An Approach to Gravity Modification as a Propulsion Technology

Scientific Paper
Title	An Approach to Gravity Modification as a Propulsion Technology
Read in full	Link to paper
Author(s)	Ben Solomon
Keywords	Gravity Modification, Electromagnetic Force, Lorentz-Fitzgerald Transformation
Published	2009
No. of pages	9
Pages	317-325

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Abstract

Gravity modification as a portable non-mass effect is feasible. Contemporary experiments such as HFGW and LIGO require mass to model gravitational acceleration and gravitational waves. A different approach to gravitational acceleration, and thus space propulsion technologies is presented here. This paper proposes that gravitational acceleration on any particle is the effect of the deformation of the shape and mass of the particle due to non-inertia transformations present in that local region of the gravitational field. The analytical formulation and numerical integration has led to the discovery of a new formula for gravitational acceleration, g = τc², that is neither a function of the mass of the gravitational source nor a function of gravitational waves; where τ is a function of the time dilation present in the local gravitational field. This formula has been tested and verified to be correct in the gravitational fields of the nine planetary bodies in our Solar System, and the Sun; mechanical acceleration, and electromagnetic fields. Thus leading to the inference that g = τc² is the generic formula for all non-nuclear force fields. The true power of this definition of gravitational acceleration lies in the fact that it now lends itself to a portable technology, as mass is no longer required to derive acceleration. This new relationship for acceleration, describes how an electron moving in a magnetic field causes a force on the electron, and explains why the electron velocity, magnetic field and resulting force relationship is orthogonal. This electron model would be the basis for future propulsion technologies.

Overview

The paper was presented at the Space, Propulsion & Energy Sciences International Forum (SPESIF) 2009, and its acknowledgements thank Paul Murad and Glen Robertson of that forum for review, together with the National Space Society and the Mars Society for earlier conference platforms. Its declared keywords are Gravity Modification, Electromagnetic Force and Lorentz-Fitzgerald Transformation.

Benjamin T. Solomon's proposal is to relocate the source of gravitational acceleration. General relativity models gravity as curvature of spacetime; Solomon proposes "an equivalent shape change on a particle" — that the change in the shape of spacetime in the particle's local region "is mirrored by an identical change in the shape of the particle", and that this deformation is the acceleration. He divides gravity into three parts, "the mass source, the field, and the field effect or acceleration", and confines the analysis strictly to the third. The technological motive is explicit: if acceleration is produced entirely by local field properties, then "any field effect technology need only be applied to the local region", and gravity modification becomes a portable, mass-free technology rather than something requiring a planet.

The claimed result is a single formula, g = τc², with τ = dt/dr the spatial gradient of time dilation. Because M does not appear, Solomon argues the formula is not merely an alternative to g = GM/r² but a more general one, applicable to mechanical and electromagnetic accelerations as well — "the universal description of acceleration for all non-nuclear forces".

The argument

Local inertia and non-inertia fields

A local inertia field is defined as a region where only inertia transformations Γ(v) are present and acceleration is absent; a local non-inertia field, "a small section (≤ 10⁻⁹ m²) of gravity", is one where non-inertia transformations Γ(a) are present. The central postulate is that the spacetime transformations are concurrently reflected as particle transformations, Γ_p(x,y,z,t) = Γ_s(x,y,z,t). Because the Lorentz–Fitzgerald transformation is linear and symmetrical about the axis of motion, the particle deformation it produces is symmetrical; gravitational transformations are non-linear in space, so the deformation along the axis of acceleration is asymmetrical.

Two transformations are compared. The Lorentz–Fitzgerald factor is written Γ(v) = 1/√(1 − v²/c²) = x₀/x_v = t_v/t₀ = m_v/m₀; the "Newtonian" gravitational factor as Γ(a) = 1/√(1 − 2GM/rc²) = x₀/x_a = t_a/t₀ = m_a/m₀. Solomon is careful to say the Einsteinian Γ(r) "is more sophisticated" and that for his discussion the Newtonian version suffices.

Table 1 then tests whether the two are equivalent, by computing for each Solar-System body the escape velocity v_e = √(2GM/r), the gravitational time dilation, and the velocity v_Γ that would produce the same dilation through Γ(v). The two agree to a fraction of a part per million (the "velocity error" column runs from 0.0000002 % for Jupiter to 0.0001586 % for Pluto), from which Solomon infers that Γ(a) and Γ(v) are equivalent at any point-sized location, and generalises to Γ(e) = x₀/x_e = t_e/t₀ = m_e/m₀ for "any local environmental transformation".

Centre-of-mass shift and the numerical model

Under gravity a particle is taken to deform so that its near side is flatter than its far side, and to become denser on the near side, shifting its centre of mass. The analytical formulation (equations 5–8) writes the shifted centre of mass CM_Φ as a double integral over slices weighted by Γ(x) = 1/√(1 − 2GM/(r + x)c²). Solomon states plainly that inverting this analytically "runs into several pages and is not presented here as it is not a simple, elegant solution", and proceeds numerically.

The numerical model slices a particle into 2,000 discs, 1,000 on each side, computing each slice's time dilation, density and thickness from Γ(a), reassembling the deformed particle and taking moments to find the new centre of mass. 1,190 numerical integrations were run: 7 particle sizes from 10⁻²¹ m up to 10⁻³ m, in 10 gravitational fields, with 17 shapes or mass distributions (cube, line, sphere, hollow sphere, crown, several multivariate normals, photon models and others). All arithmetic was carried to 250 significant digits; the computed centre-of-mass shifts range from 1×10⁻²² m down to 3.3×10⁻⁶¹ m.

Regressions on the 1,190 results, with R² between 99.998 % and 99.999 %, give three relations:

g = k_dχ/S_z² = GM/r², χ = k_mδt S_z, g = k_cδt/S_z = GM/r²

where χ is the centre-of-mass shift, δt the change in time dilation across the particle and S_z the particle size. The fitted k_c comes out to 8.9919×10¹⁶, "within 0.049 % of the numerical value of the square of the velocity of light c² or 8.9875517873681764×10¹⁶". Taking the limit δr → 0 gives the paper's headline equation

g = c² dt/dr = τc² = GM/r² (16)

Table 3 shows k_d and k_m varying by orders of magnitude across the 17 shapes while their product k_dk_m = k_c stays fixed — from which Solomon infers Internal Structure Independence: gravitational acceleration is external to and independent of the particle's shape or mass distribution. He adds the aside that "it appears that Nature has figured out how to get around Heisenberg's Uncertainty principle at the macro level", since the precision of a particle's size does not matter.

Table 4 then computes g = τc² directly for a 10⁻¹¹ m particle at each planetary surface, comparing with GM/r²; the discrepancies are listed as 0.0500250 % (Pluto, Mars, Mercury) down to 0.0493797 % (Sun).

Charged particle in a magnetic field

Table 5 extends the test to electromagnetism, comparing acceleration computed three ways for ten cases: the centripetal a = v²/r, the standard a = q(v × B)/m, and the time-dilation method using g = c²dt/dr with v = ωr. The three agree to 10⁻¹²–10⁻¹⁶ relative error, which Solomon takes as showing equation (16) is "correct for mechanical, electromagnetic and gravitational forces".

The electromagnetic force process (EMFP) model is then "reverse engineered". A charged particle moving with velocity v in a magnetic field follows an arc of radius r; the left and right sides of its own electric field, at radius dr, have velocities ω(r − dr) and ω(r + dr). With v = (q/m)Br, ω = (q/m)B and dv = (q/m)Bdr, the time-dilation difference across the electric field gives an acceleration (equation 29). Approximating an infinitesimal patch of the spherical field as a flat surface with field (q/A)/(2ε₀) yields dv = (8πε₀/m)BEdr³, and with only the E cosθ component surviving, equation (32). Solomon's interpretation: "it is the spherical shape of the electrical field that converts the perpendicular velocity of the charged particle with respect to the magnetic field into an orthogonal force. All other orientations of the particle's electric field velocity dv with respect to the magnetic field negate themselves" — so the orthogonality of the Lorentz Force is explained by geometry, and "vectors and matrices are elegant mathematical shortcuts in current electromagnetic theory".

Reconciling Laithwaite, Hayasaka–Takeuchi and Luo

The final section applies equation (16) to rotating-spinning discs. Time dilations were computed at 49,165 points across a disc and converted to acceleration. Table 6 gives averaged upward accelerations for three rotation/spin radius pairs at spins of 1,000–5,000 rpm and rotations of 0–15 rpm; the regression is

a = ω_sω_d√h (33)

with ω_s the spin rate, ω_d the rotation rate and h the hypotenuse formed by the spin and rotation radii. Solomon reads this as confirming Eric Laithwaite's demonstration that a disc spinning at 5,000 rpm and rotating at 7 rpm, with rotation radius at least 1 m and spin radius 0.3 m, "would be almost weightless (9.8 m/s² − 9.7 m/s² = 0.1 m/s²), if not rise". He stresses that it is a weight-change phenomenon: reversing the sense of spin relative to rotation reverses the sign. And he uses it to adjudicate a real experimental dispute — Hayasaka and Takeuchi (1989) reported gyroscopic weight loss, Luo, Nie, Zhang and Zhou (2002) found a null result — concluding that "given their experiments downward pointing spin vector, equation (33) shows that Lou et al were correct", because the acceleration must be orthogonal to both spin and rotation.

The conclusion lists three keys for a technology: the applied transformation Γ(s) must be non-linear along the path of the required acceleration; gravity modification consists of "field vectoring, and field modulation"; and interstellar travel "could be achieved by breaking equation (4) into two transformations, one for space Γ(s_x,y,z) and other for time and mass Γ(s_t,m) so that distances could be shrunk without altering time or mass." In a propulsion device, "the electric field holds force, while the magnetic field is used to power the non-inertia field."

Assessment

What is attractive here is the discipline of the exercise. Solomon states his scope narrowly — only the field effect at the particle, not the source or the field — computes rather than speculates, runs 1,190 integrations across seven size scales and seventeen geometries, publishes the regression quality, and tabulates enough intermediate numbers that a reader can check him. The Internal Structure Independence result, that k_d and k_m vary by orders of magnitude across shapes while their product stays constant, is a genuine and non-obvious feature of his own model. The willingness to use his formula to rule against an anomaly-friendly result — siding with Luo et al.'s null result over Hayasaka and Takeuchi's reported weight loss — is a mark of intellectual honesty rare in this literature.

The central difficulty is that equation (16) is not new. In the weak field, gravitational time dilation is dt/dt₀ ≈ 1 + Φ/c², so c² dt/dr = dΦ/dr = g identically. The relation g = c² × (gradient of the time-dilation factor) is the standard first-order consequence of the metric, and it is derivable in one line from the Γ(a) that Solomon assumes as his input. The 1,190 numerical integrations therefore do not discover the formula so much as confirm that the assumed Γ(a) was used consistently: the model was constructed from 1/√(1 − 2GM/rc²), and GM/r² comes back out. The same holds for the electromagnetic case, where equations (17)–(19) are used to define v, ω and r before the time-dilation method is applied to them; the agreement to 10⁻¹⁴ in Table 5 is a consistency check of the algebra, not an independent test. And the claim that mass has been eliminated is only true of the expression. The time-dilation field τ must still be produced by something, and in every worked example it is produced by GM/rc². The paper never shows how to generate τ without a mass, which is precisely what the technological argument requires.

Several internal inconsistencies in the paper's own numbers should be recorded. In Table 1 the gravitational-acceleration column does not follow from the mass and radius columns printed beside it: with M = 3.59×10²³ kg and r = 2.44×10⁶ m, Mercury's g is 4.02 m/s², not the 3.70 listed; Mars gives 3.82, not 3.71; Pluto 0.606, not 0.72; the Sun 280.4, not 274.98. Tables 2 and 4 use the correct values (4.0235, 3.8205, 0.6055, 280.302), so Table 1 is the outlier. Table 1 also lists Earth's escape velocity as 11,187 m/s but its "Lorentz-Fitzgerald equivalent velocity" as 1,187 m/s — a dropped digit — while still reporting a velocity error of −0.0000080 %, so the error column cannot have been computed from the printed figure. The quoted mass of Mercury, 3.59×10²³ kg, is also about 9 % above the accepted 3.30×10²³ kg.

More significantly, the discrepancy between g = τc² and GM/r² in Table 4 is 0.0500 % for every body — Pluto, Mars, Mercury, Uranus, Venus, Earth, Saturn, Neptune, Jupiter all at 0.05002 %, the Sun at 0.04938 % — and the fitted k_c misses c² by 0.049 %. A residual that is constant across four orders of magnitude in g and independent of the body is the signature of a systematic offset in the numerical scheme (a factor, a slice-thickness convention, or the finite 10⁻¹¹ m particle size), not of a physical correction. Solomon reports the number but does not diagnose it, and a formula claimed to be exact should reproduce c² exactly.

The Laithwaite application is the weakest link. A disc reduced from 9.8 to 9.7 m/s² would be a 1 % weight change readily measurable on a laboratory balance, and a device "almost weightless… if not rise" at 5,000 rpm and 7 rpm would be trivially demonstrable; no such measurement has been reported. Laithwaite's Royal Institution demonstration is generally understood in terms of ordinary rigid-body dynamics and the experimenter's own muscular effort, and the gyroscopic weight-loss claim of Hayasaka and Takeuchi failed replication not only in Luo et al. (Phys. Rev. D 65, 042005, 2002) but also in the earlier Nitschke and Wilmarth work. Solomon's own formula (33) predicts weight gain as readily as loss, which makes the absence of any confirmed effect harder rather than easier to explain. Any real anomaly of this size would also conflict with the torsion-balance tests of the Equivalence Principle, which constrain composition- and configuration-dependent deviations from universal free fall at the 10⁻¹³ level.

Two smaller asserted steps deserve flagging. The remark that "Nature has figured out how to get around Heisenberg's Uncertainty principle at the macro level" is an aside with nothing behind it — the Uncertainty Principle concerns conjugate observables, not the precision with which a modeller specifies a particle diameter. And the EMFP derivation treats the electron's own electric field as a rigid sphere of radius dr co-rotating with the particle; the classical self-field of a point charge has no such radius, and the resulting equation (32) is stated to disagree with standard theory for particle sizes above 10⁻⁴ m, which Solomon attributes to "modeling error" or to not assuming a point-like charge without resolving which.

Read as an engineering feasibility argument, the paper's honest content is narrower than its abstract: it shows that gravitational acceleration can be expressed as a local gradient of time dilation, which is true and already known, and it does not show that such a gradient can be produced without mass, which is what a propulsion technology would require.