Celebrating Einstein
E=mc^{2}
 What's the Speed of Light Got to Do With It?
"Energy equals mass times the velocity of light
squared." So what exactly does the velocity of light have to do with it?
By far, Einstein's bestknown 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 EnergyContent?" (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 JeanPhilippe Uzan, in American Journal of Physics, March 2005
(volume 73, Issue 3), pp. 240247.
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
Last Modified: 04/22/2009
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