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James N. Hanson

Does E=mc2? Probably, but no thanks to Einstein. In 1804 J. Soldner attributed to light a mass and thereby calculated light's gravitational deflection.1 Likewise, John Mitchell using Newtonian gravity acting on a corpuscle of light, deduced the escape velocity, and thereby the critical mass and radius, of a star that would emit no light [i.e., a black hole —Ed]. Mitchell reported his findings to the Royal Society of London on 27 November, 1783. One can find E=mc2 hidden in various works, usually associated with the index of refraction of light, or electrical capacitance, or both. For example, we find it in the works of Oliver Heaviside which works have been discovered in recent years; and also in James Clerk Maxwell's theoretical investigations on the electromagnetic theory of light.2

The usual derivation of E=mc2 uses the following sloppy mathematics, about which the student must not murmur. (Actually, no student dares question this most sacred amongst sacred cows.)

E = c p = c • mc(1-v2/c2)-1/2 = cmc(1+v2/2c2)= m c2 + (1/2)mv2

and what is not kinetic energy [(1/2)mv2] is something else, and that something else is the famous E=mc2. But the third quality [right-hand side, first line above — Ed.], is not an equality. Relativistic enthusiasts are always letting things move at velocities approaching the speed of light (c), but note the rotten approximation obtained from the first two terms of the appropriate binomial expansion (Taylor series, if you like):

         (1-v2/c2)-1/2    (1+v2/2c2)
v=0.5c     1.1547           1.1250
v=0.9c     2.2941           1.4050
v=0.95c    3.2026           1.4512
When E=mc2 is used, it is benignly used as an exact expression of physical fact, not rudely spoken of as an approximation; a bad one at that.

Let me add my school-boy derivation of E=mc2 which at least bears somewhat in some way on the physics of the matter and avoids the perplexing notion of converting matter to energy (whatever that means). Consider a blob of mass radiating (giving off) things called photons (Figure 1), and let them be given off by shells of thickness Dx and moving out from the blob at velocity v=c.

Assuming that Newton got F=ma right, we write the force on the shell as:

F = ma = mDx/Dt2 = m (Dx/Dt)/Dt.

But the energy associated with this motion is:

E = F Dx = m(Dx/Dt2)Dx = m(Dx/Dt)2 = mv2 = mc2.

Once relativists slip (1+v2/2c2) by you, they get clocks to slow down, measuring rods to shrink, and other Alice-in-Wonderland things by somehow replacing v2 in this expression by whatever man can grab, e.g., see Humphrey's wonders when he sticks in F (p. 101, Starlight and Time, Master Books).


1 Soldner, J. 1804. Berliner Astr. Jahrb., p. 161. Reprinted in Ann. der Phys., 65:593, 1921.

2 E.g., in A Treatise on Electricity and Magnetism, vol. 2, pp. 437-441.

Translated from WS2000 on 14 February 2005 by ws2html.