Black Holy Logic

Today, it is unanimously accepted that the Universe swarms with the so-called black holes. Astronomers report hundreds of already identified cases, while the observable behavior of many other radiation sources can only be explained by the presence of a yet undetected massive companion, presumably a black hole. The presence of black holes in galactic nuclei and quasars is considered a well-established fact beyond any discussion. For all that, the nature of black holes remains as dark as in 1915, when Karl Schwarzschild has found his second ("outer") solution of Einstein's equations, or as in 1964, when the term "black hole" was coined. So far, the only meaningful definition of a black hole can only state that it is a compact massive object that cannot be described by any existing scientific theory in a consistent manner, eliminating any unphysical singularities. Picturing black holes as monsters devouring matter is more appropriate in yellow press than in a scientific or philosophical discussion.

I am not going to delve into mathematical intricacies and the details of formal derivation. These can be found in the special literature. My goal is to draw attention back to the practical foundations of science and its logical structure; any topic will do for that purpose, black holes being no worse than anything.

The mathematical abstraction of infinity is practically useful, since it allows us to split complex problems into more tractable parts without too much computational overhead. The word "infinite" could be translated into common language as "providing enough room for action without bothering about applicability issues". In real nature, there no zeroes and infinities, as well exact equality and identity. At any moment, we choose a limited number of nearest goals, with all the rest serving as a rich background. As long as we can make yet another step forward, we have an infinity in front of us. Sometimes, this freedom would take on the sense of "too much", as we get tired of keeping on with the same activity and seek for a different occupation. This sort of behavior has been reflected in the dialectical principle of measure, indicating that quantitative changes cannot accumulate but within some natural limit; when some quantity becomes excessive, a qualitative leap will necessarily follow. Any physicist learns this attitude in school; but then some theoretically minded individuals happen to forget it and switch to a religious belief in the physical existence of mathematical abstractions.

So, let us repeat once again: there are no infinities in physics. Speaking about point particles, or material points, we only mean that some physical systems can be treated as if there were a point source somewhere far away. Speaking about singular potentials, we mean that they are like that far enough from the point of singularity. Speaking about infinitely remote regions in space and time, we only discard any influence of distant objects to the dynamics of interest. In this respect, physics is like computer science, where continuous models can be used to derive meaningful statements about essentially discrete systems. In physics, we are always to guard the limits of measure; if something grows or diminishes too fast, it's high time to change the scale.

The mystical feeling towards the power of mathematics makes people forget that mathematics too is not free from inherent logical problems, and even the most rigorous inferences are based on a shaky foundation of very strong assumptions, restricting our reasoning so that we would inevitably arrive to the desired conclusions. In the strict sense, mathematics is mere tautology, it is entirely based on logical circularity, and mathematical truths are of no use unless we break their ostentatious rigor and apply formal results to something obviously informal.

Black holes provide a vivid example of a logical hole in theoretical physics. Normally, any mathematical result will only be used within the limits of its applicability. Extrapolating any formulas beyond this area is acceptable as a loose analogy, or even a metaphor, but never a means of scientific study. Such a deliberate departure from the logic of science can be a kind of probe to mark out the boundaries of the applicability region and indicate the promising directions of development. With black holes, the sense of measure has abandoned the minds. An evident contradiction becomes an occasion for loose speculations far from any scientific standards.

Indeed, trying to obtain an exact static solution of the general relativistic equations, Schwarzschild originally assumes that there is a point mass, and its gravity is to determine the structure of the whole space. After a number of formal manipulations, he comes to a formula strictly isolating one region of space (the Schwarzschild sphere) from the rest, so that the point mass staying inside the isolated region can no longer be responsible for any physical effects at longer distances. This is an obvious contradiction, but it does not prevent us from using this theory in the space areas far from the center of gravity, with the reservation that the validity of results in the regions closer to the mass is to be verified using a more accurate approach. This kind of trade-off between mathematical rigor and physical sense is quite common in physics. For instance, the well-known Rayleigh–Jeans law describes the spectrum of blackbody radiation at low frequencies, but fails to converge for shorter wavelengths, which was known in the early 1900s as the ultraviolet catastrophe. On the contrary, the Wien's approximation works fine at high energies, but fails in the long wave domain. To reconcile the two limit cases, Max Planck has suggested a semi-empirical distribution that has been found to correspond to the picture of quantized radiation. This how the triumph of quantum physics began. The theory of black holes might reproduce that story. The inner and outer Schwarzschild solutions could be interpreted as the limit cases of a general law introducing a new physical constant (virtually equivalent to some fundamental length) and opening a new chapter in the history of physics, marked with a natural synthesis of quantum physics and relativism.

The adepts of black holy logic will certainly object. They believe that the Schwarzschild solution holds both for the outer and inner regions of the critical sphere, and the presence of singularity reflects the complex nature of gravity. Those who disagree are obviously lacking phantasy and education to acknowledge the great discovery of the XX century.

Just for comparison: there is yet another relativistic singularity, which the same believers are inclined to treat in an entirely different manner. Trying to derive the Lorentz transform from a few basic physical assumptions, we come to a singular solution dividing the whole space-time into two disjoint regions, just like the Schwarzschild solution does with gravity. However, in this case, everybody agrees that any communication between the objects separated by a space-like interval is utterly impossible, and they cannot influence each other's dynamics. Following this logic, we should admit that a point mass inside the Schwarzschild sphere were entirely hidden from the outer observer and, of course, it could not produce any physical forces.

For an unprejudiced thinker, the presence of any formal singularities in a physical theory means that its basic assumptions are not universally valid, and we need to reconsider the foundations of the theory in the regions of extraordinary behavior. This is a general principle working in classical physics as well as in relativistic and quantum theory. Thus, in classical electrodynamics, we can consider electrically charged point particles, which immediately results in a singular potential in the region around the source. This approximation seems to be experimentally proven up to very short distances, but we can predict, on the logical grounds, that there must anyway be a limit where we'll need to question the validity of the traditional notion of electric charge and reconsider the very idea of a point particle. Moving down to the center of an electron, one will eventually come to a different picture of reality, an inner space of the electron with the electron's charge being s global property of this inner structure, which may be quite different from mere spatial distribution. In a way, the Schwarzschild singularity is related to the same abstraction of a point particle, though we do not yet any experimental evidence of an essentially quantized spectrum of mass. Approaching the Schwarzschild radius, we'll get in a new space related to the usual mechanical space in a statistical manner.

It is well known that quantum physics does not remove any classical singularities, as it is entirely based on the same spatiotemporal continuum. In other words, we still stick to the abstraction of a point particle, albeit represented by a continuum of its virtual clones. The configuration space of quantum mechanics (and, of course, quantum field theory) consists of all the possible distributions in a virtual space-time, which only aggravates the malady, admitting functional spaces much more powerful than mere continuum. Modern string theories suggest a logically attractive remedy against any unphysical singularities: all we need is to add yet another spatial dimension and get over the singularity through that additional dimension. That is, a singularity appears when a regular structure gets projected onto a subspace of the whole space, as an artefact of a narrowed view. However, one could readily predict that string theories will necessarily face the same difficulties, as they fight one kind of singularity introducing yet another kind, in higher dimensions. Instead of sweeping the junk under the carpet, we rather stock in on the roof.

In the heart of the problem, we find the old philosophical question about the nature and role of discreteness and continuity in the physical world and human activity. One cannot dismiss neither of these poles; they are irreducible to each other and hence need to be treated as complementary aspects of any reality to be reflected in a consistent reasoning. As far as I can see, the only possibility of reconciliation leads to a hierarchical approach combining the opposites on a developmental basis, so that the points and infinities of one level would become extended structures on another. The formation of such a hierarchy is a natural process like any physical motion, but we can only reflect it in our notions and actions to the extent of our cultural development, that is, within the already assimilated portion of the world.

Meanwhile, let us discuss yet another logical fallacy in the theory of black holes. Modern cosmology says that black holes normally appear in the gravitational collapse of the stars that have burned off their stock of hydrogen and helium so that the weight of the outer layers of the star can no longer be countervailed by the pressure of radiation produced in thermonuclear reactions. For the stars with the mass greater than some critical value (above 3–4 solar masses), this collapse will push the star's matter into the Schwarzschild sphere, and then the newborn black hole will grow absorbing matter and radiation from the nearby space. The special literature is replete with the calculations of such accretion; the overall picture is unanimously accepted by all the specialists, and any physical models only differ in minor details to be adjusted in the future complete quantitative theory.

Here an arrogant dabbler butts in once again with his idiotic questions. Do stars really need to collapse? Well, in some cases they could behave that way, for instance, producing neutron stars that have nothing singular about them and hence are quite acceptable from the logical standpoint. But why should we admit the real existence of unphysical infinities for collapsing stars above the Chandrasekhar limit? Isn't it more natural to fancy a peculiar state of matter that would stand the gravity of the outer layers without any supernatural assumptions about adjacent worlds and wormholes?

For an illustration, let us look at our buildings, the solid constructions that do not collapse under their weight despite the fact that the atmospheric air inside then can hardly resist the pressure of stone, steel and glass. Moreover, within our houses, we do not feel the weight of all the stories above us, though, of course, we would be smashed flat if the construction lost its balance for some reasons. This is possible because different states of matter are differently organized, and, in particular, solids are very unlike liquids and gases. The distribution of pressure in solids is essentially anisotropic, and that is why an empty solid sphere would not collapse to its center, even if we put there yet another mass. The Newtonian theory of gravity tells us that the gravitational potential within a thin massive sphere is exactly zero everywhere in the whole volume. That is why we should not expect any excessive pressure inside a solid ball of any mass at all. Moreover, the gravitational potential and pressure in the center of such a ball could be close to zero.

All the theories of star evolution picture them as gas balls obeying the standard laws of gas dynamics, slightly modified to account for relativistic and quantum effects. A star is assume to be in the same phase state all over its volume during all its life. Thus, the classical Landau's paper (To the Theory of Stars, 1932) described the collapse of the cold Fermi gas; obviously, in the process of collapse, the gas cannot remain always cold (there is no such thing as adiabatic collapse), and its temperature (along with the Fermi energy) will rapidly increase with compression beyond a certain physical limit. The logic of collapse is inapplicable to very hot systems, and therefore, probably, to the absolute majority of astrophysical objects. So, what is the use of that much ado about nothing?

As everybody learns from everyday experience, even the same phase state of matter feels quite differently in different conditions. For example, the air is normally thin and impalpable, but it becomes very dense and viscous in the strong wind, and it can even kill, in the form of a shock wave. It seems natural to admit that, in the conditions of extremely high pressure, matter could develop much more efficient modes of resistance preventing any collapse. Such an assumption would be much more reasonable than the mystical realization of abstract singularities. For one possibility, very hot compressed matter could transform into pure radiation, or into any other state without any massive particles, well before reaching the Schwarzschild radius. A kind of photon gas could then be compressed to a photon liquid, or even into a solid; this is contrary to the traditional electrodynamic picture of non-confinable photons; however, the latest achievements of quantum technologies indicate that a system of photons can indeed form something like bound states. Eventually, a part of gravity force can become radiation, or conversely, radiation could become a sort of repulsion gravity preventing too much gravitational compression. Of course, these are merely fantastic hypotheses, but they demonstrate how futile the discussion of gravitational collapse, black holes and similar matters can be in the face of the real complexity of the material world. One should not trust too much all what physicists use to say. Anyway, they are certain to change their opinion if nature is insistent enough.

With more options that might come form the possible singularity-free generalizations of general relativity, the chances for the formation of black holes in gravitational collapse become practically infinitesimal. However, there is still some room for black holy logic as it comes to considering such extremely massive objects as galactic nuclei. Today, we have collected presumably convincing astronomical evidence of very compact massive bodies in the central regions of some spiral galaxies. Extremely massive compact bodies are also deemed to be responsible for the activity of quasars. Well, let us admit for a while that we need black holes to explain such observations. This will raise the same question: how that huge masses could accumulate? The traditional accretion picture obviously does not work in this case. Even if galactic nuclei were believed to exist long enough to pack so much matter in a black hole, this is certainly not the case for quasars that are deemed to be among the oldest astronomical objects, and therefore, their hypothetical black holes, in the cosmology of the Big Bang, just would not have time enough to grow to the scale observed.

Of course, there is always the magic wand of cosmic catastrophes. In the young (and more compact) Universe, the chances of galaxy collision should have been much higher, and that is how the old quasars could form. Unfortunately, in this case, we have to abandon the very convenient hypothesis of a genetic kinship between active galaxies and quasars, but one can surely afford that, to save a sheer mathematical abstraction!

But there is yet another aspect of the problem. As commonly known, the Schwarzschild radius is proportional to the mass M. Now, the average density of matter inside a sphere of radius r is defined as M / r³; this means that the densities required for the formation of a black hole are ~ 1/M². That is, for the masses of the metagalactic scale, black holes could easily form from a very cold and rare gas, without any need for collapse, or other dramatic events. In the generally accepted framework of cosmological expansion, black holes could form in piles before any galaxies and stars were born. In this picture, our Universe would certainly swarm with black holes, which were to precede stellar evolution rather than be its final stage.

All right, let us assume for a moment that this was the true mechanism of the formation of galaxies. First, many black holes of all sizes formed in the early Universe, they grew in pace with the cosmological expansion and became the centers of attraction for the surrounding gas to form stars and galaxies. This is exactly the inverse of the traditional picture, but who knows? This picture can perfectly explain the mysterious dark matter as a gas of microscopic black holes a hundred times heavier than proton. What a space for theoretical imagination!

However, if we make yet another step towards the origin of everything, we stumble over the obvious conclusion that, in a very tender age, the whole Universe should stay within a black hole! The Schwarzschild radius for the total mass of the Universe is huge, and we do not need to go too deep in the heart of the proto-matter to get drowned in the progenitor black hole. The usual laws of physics are enough to explain the material dynamics of that time. So, how could we get out of the mess?

One possible answer is that there is a hierarchy of black holes, and we still live inside a higher-level black hole, while new black holes are born as a result of gravitational collapse. In this case, we must admit that the inside of any black hole is just a Universe like ours; and our Universe too has resulted from some instance of higher-level collapse. Such a picturesque view could gain easy popularity and generously stimulate the writers of science fiction.

On the contrary, a physicist would rather prefer a less exotic, and probably too dull an explanation. The obvious objection is to indicate that in the homogeneous proto-Universe we cannot use the second Schwarzschild solution that was designed for the empty space around the gravitating mass. By why then do you apply the same solution to collapsing stars? Do you think they contain sheer vacuum? Why do you interpret the experimental evidence in terms of a mathematical abstraction that is entirely due to the insufficiency of the present physical theory? All we need is to get rid of the unphysical singularities, and the way we do it will immediately tell us how our views on the early history and the destiny of the Universe should be modified.

No doubt, having done with black holes (and all the logic behind them), we'll have next to give up the cosmological singularity, on the similar grounds. The idea of expansion from nothing is yet another illegal extrapolation; it has nothing to do with physical reality, being a mere approximation that can only be used in the regions far from singularity. Quite probably, all such relativistic singularities are due the very basics of relativity theory, to the singular character of the Lorentz transform. The future generalizations are to remove this primary singularity, thus enormously extending our view of the Universe no longer limited by the light barrier. In this greater Universe, local expansion will always be complemented by as local contraction, and the very presence of such cosmological effects will be explained by the specific choice of the reference processes and the corresponding structure of the frames of reference.

When science grows to more maturity, it will certainly drop the childish habit of stretching a special physical theory onto the whole Universe. The world is bigger than any human fantasy; it will always be a source of surprise and discoveries for any science at all, including physics. Conscious beings are eventually to get conscious of that circumstance as well.