PFE012: $E=mc^2$

Einstein's most well known formula

is most certainly $E=mc^2$. It gets tossed around as a symbol of intelligence, but what is it really about?

First, briefly, in case you don't recall from you grad-school science, there are two principles crucial to experiments and are fundamental rules to science. The first is that mass is conserved. That is, that you can't create more matter than you have and you can't destroy the stuff. If you're doing an experiment and your mass seems to change from the beginning to the end, that means you either lost some stuff or something else was added.

The second fundamental rule is that energy is conserved. The total amount of energy in the universe is a constant and all we can do is change it's form. For example, coal in the ground has chemical potential energy that we can use to turn into electricity and then into light.

These two rules will get us a long ways. In the early 20th century, however, a number of physicists were trying to find ways to relate mass and energy directly. After a few wrong turns, the famous $E=mc^2$ equation was settled upon. What this says is that energy can be converted into mass and mass into energy.

Unfortunately, this looks like another one of those times when our grade school teachers lied to us because "they didn't think we could handle the truth" or something. So energy isn't really conserved, and neither is mass, but they can be converted back and forth from one another. So then why did it take until 1905 or so to figure all of this out? Why doesn't sunlight hitting the ground just turn into a house?

It turns out that the scale of the conversion is very uneven, at least in terms of things that we're used to. For example, if we could convert a penny entirely into energy, it would cost merely $6.40 in pennies to power New York city... for a year (extra credit (pdf)).

So on the one hand if we have matter and we want energy we're totally in luck. Then again, if we suddenly need to create matter: good luck.

But of course, I've got probably nearly 640 pennies in my car, and yet we're still struggling to meet our power needs. The problem is that this conversion is quite difficult to do. It typically requires a monstrous amount of energy to happen. The initial energy isn't gone, but if it isn't there, it won't happen. The best example of something that converts mass into giant amounts of energy is our friend the sun. The sun works so well because it is so hot, and it is so hot because it works so well. Researchers have been trying to make a sustainable version of the sun for many years now unsuccessfully. News articles puts the technology as about "20 years away". As a professor of mine joked, "it has been 20 years away for about 30 years now - and is still 20 years away". Maybe this time they'll be right, maybe not.

There is another way to turn mass into energy, and that is by using different, generally rarer materials such as uranium or plutonium. They are much more willing to give up some of their mass for energy. Such a mass conversion can yield huge amounts of energy, but is also very dangerous if proper precautions aren't taken.

What does all of this mean for all of those horrible lies our K-12 teachers filled our innocent, eager brains with? Since energy and mass are both conserved anytime there isn't anything going "nuclear", and whenever something does go nuclear, there is a strict conversion ration, we can say that "mass-energy" is conserved. That is, we can add up all of the mass and the energy at any point (by converting using the handy $E=mc^2$) and it will never change.

So I guess we can all forgive our teachers on this one because, in most cases (excepted by stars, nuclear bombs, and nuclear power plants) energy and mass are each separately conserved.

That's $E=mc^2$.