, we find that from the observed dependence of the electron mass on velocity we can draw conclusions about the structure of the electron, if we assume that the mass of the electron is primarily of electromagnetic origin. On the basis of the available experimental data one may suppose that Abraham’s formula is not in agreement with experiment, and we may exclude the possibility that the electron behaves as a rigid sphere.

§ 14. What we have said here with regard to the electron can be generalized to nucleons or atomic nuclei. Although these particles are surrounded by an electromagnetic field, a much more important role in this case is played by the meson field. The reaction of the meson field under acceleration of the proton may be responsible (at least partially) for the change of the proton mass with velocity.

§ 15. Mass defect and Maxwell’s equations. Maxwell’s equations strictly lead to the conservation of energy and momentum if it is assumed, as usual, that the field possesses an energy and momentum density. Thus, when a cloud of charge—say, an electron—is accelerated, the work performed by the acting force appears in the form of accumulated electromagnetic energy of the field. If the magnitude of the acceleration is small, then practically all the work performed by the acting force is accumulated in the electromagnetic field surrounding the charge, so that in the case of braking it passes back to the particle; there is always, of course, a part of the energy that has been radiated and cannot be recovered.

Let us now consider a system consisting of positive and negative charges. Accelerating these two particles simultaneously, we must overcome both the reaction of the field of these two particles upon themselves and the mutual action of the two fields upon one another.

A direct detailed calculation shows that the force acting on a system consisting of positive and negative charges imparts to it an acceleration the greater, the closer these charges are situated to each other, i.e., the greater the (negative) binding energy; that is, in fact, a mass defect can be obtained as a direct consequence of Maxwell’s equations in a system bound by electromagnetic forces. The exact value of the mass defect can apparently be obtained if the non-electromagnetic part of the binding forces is taken into account.

§ 16. Considering, instead of electrons, for example, a system of protons and neutrons, we may expect that, at least qualitatively, we shall obtain a mass defect if we assume that the interaction between the particles is transmitted by a field. If we assume that the interaction is entirely transmitted by the meson field proposed by Yukawa, then we shall apparently obtain a mass defect in accordance with the predictions of the theory of relativity, since the Yukawa field is covariant with respect to Lorentz transformations and since it satisfies the conservation laws.

It should be emphasized, however, that at the present time it is generally considered proven that nucleons obey covariant laws. This may be so, but the number of experimental proofs seems, to some extent, disappointing. In my opinion, it is highly desirable to investigate to what extent the experimental evidence indicates the necessity (or at least testifies in favor) of the general conviction that nucleons behave relativistically. The available facts: 1) the mass of a nucleon increases with velocity, 2) the mass of a bound system of nucleons is less than the sum of the masses of free nucleons, can be explained qualitatively by any field theory, and these facts by themselves cannot be taken as decisive proof of the relativistic behavior of nucleons.

IV. LORENTZ DEFORMATIONS IN AN ARBITRARY DYNAMICAL SYSTEM

§ 17. The assumption of Lorentz deformations of a solid body may have seemed very strange at the beginning of our century, when a rather mystical view of matter was widespread. So long as a solid body is regarded as a certain structureless geometrical formation, it seems natural to suppose that the solid body is rigid and does not change under motion.

At the present time, however, we know that a solid body consists of a large number of atoms which are in a state of dynamical equilibrium with respect to one another. If such a system is accelerated, it would be strange if the initial equilibrium were not disturbed by the acceleration. We shall, in fact, show here that in a system in which the interaction between its individual parts propagates with a finite velocity, one should expect deformations of the same type as the Lorentz deformations. Thus we shall arrive at the conclusion that, under acceleration, the system is necessarily deformed. How it is deformed depends on the nature of the cohesive forces.

If we admit that a dynamical system is, as a rule, deformed in the way Lorentz and Fitzgerald assumed, then the question of whether the Lorentz–Fitzgerald hypothesis is an ad hoc hypothesis or not arises in a new form. Since on general grounds one should expect that a solid body is deformed under the influence of acceleration and that its final deformed state depends on the state of motion imparted to it, it becomes necessary to carry out experiments to determine the exact value of the magnitude of the deformation and to find mathematical expressions that would explain these deformations. The finding of these mathematical formulas, which would quantitatively describe effects that must exist on general grounds, cannot be regarded as an ad hoc procedure.

Then the following step remains: if we have found empirical formulas describing the actual deformations derived from experiment, then we must find the laws obeyed by the internal forces and which would explain the deformations observed in experiment.

Below we shall consider the dynamics of a system of point masses which form an equilibrium configuration under the action of internal forces propagating with a finite velocity.

§ 18. Equation of motion of a dynamical system. We shall consider a solid body consisting of \(N\) particles. We shall denote the vector coordinates of the particles at the time \(t\) as follows:

\[ \mathbf{r}_1(t),\ \mathbf{r}_2(t),\ldots,\mathbf{r}_N(t). \]

Selecting one particle, say particle \(k\), we may suppose that its acceleration \(\ddot{\mathbf r}_k(t)\) depends on the coordinates and velocities of all particles. However, if we assume that the action between particles propagates with a finite speed, say \(V\), then the acceleration of particle \(k\) at time \(t\) is determined by the coordinates and velocities of the other particles not at time \(t\), but at an earlier time \(t_{lk}\), such that the action of some other particle arrives precisely at time \(t\) at the point where particle \(k\) is located. We may write

\[ t_{lk}=t-\tau_{lk}, \tag{10} \]

where \(\tau_{lk}\) is the time required for the propagation of the action from particle \(l\) to particle \(k\). In fact, the quantity \(t_{lk}\) can be determined numerically from the following functional equation:

\[ \left|\mathbf r_l(t_{lk})-\mathbf r_k(t)\right|V^{-1}=t-t_{lk}\geqslant 0. \tag{11} \]

For simplicity we introduce the notation

\[ \mathbf r_l(t_{lk})=\mathbf r_{lk}(t). \tag{12} \]

Thus, \(\mathbf r_{lk}(t)\) is the retarded coordinate of particle \(l\) with respect to its action on particle \(k\). This retarded coordinate depends, generally speaking, on two indices \(l\) and \(k\). We can now write the equations of motion in the following form:

\[ \mathbf f_k\left(\mathbf r_{lk}(t),\dot{\mathbf r}_{lk}(t)\right)=\ddot{\mathbf r}_k(t), \tag{13} \]

where

\[ \left. \begin{aligned} \dot{\mathbf r}_{lk}(t)&=\left(\frac{d}{dt'}\mathbf r_l(t')\right)_{t'=t_{lk}},\\ \mathbf r_{kk}(t)&=\mathbf r_k(t). \end{aligned} \right\} \tag{14} \]

Each of the functions \(\mathbf f_k,\ k=1,2,\ldots,N\), has three components and is a characteristic of the internal forces acting between the particles. We shall try to discuss the properties of the dynamical system under acceleration while making as few specific assumptions as possible about the nature of the internal forces.

§ 19. The equations of motion (11)—(14) require some explanation.

1) We have assumed that the action between particles propagates with a constant speed \(V\); in all known cases this actual speed is equal to the speed of light \(c\). The apparent speed of propagation may, however, differ from the speed \(c\) whenever the action is propagated along a zigzag path. Thus, for example, elastic forces in bodies propagate with speeds considerably smaller than the speed of light, but elastic forces are the result of a very complex interaction of the atoms of a solid; there is no doubt that the fundamental interaction between atoms propagates with the speed of light.

2) The equations of motion (12)—(14) assume that the forces depend only on the (retarded) coordinates and velocities of the particles. Obviously, this is true only approximately; however, we shall consider only cases of small accelerations; then, if the forces in a real system do also depend on accelerations, this dependence may be neglected without thereby making significant errors in the cases that will be considered below.

We may formulate the equations of motion by assuming that the particles are sources of a certain field

\[ F(\mathbf r,t)=\mathfrak F(\mathbf r_k(t'),t) \tag{15} \]

(\(F\) is a certain functional) and that this field can accelerate particles as a result of self-action:

\[ \ddot{\mathbf r}_k(t)=G\left(F(\mathbf r_k(t),t);\ \mathbf r_k(t)\ldots\right). \tag{16} \]

We prefer to discuss the equations of motion given by formulas (11)—(14) within the framework of a simpler model. In §§ 31, 32 we shall briefly discuss the shortcomings of this simple model.

§ 20. If the rigid body is at rest, then we may put

\[ \mathbf r_k(t)=\mathbf r_{kl}(t)=\mathbf r_k^0=\mathrm{const}. \]

Therefore the equations of motion may be written (since \(\dot{\mathbf r}(t)=\dot{\mathbf r}_k(t)=0\) for all values of \(t\)) in the form

\[ \mathbf f_k(\mathbf r_l^0,0)=0,\qquad k=1,2,\ldots,N. \tag{17} \]

(17) is a system of \(3N\) equations for \(3N\) unknowns (the \(3N\) components of the \(N\) constant vectors \(\mathbf r_l^0\)). Any solution of this system gives an equilibrium configuration of the system of \(N\) particles. There are infinitely many positions of equilibrium, and the above system of equations does not determine the unknowns uniquely. For example, if \(\mathbf r_l^0\) is a system of solutions, then

\[ \mathbf r_l^1=\mathbf r_l^0+\mathbf a, \tag{18} \]

where \(\mathbf a\) is a constant vector, will also be a system of solutions. Thus, the solutions of equations (17) contain at least six arbitrary parameters, corresponding to the six (geometric) degrees of freedom of a rigid body.

In fact, equations (17) as a rule have many more solutions. A permutation of these particles gives new solutions. The most substantial ambiguity reduces to the following. If the forces between the particles are short-range, then we may divide the \(N\) particles into groups \(N_1+N_2+\cdots=N\), and we have solutions in which each of these subsystems \(N_1,N_2,\ldots\), etc., is in a state of equilibrium. These groups of particles are so separated that the interaction of particles of one group with particles of another may be neglected. In fact this means that we may divide the rigid body into a number of parts, each of which is a system in equilibrium.

In what follows we shall assume that \(\mathbf r_k^{(0)}\), \(k=1,2,\ldots,N\), represents one of such possible solutions of the system (17).

§ 21. Let us consider a system in translational motion. Then we have:

\[ \mathbf r_k^{(v)}(t)=\mathbf r_k^{(v)}(0)+\mathbf v t; \tag{19} \]

the superscript \((v)\) means that we are considering a system moving with velocity \(\mathbf v\). Substituting expression (19) into formula (11), with the aid of relation (10) we obtain:

\[ \left|\mathbf r_l^{(v)}(0)-\mathbf r_k^{(v)}(0)-\mathbf v \tau_{lk}\right|V=\tau_{lk}. \tag{20} \]

From equations (20) it is evident that, in the case of translational motion, the retardation \(\tau_{lk}\) is a function both of \(l\) and \(k\), and of the velocity \(V\), but

\[ \frac{\partial \tau_{lk}}{\partial t}=0, \tag{21} \]

i.e. \(\tau_{lk}\) is constant in time. Since we may put:

\[ \left. \begin{aligned} \mathbf r_{lk}^{(v)}(t)&=\mathbf r_l^{(v)}(0)+vt-\tau_{lk}v\\ \dot{\mathbf r}_{lk}^{\,v}(t)&=\mathbf v\\ \ddot{\mathbf r}_{k}^{\,v}(t)&=0 \end{aligned} \right\} \qquad \begin{aligned} &\text{for } l,k=1,2,\ldots,N\\ &\text{time } t \text{ arbitrary.} \end{aligned} \tag{22} \]

Then, introducing the quantities (22) into formula (17), we shall obtain, for the equilibrium configuration of a system in a state of translational motion,

the following expression \((t=0)\):

\[ \mathbf{f}_k\bigl(\mathbf{r}_l^v(0)-\tau_{lk}\mathbf{v};\,\mathbf{v}\bigr)=0 . \tag{23} \]

The solutions of equation (23), generally speaking, depend on the velocity \(v\), so that we may assert that

\[ \mathbf{r}_l^0 \ne \mathbf{r}_l^v,\qquad l=1,2,\ldots,N. \]

Except in very special cases, we may assert that

\[ \left|\mathbf{r}_l^{(0)}-\mathbf{r}_k^{(0)}\right| \ne \left|\mathbf{r}_l^{(v)}-\mathbf{r}_k^{(v)}\right| . \tag{24} \]

Thus, the equilibrium distance between particles, generally speaking, depends on the velocity \(\mathbf{v}\). Hence, a system of \(N\) particles accelerated to the velocity \(\mathbf{v}\) changes its equilibrium configuration.

More precisely, from the considerations presented in the paper\(^1\) it follows that if we apply to each of the \(N\) particles of the system such forces as accelerate each particle from a state with velocity equal to zero to a state with velocity equal to \(\mathbf{v}\), with the velocity of each particle increasing according to one and the same law, then such acceleration disturbs the state of equilibrium of the system. After the acceleration described above has ended, the system will tend to relax from the configuration it has acquired. The equilibrium configuration of a system in translational motion is a deformed equilibrium configuration of the system at rest.

§ 22. Let us proceed to compare the solutions of equations (17) and (23), i.e., the configurations taken at \(v=0\) and \(v>0\). Suppose that the velocity \(v\) is small (in comparison with the velocity \(V\)) and that the deformations, i.e., the vectors

\[ \mathbf{r}_k^{(v)}-\mathbf{r}_k^{(0)}=\delta \mathbf{r}_k, \tag{25} \]

are also small. Then we may put

\[ \begin{aligned} \mathbf{f}_k\bigl(\mathbf{r}_l^{(v)}(0)-\tau_{lk}\mathbf{v};\,\mathbf{v}\bigr) &= \mathbf{f}_k\bigl(\mathbf{r}_l^0;\,0\bigr) +\sum_l \bigl(\delta\mathbf{r}_k-\tau_{lk}\mathbf{v}\bigr)\frac{\partial \mathbf{f}_k}{\partial \mathbf{r}_l} +\sum_l \mathbf{v}\frac{\partial \mathbf{f}_k}{\partial \mathbf{v}_l} +\cdots , \end{aligned} \tag{26} \]

where by \(\dfrac{\partial \mathbf{f}_k}{\partial \mathbf{r}_l}\) we have denoted the vector with components

\[ \frac{\partial \mathbf{f}_k}{\partial r_{lx}},\qquad \frac{\partial \mathbf{f}_k}{\partial r_{ly}},\qquad \frac{\partial \mathbf{f}_k}{\partial r_{lz}}, \]

and

\[ \mathbf{v}\frac{\partial \mathbf{f}_k}{\partial \mathbf{r}_l} = v_x\frac{\partial \mathbf{f}_k}{\partial r_{lx}} + v_y\frac{\partial \mathbf{f}_k}{\partial r_{ly}} + v_z\frac{\partial \mathbf{f}_k}{\partial r_{lz}}; \]

the remaining expressions are to be interpreted analogously.

Neglecting terms of higher orders in expression (26) and noting that the first term is equal to zero, since \(\mathbf{r}_l^{(0)}\) represents the equilibrium configuration for the state with \(v=0\), we may write:

\[ \sum_l \left\{ \bigl(\delta\mathbf{r}_l-\tau_{lk}\mathbf{V}\bigr)\frac{\partial \mathbf{f}}{\partial \mathbf{r}_l} + \mathbf{v}\frac{\partial \mathbf{f}_k}{\partial \dot{\mathbf{r}}_l} \right\} =0,\qquad k=1,2,\ldots,N. \tag{27} \]

This system of equations may be used to determine the deformation \(\delta\mathbf{r}_l\) caused by translational motion with velocity \(\mathbf{v}\).

In particular, when \(V\to\infty\) and \(\dfrac{\partial \mathbf{f}_k}{\partial \dot{\mathbf{r}}_l}=0\) (forces independent of velocities), we have \(\tau_{lk}\to 0\), and the solution of equations (27) takes the form

\[ \delta\mathbf{r}_l=0,\qquad l=1,2,\ldots,N. \]

Thus, deformation cannot be expected in the case of an instantaneous propagation of actions and of forces that do not depend on velocities. If, however, the action is not propagated instantaneously, then deformation should be expected even in the case of forces that do not depend on velocities.

§ 23. Compression of a system consisting of two particles. From equation (27) one may expect, generally speaking, that the deformation \(\partial \mathbf r_l\) will be proportional to the velocity \(\mathbf v\). However, we shall show, imposing simple and natural conditions on the functions \(f_k\), that if the magnitude of the distortion is expanded in a power series in the velocity \(\mathbf v\), it will not contain a first-order term.

In order to make the above clearer, let us consider a system consisting only of two particles \(A\) and \(B\). Suppose that the particles are at rest and are situated on the \(X\)-axis of the coordinate system; then they will have coordinates \(x_A\) and \(x_B\), respectively.

Suppose that the particles interact with each other. Thus, particle \(A\) will tend to accelerate particle \(B\). The corresponding acceleration will depend only on the distance between the particles

\[ |x_A-x_B|=a. \]

Then we may write:

\[ \ddot x_B=g_B(a) \tag{28} \]

and, analogously, the acceleration of particle \(A\), caused by particle \(B\),

\[ \ddot x_A=g_A(a), \tag{29} \]

where the functions \(g_A\) and \(g_B\) characterize the forces acting between the particles, and play the same role as the functions \(f_k\) introduced above. If the particles \(A\) and \(B\) are identical, then from symmetry considerations it follows that \(\ddot x_A=-\ddot x_B\), so that

\[ g_B(a)=-g_A(a)=g(a)\quad \text{for all values of }a. \tag{30} \]

Equilibrium will be reached at that distance for which \(\ddot x_A=-\ddot x_B=0\), so that the equilibrium condition may be written in the form

\[ g(a)=0\quad \text{for }a=a_0. \tag{31} \]

We shall assume that the function \(g(a)\) is not singular at the point \(a=a_0\) and can be expanded in a power series in \(a-a_0\). If it is assumed that \(x_B>x_A\), then the equilibrium will be stable if

\[ \frac{dg(a)}{da}<0\quad \text{for }a=a_0 \tag{32} \]

(the equilibrium will be stable if all derivatives of the function \(g\) are equal to zero up to order \(2k\), while the derivative of order \((2k+1)\) is negative. We shall neglect this possibility and postulate that relation (32) holds).

We are interested in the equilibrium configuration of the particles \(A\) and \(B\) if we set them in motion. The equilibrium configuration can be characterized by

\[ x_A(t)=vt,\qquad x_B(t)=a(v)+vt, \tag{33} \]

where we have assumed that the system moves along the \(X\)-axis with velocity \(v\); the distance between the moving particles is

\[ x_B(t)-x_A(t). \tag{34} \]

The latter configuration is maintained without external intervention if only

the action of moving particles on one another is exactly equal to zero at the distance \(a(v)\).

§ 24. Let us consider the dependence of the quantity \(a(v)\) on the velocity \(v\). Or, more precisely, from general considerations we investigate the restrictions that should be imposed on the possible form of the function \(a(v)\), without making any particular assumptions about the acting forces.

If we assume that the action of the particles is completely independent of velocity, i.e., if we assume that formulas (28) and (29) are valid even in the case \(\dot{x}_A \ne 0\), \(\dot{x}_B \ne 0\), then we obtain that \(a(v)=a_0\) for any value of \(v\). However, the postulate that expressions (28) and (29) are independent of velocities implies that there is an instantaneous interaction between the particles. To see this, let us imagine that particles \(A\) and \(B\) are at rest at a distance \(a_0\) from one another. If we suddenly remove particle \(A\), then this change of distance will be felt at point \(B\) instantaneously, if we assume that the action of particle \(A\) on particle \(B\) depends only on the distance between them.

Thus, if we exclude instantaneous action, then we must assume that the forces between the particles depend on the velocities, so that

\[ \begin{aligned} \ddot{x}_A(t) &= g_A(a,v),\\ \ddot{x}_B(t) &= g_B(a,v), \end{aligned} \tag{35} \]

where

\[ g_A(a,0)=g_A(a), \qquad g_B(a,0)=-g_B(a). \]

In order to simplify the further discussion, we shall assume that we are considering purely translational motion, i.e.,

\[ \dot{x}_A(t)=\dot{x}_B(t)=v. \tag{36} \]

We can restrict the choice of the functions \(g_A\) and \(g_B\) on the basis of the following general considerations. Suppose that the action between the particles propagates with velocity \(V\); the action of particle \(A\) on particle \(B\) at the time \(t\) depends on the position \(A'\) of particle \(A\) at the time

\[ t_B=t-\tau_B, \]

so that the action propagates from particle \(A\) to particle \(B\) with velocity \(V\) in the time \(\tau_B\). We find that

\[ a_B(v)=A'B=\frac{a}{1-\dfrac{v}{V}} . \tag{37} \]

and

\[ a_A(v)=AB'=\frac{a}{1+\dfrac{v}{V}}, \tag{38} \]

where \(a_B(v)\) and \(a_A(v)\) are the retarded distances at which the particles act on one another.

In the simplest case one may assume that the action of the particles on one another depends only on the retarded distances, i.e.,

\[ \left. \begin{aligned} \ddot{x}_A(t) &= g_A(a_A(v))=g_A\!\left(\frac{a}{1+\dfrac{v}{V}}\right),\\[6pt] \ddot{x}_B(t) &= g_B(a_B(v))=g_B\!\left(\frac{a}{1-\dfrac{v}{V}}\right). \end{aligned} \right\} \]

Putting \(v=0\), we find, comparing with expressions (30) and (31):

\[ g_A(b)=-g_B(b)=g(b) \]

and

\[ g(b)=0 \quad \text{for } b=a_0 . \]

The condition of equilibrium then reduces to

\[ g\left(\frac{a}{1+\frac{v}{V}}\right) = g\left(\frac{a}{1-\frac{v}{V}}\right) =0 \quad \text{for } a=a(v); \tag{40} \]

expanding in powers of the ratio \(\dfrac{v}{V}\), we obtain:

\[ 0=g\left(\frac{a}{1+\frac{v}{V}}\right) = \left(\frac{dg}{da}\right)_{a=a_0} \left( \frac{a}{1+\frac{v}{V}}-a_0 \right) + \text{terms of higher order}. \tag{41} \]

According to formula (32), expressions (40) or (41) may be divided by the quantity

\[ \left(\frac{dg}{da}\right)_{a=a_0}; \]

then, instead of expression (40), we obtain

\[ \left. \begin{aligned} 0&=a-a_0-a\frac{v}{V}+\text{terms of higher orders},\\ 0&=a-a_0+a\frac{v}{V}+\text{terms of higher orders}. \end{aligned} \right\} \tag{42} \]

For the case \(a\ne 0\), formula (42) cannot be satisfied.

Thus, forces depending only on retarded distances, but not depending explicitly on the velocity, do not give an equilibrium configuration in a state of translational motion; more precisely, such forces do not give an equilibrium configuration that continuously passes into the original equilibrium configuration if the velocity tends to \(0\).

§ 25. In fact, forces of type (39) in general do not lead to stable configurations. Condition (32) guarantees only static stability; from expression (42) we see that any perturbation imparting velocity to the particles destroys the equilibrium.

In order to obtain dynamic stability, we must assume that the interaction between the particles depends explicitly on the velocity. Thus, we shall retain condition (49) and set:

\[ \left. \begin{aligned} \ddot{x}_A(t)&=g_A(a_A(v),v),\\ \ddot{x}_B(t)&=g_B(a_B(v),v). \end{aligned} \right\} \tag{43} \]

From formulas (37) and (38) we see that

\[ a_B(v)=a_A(-v)=\alpha(v), \]

so that, by analogy with formula (30), we may postulate

\[ g_B(\alpha(v),v)=-g_A(\alpha(-v),v). \tag{44} \]

Moreover, from considerations of symmetry we may postulate that, when the directions of the velocities of both particles are changed to the opposite ones, we also change to the opposite the direction of the acceleration of each particle caused by the other particle. Thus:

\[ \left. \begin{aligned} g_A(\alpha(v),v)&=-g_A(\alpha(-v),-v),\\ g_B(\alpha(v),v)&=-g_B(\alpha(-v),-v). \end{aligned} \right\} \tag{45} \]

From relations (44) and (45) we obtain:

\[ g_B(a(v),v)=-g_A(a(-v),v)=g_A(a(v),-v). \tag{46} \]

Thus, the equilibrium condition may be written in the form

\[ \begin{aligned} g_B(a(v),v)&=g_A(a(v),-v)=0,\\ g_A(a(v),v)&=g_A(a(-v),v)=0. \end{aligned} \tag{47} \]

Expanding the expressions in formulas (47) in a power series about the point \(a_0\), we obtain:

\[ \begin{aligned} \left(\frac{\partial g_A}{\partial a}\right)_{0,0}(a(v)-a_0) -v\left(\frac{\partial g_A}{\partial v}\right)_{00} +\text{terms of higher orders}&=0,\\ \left(\frac{\partial g_A}{\partial a}\right)_{0,0}(a(-v)-a_0) +v\left(\frac{\partial g_A}{\partial v}\right)_{00} +\text{terms of higher orders}&=0, \end{aligned} \tag{48} \]

where

\[ \left(\frac{\partial g_A}{\partial a}\right)_{00} = \left(\frac{\partial g_A(a,v)}{\partial a}\right)_{a=a_0,\ v=0} \quad \text{and so on.} \]

Dividing the expressions (48) by the quantity \(\left(\dfrac{\partial g_A}{\partial a}\right)_{00}\) and adding them, we obtain:

\[ \begin{aligned} a(v)+a(-v)-2a_0+\text{terms of higher order}&=0,\\ a(v)+a(-v)=\frac{2a(v)}{1-\dfrac{v^2}{V^2}} =2a(v)+\text{terms of higher order}. \end{aligned} \tag{49} \]

Then expression (49) reduces to the following:

\[ \delta a=a(v)-a_0=\text{terms of higher order}. \tag{50} \]

Thus, the greatest change in the equilibrium distance \(\delta a\) is of the order

\[ \delta a\sim a_0\frac{v^2}{V^2}. \]

If, in the expansion in powers of \(v\), we included terms of second order, then we would obtain:

\[ \delta a=c_2a_0\left(\frac{v}{V}\right)^2+\text{terms of third and higher orders}, \tag{51} \]

where the quantity \(c_2\) depends on the second derivative of the function \(g\). Lorentz’s proposal in connection with the Michelson–Morley experiments is equivalent to saying that

\[ c_2=-\frac{1}{2}. \tag{52} \]

Above we considered the longitudinal motion of a system of two particles. An analogous consideration can also be carried out for transverse motion or motion in any direction different from the direction \(AB\). Moreover, the consideration can be generalized to systems consisting of more than two particles. In such more general considerations it is necessary to restrict still further the possible types of forces. However, it is clear that the result of these considerations will be that a system accelerated to the velocity \(v\) must undergo a deformation of the order \(\left(\dfrac{v}{V}\right)^2\) or, in exceptional cases, a smaller deformation.

§ 26. Thus, the consideration carried out above can also be carried out for complex systems consisting of a large number of particles. The general consideration leads to the conclusion that if we consider a substance built of atoms that are in mutual dynamical equilibrium, then acceleration necessarily leads to deformations. One may expect, at least qualitatively, that these deformations will be of the Lorentz–FitzGerald type.

Let us consider, from this point of view, the Michelson–Morley experiment and suppose that for intra-atomic forces

\[ V=c. \]

Then one should expect a displacement of the fringes of the order

\[ df=a\left(c_2-\frac12\right)\left(\frac{v}{c}\right)^2 \]

upon rotation through \(90^\circ\), and not the displacement

\[ df_0=-\frac{a}{2}\left(\frac{v}{V}\right)^2, \]

which Michelson expected. Although nothing can be said a priori about the numerical value of the quantity \(c_2\), the possibility \(c_2=0\) is excluded by the dynamical consideration carried out above. Thus Michelson would not have expected to find the fringe displacement \(df_0\) if, in his time, he had taken into account the structure of matter. Historically, however, this was impossible, since in Michelson’s time what was known about the atomic structure of matter was by no means what we know at present. However, adopting the modern view of the structure of matter, we can confidently regard the experimental result

\[ df=0 \]

as an indication that the forces binding the atoms are such that

\[ V=c \quad \text{and} \quad c_2=\frac12 . \]

Considering the Michelson experiment from this point of view, we may say that, in fact, it is an experiment for determining the constant \(c_2\), i.e. an experiment investigating the nature of intra-atomic forces.

V. BEHAVIOR OF A MECHANICAL SYSTEM HELD IN A BOUND STATE BY LORENTZ-COVARIANT FORCES

§ 27. Covariant forces. We have seen that, in the case of forces whose action propagates with the speed of light, one should expect deformations of order \(\frac{v^2}{c^2}\); these deformations, however, are not identical with Lorentz deformations for all types of forces. Therefore, in order to obtain Lorentz deformations, we must impose a further restriction on the type of forces. This can be done by assuming that the forces are covariant with respect to Lorentz transformations.

Thus, we assume that the functions \(\mathbf{f}_k\) have such properties that the equations of motion are covariant. This assumption may be regarded as a restriction on the mathematical form of the equations of motion.

We consider two mechanical systems (both observed from the reference system \(K_0\)). The first system \(S\), at time \(t\), has a configuration (whether in equilibrium or not) specified by the coordinate vectors of its constituent particles

\[ S:\ \mathbf{r}_k(t), \quad k=1,2,\ldots,N. \]

The second system \(S'\) at the same instant of time is specified by the vectors

\[ S':\mathbf{r}'_{k}(t), \qquad k=1,2,\ldots,N . \]

We can construct the second system from the first by means of the following transformation formula:

\[ \left. \begin{aligned} \bigl(\mathbf{r}'_{k}(t)\bigr)_{x} &= \frac{\bigl(\mathbf{r}_{k}(t_{k})\bigr)_{x}-v t_{k}} {\sqrt{1-\dfrac{v^{2}}{c^{2}}}}, \qquad \bigl(\mathbf{r}'_{k}(t)\bigr)_{y,z} = \bigl(\mathbf{r}_{k}(t_{k})\bigr)_{y,z}, \\[6pt] t &= \frac{t_{k}-\dfrac{v}{c^{2}}\bigl(\mathbf{r}_{k}(t_{k})\bigr)_{x}} {\sqrt{1-\dfrac{v^{2}}{c^{2}}}} \, ; \end{aligned} \right\} \tag{53} \]

in the formulas given above the indices \(x,y\), and \(z\) denote the corresponding components of the vectors.

The new system is specified in a parametric representation; \(t_{k}\), \(k=1,2,\ldots,N\), are independent parameters.

We shall now require the following: if the system \(S\) satisfies the equations of motion (11)—(14), then the system \(S'\), obtained from the system \(S\) by means of the transformation (53), must also satisfy the same equations of motion. Let us note that this requirement can be fulfilled if and only if

\[ V=c. \]

Further, the requirement of covariance restricts the possible choice of the functions \(f_{k}\). As long as we are considering electromagnetic interactions, the requirement of covariance is fulfilled automatically; thus, we shall require only that the non-electromagnetic part of the forces possess the same covariance properties as those possessed by the electromagnetic forces.

§ 28. Equilibrium configuration in the case of covariant forces. In order to solve equation (23), let us consider a system moving translationally, so that

\[ \left. \begin{aligned} \mathbf{r}_{k}(t)&=\mathbf{r}^{(v)}_{k}+\mathbf{v}t,\\ \dot{\mathbf{r}}_{k}(t)&=\mathbf{v}, \qquad \frac{d^{2}\mathbf{r}_{k}(t)}{dt^{2}}=0. \end{aligned} \right\} \tag{54} \]

Then

\[ \left. \begin{aligned} \mathbf{f}_{k}\bigl(\mathbf{r}^{(v)}_{1}+\mathbf{v}t_{1k}, \mathbf{r}^{(v)}_{2}+\mathbf{v}t_{2k},\ldots;\mathbf{v},\mathbf{v},\ldots,\mathbf{v}\bigr)&=0,\\[4pt] t_{lk}&=\frac{\left|\mathbf{r}^{(v)}_{l}-\mathbf{r}^{(v)}_{k}+\mathbf{v}t\right|}{V}. \end{aligned} \right\} \tag{55} \]

To obtain a solution of equations (55), we shall use the covariance property of the equations which we have assumed, transforming the initial configuration (54) by means of the transformation (53); we obtain a new configuration

\[ \mathbf{r}'_{k}(t)=\mathbf{r}'^{(v)}_{k} \text{ is independent of } t, \]

\[ \dot{\mathbf{r}}'_{k}(t)=0, \qquad \frac{d^{2}\mathbf{r}'_{k}(t)}{dt^{2}}=0. \]

Then formula (55) takes the form

\[ \mathbf{f}_{k}\bigl(\mathbf{r}'^{(v)}_{1},\mathbf{r}'^{(v)}_{2},\ldots;0,0,\ldots,0\bigr)=0, \qquad k=1,2,\ldots,N. \tag{56} \]

The solution of equation (56), however, is the equilibrium configuration in the state of rest. Thus, we have:

\[ \mathbf{r}_k^{(v)}=\mathbf{r}_k^{(0)}. \]

Carrying out the inverse transformation to the original configuration, we obtain for the \(X\)-component:

\[ \left(\mathbf{r}_k^{(v)}+\mathbf{v}t\right)_x = \frac{\left(\mathbf{r}_k^{(0)}\right)_x+v t_k}{\sqrt{1-\frac{v^2}{c^2}}}, \]

\[ t=\frac{t_k+\frac{v}{c^2}\left(\mathbf{r}_k^{(0)}\right)_x}{\sqrt{1-\frac{v^2}{c^2}}}; \]

eliminating the parameter \(t_k\), we obtain:

\[ \left(\mathbf{r}_k^{(v)}\right)_x = \sqrt{1-\frac{v^2}{c^2}}\, \left(\mathbf{r}_k^{(0)}\right)_x \]

and, analogously,

\[ \left(\mathbf{r}_k^{(v)}\right)_{y,z} = \left(\mathbf{r}_k^{(0)}\right)_{y,z}. \]

Thus, in an equilibrium configuration moving with velocity \(\mathbf{v}\), lengths parallel to the velocity \(\mathbf{v}\) are contracted in the ratio \(\sqrt{1-\frac{v^2}{c^2}}\). Hence, accelerating the system \(S\) parallel to the axis \(X\), we disturb its internal equilibrium, and, under motion with constant velocity \(\mathbf{v}\), the system possesses a new equilibrium configuration in which all dimensions parallel to the axis \(X\) are contracted in the ratio \(\sqrt{1-\frac{v^2}{c^2}}\).

We must emphasize that the arguments given above are not sufficient to show that a system subjected to acceleration must necessarily pass into a Lorentz-contracted state. Acceleration may deform the system forever; it may force the system to rotate about an axis, and so forth. All that we have been able to show is the following: if the system is accelerated sufficiently carefully by external forces, and these forces are such that they can impart to the system only translational motion and cannot rotate it, then after the acceleration the system acquires an equilibrium configuration; the latter is a configuration with contracted lengths.

§ 29. Slowly rotating system. Consider a mechanical system rotating about an axis passing through the origin of coordinates. Denote the angular velocity by \(\omega\), introduce a unit vector \(\alpha\) in the direction of the axis of rotation and a unit vector \(\boldsymbol{\beta}(\varphi)\), perpendicular to the vector \(\alpha\); \(\varphi\) is the azimuthal angle relative to the axis. The coordinate vector of atom \(k\) in our system will then be represented by the expression

\[ \mathbf{r}_k^{(\omega)}(t)=a_k\alpha+\boldsymbol{\beta}_k(t), \qquad a_k,\ b_k,\ \varphi_k \text{ — constants.} \]

\[ \boldsymbol{\beta}_k(t)=\boldsymbol{\beta}(\omega t+\varphi_k), \]

From simple geometrical considerations we have:

\[ \frac{d\boldsymbol{\beta}_k(t)}{dt} = \omega\left(\alpha\times \boldsymbol{\beta}_k(t)\right), \]

\[ \frac{d^2\boldsymbol{\beta}_k(t)}{dt^2} = -\omega^2 \boldsymbol{\beta}_k(t). \]

Thus, the equilibrium equations in the rotating system can be written in the form

\[ f_k(a_1\alpha+b_1\beta_1(t_{1k}),\, a_2\alpha+b_2\beta(t_{2k}),\ldots,\, \omega b_1(\alpha\times \beta_1(t_{1k}))\ldots)=\omega^2 b_k \beta_k(t). \tag{57} \]

The retarded times are given by the expression

\[ t_{lk}-t=-\frac{[(a_l-a_k)\alpha+b_l\beta_l(t_{lk})-b_k\beta_k(t)]}{V}. \]

The equation given above has the solution:

\[ t_{lk}=t+\tau_{lk}, \tag{58} \]

where \(\tau_{lk}\) is constant in time.

Indeed, let us substitute this expression into equation (57). The vector on the right-hand side of expression (57) will then rotate uniformly about the axis of the system; its absolute value will be constant. The numerical values of the quantities \(\tau_{lk}\) can be written explicitly, but we shall not need them in what follows.

Since the quantities \(\tau_{lk}\) are constants, all vectors of the functions \(f_k\) rotate uniformly about the axis with one and the same angular velocity \(\omega\). The right-hand side of the expression also represents a rotating vector. Thus we see that if the solution satisfies equation (57) at the moment of time \(t=0\), then it will automatically satisfy the equation at any later moment of time. Thus rotation about an axis with constant angular velocity is in principle compatible with the equations of motion. If one starts from the system at rest, then the terms proportional to the angular velocity \(\omega\) may be regarded as a small perturbation. Then we can obtain solutions of equations (57), expressed in the following way through the solutions for the system at rest:

\[ r_k^{(\omega)}=r_k^{(0)}+\text{terms of the type }(r_k\omega^2)+\text{terms of the type }\left(\frac{r_k\omega}{c}\right)^2 . \]

The first correction term expresses the deformation introduced by centrifugal forces, the second—the retardation effect. Whereas the first term leads to a certain expansion of the cylinder in the radial direction, the second term leads to a small contraction of the system. This contraction arises as a consequence of the Lorentz contraction of the surface. The outer layers tend to contract because they possess a certain linear velocity and thus press on the inner layers. In the state of equilibrium there exists a certain elastic stress: the inner layers keep the outer layers somewhat less contracted—by the factor

\[ 1:\sqrt{1-\frac{\omega^2 r^2}{c^2}}. \]

Whether the deformed states of a freely rotating system described above actually exist can be determined depending on the exact form of the functions \(f_k\). From mechanics we know that rotation of this kind can occur only about one of the three possible principal axes of a rigid body; about any other axis more complicated forms of motion arise (nutations, etc.). In the case of these more complicated motions the quantities \(\tau_{lk}\) can no longer be regarded as constant, and therefore the system can no longer move as an ideal rigid body; the individual parts of the body will undergo periodic deformations in the course of the motion.

§ 30. Combined effects of rotation and translational motion. If we substitute the coordinate vectors \(r_k^{(\omega)}(t)\) into the Lorentz transformation, then we obtain the coordinates of the new equilibri—

configuration with coordinate vectors \(\mathbf r_k^{(v,\omega)}(t)\). If the vectors \(\mathbf a\) and \(\mathbf v\) are parallel to the \(X\)-axis, then we obtain:

\[ \left(\mathbf r_k^{(v,\omega)}(t)\right)_x = \frac{a_k+v t_k}{\sqrt{1-\frac{v^2}{c^2}}}, \qquad t= \frac{t_k-\frac{v a_k}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}} \quad \text{or} \quad t_k=t\sqrt{1-\frac{v^2}{c^2}}+\frac{v a_k}{c^2}. \tag{59} \]

Eliminating the quantity \(t_k\), we obtain:

\[ \left(\mathbf r_k^{(v,\omega)}(t)-\mathbf r_l^{(v,\omega)}(t)\right)_x = \sqrt{1-\frac{v^2}{c^2}}\, \left(\mathbf r_k^{(0,\omega)}(t)-\mathbf r_l^{(0,\omega)}(t)\right)_x . \]

Thus, in a rotating and translationally moving system the contraction of lengths is the same as in a nonrotating system. Further, for the components perpendicular to the velocity \(\mathbf v\), we obtain:

\[ \left(\mathbf r_k^{(v,\omega)}(t)\right)_\perp = \mathbf b_k^{(v,\omega)}\boldsymbol\beta_k(t_k) = \mathbf b_k^{(0,\omega)}\boldsymbol\beta_k \left( t\sqrt{1-\frac{v^2}{c^2}}+\frac{v a_k}{c^2} \right). \]

In particular, at the time \(t=0\) we have:

\[ \left(\mathbf r_k^{(v,\omega)}\right)_\perp = \mathbf b_k^{(0,\omega)}\boldsymbol\beta_k \left(\frac{v a_k}{c^2}\right) = \left[ \mathbf b_k^{(0,\omega)}\boldsymbol\beta \left(\frac{\omega v a_k}{c^2}\right) \right]. \]

From the definition of the vector \(\boldsymbol\beta(\varphi)\) we see that the rotating and translationally moving system moves as a rigid body. However, the equilibrium configuration in this case differs from that which existed in the merely rotating system. The equilibrium configuration of the rotating and translationally moving system can be obtained from the equilibrium configuration of the merely rotating system by means of a uniform rotation of the system about the axis of rotation. The angle of rotation \(\varphi_{kl}\), between the vectors \(\mathbf r_k\) and \(\mathbf r_l\) in the system which is not moving translationally, is given by the expression

\[ \varphi_{kl}=\frac{\omega}{c^2}\,v\,(r_k-r_l). \]

In the discussion carried out above we took the vector \(\mathbf v\) to be parallel to the vector \(\mathbf a\), i.e. the translational motion was parallel to the axis of rotation. We would obtain another possible mechanical system if we considered the case in which the vector \(\mathbf v\) is not parallel to the vector \(\mathbf a\). In the latter case the system will no longer be rigid, i.e. in a system in which the direction of the translational motion is not parallel to the axis of rotation,

\[ \frac{d}{dt}\left|\mathbf r_l^{(v,\omega)}(t)-\mathbf r_k^{(v,\omega)}(t)\right|\ne 0. \]

In the process of rotation certain kinds of oscillations arise, which are due to the fact that the retarded forces cannot constantly remain in equilibrium with the centrifugal force. The resulting twisting of the cylinder corresponds to the fourth type of deformations listed in § 5.

With the aid of the considerations given above we found the equilibrium configuration of the particles of a rigid body rotating about an axis parallel to the direction of translational motion. We found that the equilibrium configuration changes both when the velocity changes and when the angular velocity changes. Starting from a system having zero velocity and angular velocity \(\omega\), by means of Lorentz transformations we arrive

to a system possessing velocity \(v\) and angular velocity \(\omega'\)

\[ \omega'=\omega\sqrt{1-\frac{v^2}{c^2}}. \tag{60} \]

It does not necessarily follow from this that, when accelerated, a rotating system changes its angular velocity from \(\omega\) to \(\omega'\). From our investigation we can conclude only the following: in the process of acceleration the configuration of the rotating system will be disturbed by the accelerating force; this disturbance is determined by the manner of acceleration, i.e. by the distribution of the forces acting on the atoms of the system and by the dependence of the forces on time; after the acceleration is completed, the system, under the action of internal forces, passes into one of the possible equilibrium configurations considered above; into precisely which configuration the system passes, i.e. what the value of the angular velocity \(\omega'\) will be, cannot be determined solely from consideration of the equilibrium states—this is a dynamical problem.

§ 31. Slowing of rotation under acceleration; conservation laws. The internal forces acting in the system must be conservative. From experiment we know that the motion of a closed system is characterized by the following relations:

\[ \left. \begin{aligned} &\sum_k m_k\mathbf r_k(t)=\mathbf P \;-\; \text{constant in time},\\[4pt] &\sum_k m_k\bigl(\dot{\mathbf r}_k\times \mathbf r_k(t)\bigr)=\mathbf I \;-\; \text{constant in time}. \end{aligned} \right\} \tag{61} \]

The complete system of solutions of the equations of motion for \(\mathbf r_k\) must satisfy these conditions. One can immediately verify that the relations given above are incompatible with the covariant equations of motion (26), unless one assumes that the mass \(m_k\) depends on the velocity. If we put\(^*\)

\[ m_k=\frac{m_{0k}}{\sqrt{1-\dfrac{v_k^2(t)}{c^2}}};\qquad m_{0k}=\text{const.}, \tag{62} \]

then the relations (61) and (62) become compatible with the covariant equations of motion (11)—(14) in the following sense. Suppose that we have a system \(S\), whose coordinate vectors \(\mathbf r_k\) satisfy the equations of motion (23)—(25) and the relations (61), (62). Construct, by means of the Lorentz transformations, another system \(S'\) with coordinate vectors \(\mathbf r'_k(t)\). The latter system will also satisfy both the equations of motion (11)—(14) and the conservation laws (61), (62). However, the question remains open: for every initial configuration

\[ \mathbf r_k(0)=\mathbf r_k;\qquad \dot{\mathbf r}_k(0)=\mathbf v_k \]

do there exist solutions of equations (11)—(14) satisfying the conservation laws (61), (62)? It is easy to see that, if certain special cases are excluded, then, generally speaking, such solutions do not exist; for retarded action, in the general case, the conservation laws are not fulfilled. Indeed, let us consider, for example, the motion of particle \(k\). When at time \(t\) the particle is accelerated under the action of other particles, it changes its momentum and total angular momentum. Thus, in order that the total momentum and

\[ \text{\(^*\) The conservation laws are taken from experiment, which, generally speaking, is not so accurate as to make it possible in the macroscopic case to determine whether the mass \(m_k\) depends on the velocity \(v_k\) according to formula (62) or not.} \]

the total momentum and angular momentum have remained constant, the change that has occurred with particle \(k\) must be instantaneously compensated by corresponding changes for the other particles. Such compensation, if we neglect certain special configurations, can occur only as a result of the action of particle \(k\) on the other particles, i.e., the compensation can take place only with a certain delay.

Exceptions to this general rule are, first, systems at rest or moving translationally, and, second, systems rotating with constant velocity and moving translationally with a velocity parallel to the axis of rotation. There are also other special kinds of motion with analogous properties.

§ 32. A way out of these difficulties will be found if we assume that, under acceleration, a particle radiates; this radiation can carry momentum and angular momentum from some particles to others. In this view, the interaction between particles is carried by fields propagating with velocity \(c\). The conservation laws must be written in the form

\[ \frac{d}{dt}\left(\sum_k m_k \dot{\mathbf r}_k(t)+\mathbf P_F(t)\right)=0, \]

\[ \frac{d}{dt}\left(\sum_k m_k\bigl(\dot{\mathbf r}_k(t)\times \mathbf r_k(t)\bigr)+\mathbf I_F(t)\right)=0, \]

where

\[ \mathbf P_F(t)=\int \mathbf p_F(\mathbf r,t)\,d\mathbf r, \]

\[ \mathbf I_F(t)=\int \mathbf i_F(\mathbf r,t)\,d\mathbf r \]

are the momentum and angular momentum of the field surrounding the atoms. Fields can be radiated only by accelerated particles; therefore, if the system is in equilibrium or moves with uniform velocity, it does not radiate. Any radiation that could have been emitted in the past quickly returns, and therefore we may suppose that a system in a stationary state should contain no radiation.

A freely rotating body about an axis is not entirely free of acceleration. Therefore one may expect that such a system radiates continuously and therefore gradually slows down. This effect, if it exists, is certainly very small.

For small accelerations the effects of radiation are very small, and we shall henceforth omit them. Thus, we shall assume that the conservation laws are compatible with the equations of motion.

§ 33. Transfer of angular momentum. Let us compare the system \(S\) with the system \(S'\) (coordinate vectors \(\mathbf r_k^{(0,\omega)}\) and \(\mathbf r_k^{(v,\omega)}\), see formula (59)); we see that both systems have one and the same angular momentum about the axis of rotation (the latter is parallel to the velocity \(\mathbf v\)). Then, if we accelerate the system \(S'\) parallel to the \(X\)-axis, imparting to it additional momentum but not angular momentum, then in the final state the angular velocity will be \(\omega'=\sqrt{1-\dfrac{v^2}{c^2}}\,\omega\).

The physical content of this method of acceleration may be outlined as follows. Imagine a mechanical system of \(N=N_1+N_2\) atoms. The \(N_1\) atoms form one system, the \(N_2\) another; the distance between the systems is sufficiently large to exclude interaction between the two groups of atoms. In each group there is formed an equilib-

configuration, say, \(S_1\) and \(S_2\), they have momenta and angular momenta \(P_1\), \(P_2\) and \(I_1\), \(I_2\); as long as there is no interaction between the systems, these quantities remain constant. Suppose now that these are the following systems: \(S_1\) is a cylinder rotating about the \(X\)-axis with angular velocity \(\omega\), but not moving translationally; \(S_2\) is a system moving along the \(X\)-axis with velocity \(\mathbf{v}\) relative to the system \(S_1\), but not rotating. The systems \(S_1\) and \(S_2\) finally collide, and, if the collision is elastic, then after the collision the system \(S_1\) will move with velocity \(\mathbf{v}\). The collision can be considered with the aid of the equations of motion of a system of \(N\) particles. Particles from the systems \(N_1\) and \(N_2\) come, for a short time, sufficiently close for an interaction to take place between them. As a result of the interaction the particles separate again. If the system \(S_2\) does not begin to rotate during the collision, then the angular momentum of the system \(S_1\) does not change. Thus, after the collision the system \(S_1\) moves with velocity \(\mathbf{v}\) and angular velocity

\[ \omega' = \omega \sqrt{1 - \frac{v^2}{c^2}} . \]

Above we have considered only an example. In the general case we can say: if initially the system \(S_1\) was rotating with angular velocity \(\omega\), but was not moving translationally, then, being accelerated to velocity \(\mathbf{v}\) in the process of interaction with the system \(S_2\), the system \(S_1\) slows its rotation in the ratio

\[ \omega' = \omega \sqrt{1 - \frac{v^2}{c^2}}, \]

provided only that the interaction is such that the system \(S_2\) does not begin to rotate after the acceleration.

To make these general considerations more convincing, let us consider the special case of a rotating electrically charged system situated in an electric field parallel to the axis of rotation. Thus, the system \(S_1\) is a rotating charged cylinder, and the system \(S_2\) is a pair of capacitor plates (Fig. 3). By charging the capacitor, we introduce an interaction between the systems \(S_1\) and \(S_2\). The symmetry of the arrangement guarantees that the reaction of the rotating body on the capacitor does not impart angular momentum to it. However, if, even despite the symmetry, we were to find that the rotating cylinder, acting on the capacitor, tends to rotate it, we could always modify the apparatus so as to compensate this tendency. We shall take as an experimental fact that the masses of the particles can be written in the form

Fig. 3. Apparatus for accelerating a rotating electrically charged cylinder.

\[ m_t = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}; \qquad m_l = \frac{m_0}{\left(1 - \frac{v^2}{c^2}\right)^{3/2}}, \]

so that

\[ \frac{d\mathbf{v}}{dt} = \frac{\mathbf{F}_t}{m_t} + \frac{\mathbf{F}_l}{m_l}, \]

where \(\mathbf{F}_t\) and \(\mathbf{F}_l\) are the components of the force perpendicular and parallel to the direction of the velocity.

Suppose now that the cylinder is accelerated by forces directed along the \(X\)-axis and acting on each charged atom of the cylinder separately. The system moves in the direction of the \(X\)-axis with velocity \(\mathbf{v}(t)\), parallel to the \(X\)-axis. Owing to the rotation, each particle has a transverse component of velocity. Consider an atom with velocity \(\mathbf{w}(t)\), perpendicular

axis. Then the total velocity of this particle is

\[ \mathbf V=\mathbf v(t)+\mathbf w(t). \]

Longitudinal and transverse components of the force act on the system, so that

\[ \mathbf F_l=\frac{\mathbf v(t)}{V(t)}\,\mathbf F, \]

\[ \mathbf F_t=\frac{\mathbf w(t)}{V(t)}\,\mathbf F. \]

The acceleration in the direction of \(\mathbf w\) will be:

\[ \frac{d\mathbf w(t)}{dt} = \frac{\mathbf w(t)}{V(t)}\cdot\frac{\mathbf F_l}{m_l} - \frac{\mathbf v(t)}{V(t)}\cdot\frac{\mathbf F_t}{m_t} = \frac{\mathbf F}{m_0}\, \frac{\mathbf v(t)\mathbf w(t)}{V(t)^2} \left[ \left(1-\frac{V^2}{c^2}\right)^{3/2} - \left(1-\frac{V}{c^2}\right)^{1/2} \right]; \]

neglecting the term of order \(\dfrac{V^4}{c^4}\), we obtain:

\[ \frac{1}{\mathbf w(t)}\,\frac{d\mathbf w(t)}{dt} \approx -\frac{1}{2}\,\frac{F\mathbf v(t)}{m_0}. \]

Integrating with respect to the variable \(t\) from \(0\) to \(T\) and neglecting relatively small terms, we obtain:

\[ \int_0^T \frac{F}{m_0}\,\mathbf v(t)\,dt = \frac{1}{2}\,\mathbf v(T)^2 \]

and, consequently,

\[ \frac{\mathbf w(T)-\mathbf w(0)}{\mathbf w(0)} = -\frac{1}{2}\,\frac{\mathbf v(T)^2}{c^2}. \]

If we substitute \(\mathbf w(t)=r\omega(t)\) into the last expression, then it can also be written in the form

\[ \omega(T) = \omega(0) - \frac{1}{2}\,\omega(0)\cdot\frac{v^2(T)}{c^2} \approx \omega(0)\sqrt{1-\frac{v^2(T)}{c^2}}. \]

The exact consideration, based on general principles, was carried out by us in the preceding section. The derivation given above is only a simple example showing how the rotation slows down in a special case.

§ 34. The considerations given above are easy to use for explaining the transverse Doppler effect. Consider, for example, the ion \(\mathrm{He}^{+}\), i.e. an \(\alpha\)-particle with one electron moving around it. Suppose that an electric field acts on the ion and that the field strength is perpendicular to the orbit of the electron. The ion undergoes acceleration; the accelerating force has (according to the considerations given above) a component that tends to slow down the motion of the electron. From the calculations carried out above it can be shown that the decrease in the frequency of the electron is

\[ \Delta\omega= \frac{1}{2}\,\omega\,\frac{m_l-m_t}{m_0}\,\frac{V^2}{c^2}; \tag{63} \]

if for the quantities \(m_l\) and \(m_t\) we take the values according to Lorentz’s formulas, then we obtain that the decrease in the rotational frequency of the electron is in agreement with the transverse Doppler effect.

Let us note, however (and as far as we know, this has never been pointed out before), that if for \(m_l\) and \(m_t\) we take the quantities proposed by Abraham, then we again obtain an effect of frequency decrease

\[ \Delta\omega_{AB}=0.8\,\Delta\omega. \]

Thus, we may conclude that the transverse Doppler effect is a consequence of the change in the mass of the electron with change in velocity. The numerical value of the magnitude of the effect depends on the law of the variation of mass with velocity; the observed magnitude speaks more in favor of Lorentz’s formula than of Abraham’s formula.

Thus, the transverse Doppler effect could have been predicted before the theory of relativity on the basis of classical theory, as a consequence of the difference between the masses \(m_l\) and \(m_t\) of a fast electron.

§ 35. Change in the state of translational motion as a physical change. From the preceding discussion we see that, under the action of acceleration, a system passes into a new physical state; the changes are caused chiefly by retardation effects. In this new state the internal forces (if they are left free) will continuously deform the system until a new equilibrium configuration is reached. This new configuration is the old equilibrium configuration subjected to Lorentz transformations.

This change of state caused by acceleration may be compared with the change of state under a change of temperature. We shall examine this analogy in more detail.

If a rod is heated, it expands. This expansion occurs in the following way. At the initial temperature \(T_1\), the internal forces in the rod are in equilibrium with the thermal motion; in this state the length of the rod, say, is equal to \(a_1\). By the application of external forces the rod may be stretched or compressed, but with this change of external form the internal equilibrium is disturbed and stresses arise which tend to restore the old equilibrium state.

If we change the temperature of the rod by an amount \(\Delta T\), but by means of external forces prevent the expansion of the rod to the length \(a_2\), then the change of temperature disturbs the internal equilibrium in the rod and internal stresses arise. If we then release the rod, removing the external forces, these stresses will establish a new equilibrium configuration, in which the length of the rod becomes equal to \(a_2\), and as soon as the new configuration is established, we shall again have a state of equilibrium, and the internal stresses will disappear.

The internal forces in a rod whose temperature has been raised to the temperature \(T_2\), but whose length is held by external forces equal to \(a_1\), will be exactly the same as in the case when a rod initially in equilibrium at the temperature \(T_2\) was compressed by external forces to the length \(a_1\). If we then free the rod from the action of external forces, the manner of reaching the equilibrium configuration will not depend on the preceding history of the deformation. Thus, the relaxation process is completely determined by the mechanical properties of the rod, and one and the same relaxation will take place in the following three cases:

1) we stretch a rod of length \(a_2\) at temperature \(T_2\) by means of external forces to the length \(a_1\), and then instantaneously remove these forces;

2) we cool a rod of length \(a_1\) from the temperature \(T_1\) to the temperature \(T_2\), preventing the contraction of the rod to the length \(a_2\) by means of external forces, and then instantaneously remove these forces;

3) we accelerate a rod of length \(a_1\) to velocity \(v\) (so that

\[ \frac{a_2}{a_1}=\sqrt{1-\frac{v^2}{c^2}} \]

) and, by means of external forces, prevent the Lorentz contraction, and then instantaneously remove these forces.

In all these three cases the result is one and the same: compression of the rod from length \(a_1\) to length \(a_2\). Example 1), obviously, shows that the manner and time of relaxation depend on the mechanical properties of the rod. In case 2)

a change in temperature is simply responsible for a change in the internal state of the rod, which ultimately leads to a change in the dimensions of the rod; and, in complete analogy with this, the acceleration of the rod leads to the same kind of change in the internal state of the rod, which likewise leads to a change in the dimensions of the rod.

It is especially interesting to consider a sudden change of state. Suppose that the relaxation time of the rod is \(t_R\); it is determined by the elastic properties of the rod. If we heat the rod from temperature \(T_1\) to temperature \(T_2\) during a time interval

\[ \delta t \ll t_R, \]

then, even without the action of external forces, during the time \(\delta t\) deformation cannot occur merely as a consequence of the inertia of the rod. In this case, after the rod has acquired the temperature, it will gradually relax and, finally, pass into an equilibrium state. This case occurs in practice: during an explosion the substance instantaneously (by chemical means) acquires a high temperature, but its expansion takes place only afterward.

By analogy with this thermal process we may assert that, if the rod is uniformly accelerated during a time \(\delta t\) up to a velocity \(v\), then it will not have time to pass into the Lorentz-contracted state during this short interval of time, and it will contract only afterward, passing into the contracted state after a time \(t \sim t_R\), which depends only on the material structure of the rod.

If, on the other hand, we heat the rod gradually, so that its temperature is some function of time \(T(t)\) such that

\[ T(t_1)=T_1,\quad T(t_2)=T_2, \tag{64} \]

then we shall have the approximate relation

\[ a(t)\simeq \bar a(T(t)),\quad t_1 \leq t \leq t_2, \tag{65} \]

where

\(a(t)\) is the actual length of the rod at the instant \(t\), and \(\bar a(T)\) is the equilibrium length of the rod at temperature \(T\).

Relation (65) is satisfied the better, the slower the heating proceeds, although there must always be some lag in attaining the equilibrium state.

The same considerations can also be applied to the case of a slowly accelerated system. Let us consider the acceleration of a rod by a force uniformly distributed over its volume.*) The velocity acquired by time \(t\) will then be \(v(t)\), and the equilibrium length of the rod at this time will be

\[ a(v(t))=a\sqrt{1-\left(\frac{v(t)}{c}\right)^2}. \]

The actual length of the rod at the instant \(t\) will satisfy this relation the better, the smaller the magnitude of the acceleration.

*) A uniformly distributed accelerating force is the analogue of a uniformly heated rod. We could accelerate the rod by applying a force only to one of its points. However, in this case we would be dealing with further complications, because the action of this force must still propagate throughout the entire rod. The action of a force at one point is analogous to heating the rod at one point. In the latter case, in order to establish equilibrium, it is necessary both for heat to be distributed throughout the entire body and for the mechanical adjustment of the different parts of the body to the final temperature.

Similarly, if we accelerate a cylinder rotating about its axis, we shall find that the cylinder gradually contracts in the direction of motion and twists about the axis of rotation. If the magnitude of the acceleration is sufficiently small, the Lorentz-contracted state will be a good approximation for describing the state of the cylinder at any moment of time.

§ 36. Let us consider a real example. The speed of sound in steel is of the order of \(5\cdot 10^5\) cm/sec, so that the relaxation time in a rod of length \(a=1\) m will be of the order

\[ t_R=\frac{100}{5\cdot 10^5}=2\cdot 10^{-4}\ \text{sec}. \]

The acceleration of the Earth in the gravitational field of the Sun is of the order

\[ A=\frac{v^2}{R}\sim 0.6\ \text{cm/sec}^2 \quad (v=3\cdot 10^6\ \text{cm/sec};\ R=1.5\cdot 10^{13}\ \text{cm}). \]

The change in the length of the rod upon acceleration from velocity \(-v\) to velocity \(+v\) will then be

\[ -\frac{2avAt_R}{c^2}=4\cdot 10^{-17}\ \text{cm}. \]

It is by precisely this amount that the arm of the Michelson interferometer made of steel should have shortened, as compared with the other arm, during a quarter of a year. We expect this contraction as a consequence of the finiteness of the relaxation time in the steel rod. The effect is many orders of magnitude smaller than the possible accuracy of measurements.

The effect indicated above, although unobservable because of its smallness, points to the fact that the fringes in a rigid Michelson interferometer are slightly displaced over the course of a year. A somewhat larger periodic effect may be caused by the rotation of the Earth about its axis.

In order to avoid misunderstandings, we emphasize that the effects indicated above are not “new effects” depending on our interpretation of the theory of relativity; they can be derived by means of ordinary relativistic mechanics. In particular, we must emphasize that the effects indicated above cannot be used to determine absolute velocity in the sense of a limitation of the theory of relativity. We can assert only the following: if the interferometer is adjusted at a certain time of year, then this adjustment will be partially disturbed during the following six months as a consequence of the accelerated motion of the Earth. In fact, the effect considered here is closely connected with the well-known “clock paradox.”

VI. CONCLUSIONS

§ 37. The dynamical considerations presented in the preceding sections allow us to explain the observed relativistic effects.

The Michelson–Morley experiment and all analogous experiments can be explained by the change, under acceleration, of the equilibrium state of the atoms of which the material of the apparatus consists.

The decrease of atomic frequencies in the transverse Doppler effect can be understood in exactly the same way as the slowing of the rotation of a cylinder caused by internal atomic forces.

The change of the mass of the electron and proton with velocity follows necessarily from the postulate of covariance of the internal forces and the mass defect, if we consider the classical picture of the electron.

These results are based on the essential assumption that intra-atomic forces and the forces acting between atoms have, chiefly,

electromagnetic nature; non-electromagnetic forces, which exist in addition to them, must have the same covariant properties as electromagnetic forces; in particular, their action must propagate with the speed of light.

These assumptions are analogous to the assumptions of the special theory of relativity, but they do not go so far; moreover, our formulation of the assumptions is much more closely connected with physical facts than are the postulates of the special theory of relativity. The following comparison of the two points of view may be given:

1) In the theory of relativity the question considered is: how will a given system appear when observed from another frame of reference.

From our point of view, it is much more important to investigate what happens in a physical system when it is accelerated and, consequently, changes the state of its translational motion.

The fact that actual experimental results always deal with the effects of acceleration of physical systems was discussed in § 3.

2) In the theory of relativity it is postulated that all laws of nature are covariant with respect to Lorentz transformations.

We have found that it is sufficient to admit that the Maxwell–Lorentz equations have a covariant form. Moreover, since purely electromagnetic forces are not sufficient to keep matter in a stable state, we postulated that the atomic forces which, together with electromagnetic forces, keep matter in a stable state must obey laws that have a covariant form.

There is no need to generalize this postulate to all existing types of forces, because we have very little information about the velocity of propagation and the covariant properties of forces different from those that keep matter in a bound state (for example, there is no real experimental evidence for the velocity of propagation of gravitational forces).

Moreover, although it is probable that nuclear forces are covariant, on the basis of the available experimental data this has not yet been precisely proved.

It is often assumed that there is “convincing negative evidence” that all laws of nature are covariant with respect to Lorentz transformations. This evidence seems less convincing if one takes into account that in all observations the influence of translational motion on matter was investigated; then, analyzing all the experimental evidence for the covariance of the laws of nature, we arrive again at the same conclusion—that the forces which keep matter in a bound state are covariant in nature. Furthermore, this covariance is needed only near equilibrium configurations; there is practically no evidence as to how matter behaves far from states of equilibrium.

3) It is assumed that the point of view of the theory of relativity is more satisfactory than Lorentz’s point of view, because it derives everything from a single general principle. I cannot agree with this assertion for the following reasons.

The theory of relativity asserts: “All laws of nature are such that they have one and the same form when observed from different Lorentzian frames of reference.” In my opinion, one need assume only the following: “The forces which keep matter in a bound state obey laws which, for stationary or nearly stationary systems, can be formulated in a Lorentz-invariant manner.”

The first assertion is much more general than the second, since the first assertion concerns many uninvestigated questions which, however, may be investigated at some time in the future—for example, the velocity of propagation of gravitation, nonstationary processes.

The first assertion, despite the fact that it is much broader than the second, gives us no information about the behavior of systems under acceleration. According to Einstein, in order to consider systems moving with acceleration, it is necessary to introduce the general theory of relativity. Then accelerations may be considered in terms of apparent gravitational fields; the latter procedure seems to me rather artificial, if one takes into account that the accelerated motion of a rod or of a rotating cylinder can be considered directly, without reference to a gravitational field, if by means of the method we are using we modify the covariance hypotheses of the special theory of relativity.

4) There remains the question of the distinguished frame of reference \(K_0\). One of the greatest successes of the theory of relativity is considered to be the elimination of the necessity for such a frame. This is justified by the fact that such a frame cannot be singled out “on general grounds,” and therefore the assumption of its existence is unsatisfactory. These arguments do not seem convincing. The “general ground” from which the conclusion is drawn that such a frame cannot be singled out is the assertion of the theory of relativity concerning the laws of nature, which we cited in item 3. Since the experimental verification of this assertion is rather limited, we must be more cautious. In general it is dangerous to make broad assertions about what can and what cannot be discovered in the future.

§ 38. Remarks on the concept of a distinguished frame of reference. In all the considerations carried out above we have used a single frame of reference, but we have not discussed what is meant by such a frame.

When we use the concept of the frame \(K_0\), we by no means require that it be an all-embracing “absolute frame” associated with the ether. When we introduce the frame \(K_0\), we consider a frame that plays a distinguished role in the neighborhood of the solar system (or of our galaxy); but in any case this frame is closely connected with the motion and distribution of matter in our surroundings.

Let us note that we may postulate (as a definition) that the inertial frame in which the center of gravity of the solar system is at rest is the frame \(K_0\). We could also choose any other inertial frame that moves with respect to this frame. None of these assertions is in contradiction with the experimental data known at the present time. The difficulty that has been so much emphasized consists in the ambiguity of such a choice: our contemporary knowledge does not allow us to choose unambiguously one of the possible inertial frames. But the fact that today we do not know the difference between possible inertial frames does not mean that such a difference does not exist in reality. It is possible that we shall succeed in clarifying the difference between them, if it really exists, after some time, and one should not from the very beginning completely exclude this possibility.

In fact, we cannot single out the absolute motion of the Earth (i.e., motion with respect to the frame \(K_0\)) by means of the Michelson interferometer or instruments of an analogous type. Let us add to this that there are weighty reasons for asserting that, even if the frame of reference \(K_0\) exists, the Michelson interferometer is an unsuitable instrument for singling out motion with respect to it. Failures to single out the frame by means of the interferometer should not completely discourage us; indeed, in observing the universe we find that a frame connected with the fixed stars could apparently be chosen as the distinguished frame (a frame in which their mean translational motion is small).

The stars have, of course, a complex proper motion, and much labor will be required to determine the exact nature of this motion. What is important is that the surroun-

the stars around us can serve for a very clear selection of the system \(K_0\). There may be some disagreements concerning the exact definition of this system, but it is quite clear that an inertial system moving with a constant velocity, say \(100{,}000\ \mathrm{km/sec}\), relative to the majority of the stars differs sharply from the system selected by these stars. Thus there is no doubt that, to within the uncertainty in velocity \(\Delta v\) (where \(\Delta v \ll c\)), a particular inertial system is singled out most distinctly with the aid of the stars. It is quite possible that the system thus chosen will turn out to be physically distinguished in some way.

§ 39. We shall present further arguments in favor of the view that one may suppose that the distribution of fixed stars (and of matter around us) physically singles out one reference system from the aggregate of inertial systems.

Before the appearance of the general theory of relativity, the possibility of absolute accelerations was considered a defect of the special theory of relativity. It was assumed that inertial systems are systems that move without acceleration. This unsatisfactory feature was removed by the general theory. It was pointed out that in empty space, in the absence of matter, we cannot single out any inertial system. The inertial systems of the special theory of relativity are systems that move without acceleration relative to the stars. We single out accelerated motion when centrifugal forces or, in general, forces of inertia appear. The latter arise as a reaction of distant masses to the accelerated system.* With the aid of the general theory of relativity it proved possible to get rid of the metaphysical concept of absolute acceleration that is used by the special theory of relativity.

In my opinion, analogous considerations can be used in considering the question of absolute motion. If one postulates that there exists an absolutely resting system \(K_0\), irrespective of the existence of matter, then this postulate will be metaphysical and cannot be accepted. If, however, we assume that the system \(K_0\) is at rest relative to the greater part of the matter surrounding us, and that it may be taken as a locally resting distinguished system, then the latter hypothesis seems correct both philosophically and physically.

From the preceding discussion it follows that a rod that was initially at rest and was then accelerated to a velocity \(v\) relative to the system \(K_0\) undergoes a contraction of length; we shall suppose that this contraction of length is caused by the motion of the rod relative to the system \(K_0\), or, what is more important, relative to the surrounding stars. Thus a system undergoes Lorentz deformations if it is set into translational motion relative to the local system \(K_0\). They may be regarded as a reaction of the distant stars to the moving system. This reaction may also be represented qualitatively as the reaction of distant masses to accelerated systems.

Let us note that the mechanism of this reaction is also clear from the preceding sections. If we postulate that the speed of propagation of our Lorentz-covariant forces must be taken relative to the stationary gravitational field, then Lorentz contractions will automatically have

* In order to avoid confusion, we shall assume that this reaction is transmitted by the gravitational field of the masses. This reaction of distant masses should be understood as follows: these distant masses, acting on the distant parts of space, create an approximately stationary gravitational field in our part of space. Translational motion relative to this field causes Lorentz contractions; the reaction of distant masses does not imply an “instantaneous” transmission of action.

take place in systems that move with respect to this gravitational field. If we take, in a purely qualitative way, electromagnetic waves “scattering” the gravitational field, then we can automatically explain the Lorentz deformations. In order to determine whether this point of view is correct or not, it is necessary to observe substantially nonstationary gravitational fields. This lies beyond the limits of practical possibility, at least at the present time.

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