Bohr’s Theory and Consequences of the Principle of Relativity

K. Försterling. Bohr’sches Atommodel und Relativitätstheorie. Zeitschrift für Physik, 3, 404, 1920.

The inertia of energy and the subordination of the “mass of energy” that follows from this to the general laws of gravitation can be derived independently of the theory of relativity, or, in any case, can be regarded as independent empirical facts. The inertia of energy makes possible a very simple derivation of the dependence of mass on velocity. The gravitation of energy explains, at least qualitatively, the deflection of light rays in a gravitational field. In the note under review, the author shows that the inertia and gravitation of energy, in connection with the fundamental “frequency condition” of Bohr’s theory of the atom, make it possible to obtain, in an unusually simple way, two further consequences that coincide with the consequences of the theory of relativity. The frequency condition of Bohr’s theory may be expressed as follows: if an atom emits energy \(E\), then the frequency of the emitted light \(\nu\) is determined in the following way:

\[ \nu = \frac{E}{h} \tag{1} \]

where \(h\) is Planck’s constant, having one and the same value under all circumstances. Let the atom move with some velocity \(v\). In that case, by virtue of the property of inertia of energy, the mass of the atom increases by the amount

\[ M = \frac{E}{c^2}. \]

where \(c\) is the velocity of light. The presence of an additional mass is accompanied also by an additional kinetic energy:

\[ e=\frac{1}{2}\frac{E}{c^{2}}v^{2}. \]

The internal energy of the atom \(E\) will change and take the value:

\[ E' = E+e = E\left(1+\frac{1}{2}\frac{v^{2}}{c^{2}}\right) \tag{2} \]

A more exact derivation leads, as is not difficult to verify, to the following formula:

\[ E'=\frac{E}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \tag{3} \]

Substituting, in place of \(E\), the value \(E'\) from formula (3) into the frequency condition (1), we obtain a new frequency:

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

or

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

This phenomenon coincides with the so-called “transverse Doppler effect,” which follows from the theory of relativity but is inaccessible to experimental observation because of its smallness.

The mass of the energy of the atom, \(\frac{E}{c^{2}}\), is, by the original assumption, gravitating; therefore, when the atom is displaced from a point with gravitational potential \(0\) to a point with gravitational potential \(\Phi\), the energy \(E\) changes by the amount

\[ e=\frac{E}{c^{2}}\cdot \Phi, \]

and hence the energy of the atom is

\[ E' = E+e = E\left(1+\frac{\Phi}{c^{2}}\right) \tag{5} \]

The frequency of the emitted light, according to condition (1), must therefore change:

\[ \nu'=\nu\left(1+\frac{\Phi}{c^{2}}\right) \tag{6} \]

Formula (6) coincides with the “red shift” necessarily following from the general theory of relativity.

S. Vavilov.