So, well, the simple version is just to tell you what the terms in the equation mean
which is that E means energy, and that's equal to m, mass, times c,
the speed of light, squared.
Energy can be measured in joules, mass of a particle could be measured in kilograms.
so if you've going to relate them, you have to relate them with some factor
and the factor that does it is this...
square of the speed of light.
Taken at face value, it means that energy is equivalent to mass
so that if you were to look at Newton's laws of motion and say that masses attract
this now implies that energy is attracted by another mass.
So, if a light wave with energy goes near a planet or the sun, it's actually bent by it
because it's attracted. So it's saying that energy is equivalent to mass.
The consequence of it is quite dramatic, because the speed of light squared is a huge number.
The speed of light in meters per second is...
... Uh... three times ten to the eight (power). Three hundred million meters per second.
You square that, and you've got...
... Uh...
... Oh, rubbish...
... nine times ten to the sixteenth. So, you've got an order of ten to the seventeeth as a conversion factor.
So, a very small amount of mass
is equivalent to a hugh amount of energy.
In physical terms, what it means is that there's actually an energy associated with mass.
It's something that just sort of falls out, sort of almost automatically from special relativity.
That is just something that comes out of the equations.
But, obviously, it's an incredibly powerful result, because it basically means that whenever you've got a mass
you have a source of energy.
And, so, you know, if you...
have a... fusion bomb, for example, you actually convert some of that mass directly into energy.
But it basically is another way of saying that mass is just a way of storing energy.
But it's not even the right equation!
The proper equation is E squared equals m squared (times) c to the fourth
plus another term involving the momentum, which is an extra p squared (times) c squared.
The mass in E equals m (times) c squared can either be the rest mass,
and that's what is usually thought of, that's where the object isn't moving,
and so it's actually just got an intrinsic mass, and so it's got an intrinsic energy associated with it,
or, it could be what's known as the relativistic mass.
And if it's the relativistic mass, then it's including the momentum
and I think I'm probably agreeing with Roger.
In the textbooks you often see the bare masses; m with a little subscript zero beside it
just to make it clear that that's not including the momentum of the particle.
The whole equation is never explained because people like you don't want us to write down equations
'cause they're boring *pretends to yawn*.
But if you do it properly, that's what it is. And if you don't do it properly
then you can't explain why energy is attracted to (mass), gravitating to objects,
nor can you explain how a photon, which doesn't have any mass, behaves the way it does.
I think an interesting aspect is, why is it the squared speed of light?
Why isn't it the square of the speed of sound, which is a much lower number?
For that you have to look at what Einstein did
when he was coming up with this theory of special relativity.
Without any extra work, it sorta falls out of the analysis of special relativity,
that's just a result that comes along.
And, it's very nice when you just get a result that just sort of falls out without any extra work
but actually, on its own, has huge, amazing implications for the laws of physics.
The people I have spoken to, and when I gave public talks, the people who attended,
certainly have an idea that you've got energy and mass.
I don't think they often realized quite how significant it is, and quite how broad that equation is
at explaining phenomena that we experience.
They just tend to think of it in the world of particle physics, or something rather extreme.
But, in fact, it accounts for the binding of atoms.
If you've got the constituents of atoms, and you just measure that mass
then you put them together, the mass drops.
And the mass drops because of the binding energy, and the thing that explains that binding energy
is E equals m (times) c squared.
The correct equation, even if the momentum is zero,
is E squared equals m squared (times) c to the fourth.
And if you take the square root of that, you can get negative energies coming out.
And Dirac noticed this, and interpreted the negative energy states as the antiparticles.
So you now get something for nothing. If you use the correct equation, you can take the square root,
come out with a negative energy solution, and then interpret it in quantum mechanics as the antiparticle.
And he predicted this well before anybody did the experiment that discovered the positron.
So, the proper equation is much richer.
The public can't cope with anything more complicated than E equals m (times) c squared
and even that's so full of meaning.