In my last post I introduced you to dimensional analysis, an essential trick for theoretical physicists, and showed you how you could address and sometimes solve interesting and important problems with it while hardly doing any work. Today we’ll look at it differently, to see its historical role in Einstein’s relativity.
Energy and Mass Before Einstein
One learns in first-year physics that the motion-energy (“kinetic energy”, in physics-speak) of a moving object of mass m and speed v is defined to be
- Ekinetic = ½ m v2
The concept developed gradually over 250 years, and reached an recognizably modern form by 1850.
From this equation, knowing that m has “dimensions of mass” (see the previous post if that terminology doesn’t make sense to you) and v has dimensions of length divided by time, we can learn the dimensions of energy:
- dimensions of energy = dimensions of (mv2) = (mass multiplied by length2 divided by time2).
Once energy and its definitions for moving objects became established within physics, it was natural for physicists to ask what other equations can potentially be written for one kind of energy or another. Dimensional analysis is quite useful in that regard.
We can easily get a new possible equation by simply replacing the velocity v of the moving object with the speed of something else! For instance, sound was known centuries ago to have a fixed speed cs. An equation such as
- E = # m cs2
where # is some unknown number, is clearly just as dimensionally consistent as the one for kinetic energy.
But so what? Sure, the equation isn’t inconsistent. But there are far more potentially consistent equations then there are physically sensible ones. If you wanted to believe this equation had some meaning, you’d have to think of a context in which some kind of energy has something to do both with sound and with the mass of some kind of object.
In the same way, knowing that light has a constant speed c, you could wonder, even in the 18th century, if an equation such as
- E = # mc2
could be meaningful in physics. But again, why would you expect one? You’d have to think of some context in which some kind of energy has something to do both with the mass of some object and with light. Why would there be such a relationship?
In the 1860s, thanks to the work of Maxwell, it was learned that light is an electromagnetic wave — i.e. a wave in the electric and magnetic fields. That’s when relations between energy and mass and light stopped seeming entirely bizarre, at least for particles that have electric charge (i.e. they are affected by electric forces.) Great physicists of the age occasionally considered the question. [The ensuing history is controversial, but has been recently reconsidered in papers by my old friend Steve Boughn and his colleague Tony Rothman (see here and here), and in a number of public articles by Rothman, such as this one.]
By the 1880s, some general calculations suggested that an electrically charged sphere would get extra mass mextra because of its electric fields, which carry energy Efield. These led to the equation
- Efield = # mextra c2
(The subscript notation is my own, for clarity.) First, in 1881, J.J. Thompson found an equation which can be reinterpreted as this equation with # = 15/8. In 1889, Oliver Heaviside found it more directly, with #=3/4.
The situation became more urgent when the electron was discovered in the very late 1800s. Each electron was found to have the same electric charge and the same mass. It immediately occurred to various physicists that perhaps the reason for this uniformity is that the electron’s mass comes entirely from its charge, via the electric fields around it. In 1902, Max Abraham applied reasoning similar to Heaviside’s, and concluded that perhaps
- Efield = # melectron c2
with #=3/4. This is a very nice idea. But it is wrong, and not because the # isn’t the familiar one. We know today that the electron’s mass comes not from its electric field but from a Higgs field, probably the recently-discovered one. So this is almost the right equation, but it’s definitely based on the wrong reasons.
Two other great physicists of the day considered a more subtle question: rather than focusing on the masses of objects, they asked whether electromagnetic waves (let’s call them “em-waves”) could themselves act as though they have mass, via formulas such as
- Eem-waves = # mem-waves c2
Henri Poincare’ found some evidence in favor, with #=1, in 1900; and Friedrich Hasenöhrl, in a tour de force of challenging calculations, found stronger evidence, with #=3/4, in 1904 and 1905.
By the way, you should notice that, as last time, everyone’s # was close to 1. Nobody got 482 billion, or six trillionths. We’ll see this again in future posts.
Einstein and E=mc2
Clearly, when Einstein came onto the scene as a graduate student in the early 1900s, he and all the other graduate students around him were well aware of all this ongoing, published research. They knew that the dimensionally consistent equation
- E = # m c2
could potentially be meaningful in certain contexts, even though none of the ideas mentioned above had been experimentally tested.
But it is equally clear that no one yet had any idea how general this equation could be. Even if Abraham’s explanation of the electron’s mass had been right, it couldn’t have worked for everything else. [The remainder of a hydrogen atom, whose charge is the same as that of an electron, has a field-energy that is the same too, but its mass is thousands of times larger than an electron’s.] Meanwhile Poincare’s and Hasenöhrl’s work was overly focused on electromagnetic waves.
Einstein’s breakthrough, therefore, had nothing to do with inventing the idea that mass and energy could be related, and/or that the equation relating them might involve c2. Any physicist thinking about electromagnetism (a substantial fraction of the physics community) and familiar with dimensional analysis could have imagined there was some connection.
Nevertheless, Einstein came at this problem in a unique way. In June 1905, he proposed his update to Galilieo’s notion of relativity, in which c is no longer merely the speed of light waves, in the way that cs is merely the speed of sound waves. He proposed that c is related to properties of space and time themselves, and represents a cosmic speed limit — a completely universal limit on the relative speed of all objects, no matter what they are made from. And this, in turn, suggested that a formula such as E = # mc2 might not just be about light or about electrically charged objects. It could be far deeper than that: it could potentially apply to everything in space and time. It could be a universal equation.
This is why 26-year-old Einstein, surrounded by great senior physicists who knew that E = # mc2 might be true in some contexts, was the first and only one to propose this equation as a universal statement. His little paper on the subject, sent to a journal in September 1905, provides evidence and intuition (though, some would argue, not definitive proof from the principles of relativity) through a simple argument. He merely considers an object at rest that emits light, as viewed by two observers. From this he concluded that the mass m of any stationary object is related to some internal form of energy by the formula we all know:
- Eobject = mobject c2 (for stationary objects)
Again, in sharp contrast to all previous formulas of this type, there’s nothing electromagnetic about this one. The relation is general. It applies to all objects, no matter what their electric charge. And Einstein got the # right, too.
History is Quietly Made
Incidentally, Einstein didn’t actually write the formula E = mc2 until a few years later. In his famous 1905 article, he referred to energy as L, called the speed limit by the name V, and only stated his famous relation in words (translated here from German of course):
If an object gives off energy L in the form of radiation, its mass decreases by L/V2. The fact that the energy withdrawn from the object becomes energy of radiation obviously makes no difference, so we are led to the more general conclusion that
The mass of an object is a measure of its energy-content; if the energy changes by L (in ergs), the mass (in grams) changes in the same sense by L/(9 × 1020).
Through this terse, unadorned comment, the third-to-last sentence in a paper not even three pages long, Einstein taught us that the energy in a human finger could obliterate a city.