by

Stephen J. Crothers

A number of quite malicious proponents of black holes and Big Bang Cosmology, adducing no arguments of their own devise, have resorted to merely citing the following equally feckless quintet, on a number of blogs and other websites, in their unreasonable protestations to my proofs that Black Hole universes and Big Bang universes are nonsense:

1. Gerardus 't Hooft, Nobel Laureate (physics)

5. G. W. Bruhn

I have dealt thoroughly with the quintet elsewhere:

**1.** 't Hooft,

**2.** Sharples (a),
Sharples (b),
Sharples (c)

**3.** Bruhn

**4.** Corda (a),
Corda (b)

**5.** Clinger

A common mathematical issue of the 'quintet' is the alleged 'extension' of Droste's solution to Hilbert's solution. It is from the latter that the black hole was first conjured. Cosmologists always and incorrectly call Hilbert's solution "Schwarzschild's solution". However, it is an irrefutable fact that Hilbert's solution is not Schwarzschild's solution, which can be easily verified by reading Schwarzschild's original paper and comparing it to Hilbert's scribblings. Droste's solution is equivalent to Schwarzschild's solution but Hilbert's 'solution' is not.

The equivalence of the Schwarzschild and Droste solutions is easily established. Here they are (in both cases the speed of light in
vacuum, *c*, is set to unity):

__Schwarzschild__

ds^{2} = (1 - α/R)dt^{2} - (1 - α/R)^{-1}dR^{2} -
R^{2}(dθ^{2} + sin^{2}θ dφ^{2})

R = (*r*^{3} + α^{3})^{1/3}

0 ≤ *r*

__Droste__

ds^{2} = (1 - α/*r*)dt^{2} - (1 - α/*r*)^{-1}d*r*^{2} -
*r*^{2}(dθ^{2} + sin^{2}θ dφ^{2})

α ≤ *r*

The constant α is positive but otherwise indeterminable. Note that Droste's *r* = α corresponds to Schwarzschild's *r* = 0. In both cases ds^{2} is then undefined (i.e.
'singular') because the coefficient in the second term on the right side produces -1/0. Contrary to the practice of cosmologists (who claim that
1/0 = ∞), division by zero is undefined. Compare now to Hilbert's 'solution' (here *c* = 1 and *G* = 1 in the 'Schwarzschild radius'
*r _{s}* = 2

__Hilbert__

ds^{2} = (1 - 2*m*/*r*)dt^{2} - (1 - 2*m*/*r*)^{-1}d*r*^{2} -
*r*^{2}(dθ^{2} + sin^{2}θ dφ^{2})

0 ≤ *r*

Note that Hilbert's *r* = *r _{s}* = 2

The solutions obtained by Schwarzschild and Droste are not only equivalent, they are elements of an infinite equivalence class, i.e. an infinite set of equivalent solutions. All elements of this infinite equivalence class describe the very same geometry, so they are effectively the same solution. Just as something can be described in different words, so too the same mathematical solution can be described in different symbols. That being so, if any element of this equivalence class is extendible in the fashion of Hilbert's solution, then all must be extendible. Conversely, if any element of the equivalence class is not extendible then none are extendible. What then is the ground-form or generator of the equivalence class? This was adduced in my very first paper on the subject (in 2005), yet none of my critics, it seems, have noticed it. So here it is again;

__Crothers__

ds^{2} = (1 - α/R_{c})dt^{2} - (1 - α/R_{c})^{-1}dR_{c}^{2} -
R_{c}^{2}(dθ^{2} + sin^{2}θ dφ^{2})

R_{c} = (|*r* - *r*_{o}|^{n} + α^{n})^{1/n}

*r*, *r*_{o} ∈ **R**, *n* ∈ **R**^{+}

Here the constant *r*_{o} is any real number and the constant *n* is any positive real number (take your pick). Note that R_{c} is
defined for all values of *r* and all values of *n*, and that R_{c}(*r*_{o}) = α for all values of
*r*_{o} and all values of *n*. Similarly, ds^{2} is 'singular' only when *r = r*_{o}. If *r*_{o} = 0, *n* = 3, *r* ≥ *r*_{o}, then Schwarzschild's solution is obtained. If
*r*_{o} = α, *n* = 1, *r* ≥ *r*_{o}, then Droste's solution is obtained. It is clear from the metric
ground-form that no (equivalent) solution generated by it can be extended in the fashion of Hilbert, to thereby produce a black hole. This is
amplifed by taking *r*_{o} = 0 and *n* = 2, to yield,

ds^{2} = (1 - α/R_{c})dt^{2} - (1 - α/R_{c})^{-1}dR_{c}^{2} -
R_{c}^{2}(dθ^{2} + sin^{2}θ dφ^{2})

R_{c} = (*r*^{2} + α^{2})^{1/2}

*r* ∈ **R**

This metric is defined for all values of *r* except *r* = *r*_{o} = 0. It can't be extended to -α^{2} ≤ r^{2} to
produce a black hole, because *r*^{2} can never have values less than 0, and hence R_{c} can never have values less than
α. Thus, on account of equivalence, no element of the infinite equivalence class can be extended to produce a black hole. Hence, Droste's
solution cannot be extended to Hilbert's 'solution' and so there is no possibility of a black hole 'solution'. Similarly, Schwarzschild's
solution cannot be extended. Consequently, there is no black hole universe. All other black hole universes rely upon Hilbert's, and so they are
all false. The black hole requires, in the specific example above, that -α^{2} ≤ r^{2}, which is a violation of the rules
of pure mathematics. In general, the mathematical theory of black holes
requires that -α^{n} ≤ |*r* - *r*_{o}|^{n}. The mathematical theory of black holes violates the
rules of pure mathematics. Consequently, it is certainly false.

__A Few Other Closely Related Issues__

Einstein's field equations are:

*λ* is the 'cosmological constant'. *T _{μν}* is the energy-momentum tensor that describes the material sources of
Einstein's gravitational field. The left side of the equation gives spacetime geometry, which is curved due to the presence of material sources,
and this spacetime curvature is Einstein's gravitational field. According to Einstein, matter must be present to produce his gravitational field,
i.e. to induce the curvature of his spacetime. According to Einstein and his followers, if

Einstein and his followers assert that *R _{μν}* = 0 describes his gravitational field '

That *R _{μν}* = 0 contains no matter whatsoever, and hence cannot lead to a black hole at all, is reaffirmed by the case,

The solution for these equations is de Sitter's empty universe, which is empty because it contains no material sources
(*T _{μν}* = 0), even though the solution has a non-zero curvature. Matter is the source of Einstein's gravitational field,
and, according to Einstein, matter is everything except his gravitational field. Without material sources no gravitational field can be produced.
So 'spacetime curvature' alone, without material sources to induce it, does not produce Einstein's 'gravitational field'. By
virtue of

The quantity *r* in Hilbert's solution is not even a distance let alone the radius therein. Nonetheless the cosmologists always treat it as the
radius. This is most evident in the so-called '*Schwarzschild radius*' of their black holes, and even of stars and planets. According to the
cosmologists their 'Schwarzschild radius' *r _{s}* = 2

According to the cosmologists their black hole has a finite mass. This mass is inserted *post hoc* into Hilbert's 'solution', by insinuating the Newtonian
expression for escape speed, in order to satisfy the false assertion that *R _{μν}* = 0 contains a material source, '

This is immediately recognised as the Newtonian expression for escape speed. It is from this equation that the cosmologists obtain the speed of light for the 'escape speed' at their black hole 'event horizon' (i.e. at their 'Schwarzschild radius'). No cosmologist even understands the meaning of 'escape velocity'. On the one hand, according to the cosmologists, their black hole has an escape velocity. At their black hole 'event horizon' they assert that the escape speed is the speed of light. On the other hand they also assert that nothing can even leave their event horizon. The event horizon is a one-way membrane: things can go into the black hole but nothing can emerge. Light, they say, hovers forever at the 'event horizon' when trying to 'escape' from there. However, escape speed does not mean that nothing can leave, only that physical things cannot escape if they do not achieve the escape speed. Thus, on the other hand the cosmologists assert that their event horizon has no escape speed, since nothing can even leave it. So their black hole has the schizophrenic properties of having and not having an escape speed simultaneously at the same place (at the 'event horizon'). But nothing can have and not have an escape velocity simultaneously at the same place. Furthermore, since light travels at the speed of light, and the escape speed at the event horizon is the speed of light, then, by definition of escape speed, light must escape! But not according to the cosmologists; light hovers forever at their event horizon.

According to the cosmologists, the finite mass of their black hole is concentrated at the 'physical singularity' of their black hole (i.e. at Hilbert's *r* = 0),
where, they say, volume is zero, density is infinite, and spacetime is infinitely curved. There are forces in General Relativity, but gravity is
not one of them, because it is spacetime curvature. Thus, according to the cosmologists, at their 'physical singularity', a finite mass produces
infinite gravity! However, no finite mass has zero volume, infinite density, infinite gravity, anywhere. Once again, the theory of black holes
flase.

All this and more has been explained in detail in my papers and video recordings.

Stephen J. Crothers

My email address: thenarmis@yahoo.com

Page established: 9^{th} June 2015