Bringing The Black Hole Fallacy Into Focus
Sep 07, 2008
[Editor's Note: The following article is a copy of a letter sent by
Stephen J. Crothers to the author of an article which was published in
the journal Nature on
(subscription required) which offers no
proofs of the outrageous claims it makes regarding black holes. Anyone
who has any doubt about the validity of black hole claims should read
this as it offers mathematical proof
(in conjunction with referenced material) that black holes cannot exist.
Crothers has offered his math for scrutiny to many scientists and
theorists and to date none have offered a mathematical refutation
of his claims. Handwaving and derision (the usual response) does not a refutation make.
Only minor edits have been undertaken to assist flow and readability.]
Mr. Christopher S. Reynolds
Department of Astronomy,
University of Maryland, College Park,
Maryland 20742, USA.
I write in relation to your recent article in the journal Nature, "Bringing
black holes into focus". There are a number of anomalies in your
article to which I must draw your attention.
1. You remark;
It is believed that the centre of essentially every galaxy, including our
own, plays host to a supermassive black hole. In a small fraction of
galaxies, large quantities of gas rain down into these giant black holes,
causing the black hole to grow while releasing enough energy within the
central few light hours of the galaxy to outshine all of the galaxy's stars
thousands of times over.
And of an alleged "spinning black hole" you say;
Did it grow through the successive mergers of smaller black holes as
galaxies came crashing together? Or did it grow through the accretion of gas
and, if so, did it snack on gas hundreds of times or feast just once or
By what means can a black hole interact with other matter? You have not
even addressed this issue. First, the fundamental black hole (a so-called
"Schwarzschild" black hole) is allegedly obtained from a solution for Ric =
Rij = 0 (subscripts i,j = 0,1,2,3), which is a spacetime that, by
definition, contains no matter. So the alleged black hole can interact with
nothing because its associated spacetime is empty by definition - it
precludes the presence of any matter by virtue of Ric = 0. So there is no
matter outside the black hole by initial hypothesis
Indeed, there is no
matter present by initial hypothesis to even cause the gravitational field.
Furthermore, Einstein's theory of gravitation is non-linear and so the
'Principle of Superposition' does not apply. It does apply in Newton's
theory. These are fundamentally different theories, and so one cannot simply
insert lumps of mass or electromagnetic radiation into any spacetime of
Einstein by an
analogy with Newton's theory. Now the "supermassive black hole" allegedly
associated with Sgr A*, and the centres of galaxies in general, supposedly
interacts with matter external to it, including other black holes, as you
have claimed in your article, But this is impossible since the spacetime
associated with the alleged black hole is devoid of matter by hypothesis in
the writing of and the solution for Ric = 0. So the notion of black holes at
the centres of galaxies is nonsense. The addition of angular momentum makes
You also assert that not only does the "supermassive black hole" interact
with "large quantities of gas" but also suggest that it has grown by "the
successive mergers of smaller black holes". But Ric = 0 is also why alleged
black hole collisions, mergers and binaries are also nonsense. Each black
hole is obtained separately as a solution to Ric = 0. The one black hole
cannot therefore be in the spacetime of another black hole and mutually
interact in a mutual spacetime that by definition contains no matter!
Furthermore, before one can talk of black hole interactions it must first be
proven that even the two-body problem is well-defined within General
Relativity. This can be done in only two ways, (a) derivation of an exact
solution to Einstein's field equations for two bodies, or (b) proof of an
existence theorem by which it can be shown that Einstein's field equations
contain latent solutions for such a configuration of matter. There are
no known solutions to the field equations for the interaction of two or more
bodies, so option (a) has never been fulfilled, and no existence theorem has
ever been proven, so option (b) has never been fulfilled either. Moreover,
General Relativity has not been able to account for the simple experimental
fact that two fixed bodies will approach one another upon release. The post
hoc introduction of mass into the so-called "Schwarzschild solution" for Ric
= 0 by an association with the Newtonian gravitational potential is clearly
inadmissible. So all talk of black holes interacting is also nonsense.
there is no known solution for two or more bodies and no existence theorem
for such, by what solution to the field equations do you and the relevant
astrophysical scientists allege that black holes can interact with one
another or with other matter?
But We Have Images!
2. You say in the final paragraph of your article,
We have entered a new era, one in which we can now directly image
structure at the event horizon of a black hole. As the VLBI array capable of
millimetre resolution is expanded and its sensitivity increased, the world
at the edge of the black hole will literally come into focus.
The signatures of the alleged black hole are (a) an infinitely
dense-point-mass singularity and (b) an event horizon. Nobody has ever found
a black hole, despite the many claims for their discovery here and there and
everywhere, because nobody has ever found an infinitely dense point-mass
singularity and nobody has ever found an event horizon. Moreover, according
to the mathematical theory of black holes, it takes an infinite amount of
time for an observer to establish the presence of an event horizon, but
nobody has been and nobody will be around for an infinite amount of time, so
it is impossible to resolve anything at the alleged event horizon. All
claims for the discovery of black holes are thus patently false.
provide the coordinates of just one infinitely dense point-mass singularity?
Can you provide the coordinates of just one event horizon? Nobody has ever
Well What About The Event Horizon?
3. You also remark,
But the strong bending of light rays within the gravitational field of the
black hole will double the apparent size of the event horizon, the boundary
of the area around the black hole from which nothing, not even light, can
escape. Thus Doeleman and colleagues' observations have finally brought us
to the threshold of imaging horizon-scale structures - a holy grail of radio
The so-called "Schwarzschild radius" is alleged to be the radius of the
event horizon, beneath which is a region of spacetime. However, there is no
interior region, i.e. the alleged event horizon does not mark a boundary
between two regions of spacetime. The alleged event horizon does not contain
a volume. Indeed, the alleged radius of the event horizon, the
"Schwarzschild radius", is not a distance in the spacetime manifold, let
alone a radial distance. The astrophysical scientists have asserted this in
ignorance of even elementary differential geometry
Black holes are truly bizarre objects. Einstein's theory of general
relativity tells us that they are objects in which gravity has run amok,
cutting off a region of space (inside the event horizon) from the
outside Universe. Inside the event horizon, theory predicts the existence of
regions in which densities laws of physics break down.
geometric fact is that the quantity denoted by 'r' in the line element of
the so-called "Schwarzschild solution", a particular value of which they
call the "Schwarzschild radius", does not directly determine any distance at
all in the spacetime manifold because it is in fact the inverse square root
of the Gaussian curvature of any spherically symmetric geodesic
surface in the spatial section of the spacetime manifold. As such it is not
the geodesic radial distance from the centre of spherical symmetry of the
spatial section. Only in this sense can the said quantity 'r' be called a
radius: it is the inverse square root of the Gaussian curvature of a
spherically symmetric geodesic surface in the spatial section - the radius
of Gaussian curvature thereof. It does not directly define any distance
whatsoever in the spatial section of the so-called "Schwarzschild" spacetime
Gaussian curvature is an intrinsic property of any surface, as
Gauss proved long ago by his Theorema Egregium. The geometry of the line
element for Ric = 0 is non-Euclidean, and in consequence of this any point
in the associated spatial section has the property that it has a finite
non-zero surface area, but a geodesic radius of zero and a volume of zero.
This is odd, but inevitable. The indefinite metrics associated with
Einstein's General Theory of Relativity admit of other oddities, such as
null vectors, i.e. non-zero vectors that have zero length, or equivalently,
non-zero vectors that are orthogonal to themselves. These are also
inevitable geometric consequences of the associated pseudo-Riemannian
So it is demonstrably false that the alleged event horizon contains
anything. It in fact describes a geometric point having the properties
described above, in a pseudo-Riemannian metric space. No proponent of the
black hole is even aware of the fact that their quantity 'r' in their
so-called "Schwarzschild solution" relates to the Gaussian curvature of a
spherically symmetric geodesic surface in the spatial section of the
spacetime manifold, not to distances in the spatial section of the spacetime
Can you provide a mathematical proof that the quantity denoted by
the symbol 'r' in the so-called "Schwarzschild solution" is a radial
distance, or even a distance, in the spatial section of the spacetime
manifold? No proponent of the black hole has ever done this, and for good
reason, it is impossible, because it is false.
Rigorous mathematical proof of the Gaussian curvature is given here:
Can you provide a proof that the spacetime of Ric = 0 does not violate
Einstein's Principle of Equivalence? Can you provide a proof that the laws
of Special Relativity can manifest in the spacetime of Ric = 0, a spacetime
that by definition contains no matter? Can you provide a proof that a freely
falling inertial frame can manifest in the empty spacetime of Ric = 0?
4. At the start of your article you say:
Do black holes exist? Observations at the finest resolution so far indicate
that only gross deviations in the behaviour of gravity from that predicted
by general relativity can invalidate the case that they do.
The geometrical facts already enunciated above are sufficient to prove the
black hole a fallacy. On a much simpler level the black hole is inconsistent
with the Theory of Relativity. The alleged singularity of the black hole is
infinitely dense. Now Special Relativity forbids infinite density because
infinite density implies that a material body can acquire the speed of light
in vacuum (or equivalently that there is infinite energy), which violates
the fundamental premise of Special Relativity. General Relativity, by
definition, cannot violate Special Relativity, and so it too forbids
infinite density. Thus, the Theory of Relativity forbids infinitely dense
point-mass singularities and hence forbids black holes. Consequently, all
alleged black hole phenomena are meaningless.
Let's Ask Newton Then
Black holes are not predicted by Newton's theory of gravitation either,
despite the claims of the astrophysical scientists. The hypothetical
Michell-Laplace dark body of Newton's theory is not a black hole because it
possesses an escape velocity, whereas the black hole has no escape velocity;
it does not require irresistible gravitational collapse, whereas the black
hole does; it has no infinitely dense point-mass singularity, whereas the
black hole does; it has no event horizon, whereas the black hole does; there
is always a class of observers that can see the dark body, but there is no
class of observers that can see the black hole; the Michell-Laplace dark
body can persist in a space which contains other matter and interact with
that matter, but the black hole's spacetime is devoid of matter and so it
cannot interact with any matter. Thus the Michell-Laplace dark body does not
possess the signatures of the alleged black hole and so it is not a
Schwarzschild's Solution/Hilbert's Corruption
The so-called "Schwarzschild solution" from which the "Schwarzschild black
hole" is alleged, is not even Schwarzschild's solution. This is easily
verified by simply reading Schwarzschild's first paper on the subject (he
wrote two papers), available here:
Schwarzschild's actual solution does not admit of the black hole. There is
no event horizon associated with his actual solution. The so-called
"Schwarzschild solution" is a corruption, by David Hilbert (Dec 1916,) of
the solution obtained by Schwarzschild (Nov/Dec 1915, published 1916) and
also a corruption of the solution obtained independently by Johannes Droste
(May 1916, published 1917). Droste's original paper can be obtained here:
Droste's solution is consistent with Schwarzschild's solution, and so it
accordingly does not admit of the alleged black hole. Hilbert's corruption
is inconsistent with Schwarzschild and hence with Droste. Marcel Brillouin
also obtained a solution in 1923, consistent with Schwarzschild and Droste.
His paper can be obtained here:
It was from Hilbert's corruption that the black hole was originally
Thus, General Relativity does not predict the existence of black holes. I
shall amplify this even further.
So What Did
According to Einstein, in his gravitational field, gravitational mass and
inertial mass are equivalent, and also, in a sufficiently small region of
his gravitational field his laws of Special Relativity must hold. Here is
what Einstein himself expounded (see his book 'The Meaning of Relativity',
Science Paperbacks and Methuen & Co. Ltd., 1967, pp. 56-57, which Einstein
revised in 1954, the year before his death):
God er... Bertie Have To Say About It?
Let now K be an inertial system. Masses which are sufficiently far from
each other and from other bodies are then, with respect to K, free from
acceleration. We shall also refer these masses to a system of co-ordinates
K', uniformly accelerated with respect to K. Relatively to K' all the masses
have equal and parallel accelerations; with respect to K' they behave just
as if a gravitational field were present and K' were unaccelerated.
Overlooking for the present the question as to the 'cause' of such a
gravitational field, which will occupy us later, there is nothing to prevent
our conceiving this gravitational field as real, that is, the conception
that K' is 'at rest' and a gravitational field is present we may
consider as equivalent to the conception that only K is an 'allowable'
system of co-ordinates and no gravitational field is present. The
assumption of the complete physical equivalence of the systems of
K and K', we call the 'principle of equivalence'; this principle is
evidently intimately connected with the law of the equality between the
inert and the gravitational mass, and signifies an extension of the
principle of relativity to co-ordinate systems which are in non-uniform
motion relatively to each other. In fact, through this conception we arrive
at the unity of the nature of inertia and gravitation. For, according to our
way of looking at it, the same masses may appear to be either under the
action of inertia alone (with respect to K) or under the combined action of
inertia and gravitation (with respect to K').
Now Einstein's field equations for the static vacuum gravitational field,
i.e. Ric = 0, violate his 'Principle of Equivalence' because the equivalence
of gravitational and inertial mass, and the laws of Special Relativity,
cannot manifest in a spacetime which by definition contains no matter! QED.
Clearly, Einstein's writing of Ric = 0 was a major blunder. Consequently, if
his energy-momentum tensor is zero there is no Einstein gravitational field.
Hence his field equations must take the following form:
Stated more exactly, there are finite regions, where, with respect to a
suitably chosen space of reference, material particles move freely without
acceleration, and in which the laws of special relativity, which have been
developed above, hold with remarkable accuracy.
Gij/k + Tij = 0, (subscripts) i,j = 0,1,2,3, k = constant,
wherein the Gij/k are the components of a gravitational energy tensor. Thus
the total energy of the gravitational field is always zero; the Gij/k and
Tij must vanish identically; there is no possibility for the localisation of
gravitational energy (i.e. there is no possibility for Einstein's
gravitational waves). Moreover, this means that Einstein's General Theory of
Relativity violates the experimentally well established conservation of
energy and momentum, so if the usual conservation of energy and momentum is
valid (bearing in mind that there is no experimental evidence to refute it)
then Einstein's General Theory of Relativity is invalid.
aware that his theory violated the usual conservation of energy and
momentum, simply invented his pseudo-tensor to save it, and by which he and
his followers claim that his gravitational energy can be localized. However,
Einstein's pseudo-tensor is a meaningless concoction of mathematical
symbols for the following reason - it implies the existence of a 1st-order
intrinsic differential invariant which depends only upon the components of
the metric tensor and their 1st-derivatives (to see this just contract his
pseudo-tensor and apply Euler's theorem). But the pure mathematicians G.
Ricci-Curbastro and T. Levi-Civita proved in 1900 that such invariants do
not exist! So Einstein committed another major blunder when he invented his
In addition, Einstein and his followers resort to
linearisation of his field equations to localize his gravitational energy
and to obtain a "Newtonian approximation". This too is nonsense, because
linearisation implies the existence of a tensor which, except for the
particular case of being precisely zero, does not otherwise exist, as proven
by H. Weyl in 1944. So LIGO and its international counterparts such as the
AIGO in Australia and VIRGO in Europe, are all destined to detect nothing;
and the black hole is not predicted by General Relativity.
Go here for a proof that Einstein's pseudo-tensor violates pure mathematics:
Go here for H. Weyl's 1944 proof that linearization of Einstein's field
equations is erroneous:
Time To Straighten Out "Spacetime"
It is also alleged by most astrophysicists and astronomers that spacetimes
described by the field equations
Ric = λgij, (subscripts) i,j = 0,1,2,3
where λ is the 'cosmological constant', describe gravitational fields in the
absence of matter; that the spacetimes are curved by themselves, without the
causative influence of matter; in other words that a gravitational field can
exist in the complete absence of matter as a causative agent. However, there
is not a shred of physical evidence to suggest that a gravitational field
can exist without a material cause.
Curiously, the astrophysical scientists
allege on the one hand that although this expression contains no sources for
the gravitational field, because the energy-momentum tensor is zero, on the
other hand they also allege that Ric = 0 contains a source even though the
energy-momentum tensor is zero there too. In the latter case the massive
source is inserted post hoc into the solution, and hence inadmissible.
Furthermore, according to Einstein, matter is the cause of the curvature of
spacetime, i.e. of the gravitational field, and the
causative matter must manifest mathematically in a non-zero energy-momentum
tensor in his field equations.
The late theoretical physicist John A.
Wheeler has reasserted Einstein's geometrodynamics thus, "Matter tells
spacetime how to curve and spacetime tells matter how to move". The fact
that Einstein's field equations violate the usual conservation of energy and
momentum also means that Ric = λgij is a physically meaningless expression
to begin with.
Can you provide a proof that Einstein's pseudo-tensor is not mathematical
gibberish? Can you provide a proof that Einstein's field equations do not
violate the usual conservation of energy and momentum? Can you provide or
otherwise cite experimental evidence that a gravitational field can have no
Stephen J. Crothers.
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XMM-Newton Spectrum & Illustration of RX J1242-11 or Graph and a cartoon of a black hole...
Credit: Spectrum: ESA/XMM-Newton/S. Komossa et al. Illustration: NASA/CXC/M. Weiss
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