A Staff Report from the Straight Dope Science Advisory Board

In E=mc2, what units of measurement was Einstein using?

May 10, 2000

Dear Straight Dope:

Recently I have been studying Einstein's theories of relativity. I am not a physicist or mathematician, but they are obviously among the greatest concepts in human history, and I think it behooves any thinking person to try and understand them at least once.

So, I'm frowning along, my head hurting a bit, and I come inevitably to the famous E=mc2, the only math equation to be a worldwide household word, and I find that just the sight of this equation irritates hell out of me. Why? Because for goodness' sake, what units of measure are being used here? What exactly does he mean by "energy": Ergs? Volts? Footpounds? Watts? How about "mass": Pounds? Grams? Tons? Ounces? And what about "speed of light": Miles an hour? Metres per second? Inches per century? Just what units of measurement are being plugged into these variables?

The atomic bomb has shown us that practical application of the equation yields tangible, predictable, woeful results, so someone must know what he's talking about, but I haven't seen it explained yet. To me the equation might as well read Grapefruit=Acorn Squash x Nectarine2.

Obviously I'm missing something; but what the hell is Einstein talking about? Any energy you can shed on this for me would be greatly appreciated.

SDStaff Karen replies:

Oh, for crying out loud. Energy in watts?

Equations such as E=mc2 are independent of units of measurement. The units attach themselves to the values, which is why physicists and mathematicians prefer to think and write equations in terms of concepts instead of numbers. For example, let's say I have a piece of string. You and I can agree that the string has a length l, a fundamental attribute independent of its value. Given that the string has the attribute length, I can answer questions such as: what is half the length, or the area of a circle enclosed by the string, and express those answers in terms of the original concept, the length l.

If I do wish to give a value to the length of the string, I could say it's 12 inches, or 0.3048 meters or 1.52 millifurlongs. Those are all different expressions of the same length. The choice of units+value is based on convenience, personal preference, or local law. [Aside: computers, like freshman physics students, are notorious for only expressing numbers and not attaching the units. This is why NASA can get screwed when one part of the computer program says the length is "12" and another part is expecting "0.3048". While freshman physics students lose a couple points on their exams, NASA loses an entire mission, so I say the freshman should stop whining. You've got to attach units to make the numbers meaningful.]

Energy has dimensions of mass*length2/time2.  Some popular units for energy are joule, erg, calorie, and electron-volt. You buy energy from the power company in units of kilowatt-hours. (It should be called the energy company — power in physics is a distinctly different concept.)

As for the meaning of E=mc2, it basically says mass is a form of energy.  To put it another way, there is a form of energy that is associated with mass, just as kinetic energy is associated with speed, potential energy is associated with location in a gravitational or electric field, heat energy is associated with temperature, etc. The c2 is a conversion constant just as m/2 is the conversion constant between kinetic energy and speed2, mg is the conversion constant between potential energy and location, and Boltzmann's constant k is the conversion constant between heat energy and temperature. The speed of light of course is a very large number, so E=mc2 tells you mass contains an large amount of energy. The atomic bomb is testimony to that.

The mass of an object is a dense, difficult-to-convert form of energy. Think of it as "frozen concentrate" energy. In most of your daily, mundane interactions, mass is conserved, so when you add up all the forms of energy, you have an mc2 on each side of the equation and they cancel each other out. But as you point out, E=mc2 can be used for practical purposes by converting some atomic bomb mass into explosion (=kinetic+heat) energy. Also for some impractical purposes like annihilating high kinetic energy electrons and positrons to create high mass B and anti-B mesons.

In fact, in particle physics E=mc2 is used so frequently that particle physicists end up measuring mass in units of energy. For instance, an electron has a "mass" of 511 keV (kilo electron volts). It's like saying San Jose is 60 minutes south of San Francisco — we both know I'm not measuring distance in units of time, we both understand an implicit conversion factor "at freeway speeds of around 60 miles/hour." To be totally accurate, particle physicists should say 511 keV/c2, but the c2 conversion factor is typically implicitly understood, and it's very handy to have things pre-multiplied by c2.  Particle physicists also pretend that Planck's constant, h/2p, is exactly equal to 1 — no units whatever. But perhaps that's best left for your next physics lesson.

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