Gravitation - A New Theory. Further Kinetics of Gravitational Motion
| Scientific Paper | |
|---|---|
| Title | Gravitation - A New Theory. Further Kinetics of Gravitational Motion |
| Read in full | Link to paper |
| Author(s) | Peter G Bass |
| Keywords | Gravitation, Relativity, Special, General, Minkowski, Space-Time, Temporal, Acceleration, Potential, Time Dilatation. |
| Published | 2004 |
| Journal | Apeiron |
| Volume | 11 |
| Number | 1 |
| No. of pages | 25 |
Read the full paper here
Abstract
Following the presentation of a new theory of gravitation in [1], this short paper discusses three further aspects concerning kinetics within the gravitational Space-Time Domain D1. They are (i) the spatial-temporal distribution of the internally generated accelerative force, (ii) the relationship between gravitational and inertial mass, and (iii) kinetic energy.
Overview
This is the third instalment of Peter G. Bass's programme in Apeiron, following his classical reconstruction of special relativity and his first gravitation paper of 2003. Bass works with two "Relativistic Space-Time Domains": D0, pseudo-Euclidean space-time, in which motion is produced by an externally applied force, and D1, the gravitational domain, in which motion is produced by what he calls the Acceleration Potential acting on the energy mass of a body and generating a force inside it. The whole paper turns on that distinction between a force applied at a surface and a force generated within the fabric of a body.
Two conclusions follow that put the paper squarely against the standard account. First, gravitational and inertial mass are not equal in D1: gravitational mass is simply energy mass, while inertial mass is "an artefact of only the artificially accelerated state" and, in purely gravitational motion, "does not exist." Second, a freely falling body accumulates no kinetic energy at all — its total energy is constant throughout the fall, so the accelerative force "is not therefore the cause of the motion but the consequence of it." Bass regards gravitational free fall as the natural state of existence in D1, exactly as being at rest is the natural state in D0, and appends a critique of the standard proof of the equality of the two masses as given in Fock's The Theory of Space, Time and Gravitation.
The argument
Four reaction terms of the gravitational force
Taking the rectilinear case (ω = 0) of his earlier equation of motion, Bass writes the internally generated gravitational force Fg as a sum of four reaction terms — two spatial and two temporal — and resolves them against the Existence Velocity Vector of the gravitating mass, whose magnitude is cu. The component along that vector, Fe, relates the vector to the time rate of change of energy mass; the transverse component, Fa, relates the energy mass to the time rate of change of the vector. The four terms are then read as:
- mσ̈ — the reaction of the energy mass to spatial acceleration;
- a temporal term, the reaction of the energy mass to temporal deceleration;
- a temporal term generated by the combination of energy-mass rate and temporal velocity, opposing (ii);
- ṁσ̇ — a further spatial reaction from the combination of energy-mass rate and spatial velocity.
This is deliberately parallel to the decomposition Bass had obtained in D0 for a force applied from outside; setting u = 1 in Appendix B reduces all four to their special-relativistic counterparts, and setting c = ∞ reduces them to the classical ones. The one substantive difference, he stresses, is how the motion is driven: externally in D0, internally by the Acceleration Potential in D1.
Gravitational mass is energy mass
Using σ̈ = −c2u du/dσ + (2σ̇2/u)du/dσ together with σ̇ = cu(1 − u02/u2)1/2 and the energy mass m = m0u02/u2, Bass evaluates the two spatial reaction terms separately and adds them. The cross terms cancel and the sum collapses to
- Fg = mσ̈ + ṁσ̇ = −mc2u(du/dσ),
the expression already derived in the first gravitation paper. He also shows explicitly that the two temporal terms are equal in magnitude and opposite in sign, confirming "the purely spatial nature of gravitation." The conclusion drawn is that what the literature calls gravitational mass is exactly the energy mass m = m0u02/u2, with no inertial-mass factor appearing anywhere.
Inertial mass in D1
To obtain a like-for-like comparison inside one domain, Bass applies an artificial force F opposing the gravitational motion. Solving for σ̈ gives the gravitational acceleration plus an extra term F(1 − σ̇2/c2u2)/m, and the mass multiplying that force — the true inertial mass of D1 — is
- ma = m0u02/{u2[1 − σ̇2/(c2u2)]3/2}.
This has the same form as the D0 inertial mass but carries the extra factor u0/u, which Bass attributes to motion through the varying temporal rate generated by the source. Since ma ≠ m, "a hitherto basic belief of gravitational theory, the equivalence of gravitational and inertial mass, does not apply in D1." Two limiting cases are extracted: setting σ̇ = σ̈ = 0 recovers the weight of a supported mass, and setting σ̈ = 0 with negligible F gives a maximum free-fall velocity σ̇ = cu/√2 — far below the terminal velocity of the domain — because the positive acceleration from motion through the reducing temporal rate eventually balances the negative acceleration of the Acceleration Potential.
Kinetic energy does not exist in free fall
Because dE/dσ = 0 throughout gravitational motion, the total energy of a falling body never changes; "in purely gravitationally accelerated motion, kinetic energy does not exist." Under an externally applied force, however, kinetic energy behaves as in D0. Substituting σ̇ = cu sin φ, Bass integrates Fdσ by inspection to obtain Ek = m0c2u0 sec φ + k, and with Ek = 0 at σ̇ = 0 this becomes
- Ek = mc2u2 − m0c2u02,
the difference between the total energy at the point of observation and at the point where motion began — formally identical to D0, to which it reduces when u = u0 = 1. Rearranged, m = Ek/(c2u2) + m0u02/u2, so the energy mass splits into a gravitational part and a stored-kinetic-energy part.
The apparent dissipation when a falling body is stopped is then explained in two ways: the arresting mechanism itself dissipates energy, and, more importantly, decelerating the body against its natural state extracts energy from it by reducing its mass. Carrying the calculation through gives Ek = m0c2u0(u0/u − 1), which is negative since u/u0 < 1, and a mass loss Δm = m0(u0/u)(1 − u0/u), absorbed as mechanical deformation in the arresting and gravitating bodies.
Critique of the equality of the two masses
Appendix A reviews the proof in Fock. It sets F = mgr grad U for the gravitational force and F = minw from Newton's second law, equates them, and invokes Galileo's law to conclude min = mgr. Bass's objection is to the unstated assumption that Newton's second law applies to gravitationally induced motion in the same way it applies to artificially produced motion. A gravitational force acts on "each and every atom simultaneously and equally," stressing the atomic bonds only through the small differences of position within the field; an applied force acts over a contact area and is transmitted through those bonds. The mechanical experiments that established the second law used the latter kind of force, and — however precise — "would not have been able to distinguish between the various mass values," rest, energy and inertial. He concludes that no proof of the applicability of F = m'w to gravitational motion has since been supplied, so the equality rests on an untested step.
Assessment
The paper is carefully and consistently executed on its own terms. Every result is checked against two limits — special-relativistic (u = 1) and classical (c = ∞) — in an explicit appendix, and each reduces correctly, which is more internal discipline than most alternative gravitation papers display. The physical instinct behind the whole argument is also a real one and worth stating: there is a genuine difference between a body accelerated by contact forces, which is internally stressed, and a body in free fall, which is not, and that difference is exactly what makes free fall locally indistinguishable from rest. The observation that a freely falling body exchanges no energy with anything, and that the "kinetic energy" of a falling object only becomes manifest when something stops it, is not wrong; it is close to the standard statement that gravitational potential and kinetic energy are not separately meaningful in general relativity, only their sum along a geodesic. Bass's critique of the Fock derivation is likewise fair as a matter of logic — that argument does assume rather than prove the applicability of the second law across the two cases.
The difficulties are substantial. The paper is not self-contained: the Acceleration Potential, the Existence Velocity Vector, the function u and the whole D0/D1 apparatus are all defined in the two earlier papers, so the derivations here can be followed algebraically but not assessed physically without them. More seriously, the central conclusion is stated as a consequence of the formalism rather than tested against anything. Bass never says what measurement would distinguish his inequality of gravitational and inertial mass from the standard equality, and this is the one place where an answer is indispensable, because the equality is among the most precisely tested propositions in physics. Torsion-balance experiments of the Eötvös type — the Eöt-Wash results, and the space-based MICROSCOPE mission — constrain the differential acceleration of test bodies of different composition to a few parts in 1015, and lunar laser ranging constrains the equality for bodies with substantial gravitational self-energy at comparable precision. Bass's ma differs from m by the factor u0/u together with a (1 − σ̇2/c2u2)3/2, so the inequality he asserts is not a formal relabelling but a physical difference, and no estimate of its size in any laboratory or solar-system situation is given. Since u → u0 and σ̇ → 0 in the weak-field, low-velocity regime where the Eötvös tests operate, the discrepancy may well be unobservably small — but the paper does not say so, and demonstrating that would be the necessary first step.
The claim of a maximum free-fall velocity cu/√2 is a genuine, in-principle testable prediction and the most interesting number in the paper, yet it is asserted in a single paragraph and never compared with anything — not with the measured velocities of infalling matter around compact objects, not with the escape-velocity relations of the standard theory. Similarly, the mass loss Δm = m0(u0/u)(1 − u0/u) when a falling body is arrested is offered without an order-of-magnitude estimate for any real case; a reader cannot tell whether it is a laboratory-scale effect or one suppressed by c2. Finally, Appendix A shows only that one textbook derivation is logically incomplete. That is a legitimate point against a proof, but it is not evidence against the proposition, which is supported empirically rather than by that derivation.
Read as one step in a longer construction — Bass closes by announcing that the mechanism by which a source generates its Acceleration Potential will follow in the next paper — the work is coherent, honest about its dependence on the earlier instalments, and free of the arithmetic and dimensional slips common in this literature. Read as a challenge to the equivalence of gravitational and inertial mass, it does not yet engage the measurements that would decide the question.