The Black Hole, the Big Bang: A Cosmology in Crisis
| Scientific Paper | |
|---|---|
| Title | The Black Hole, the Big Bang: A Cosmology in Crisis |
| Read in full | Link to paper |
| Author(s) | Stephen John Crothers |
| Keywords | black hole, Big Bang, Cosmology |
| Published | 2010 |
| No. of pages | 38 |
Read the full paper here
Abstract
The paper carries no separate abstract. Its opening statement of purpose, in the author's own words, reads:
It is often claimed that cosmology became a true scientific inquiry with the advent of the General Theory of Relativity. A few subsequent putative observations have been misconstrued in such a way as to support the prevailing Big Bang model by which the Universe is alleged to have burst into existence from an infinitely dense point-mass singularity. Yet it can be shown that the General Theory of Relativity and the Big Bang model are in conflict with well-established experimental facts.
Black holes are not without cosmological significance in view of the many claims routinely made for them, and so they are treated here in some detail. But the theory of black holes is riddled with contradictions and has no valid basis in observation. Nobody has ever found a black hole, even though claims for their discovery are now made on an almost daily basis. Nobody has ever found an infinitely dense point-mass singularity and nobody has ever found an event horizon, the tell-tale signatures of the black hole, and so nobody has ever found a black hole. In actuality, astrophysical scientists merely claim that there are phenomena observed about a region that they cannot see and so they illogically conclude that the unseen region must be a black hole, simply because they believe in black holes. In this way they can and do claim the presence of a black hole as they please. But that is not how science is properly done.
Overview
Crothers argues that the black hole and the Big Bang are not merely unconfirmed but internally incoherent — that they contradict the very theory they are said to follow from, and that the mathematics from which they were first extracted has been persistently misread. The paper is structured as a sequence of short, largely independent objections, each supported by extended verbatim quotation from standard textbooks and from Hawking, Chandrasekhar, Dirac, Misner–Thorne–Wheeler, Schutz, Carroll, Taylor and Wheeler, so that the position under attack is stated in its proponents' own words before being answered. A mathematical appendix develops the differential geometry from first principles.
The departure from the mainstream account is total rather than partial. Crothers does not propose an alternative black hole or a modified Big Bang; he argues that neither object is predicted by general relativity at all, that Rμν = 0 cannot describe a gravitational field, that Einstein's gravitational pseudo-tensor is mathematically meaningless, that gravitational waves therefore do not exist, and that the Cosmic Microwave Background is terrestrial in origin. He also locates the cause institutionally: these are "products of a peer review system that has gone awry, having all the characteristics of a closed academic club of mutual admiration and benefit into which new members are strictly by invitation only."
The argument
Infinite density is forbidden by special relativity
The first objection is elementary in form. Take two rest masses Mo and mo, the latter cuboid of side Xo. At relative speed v, Mo sees the mass increased by the factor (1 − v2/c2)−1/2 and the side along the motion contracted by (1 − v2/c2)1/2, so the observed density is
- D = mo / [Xo3(1 − v2/c2)]
As v → c, D → ∞. Since no material body can reach c, infinite density is unattainable and therefore forbidden. Because special relativity must hold in sufficiently small regions anywhere in the gravitational field, general relativity inherits the prohibition. Crothers presses the point against the standard escape clause: if relativity "breaks down" at the singularity, then it cannot simultaneously be asserted that the singularity is infinitely dense — "It can't be both, either at the same time or at different times, according to fancy." He adds that a point is a mathematical object and a mass a physical one: "One cannot go to a shop and buy a bag full of points, but one can buy a bag full of marbles."
Escape velocity and the Michell–Laplace dark body
Textbooks routinely define a black hole as an object whose escape velocity equals or exceeds c. Crothers takes the definition literally. If the escape velocity is c, light escapes by definition. If it exceeds c, light can still leave — rise, stop and fall back — and so there is always a class of observers who see it. Escape velocity never means that nothing can leave, only that nothing leaves permanently below that speed. So the two standard claims — escape velocity ≥ c, and nothing whatever can leave — are mutually contradictory, "nothing but a meaningless play on the words 'escape velocity'." He adds that escape velocity is intrinsically a two-body concept, while the black hole is derived from a one-body (in his view no-body) configuration.
The same distinction disposes of the frequent claim that John Michell's 1783 dark star anticipated the black hole. The Michell–Laplace body has an escape velocity; the black hole has none. Objects can leave the M–L body; nothing leaves a black hole. The M–L body requires no irresistible collapse, has no singularity, has no event horizon, can coexist and interact with other matter, and is always visible to some class of observers. "Thus, the M-L dark body does not possess the characteristics of the hypothesized black hole and so it is not a black hole."
r is not a radius: the Gaussian curvature argument
The mathematical core of the paper concerns the meaning of r in what is universally called the Schwarzschild solution,
- ds2 = (1 − 2Gm/c2r)c2dt2 − (1 − 2Gm/c2r)−1dr2 − r2(dθ2 + sin2θ dφ2)
Crothers first notes that this is not Schwarzschild's own solution but Hilbert's version of it; Schwarzschild wrote R = (r3 + α3)1/3, r > 0, and "there is no black hole in Schwarzschild's solution. Indeed, his solution precludes the black hole, and for this reason he never spoke of the black hole."
He then lists a dozen incompatible names the literature gives to r — distance, the radius, radius of a 2-sphere, coordinate radius, radial coordinate, areal radius, reduced circumference, a gauge choice — and argues that all are wrong. Applying the standard formula for Gaussian curvature K = R1212/g to the spherically symmetric geodesic surface ds2 = r2(dθ2 + sin2θ dφ2) gives K = 1/r2. Therefore r is the inverse square root of the Gaussian curvature of that surface — an intrinsic bending invariant, by Gauss's Theorema Egregium independent of any embedding — and not a radial geodesic distance in the manifold at all. Since r is constant on the surface while arc-length s and area Ap are functions of the angular coordinates, neither "areal radius" √(Ap/4π) nor "reduced circumference" Cp/2π identifies what r geometrically is; both are in fact the same quantity, the radius of Gaussian curvature.
On this reading the "Schwarzschild solution" is one member of an infinite equivalent family with Rc = (|r − ro|n + αn)1/n, with n and ro arbitrary. Choosing n = 3, ro = 0 recovers Schwarzschild's form; n = 1, ro = 0 gives Brillouin's; n = 1, ro = α gives Droste's, which Crothers holds is the correct form of Hilbert's metric. In every case Rc(ro) = α — "a scalar invariant" — and the proper radial distance Rp vanishes there. "In no case can a black hole result."
He adds a signature argument: the metric is constructed with fixed signature (+,−,−,−), but for 0 < r < 2m the signs of g00 and g11 reverse, giving (−,+,−,−) and interchanging the roles of t and r. Relabelling t = r* and r = t* makes every metric component a function of the timelike t*, so the interior is "a non-static solution to a static problem" and bears no relation to the original problem. Hence 0 < r < 2m is meaningless.
Matter smuggled in post hoc, and the failure of superposition
Einstein's equations couple geometry to matter through the energy-momentum tensor: Gμν = Rμν − ½gμνR = κTμν. Setting Tμν = 0 gives Rμν = 0, which by construction contains no matter. Crothers's charge is that mass is then reinserted by hand: the constant of integration m is identified with the source mass by "a contrived analogy with Newton's theory and his expression for escape velocity" — a two-body relation imported into what is claimed to be a one-body problem. "The astrophysics community removes all matter on the one hand by setting Rμν = 0 and then puts it back in at the end with the other hand by means of Newton's theory."
He sharpens this with de Sitter's empty universe. For the Schwarzschild–de Sitter line element, Tμν = 0 is said to permit a material cause (the post hoc m); for de Sitter's empty world, obtained by setting m = 0, the same Tμν = 0 is said to preclude one. "Tμν = 0 therefore includes and excludes material cause. This is not possible."
The related objection concerns multiplicity. Einstein's equations are non-linear, so the Principle of Superposition does not hold; each configuration of matter requires its own energy-momentum tensor and its own solution. But no exact solution for two or more masses exists, and no existence theorem has been proven for one. Therefore binaries, mergers, collisions and populations of black holes — all of which tacitly assume superposition — are, on his account, invalid concepts. He notes as an aside that general relativity "has to date been unable to account for the simple experimental fact that two fixed bodies will approach one another upon release." The Oppenheimer–Snyder collapse calculation is faulted on the same grounds: it assumes many mass elements a priori, in a theory where superposition fails, and attributes their collapse to a self-gravity for which no general-relativistic mechanism is given. "A Newtonian universe cannot 'collapse' into a non-Newtonian universe."
The pseudo-tensor, conservation of energy, and gravitational waves
Crothers concludes from the foregoing that Rμν = 0 cannot describe a gravitational field, so the field equations must instead read Gμν/κ + Tμν = 0. On this reading the total energy of the gravitational field is always zero, gravitational energy cannot be localised, and gravitational waves do not exist — while the usual conservation of energy and momentum is violated, which he takes as decisive against the theory.
He then attacks Einstein's gravitational pseudo-tensor, reproducing a proof he attributes to Levi-Civita (1917). Contracting the pseudo-tensor gives a linear invariant; since L is quadratic and homogeneous in the Christoffel symbols and hence in gμν,σ, Euler's theorem yields gμν,σ ∂L/∂gμν,σ = 2L, so the contraction reduces to a first-order intrinsic differential invariant depending only on the metric components and their first derivatives. Ricci-Curbastro and Levi-Civita proved in 1900 that no such invariants exist. "This is sufficient to render Einstein's pseudo-tensor entirely meaningless, and hence all arguments relying on it false." He adds Weyl's 1944 result that linearisation of the field equations implies a tensor which, save for the trivial zero case, does not exist — and quotes Eddington's own remark that the speed-of-light propagation of gravitational waves "is only true in a very conventional sense" and "follows a vicious circle."
The CMB and the Big Bang
The final section disputes the interpretation of the 2.7 K background. Crothers observes that Hubble and Humason proposed a redshift–distance relation, not a redshift–recessional-velocity relation, "reinterpreted as the latter in order to forge a correspondence with theory." He recites the chequered history of predicted background temperatures — Dicke 20 K, later 40 K, then 45 K; Peebles ~10 K then ~3 K; Gamow 50 K; Alpher and Herman 5 K then 28 K — against non-Big-Bang stellar-background estimates that were closer to the measured value: Nernst 0.75 K (1938), Eddington 3.2 K (1926), Regener 2.8 K, McKellar 2.3 K (1941). He then endorses Pierre-Marie Robitaille's position that COBE and WMAP have not in fact measured a cosmic signal, and that the emission originates in the Earth's oceans — water being a strong microwave absorber and emitter through hydrogen bonding — scattered by the atmosphere into an isotropic signal from an anisotropic source.
Assessment
The strongest and most durable part of this paper is the geometric argument about r. Crothers is correct that r in the Hilbert form of the metric is not a proper radial distance, that √(gθθ) is the inverse square root of the Gaussian curvature of the spherically symmetric surface, and that proper radial distance must be obtained by integrating (1 − 2m/rc)−1/2drc. That point is not in dispute among relativists — it is why the coordinate is called the areal radius rather than the radius — and Crothers is right that textbook language about it is often careless. His demonstration that the "areal radius" and the "reduced circumference" are the same object is likewise sound. The insistence that the Principle of Superposition fails in a non-linear theory is correct in itself, and his demand that the two-body problem be shown to be well-posed before binaries are discussed is a legitimate question rather than a rhetorical one. The paper is also unusually scrupulous in quoting its opponents at length and in the original, which lets a reader check the argument against the sources.
The difficulties are correspondingly serious, and several of the paper's central steps do not survive inspection.
The special-relativistic proof against infinite density does not establish what it claims. The formula D = m/[X3(1 − v2/c2)] shows only that kinematically observed density diverges in the limit v → c — a statement about a limit never attained by a body of finite rest mass. It says nothing about whether a rest-frame density can grow without bound under gravitational collapse, which is the actual claim at issue. The step from "no observer sees infinite density by boosting" to "infinite density is forbidden anywhere" is asserted, not derived.
The escape-velocity argument attacks a popularisation rather than the theory. Textbook prose does often say "escape velocity exceeds c", and that phrasing is genuinely misleading; but the technical definition of the horizon in general relativity is the null surface bounding the causal past of future null infinity, which involves no escape velocity at all. Crothers refutes the loose gloss and treats the result as a refutation of the object. His own quotations from Misner–Thorne–Wheeler and Chandrasekhar in fact state the causal-structure version, not the escape-velocity version.
The signature argument mistakes a coordinate pathology for a physical one. That g00 and g11 change sign at r = 2m in Schwarzschild coordinates is exactly the coordinate breakdown removed by the Eddington–Finkelstein and Kruskal–Szekeres charts, in which the metric is smooth across the surface, the metric signature (−,+,+,+) is preserved everywhere, and the curvature invariant RabcdRabcd = 48m2/r6 — a scalar, hence coordinate-independent — remains finite at r = 2m while diverging at r = 0. Crothers nowhere addresses the Kruskal extension or the curvature invariants, which is a substantial omission given that his conclusion is precisely that the interior region does not exist. His "infinite family" of solutions Rc = (|r − ro|n + αn)1/n is likewise a family of relabellings of one geometry: the different choices of n and ro are coordinate transformations, and his own observation that Rc(ro) = α invariantly is the demonstration of that, not a refutation of the horizon.
Three claims collide directly with measurement, and in each case the measurement postdates or is not engaged by the paper. First, gravitational waves: Crothers writes that "over a period of some 40 years and at great monetary expense, the international search for Einstein's gravitational waves has detected nothing," which was true in 2010 for direct detection but already false for the indirect evidence — the orbital decay of the binary pulsar PSR B1913+16 has tracked the quadrupole-radiation prediction to better than 0.3% since the 1980s, and the paper does not mention it. Since 2015 LIGO and Virgo have observed the coalescence of black-hole and neutron-star binaries directly, with waveforms matched to numerical-relativity templates and, in GW170817, an electromagnetic counterpart. Second, the two-body problem: numerical relativity solved the binary black-hole inspiral–merger–ringdown in 2005, without any appeal to superposition, which answers the specific challenge Crothers raises. Third, the horizon: the Event Horizon Telescope has since imaged the shadow of the compact object in M87 and in Sgr A*, at the angular size predicted for a horizon of the dynamically measured mass, while the Keck and VLT stellar-orbit programmes had already constrained the Galactic-Centre mass to ~4 × 106 solar masses within a region smaller than the orbit of S2's pericentre. None of these is an "indirect" inference of the kind Crothers dismisses as licensing "deep space unicorns"; they are measurements of a specific predicted geometry.
The Levi-Civita pseudo-tensor argument, though genuinely a real historical objection, proves less than Crothers takes it to prove. That the pseudo-tensor is not a tensor and that gravitational energy cannot be localised are standard results — Dirac's quotation, which Crothers reproduces approvingly, says exactly this — but they do not entail that the total energy is zero or that radiative energy loss is unphysical. The Bondi–Sachs mass at null infinity and the ADM mass at spatial infinity are coordinate-independent quantities, and it is the measured decrease of the former, not the pseudo-tensor, that underwrites the binary-pulsar result.
Finally, the CMB section relies on the temperature-prediction history as evidence of arbitrariness, but the modern case for a cosmological origin does not rest on the mean temperature at all. It rests on the spectrum, measured by COBE/FIRAS to be a blackbody to within 50 parts per million — the most precise blackbody ever measured — and on the acoustic peak structure of the anisotropies, whose positions and relative heights were predicted before measurement and fit a six-parameter model. An oceanic origin would have to reproduce both, and Robitaille's proposal does not attempt the peak spectrum.
What remains, after these deductions, is a paper that is right about a real sloppiness in how the Schwarzschild coordinate is described, right that superposition cannot be assumed in a non-linear theory, and right that "black hole discovered" headlines frequently outrun what has been measured. It does not, however, establish its central negative claims, because the arguments that would have to do the work — against the interior region, against radiative energy loss, against the cosmological interpretation of the microwave background — each stop short of the mathematics or the data that bear on them most directly.