Straightforward Derivation of Mie's Gravitational and Inertial Masses

Scientific Paper
Title	Straightforward Derivation of Mie's Gravitational and Inertial Masses
Read in full	Link to paper
Author(s)	Tolga Yarman
Keywords	gravitational mass, inertial mass, mass deficiency, energy conservation, Newton's law of gravitation, equivalence principle, Gustav Mie, general relativity
Published	2006

"Straightforward Derivation of Mie's Gravitational and Inertial Masses Via Just Newton's Law of Gravitation and Energy Conservation Law: A New Approach to the End Results of the General Theory of Relativity" is a 2006 paper by Tolga Yarman. It is a foundational statement of the research programme he has pursued for decades: an attempt to reproduce the observable predictions of general relativity — gravitational time dilation, the redshift, the perihelion advance — starting from Newton's law of gravitation and the conservation of energy, and without invoking the principle of equivalence on which general relativity is built.

(The page has been moved to a shorter title; the full original title is retained in the citation.)

Overview

Yarman's central result is a relationship between an object's gravitational mass and its inertial mass when the object is in motion:

m_{gravitational} = m_inertial / γ²

where γ is the usual Lorentz dilation factor. In other words, motion (or, more generally, any energy an object acquires) is held to increase its inertial mass but decrease its gravitational mass, so that the two are no longer equal in a moving frame. This is the point at which the paper parts company with general relativity, whose foundation is precisely the equality — the equivalence — of gravitational and inertial mass.

Yarman's claim is that this apparent break with the equivalence principle is not a defect but a feature: once the two masses are allowed to differ in the way he derives, one can recover all the tested consequences of general relativity by an alternative and, he argues, simpler route.

The connection to Gustav Mie

The paper's title credits Gustav Mie, the German physicist who, in 1912, in the course of building an ambitious Lorentz-invariant electrodynamic theory of matter, reached the same relationship — that under a Lorentz transformation the gravitational and inertial masses of an object diverge, invalidating the Newtonian assumption of their equality in a moving frame. Mie, however, did not postulate an alternative principle of equivalence to take the place of the one he had undermined, and abandoned that line of work — which, Yarman notes, presumably encouraged Einstein to set Mie's theory aside.

Yarman's contribution, as he frames it, is to have reached Mie's 1912 result by a completely different and more physical route — through ordinary energy conservation rather than Mie's "cumbersome" universal Hamiltonian energy density — and then to have carried the theory to the finish line that Mie could not, by supplying the missing ingredient in the form of a quantum-mechanical account of "mass deficiency."

The mass-deficiency argument

The physical picture underlying the derivation is that a body bound in a field develops a "mass deficiency" equal to the binding energy it acquires there — the same idea as the familiar mass defect of a bound nuclear or atomic system, here applied to gravitation. Yarman then argues, via quantum mechanics (the wave equation together with the de Broglie relationship, which he stresses is itself already consistent with special relativity and energy conservation), that this mass deficiency:

slows the internal "clock" of a wave-like object in a gravitational field — reproducing gravitational time dilation and the redshift;
stretches the spatial size in which the object sits — reproducing the metric change of general relativity.

A recurring formal device is the observation that certain combinations of quantities are Lorentz invariant — Yarman uses [energy] × [mass] × [length]² (dimensionally the square of Planck's constant, hence invariant) as an organising constraint on how mass, size and internal frequency must scale in a field. From this he claims to obtain the "end results" of general relativity using only Newtonian gravitation restricted to static masses, energy conservation, and quantum mechanics.

Assessment

This is serious, mathematically literate work by a credentialed physicist — Yarman holds an MIT doctorate in nuclear engineering and has developed this programme across dozens of papers and two books — and it deserves to be read as such rather than dismissed. The individual ingredients are all real physics: gravitational binding energy really does show up as a mass deficiency; gravitational time dilation and redshift really can be obtained from energy conservation alone (the standard "photon climbing out of a potential well" argument makes exactly this point); and Mie's 1912 theory is a genuine, if now obscure, episode in the pre-relativistic history of gravitation, discussed by Pauli among others. Building an alternative derivation of GR's weak-field predictions from these pieces is a legitimate theoretical exercise.

The strong and weak points should nonetheless be kept distinct.

Reproducing the "end results" of general relativity is not the same as replacing it. The redshift, time dilation and perihelion advance are the weak-field predictions, and several independent frameworks recover them; general relativity is a full nonlinear field theory whose content extends well beyond them — frame dragging, gravitational lensing in the strong field, the detailed dynamics of merging black holes, and the gravitational waves detected since 2015 with waveforms matching GR to high precision. An alternative that matches the classic tests still has to confront that larger and now very well-measured body of evidence.

The claim that the equivalence principle is violated needs care, because two different statements are in play. The weak equivalence principle — that all test bodies fall alike in a gravitational field, independent of composition — is among the most stringently confirmed facts in physics, verified to about one part in 10¹⁵ by the MICROSCOPE satellite and by lunar laser ranging. Yarman's m_grav = m_inertial/γ² is a velocity- and energy-dependent statement, not a composition-dependent one, so it does not straightforwardly contradict those null results; but by the same token it is not the dramatic overthrow of the equivalence principle it is sometimes read as. Standard general relativity already predicts that the gravitational behaviour of a fast-moving body differs from the naïve Newtonian expectation, without abandoning the equivalence of gravitational and inertial mass for a body at rest.

The fair verdict is that the paper offers a genuine and internally coherent alternative derivation of gravitation's classic tests, resting on respectable physics and an honest reconstruction of a neglected historical idea — but that "the same end results by a different route" is a claim about the weak-field limit, and does not by itself establish that the equivalence principle is dispensable or that general relativity is wrong. It is best read as one of the more substantial entries in the energy-conservation-first tradition of gravitational theory, and as the seed of Yarman's later, more fully developed work.

Overview

The connection to Gustav Mie

The mass-deficiency argument

Assessment

See also