The Graviton Equations
| Scientific Paper | |
|---|---|
| Title | The Graviton Equations |
| Read in full | Link to paper |
| Author(s) | Bob de Hilster |
| Keywords | graviton |
| Published | 2008 |
| No. of pages | 12 |
Read the full paper here
Abstract
Isaac Newton's Law of Universal Gravitation has no mechanism although Newton did propose that gravity could be caused by a particle called the fluxion. Einstein proposed that gravity was caused by the bending of space-time but also gives no mechanism. This paper proposes that there is a particle or a quantum called the graviton that is not an infinitely divisible field as proposed by the equations of Newton and Einstein, but a finite force which has effects that have a defined limit. Four postulates are proposed that lead to a step by step development of the graviton equation. The curves for gravitational acceleration are compared using measured data, Newton's equations, and the graviton equation.
Overview
This is a working paper, written up from a wiki work area and dated 1 April 2008, in which Bob de Hilster builds a push-gravity calculation from four postulates and then compares its output numerically against Newton's law and against measured values of g. The idea is the classical shadow mechanism associated with Le Sage: an isotropic flux of particles pushes on every body from all directions, so an isolated body feels no net force; but a second body absorbs some of the flux, casts a shadow, and the two are pushed together. What is new here is not the mechanism but the attempt to compute with it — to write down an explicit summation, feed in real Earth–Moon data, and see what curve comes out.
The paper's stance towards its own result is unusually modest and worth quoting, because it sets the terms on which it should be judged. "The equations are not gravity. They only attempt to calculate the force between two objects. The apple dropping on Newton is an example of gravity. Gravity does its own thing and ignores any equation." And in closing: "if the theory of gravity as a particle proves false it will fade away. If it proves true, it would be an exciting day for physics… Theory without experimentation, is just talk." The companion papers "An Equation for G" and "The Graviton Experiment", both presented at the NPA in 2008, are named as the places where the constant and the experimental test are pursued.
The argument
The four postulates
- The graviton — a particle with mass and velocity that "imposes itself on an object from all directions". Its source "is not known".
- Absorption rate — a small fraction of gravitons are absorbed in passing through an object, the rest pass on; the absorption rate is the percentage absorbed per unit distance travelled, "initially… assumed to be linear".
- Pushing force — an absorbed graviton, having mass and velocity, imparts a push in its direction of travel.
- Absorption constant — the absorption rate is proportional to the density of the material: Abe = Abc × De, with Abc "different but similar to G".
Building the equation
The graviton flux at the Moon is discretised into paths: 180 planes, 360 angles per plane, giving 64,800 paths, with one graviton per path per unit of time. Along a path that crosses the Earth for a chord length Ze, the surviving fraction is (1 − Ze·Abe); if the path misses the Earth, Ze is set to zero. Summing the surviving gravitons over angles with a sin(a) factor for the vertical component, and over planes with a cos factor for the plane's tilt, gives the number imposed on the Moon, Ngi. The force is then Ngi multiplied by the Moon's own absorption Zm·Abm and by Fg, the force one graviton delivers. Substituting the density relation gives equation 9, the paper's "basic building block", into which different geometries, densities and chord lengths can be inserted, and in which cos(a) replaces sin(a) if a horizontal component is wanted.
De Hilster checks the units: for (1 − Ze·Abc·De) to be dimensionless, Abc must carry units of m2/kg. He also notes the choice between particle and continuum descriptions, observing that the double summation over a finite set of paths is what makes his account a particle theory, and that it would become a continuum theory if the summations became integrals — though that "would have to account for the discontinuities of the physical configuration."
The Earth–Moon force curve
The calculation uses mass 7.475×1022 kg and radius 1738 km for the Moon, 5.98×1024 kg for the Earth, and a mean separation of 384,400 km. The absorption constant is set to 10−25, explicitly "an arbitrary number, but it is used as a base for other calculations". The Moon is then placed at eight distances obtained by repeated halving and doubling of the true separation — 24,025; 48,050; 96,100; 192,200; 384,400; 768,800; 1,537,600; 3,075,200 km — of which only the fifth is physical, the rest being included so the curves can be compared.
Because both Abc and the graviton count are arbitrary, Fg is chosen to make the first point coincide with the Newtonian value (9.94×1059 for this configuration). De Hilster is explicit that "since the magnitude has been forced, the only comparison to be made is in the shape of the curve", and he therefore compares slopes rather than values. His finding: "The slope is 31% high for the short distances and gets closer at long distances."
Gravitational acceleration
The second test replaces the Moon with a lead ball at altitude and compares against a table of measured g taken from Shortley and Williams, Elements of Physics — values at 40° latitude running from 9.80171 m/s2 at sea level to 9.70296 m/s2 at 32 km. Again Fg is forced at the first point (7.23×1035 here), and again De Hilster reports the outcome against himself: "Since Fg is forced, the value of the graviton g is invalid. Further, the slope of the graviton curve is greater than the measured values." Extending to twenty points, out to 131,072,000 m, he concludes only that "these curves show that the curves are similar, but different", and refers the question of why they are similar to the companion paper.
Extensions and geometry
Equation 9 computes the force at one point only — the Moon's centre of gravity. De Hilster notes that a matching equation 12, with Earth and Moon terms interchanged, gives the force at the Earth's centre, and that in principle the process "could go on for every point in the earth and the moon". He also observes that the same pair of gravitons produces a force on both bodies, pushing them toward each other, and flags this as work still to be done. The final section derives the chord lengths Z geometrically: plane #80 is the plane 80° from horizontal cutting the Moon's centre of gravity, angles step in 1° increments from 0.5°, and equations 13–20 give the distances.
Assessment
What is genuinely attractive here is the discipline. De Hilster does not claim a result he has not got. He states which numbers are arbitrary, states that forcing Fg invalidates the magnitude, reports the sign and size of his own disagreement with measurement, and ends by asking for an experiment rather than for assent. That is a rarer posture than the underlying idea, and it makes the paper easy to evaluate. The mechanical arithmetic is also sound where it can be checked: 180 × 360 is indeed 64,800; the dimensional analysis is right, since metres × (m2/kg) × (kg/m3) is dimensionless; the eight distances are exact halvings and doublings of 384,400 km; 131,072,000 m is 81,445 miles as stated, and is exactly 500 m doubled eighteen times, so the twenty-point set is internally consistent; and the quoted g table is itself consistent with an inverse-square fall-off from 6,368 km to within about one part in 104 at every altitude listed.
Two data entries do not survive checking. The Earth's radius is given as "6.3677E+6 Km", which is the value in metres carrying the wrong unit — a slip, though the calculations evidently used the correct figure. The Moon's mass is given as 7.475×1022 kg against the accepted 7.342×1022 kg, about 1.8% high; since the Moon's mass enters the force linearly, that is a systematic error comparable to some of the differences being examined.
The substantive difficulty is the one the paper itself reports. A 31% slope error at short range is not a refinement issue: it means the model is not reproducing an inverse-square law, and the inverse-square law is verified to extraordinary precision by planetary ephemerides — the residual anomaly in the precession of Mercury's perihelion is 43 arcseconds per century out of 5,600, roughly one part in 107. Whether the 31% is a defect of the physical model or of the 64,800-path discretisation and the single-point treatment of extended bodies is exactly what the paper cannot yet say, and it says so. Until that is settled the comparison remains a comparison of shapes with a free multiplicative constant fitted at one end.
Beyond the numerics, the postulates carry the classical costs of shadow gravity, none of which is addressed here. Linear attenuation, (1 − Z·Abc·D), is only the first term of an exponential and goes negative for large optical depth; more importantly, any absorption model makes the attraction sub-linear in mass for large bodies, because the far side of a body is shadowed by the near side. That is a prediction of measurable shielding and of a departure from the strict proportionality of gravitational to inertial mass — a proportionality confirmed by Eötvös-type torsion balance experiments to about one part in 1013, and by lunar laser ranging for bodies as different as the Earth and Moon themselves. Since Postulate 4 makes absorption depend on density, the model predicts composition-dependent gravity, which is precisely what those experiments exclude. Postulate 3 also has a thermodynamic price: gravitons that are absorbed deposit their kinetic energy, and Poincaré's calculation of exactly this mechanism found that the Earth would be heated to incandescence; the same momentum transfer produces a velocity-dependent drag that would decay planetary orbits. And to avoid an observable aberration in the direction of the force, the flux would need to travel very much faster than light — whereas binary pulsar timing constrains the propagation speed of gravity to be close to c. The paper does not offer a flux, a graviton mass or a velocity, so none of these can be evaluated against it; its Fg of 9.94×1059 is a fitting parameter, not a physical prediction.
One small historical correction is worth recording. The abstract states that Newton "did propose that gravity could be caused by a particle called the fluxion". A fluxion in Newton is a time derivative in his calculus, not a gravitational particle; his speculations about a mechanical cause of gravity were framed in terms of a subtle aether, in the letters to Bentley and in Query 21 of the Opticks. The shadow mechanism the paper actually develops is Le Sage's, from 1748, and the paper would be strengthened by naming that lineage and engaging with the objections raised against it over the following two and a half centuries.