This is a preliminary examination of some of Mr. Crothers' papers. It is by no way complete, but there are enough interesting errors in the works I've examined so far to make a few comments. I've concentrated on the paper “On Certain Conceptual Anomalies in Einstein's Theory of Relativity”, with some supplemental reading of references [1,2,3].
Crothers seems to take particular issue with the concept of black holes in general relativity. However, it appears most of his complaints are non-issues if it is impossible for an infalling observer or particle to actually cross the event horizon. I know of no astrophysical processes involving black holes that actually require infalling material to cross the event horizon, though I have seen this description used in press releases. If the singularities in the Schwarzschild metric at the center and at the event horizon are not reachable by an observer, then Crothers issues are largely moot. In some older literature, stars collapsing to the Schwarzschild radius were often called 'collapsed stars' which might be a more technically correct way to describe these objects. I have my own favorite 'black hole paradox' which demonstrates that the idea popularized in the media and science fiction of falling through the event horizon probably cannot occur.
Each section of "On Certain Conceptual Anomalies in Einstein's Theory of Relativity" states some conclusion backed by claims which are at best weak, and in a number of cases, scientifically wrong. Here's a few I found that can be described with a minimum of mathematics.
[4, Section 2]: “Misconception: that Ricci=0 fully describes the gravitational field”
I find several problems in this section, as Mr. Crothers attempts to justify this statement.
1) The issues attributed to the statement describing "the perceived source of the field in terms of its center of mass" is a red herring. Newtonian gravity has a similar issue in that the exterior field of a spherical shell of mass is identical to the field of a point mass at the center of the spherical shell, a location where there is no actual mass. So what is Mr. Crothers' point?
2) The statement that one needs two line elements to describe the metric for the interior and exterior of the mass is somewhat strange. The 'two' line elements cover two non-overlapping, but continuous ranges of the independent variable, r, and therefore the elements are considered to be a single function, the function being described as piecewise. This is standard knowledge, in which case, why does Crothers mention it at all. Perhaps he does not understand it?
3) Mr. Crothers makes a fair number of his claims based on the concept of an 'incompressible fluid' in relativity. However, an 'incompressible sphere of fluid' is not a valid concept in relativity. In terms of the equation of state, incompressible means
where \rho is the fluid density and P is the fluid pressure. This means it no amount of change in fluid pressure will change the density of the fluid. This could also be inverted to read
But this term is directly related to the speed of propagation of disturbances in the matter, AKA the speed of sound. The general expression for the speed of sound in a fluid, c_s (not to be confused with the speed of light), is
Therefore, it is trivial to see that a truly incompressible fluid must have an infinite speed of sound. This is in violation of the principle that there is a limiting speed for signal propagation in relativity.
The mathematics does not stop one from combining these contradicting concepts, but the results and conclusions from such an analysis cannot be trusted. I found the reference to an early paper where Schwarzschild explored this problem historically interesting, but the paper was largely ignored after the realization of this contradiction. Modern researchers use more realistic equations of state, derived from nuclear experiments or theory, when dealing with compact objects. There have been some examinations of ways to treat incompressibility in GR, but Mr. Crothers does not use this.
[4, Section 3]: “Misconception: that General Relativity permits point-masses”
This section seems to rely on the bizarre claim that things permitted in General Relativity do so in violation of Special Relativity. This is backwards and misses the point of the distinction.
1) Special relativity is 'special' because it is limited to the 'special 'case of observers in uniform relative motion. General Relativity is the more 'general' theory, permitting non-uniform motion (accelerating) observers. Special relativity is a subset of General Relativity. In general relativity, the metric can have the very general form:
where x_i represents generalized coordinates. Due to the symmetry of the metric tensor, these 16 functions reduce to 10 independent functions.
Special relativity is limited to the much simpler Minkowski metric,
which simply means that the g_mn terms are constants limited to values of 0, 1, and -1. All solutions in special relativity are solutions in general relativity.
This suggests that Mr. Crothers does not understand relativity at all. Any conclusions based off the notion that SR is a more fundamental theory than GR, is immediately suspect.
2) Point masses are used in Newtonian gravity and point charges are used in electromagnetism. This idealization is convenient for modeling systems where the range of motion is much larger than the dimensions of the objects. This assumption essentially says that tidal effects, due to a finite object size, are negligible.
In “A brief history of black holes”, Crothers makes a related bizarre claim:
Even the electron has spatial extent, according to experiment, and to quantum theory.Crothers gives no reference for this claim and it is simply wrong. I haven't done quantum electrodynamics (QED) since the late 1970s, but QED treats electrons as point particles. An experimental upper limit of about 1e-22 m has been placed on the electron's size from scattering experiments. This is far smaller than the classical electron radius of 2.8e-15 m obtained by equating the total electrostatic energy to the electron rest mass. References to the papers on this work can be found at Wikipedia: Electron-Fundamental Properties
But even mathematically, this makes no sense. Infinitesimals, a mathematical entity that is as small as it needs to be, down to a point, are the foundation of calculus which forms the mathematical underpinnings of modern physics & engineering. One technique, Green's Functions, builds the fields of extended objects by integrating (essentially summing) the fields of point sources, represented as a Dirac Delta Functions. Point sources are the foundation of much of mathematical physics.
[4, Section 7]: "Misconception: that "Schwarzschild's solution" is Schwarszshild's solution"
I found this historically interesting, and I'll probably examine that aspect of it further. However, it is irrelevant to the physics whether Schwarzschild said anything about black holes.
It is not clear whether Crothers totally denies the validity of relativity or is promoting an alternative interpretation or another theory entirely.
I found nothing in Crothers writing on the experimental implications of his claims beyond his claimed non-existence of black holes. Mr. Crothers needs to explain why we get experimental agreement in spite of his claimed flaws in relativity? There is plenty of experimental evidence of the validity of general relativity (pulsar timing, accretion disk doppler profiles, GPS) [7,8].
- S.J. Crothers. On the vacuum field of a sphere of incompressible fluid. Progress in Physics, 2:76–81, July 2005.
- S.J. Crothers. A brief history of black holes. Progress in Physics, 2:54–57, April 2006.
- Stephen J. Crothers. On line-elements and radii: A correction. Progress in Physics, 2:25–26, April 2007.
- Stephen J. Crothers. On certain conceptual anomalies in einstein’s theory of relativity. Progress in Physics, 1:52–57, January 2008.
- F. I. Cooperstock and R. S. Sarracino. General relativistic incompressibility. Nature, 264:529–531, December 1976. doi: 10.1038/264529a0.
- K. Thorne, R. Price, and D. MacDonald. “Black Holes: The Membrane Paradigm”. Yale University Press, 1986.
- Clifford M. Will. The confrontation between general relativity and experiment. Living Reviews in Relativity, 9, 3 2006.
- Scott Rebuttal. I. GPS & Relativity, April 3, 2009