ON THE ENERGY-MASS EQUATION
John Byl, Ph.D.
In a recent issue Dr. Hanson
The derivation that Hanson cites is
E = cp = cmc(1 - v^{2}/c^{2})^{-1/2} = mc^{2} + (1/2)mv^{2} (1)Here 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 Yet, it should be noted first that in relativity the mass 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
Unfortunately, Hanson's own derivation falls short of the mark. He
considers a blob of mass, radiating photons in shells of thickness
F = ma = m(x/t) = m (x/t)/ t (2)Here the first
E = F x = m(x/t2) ^{}x = m (x/t)^{2} = mc2
(3)^{}
But this derivation has a number of shortcomings. First, the expression
for the acceleration is wrong: it should be
F = ma = m (x/t)/t (4)
Second, equation (3) refers not to the energy
E = F x = m [ (x/t)/t] x = m v v (5)
Upon integrating this we find that 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
One could derive the equation
p = Mv = mv (1 -(v/c)^{2})^{-1/2} (6)
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. If a force is applied in the direction of v, the change in energy E is:
E = F dx = [dp/dt] dx = dp v (7)
Dividing by d
E/dv = v dp/dv = v m (1 -(v/c)^{2})^{-3/2} (8)
Integrating, we get:
E = mc2(1 -(^{}v/c)^{2})^{-1/2} = Mc2 (9)
^{}which yields the rest energy v = 0. If we take the relativistic
kinetic energy T to be just E minus the rest mass energy, we find that:
T = mc2 [(1 - (^{}v/c)^{2})^{-1/2} - 1] (10)
For small speeds (i.e.,
T ÷ 1/2 mv2 (11)^{}
Who first asserted the mass-energy equation? Dr. Bouw 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
others 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 ^{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.
Hanson concludes by asserting “once relativists slip (1 + 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 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.
REFERENCES
1. Hanson, James N, “Does 2. for a discussion of the history of the notion of relativistic mass, see
Carl G. Adler, “Does Mass Really Depend on Velocity, Dad?”, 3. Bouw, Gerardus D., “Editorial”, 4. for references and detailed discussion see Max Jammer 5. H.E. Ives, “Derivation of the Mass-Energy Relation”, |