**E = Mc ^{2}**

In the same issue of Annalen der Physik (Annals of Physics) in which he published his famous article "On the Electrodynamics of Moving Bodies", Einstein also published a short little paper titled "Does the Inertia of a Body Depend upon its Energy-Content?" In that little paper he deduced the most famous equation in all of science, the one that I have used as the title of this essay. Before I show you how Einstein made that deduction I will show you an alternate way to do it.

Imagine that we have a freight train moving on a straight track in a world where light moves only one hundred miles in an hour. In that freight train we have a flatcar on which we have mounted two massive wheels whose common axis points in the direction of the train's motion. The wheels will spin in opposite directions and we have equipped them with motors and a battery or some other suitable energy source. Once the train leaves the station and gets up to speed a brakeman uses the motors to spin up the wheels, spinning them in opposite directions at the same speed so that the two wheels together gain no net angular momentum (that is, each wheel gains angular momentum that equals and opposes that gained by the other wheel). Due to that extra motion the wheels become more massive than they were when they were not spinning.

We can calculate the amount of energy that the motors must draw from the battery to give the wheels enough spin to increase their masses by a given amount. To carry out the calculation we must resort to the calculus: even so we must make simplifying assumptions lest the calculation get out of hand. One assumption, for example, tells us that we have made the wheels' spokes from some magical material that has zero mass, so we can represent the wheels as if they have all of their masses in their rims. Next we take two steps to set up the process of integration that comprises the actual calculation of the energy that we put into the wheels.

First we describe the increment of work that the motors do in changing the wheels' speeds by some minuscule amount. We represent that increment of work as the product of the force acting on a piece of the rim and the minuscule distance that the piece moves in a suitably brief increment of time. By Newton's second law of motion, the force equals the mass of the piece multiplied by its acceleration. So we have multiplied together a mass, an acceleration, and a minuscule increment of distance.

Now we apply a mathematicians trick. We know that acceleration equals a minuscule increment of velocity divided by the minuscule increment of time in which the velocity changed by that increment. And we know that a minuscule increment of distance equals a velocity multiplied by a minuscule increment of time. So in the multiplication above we allow the minuscule increments of time to cancel each other out and we have the minuscule increment of work done on the wheels as the product of the wheels' masses, the speed at which they turn at some given instant, and the minuscule increment by which that speed changes. That change may look a little fishy, but mathematicians have proven and verified the legitimacy of it and physicists have found that it does, indeed, give a correct description of Reality.

If we were to carry out the integration now, we would find that the work done on the wheels equals the wheels' masses multiplied by one half the square of their speed of rotation, the classical expression of kinetic energy. But because we invoke Relativity in this case we must take one more step before we carry out the integration.

And in that second step we must accommodate the fact that our formula for work done includes the Lorentz factor, the factor by which the wheels' masses increase as they move ever faster. We represent the Lorentz factor by a mathematical function that includes the square of the velocity between inertial frames (in this case the speed at which the wheels' rims turn) divided by the square of the speed of light. In the integration before us that fact necessitates that we divide the product of velocity and the increment of velocity by the square of lightspeed in order to make that product compatible with the Lorentz factor. But if we divide our formula by the square of lightspeed, we must also multiply it by the square of lightspeed so that we don't change it. And that's how the "cee-squared" gets into Einstein's formula; for when we carry out the integration the square of the velocity between the inertial frames gets absorbed into the Lorentz factor, leaving "em-cee-squared" to represent the work done in spinning up the wheels.

In this calculation we see how mathematical logic can augment physical logic. When we carry out the calculation, we find that the work comes out equal to the increase of mass multiplied by the square of the speed of light. Thus we obtain Einstein's equation, but we don't seem to have done anything more significant than calculate the work that we put into the wheels.

Now we come to the funny little trick of physical logic that Einstein used. Consider how our experiment looks to an observer watching it from a platform beside the track as our train goes by. The increasing mass of the wheels combines with the forward velocity of the train to yield an increase in the train's linear momentum. On that basis the trackside observer infers that the spin-up of the wheels exerts a force upon the train, in accordance with the rocket formula.

But, the observers on the train object, they see no basis for such a force, since the wheels have no forward velocity in their frame. And, in any case, they don't measure any acceleration of the train. Surely an unbalanced force would make the train change its velocity, again in accordance with the rocket formula. Our observers seem to have come upon a solid contradiction.

No, our trackside observer says, we don't have a contradiction. True, the wheels generate a force by gaining mass, but that force must be negated by something losing mass at the same rate. That something must be closely associated with the spinning up of the wheels in order to match the wheels' increase of mass perfectly and it must correlate with a loss of something. We have only one candidate for that something - the energy draining out of the battery to drive the motors. Now we see the full depth of what Einstein deduced.

The mass-energy equivalence equation goes beyond stating that it takes energy to increase the mass of a body. It also tells us that energy, in whatever form it may take, possesses mass. Energy stored in a battery confers mass upon the battery. Energy in a forcefield, such as the magnetic field surrounding a refrigerator magnet, carries mass. But fundamentally energy expresses a relationship among bodies, so Einstein's equation claims that a relationship exists as firmly in space as do bodies.

We commonly think of the mass-energy relation as it applies to nuclear energy. We conceive it in the kinetic energy that the fragments of a fissioned Uranium nucleus carry as they fly apart and in the radiation that comes with them. And we conceive it in the fusion of hydrogen nuclei that heats the stars.

But apply Einstein's mass-energy theorem to forcefields and you will conceive new theorems that will take your breath away. In one fell swoop you will have cleared up two mysteries of Nature. You will have revealed the full nature of the magnetic field. And you will have answered the question Why do bodies resist acceleration?; that is, you will have explained inertia. I will show you how to work out those explanations in the essays on the relativity of force and in the section on General Relativity.

And how did Einstein himself deduce his mass-energy theorem? He looked at the relationship between light and matter and he focused his attention on one particular aspect of that relationship through a clever imaginary experiment.

In the last half of the Nineteenth Century physicists discovered, to their astonishment, that light exerts a force upon any body that emits it, absorbs it, or reflects it. James Clerk Maxwell's electromagnetic theory provided the explanation of that force by way of pointing out that a ray of light consists of crossed electric and magnetic fields, each field generating the other continuously as they propagate together through space. Consider what that means, for example, in the reflection of a ray of light from a flat metal surface. When the ray strikes the metal the ray's electric field makes the metal's conduction electrons move and those electrons, thus moving across the ray's magnetic field, are pushed in the direction in which the ray is traveling. At the same time the acceleration of the electrons produces an electromagnetic wave that cancels the ray and yields one traveling in the direction defined by the angle at which the ray strikes the metal's surface. A deeper mathematical analysis indicates that a ray that falls perpendicularly onto a flat surface exerts twice as much force when it is reflected as it does when it is completely absorbed (as it is by materials whose electrons are not completely free to move). Because reflection can be represented as an absorption and re-emission of a wave, we can infer that the emission of a ray of light exerts as much force upon the emitter as would absorption of an identical ray traveling in the opposite direction. If a body emits light in a narrow beam, then the force that the light exerts upon the body (in newtons) is equal to the power in the beam (in watts) divided by the speed of light (in meters per second). The light acts much like the exhaust plume of a rocket and in some older science fiction stories the explorers fly through outer space in "photon rockets".

That last fact inspired Einstein to devise the imaginary experiment that he published in 1905. I want to show you a slightly simpler version of the same experiment, one without the trigonometric flourish that Einstein gave his.

Imagine that the conductor of our train has mounted a pair of spotlights atop the cupola on his caboose. He has so mounted them that one points straight forward and the other points straight backward and he energizes them with electrical power that is divided equally between them. Thus the beams emitted from the spotlights are of equal intensity and the forces that they exert upon the train through the spotlights cancel each other out. The net force exerted upon the train equals zero, so the train suffers no acceleration. Now imagine that the train passes an observer standing by the track. That observer will see the train and its spotlights in a rather different light.

When the train approaches the observer, the light from the forward-facing spotlight appears blue to the observer. And when the train has passed that observer and recedes into the distance, the light from the backward-facing spotlight appears red to the observer. Those observations puzzle the observer, because someone told him that the spotlights emit yellow light. A second observer comes to the rescue and explains to the first observer the reasons behind his observation: she tells him that there are two reasons and taken together they are called the Doppler shift.

First, the electromagnetic vibrations in the light constitute the essential part of a clock, so their period (the time required for one complete cycle of vibration) is subject to time dilation: to calculate the period of the light waves received by the observers we must first multiply the period measured in the train's frame by the Lorentz factor between the train's frame and the trackside observers' frame and then multiply that by a factor derived from the second reason.

And that second reason is this: between the instant that the first crest of a given wave cycle is emitted and the instant that the cycle's second crest (which is also the next cycle's first crest) is emitted, the train moves slightly closer to the observers or farther from them. That little span of distance, which equals the train's velocity multiplied by the wave's time dilated period, represents an increment of time that the second crest saves or gains relative the first crest in going to the trackside observers, so divide that little distance by the speed of light and either subtract the result from (for the forward-facing spotlight) or add the result to (for the backward-facing spotlight) the time dilated period of the wave to obtain the period of the wave as it is received by the observers.

Just as physicists
have defined the Lorentz factor as part of the description of the relationship
between two inertial frames, so too have they defined two Doppler factors as
part of that description. The negative Doppler factor is calculated by dividing
the velocity between the frames by the speed of light, subtracting the result
from the number one, and then multiplying the result by the Lorentz factor
between the two frames. We can thus say that the period of the wave received by
the observer (measured in the observer's frame) equals the period of the wave
emitted by the train (measured in the train's frame) multiplied by the negative
Doppler factor between the frames. The positive Doppler factor is also defined
and it is calculated by dividing the velocity between the frames by the speed of
light, __adding__ the result to the number one, and then multiplying the
result by the Lorentz factor. We would use the positive Doppler factor after the
train has passed the observer and part of the beam from its backward-facing
spotlight reaches the observer: because the train is at that time moving away
from the observer the period of the wave received by the observer equals the
period of the wave as it is emitted from the train multiplied by the positive
Doppler factor; that is, the period of the wave appears to the trackside
observers to be longer (i.e. redder) than the period of the light as the
spotlight emits it.

But it's not the period of the waves in which we are interested. It's the frequency that we want and it's easy to get: the frequency is merely the inverse of the period; that is, to obtain the frequency of a wave divide its period into the number one. Thus, when the train is coming toward the observers, the frequency of the wave as received by the observers equals the frequency emitted by the spotlight multiplied by the inverse of the negative Doppler factor. By an astonishing mathematical coincidence, the inverse of the negative Doppler factor is just the positive Doppler factor (and, of course, vice versa). So the frequency of the forward beam equals the frequency emitted by the spotlight multiplied by the positive Doppler factor and the frequency of the backward beam equals the frequency emitted by the spotlight multiplied by the negative Doppler factor; that is, the motion of the train relative to the observer makes the frequency of the forward beam higher (and the wavelength bluer) and the frequency of the backward beam lower (and the wavelength redder).

All other things being equal, the amount of power propagating in a light beam is proportional to the frequency of the light in the beam. That means that the power in the forward beam is greater than the power propagating in the backward beam in the trackside observers' frame, even though the spotlights are being fed equal amounts of power in the train's frame. The difference in the power in the two beams is equal to the total amount of power that is being fed to both spotlights in the train's frame, multiplied by the Lorentz factor, multiplied by the velocity of the train, and divided by the speed of light. That difference, divided by the speed of light, represents a force being exerted backward upon the train.

This is where Einstein got truly clever in applying his little trick of physical logic. The effect of a force equals the rate at which that force makes the linear momentum of a body change. We know two ways in which that rate can be manifested. The first and more familiar is described by the product of the body's mass and the rate at which the body's velocity is changing. But the velocity of our train is not changing: the train is not accelerating in its own frame, so it can't be accelerating in any other inertial frame. Einstein was thus left with the second way of describing the effect of a force, the one that describes the thrust of a rocket engine, the one that's equal to some velocity multiplied by the rate at which mass is being associated with or dissociated from that velocity. In our case we must describe the effect of the force exerted by the spotlights upon the train as being equal to the product of the train's velocity and the rate at which the train's mass is changing: in essence we are treating the spotlights as if they were oppositely facing rocket engines with one exerting more thrust than the other in the observers' frame. Further, because the force that the spotlights are exerting upon the train is oriented in the direction opposite the train's velocity, the rate at which the train's mass is changing is represented by a negative number, which means that the train's mass is decreasing.

Now we can combine all of that into one statement: the product of the total power being fed to the spotlights (in the train's frame), the Lorentz factor, the train's velocity, and the inverse of the square of lightspeed is equal to the product of the rate at which the train's rest mass is decreasing, the Lorentz factor, and the train's velocity. The Lorentz factor and the train's velocity appear in both products in the same way, so we can cancel them out without affecting the truth of the statement. We have then the statement that the rate at which power is fed to the spotlights (in the train's frame) divided by the square of the speed of light is equal to the rate at which the train's mass is decreasing. We have to be careful with rates (because they involve the calculus, which is a step up in complexity from algebra), but if we assume that the power is being fed to the spotlights at an unvarying rate, we can represent the rates as simple divisions: the rate at which power is being fed to the spotlights is equal to the energy emitted in a given time divided by that time interval and the rate at which the train's mass is decreasing is equal to the amount by which the mass decreases in a given time divided by that time interval. We can then cancel the divisions by time and obtain the statement that the energy emitted from the train in a given time divided by the square of lightspeed is equal to the amount by which the train's mass decreases in that time. That statement is one form of the most famous equation in science, the one that I used as this chapter's title.

What does it mean? What did Einstein have in mind when he said, in his thick German accent, "Ee is eqval to em cee sqvare"? A contemplation of the particulars of the imaginary experiment used to deduce the equation makes clear that the first thing we can say is that energy confers inertia upon bodies that possess it. In the freight train of our imagination the energy stored in batteries or in the coal that fuels the train's firebox prior to being fed to the spotlights gives the train more mass than it would have without that energy on board. Indeed, any form of energy confers mass. If you had a box whose inside walls were perfectly reflective and if you filled that box with light, the energy content of the light would add mass to the box. And if the box's walls were to be warmed by the light from the sun, the heat would also add mass to the box. The mass thus added is not great: one kilowatt-hour confers a mass of about four hundred-millionths of a gram, about the mass of a droplet of mist with a diameter of one tenth of a millimeter.

Now turn that proposition around and say that mass is equivalent to energy, at the rate of 25 million kilowatt-hours per gram (a volume of water equal to that of a sugar cube ponders about one gram). What kind of energy is it? Certainly kinetic energy is part of it and potential energy as well: the motions of matter and matter's relationship with forcefields add to the matter's inertia. But what of matter at rest in fieldfree space?

Recall our
calculation of the work done spinning up a pair of counter-rotating wheels. At
the end of the calculation we had to subtract from the total energy of our
integration the product of the wheels' rest masses and the square of lightspeed;
that is, the rest mass of a body is also assigned an em-cee-squared energy. That
mass-times-velocity-squared form of the calculation implies a kinetic energy,
but the body is not moving, so it can't be a kinetic energy. Is it, then, a kind
of potential energy? If it is, then matter must be structured as an assembly of
parts held together by forcefields. Does Existence make that proposition true to
Reality and, if so, do we have a way in which we might extract energy from
matter by disassembling its parts? How could we possibly answer that question?
In the next essay, Relativistic Alchemy, I offer a brief description of how
chemists and physicists did, indeed, answer it.

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