Development of Einstein's Equation

Date: 06/02/99 at 02:02:18
From: Aaron Brown
Subject: Einstein's Equation Derived

We've been doing a lot with quantum theory and nuclear interaction in 
my physics class this marking period. One of the equations we covered 
was Einstein's e = mc^2, but for all of our searching we couldn't find 
a source that showed how it was derived. What equations and principles 
did Einstein specifically use to come up with such an equation?

Date: 06/02/99 at 10:04:42
From: Doctor Rick
Subject: Re: Einstein's Equation Derived

Hi, Aaron, thanks for the question!

I will include below something I wrote to another, younger student who 
wanted to know where E = mc^2 came from. I couldn't go into the 
equations, but I discuss the principles involved. If you have some 
background in relativity, you may want to know more details than I 
give here; if so, go ahead and ask questions.

========

It's hard to really explain this formula until you've had several more 
math and science classes. But I'll try to explain a little about what 
it means and how Einstein (with the help of others) came up with it.

In the equation

        2
  E = mc

E stands for energy and m is mass (in relativity theory, it's called 
the rest mass). The letter c is the speed of light (from the Latin 
word for speed, celeritas), which is 300,000,000 meters per second.

Einstein started out to find a theoretical explanation for some 
puzzling experiments. Prior to those experiments, people had figured 
that light had a constant speed relative to some medium in which it 
moved. But the earth must be moving relative to that medium. (Even if 
it's stationary at some time, then it must be moving 6 months later 
when the earth moves in the opposite direction relative to the sun.) 
Light should appear to move more slowly in the direction the earth is 
moving, the way the wind seems gentler if you are moving with it in a 
sailboat. But the experiments seemed to show that the speed of light 
was the same in all directions.

Einstein found that he couldn't just patch up Isaac Newton's theory of 
motion. He came up with a whole new theory that gave the same results 
as Newton's theory when things are moving slowly, but very different 
results at high speeds (approaching the speed of light).

Newton's laws included a law that momentum (which is mass times 
velocity, or speed) is conserved (always stays the same) unless an 
outside force changes it. You probably have heard of this as "An 
object at rest stays at rest..." Well, Einstein's laws mean that time 
and space are connected, and that means that momentum and energy are 
connected. So Einstein found that his laws made the entirely 
unexpected prediction that the total energy of a system (for instance, 
a group of atoms) is conserved just like momentum. Newton's laws had 
said that energy is conserved under certain conditions (such as an 
"elastic collision"), but Einstein's laws said that energy is 
conserved always - even if, for instance, atoms combine or split.

There is a catch, though - for this energy law to work, it must 
include the rest masses of the atoms. This is where the famous 
equation comes in: even if an atom is standing still, it has energy 
equal to its mass times the speed of light squared. No one had thought 
of mass as a form of energy before. The equation meant that it was 
possible for mass to be converted into energy. 

Quickly people noticed that this was just what happened in nuclear 
reactions. If you take the mass of a uranium atom, and compare it with 
the masses of the atoms and other particles that come out of a nuclear 
fission reaction (when the uranium nucleus breaks apart), you find 
that a little bit of mass is missing. Einstein's law explains that the 
mass has become energy - the kinetic energy (energy of movement) of 
the new particles speeding away from the explosion. And because the 
speed of light is so high, a little mass becomes a LOT of energy. This 
energy becomes heat that can be used in nuclear reactors to make 
electricity.

I hope this at least gives you an appreciation of the kind of work 
that went into developing this equation. It looks very simple in the 
end, but the math involved includes things like a different form of 
geometry (non-euclidean) in four dimensions. Other people before 
Einstein had developed the math without knowing that it described the 
real world; Einstein saw that this math could answer the questions he 
was asking, and he ended up changing the world.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/