The Momentum of Light

In the essay on relativistic energy I showed you how to deduce the fact that energy in whatever form ponders mass. A body moving at about 87% of the speed of light, just because it moves and carries kinetic energy, ponders twice as much mass as it does when it does not move at all. Now I want to show you what that fact tells us about the nature of light.

Suppose that we have two rays of light of equal intensity (energy flowing at the rate dE/dt in each ray) moving in opposite directions along the x-axis in our arbitrarily established coordinate grid and that a black body absorbs both of them completely. We know that light carries energy, that it does work on things that absorb it: in this example it makes the black body grow warmer. It also, as we have seen, makes the body grow heavier. In fact, if the energy, once absorbed, remains trapped in the body, then we can calculate the rate at which the body's mass increases; if energy flows into the body at the rate 2dE/dt, then we have

(Eq'n 1)

If our black body does not move, then we have nothing more to say.

Assume, then, that the body moves in the positive x-direction with a speed v. In this case the increase in mass represents an increase in the body's linear momentum at the rate

(Eq'n 2)

But for the body to gain or lose linear momentum something must exert a force upon it. That something can only be the light that the body absorbs; light must carry linear momentum as well as energy. Now we want to figure out how much momentum.

We know that

(Eq'n 3)

so we also know that

(Eq'n 4)

Rather, we believe that we know it. Certainly Equation 4 must have some truth in it, but, as usual, we seem to have missed something.

We have tacitly assumed that in shifting our point of view into a moving frame we have imposed identical changes upon the rays of light, that we have increased the intensity of both of them by the Lorentz factor between the moving frame and our original non-moving frame. But in the non-moving frame the rays, of equal intensity, exert no net force upon the body, while in the moving frame they exert the force represented in Equation 2 upon the body. Clearly in the moving frame the rays must differ from each other in a way that subtracting the momentum carried by one ray from the momentum carried by the other leaves a non-zero amount that the body manifests. Thus we know that we must modify Equation 4 by adding an extra term to it for the ray coming from the negative x-direction and by subtracting the same term from it for the ray coming from the positive x-direction. When we add the energy contents of the rays together and divide by cee-squared to calculate the mass added to the body those extra terms will cancel each other out and we get Equation 3.

Further, when we subtract the energy content of one ray from the energy content of the other, we must get a difference proportional to the momentum gained by the body, as expressed in Equation 2. Thus we know that, at minimum, the new term must be proportional to the velocity v.

We also know that the new term must be proportional to the ray's proper energy content. If Existence did not make that statement true to Reality, then we could violate the law pertaining to conservation of energy by adding energy to a moving body in a certain number of steps and then withdraw it in a different number of steps with light.

We also know that the new term must be proportional to the Lorentz factor between the non-moving frame and the moving frame. If Existence did not make that statement true to Reality, then the mass that the body gains as it absorbs the ray would not conform to Equation 3.

And finally we know that the new term must have the units of energy, so we know that we must divide the velocity factor by some other velocity. An examination of Equation 4 shows us that the speed of light appears as a divisor, so we will use that.

Now we can apply the new terms to Equation 4 to obtain the correct descriptions of the energy contents of our two rays. For the ray coming from the negative x-direction we have

(Eq'n 5)

And for the ray coming from the positive x-direction we have

(Eq'n 6)

Comparing those equations with Equation 2 allows us to infer, at last, that if a ray of light carries an amount of energy E, then it also carries an amount of linear momentum

(Eq'n 7)

Historically Albert Einstein worked that derivation the other way around. He began with the fact that light exerts a force and deduced the mass-energy equivalence law, using a trigonometric flourish that I didn't need here (he had his rays of light coming into the body from a direction that made an arbitrary angle with the direction of relative motion).

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. It was Maxwell's electromagnetic theory that provided the explanation by pointing out that a ray of light consists of crossed electric and magnetic fields. Consider, for example, 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 we can represent reflection 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) equals the power in the beam (in watts) divided by the speed of light (in meters per second), in accordance with Equation 7 (or, more properly, its time derivative).

That last fact inspired Einstein to devise an imaginary experiment that he published in 1908. And this is where Einstein got truly clever.

We have seen that the effect of a force equals the rate at which that force makes the linear momentum of a body change. There are two ways in which that rate can be manifested. We describe the first and more familiar way by the product of the body's mass and the rate at which the body's velocity changes. But the velocity of our light-absorbing body does not change: the body does not accelerate in its own frame, so it can't accelerate 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 equals 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 rays upon the body as equal to the product of the body's velocity and the rate at which the body's mass changes: in essence we treat the rays as if they were oppositely facing rocket engines with one exerting more thrust than the other in the moving frame. Further, because the net force that the rays exert upon the body is oriented in the direction of the body's velocity, the rate at which the body's mass changes is represented by a positive number, which means that the body's mass is increasing.

What does that result 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. Indeed, any form of energy confers mass upon bodies that contain it. 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 field-free space? 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. Is that proposition true to Reality and, if so, is there a way in which we might extract energy from matter by disassembling its parts? How could we possibly answer that question? In another essay I offer a brief description of how chemists and physicists did, indeed, answer it.

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