A note on Black Stars

The existence of “Black Holes” and “Neutron Stars” as real physical entities are accepted almost universally nowadays. A black hole is understood to be an enormous mass concentrated at a mathematical point, i.e. a singularity in space-time. A neutron star (observed as a pulsar) is to be a star (a pre- black hole) which has collapsed due to terrific gravitational force. It supposedly consists entirely of nucleic matter, specifically neutrons. This implies terrific density- comparable to that of an atomic nucleus.

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Are there really good grounds to believe in such stellar bodies?

Of course these ideas are fascinating and appeal strongly to the imagination, they were readily accepted by the general public. But for a start (in my ignorance) one certainly would not be able ever to see such a black hole. Outside effects would however be observed for instance the absorbing of matter and gravitational bending of light rays.

As for the existence of possible singularities? Intuitively I can only think that such must have catastrophic implications for space-time as a whole. However as I now show, there might be other ways of explaining things.

It will be demonstrated that a very large body- a gigantic star- must of necessity be a “Black Star” provided it is large enough. Meant by this, neither light nor matter can escape the gravitational force on the surface consequently making the star invisible and completely dark.

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We will work in classical (Newtonian) mechanics although the modelling rightly belongs to General Relativity- this because of the immense gravitational fields involved, as well as velocities of the order of light-speed.

To simplify things further the star is modelled as a sphere with uniform density.

If the escape velocity exceeds the speed of light then no radiation and no mass can escape from the body. We need to calculate it. One can do so with Newton’s universal law of gravitation and the conservation of energy. It is a routine exercise I’ve provided it as an attachment.

The result shows the escape velocity is a function of the star radius and is higher if the star is larger, and if the star is large enough it must exceed the speed of light.

So this would settle the “darkness” issue. However there are further considerations such as the internal pressure. At what stage would it be so high as to overcome nuclear forces for total collapse thus creating a singularity? You can only estimate this theoretically obviously you cannot measure it in a laboratory.

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In conclusion a “Black Star” doesn’t necessarily have to be a “Black Hole”. This insight is probably not original. The calculations in the Newtonian setting were elementary. General Relativity is the best framework and it will be very interesting to see the same calculation for the same model.

Personally I feel there is really not enough scientific justification to believe in these “Black Holes” and “Neutron Stars”.

During theoretical research in 2006 - 2007 related to the observed “glitches” in Pulsar spin I studied some seminal research articles in respectable Astrophysics journals. More recent papers especially seem to be a mixture of speculation and fantasy. Still, mathematical investigations in Astrophysics must remain a worthwhile endeavour. My main objection really is simply this: Theories and possible explanations are often presented to the public as truth and as facts.

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There is another altogether different way of thinking about black holes which is purely geometric in nature and disregards all other considerations of physics: Time, motion, gravity and so on.

We imagine a static state, a configuration at a moment in time, and look at space as simply three-dimensional (Euclidian space) i.e. the ordinary space of everyday experience.

Under certain conditions I propose to demonstrate the existence of a black hole.

Assume that mass exists only in the form of particles, as well as there is a certain minimum mass for particles. With which I say that there is a “smallest” particle and none is lighter.

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Suppose there are infinitely many particles. Consider two possibilities:

First: The physical universe is enclosed in a bounded region of space.

So, postulate the existence of infinitely many particles and that the whole (all matter / every particle) of the universe is contained in some finite volume. From a geometric (spatial) viewpoint it then has to follow that there is an “accumulation” point.

Mathematically this is a point “c” in space where infinitely many points accumulate (cluster) infinitesimally close to c. Technically this point c is a black hole since its mass would be infinite.

The existence of such a c follows from a result in elementary real analysis, the Bolzano-Weierstrass (BW) Theorem.

The theorem states that if infinitely many points are contained in a bounded subset of (finite-dimensional) space, then of necessity there has to be such a accumulation point.

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Second: The physical universe is unbounded in space but the order of the infinity of particles is higher.

There are orders of infinity. The “smallest” infinity is “denumerably infinite”. This would be that of for example the positive integers, the odd numbers, the prime numbers and any set for which the members can be written out (“exhausted”) as a sequence. Surprisingly, Cantor’s diagonal method shows that the rational numbers (fractions) are denumerable as well. A simple and elegant argument.

However for example the set of all real numbers cannot be written as a sequence. The real numbers are more “prolific” than denumerably infinite. They are of a higher order of infinity.

Suppose now there are non-denumerable infinitely many points spread / scattered in space but which cannot be enclosed in any bounded region. Then there still has to be an accumulation (cluster point).

The fact is a direct consequence of the BW Theorem. The proof is easy and a good third-year student shouldn’t have difficulty. One can prove it by contradiction using only very elementary considerations of set theory. The claim does not hold for denumerable sets.

So for this case too, we may conclude that there must be a black hole.

Our discussion was limited to the real numbers, that is one-dimensional space. From here BW can be very easily deduced for the plane (two dimensions) and in fact any finite-dimensional space. One only needs to think of points in terms of the coordinates. The result then comes straight from BW (for real nr’s) and the definitions.

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One has to have the completeness property for the real numbers in order to prove the BW theorem. In fact the two are equivalent.

Note that the crucial assumption in the Axiom of Completeness is not so much the existence of a least upper bound, but the fact that the lub itself is again a real number.

I’ve written out the necessary definitions and an exact formulation of the BW Theorem. The material as it is presented should be within the grasp of a first-year university student. My attachments are meant as an explanation of how the ideas work but they are really quite sound from the mathematics’ viewpoint. Proofs could be found in any standard textbook on real analysis, e.g. Serge Lang, “Undergraduate Analysis”.

∞ + ∞

The idea of a universe of an infinite mass is not as blasphemous as one might think and if I may, I refer the reader to the number that is named in Revelation 7:9 (KJV)