In a recent issue Dr. Hanson¹ discusses the famous energy-mass equation E = mc². He critiques one derivation of this equation, gives his own simple alternative derivation, and makes some far-reaching conclusions regarding relativity. In doing so he raises a number of interesting points that merit further discussion.

A Derivation of E = mc²?

The derivation that Hanson cites is

Here E is the energy, p is the momentum, c is the speed of light, m is the rest mass and v is the speed of the object.

Hanson disputes only the portion following the last equal sign. Since the last term in the last equation is the classical kinetic energy, it seems that the total energy E consists of the rest energy mc² plus the kinetic energy. As Hanson points out, however, this derivation is not correct because the last equation is just an approximation, consisting of only the first two terms of the Taylor expansion of the previous equation.

Yet, it should be noted first that in relativity the mass M is considered to vary with speed according to the relation M = m(1 - (v/c)²)^-1/2, where m is the rest mass. Thus, as far as relativity is concerned, the left hand side of equation (1) already reads E = Mc2, which reduces to E = mc2 for an object at rest. Thus the approximation that follows is not needed to derive E = mc2.

Indeed, this approximation is generally made merely to demonstrate that for small speeds v, when higher-order terms can be neglected, the equation yields the rest energy mc2 plus the usual classical kinetic energy. Moreover, whenever this is done it is generally made very clear that it is only an approximation.

Hanson's Derivation of E = mc²

Unfortunately, Hanson's own derivation falls short of the mark. He considers a blob of mass, radiating photons in shells of thickness x moving out with a speed c. He calculates the force on the shell to be:

Here the first x (after the second equal sign) is no doubt a typo, and should read v. The energy associated with this motion he asserts to be:

But this derivation has a number of shortcomings. First, the expression for the acceleration is wrong: it should be a = v/t = (x/t)/t. Thus equation (2) should read:

Second, equation (3) refers not to the energy E but to the change in energy E. Thus the equation should be

Upon integrating this we find that E = 1/2 mv2 = 1/2 mc2, which is just the classical kinetic energy.

A further difficulty is that Hanson assumes implicitly that the mass m in equations (2) and (3) to be independent of speed (which poses a problem for the photons he considers, since they are generally considered to have no rest mass). In relativity the effective mass depends on velocity, and thus also on time t if v is changing. Hence Newton's law should take the form F = dp/dt and not F = ma, which holds only if the mass is independent of t.

An Alternative Derivation

One could derive the equation E = Mc2 as follows. Start with the equation for the momentum of an object:

This equation for the momentum can be derived from Maxwell's equations and is experimentally confirmed by cyclotron experiments. I take it to be well established; Hanson seems to have no difficulty with it either. I note in passing that Einstein's original formula for the relativistic mass was incorrect and that the above equation for the momentum was first introduced by Planck in 1906.²

If a force is applied in the direction of v, the change in energy E is:

Dividing by dv this becomes:

Integrating, we get:

which yields the rest energy E = mc2 when v = 0. If we take the relativistic kinetic energy T to be just E minus the rest mass energy, we find that:

For small speeds (i.e., v << c) this reduces to

The Origin of the Energy-Mass Equation

Who first asserted the mass-energy equation? Dr. Bouw³, in his editorial discussion of Hanson's paper, states that the formula antedates Einstein by more than 100 years. Hanson himself claims that it can be found hidden in the works of Maxwell (1860's) and Heaviside (1880's).

To this it must be replied first that, even if the mass-energy formula were implied in Maxwell's electromagnetic equations, Maxwell himself did not take the further step of explicitly deriving the formula. Moreover, when this was done, by J.J. Thompson (in 1881), Heaviside (in 1889) and others⁴, the formula found was E = (3/4) mc2, rather than E = mc2. Poincare, using the electron theory of Lorentz, came very close in 1900: his work implies E = mc2, but he stops just short of taking the final step.

It seems that the first explicit derivation of the equation was indeed by Einstein in 1905. It is interesting to note, however, that Einstein's own derivation of the formula E = Mc2 was fallacious, as was pointed out by Ives.⁵ According to Ives, a correct derivation of the formula was not published until 1907, by Planck.

It should be noted that this formula can be validly derived via a number of different methods: via special relativity, via Ives' approach, or from Maxwell's equations.

Implications for Clocks and Measuring Rods

Hanson concludes by asserting “once relativists slip (1 + v2/c2) [the approximation used in equation (1) JB] by you, they get clocks to slow down, measuring rods to shrink...”. Bouw, too, writes that ”Hanson...shows that much of the formulae used to explain phenomena at speeds close to that of light are derived on the assumption that the speeds are very much less than the speed of light”.

This makes it seem as if time dilation and length contraction are based on erroneous derivations E = mc2, involving low-speed approximations.

But this is hardly the case. Time dilation and length contraction are not mere consequences of any such approximations. Rather, they follow inevitably follow Maxwell's equations for electro-magnetism. This was shown already in 1889 by Fitzgerald, who deduced length contraction from Heaviside's (1888) formula for the field of a moving charge; by 1900 Larmor and Lorentz had derived the time dilation formula by similar means. Later, the same effects were obtained also viaspecial relativity. These phenomena can be interpreted in a variety of ways, depending on whether one wishes to retain an ether or not. Yet the existence of these effects can be disputed only if one is prepared to challenge also Maxwell's electro-magnetic equations.

My recommendation is that we accept the reality of shrinking lengths and slowing clocks for objects that are moving with respect to the aether. I support an interpretation of the Lorentz transformations, not in terms of Einstein's initial relativism, but in the sense of motion with respect to an aether, as taken by Lorentz, Ives, and others (including the later Einstein). This strikes me as much the simpler and more plausible course than that of attempting to revamp electro-magnetism so as to maintain speed- independent clocks and lengths, and then making suitable modifications in order to square the new theory with the observational data.

1. Hanson, James N, “Does E Equal M C Squared?”, Biblical Astronomer Vol.6 No.76 (1996):13-14.

2. for a discussion of the history of the notion of relativistic mass, see Carl G. Adler, “Does Mass Really Depend on Velocity, Dad?”, Am. J. Phys. 55(8):739-43 (1987).

3. Bouw, Gerardus D., “Editorial”, Biblical Astronomer Vol.6 No.76 (1996):3.

4. for references and detailed discussion see Max Jammer Concepts of Mass in Classical and Modern Physics (New York: Harper & Row, 1961):136-144.

5. H.E. Ives, “Derivation of the Mass-Energy Relation”, Journal of the Optical Society of America 42:540-543 (1952).