Stellar Collapse
| Scientific Paper | |
|---|---|
| Title | Stellar Collapse |
| Read in full | Link to paper |
| Author(s) | Richard A Waldron |
| Keywords | stellar collapse, ballistic theory of light |
| Published | 1990 |
| Journal | Apeiron |
| Volume | 1 |
| Number | 7 |
| No. of pages | 12 |
| Pages | 4-7 |
Read the full paper here
Abstract
The phenomenon of stellar collapse is considered from the viewpoint of the neo-Newtonian ballistic theory of light. The restrictions of the special theory of relativity are thus removed. The theory predicts that a collapsing star will expand again, and continue to alternately expand and collapse at a rate depending on the mass and greatest radius of the star. On each cycle material will be lost, including photons which appear to distant observers as the emissions of a pulsar. It is concluded that pulsars are oscillating stars, that these eventually "evaporate" away completely, and that there is no such object as a black hole.
Overview
Waldron's paper, published in Apeiron No. 7 (1990), continues an argument he began in "Do Black Holes Exist?" (Speculations in Science and Technology, 1983). His original objection was that the gravitational energy released by a collapsing burnt-out star would blow the star apart before it reached the Schwarzschild radius. Correspondence with "a number of leading cosmologists" prompted a rethink, and this paper is the result: instead of an explosion, he now argues for an oscillation. The star collapses, turbulence and rotation halt the infall, it re-expands to nearly its original radius, and the cycle repeats, shedding matter and radiation each time. What a distant observer sees is a pulsar. There is no black hole.
The framework is what Waldron calls the "neo-Newtonian ballistic theory of light" — an emission theory in which Newton's three laws are "inviolable", Einstein's invariance postulate is rejected as "incompatible with the principle of relativity", and light consists of material particles. The decisive consequence for this paper is that with special relativity abandoned there is no speed limit, so the escape-velocity argument that defines a black hole loses its force: matter and light leaving the surface faster than c are permitted. The mainstream picture Waldron opposes is not the collapse itself — he agrees a burnt-out star above roughly 1.4 solar masses will collapse inside its Schwarzschild radius — but everything that is said to happen afterwards.
The argument
The ballistic theory in brief
Waldron sketches the background theory before applying it. Coulomb's law is replaced by a law carrying a factor in the relative velocity of the two charges, reducing to Coulomb's law when that velocity vanishes; he emphasises that this "does not contradict Coulomb; it goes into a region about which Coulomb was silent". An analogous modification is made to Newton's law of gravitation. No reason is offered for the forms of the force laws: "they have been arrived at simply by requiring that they lead to correct calculations of experimental results."
The photon is a material particle ejected from its source at c relative to that source, thereafter moving in straight lines under Newtonian mechanics. It has mass, is overall neutral but carries equal positive and negative charge giving it a dipole moment, and it rotates — the rotating dipole supplying the phase that accounts for wave-like behaviour. Because the emission model cannot produce the cosmological redshift by Doppler shift, Waldron attributes the redshift to photon decay, giving "a stable, non-expanding universe, without beginning or end" in which the Hubble constant measures the photon decay rate rather than an expansion rate.
The collapse calculation
For the collapse itself the treatment is purely Newtonian. Taking the density ρ(r,t) as uniform in r for simplicity, the equation of motion for a shell at radius r2 reduces to r2″ = −Gm2/r22. Integrating gives the collapse time from the initial radius R0. Writing RS = 2GM/c2 for the Schwarzschild radius, n = R0/RS and M = μ solar masses, the time to fall from R0 to RS becomes, for large μ,
t0 ≈ 9.854×10−6 × (π/2) × n3/2 × μ seconds
The first integral gives the infall speed, R' = −c√(1/n)√(n − 1) in Waldron's notation, so that on reaching RS the surface is moving at very nearly c. Table I tabulates t0 for representative cases — among them Betelgeuse (μ = 15, n = 1.7×106, t0 = 5.15×105 s), the Sun (μ = 1, n = 2.36×105, t0 = 1774 s) and Van Maanen's star (μ = 0.14, n = 2.9×104, t0 = 10.7 s) — and the range of t0 runs from years down to well under a millisecond.
He also notes a floor: the radius cannot fall to zero because the constituent particles (he suggests neutrons) come effectively into contact. Denoting this minimum by Rc, collapse inside RS requires RS > Rc, satisfied for μ above a critical value "lying somewhere between about 1 and 10" depending on the assumed particle radius — a figure he presents as agreeing with the orthodox 1.4-solar-mass threshold.
Energy and temperature
A Newtonian calculation of the released binding energy gives W ≈ 0.3(1 − R/R0)Mc2, and using Knudsen's Maxwell–Boltzmann relation between root-mean-square speed and temperature, the corresponding temperature at the Schwarzschild radius is about 2.16×1012(1 − R/R0) K.
Why the collapse reverses
This is the paper's central physical claim, and it is argued qualitatively. Waldron attacks the orthodox conclusion that the collapse energy cannot be radiated away in time t0 by pointing out that the calculation "depends on the assumption that the radiation is electromagnetic, according to Stefan's law. No allowance is made for turbulence in the star, with the possibility of ejection of ordinary matter."
Three sources of turbulence are offered. The density is not really uniform, so collapse is not uniform. Particle collisions become far more frequent as the material is compressed. And conservation of angular momentum, applied to "reasonable values" for the angular momentum of an expanded star, gives an equatorial speed at R = RS in the range c/√10 to c√10 — a superluminal figure that the ballistic framework permits. Compounded with the infall velocity and some random motion, Waldron argues, the state at R = RS is nothing like uniform inward motion: "as much matter is moving outwards as inwards. At this point expansion will start."
Some material in the turbulent velocity distribution exceeds escape velocity and leaves, as ordinary matter or as photons. The remainder expands to nearly R0 and collapses again. The result is "a spherically symmetrically pulsating object, which on every contraction radiates a burst of material and electromagnetic energy" — observed at a distance as a pulsar, with period somewhat less than 2t0. Waldron regards this as an advantage over the orthodox lighthouse model, where the required surface irregularity "is problematical", and notes that the pulsar must eventually "evaporate" away entirely.
Assessment
The paper has real virtues of construction. The collapse arithmetic is straightforward Newtonian free-fall and it is correct: the constant 9.854×10−6 s is exactly RS/c for one solar mass (2953 m divided by c), the free-fall formula (π/2)n3/2RS/c follows properly from the equation of motion, and spot-checks of Table I reproduce the tabulated values (μ = 1, n = 2.36×105 gives 1775 s against the listed 1774 s; μ = 0.14, n = 2.9×104 gives 10.70 s against 10.7). The solar value of n is right, and the 2.16×1012 K figure reproduces correctly from 0.3c2 per neutron mass. The paper is also honest about the status of its central step, presenting the turbulence-driven bounce as a plausibility argument rather than a calculation.
That candour is also the difficulty. The bounce is the whole paper, and it is asserted, not derived. No estimate is given of how much kinetic energy the turbulence can convert into outward motion, nor of what fraction of the infalling mass must reverse, nor of whether the star can in fact return to "approximately the initial value R0" — which for the Sun would mean expanding from 3 km back to 700,000 km on every cycle, an oscillation of extraordinary violence with no energy accounting offered. The angular-momentum estimate, which is the only quantitative element in the reversal argument, has an unstated range of a factor of ten and depends on "reasonable values" that are never specified.
There is a direct internal contradiction between the Discussion and the Conclusion. The Discussion states that "the periodic time will increase as the mass falls"; the Conclusion states that "as it loses mass, the period decreases." The paper's own eqn (7) settles it: since n = R0/RS and RS ∝ μ, the period scales as R03/2μ−1/2 at fixed R0, so the Discussion is right and the Conclusion is wrong. A related slip: eqn (11) is written as a function of general R, but the coefficient 0.3Mc2 is the value the binding-energy expression takes specifically at R = RS, and the same restriction is inherited by the temperature formula. Neither error is fatal to the picture, but both are in the load-bearing arithmetic.
The confrontation with observation is where the model is weakest. Reproducing millisecond periods requires n ≈ 10, that is a "fully expanded" star only about 30 km across — not a star in any recognisable sense, and it is not explained how such an object arises from stellar evolution. The predicted emission is explicitly "spherically symmetrical", but pulsar radio emission is strongly beamed and strongly linearly polarised, with a position-angle sweep across the pulse that the rotating-vector model of a magnetised, obliquely rotating neutron star reproduces in detail; a breathing sphere gives no account of that sweep, nor of why most neutron stars are not seen as pulsars at all. Observed pulsar periods are also extraordinarily stable and slowly increasing, with timing residuals at the microsecond level over decades — hard to reconcile with a mechanism driven by turbulence and mass ejection. The Hulse–Taylor binary pulsar's orbital decay, matching the general-relativistic gravitational-wave prediction to better than a percent, and the gravitational-wave detections of black hole mergers with the ringdown spectra of horizons, both bear directly on the paper's conclusion, though they postdate it. Finally, the underlying emission theory carries the classic difficulty that Waldron does not address here: the constancy of the speed of light emitted from fast-moving sources, measured directly by Alväger and colleagues in 1964 from the decay of pions travelling at 0.99975c, and the absence of the ghost-image effects de Sitter's binary-star argument predicts. A reader sympathetic to the anti-black-hole conclusion still has to supply the physics that this paper leaves as a sketch.