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The mass-energy equation is derived in general from Newton’s equation of motion without use of electrodynamics, or Einstein’s Postulates which were presented in his superb 1905 paper on Special Relativity (SR). This was previously not thought to be possible. This novel derivation of an accelerated body of rest mass m0 is compared with the traditional SR inertial derivation. A discussion is given of pre-1905, electrostatic and electrodynamic derivations of the mass-energy relation yielding , as well as more recent ones. A concise pre-relativity history of the mass-energy relation is traced back to Newton in 1717.

In 1687 when Newton introduced mass in his 2^{nd} law of motion [

There have been many electrostatic and electromagnetic derivations of the mass-energy relation. The electrodynamic derivations primarily relied on the work of J. H. Poynting [

where S is the Poynting vector, E is the electric field vector, and H is the magnetic field vector. This is the commonly used Abraham form of the Poynting vector in terms of E & H rather than the Minkowski form in terms of D & B. Although this makes no difference in free space, it makes a difference in electromagnetic energy flux transfer in a solid.

One historical approach uses the Poynting vector to yield the electromagnetic momentum of a point-like charge moving with velocity v:

where V is the volume occupied by the electromagnetic field. If the point-like charge (such as an electron) is moving in the x-direction, by symmetry

Combined with Newton’s mechanics, these approaches led to the mass-energy relation

More than a century ago prominent physicists combined Maxwell’s equations with Newton’s equation of motion to obtain velocity dependent masses before Einstein’s awesome 1905 paper [

Longitudinal mass

Transverse mass

In 1889 George Fitz Gerald explained the null result of the 1887 Michelson-Morley experiment by introduc-

ing a length contraction factor

which he earlier recast from the original cumbersome 20 equations with 20 unknown variables, into 4 differential vector equations with 2 variables) compared the electric and magnetic fields of a point-like charge and an equally charged ellipsoid. Remarkably, Heaviside found that the fields of a steadily moving (mathematically

constructed) contracted ellipsoid having axes in the ratios

moving point charge, v being the velocity of both the ellipsoid and the point charge in free space.

Lorentz found that experiments on fast moving electrons gave results that were not consistent with Abraham’s model of a rigid charged spherical conductor. In 1904, Lorentz, assuming an ether throughout absolute space, found similar results to FitzGerald from what is known as the Lorentz Transformation which is integral to Einstein’s 1905 [

Einstein used the concepts of longitudinal and transverse mass in his 1905 paper [

The electrodynamic masses of Equations (3) and (4), as well as Einstein’s 1905 longitudinal and transverse masses all have been abandoned, although some would argue that Equations (3) and (4) are valid relativistic mass concepts. Subsequent to his 1905 paper, Einstein showed by means of his postulates that the velocity dependence of mass does not depend on electromagnetics or structural details of a charged body such as an electron, but is totally general. He also eliminated the ether and absolute space, but may have inadvertently reintroduced the ether in his theory of General Relativity by imbuing space with properties such as curvature and metric expansion.

So, there is now only the Einstein “relativistic mass” which is the same form as

From the past to the present, the Energy-Mass relation

and Lorentz got

Despite Einstein’s 1948 comments, many distinguished relativity experts such as M. Born, V. Fock, F. Klein, M. Laue, and R. Tolman in the past; and W. Rindler, and R. Penrose in relatively recent times use the concept of relativistic mass, and it is still taught. The intent of the present paper is not to take sides on the dispute of whether one should or should not use the concept of relativistic mass. Both sides will agree that in Section 2.3

the basic undisputed finding of Equation (14)

momentum four-vector has a fixed magnitude

can be derived simply from Newton’s equation of motion. The significance of this finding means this result is more general than previously thought; and may even have relevance to accelerated systems and quantum mechanics where the concept of mass at times appears nebulous as it does to some degree in zitterbewegung.

Choosing the Feynman route (approach) to investigate the available options for the mass-energy relation, we begin with the classical Newtonian equation of motion for single, isolated particles. Although deserving of attention, to the author’s knowledge this simple option has not yet been treated in the literature. The main goal of this derivation is to show that the mass-energy relation can simply be derived from Newton’s equation.

Let us derive the mass-energy relation m of a body of rest mass

discussed above [or for Newton’s light corpuscles with the dimensionless number

assume in general that

is the energy of the body that can increase with acceleration. The constant c is the speed of light now known to be invariant in all frames. If this derivation were done long before Einstein’s 1905 SR paper [

The derivation here uses Newton’s equation of motion cast in the form of a power equation. Thus it differs from Einstein’s approach [

With

This author first derived

Using Equation (8) for

with

Therefore the assumption

Relativity. Note that the low velocity equation for kinetic energy,

the transverse mass of Equation (5) obtained by Einstein [

We will briefly sketch the traditional SR approach to point out the differences between it and the derivation given above using Newton’s equation.

In a space-time of 3 spatial dimensions and 1 dimension of time, the 4-momentum is

where the 4-velocity

The dot product of the vector velocity is the scalar

Using the Einstein summation notation

Combining Equations (11) and (13), we have the well-known equation:

Dividing Equation (14) by

Equation (15) gives the familiar increase of mass with velocity. Although this standard Special Relativity approach is strictly just for inertial frames, it is used for accelerating frames such as with cyclotrons.

In the 1905 paper [

“They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the ‘Principle of Relativity’) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.”

From context, the first postulate has been interpreted as “inertial frames”. The present paper derived the mass- energy relation in general terms without recourse to Einstein’s postulates.

Reference [

Newton had for his light corpuscles (photons):

dies and light convertible into one another…?” [

energy-mass relationship

1717, as demonstrated in Section 2.2. However, he did not do so.

In simplest terms the derivation of this paper comes from the conservation of total power, which is a consequence of the detailed balance of the conservation of mass-energy. For those who prefer to restrict the concept of mass to rest mass

This paper derived the generalized mass-energy relationship from Newton’s equation of motion cast in the form of a power equation. The result is fully consistent with special relativity. Einstein’s postulates were not used regarding the velocity of light, nor that the laws of physics are the same in all inertial frames. If an incorrect mass such as either the longitudinal mass of Equation (3) or the transverse mass of Equation (5) obtained by Einstein [

It is the great genius of Albert Einstein who accomplished the monumental task of constructing a complete mechanics within the framework of relative space and relative time, and described both mechanical and electrodynamic phenomena in a single stage within that structure.

I wish to acknowledge the countless generous, hard-working, and tireless beings from whom I have learned and gleaned knowledge both directly and indirectly; and upon whose unselfish shoulders I have stood.

MarioRabinowitz, (2015) General Derivation of Mass-Energy Relation without Electrodynamics or Einstein’s Postulates. Journal of Modern Physics,06,1243-1248. doi: 10.4236/jmp.2015.69129