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Physics, Mathematics and Cosmology: A Brief Informative Note
Spyrou NK*
Professor of Astronomy, Aristoteleion University of Thessaloniki, Thessaloniki, Greece
Submission:January 17, 2025;Published:February 06, 2025
*Corresponding author:Spyrou NK, (Retired 2011), Professor of Astronomy, Aristoteleion University of Thessaloniki, Thessaloniki, Greece Email: spyrou@auth.gr
How to cite this article:Spyrou NK. Physics, Mathematics and Cosmology: A Brief Informative Note. Academic Journal of Physics Research. 2025; 1(2): 555556.DOI:10.19080/AJPN.2025.01.555556
Abstract
Keywords:Dark Matter; Dark Energy, Cosmic Acceleration; Thermodynamic Pressure; Mass-Energy Density; Gravitational Fluid; Internal Thermodynamic Energy; Kinetic Energy; Negative Pressure; Tensor Mathematics; Cosmological Research Philosophy; Cosmic Fluid Dynamics; Cosmic Thermodynamics
Mini Review
Based on the closing lecture by the author, at the “First Balkan Student Summer School on Astronomy and Astrophysics-A child from the Balkans counts the stars”. The school was organized by the Union of Greek Physicists, the Balkan Physical Union, and the European Physical Society, and the lecture was delivered at the host American Farm School in Thessaloniki, Greece, on 30 August 2024.The main objective of the school was to promote the idea of the necessity of teaching cutting edge Physics and Astronomy in Balkan schools.
Summary
Realistic Cosmology is neither simply Mathematics?, nor simply impressive photographs, nor simply a powerful electronic computer, nor a particle accelerator but, beyond and above all these, firstly and mainly, it is physical thought, physical way of thinking, physical intuition, physical realism, namely and in short, it is Physics.
Philosophy of Cosmological Research
The purpose of this short essay is to offer to the younger generation of cosmologists and their advisors and supervisors, some interesting and useful messages, concerning the appropriate method for active study and productive research, particularly in Cosmology, and, more generally, in Astronomy and in Astrophysics.
The useful message I want to pass here also refers to the way physics and mathematics, as two entirely different areas of knowledge, should be used in a realistic description of the cosmological Universe, namely, a description that must be also physically oriented, not mathematically oriented only.
Cosmology, as it is known, is the science aiming at the study of the Universe as a whole, its beginning, its evolution and its possible end. And it is truly interesting that working with Cosmology is not something to choose between other possible choices. Working with Cosmology is something “coming from one’s inside”, and it is not possible to abandon Cosmology, even if (for various reasons) he/she is forced to do so.
The study of Cosmology, as of every other physical scientific area, is based on the mutual interaction between theory and observations/experiment.
Suppose that we wish to study the dynamic behavior of a physical system. This system can be, e.g., the atmosphere of the Earth, or a star and its evolution, or a galaxy and its properties, or a system of galaxies, or, even, the Universe as a whole. In the dynamical description of each one of these physical systems, in simple words, we use an underlying, well-established theory, that makes certain theoretical predictions. In Cosmology, this theory is Einstein’s General Relativity Theory (GRT).
Now, the predictions of the theory used are either verified or disproved by the experimental or/and observational results. If a theoretical prediction is verified, the theory is enhanced and proceeds to another theoretical prediction, and another one, and so on, with this procedure being applied to all (not only one or a few!) of the available experimental or/and observational data.
However, even if just only one reliable theoretical prediction does not agree with the reliable experimental or/and observational results, then the whole theory collapses, it is rejected and, if possible, it must be improved, as regards, e.g., its main and basic assumptions.
This is and this should be the usual way scientific progress proceeds.
Notice that, so far, no astronomical-cosmological theoretical prediction of the GRT has been disproved. Up to now, all of them have been verified by observations.
Furthermore, it is obvious that, necessarily, for the research to be realistic and reliable, the theoretical predictions and the corresponding experimental or/and observational data themselves must be reliable.
What “reliable observational data” means? It means that the presented observational data are not, say so, “cooked”, namely, they, truly, express what the observations/experiments have revealed.
But, also, the underlying theory and the corresponding theoretical predictions must be reliable from both the mathematical point of view and the physical point of view.
And at exactly this point lies the most serious problem concerning the way cosmological research is done. Namely, up to now, cosmological research has been only mathematically oriented, but, not, at the same time, also physically oriented. This necessity is known since long ago, as, Einstein himself had emphasized, “As far as the laws of mathematics refer to reality, they are not certain, and, as far as they are certain, they do not refer to reality”!
Being a little more precise and detailed on this, in describing the Universe as a whole, we have many reasons to assume that, on very large scales, the content of the Universe is in the form of a continuous medium, not a system of galaxies, each one of linear dimensions small, almost negligible compared to the mutual distances of neighboring galaxies. This continuous medium, in general, is composed of various forms of mass and of various forms of energy.
The mass, both baryonic and dark, as we all are used to know, is gravitational, namely it creates a field of the familiar attractive gravitational forces, which tend to shrink the physical system under consideration.
Also, it is known that, according to the well-known formula E=mc2, any amount of energy, E, is dynamically equivalent to a certain amount of mass, m. Therefore, every form of energy creates, like mass, a gravitational field, and so both mass and energy in the Universe are responsible for the universal gravitational field, responsible for and producing all the observed motions and phenomena in the Universe.
Consequently, the cosmological physical medium, dynamically, can be treated as a continuous gravitational physical system, in other words as a gravitational fluid, and this fluid is called the cosmic fluid.
Notice, however, that, until very-very recently, in cosmological research, all forms of energy of the cosmic fluid have been ignored and omitted as sources of the universal gravitational field. In all theoretical cosmological treatments and theoretical results, all these forms of energy have been tacitly omitted and only the mass has been taken into account as the source of the universal gravitational field.
Now, what “taken into account” means? Well, the dynamicalthermodynamical description of such a gravitational fluid is very well known, and in this context, the cosmic fluid, is characterized by many physical thermodynamical characteristics, such as (beyond mass density) pressure, temperature of matter, heat quantity, heat capacity, thermodynamic transformations, polytropic transformations, internal thermodynamic motions, internal thermodynamic energy, entropy (as well as other statistical- physics characteristics) et c.
Furthermore, the relativistic dynamical-thermodynamic description of the cosmic fluid is based on the use of the socalled Field Equations of the GRT. The important feature of these equations is that, beyond mathematics (namely, the tensorial mathematics used), they contain two important physical parameters of the cosmic fluid, the thermodynamic pressure, p, and the total mass-energy density, ε.
At exactly this point lies the above-mentioned problem with the research in Cosmology, and, since long ago, this mostly serious problem, appeared as follows:
About a century ago (in 1932), two great scientists (Albert Einstein and Willelm de Sitter) proposed a solution to the field equations describing a cosmological model of the Universe, the so-called Einstein-de Sitter cosmological model. This model, in accordance, however, with the prevailing cosmological ideas a century ago, is composed only of mass, namely, it has no thermodynamic content at all. In other words, and in physical terms, in the Einstein-de Sitter model, the thermodynamic pressure of the cosmic fluid is zero, its thermodynamic content is totally ignored and, so, quite unphysically, this cosmic fluid-Universe is, simply, dust.
In their two-page long article, Einstein and de Sitter argued
“It seems certain that the pressure p in the actual universe is
negligible as compared with the material density ρ0c2. The same
reasoning, however, holds good if the pressure is not neglected”.
And, furthermore, they concluded
“…for the sake of simplicity, we neglect the pressure p…” [1].
These assumptions, a century ago, were not strange and they were in accordance with the then prevailing mathematically oriented philosophy in Cosmology. And they expressed the agony of the theoreticians, at last, to find a cosmological model of the Universe!
The strange thing, however, is that, after whole a century, this model, based on the above assumptions, has been in use until very recently. And, especially, the Universe was considered as dust, with, however, not any research problematization at all for this and for its consequences.
I could say that this is like assuming that the Earth’s atmosphere has no pressure (it is dust), and we are not interested in its consequences in the preparation of a reliable weather forecast. Or that it is like assuming that the air in this beautiful lecture hall is composed of only particles with mass, nothing else. However, based on the simple law of perfect gases, one can imagine what, really, a vanishing thermodynamic pressure in the lecture hall would mean? Simply, it would mean a vanishing temperature, whence one could ask “how everybody in the lecture hall remains alive, although frozen at absolute zero?”.
From a physical point of view, it is truly disappointing and curious that, until very-very recently, really very-very recently, to the above note not any special attention, either physical or mathematical, was given at all, especially to the fact that the cosmic fluid had no thermodynamical pressure.
The consequences of these assumptions are very important and have prevented cosmologists from solving many current cutting-edge cosmological problems.
I shall mention only one, widely known problem, that is related to the composition of the cosmic fluid: In the context of the Einstein-de Sitter Cosmological Model, among many others, like the vanishing pressure, only the (then known) “baryonic or luminous matter”, was known and was taken into account in solving the field equations of GRT.
Later, with the discovery of dark matter and given many other observational data, the Einstein-de Sitter model took the form of the currently prevailing cosmological model. Namely, in this prevailing model of generally vanishing pressure, 5% of the mass-energy content of the cosmic fluid is our familiar “baryonic or luminous matter”, 25% is the so-called and unknown “dark matter”, and the rest 70%, exotic and totally unknown, was given the name “dark energy”.
The dark energy has been denoted by Λ, the (resurrected) unknown cosmological constant, and so, it has characterized the currently prevailing ΛCDM (Λ Cold Dark Matter) Cosmological Model.
It is noticeable, also, that in the context of the ΛCDM Model, among many others, 95% of the Universe’s mass-energy content remains still unspecified! And, additionally, that in the ΛCDM Model, an unknown, the cosmological constant, is used to explain another unknown, the dark energy, a, really, absurd situation!
The way out of this dead end was revealed, about 14 years ago (in 2011), only when all the above mentioned physical thermodynamical characteristics of the cosmic fluid had been consistently taken into account as sources of the universal gravitational field.
Particularly, the matter, baryonic and dark, is not there standing immovable and frozen, as in the ΛCDM model, but, quite physically, it is characterized by collisions of their moving constituent particles, so that the kinetic energy of these particles is distributed, with, as a result, the Universe to acquire the properties of a continuous thermodynamic medium.
This is the polytropic model with collisions and thermodynamic content, and, in the international bibliography, it is known as the Polytropic Collisional Cold Dark Matter (PC/CDM) Model./
Obviously, the PC/CDM model is a much more physical fluid, but also a much more complex and difficult fluid to treat with (in the sense, also, of the much more difficult and time-consuming preparation of original scientific publications!).
In the PC/CDM model, however, almost all the current cuttingedge cosmological problems were confronted successfully, in a self-consistent and classical way with the use of only Classical General Relativity Theory and Classical Thermodynamics. In other words, neither Quantum Theory, nor Modified Gravity are necessary for the dynamical description of the observationally approachable Universe.
Such problems, definitely solved in the PC/CDM model, are, e.g., the spatial flatness of the Universe, the dark energy (as the thermodynamic internal energy of the polytropic cosmic fluid), the cosmic acceleration, the transition from deceleration to acceleration, the deceleration parameter, the age of the Universe, the cosmological constant, the Hubble Diagram of the distant supernovae, the supernova dimming, the coincidence problem (why the cosmic acceleration happened “now”), et c.
On all the above, further information and international bibliography can be found at the quite recent (2023) article [2],In Special Issue Recent Advances in Dynamic Phenomena [2]. with further extended international bibliography of the authors in it.l
Closing this short essay-lecture, I suggest that, in the future, the ambitious cosmologists, especially, the younger ones should consistently not forget four necessities:
The first, as already mentioned, is that, essentially working with Cosmology is not something to choose between other possibilities, because it is something “coming from one’s inside”, and, even if he/she is forced to do so, it is not possible to abandon Cosmology.
The second is that, always, the researcher cosmologists, especially, the younger ones, must have the courage to say and defend their free and, absolutely and well, established opinion, both scientific and nonscientific.
The third, as already mentioned, is that, according to Einstein, “As far as the laws of mathematics refer to reality, they are not certain, and, as far as they are certain, they do not refer to reality”!
And the fourth necessity, I wish them to remember always in their future cosmological research, is the above-mentioned Summary of this essay:
Realistic Cosmology is neither simply Mathematics, nor simply impressive photographs, nor simply a powerful electronic computer, nor a particle accelerator, but beyond and above all these, firstly and mainly, it is physical thought, physical way of thinking, physical intuition, physical realism, namely and in short, it is Physics!!!
References
- Einstein A, Sitter WD (1932) On the relation between the expansion and the mean density of the Universe. Proceedings of the National Academy of Sciences 18(3): 213-214.
- Kleidis K, Spyrou NK (2023) The dark energy as a natural property of cosmic polytropes - A Tutorial. Dynamics 3(1): 71-95.
