Mathematical quests for cosmology and astrophysics
Since in science we know what we think we know (see relevant previous publication), when we talk
about scientific issues, we must, in general, bear in mind that every theory
structured to explain the hitherto unexplained phenomena is based on a model or
scenario, through which answers to unanswered questions are sought.
In particular, with regard to sciences such as physics/astrophysics and cosmology, the majority of experts set out as a starting point a fundamental idea of how to achieve answers/solutions to emerging problems for which explanation is sought.
One could begin to describe suggestions and ideas that man formulated in one way or another in a communicative way, even from the time when he took those first steps as a rational being, in his attempt to understand the phenomena around him.
In modern physics, discoveries were always supported by observation and experiment. But what is the experiment? Well, nothing but a repeated remark. Let us take as an example the knowledge we have about the problem of the establishment of our material world. Starting, therefore, the "scientist" man, in the above explained concept (observation + experiment), from the well-known Mendeleev’s Periodic Table of Elements, he then proceeded to interpret the idea of the existence of atoms. Thomson's plum pudding model proved to be incorrect and we moved to the Ratherford's atom with the nucleus in the center and electrons rotating around it. However, this also led to distortion, since the electrons with the loss of energy (as charges in accelerated motion) eventually, even very soon, fall towards the central core. That way no atom could survive for long. Then Bohr came, who with his well-known quantum theory, offered a solution to this problem, somewhat. This was followed by the rapid development of Quantum Mechanics, which provides answers and a better understanding of the function of the atom.
Then another idea came: the unification of the four fundamental dynamic fields of nature. This idea has become the central problem of physics worldwide. The theories developed to find answers to this problem are based on increasingly advanced and complex mathematics, such as the so-called "Standard Model", versions for the "Great Unified Theory (GUT)" and the "Theory of Everything (TOE)". However, gravity could not be unified with the other three fundamental forces/interactions (strong nuclear, weak nuclear and electromagnetic) and therefore new models came as suggestions from the scientific community, such as "Supersymmetry" and "Supergravity". It should be noted that, even through these theories, the unification of Quantum Mechanics with the General Theory of Relativity has not yet been achieved. Thus, even through these theories, the ultimately sought explanation has not yet been obtained.
Not to appear completely pessimistic, let us point out that the so-called "Superstrings" is a new model, which is considered that in some version it could give the final answer. In essence, as scientists have not been able to calculate, in theory, what will happen when two gravitons (the carrier particles of gravitational interaction) collide with each other, since the calculations showed energy tending to infinity in a small space – a sign of weakness of the hitherto known mathematics – they turned to the invention of the strings. Thus, according to the models, where the strings are assumed, only strings can collide and be reflected in a clear and mathematically explainable way, without resulting in weak, from the physics point of view, infinitations. However, so far, even though strong mathematical minds worldwide are working on this subject, the results are not yet successful. Some argue that this theory should be waiting for the discovery of new, more advanced mathematics to solve its equations.
That being the case in the context of mathematical quests to support the models of cosmology and physics/astrophysics and stating from the outset our humility towards the strong minds of the planet, we quote, below, some thoughts, with the aim of enriching the approach of unresolved problems of physics. These thoughts come in the wake of the above review of the progress of the searches made to date to date and are based on mathematical documentation.
Question 1 and "subversive" for this approach: Is it possible that the cosmological models to date are not sufficient? Does mathematical research to support such theoretical models have no future?
Where were we left in the above review? In the string-dependent models. Well, are these invented entities, the strings, which resemble their physical properties with the normal strings of our real world, but only that these ones grow in more than four dimensions, not constitute the right choice? Therefore, given the mathematical difficulties of solving string equations, would another entity be a more solid but also mathematically more accessible basis for supporting a cosmological model?
Our personal thought goes to the idea of White Holes and even for the needs of mathematicalisation of the model but also of giving further degrees of freedom regarding the development of an integrated cosmological model, let us consider that these are not holes of large dimensions. On the contrary, let's imagine White (not Black) Holes in the sub-Planck space, i.e. for scales (where, ℏ ~ 6,6 x 10-34 m2kg/s is the known Planck's constant), in other words for lengths of ≪1,6 x 10-35 m, or times ≪5,4 x 10-44 s, or masses ≪2,2 x 10-8 kg, or temperatures of ≪1,4 x 1032 Κ (~ -272 x 1032 oC), or electrical charge ≪1,9 x 10-18 C. These holes can be called Mini White Holes (MWH).
Perhaps, clarification is also needed on the distinction between White and Black Holes. In the sense of the Theory of Relativity, Black Hole in space-time is the area where for each entity (including light) entry is possible but escape impossible. Correspondingly, White Hole is the area of space-time where for each entity (including light) entry is impossible but escape possible.
The advantage of the White Holes approach instead of Strings may have already been realised and this is the fact that the White Holes and indeed those of the sub-Planckian space, i.e. the Mini White Holes, could be the most probabilistic generators of the universes/universe.
There is one more additional key advantage in the approach involving these entities (MWH), that the mathematical search for the behavior of matter within them is perfectly workable by solving differential equations by replacing the numerical values of the sub-Planckian space. The aim is to determine the probability of an event within such a hole, i.e. within the sub-Planckian space.
For this event to be evidence of the generator status we assumed for the MWH it is sufficient to mathematically determine the probability of a quantity of mass emerging through such a hole and then being transited to the quantum level of the normal (i.e. the post-Big Bang) of the universal space, which we experience.
Well, the first mathematical calculations (i.e. solving the one-dimensional Schrödinger's relativistic equation
with initial assumption P = a2|Ψ|2) indicate that the probability (let it be P) of an event such as the above (emergence of a mass quantity from MWH and transition to the normal universal space) is over time of exponential form with high adaptability control R2 = 0,999, i.e.
P = f(t) = a tb
where
a ~ 3,66
b ~ 1,96
Is it a coincidence or possibly an unknown law determines the development of the probability in the sub-Planckian space and obliges it to follow a regression of an exponential form? It remains to be answered with further study and perhaps at some point in the future with experimental data.
Though this approach does not only have positive features. It has also got unanswered questions. For example, would it not be reasonable to wonder that, if the above apply and various small (partial) universes emerge from the MWH, does this mass insertion not put a problem in the Schwarzschild radius of the Great (known) Universe? This is, therefore, about a continuous expansion (not inflation).
But, so be it, this is a first approach, which can duly be subject to future amendments and revisions, combined with any experimental data, if and when this is possible by the science of the future.
Literature Review
- ‘The Aspect of Information in the Process of Observation’, H. Atmanspacher, Foundations of Physics, 1989
- ‘The First Three Minutes’, S. Weinberg, Basic Books Inc., N.Y., 1977
- ‘Introduction to Elementary Particles’, D. Griffiths, N.Y., 1987
- ‘Six Impossible Things’, J. Gribbin, Icon Books Ltd, 2019
- ‘Superstrings – A theory of everything’, P. Davies & J. Brown, Cambridge University Press, 1988
- ‘Principles of Cosmology and Gravitation’, M. Berry, Cambridge University Press, 1976
- ‘Black Holes’, J. Taylor, Fontana-Collins, 1974
- ‘A Brief History of Time – From the Big Bang to Black Holes’, S. Hawking, Bantam Books, 1988
- ‘Lost in Math: How Beauty Leads Physics Astray’, S. Hossenfelder, Basic Books, N.Y., 2020