Our analysis of the [Newtonian mass] vs. [relativistic mass] debate thus leads us to the conclusion that the conflict between these two formalisms is ultimately the disparity between two competing views of the development of physical science.
Max Jammer1
Needless to say, in the years since its publication, many physicists have picked over Einstein’s derivation of this, his most famous equation. Some have criticized the derivation as circular. Others have criticized the critics. It seems that Einstein himself was not entirely satisfied with it and during his lifetime developed other derivations, some of which were variations on the same theme, others involving entirely different hypothetical physical situations.
Despite his efforts, all these different approaches seemed to involve situations that, it could be argued, are rather exceptional or contrived. As such, these different approaches are perhaps insufficiently general to warrant declaring E = mc2 to be a deep, fundamental relationship—which Einstein called an ‘equivalence’—between mass and energy.
This equation has by now become so familiar that we’ve likely stopped thinking about where it comes from or what it represents. So, let’s pause to reflect on it here. Perhaps the first question we should ask ourselves concerns the basis of the equation. If this is supposed to be a fundamental equation describing the nature of material substance, why is the speed of light c involved in it? What has light (or electromagnetic radiation in general) got to do with matter? The second question we might ask concerns what the equation is actually telling us. It says that mass and energy are equivalent. But what does that mean, exactly?
Let’s tackle these questions in turn. The basic form of E = mc2 appears to tie the relationship between mass and energy to electrodynamics, the theory of the motion of electromagnetic bodies. Indeed, the first of Einstein’s 1905 papers on special relativity was titled ‘On the Electrodynamics of Moving Bodies’, which is a bit of a giveaway.
For sure, any kind of physical measurement—hypothetical or real—will likely depend on light in some way. After all, we need to see to make our observations. But if this relationship is to represent something deeply fundamental about the nature of matter, then we must be able to make it more generally applicable. This means separating it from the motions of bodies that are electrically charged or magnetized and from situations involving the absorption or emission of electromagnetic radiation. Either we find a way to get rid of c entirely from the equation or we find another interpretation for it that has nothing to do with the speed of light.
This project was begun in 1909. American physicists Gilbert Lewis and Richard Tolman set out to establish a relativistic mechanics—meaning a mechanics that conforms to Einstein’s special theory of relativity—that is general and universally applicable to all matter. They were only partially successful. They didn’t quite manage to divorce c completely from its interpretation as the speed of light.
Their project was arguably completed only in 1972, by Basil Landau and Sam Sampanthar, mathematicians from Salford University in England. Their derivation requires what appears to be a fairly innocent assumption; that the mass of an object depends on its speed. Landau and Sampanthar made no further assumptions about the precise nature of this dependence, and we can express it here simply as m = fvm0, where m is the mass of an object moving at velocity v, m0 is the mass of the object when stationary (called the ‘rest mass’), and fv is some function of the velocity that has to be figured out.
What Landau and Sampanthar discovered as a consequence of their mathematical manipulations is that a quantity equivalent to c appears as a constant,* representing an absolute upper limit on the speed that any object can possess. The function fv then becomes nothing other than the (by now hopefully familiar) Lorentz factor, γ.
What this suggests is that nothing—but nothing—in the universe can travel faster than this limiting speed. And for reasons that remain essentially mysterious, light (and, indeed, all particles thought to have no mass) travels at this ultimate speed. This answers our first question—we don’t need to eliminate c from the equation connecting mass and energy; we just reinterpret it as a universal limiting speed.
But, as it so often turns out, solving one problem simply leads to another. The logic that Landau and Sampanthar applied would suggest that, just as space and time are relative, so too is the mass of an object. If fv = γ, this means that m = γm0, where m is now the mass of an object in a frame of reference moving at velocity v and m0 is the rest mass or ‘proper mass’, measured in the rest frame. An object moving at 86.6 per cent of the speed of light will appear to have a ‘relativistic mass’ m equal to twice the rest mass. It is measured to be twice as ‘heavy’.
The object is not literally increasing in size. The m in m = γm0 represents mass as a measure of the object’s inertia, which mushrooms towards infinity for objects travelling at or very near the limiting speed.* This is obviously impossible, and often interpreted as the reason why c represents an ultimate speed which cannot be exceeded. To accelerate any object with mass to the magnitude of c would require an infinite amount of energy.
But Einstein himself seems to have been rather cool on the idea of relativistic mass, and in certain physics circles the notion remains very dubious. In a 1948 letter to Lincoln Barnett, an editor at Life magazine who was working on a book about Einstein’s relativistic universe, Einstein suggested that Barnett avoid any mention of relativistic mass and refer only to the rest mass.2
In an influential paper published in 1989, the Russian theorist Lev Okun reserved particular ire for the concept of relativistic mass. As far as he was concerned there is only one kind of mass in physics, the Newtonian mass m, which is independent of any frame of reference, whether moving or stationary.3
The simple truth is that even today there appears to be no real consensus among physicists on the status of these concepts. I have textbooks on special relativity sitting on my bookshelves which happily explore the consequences of the relativity of mass. I have other books and some papers stored on my computer which decry the notion and declare that there is only Newtonian mass, and special relativity is simply an extension of classical mechanics for the situation where objects move at speeds close to the ultimate limiting speed, c. The author of one textbook suggests that relativistic mass is a convenient construct and the decision whether or not to use it is a matter of taste.4 Let’s park this for now and move on to our second question.
Today, nobody questions the fundamental nature of E = mc2, or its essential correctness and generality. But, just as arguments have raged about the importance—or otherwise—of relativistic mass—so arguments have raged for more than 100 years about what the equivalence of mass and energy actually means. How come? Isn’t it rather obvious what it means?
The default interpretation—one firmly embedded in the public consciousness, expressed in many textbooks and shared by many practicing physicists—is that E = mc2 summarizes the extraordinary amount of energy that is somehow locked away like some vast reservoir inside material substance. It represents the amount of energy that can be liberated by the conversion of mass into energy.*
This was very much my own understanding, working as a young student in the 1970s, then as a postgraduate student and subsequently a university lecturer and researcher in chemical physics in the 1980s. To me, the fact that a small amount of mass could be converted into a large amount of energy just seemed to make matter—whatever it is—appear even more substantial.
In 1905, Einstein was quite doubtful that his ‘very interesting conclusion’ would have any practical applications, although he did note: ‘It is not excluded that it will prove possible to test this theory using bodies whose energy content is variable to a high degree (e.g., radium salts).’5
The next few decades would provide many examples of bodies with highly variable energy content. Physicists discovered that the atoms of chemical elements have inner structures. Each atom consists of a small central nucleus containing positively charged protons (discovered in 1917) and electrically neutral neutrons (discovered in 1932), surrounded or ‘orbited’ by negatively charged electrons (1895). It is the number of protons in the nucleus that determines the nature of the chemical element. Different elements, such as hydrogen, oxygen, sulphur, iron, uranium, and so on, all have different numbers of protons in their atomic nuclei. Atoms containing nuclei with the same numbers of protons but different numbers of neutrons are called isotopes. They are chemically identical, and differ only in their relative atomic weight and stability.
Physicists realized that the neutron could be fired into a positively charged nucleus without being resisted or diverted. Italian physicist Enrico Fermi and his research team in Rome began a systematic study of the effects of bombarding atomic nuclei with neutrons, starting with the lightest known elements and working their way through the entire periodic table. When in 1934 they fired neutrons at the heaviest known atomic nuclei, those of uranium, the Italian physicists presumed they had created even heavier elements that did not occur in nature, called transuranic elements. This discovery made headline news and was greeted as a great triumph for Italian science.
The discovery caught the attention of German chemist Otto Hahn at the prestigious Kaiser Wilhelm Institute for Chemistry in Berlin. He and his Austrian colleague Lise Meitner set about repeating Fermi’s experiments and conducting their own, much more detailed, chemical investigations. Their collaboration was overtaken by events. When German forces marched into a welcoming Austria in the Anschluss of 12 March 1938, Meitner lost her Austrian citizenship. This had afforded her some protection from Nazi racial laws, but overnight she became a German Jew. The very next day she was denounced by a Nazi colleague and declared a danger to the Institute. She fled to Sweden.
Meitner celebrated Christmas 1938 with some Swedish friends in the small seaside village of Kungälv (‘King’s River’) near Gothenburg. On Christmas Eve she was joined by her nephew, the physicist Otto Frisch. As they sat down to breakfast, Frisch had planned to tell his aunt all about a new experiment he was working on. However, he found that she was completely preoccupied. She was clutching a letter from Hahn, dated 19 December, which contained news of some new experimental results on uranium that were simply bizarre.
Hahn and another colleague Fritz Strassman had repeated Fermi’s experiments and concluded that bombarding uranium with neutrons does not, after all, produce transuranic elements. It produces barium atoms. The most stable, common isotope of uranium contains 92 protons and 146 neutrons, giving a total of 238 ‘nucleons’ altogether (written U-238). But the most common isotope of barium has just 56 protons and 82 neutrons, totalling 138. This was simply extraordinary, and unprecedented. Bombarding uranium with neutrons had caused the uranium nucleus to split virtually in half.
Meitner did a little energy book-keeping. She reckoned that the fragments created by splitting the uranium nucleus must carry away a sizeable amount of energy, which she estimated to be about 200 million electron volts (or mega electron volts, MeV).* The fragments would be propelled away from each other by the mutual repulsion of their positive charges. Now, energy had to be conserved in this process, so where could it have come from?
She then recalled her first meeting with Einstein, in 1909. She had heard him lecture on his special theory of relativity, and had watched intently as he had derived his famous equation, E = mc2. The very idea that mass could be converted to energy had left a deep impression on her. She also remembered that the nuclear masses of the fragments created by splitting a uranium nucleus would not quite add up to the mass of the original nucleus. These masses differed by about one-fifth of the mass of a single proton, mass that appeared to have gone ‘missing’ in the nuclear reaction. The sums checked out and it all seemed to fit together. A neutron causes the uranium nucleus to split almost in two, converting a tiny amount of mass into energy along the way. Frisch called it nuclear fission.*
The practical consequences of E = mc2 would become all too painfully clear just a few years later. It turned out that it is the small quantity of the isotope U-235 present in naturally occurring uranium that is responsible for the observed fission, but the principles are the same. When scaled up to a 56-kilogramme bomb core of ninety per cent pure uranium-235, the amount of energy released by the disintegration of just a small amount of mass was sufficient to destroy utterly the Japanese city of Hiroshima, on 6 August 1945.†
The destructive potential of E = mc2 is clear. But, in fact, our very existence on Earth depends on this equation. In the so-called proton–proton (or p–p) chain which operates at the centres of stars, including the Sun, four protons are fused together in a sequence which produces the nucleus of a helium atom, consisting of two protons and two neutrons. If we carefully add up all the masses of the nucleons involved we discover a small discrepancy, called the mass defect. About 0.7 per cent of the mass of the four protons is converted into about 26 MeV of radiation energy, which when it comes from the Sun we call sunlight.*
This all seems pretty convincing. We import our classical preconceptions concerning mass largely unchanged into the structure of special relativity. We view the theory as an extension of classical mechanics to treat situations in which objects are moving very fast relative to the ultimate speed, which also happens to be the speed of light.
We discover that under exceptional circumstances, the mass of one or more protons can be encouraged to convert into energy. But this is still Newtonian mass. If we hang on to the concept of Newtonian mass (despite the fact that Newton himself couldn’t really define it), then it seems we must reject the notion of relativistic mass. But if we reject relativistic mass, then Landau and Sampanthar’s derivation is no longer relevant and we’re stuck with the problem of how to interpret c. Einstein’s most famous equation E = mc2 seems so simple, yet we’ve managed to tie ourselves in knots trying to understand what it’s telling us. Why?
We have to face up to the fact that we’re dealing here with two distinctly different sets of concepts for which we happen to be using the same terminology, and a degree of confusion is inevitable. In a Newtonian mechanics with an absolute space and time, mass is a property of material substance—perhaps an absolute or primary property, as the Greek atomists believed and as Locke described—and as such must indeed be independent of any specific frame of reference. In this conceptual framework we hold on to the dream of philosophers in its original form: material substance can be reduced to some kind of ultimate stuff, and the particles or atoms of this stuff possess the primary property of mass.
But in special relativity, in which space and time are relative, mass doesn’t seem to behave like this. Mass is also relative. It does seem (or can at least be interpreted) to depend on the choice of reference frame and it appears to be intimately connected with the concept of energy, m = E/c2.
So let’s chart a different path, and see where it takes us. Einstein chose to title the 1905 addendum to his paper on special relativity thus: ‘Does the Inertia of a Body Depend on its Energy Content?’ This seems to me to be a very interesting choice of words.
As I explained in Chapter 5, Einstein offered a derivation which suggested that the energy E carried away by two bursts of light derives from the mass of the moving object, which decreases by an amount E/c2. Now we can try to interpret this as something that happens to classical Newtonian mass under certain circumstances, involving a conversion of mass to energy, or we can instead assume that this is telling us something completely new and fundamentally different about the nature of inertial mass itself.
In this alternative interpretation, mass is not an intrinsic primary property of material substance; it is, rather, a behaviour. It is something that objects do rather than something that they have. Material substance contains energy, and it is this energy content which (somehow) gives rise to a resistance to acceleration, which we call inertia. We happen to have a tendency derived from a long association with classical mechanics (and, for that matter, everyday experience) to say that objects exhibiting inertia possess inertial mass.
In this conceptual framework we revise the dream of philosophers: material substance can be reduced to some form of energy, and this form of energy exhibits behaviour that we interpret as a resistance to acceleration. We could choose to eliminate mass entirely from this logic and from our equations and just deal with energy (though I’m confident that won’t happen any time soon). The result would be ‘mass without mass’—the behaviour we interpret as mass without the need for the property of mass. For sure, we still have some explaining to do. We would need to try to explain how energy gives rise to inertia.
But after working your way through the opening chapters of this book I suspect you’re under no illusions. Trying to explain how energy creates inertia should be no more difficult in principle than doing the same for mass itself. After all, to say that mass is a measure of an object’s inertia and that measurements of an object’s inertia give us a handle on its mass is no explanation at all.
We still have conversion, but this is simply conversion of one form of energy into another. Of course, this just shifts the burden. Though it can take many forms, our intuition tells us that energy is a property: it is possessed by something. In this sense we tend to think about energy much like we think about temperature—not something that can exist independently of the things that possess it. The question now becomes: what is this something that possesses energy and exists in all material substance? Could this something still be called matter? We’ll have the answers by the end of this book.
This is sometimes how science works. Very occasionally, we experience a revolution in scientific knowledge and understanding which completely changes the way we attempt to interpret theoretical concepts and entities in relation to empirical facts. Inevitably, the revolution is born within the old conceptual framework. Equations constructed in the old framework spring a few new surprises (such as E = mc2), and the revolution begins. But through this process the old concepts tend to get dragged into the new framework, sometimes to serve the same purpose, sometimes to serve a different purpose.
The result is what Austrian philosopher Paul Feyerabend and American philosopher Thomas Kuhn called ‘incommensurability’. Concepts dragged from the old framework are no longer interpreted in the same way in the new framework—strictly speaking they are no longer the same concept, even though the name hasn’t changed. Some scientists hang on to the older, cherished interpretation. Others embrace the new interpretation. Arguments rage. The concepts are incommensurable.
And this, it seems, is what happened with mass. The old Newtonian concept of mass was dragged into the new framework of special relativity. Classical Newtonian mechanics is then seen as a limiting case of special relativity for situations involving speeds substantially less than that of c, the ultimate speed. But, for those willing to embrace the new interpretation, mass loses its primacy. It is just one (of many) forms of energy, a behaviour, not a property. Feyerabend wrote:6
That the relativistic concept and the classical concept of mass are very different indeed becomes clear if we consider that the former is a relation, involving relative velocities, between an object and a co-ordinate system, whereas the latter is a property of the object itself and independent of its behavior in co-ordinate systems.
This would seem to make mass rather mysterious, or at least much more nebulous. But the simple truth is that—as we’ve seen—we never really got to grips with the classical conception of mass in the first place. Hand on heart, we never really understood it. Now we discover that it may not actually exist.