Maximum Force Derived from Basic Physical Principles
| Scientific Paper | |
|---|---|
| Title | Maximum Force Derived from Basic Physical Principles |
| Read in full | Link to paper |
| Author(s) | Richard Benish |
| Keywords | Maximum Force, Gravity, Equivalence Principle, Singularity-Free, Laboratory Experiment |
| Published | 2009 |
| No. of pages | 18 |
Read the full paper here
Abstract
Based on the work of Jacobson [1] and Gibbons, [2] Schiller [3] has shown not only that a maximum force follows from general relativity, but that general relativity can be derived from the principle of maximum force. In the present paper an alternative derivation of maximum force is given. Inspired by the equivalence principle, the approach is based on a modification of the well known special relativity equation for the velocity acquired from uniform proper acceleration. Though in Schiller's derivation the existence of gravitational horizons plays a key role, in the present derivation this is not the case. In fact, though the kinematic equation that we start with does exhibit a horizon, it is not carried over to its gravitational counterpart. A few of the geometrical consequences and physical implications of this result are discussed.
Overview
Christoph Schiller had shown that a maximum force FMAX = c4/4G follows from general relativity, and that general relativity can be recovered from it — the maximum force playing "the same role for general relativity as the maximum speed plays for special relativity." Richard Benish's paper, submitted to the International Journal of Theoretical Physics in 2009, reaches exactly the same number by a different route: a modification of the special-relativistic equation for velocity under uniform proper acceleration, combined with the Equivalence Principle and the inverse-square law.
The point of the exercise is not the number, which is not in dispute, but what comes with it. In Schiller's derivation gravitational horizons are essential. In Benish's they are impossible. His version of the metric adds 2GM/rc2 to unity where the Schwarzschild solution subtracts it, and the resulting geometry is well-behaved everywhere: no horizons, therefore no Black Hole singularities — what are now called black holes would be better described as "dim compact massive objects." The paper's real ambition, however, is its last move: it argues that the difference from general relativity shows up most cleanly inside matter, where nothing has ever been measured, and it proposes a concrete laboratory test.
The argument
Hyperbolic motion and its horizon
Benish starts from the standard result for a body under uniform proper acceleration a with respect to an inertial system I:
- v = at / √(1 + a2t2/c2)
As t → ∞, v → c. On a spacetime diagram the track is a hyperbola whose asymptote is both a light cone and a horizon: the accelerating observer B will never receive signals emitted by the stationary observer A after the time c/a. Benish is explicit that this horizon is an elementary consequence of Special Relativity, and that his gravitational adaptation will not inherit it.
Equivalence, and where it stops working
The frequency shift between the leading and trailing ends of an accelerating body of extent h, fB2 ≈ fB1(1 − ah/c2), is the usual bridge to gravitational clock-rate variation. Benish notes carefully that the analogy is imperfect. In the kinematic case the shift is a genuine Doppler effect; in a stationary field "the observed frequency difference isn't due to a spectral shift caused by a change in motion between emitter and receiver," but to a difference in the rates of two clocks neither of which moves while the signal is in flight. A second non-equivalence: an accelerating system would see light from ahead grow steadily hotter and light from behind colder, which does not happen on a gravitating body. "The spirit of the equivalence principle is thus to deduce this curvature and to not worry too much about the differences."
The substitution
The modification is a single replacement. What makes B's case unlike life on a planet is that the speed limit is approached with increasing time. So Benish replaces (at) by √(2GM/r), giving a "stationary surface velocity"
- VS = √(2GM/r) / √(1 + 2GM/rc2) = √[2GM / (r + 2GM/c2)]
He is candid that this is a substitution rather than a derivation: "If not clearly analogous, this is at least mathematically permissible." Its physical reading is "(at least approximately) the relative speed of the surface at r, with respect to a geodesic trajectory 'from infinity'." Two features recommend it: VS < c for all physical M and r, and it yields the maximum force.
Squaring gives a denominator rγ = r + 2GM/c2, so a body possesses an additional spatial extent by virtue of its mass. Benish then treats (1 + 2GM/rc2) as playing the role that (1 − 2GM/rc2)-1 plays in Schwarzschild, and shows the two differ by 4G2M2/r2c4(1 − 2GM/rc2) — negligible in weak fields, which is why the two schemes agree wherever anything has been measured.
Maximum force
Taking the surface acceleration as gS = GM/rγ2 and letting r → 0 gives gMAX = c4/4GM, and hence
- FMAX = c4/4G = 3.0256 × 1043 N
Benish plots gMAX against masses from the electron to a galaxy cluster and stresses that because real bodies have finite radii these maxima are never attained. Multiplying (6) by any mass M′ always yields less than FMAX, because the numerator's increase is accompanied by an increase of at least 2GM′/c2 in the denominator.
Singularity-free geometry
Since M must vanish when r does, any physical M/r leaves VS finite and below c. A gravitational horizon can never form. Building a body up by adding shells of constant density produces a family of embedding parabolas whose tangent lengths RPT = R√(1 + 2GM/Rc2) trace a hyperbola of asymptotic slope 2 — what Benish calls "hyperbolic stationary motion," increasing not with time but with M/R. The case R = 2, M = 1 that would be a Schwarzschild black hole is here "just one unexceptional case in a continuous series."
The interior, Tangherlini, and the rotation analogy
The observationally decisive difference is inside matter. In the Schwarzschild interior solution the spatial coefficient returns to unity at the centre (space is flat there) while the inverse temporal coefficient keeps increasing, so the central clock is the slowest in the field. Benish emphasizes that this has never been checked, directly or indirectly.
He notes Tangherlini's 1962 postulational treatment, built from the same two starting assumptions, in which clocks inside a spherical shell run at maximum rate and an object dropped into the shell would never enter the cavity — "extremely unlikely to be physically true," but proof that exterior agreement with Schwarzschild does not fix interior behaviour. Benish's own proposal is a third option: inside matter the inverse temporal coefficient decreases along with the spatial coefficient, both returning to unity at the centre.
His justification is an analogy with rotation, which guided Einstein almost as much as the equivalence principle. On a rotating body four effects vanish at the axis by symmetry and grow with radius: inward acceleration, tangential velocity, rod shortening and clock slowing. Einstein subsumed the effects of motion under curvature; Benish proposes the reverse — acknowledge rotation as absolute motion, then, on finding the same effects near a gravitating body, attribute them to the same cause. This yields three explicit propositions: (1) gravitational spacetime curvature is caused by stationary motion; (2) accelerometer readings and clock-rate variation establish that motion; (3) therefore gravitating bodies do not induce geodesic motion through their centres. The crucial distinction is that rotation is motion through space while gravitational stationary motion is motion of space — omnidirectional and volumetric (a surrounding array of accelerometers measures the product 4πGM), which he says requires a fourth spatial dimension, the familiar three being a "compactified" state and the fourth an "expandification" of them.
The proposed experiment
The prediction is sharp: a test object dropped into a hole through a massive body will not pass the centre. Benish proposes a modified Cavendish balance in which the arm is free to swing through the centre of the large source masses. The obstacle is the suspension: conventional Cavendish balances have a restoring force and a short range of motion. A magnetic suspension of the kind Faller and Koldewyn used in 1976 to measure G would, he argues, be adaptable. He also sketches interior expressions for stationary acceleration and velocity for a uniform sphere, and notes that at very high density the interior acceleration rises and then falls — a manifestation of remaining below the maximum force.
He is scrupulous about the stakes: "If the results of the modified Cavendish experiment should confirm the standard prediction, then our derivation of the maximum force would be proven to be an inconsequential coincidence. The novel conceptions of matter, space, time, and gravitation presented in this paper should then all be discarded."
Assessment
What is genuinely attractive here is the structure of the claim. Benish has identified a regime — the interior of matter, the region below the surface of a source mass — where general relativity makes a confident prediction that has in fact never been tested, and he has proposed a table-top experiment that would settle it. That combination is rare in dissident gravitation, and the falsifiability is stated by the author himself in the strongest possible terms. The arrival at Schiller's c4/4G from an independent starting point is a real result, whatever one makes of the surrounding interpretation, and the explicit computation of the deviation from Schwarzschild shows honestly why the scheme is not already excluded by solar-system data.
The difficulties are equally clear. The central step is asserted, not derived. Replacing (at) by √(2GM/r) is a formal substitution licensed by nothing stronger than "mathematically permissible," and the subsequent decision to read (1 + 2GM/rc2) as the metric coefficient — reversing the sign of the Schwarzschild term rather than deriving a field equation — has no dynamical basis. There is no action, no field equation, and no demonstration that the geometry is a solution of anything; the paper offers a modified metric coefficient and consequences drawn from it. Nor is the fourth spatial dimension developed: it is introduced because omnidirectional stationary motion cannot be motion through pre-existing three-space without material bodies "rapidly disintegrating," and then left as a requirement rather than a construction.
Two conflicts with measurement deserve naming. First, the model predicts no horizons at all, so the shadow imaged by the Event Horizon Telescope around M87* and Sgr A*, and the ringdown waveforms of the Gravitational Waves events detected by LIGO — both of which match Kerr predictions closely — would have to be reproduced by "dim compact massive objects" in this scheme. The paper, written in 2009, does not address either, and no mechanism in it obviously supplies them. Second, the non-oscillation prediction implies that a body falling below a surface eventually slows and does not reach the centre, which conflicts with the ordinary Newtonian interior solution that underlies, among other things, the standard interpretation of terrestrial and stellar interior structure. Benish's answer is that this has never been directly tested, which is true of the free-fall trajectory through a centre but not of the interior gravitational field inferred from seismology and stellar models.
The deeper implications the paper reaches for — inertia as resistance to changing linear motion within an omnidirectional stationary motion, and time's arrow as a consequence of "space's arrow and matter's arrow" — are stated as conditional consequences of the experiment succeeding, which is the right way to state them, but they rest on the same unsupported substitution as everything else.
On its own terms the paper is honest and self-limiting. It does not claim to have overthrown general relativity; it claims to have found the same maximum force by another route, noticed that the route implies no horizons, and identified the one place where the two schemes diverge measurably. The right response to it is the one Benish asks for: perform the modified Cavendish experiment.