## Celebrating Einstein

### E=mc^{2} - What's the Speed of Light Got to Do
With It?

By far, Einstein's best-known equation is "E=mc^{2} - energy equals
mass times the velocity of light squared." According to this
equation, any given amount of mass is equivalent to a certain amount of
energy, and vice versa.

We all have some idea of what mass and energy are, and can appreciate that either can be converted into the other. Einstein's equation even tells us how much of one potentially converts into how much of the other. But what exactly does the speed of light have to do with either matter or energy? How does the speed of light, of all things, come into the picture at all?

The answer turns out to be one of the easiest to follow of all Einstein's derivations. When Einstein derived the relation of mass to energy, he had already demonstrated how time is a direction much like the directions of space, and how the distance and time intervals between events depend on one's frame of reference, which changes as one changes velocity. He also found that other things depend on one's frame of reference in a similar manner, such as the strengths of electric and magnetic force fields. The way electric and magnetic fields depend on frames of reference gave Einstein a road to what he called "a very interesting conclusion."

We can follow Einstein's road by considering a simple physical situation. Suppose we have an object of some kind that can emit energy. The kind of energy it emits doesn't matter. The object could be a cup of hot chocolate that heats up the surrounding air. Or it could be something that can radiate energy in the form of sound, like a telephone, an alarm clock, or a radio. To keep the situation simple, as Einstein did, we will stipulate that in our own frame of reference the object is stationary. For the object to remain stationary throughout the process, it will need to emit the energy evenly in different directions; otherwise, the object would move as it recoils away from the direction most of the energy would be going.

To keep things even simpler we'll further specify, again like Einstein,
that the object always emits the *same* amount of energy at the same
time in *exactly opposite* directions. Furthermore, we will
assume that the object emits the energy in a finite amount of time.
Then we can consider what happens before and after the energy is emitted,
without being concerned with what happens during the process itself.

Since natural processes only change the form of energy without changing the total amount, the energy the object has before it emits any is equal to the energy it has afterward, plus the amount it emits.

object's energy before = object's energy after + emitted energy

That's the situation as we see it from our own reference frame. The situation in any other reference frame differs only in the amounts of energy involved: the energy of an object is greater when it is moving that when it is stationary, and anyone moving past us will see the object as moving past them. But even in other frames of reference, the law of energy conservation still holds - total energy before equals total energy after.

We can summarize the energy situation in both reference frames with two simple equations, one for the moving observer (m.o.) and one for us, the stationary observers (s.o.):

object's energy before (m.o.) = object's energy after (m.o.) + emitted energy (m.o.)

object's energy before (s.o.) = object's energy after (s.o.) + emitted energy (s.o.)

As we just mentioned, the energy of an object that's moving in one
reference frame is greater than the energy of the same object in a different
reference frame in which it is stationary. The same energy difference
exists when the object goes from moving to being stationary in any *single*
frame of reference. This difference is called the object's kinetic
energy, or energy of motion. If we subtract everything in the equation
just above from the corresponding items in the equation just before it, we
will find what the moving observer sees as the kinetic energy of our object,
both before and after it emits energy:

object's energy before (m.o.) - object's energy before (s.o.)

= object's energy after (m.o.) - object's energy after (s.o.)

+ emitted energy (m.o.) - emitted energy (s.o.).

Put another way,

object's kinetic energy before (m.o.)

= object's kinetic energy after (m.o.)

+ emitted energy (m.o.) - emitted energy (s.o.).

The only energy difference we haven't figured out at this point is the difference between the energy emitted as seen in the other observer's reference frame and as seen in ours.

It is at this point that Einstein's earlier discovery about how electric
and magnetic fields are different in different reference frames gave him a
road to his interesting conclusion. Einstein realized that the form in
which the object emits energy is not important. It could be sound, it
could be heat, it could be something else, but whatever form it is, the
change that the moving observer sees in the object's kinetic energy will
equal the difference between the energy that he sees the object emitting and
the energy *we* see it emitting.

What Einstein did was to consider emission of *electromagnetic*
energy, which he had already figured out how to calculate for two different
frames of reference. In particular, he considered energy in the form
of light, which is a type of electromagnetic force field. So instead
of a cup of hot liquid heating its surroundings, or a bell making a sound,
we can imagine a light bulb shining equally in all directions in our
reference frame. In this case, if the emitted light has (to us) an
energy L:

emitted energy (s.o.) = L

the emitted light has, to the moving observer, a higher energy:

emitted energy (m.o.) = ,

which is times greater than L. The "v" stands for the velocity of our moving observer (or the velocity that he sees the object moving), and the "c" stands for the speed at which light travels in a vacuum.

When we use these expressions for the emitted energy in the equation preceding them, we find the change that our moving observer sees in the object's kinetic energy:

object's kinetic energy before (m.o.)

=
object's kinetic energy after (m.o.)

+
- L.

Two more facts and we are there.

First, as long as the velocity of our moving observer is not very large compared to the speed of light in a vacuum (our usual experience), the difference between the emitted energy as seen by us and the moving observer is approximately

½
(L/c^{2})
v^{2}.

Second, under those same conditions, the kinetic energy of a moving object is approximately

½
(mass of the object)
v^{2}.

Since the velocity of the object as seen by the moving
observer, "v", is the same after it emits the energy as it was
before, the only way its kinetic energy can change is if its mass
changes. Evidently, the mass changes by L/c^{2} - by the energy the
object emits (in our frame of reference), divided by the speed of light in a
vacuum squared. Since, as Einstein pointed out, the fact that the
energy taken from the object turns into light doesn't seem to make any
difference, he concluded that whenever an object emits an amount of energy L
of * any* type, its mass diminishes by L/c^{2}, so that the mass of an object is a
measure of how much energy it contains.

If we go back to Einstein's first paper on relativity, we find that the speed "c" is involved, not because we considered light instead of some other energy form, but because "c" is the speed at which time becomes, in a sense, equivalent to space, as the preceding article in this series illustrates. The fact that "c" is also the speed of light in a vacuum is coincidental. We would have found the same relation between mass and energy even if we had considered energy emitted in a form other than light, although it might have made the math more difficult.

Interestingly enough, Einstein first expressed his
conclusion in about the same way we did above, without actually using the
equation "E=mc^{2}". He only expressed the result that way
later on.

*Next article:* **"Seeing
the Wind"**

**References, Links, and Comments:**

"Does
the Inertia of a Body Depend upon Its Energy-Content?" (at
www.fourmilab.ch) [exit federal site]

An English translation of the original paper in HTML. Links at the bottom fo the page to PDF version and to zipped PostScript and LaTeX versions.

The fact that is a close approximation of L – L if is much smaller than was crucial to Einstein’s argument; in his paper, he alludes to the fact that, or equivalently , is a binomial expression (a sum or difference of two terms, raised to a power). Such approximations of binomial expressions are described in this Wikipedia article, using a generic binomial expression (1+x)^{α}. In Einstein’s expression ), the generic x is represented by () while the generic α is represented by ().

"c is the speed of light, isn't it?" by George F.R.
Ellis and Jean-Philippe Uzan, in *American Journal of Physics*, March
2005 (volume 73, Issue 3), pp. 240-247.

In Einstein's paper about the relation of mass to energy, the speed of light
appears in two different roles. Ellis and Uzan point out that the
speed of light plays multiple roles in physical equations generally, and
analyze the significance of this in relation to recent proposals that the
speed of light may have changed over time.

Prepared by Dr. William Watson, Physicist

DOE Office of Scientific and Technical Information