On the Origin of Inertial Force
| Scientific Paper | |
|---|---|
| Title | On the Origin of Inertial Force |
| Read in full | Link to paper |
| Author(s) | C Johan Masreliez |
| Keywords | inertial mass, gravitational mass, Minkowskian line element |
| Published | 2006 |
| Journal | Apeiron |
| Volume | 13 |
| Number | 1 |
| No. of pages | 35 |
| Pages | 43-77 |
Read the full paper here
Abstract
According to general relativity inertial mass and gravitational mass are equivalent. Therefore, inertial acceleration might, like gravitational acceleration, be associated with changing metrical coefficients in the line element of general relativity. This possibility is investigated. If a cosmological reference frame exists, the Minkowskian line element may be modified by a velocity dependent scale factor for which all accelerating motion becomes motion on spacetime geodesics. This might model inertia as a gravitational-type phenomenon.
Overview
This 2006 Apeiron paper by C Johan Masreliez takes up a question that Masreliez regards as still unanswered: what physically resists acceleration? Newton's spinning-bucket argument, he notes, points to a frame of absolute rest — the water climbs the wall of the bucket, but "spinning relative to what?" It cannot be the Earth, since the planets feel the same effect in their orbits, and it cannot be the Sun, since stars in a galaxy feel it too. Masreliez revives the Newton–Leibniz question and answers it in Newton's favour, but with a twentieth-century mechanism: an inertial reference frame (IRF) supplied not by distant matter, as in Mach's argument, but by the cosmic drag predicted by his own Scale Expanding Cosmos (SEC) theory, which he says gradually damps relative velocities and angular momenta until free motion converges on a common frame.
The proposal itself is compact. If the equivalence of inertial and gravitational mass is taken seriously, and if gravitational acceleration in general relativity arises from gradients in the metrical coefficients, then forced acceleration ought to arise from metrical gradients too. Masreliez therefore postulates an "inertial field": a gravitational-type field generated not by mass but by acceleration relative to the IRF. He then looks for a velocity-dependent scale factor on the Minkowskian line element such that the geodesic equation of the resulting metric reproduces exactly the acceleration that produced it. He reports that such a factor exists and is unique, and that with it inertia becomes a species of gravitation. This departs from the standard account in two ways: it denies that all inertial frames are philosophically equivalent (absolute velocity matters), and it treats spacetime as not strictly a continuous manifold.
The argument
Scale equivalence and the cosmological frame
The SEC theory's founding idea is that space and time expand together: when space expands the pace of proper time slows, and all four metrical coefficients of the line element change by the same factor. Because Einstein's field equations are unchanged by a constant scale factor, the universe is "scale equivalent" — no particular scale takes preference, and Masreliez likens this to "a palace with many identical rooms," each experienced identically from inside yet obviously not the same room. On this view the universe expands without aging, so "on the average the universe has always been the same as it is today," which eliminates the big-bang creation event. Cosmic drag is offered as an independent consequence, said to explain the low relative velocities of galaxies, reported discrepancies between optical planetary observations and the ephemerides, and the Pioneer anomaly.
The scaled Lorentz transformation
Scale equivalence permits a coordinate frame moving at velocity v relative to the IRF to carry the scaled Minkowskian line element
- ds2 = φ2(v)(dt2 − dx2 − dy2 − dz2)
with the ordinary Lorentz transformation multiplied throughout by φ(v). Masreliez points out that Einstein considered exactly this transformation in 1905 and set φ = 1 on the strength of his postulate that all inertial frames are equivalent. Where a cosmological reference frame exists, Masreliez argues, that step is no longer forced: φ may depend on the absolute velocity while the line element stays Minkowskian in character.
The inertial field and the inertial scale metric
If φ depends on velocity and velocity varies with position along an accelerating trajectory, then the metrics acquire a positional gradient — and a gradient in the metrics is, in general relativity, a gravitational field. Masreliez demands one property of this field: "the geodesic acceleration given by the inertial field metrics is always identical to the forced acceleration that generates the inertial field." In Appendix I he derives the scale factor for which the geodesic equation becomes an identity:
- φ2(v) = 1 − v2
He calls this the "inertial scale metric" and the resulting line element the "inertial line element." Because a constant v makes the scaling a mere constant, all of special relativity survives untouched for uniform motion; the new content appears only during acceleration, which special relativity does not model. He further reports that the inertial metric yields the relativistic energy and momentum relations directly, though with v read as absolute rather than relative.
Notably, the inertial field is in some respects the mirror image of the gravitational field: spacetime is curved for an accelerating particle and Minkowskian for one at rest or in uniform motion, whereas a gravitational field curves spacetime for a particle at rest and vanishes in free fall. The gravitational field is static if its source is; the inertial field is "dynamic" and exists only while acceleration lasts.
A heuristic particle model
Section 5 offers an ontological picture. Drawing on SEC paper IV, Masreliez treats particles as three-dimensional standing waves in oscillating spacetime metrics at the Compton frequency. He then models a particle as light bouncing inside a moving box of side L, and assumes that a moving particle oscillates synchronously with a stationary one. Transverse to the motion, the zigzag light path forces the perpendicular scale to shrink by φ = √(1 − (v/c)2); along the motion, the out-and-back time L/(c − v) + L/(c + v) is longer by 1/(1 − (v/c)2), requiring a longitudinal factor of φ2. The moving box therefore carries spatial metrics a factor φ smaller than special relativity's — the extra factor appearing precisely as the perpendicular contraction that special relativity forbids — and the line element is scaled by φ2 = 1 − (v/c)2: the inertial line element again. "During acceleration moving particles may adjust their scale in order to preserve their oscillation frequencies."
The co-moving observer and the temporal puzzle
A co-moving observer cannot detect the scale change, since all her spatial and temporal references change with it; by a suitable redefinition of the time coordinate — "a trick employed by Lorentz, Poincaré and Einstein" — the speed of light appears isotropic in her frame, and the transformation back to rest is the ordinary Lorentz transformation. Masreliez suggests that this is why Einstein reached φ = 1, and that "the Minkowskian coordinate frame might be required in order to preserve the structure of particles during motion."
Section 7 confronts an apparent contradiction: the inertial line element makes all moving particles oscillate at the same absolute frequency, yet the time dilation of moving clocks is experimentally established (he cites Hafele and Keating 1972). His resolution is that acceleration proceeds in cycles — a continuous phase modelled by the inertial metric, terminated by a discrete scale adjustment that restores the Minkowskian line element and resets the pace of proper time, with ds(v) = (1 − v2)·ds(0). General relativity is "blind" to the discrete step, since constant rescaling leaves the field equations unchanged, so the change in the pace of time stays hidden inside the Lorentz boost. The two segments together implement the boost. This cycle deliberately mirrors the SEC cosmological expansion cycle, with one difference: in spatial acceleration the SEC's "fictitious observer" becomes a real observer in the cosmological rest frame.
Kinetic energy as inertial potential
Section 8 draws the parallel out formally. Against the Newtonian gravitational acceleration ag = −mG/r2 and potential −mG/r, the inertial geodesic acceleration is ∇(v2/2), with potential V2/2. Kinetic energy is thus reinterpreted as an "inertial potential," and forced acceleration as its spatial gradient. Similarly, Schwarzschild time dilation dτ = dt√(1 − r0/r) is set beside the inertial dτ = dt√(1 − v2). In the closing speculation Masreliez suggests the discrete energy increments would explain quantization in quantum theory and, following Tifft's 1978 reports, quantization of the cosmological redshift.
Assessment
What is genuinely attractive here is the economy of the central move. Rather than positing a new force or a new field of matter, Masreliez asks whether inertia can be absorbed into the same geometric language that already accounts for gravitation, and he identifies a concrete, checkable condition — that the geodesic acceleration of the scaled metric equal the acceleration that generated it — under which a unique scale factor emerges. That the result φ2 = 1 − v2 reproduces the relativistic energy and momentum relations without being fitted to them is a real internal consistency, and the discussion of Einstein's discarded φ(v) is historically accurate: that factor is indeed eliminated in 1905 by an equivalence postulate rather than by measurement. The paper is also unusually candid about its own cost, admitting that abandoning the continuous manifold "implies a leap of faith into new territory."
The difficulties are correspondingly serious. The largest is that the paper's explanatory work is done almost entirely by assertion at the junctions. That an inertial field exists is postulated, not derived; the demanded identity between forced and geodesic acceleration is imposed as a requirement and then a scale factor is found to satisfy it, which establishes consistency rather than causation. Nothing in the argument says why acceleration should generate a metrical gradient in the first place. The discrete "scale adjustment" that closes each acceleration cycle is introduced precisely because the continuous phase cannot reproduce observed clock behaviour, and it is placed exactly where general relativity is said to be blind to it — a step that is unfalsifiable by construction, since it is defined as producing no effect the field equations can see. The particle-box model is offered as "rudimentary, provisional and heuristic," and it is: it assumes synchronous oscillation of moving particles, which is the conclusion it is used to support.
The clearest conflict with measurement is transverse contraction. The inertial line element requires lengths perpendicular to the motion to shrink by φ, whereas special relativity requires them to be unchanged — and the transverse case is not a matter of interpretation, since a transverse contraction is what makes the standard reciprocal-rod argument consistent. The paper does not confront this directly, resting instead on the claim that a co-moving observer cannot notice the scaling. Relatedly, the whole scheme requires a detectable absolute velocity, yet the only velocity that appears in any experimental prediction is the relative one; no measurement is proposed that would isolate v with respect to the IRF, which leaves the IRF doing heavy conceptual work while remaining operationally inaccessible. The supporting cosmological claims — cosmic drag explaining the Pioneer anomaly and planetary ephemeris drifts — are asserted by reference to earlier papers rather than defended here; a thermal-recoil account of the Pioneer anomaly has since been advanced, and the ephemerides have tightened considerably. Finally, the redshift-quantization appeal to Tifft is the weakest link in the chain: subsequent large redshift surveys have not sustained the effect, so invoking it as confirmation adds risk rather than support.
Taken on its own terms — as a demonstration that inertia can be written as a metrical effect if an absolute frame is granted — the paper does what it claims. Whether the frame should be granted is the question it does not settle.