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.

Colossal stars

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 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 would 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. A colossal black star.

Escape velocity and light-speed

We will work in classical mechanics although the modelling rightly belongs to General Relativity- this because of the immensely powerful 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 a fixed uniform density.

If the escape velocity exceeds the speed of light then no radiation and no mass can escape from the body. It needs to be calculated. 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 but of the same density, 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.

In ordinary Euclidean space

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 Cartesian 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, none has less mass.

Infinite mass spatially contained

Suppose there are infinitely many particles. Consider two possibilities, in the first assume that the physical universe is enclosed in a bounded region of space. Postulate then 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 basic result in real analysis, the Bolzano-Weierstrass 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 an accumulation point.

Higher order infinities

On the other hand imagine the physical universe to be unbounded in space but the order of the infinity of particles is higher.

There are orders of infinity. The “smallest” infinity is “denumerable 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 and exhausted as an endless sequence. Surprisingly Cantor’s diagonal method shows that the “rational numbers”, i.e. all fractions, are denumerable as well. A simple and elegant argument.

However the set of all “real numbers” for example cannot be written out as a sequence. The real numbers are more prolific than denumerable infinite. They are of an higher order of infinity. Suppose now there are non-denumerable infinitely many points spread or scattered in space but which cannot be enclosed in any bounded region. Then there still has to be an accumulation, a cluster point.

The fact is a direct consequence of the Bolzano-Weierstrass 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 we may also conclude that there must be a black hole.
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In the illustrations our discussion is limited to the real number line, that is one-dimensional space. From here the theorem can be very easily deduced for the plane i.e. two dimensions, and in fact any finite-dimensional space. We need only think of points in terms of the coordinates. The result then comes straight from the theorem for the real line and the definitions.

The property of completeness, and cluster points

One has to have the completeness property of the real numbers in order to prove the the Bolzano-Weierstrass 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 l.u.b. is again a real number.

The difference between the axiom and the theorem is a bit subtle. The definition we used for completeness states that a set of real numbers that is bounded above has a least upper bound, (even if unbounded below!) That would apply to finite sets too.

On the other hand the Bolzano-Weierstrass Theorem states that a bounded infinite set of real numbers has to have a cluster point. These are not the same. For example, the cluster point could lie inside the set’s bounds but now the numbers have to be infinitely many.

I’ve written out the necessary definitions and an exact formulation of the theorem. The material as it is presented should be within the grasp of a second-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 may 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.

∞

“Two men look out through the same bars: One sees the mud, and one the stars.”

–Frederick Langbridge