The Schwarzschild Solution and its Implications for Gravitational Waves

Scientific Paper
Title	The Schwarzschild Solution and its Implications for Gravitational Waves
Read in full	Link to paper
Author(s)	Stephen John Crothers
Keywords	gravity, Schwarzschild Solution, Gravitational Waves, black hole
Published	2009
No. of pages	28

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Abstract

The so-called 'Schwarzschild solution' is not Schwarzschild's solution, but a corruption, due to David Hilbert (December 1916), of the Schwarzschild/Droste solution, wherein m is allegedly the mass of the source of a gravitational field and the quantity r is alleged to be able to go down to zero (although no valid proof of this claim has ever been advanced), so that there are two alleged 'singularities', one at r = 2m and another at r = 0. It is routinely asserted that r = 2m is a 'coordinate' or 'removable' singularity which denotes the so-called 'Schwarzschild radius' (event horizon) and that the 'physical' singularity is at r = 0. The quantity r in the so-called 'Schwarzschild solution' has never been rightly identified by the physicists, who, although proposing many and varied concepts for what r therein denotes, effectively treat it as a radial distance from the claimed source of the gravitational field at the origin of coordinates. The consequence of this is that the intrinsic geometry of the metric manifold has been violated. It is easily proven that the said quantity r is in fact the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section of the 'Schwarzschild solution' and so does not in itself define any distance whatsoever in that manifold. With the correct identification of the associated Gaussian curvature it is also easily proven that there is only one singularity associated with all Schwarzschild metrics, of which there is an infinite number that are equivalent. Thus, the standard removal of the singularity at r = 2m is erroneous, as the alleged singularity at r = 0 does not exist, very simply demonstrated herein. This has major implications for the localisation of gravitational energy, i.e. gravitational waves. Schwarzschild's actual solution forbids black holes !

Overview

Presented at the 16th Natural Philosophy Alliance Conference at the University of Connecticut in May 2009, this is Stephen Crothers' fullest statement of a case he has pressed for two decades: that the black hole is not a prediction of General Relativity at all but the product of a misreading of a coordinate. The paper is not an attack on differential geometry, nor an attempt to replace Einstein's field equations with a rival theory. It works entirely inside the standard formalism, and its central claim is a claim about identification: that the symbol r in the line-element universally taught as "the Schwarzschild solution" is the inverse square root of the Gaussian curvature of a spherically symmetric geodesic surface in the spatial section, and therefore is not a radius, not a distance, and cannot be allowed to "go down to zero."

From that single point Crothers builds outward. If r is a curvature parameter rather than a distance, then the interval 0 ≤ r < 2m is not a region of the manifold to be explored with better coordinates but a stretch of parameter values with no geometric referent; the "removal" of the singularity at r = 2m by Kruskal–Szekeres or Eddington–Finkelstein coordinates is then not a repair but a violation of the geometry fixed by the form of the line-element. And if there are no black holes, there are no black-hole mergers, and therefore — combined with a separate argument that Einstein's gravitational pseudo-tensor is mathematically meaningless — no gravitational waves for LIGO to find. The paper was written in November 2008, six and a half years before the LIGO detection of GW150914.

The argument

Schwarzschild's solution versus Hilbert's

Crothers opens by setting the textbook metric

ds² = (1 − 2m/r)dt² − (1 − 2m/r)⁻¹dr² − r²(dθ² + sin²θ dφ²)

against what Karl Schwarzschild actually published in January 1916, in which the same form appears but with R = (r³ + α³)^1/3, 0 < r < ∞, and α an undetermined constant. In Schwarzschild's own solution the singularity sits at r = 0, where R = α, so R can never fall below α. Crothers lists what Schwarzschild did not do: he did not set α = 2m; "he did not breathe a single word about the bizarre object that is called a black hole"; he did not name a Schwarzschild radius or an event horizon. The modern form he attributes to David Hilbert's December 1916 recasting, calling it "a corruption" of the Schwarzschild/Droste solution.

He then catalogues, with citations to roughly two dozen textbooks, the many incompatible names given to r in the literature — "a distance," "the radius," "the radius of a 2-sphere," "the coordinate radius," "the areal radius," "the reduced circumference," even "a gauge choice" — and argues that this variety is itself a symptom of the problem.

What r actually is

The mathematical core invokes Gauss's Theorema Egregium: the Gaussian curvature of a surface is a bending invariant, determined by the first fundamental form alone and independent of any embedding space. For the two-dimensional surface ds² = R_c²(dθ² + sin²θ dφ²), computing K = R₁₂₁₂/g from the Christoffel symbols gives K = 1/R_c². Hence R_c is the radius of Gaussian curvature — and nothing else. In flat Euclidean 3-space the radius of Gaussian curvature happens to coincide with the radial distance, which Crothers argues is precisely why the confusion arose: physicists carried a Euclidean coincidence into a non-Euclidean manifold.

He is scathing about the standard fallbacks. Since C_p = 2πr and A_p = 4πr² hold with r a constant, neither the "reduced circumference" r = C_p/2π nor the "areal radius" r = √(A_p/4π) identifies r geometrically; they are "platitudinous expressions containing the constant r," and they are in fact "one and the same." The quantity that does measure distance is the proper radius R_p, obtained by integrating √Ψ dR_c, and for the Schwarzschild form this gives R_p = √(R_c(R_c − α)) + α ln[(√R_c + √(R_c − α))/√α].

The infinite family of equivalent metrics

Since r can be replaced by any analytic R_c(r) without disturbing spherical symmetry or violating R_μν = 0, satisfying the field equations is "a necessary but insufficient condition." Crothers imposes the boundary condition R_p(r_o) = 0 and obtains the general admissible form

R_c(r) = (|r − r_o|ⁿ + αⁿ)^1/n = 1/√K(r)

with r_o and n entirely arbitrary. This single expression reproduces the historical solutions as special cases: r_o = 0, n = 1 gives Brillouin's; r_o = 0, n = 3 gives Schwarzschild's own; r_o = α, n = 1 gives Droste's, which is the textbook metric restricted to α < r < ∞. All are asymptotically Minkowski; all have exactly one singularity, at r_o, where the scalar invariants R_p(r_o) = 0, R_c(r_o) = α and K(r_o) = α⁻² hold for every r_o and every n. The Kretschmann scalar likewise evaluates to f = 12α²K³, giving f(r_o) = 12/α⁴ — finite, and hence, on Crothers' reading, not an independent curvature invariant capable of adjudicating what is a "true" singularity. Doughty's radial geodesic acceleration diverges at r_o for every member of the family. The Kerr–Newman geometry is given the same treatment.

Ancillary arguments

Several independent lines are then piled on. Because the construction starts from e^λ > 0 and e^β > 0, neither can change sign, so the signature cannot flip from (+,−,−,−) to (−,+,−,−); the standard claim that t and r exchange roles inside 2m therefore turns a static problem into a time-dependent metric — "a non-static solution to a static problem: contra hyp." Infinite densities are argued to be forbidden by Special Relativity, since D = m₀/[L₀³(1 − v²/c²)] diverges only as v → c; hence point-mass singularities are forbidden in General Relativity too. Escape velocity is shown at length to be an irreducibly two-body concept, so a one-body solution to R_μν = 0 cannot have one — and the Michell–Laplace dark body, which does have an escape velocity and from which objects can depart, is therefore not a black hole. Because the field equations are non-linear, the principle of superposition fails, and there is neither an exact two-body solution nor an existence theorem; so binaries, collisions and mergers of black holes are "invalid concepts," and the Oppenheimer–Snyder collapse calculation is said to smuggle superposition in from the outset.

The gravitational-wave conclusion rests on two further claims. Crothers repeats T. Levi-Civita's 1917 proof that contracting Einstein's pseudo-tensor and applying Euler's theorem to the quadratic homogeneous L yields a first-order intrinsic differential invariant of the metric — an object Ricci and Levi-Civita had proved in 1900 does not exist. He also quotes Eddington at length on the coordinate-dependence of wave propagation ("we can 'propagate' coordinate-changes with the speed of thought") and Dirac on the non-localisability of gravitational energy, and cites Weyl's 1944 result to argue that linearisation of the field equations is inadmissible. Since R_μν = 0 describes a spacetime containing no matter, in which neither the Equivalence Principle nor Special Relativity can manifest, he concludes the field equations must instead read G_μν/κ + T_μν = 0, so that total gravitational energy is always identically zero and no localisation — hence no wave — is possible.

Assessment

The paper's most valuable contribution is historical and pedagogical, and it is largely correct on that ground. Schwarzschild's 1916 paper really does use R = (r³ + α³)^1/3, really does not mention event horizons, and really is not the metric that bears his name; Droste's independent derivation and Hilbert's role are genuine and under-acknowledged history. Crothers is also right that r in the standard line-element is not a proper radial distance, that C/2π and √(A/4π) are the same quantity, and that the areal radius is tied to the Gaussian curvature of the 2-sphere. Sharpening the loose textbook talk of "the radius" is a real service, and the generalised family R_c(r) = (|r − r_o|ⁿ + αⁿ)^1/n is a neat way to display the coordinate freedom explicitly.

The inference from that to "no black holes" is where the argument does not hold. Everyone in the field agrees r is not a proper distance; that is exactly why the Kretschmann scalar, and not r, is used to distinguish a curvature singularity from a coordinate one. Crothers' reply — that the Kretschmann scalar is "not an independent curvature invariant" because it is a function of the metric components — proves too much, since every curvature invariant is a function of the metric components; what matters is that f = 48G²M²/c⁴r⁶ is finite at r = 2M and unbounded as r → 0, and that is a statement about a scalar, invariant under precisely the coordinate changes the paper objects to. His own calculation gives f = 12α²/R_c⁶, which is the same function; what differs is only the stipulated range of R_c, imposed by his choice of boundary condition R_p(r_o) = 0 rather than derived. That boundary condition — that proper radial distance vanish at R_c = α — is the load-bearing assumption of the whole paper, and it is asserted, not shown to be required. Assume it and the horizon becomes the origin by construction; the conclusion is in the premise.

The signature argument fails for a related reason: g₀₀ and g₁₁ changing sign together preserves the metric signature (the count of positive and negative eigenvalues) even as the coordinate labels t and r swap timelike and spacelike character. That is a well-understood feature of the extension, not a contradiction. The special-relativistic prohibition of infinite density conflates the density of a body accelerated to c with the density at a curvature singularity, where the classical description is expected to break down and no material object is being accelerated to c in any frame. And the claim that escape velocity is two-body-only is answered by noting that 2GM/c² in the metric is not derived from a Newtonian escape argument but fixed by matching to the weak-field limit, a matching Crothers rejects but does not replace with any alternative determination of α.

Most decisively, the paper stakes clear empirical claims that have since been tested. "The LIGO project and its international counterparts have not detected gravitational waves ... They are destined to detect nothing" was falsified on 14 September 2015 by GW150914, whose waveform matched a numerical-relativity binary black hole merger template through inspiral, merger and ringdown, was seen by two independent detectors 7 ms apart, and has since been followed by dozens more events and by the neutron-star merger GW170817 with an electromagnetic counterpart. The assertion that no event horizon has been imaged is answered by the 2019 and 2022 Event Horizon Telescope images of M87* and Sgr A*, and the assertion that no irresistible collapse has been observed sits against the measured orbits of stars around Sgr A*, which confine roughly 4 × 10⁶ solar masses within a region smaller than Mercury's orbit. Crothers' claim that black-hole binaries are impossible because there is no exact two-body solution also underestimates what numerical relativity does: it solves the full non-linear field equations for two-body initial data without superposition, and the waveforms it produces are what LIGO matched.

None of this touches the historical case, which stands on its own. But the paper's argument runs from a stipulated boundary condition to a sweeping empirical prediction, and it is the prediction that the last decade of observation has tested.