What is the significance of Newton's equation E= MC square?
Not to be picky, but your subject (Einstein) does not agree with
the subject of your question (Newton) - E = m c^2 (squared) is
indeed due to Einstein (early years of this century) and not
Newton (who lived over 300 years ago).
This equation is often taken to symbolize Einstein's relativity
in the popular press, but in fact it is not really a consequence
but more a separate fact of nature that fits very well into the
initial assumptions of relativity (that the speed of light is
a universal constant, independent of reference frame).
The E refers to energy. Throughout the 18th and 19th centuries discoveries
were made about energy - that various forms of energy were really
interchangeable, for example mechanical energy (produced by applying
a force to an object over a distance) could be converted into
thermal energy (associated with heating an object) or electrical
energy, and that there was also chemical energy in various reactions
that could be measured. The units of energy are mass * acceleration *
distance (using the force definition) or mass * velocity^2 (using
the thermal definition) - in metric that is kg * m^2/s^2.
The m refers to mass. The famous equation (E = m c^2) says that an
object with a mass m has an associated total energy E that is proportional
to that mass, and the ratio of proportionality (which has to be a velocity
squared because of units) is the maximum velocity possible - speed of light.
This relation between mass and energy was made most obvious
in the case of radioactivity, when one element changes into
another - the masses tend to be different, and the energy released
can be very large. The same applies to the nuclear fission and
fusion reactions, where the energy released can be determined directly
from the mass difference of the reactants. Now, this is actually
not really much different from the way in which the chemical energy
of molecules changes when molecules join and break up - there would
be an associated mass change as well. But the mass change for chemical
energy changes is so tiny that nobody had ever observed it, while
it is relatively easy to measure in the nuclear case.
So the consequence of E = mc^2 is that two quantities that were
previously thought to be quite different (energy and mass) were
shown to be very closely related and interchangeable, just as
in previous centuries the various forms of energy itself had
been shown to be really the same kind of quantity.
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Update: June 2012