A new experimental approach to Mach's principle and relativistic gravitation
James F. Woodward
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In one page
Why does a shove meet resistance? James Woodward’s answer, following Ernst Mach, is that inertia is not something an object carries by itself — it is all the distant matter in the universe answering back. In this 1990 paper he does something new with that old idea. He asks what follows if the answer arrives as a field and if special relativity must hold in every small enough patch of spacetime. Working the four-force through in the object’s own instantaneous rest frame, he arrives at a wave equation for the gravitational potential with extra terms in it that nobody had had a use for. Those terms say that while an object’s internal energy is being changed, its rest mass changes too — transiently, by an amount set by how fast the rate of energy change is itself changing. And Woodward’s point is that this is testable on a bench: drive energy in and out of a capacitor fast enough and the fluctuation should be large enough to weigh.
Why it matters hereChapter 3 argues that inertia is a response of the universe rather than a property glued to matter, and this is the paper that turned that argument into an equation with a laboratory-sized number attached. Chapter 8 gains its origin document: every Mach-effect thruster built since, and every attempt to measure one, traces back to the transient mass fluctuation derived here.
What it claims
01The derivation rests on two premises and nothing else. First, that inertial reaction forces in an accelerated object are produced by the object’s interaction with a field, rather than being the immediate consequence of some property inherent in the object. Second, that any acceptable physical theory must be locally Lorentz invariant — in a small enough region of spacetime, special relativity must hold.Woodward’s own restatement, section 2, An Inertial Reaction Effect, in the NASA Breakthrough Propulsion Physics Workshop Proceedings
Published and peer-reviewed02Those two premises, worked through in the instantaneous rest frame of the accelerated object, give a wave equation for the gravitational potential whose source density carries extra time-dependent terms. The leading one is a transient change in the proper matter density equal to one divided by four pi times Newton’s constant times the density times the square of the speed of light, multiplied by the second time derivative of the proper energy density. Woodward writes the same relation with the gravitational potential in place of the square of the speed of light, which is the same thing whenever that potential is about the speed of light squared.Haisch and Rueda, Box 1, Derivation of the Woodward Effect, and their equation 5, which cites this paper as reference 5
Published and peer-reviewed03The Machian condition is recovered, not assumed. In the stationary case, where every time derivative vanishes, the equation reduces to Laplace’s equation and the potential is just the sum of contributions from all the matter inside the particle horizon — which, for reasonable values of the mass of the universe and its radius, comes out at roughly the square of the speed of light. The new terms are therefore not a new force. They are the time-dependent part of the gravitational field equation that was always there.Woodward’s own restatement, section 2, the paragraph following the field equation
Published and peer-reviewed04This is what makes the approach experimental, and it is what the title means. Because the effect scales with the rate of change of the rate of change of internal energy, it can be driven electrically: Woodward argues that rapid energy changes of the order of ten to the tenth through ten to the twelfth erg per cubic centimetre per second can be induced by charging and discharging capacitors, which would produce mass fluctuations at the milligram level once the change in density is integrated over the device.Haisch and Rueda, the paragraph immediately following their equation 5
Published and peer-reviewed05The transient terms either are negative, or in some circumstances can be made negative. Woodward flags this as the property that matters most: the same nonlinearity that lets the mass fluctuate is what, at large amplitude, would carry the proper matter density toward zero and past it, which is the regime traversable wormholes and the Alcubierre metric are written in.Woodward’s own restatement, section 2, closing paragraphs, and his section 1
What to watch06What to watch: Woodward states plainly that the equation is not validated by the existence of analogous effects in general relativity — its validity is a matter of fact determined by experiment. That measurement is now an instrument problem, and the instrument exists: Martin Tajmar’s SpaceDrive balance at TU Dresden resolves one picometre with a noise floor near eleven nanonewtons and can turn a thruster through 180 degrees inside the vacuum chamber, so a real Mach effect reverses with the device while a thermal artefact does not.Woodward’s own restatement, section 2, final sentence; the instrument is described in the SpaceDrive paper on this site
What to watch
The way in
https://doi.org/10.1007/bf00665932THE TITLE. The Crossref record, and every index that copies it, prints the last word as ‘graviation’. That is the publisher’s misprint; the word is gravitation, and the correct spelling is used above. The author’s own corrigendum, published the following year, spells it correctly in its title. THE CORRIGENDUM. Foundations of Physics Letters volume 4, issue 3, page 299, June 1991, doi 10.1007/bf00665763 — a one-page correction to this paper by the same author. A reader working from the 1990 text should read the two together. LICENCE. Published as Foundations of Physics Letters volume 3, issue 5, pages 497 to 506, October 1990. Crossref carries only Springer’s text-and-data-mining licence, Unpaywall and OpenAlex both report the article closed with no repository copy, and no Creative Commons statement exists — checked 2026-09-08. The publisher’s own pages decline automated retrieval behind an authentication redirect and a bot challenge, so the paper itself could not be read for this sheet, and no text of it is reproduced here. WHAT THIS PAGE IS WRITTEN FROM. Two open restatements of this paper’s derivation, both already in this library and both read in full on 2026-09-08. First, the author’s own: James F. Woodward, ‘Mach’s Principle and Impulse Engines: Toward a Viable Physics of Star Trek?’, in the NASA Breakthrough Propulsion Physics Workshop Proceedings, NASA/CP-1999-208694, which sets out the same two premises and the same field equation and cites this paper as its source. Second, an independent one: Bernard Haisch and Alfonso Rueda, ‘Inertia and Gravitation in the Zero-Point Field Model’, the final report on NASA contract NASW-5050, whose Box 1 is headed ‘Derivation of the Woodward Effect’ and whose equation 5 is the result, citing this paper as reference 5. Each locator below names which of the two carries the step. The paper’s own section and equation numbers are therefore not used, because they were not read. CONTEXT. Woodward wrote from the Departments of History and Physics, California State University, Fullerton. The measurement he proposes here was reported the following year in ‘Measurements of a Machian Transient Mass Fluctuation’, Foundations of Physics Letters volume 4, pages 407 to 423, 1991, and the method was patented in 1994 as US 5,280,864.
How to cite it
James F. Woodward (1990) A new experimental approach to Mach's principle and relativistic gravitation. doi:10.1007/bf00665932
Where it sits in the curriculum
Inertia and gravity from the vacuumInertial mass reduction and transmedium craftThe metric, warp drives and wormholes