Abstract
A report delivered at the joint memorial meeting for P. N. Lebedev of the Society of Devotees of Natural Science, Anthropology, and Ethnography and the colloquium of the Moscow Physical Institute of the Society of the Scientific Institute on March 18, 1922.
Full Text
The Pressure of Light, Mass, and Energy ^1)
(In memory of P. N. Lebedev.)
S. I. Vavilov.
In the physics of our day, alongside the astonishingly rapid accumulation of new experimental facts, there is taking place a profound organic process of changing the original point of view, a revision of basic conceptions and notions. The old, familiar Newtonian concepts of mass, force, and acceleration are beginning to be replaced, still timidly and tentatively, by the unfamiliar “curvature of four-dimensional space,” “world lines,” and the “energy tensor.” At times it seems to the modern physicist that the ground is slipping from under his feet and that all support has been lost. The dizzying sensation experienced in this case is probably akin to that which an astronomer—an old believer in the time of Copernicus—had to undergo when trying to grasp the immobility of the moving celestial vault and of the sun. But this unpleasant sensation is deceptive: the ground beneath the physicist’s feet is firm, because this ground is facts. In the raging ocean of theories and “points of view,” facts remain immobile and solid as before, and the experimental physicist can calmly look out from his stronghold upon the surging waves, judiciously choosing what is needful and guiding for himself.
P. N. Lebedev was an experimenter par excellence. The chief work of his life—the pressure of light, discovered and measured by him—was for him first of all and most importantly an experimental fact. He knew that “Maxwell–Bartoli forces of pressure of rays may in time acquire great significance in questions of physics and astronomy” ^2), but this significance was again wholly determined by new facts, chiefly by the application of light pressure to the physics of the heavens.
Meanwhile, in the domain of theoretical physics, where the ground is unsteady and where “revolutions” and changes of point of view are possible at every moment, the fact discovered by Lebedev—the pressure of light—has acquired enormous significance, in some cases quite equivalent to the significance of the first and
^1) Report read at the joint meeting in memory of P. N. Lebedev of the Society of Lovers of Natural Science, Anthropology, and Ethnography and of the annual meeting of the Moscow Physical Institute of the Society of the Scientific Institute, March 18, 1922.
^2) P. N. Lebedev. Collected Works, p. 123. Moscow, 1913.
of the second law of thermodynamics. It should not be forgotten that both laws are essentially empirical facts—the impossibility of realizing in nature mobili perpetui. The pressure of light served as just such a fundamental fact-principle for the thermodynamics of radiant energy. The necessity of the existence of light pressure follows, to be sure, already from the second law, as was shown by Bartoli, but its magnitude and its dependence on various quantities still remain undetermined from this. For this, a concrete conception of the nature of light is required. Different views lead to different results. In the emission theory, the pressure of light must be twice as great as in the wave theory of light. Lebedev’s measurements are decisive in this respect and provide a new argument in favor of the wave theory of light. By a purely thermodynamic route, taking the pressure of light as an additional principle, Boltzmann derives the law of the energy of integral black radiation as a function of temperature, the so-called Stefan–Boltzmann law.
Using the pressure of light and Doppler’s principle, Wien, again in a purely thermodynamic manner, takes the next step toward determining the dependence of black radiation on temperature and wavelength.
Thermodynamics is the most unshakable branch of theoretical physics, being in essence a logical-mathematical development of the basic initial facts-principles. From this point of view, the pressure of light may also be regarded as a principle of thermodynamics.
But the significance of the fact discovered by Lebedev is not limited to the domain of thermodynamics. The pressure of light plays a primary role in that revisionist movement of contemporary physics to which I pointed at the beginning. I shall allow myself to dwell in somewhat greater detail precisely on this application of light pressure in theoretical physics.
Only at the present time can one appreciate the full caution of Newton, who in the second law of motion defined force through the quantity of motion, or impulse. The concepts of mass and force, with which we have become accustomed to operate, are not primary. We do not perceive directly forces and masses; what is given to us primarily are impulses-pressures; precisely pressure-impulse must serve as the true foundation of mechanics. Newton’s second law essentially contains two parts: first, the definition of force:
\[ J = f \cdot dt, \tag{1} \]
where \(J\) is impulse, \(f\) is force; secondly, impulse is defined as the change of the quantity of motion:
\[ J = \Delta(m \cdot v), \tag{2} \]
i.e. the concept of mass is introduced; the change of the product of two functions \(m\) and \(v\) is equal to the impulse. In the more “popular” form of the second law
\[ f=m\alpha, \tag{3} \]
where \(\alpha\) is acceleration. Such a form is less rational, since in both parts of the equality the unknowns “force” and “mass” figure, while one proceeds from the approximately established experimental fact of the constancy of mass.
Of primary importance in physics is the secondary concept of energy, which, with the aid of impulse, can be defined as follows:
\[ dE=J\,ds=v\,d(mv). \tag{4} \]
Without making any hypotheses concerning mass, we are entitled, expanding the meaning of the total differential \(d(mv)\), to rewrite equality (4) as follows:
\[ dE=v^{2}\frac{\partial m}{\partial v}\,dv+mv\,dv. \tag{5} \]
Let us suppose that in nature we detect the presence of a certain pressure-impulse, observing it, for example, with the aid of Lebedev’s pressure apparatus. If we measure the magnitude of the pressure and the velocity of the pressing agent, then, by Newton’s definition, we shall find the change in the quantity of motion. In Lebedev’s experiments with a black absorbing surface the pressure is
\[ P=\frac{E}{c}, \tag{6} \]
where \(E\) is the energy incident in 1 second, and \(c\) is the velocity of light. A light flux, having the initial velocity \(c\), being absorbed, comes to rest, i.e. its final velocity is zero. The change in the quantity of motion will therefore be
\[ M.c-M'.0=Mc, \tag{7} \]
where \(M\) and \(M'\) are the initial and final mass of the light flux, which we have the right to ascribe to it according to the basic definition (2). Equating (6) and (7), we find
\[ \frac{E}{c}=Mc, \]
whence we can determine the “mass” of light:
\[ M=\frac{E}{c^{2}}, \tag{8} \]
In the emission theory of light, as is easy to see, the corresponding mass is
\[ M=\frac{2E}{c^2} \]
but there this mass is quite understandable and customary; whereas the mass (8), connected with radiant energy, in which we are accustomed to seeing only energy, seems to us strange and unexpected. We are compelled to recognize it, as we have seen, by light pressure. We are accustomed, in the objects of nature, to assume the presence of entirely separate “mass” and “energy”; in the light flux we encounter a strange object, in which one has to speak of the “mass of energy.” Light pressure compels us to revise the customary concept of mass.
One may also take the following bold, though somewhat hypothetical, step1. We know that one form of energy is transformable into another, while the magnitude of the energy remains unchanged. We therefore have sufficient grounds to try to extend relation (8) to every form of energy, in particular to the energy of motion. Let us use the differential relation (5), substituting instead of \(dE\), on the basis of (8),
\[ dE=c^2dm. \tag{9} \]
Then we shall find
\[ c^2dm=v^2dm+mv\,dv. \]
Separating the variables \(m\) and \(v\), we find
\[ \frac{dm}{m}=\frac{v\,dv}{c^2-v^2}. \tag{10} \]
Equation (10) is easily integrated. Let us suppose that, when the velocity is zero, the mass is \(m_0\), and when the velocity is \(v\), the mass is \(m\); then we find
\[ m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} =m_0\left\{1-\frac{v^2}{c^2}\right\}^{-\frac12} =m_0\left\{1+\frac12\frac{v^2}{c^2}-\ldots\right\}. \tag{11} \]
This remarkable formula can be derived approximately in an entirely simple way. Indeed, let us imagine that a certain stationary mass \(m_0\) has begun to move with velocity \(v\), thereby acquiring the kinetic energy
\[ E=\frac{m_0v^2}{2}, \]
but a mass possessing energy \(E\), according to our initial assumption, increases by the amount
\[ M=\frac{E}{c^2}=\frac{m_0}{2}\frac{v^2}{c^2}. \]
Thus the total moving mass \(m\) will be
\[ m=m_0+\frac{1}{2}m_0\frac{v^2}{c^2} =m_0\left(1+\frac{1}{2}\frac{v^2}{c^2}\right). \tag{12} \]
The inaccuracy that we have allowed consists in the fact that, in the second additional term, we considered the mass \(m_0\) to be at rest. However, for those velocities \(v\) with which one can operate in experiment, formulas (11) and (12) give practically one and the same result.
As is known, formula (11) can be derived on the basis of the electron theory of matter, and also of the theory of relativity. The derivation given is the simplest. The study of the change in the mass of electrons emitted by radium with enormous velocities confirmed the validity of the relation found. Thus we have every right to speak of the “mass of energy” for energy of any form. The popular concept of mass has completely lost all clarity.
According to Newton, masses gravitate toward one another. Does this property extend also to the “mass of energy”? Mechanically the two masses do not differ in any way, but perhaps there exists a difference with respect to gravitation. An experiment with a radioactive pendulum, carried out by Sautherns (Southerns),1 shows that no such difference exists; the mass of energy is at the same time gravitational mass. Hence the next step is quite natural. A stream of light possesses energy and “mass of energy”; consequently, in the gravitational field of some mass a ray of light must deviate from its rectilinear path. The observation of the total solar eclipse of May 29, 1919 confirmed this conclusion as well: the light of fixed stars passing near the sun is deflected. It is not difficult to calculate the magnitude of this deflection, whereupon the observed value proved to be approximately twice as large as the calculated one. The exact result, as is known, is given by the general theory of relativity; however, for a definite judgment on this question, further new observations are still necessary.
Thus, we have seen that Lebedev’s light pressure leads to the necessity of revising the concept of mass and to new remarkable facts. Thus, around the fundamental, firmly established fact by way of
of temporary constructions of theories ever newer facts begin to appear; the ground beneath the feet of the experimental physicist grows firmer, and he can cheerfully listen to the crack and crash of collapsing worldviews: the facts will not be affected. In our day, when in connection with this collapse of theories people begin even to speak of “the end of physics,” it is especially instructive to recall the experimental physicists who laid the cornerstones in the edifice of physics, among whom P. N. Lebedev also belongs.