Einstein started from the assumption that for small velocities (i.e., much smaller than the speed of light, c) every mass moves according to Newton’s laws of motion. By applying the procedure mentioned above, Einstein succeeded in deriving from the Newtonian laws the laws of motion for high speed. The chief result obtained in this way is the rather startling fact that the mass of a body is not constant; like the duration of time and the length of a measuring rod, it is dependent on its velocity. The mass increase’s with velocity in such a way that as the velocity becomes very great, the mass also becomes very great. A given force will produce smaller and smaller change in the actual velocity the more it approaches the velocity of light. For this reason no particle can ever actually attain the velocity of light, no matter how great a force acts on it and for how long a time.
Proceeding to the domain of electromagnetic phenomena, Einstein was again led to the conclusion that electric and magnetic field strengths are also “relative quantities.” Every helpful description of electric or magnetic field strength must contain not only their magnitude but also the system with respect to which they are measured.
The necessity of this is easily seen. When an electric charge is at rest in L, it possesses an electric field only “relative to L.” There is no magnetic field relative to L since an electric charge at rest exerts no magnetic force. However, when this same situation is described relative to F, the electric charge is moving with a velocity v; this means that there is an electric current. Since every electric current exerts a magnetic force, it is appropriate to say that there is magnetic field “relative to F.” The existence of these fields is, of course, a physical fact. But their descriptions “relative to L” and “relative to F” are different.
7. Equivalence of Mass and Energy
From the same hypotheses Einstein was able to draw still another conclusion that at first one can hardly believe is contained in them. If an agglomeration of masses is formed or falls apart under production of kinetic energy or radiation the sum of the masses after the agglomeration or disintegration is smaller than before. The produced energy is given by E = mc2, where m is the loss of mass. This statement may be considered as a law about the “transformation of mass into energy.” In a process where there is such a transformation from mass to energy or vice versa, the energy of the system will not be conserved unless account is taken for the gain or loss due to the change in mass.
This law has proved to be of immense significance in the development of our knowledge of the interior of the atom. According to our modern conception of the atom, it consists of a massive central core with positive charge, which is called the nucleus, and around it a number of negatively charged particles, called electrons, circulating at great speed. The nucleus itself is a complex structure built up of two kinds of particles, the positively charged protons, which are the nuclei of the simplest atom, hydrogen, and neutrons that are exactly like protons except for the lack of any electric charge. The various atoms found in nature differ only by the difference in the number of protons and neutrons they possess in the nucleus, the heavier atoms containing more particles and hence being of more complex structure. As stated already, hydrogen, the lightest atom, has a nucleus that is simply a proton. The next lightest atom is helium, whose nucleus contains two protons and two neutrons. These four particles are bound very tightly together in the nucleus by certain nuclear forces. It is one of the most important problems of modern physics to investigate the strength, character, and quality of these nuclear forces which bind the atomic nuclei together.
A measure of the strength with which particles in the nucleus are packed together can be obtained by considering how much energy is necessary to pry the particles loose and separate them so they are all a large distance apart from one another. This energy is known as the binding energy of a nucleus. Now, according to Einstein’s theory, this energy (E) which is produced by the formation of the nucleus must appear as loss of mass due to the agglomeration. This means that the masses of the individual protons and neutrons added together are by E/c2 greater than the mass of the nucleus where these particles are bound together. Thus by measuring the masses of the protons and neutrons while they are free and the mass of the nucleus, it is possible to obtain the binding energy of the nucleus. Such measurements have been carried out for many of the atoms found in nature, and we are now able to classify according to how strongly the particles in the nuclei are bound together. These results have been of immense value in the planning and interpretation of recent researches on the artificial transmutation of atoms, where by bombarding various atomic nuclei with protons, neutrons, and other similar particles, new atoms have been produced.
Einstein’s mass-energy relation has also for the first time in history made possible the solution of the problem of the source of the sun’s energy. The sun has been radiating heat and light at the same rate as it is now doing for billions of years. If that energy had come from ordinary combustion, such as the burning of coal, the sun would have cooled off by now. The problem had the scientists completely baffled until Einstein’s equation E = mc2 appeared. The velocity of light (c) is a very large number, and with this squared, the formula states that a small quantity of mass can transform into a very large amount of energy. For this reason, by losing only an immeasurable amount of mass, the sun has been able to continue radiating for so long, and will continue to do so for billions of years to come. The actual mechanism of the transformation of mass to energy occurs in nuclear reactions that are going on in the interior of the sun. It is believed now that they ultimately boil down to the formation of helium nuclei from hydrogen. In this “packing effect,” as we have learned already, mass is lost and radiation emitted.
This possibility of using mass as a source of energy has aroused very optimistic hopes that methods of liberating the energy stored in the atom as mass for practical use might be found. There has also been, on the other hand, a very frightening prospect that such a process might be used to produce an explosive so devastating that a pound of it would completely annihilate everything within a radius of many miles. This foreboding was fulfilled forty years later when the first atomic bomb destroyed Hiroshima.
To Einstein, however, the main value of his result was not in the applications, no matter how numerous or important. To himself his principal achievement was to have deduced the law E = mc2 from the relativity principle. It was in accord with Einstein’s conception of the universe to strive continually for the discovery of simple, logical bridges between the laws of nature. The wealth of conclusions derived from his two hypotheses constitutes what has been known since as the “theory of relativity.” Einstein had struck a rich well of information about nature, which would yield knowledge for many decades to come.
8. Theory of Brownian Motion
In the same year (1905) Einstein discovered new fundamental laws in two fields outside the theory of relativity. At the time when Einstein came to Bern, he was intensely occupied with the problem of light and motion, But he saw that the final goal could be attained only by attacking the problems from various angles. One of the paths to the goal, he realized, was to investigate the relations between light and heat, and those between heat and motion.
It had been known for some time that heat is connected with the irregular motion of molecules. The higher the temperature, the more violent is this motion. The statistical behavior of particles in such irregular motion had been investigated chiefly by the Scottish physicist James Clerk Maxwell (1831–79) and the Austrian Ludwig Boltzmann (1844–1906). It had been assumed even before, that the kinetic energy of the molecules is proportional to the absolute temperature. At the time of Maxwell and Boltzmann, however, the molecular constitution of matter was still a hypothesis which could be doubted. It enabled many different phenomena to be explained very simply, but there was as yet no very direct proof of the existence of the molecule. Furthermore, it had not yet been possible to obtain an accurate value of such a significant quantity as th
e number of molecules in a unit volume of matter. Estimates of this number had been made by such men as the Austrian physicist Loschmidt (1865), but they were based on involved and rather indirect methods. Einstein strongly felt the necessity of investigating this matter more thoroughly and obtaining a more direct proof of molecular motion.
It had long been known that small but microscopically visible particles, when suspended in a fluid with approximately the same density, exhibit a constant, apparently irregular zigzag motion. It had been discovered by the Scottish botanist Robert Brown for pollen dust suspended in water, and for this reason it is known as Brownian motion. It is not caused by any external influence jarring the vessel or by currents of water in the vessel, and the agitation increases in intensity when the temperature of the water is raised. For this reason it had been conjectured that the motion is connected with the heat motion of the molecules. According to this view, the kinetic energy of the water molecules in constant collision with the microscopic particles produces irregular forces in random directions, which give rise to the observed motions.
In 1902 Einstein had restated Boltzmann’s theory of random motion in a simplified form. He now treated the Brownian motion with this method and arrived at a surprisingly simple result. He showed that the results of the kinetic theory of molecules should also hold for particles visible by microscope — for instance, that the average kinetic energy of the particles in the Brownian motion should have the same value as that for the molecules. Hence, by observing the motion of the microscopically visible particles, much valuable information could be obtained about the invisible molecules. In this way Einstein was able to derive a formula which stated that the average displacement of the particles in any direction increases as the square root of the time. He showed (1905) how one can determine the number of molecules in a unit volume by measuring the distances traveled by the visible particles.
The actual observations were later made by the French physicist Jean Perrin, who completely verified Einstein’s theory. The phenomenon of Brownian motion has subsequently always been included among the best “direct” proofs of the existence of the molecule.
9. Origin of the Quantum Theory
To Einstein it was always clear that his theory of relativity could not claim (and, indeed, it never did claim) to solve all the mysteries of the behavior of light. The properties of light investigated by Einstein concerned only a certain group of phenomena dealing with the relation between the propagation of light and moving bodies. For all these problems light could be conceived along the lines of traditional physics as undulatory electromagnetic processes which filled space as a continuum. By the theory of relativity it was assumed that some objects can emit light of this nature, and no attempt was made to analyze the exact process by which light is emitted or to investigate whether it sufficed for a derivation of all the laws for the interaction of light with matter.
The investigations on the nature of light and its interaction with matter, however, were to lead to the rise of the “quantum theory,” a revolution in physical thought even more radical than the theory of relativity. And in this field, too, Einstein’s genius had a profound influence on its early development. In order to make understandable the nature of Einstein’s contributions, I shall describe briefly the situation prior to his researches.
The simplest way of producing light is by heating a solid body. As the temperature rises, it begins to glow from a dull cherry red to a brighter orange, and then to blinding white light. The reason for this is that visible light consists of radiations of different frequencies ranging from red at the low end through the colors of the spectrum up to violet at the high end. The quality of light emitted by a solid body depends solely on its temperature; at low temperatures the low-frequency waves predominate and hence it looks red; at higher temperatures the shorter wave lengths appear and mingle with the red to give the white color.
Attempts to explain this change in quality of light with temperature on the basis of nineteenth-century physics had ended in failure, and this was one of the most important problems facing physicists at the beginning of the twentieth century. At that time the emission of light was thought to be produced by the oscillations of charged particles (electrons), the frequency of light emitted being equal to the frequency of the vibration. According to Boltzmann’s statistical law, already mentioned, the average energy of oscillation of an electron should be exactly equal to the average kinetic energy of gas molecules, and hence simply proportional to the absolute temperature. But this led to the conclusion that the energy of vibrations is independent of the frequency of oscillation, and hence light of different frequencies will be emitted with the same energy. This conclusion obviously was contradicted by the observations on light emitted by heated bodies. In particular, we know that light of very short wave lengths is not emitted to any great extent by hot bodies. As the temperature increases, rays of increasingly higher frequencies appear, but yet at a given temperature there is no perceptible radiation above a certain definite frequency. Consequently it appeared that somehow it must be difficult to emit light of very high frequencies.
Since all arguments based on the mechanistic theory of matter and electricity led to results conflicting with experience, the German physicist Max Planck in the year 1900 introduced a new assumption into the theory of light emission. At first it appeared to be rather inconsequential, but in the course of time it has led to results of an increasingly revolutionary character. The turn in physics coincided exactly with the turn of the century. I shall sketch Planck’s idea in a somewhat simplified and perhaps superficial form.
According to Boltzmann’s statistical law, the average energy of oscillation of an electron in a body is equal to the average kinetic energy of the molecules. The actual energies of the individual atoms or molecules can, of course, have very different values; the statistical law only relates the average energy with the temperature. Boltzmann, however, had been able to derive a second result which determined the distribution of the energy of the particles around the average value. It stated that the number of particles with a certain energy depends on the percentage by which this energy differs from the average value. The greater a deviation, the less frequent will be its occurrence.
As Planck realized, the experimental results indicated that the oscillating electrons in a body cannot emit radiation with an arbitrary frequency. The lack of high-frequency radiation shows that the mechanism of radiation must be such that it is somehow difficult to emit light of high frequency. Since no explanation of such a mechanism existed at that time, Planck was led to make the new assumption that, for some reason as yet unknown, the energy of oscillation of the atoms cannot have just any value, but can only have values that are integral multiples of a certain minimum value. Thus, if this value is called , then the energy of the oscillations can only have the discrete values 0, , 2 … or n, whose n is zero or an integer. Consequently the radiation emitted or absorbed must take place in portions of amount . Smaller amounts cannot be radiated or absorbed since the oscillation cannot change its energy by less than this amount. Planck then showed that if one wants to account for the well-known fact that a shift to higher temperatures means a shift to higher frequencies, one has to take values for that vary for different values of the frequencies of the oscillations, and in fact has to be proportional to the frequency.
Thus he put = hv, where v is the frequency and h is the constant of proportionality, which has since then been called Planck’s constant and has been found to be one of the most fundamental constants in nature. With this assumption Planck was immediately able to derive results in the theory of radiation that agreed with observations and thus removed the difficulties that had confronted the physicists in this field.
10. Theory of the Photon
Planck thought he was only making a minor adjustment in the laws of physics in formulating his hypothesis, but Einstein realized that if this idea was developed consistently it would lead to a rupture of the f
ramework of nineteenth-century physics so serious that a fundamental reconstruction would be necessary. For if the electron can oscillate only with certain discrete values of energy, it contradicts Newton’s laws of motion, laws which had been the bases for the whole structure of mechanistic physics.
Planck’s hypothesis dealt only with the mechanism of radiation and absorption of light and stated that these processes could take place only in definite amounts. He said nothing about the nature of light itself while it is propagated between the point of radiation and that of absorption. Einstein set out to investigate whether the energy transmitted by light retained this discrete character during its propagation or not. He once expressed this dilemma by the following comparison: “Even though beer is always sold in pint bottles, it does not follow that beer consists of indivisible pint portions.”
Retaining the analogy, if we wish to investigate whether the beer in a barrel actually consists of definite portions or not, and if so whether this portion is a pint, two pints, or ten pints, we can proceed as follows: We take a number of containers, say ten to be definite, and pour the beer from the barrel at random into these containers. We measure the amount in each container and then pour back the beer into the barrel. We repeat this process a number of times. If the beer does not come in portions, the average value of the beer poured into each container will be the same. If it consists of pint portions, there will be variations in the average values. For two pints the variations will be greater, and for ten they will be still greater. Thus by observing the distribution of beer among the ten vessels, we can tell whether the barrel of beer consists of portions and what size they are. We can realize it easily by imagining the extreme case that the whole content of the barrel is one portion.
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