Unification of Space-Time-Matter-Energy
| Scientific Paper | |
|---|---|
| Title | Unification of Space-Time-Matter-Energy |
| Read in full | Link to paper |
| Author(s) | Tolga Yarman, Garret Sobczyk |
| Keywords | Binding energy, Elementary forces, Energy conservation, Equivalence principle, Minkowski |
| Published | 2008 |
| Volume | 7 |
| Number | 2 |
| No. of pages | 14 |
| Pages | 255-268 |
Read the full paper here
Abstract
Appl. Comput. Math. 7(2) (2008), pp. 255-268. A complete description of space-time, matter and energy is given in Einstein's special theory of relativity. We derive explicit equations of motion for two falling bodies, based upon the principle that each body must subtract the mass-equivalent for any change in its kinetic energy that is incurred during the fall. We find that there are no singularities and consequently no blackholes.
Overview
Published in Applied and Computational Mathematics under the heading POLEMICS — with an editorial note conceding that "the subject of this paper does not correspond the profile of our journal" but that the Board judged the topic important enough to publish and hoped it would provoke discussion — this paper by Garret Sobczyk and Tolga Yarman attempts to obtain gravitational dynamics from special relativity alone. Its single organising postulate is a strict local conservation of energy: every body must pay for any change in its kinetic energy by subtracting the equivalent from its own mass. Nothing is added to relativity; a constraint is imposed on it, and the authors argue that under that constraint "the theory takes on a new elegance and universality".
The consequences are dramatic and are the reason the paper appears on this wiki. Because a falling body continually converts its own rest-mass into motion, the smaller of two gravitating bodies self-annihilates at exactly the radius where the Newtonian singularity would otherwise appear. There is therefore no singularity even for point masses, no critical mass, no Schwarzschild radius, and consequently no black holes. The authors' departure from the mainstream is doubly explicit: they reject the equivalence principle as the foundation of gravitation, arguing that "there is a clear asymmetry between an accelerating elevator and a gravitational field", and they replace the invariant rest-mass with an instantaneous rest-mass that varies with the body's position in a non-homogeneous field. In place of black holes they predict very dark but not black objects at galactic centres, and they suggest that total collapse into the regime of the other elementary forces could produce annihilation with a large energy burst.
The argument
The case against the equivalence principle
The introduction assembles the authors' grounds for setting the equivalence principle aside. They cite claims that equating acceleration with gravitation breaks conservation of energy and momentum and undermines mass–energy equivalence; that Yilmaz and Logunov have each proposed exponential metrics yielding no black holes in place of the Schwarzschild metric; that Yilmaz went so far as to ask whether Newton's apple would fall in Einstein's theory; and Santilli's theorems of inconsistency in general relativity. They also report a Mössbauer rotor experiment by Kholmetskii's group at Belarusian State University, which they say confirmed a prediction of Yarman's — that a nuclear clock on a rotor edge is affected not only by its tangential velocity but by its binding to the acceleration field, with an overall time dilation "practically twice as much as predicted classically". Their own argument for asymmetry is physical rather than formal: an observer must accelerate to catch an accelerating elevator but must decelerate to land on a celestial body; the first yields a mass increase, the second a mass decrease.
Instantaneous rest-mass in spacetime algebra
The formalism is Hestenes's spacetime algebra, in which an inertial frame is a constant Minkowski timelike unit vector u. The rest-mass m∞ is defined as the mass a body would have infinitely far from all other bodies and forces. Expanding the product of the energy–momentum vector with an observer's instantaneous frame v gives the familiar relative energy Ev = γvm∞c2.
The paper's decisive move is to insist that the work of boosting be taken out of the body itself, leaving the residual rest-mass m = m∞/γv. Two consequences follow immediately. First, m → 0 as |v| → c: "the energy content of each material body is exactly the energy which would be required to accelerate the body to the speed of light", so that a perfectly efficient photon drive would exhaust the last of the body's mass precisely as it reached c. An elementary particle, on this view, "annihilates if and only if it reaches the speed of light". Second, the energy actually consumed, ΔE2 = m∞(1 − 1/γv)c2, has a Taylor expansion beginning with the classical (1/2)m∞v2 but tracking it far more closely over the sub-luminal range than the usual (γv − 1)m∞c2, the two differing by a factor of 2 at |v| = c.
The resulting conservation law p(τ)·u = m∞c2 holds along the whole worldline, and the relative force reduces to F = γvm∞a, recognisable as the relativistic form of Newton's second law, together with the mass-loss rate dm/dt = −F·v/c2. The authors are explicit about the ontology this entails: a particle's field "carries only information about the location of that elementary particle, but does not magically transfer energy across spacetime", so each particle funds its own motion from its own mass, "guided by the information supplied by the four elementary forces of Nature".
Binding energy and the two-body solution
For two isolated bodies starting at rest at infinite separation, the quantity Ebi(r) = m∞ic2(1 − 1/γi) — named by Sobczyk "Tolga's binding energies" — measures the work done on each body by whichever of the four forces acts, and the instantaneous rest-masses are simply mi = m∞i − Ebi/c2. Writing m∞2 = s m∞1 with s ≥ 1 and imposing conservation of total energy and of linear momentum yields a closed pair of equations for the velocity and for the fraction f1 of the binding energy paid by the lighter body, both solved explicitly.
From these the authors compute the critical binding energy at which the lighter mass vanishes: Ebc = c2m∞1(s + 1 − √(s2 − 1)). At that point m1 = 0, v12 = c2, and the survivor has mass m∞1√(s2 − 1). Crucially, none of this yet assumes a particular force law.
Specialising to Newtonian gravity turns the problem into a Riccati-like differential equation for Eb(r), solved numerically in general but in closed form in two important cases. For a test body falling toward an effectively immovable mass, the solution is the strikingly simple exponential m1(r) = m∞1exp(−Gm∞2/c2r), with velocity |v1(r)| = c(1 − exp(−Gm∞2/c2r)) — an exponential rather than a Schwarzschild form, consistent with the Yilmaz and Logunov metrics they cited. For two equal masses the binding energy is Eb(r) = 2c2G m∞12/(Gm∞1 + 2c2r), and both bodies reach c and self-annihilate together. A restricted three-body case — two equal masses on a line either side of a third — also admits a closed-form solution, and the authors note that for m∞3 much larger than the others it reproduces the one-body-falling formulas.
Why there are no black holes
The conclusion the paper draws from these solutions is that "whenever a less massive object approaches a very massive object, depending upon initial conditions, it will necessarily self-annihilate or coalesce. There cannot be any critical mass which would define the Schwarzschild radius of a black hole." The mass that would have to be concentrated inside a horizon has instead been spent on the fall. In the discussion the authors soften the r = 0 limit for macroscopic bodies by appealing to quantum mechanics — an object's dimensions "effectively become that of space itself at r = 0", so r = 0 is never reached — while allowing genuine self-annihilation for elementary particles where the other forces operate. In place of black holes they predict objects made nearly invisible by extreme redshift at galactic centres, and they speculate that a total collapse with a large energy burst might account for the reported billion-light-year void, "the biggest expanse of nothing".
Yarman's related work is cited for the claims that, once the quantum-mechanical stretching of unit lengths in a gravitational field is taken into account, the perihelion precession of Mercury and the deflection of light both follow; that the approach lends itself to quantization of gravitation; and that it yields the de Broglie relation directly. The authors state plainly that their results agree with general relativity "only ... up to a third order Taylor expansion", and close by conceding that the value of any theory rests "not upon the conviction or authority of its authors, but on the fruits of its predictions".
Assessment
This is a carefully executed piece of work and considerably more disciplined than most anti-black-hole arguments. The postulate is stated once, in one sentence, and everything else is derived from it; the geometric-algebra formulation is clean and the derivations are checkable; the authors solve the general two-body case before committing to a force law, so the reader can see exactly which results depend on gravity being inverse-square and which do not; and they give closed-form solutions in the limits where they exist and honest numerics where they do not. The observation that ΔE2 = m∞(1 − 1/γ)c2 approximates the Newtonian kinetic energy over a far wider range than the standard expression is a genuinely interesting formal fact, and the emergence of an exponential mass profile — matching the exponential metrics of Yilmaz and Logunov, reached by a completely different route — is a real point in the paper's favour. The authors are also candid where it matters: they concede agreement with general relativity only to third order, and they end by inviting empirical judgement rather than claiming victory.
The difficulties begin with the central postulate, which is asserted rather than argued. That a body must fund its kinetic energy from its own rest-mass is not a consequence of special relativity; in ordinary relativistic mechanics the work is done by the external field and rest-mass is invariant. The authors' motivation — that a field should carry "only information ... but does not magically transfer energy across spacetime" — is a metaphysical preference, and the paper does not confront the classical evidence that fields carry energy and momentum, from radiation pressure to the momentum balance in electromagnetic scattering. Related to this, the treatment is quasi-static and radial throughout: every solution has the bodies starting at rest at infinity and falling directly together. No orbit is computed, which is a serious gap, since orbital dynamics is where the theory would most easily be distinguished from the Newtonian and Einsteinian ones — and the paper's own claim to recover Mercury's perihelion is not demonstrated here but referred to Yarman's earlier work.
Against established measurement the exposed claims are the ones about compact objects. The prediction that no horizon exists and that galactic centres host "very dark objects, but no black holes" must now be set against evidence the paper could not have addressed in 2008 but which any current reader will apply: the orbits of stars around Sgr A*, which constrain the central mass to lie within a radius incompatible with any ordinary object; the ringdown waveforms of merging compact binaries observed by LIGO and Virgo; and the Event Horizon Telescope images of M87* and Sgr A*, whose shadow diameters match the Kerr prediction. Equally, the mechanism itself predicts something not observed: if infalling matter converts its rest-mass into motion and self-annihilates on reaching c, accreting systems should exhibit a characteristic total-conversion energy release, whereas measured accretion efficiencies onto compact objects are a few to a few tens of percent of mc2. The suggestion that such annihilation explains the "biggest expanse of nothing" is offered as speculation and rests on a single newspaper citation, with no estimate of the energy or mass involved. Finally, the appeal to quantum mechanics to prevent macroscopic bodies from reaching r = 0 is invoked in a sentence and does no calculational work, so the paper's headline result — the disappearance of the singularity — is secured classically by self-annihilation but then quietly withdrawn for exactly the macroscopic bodies whose collapse is at issue. The framework is internally consistent and mathematically competent; what it lacks is any prediction that has been tested and confirmed independently of the assumption that produced it.