Of Particular Significance

# E = m c-Squared: The Simple Dimensions of a Discovery

#### POSTED BY Matt Strassler

ON 06/23/2022

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.

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.

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### 16 Responses

1. klaus says:

Sorry, but the last sentence “obliterate …” is not what Einstein ever taught “us”. The content is correct – but it is not what Einstein taught us.

2. Jason Stanidge says:

Thanks for the article, a quick question: what are the units of space-time?

I ask this because on the one hand SR combines space and time so that space-like and time-like vectors can be added together making c a dimensionless constant. Yet every vector can be uniquely defined as space-like, time-like or light-like which may influence whether space or time units are usefully retained to emphasize this.

1. Space remains a length, time remains a time, as long as c is a velocity. Then the coordinates of spacetime are not t and x, but either t and x/c (time), or ct and x (length). Since c is constant, which one we choose is up to us. If we choose units where c is 1, then people usually measure lengths using “light-seconds”, or just “seconds”; it’s the same thing..

3. Ronald Sechler says:

The universe is energy, and it exists as energy and mass. In the beginning the universe was energy, and it existed at an ultra-extreme level where mass popped into existence all through the universe at the same time. Energy acting on mass creates heat, pressure, movement, speed, gravity, time, information, and evolution. There is an upper and a lower limit to everything. The upper limit of energy is mass. The lower limit of energy is nothing. The upper limit of mass is fission. The lower limit of mass is energy. At the speed of light mass will convert into energy. At the speed of light energy will convert into mass. At the speed of light mass and energy are unstable. Not only E=MC², but M=E/ C², C²=E/M, and C=√(E/M). The speed of light is not simply some kind of constant, it is the relationship between energy and mass. E=MC² is, and has always been the theory of everything.

1. Energy is not a thing, and there’s no thing that *is* energy. Energy is something that things can possess, not something they can be.

1. /df says:

“Energy is something that things can possess, not something they can be.” This is a great distinction .

What might be the thing(s) that possess(es) “dark energy”?

1. It could be intrinsic to spacetime, an effect of a field and its potential energy, or an effect of the quantum fluctuations of fields, and quite possibly a combination of all of them.

But note: dark “energy” is just shorthand. It means “positive energy density and negative pressure.”

4. Andy says:

Ooops…I just saw your post from July 2013 on exactly this point

1. Andy says:

Ooops again. now that I have read the July 2013 post, I see that I was misled by the title. That post is about something else. Maybe I’ll just go back to my seat now

5. Andy says:

And when we use the word “mass”, lots of us immediately think we’re talking about how heavy something feels in our hands; that is, the object’s interaction with the earth’s gravitational field. Had we talked to Einstein in 1905, he would have said “Nein. That’s not what I’m talking about at all. I’m talking about how much the object resists being accelerated. Its inertia, ja?” and we would have had to wait a few years for him to call up to say “ooops….never mind. They’re the same thing”

(I am sure most of your readers already know all about this so I am commenting just to have a little fun. I remember clearly being asked in high school to think about whether gravitational mass and inertial mass are connected and not really understanding the distinction the teacher was trying to make. Still a bit embarrassed, all these years later)

1. Einstein took a little while to figure it out. The point is that rest mass (aka invariant mass) is not the same thing as inertial mass, which in Newtonian and Einsteinian gravity is the same as gravitational mass. In retrospect, what Einstein showed in 1905 is that rest mass is a measure of the energy carried by a stationary object.

1. veeramohan says:

Localization refers to the fact that the wave functions of neighboring quantum states are not extended enough (or not extended at all) to give significant overlap (or no overlap at all).
/What Einstein showed in 1905 is that rest mass is a measure of the energy carried by a stationary object./

“Stationary object” is a Localization?

6. Chip Brock says:

Hi. Something is wrong with the Rothman link? I didn’t know about the moving black body problem. Thanks!

1. I fixed the link… You should thank Steve Boughn and Tony Rothman for doing the hard work on Hasenhorl’s efforts.

7. Joseph says:

Great write up. When you look at a electron by itself, and a proton by itself, with their electric fields around them, they have a certain mass. When you bring the electron together with the proton (forming a hydrogen atom) the “whole is less than the some of its parts”. In a sense the positive & negative fields cancel each other out, I believe energy is released in the form of a photon, accounting for the missing mass.
If I do “work” pulling apart the electron and proton, the missing mass of each part returns, as they now have their electric fields back. So, when you pull them apart, is the work you are doing actually going into the mass associated with their respective electric fields?

1. The answer to your question is “sort of, but I wouldn’t say it that way.” There is only one electric field; there isn’t one for the electron and one for the proton… so it’s not really accurate to say things like “the positive and negative fields cancel each other out” or “mass associated with their respective electric fields.” It’s not entirely wrong (due to the superposition principle which works for electromagnetism at long distances) but it builds the wrong intuition, which causes problems in other contexts. It is better to say that when the electron and proton bind, it’s not so much that that their fields cancel but that their equal but opposite *charges*, which create their fields in the first place, mostly cancel the long-distance electric field. This is what reduces the total energy of the electron-proton atom.

As you say, conservation of energy requires that the extra energy now go somewhere, and it will inevitably be carried off, typically by one or more photons (though in principle neutrinos or even gravitons would do too, they would just be extremely rare.) You can then reverse the process: by adding energy in some way, most easily by pounding the system with photons, you can cause the electron and proton to separate, and indeed, by energy conservation, some of the energy you added will then be carried by the separated electron and proton. You can view this energy as having been added to the electric field if you like, though that’s not a unique viewpoint. But you need to use “field”, singular, in that statement.

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