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E = mc².        Or is it      E = m ?

A surprising clarification

October 2005

Einstein's famous equation was dressed up for its debut.  It works just as well in its simpler form.
Toward the end of 2005 we're celebrating the 100th anniversary of Albert Einstein's publication of his famous equation, E = mc², which expresses the equivalence of matter (mass) and energy.  In the equation, "E" stands for energy, "m" stands for mass, and "c" is the speed of light, about 300,000 km (or 186,000 miles)  per second.  In other words, the energy contained in a bit of matter is found by multiplying its mass by the square of the speed of light, which yields the substantial-looking number 9x1016 (that's 90,000,000,000,000,000)  m²/sec².  An impressive result!

It's often heard that the enormity of the speed of light gives this equation its power, and illustrates the vast amount of energy that is contained in a small amount of matter.  But the speed of light, though absolute, is a relative matter.  That is, it's relative to the scale we choose.  We represent it as a large number (let go for now the point that there are no "large numbers")  because we choose to measure its speed in meters or kilometers per second.  If we chose instead to express the speed of light in terms of, say, light-seconds (an absolute measure of distance)  per second, the speed of light (c)  – and its square c2 –  would appear in the equation as exactly 1.0, and the equation would read simply as E = m.  But Einstein wisely chose not to do that.  The well-known form has a lot more mystery and pizzazz.

In fact, the term is of course a constant, and thus is merely a scaling factor.  We could as well use any other constant, say the top speed of a '62 VW beetle, and get the same result, in appropriate units.  Or we could adjust the units of either the energy term or the mass term in the equation, or both, so that the term falls out altogether, and we would get the same result (expressed as E = m)  without it.  The term turns out to be unneeded – a decoration for effect.  (The equation, incidentally, gives the result that 1 kg of matter will completely convert to yield 89,875,517,873,681,764 joules of energy.  By defining 1 "baruba" (b)  to equal 89,875,517,873,681,764 joules, we have the handy solution that 1 kg of matter yields exactly 1 b of energy.)

But truly Einstein had more in mind than mystery and pizzazz when he introduced the term .  He had an insight and a solution to his (and Maxwell's)  nagging equations that tested out, and that has been subsequently verified.  I think we can assume that Einstein was fully aware of the tautological, self-recursive nature of his solution.  Using the Lorenz transform, he defined time in terms of light speed and light speed in terms of that same time.  No wonder his solution (and all of special relativity)  worked out.  One day we (perhaps even relativity physicists)  will have a good laugh at Einstein's little joke.

But Einstein was right:  E = m.  No doubt about it.

© 2005 H. Paul Lillebo

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