How a Total Mass, Equivalent to the Gravitational Binding Energy Should be Dumped, from the Rest Masses of Two Bodies Falling into Each Other?
| Scientific Paper | |
|---|---|
| Title | How a Total Mass, Equivalent to the Gravitational Binding Energy Should be Dumped, from the Rest Masses of Two Bodies Falling into Each Other? |
| Read in full | Link to paper |
| Author(s) | Tolga Yarman |
| Keywords | energy, mass, Special Theory of Relativity, gravitational field, conservation |
| Published | 2005 |
| No. of pages | 14 |
Read the full paper here
Abstract
Previously, based on the energy conservation law (in the broader sense of the concept of energy, thus embodying mass), as imposed by the special theory of relativity, we had proposed to alter the rest mass of a an object of mass minfinity (measured at a place free of gravitational field), gravitationally bound to a host celestial body of mass Minfinity (still measured at a place free of surrounding gravitational field), practically infinitely more massive as compared to the rest mass minfinity . Accordingly, minfinity was to be decreased, as much as the binding energy coming into play, in between the two masses of concern. This manipulation, together with a quantum mechanical theorem we had established (indicating that if the mass of a wave-like object is decreased by a given amount, its internal energy is concomitantly decreased as much), did essentially yield the end results of the general theory of relativity, though through a completely different set up than that of this latter theory.
Herein we remove the restriction that the host body is infinitely more massive that the test mass, and we end up with a generalized expression for the total binding energy, coming into play.
Were the original masses minfinity and Minfinity set free to fall onto each other, our approach yields the classical linear momentum conservation law.
Overview
This is a technical paper in Yarman's long programme of replacing the geometrical apparatus of general relativity with a bookkeeping rule about mass. The rule is simple to state: when a body of rest mass m0, measured far away where there is no gravitational field, becomes gravitationally bound to a host body of rest mass M0, its rest mass is physically reduced by an amount equal to the binding energy divided by c02. Yarman couples this to a quantum-mechanical theorem he claims to have established earlier — that if the mass of a "wave-like object" is decreased by a given amount, its internal energy is decreased in the same proportion — and reports that the combination reproduces "the end results of the general theory of relativity, though through a completely different set up".
Earlier instalments of the programme assumed the host to be effectively immovable, m << M, so that the whole mass deficit could be charged to the light body. The present paper removes that restriction. It asks how the mass equivalent of the binding energy should be shared between two comparable bodies falling into each other, derives a differential equation for the total binding energy, solves it approximately in closed form, writes the resulting position-dependent static masses, and then differentiates them to obtain equations of motion. The declared payoff is consistency: the scheme is shown to reproduce the classical conservation of linear momentum, which had been assumed rather than proved in the earlier work.
The argument
Sharing the mass deficit
Two bodies of rest mass m0 and M0 are released from rest at a great separation and accelerate toward each other, acquiring speeds v and V and kinetic energies Km and KM. In Yarman's picture the kinetic energy is not supplied by an external potential but is "fueled by the mass deficiencies coming into play", so that the overall mass of each body — rest mass reduced by its share of the binding energy, then multiplied by its Lorentz factor — stays equal to its original value at infinity. If linear momentum is conserved, M0V = m0v.
The fraction f of the total binding energy to be retrieved from the small body is defined as Km/(Km + KM). In the slow-motion limit this gives f = M0/(m0 + M0) and F = 1 − f = m0/(m0 + M0): the lighter body gives up the larger share, and in the limit m << M it gives up essentially all of it, recovering the earlier work. At relativistic speeds Yarman equates the two relativistic momenta and obtains an exact expression for f in which f itself depends on the binding energy — the sharing ratio is no longer a fixed number but a function of how deeply bound the pair already is.
The binding-energy differential equation
Because each body's instantaneous rest mass is its infinity value reduced by its share of EB, the Newtonian attraction integral must be written with the reduced masses inside it. Differentiating that integral yields a nonlinear ordinary differential equation for dEB/dr containing both EB and EB2. Yarman calls this "the rigorous differential equation that will furnish the static binding energy" and concedes it "does not seem an easy one". His attempt to keep an approximate EB2 term is reported as "a critical result" which, when substituted back, "one gets unacceptable solutions". He therefore drops the quadratic terms altogether, justifying this in a footnote: the bracketed correction factor is close to unity both when m << M and when m ≈ M, and even at the extreme case he identifies with "the classical singularity" it falls only to about 0.85.
The closed-form result
With the quadratic terms dropped the equation integrates immediately to the paper's central result: for two bodies that have fallen to a separation R,
EB(R) = (m0c02/α) [ 1 − exp( −α GM0 / Rc02 ) ]
with γ(r) = GM0/rc02 and α = (1 + m02/M02)/(1 + m0/M0). The only differences from the earlier fixed-host result are the appearance of α inside the exponent and of 1/α outside the bracket. Yarman notes that α equals unity both when m is negligible against M and when the two masses are equal; in between it dips to a minimum of about 0.828 at m/M = √2 − 1. The mass-ratio correction is therefore never large.
The exponential form is the signature of the whole programme. It agrees with the Newtonian binding energy to first order in γ, but it saturates rather than diverging: however small R becomes, EB approaches m0c02/α and no more. The condition γ = 1/2, which Yarman calls the classical singularity, is the Schwarzschild-radius condition r = 2GM/c2; there the binding energy has consumed roughly 0.6 of the rest energy, and the corresponding free-fall speed from infinity is about 0.8c0.
Static masses and equations of motion
Substituting f, F and EB(r) gives explicit expressions for the two position-dependent static masses m(r) and M(r). Yarman checks the two obvious limits: they reduce to the same quantity when m = M, and the heavy body's mass is left unaltered by the binding when m is negligible. Orbital motion is then constructed in two steps — bring the bodies quasi-statically from infinity to separation r, then assign them their orbital velocities — and the requirement that each body's total relativistic mass remain constant on the orbit yields two constants Dm and DM, both equal to unity for a free fall.
Momentum conservation
Differentiating the two constancy relations produces equations whose right-hand sides are the changes in kinetic energy, hence the work done by what Yarman calls the "effective gravitational forces". He argues that the effective force exerted by M on m and that exerted by m on M must be equal, so the rates of change of the two momenta are equal, and the total linear momentum is conserved. A footnote adds an important caveat: the centre of mass must be defined using the relativistic masses, not the static ones, for this to work. The vectorial equations of motion are written down but explicitly not solved: "this is not the task we propose to undertake, herein". A closing remark suggests the result "may point to a clue to the quest of inflation", since mass equivalent to binding energy would have had to be manufactured out of energy as the universe expanded after the Big Bang.
Assessment
The attractive feature of this approach is its economy. It uses only special relativity's mass–energy equivalence, Newton's inverse-square law between static masses, and one bookkeeping rule, and from these it obtains an exponential law that agrees with the standard weak-field results to first order while behaving quite differently in the strong field. The exponential never reaches zero at finite radius, so on Yarman's account there is no horizon and no black hole in the usual sense — the deep-field difference is a genuine, in-principle-testable consequence rather than a restatement. The internal-consistency goal of this particular paper is also legitimate and non-trivial: having replaced the potential by a mass deficit, one really does owe a demonstration that momentum is still conserved, and that demonstration is attempted rather than waved at.
The difficulties are substantial. The central formula is not a solution of the paper's own governing equation but of a truncation of it, and the truncation is worst exactly where the result is most interesting: the paper admits that the exact solution "remains yet to be pinned down, especially, nearby heavily dense objects", and that its one attempt to retain the quadratic term produced "unacceptable solutions". The strong-field claims therefore rest on an approximation that is only argued, in a footnote, to be tolerable at the very boundary of its validity.
The momentum proof is partly circular. Equation (1) is written with masses at infinity on the strength of the assertion that total masses stay constant, which is what the scheme is designed to make true; the later argument that the two effective forces are equal is Newton's third law imported into a dynamics that has already been modified. The claim, carried over from earlier work, that inertial mass exceeds gravitational mass by a factor of the squared Lorentz factor is the one place where the model departs from the Equivalence Principle in a way that ought to be measurable; no bound is offered, and no comparison is made with the tests that constrain it — Eötvös-type experiments, lunar laser ranging, or the orbital decay of binary pulsars. Since the departure is velocity-dependent it may well evade the laboratory tests, but this needs showing, and it is not shown.
More generally, the mass-deficit rule is asserted for celestial bodies on the authority of a theorem about "wave-like objects". Whether the internal energy of a planet is entitled to behave like that of a bound quantum system is precisely the question, and it is not addressed here. Nothing in the paper is compared with an observation: no perihelion advance, no light deflection, no gravitational redshift figure is computed for the two-body case, and the orbital equations that would allow such a comparison are set up and then abandoned. The concluding gesture toward inflation is a suggestion without a calculation. As a consistency exercise within Yarman's own framework the paper does what it sets out to do; as a case for that framework against general relativity it does not yet engage.