It's professor Dave, let's discuss
mass-energy equivalence.
Time dilation and length contraction are two
of the bizarre ramifications of
Einstein's special relativity, but we
have one more to get through, and it
comes in the form of his most famous
equation, which is arguably the most
famous equation in all of science.
E = mc^2. Almost everyone can
recite this equation from memory, but far
fewer know precisely what it means.
It outlines the concept of mass-energy
equivalence, stating that energy is equal
to mass times the speed of light squared.
One may wonder what the speed of light
has to do with energy and mass,
especially since light has no mass.
Nevertheless, we will find that the two
are inextricably intertwined. We know
that momentum is equal to mass times
velocity, so as velocity increases
momentum increases, but as velocity
approaches the speed of light
relativistic effects take hold, and we
have to look at the relativistic
momentum. That will involve this
additional term here, which should be
familiar from the time dilation and
length contraction equations. As velocity
increases, this term increases, which
makes this term decrease, and momentum
will increase overall. It is the square
of this relativistic momentum, known as a
Lorentz invariant, that is conserved in
all inertial reference frames. At speeds
much less than the speed of light, there
is only negligible difference between
the relativistic and non-relativistic
momenta, which is one of the reasons
that special relativity is imperceptible
in our everyday lives. But as an object
approaches the speed of light, it gains
momentum exponentially, and the momentum
approaches infinity as the velocity
approaches the speed of light. This gives
us possibly the best explanation for why
the speed of light truly is the
universal speed limit,
because it would require literally
infinite energy to accelerate anything
with mass to the speed of light. We can
find similar logic in the other
equations as well. With time dilation, we
can see that time slows down as you go
faster, with the limit of something
moving the speed of light experiencing
zero time. So if you could have the
perspective of a photon, time wouldn't
exist. To move faster than this would
mean experiencing imaginary time, because
you would then be taking the root of a
negative number, and this can have no
correlation with physical reality.
The same goes for length. In the limit of the
speed of light, distances contract to
zero, so a photon experiences zero space
as well. To go faster than this would
mean measuring imaginary length, which
again is absurd.
Beyond this, faster than light travel
implies problems with simultaneity
whereby an effect could happen before
its own cause, which violates causality,
so we must accept that for all these
reasons, the speed of light absolutely
can never be attained or surpassed by
any massive object, for reasons that are
fundamental not just to physics, but to
mathematics and logic as well.
E = mc^2 has many
implications given the incredible
conclusion that energy and mass are
equivalent. Mass is energy, and incredibly
dense energy at that, since to get the
energy contained in some amount of
matter, you multiply its mass by the
speed of light squared, which is a huge
number, so to the list of the types of
energy we can add one more: matter.
Mass-energy equivalence
elucidates the incredible source of
energy that has been exploited during
the atrocities of nuclear warfare in
World War II. The conversion of matter to
energy via a nuclear process is a mighty
force indeed, one that must be met with
great maturity. While the atomic nucleus
could hold the key to energy needs of
future civilization, its misuse
can also bring about our destruction.
And so as we look at these four quadrants
along axes that represent size and speed
we can see that the realm of the very
large and the very slow represents the
Newtonian mechanics that we took for
granted for hundreds of years as being
the rules of the universe. As we explored
the realm of the very small,
we found that quantum mechanics had to
be developed, from which Newtonian
mechanics emerges when objects become
large enough that their wavelengths
become negligible. In the realm of the
very fast, special relativity proved to
be an adequate description, while again
Newtonian mechanics emerge as velocity
slows down enough that relativistic
effects become negligible. But that
leaves this fourth quadrant, which
requires some marriage of quantum
mechanics and relativity. This is
perplexing territory, and it contains the
current frontier of physics. Later in
this series we will outline our current
progress towards a grand unified theory
that can explain all known phenomena, but
let's not get ahead of ourselves,
first let's check comprehension.
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