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# On some analogues of gravitational black holes

Daniel Parrochia

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# On some analogues of gravitational black holes

Daniel Parrochia

University of Lyon (France)

"There is a river called the river of no return."

Ken Darby and Lionel Newman.

## Abstract.

In this article, after reviewing some elements of the theory of gravitational black holes and then showing how the concept of "black hole" could find an analogy in the acoustic field, we attempt to develop the notion of an "informational black hole," which we introduced in a book written in French in 1994. This is an opportunity for us to revisit the idea of a "cosmology of information" that we developed at the time, and of which the third part of this article can be a brief summary.

**Keywords.** Black holes, acoustic black holes, signal theory, relativity of information, cosmology of information, quantum mechanics of information, Dirac equation.

## 1 Reminder on gravitational black holes

The idea that, under certain conditions, the gravitational attraction of a star can be so intense that no object can escape, not even tiny photons, led to the physical notion of the "black hole"<fnref n="1" />. For these conditions to be met, as we know, it is sufficient for the star in question to be dense enough for the escape velocity of any object to be greater than the speed of light. Very precisely, we must have:

$$R_S = \frac{2GM}{c^2} \approx \frac{3M}{M_{\oplus}} km, \qquad (1)$$

with M, mass of the body, $M_{\oplus}$, solar mass, $R_S$, critical radius.

<footnote n="1">¹For a historical and epistemological study of this notion, see [Parrochia 25], [Parrochia 24].</footnote>
In this case, obviously, even light cannot overcome the gravitational force. Anticipated by the remarks of Michell and Laplace at the end of the 18th century, then popularized by Wheeler in the 1960s, this notion of a "black hole" has long been accepted in physics.

Following the work carried out in the 20th century by Schwartzschild, Weyl, Oppenheimer and Snyder, Kruskal and Szekeres, Hawking and Penrose, and many others, the theory of gravitational black holes reached a certain maturity, which allowed, over time, to distinguish four forms of these objects, depending on the zero or non-zero nature of two characteristic variables: their charge and their angular momentum.

1. When the electric charge and angular momentum are zero, we speak of a Schwarzschild black hole, named after Karl Schwarzschild, who first discovered these objects as solutions to the equations of general relativity (Einstein's equations) in 1916.

2. When the electric charge is non-zero and the angular momentum is zero, we speak of a Reissner-Nordström black hole; but these fleeting objects – because it is not clear how a compact object could retain an electric charge, which is generally absorbed by the opposite electric charges located in the environment – are improbable and of little astrophysical interest.

3. When the black hole has angular momentum, but no electric charge, we speak of a Kerr black hole, named after the New Zealand mathematician Roy Kerr who found the formula describing these objects in 1963. Unlike Reissner-Nordström and Schwarzschild black holes, Kerr black holes are of considerable astrophysical interest, because they are practically the only ones that we really expect to encounter in astronomy.

4. An electrically charged version of the Kerr black hole, which like it has rotation, is known as a Kerr-Newman black hole and, like the Reissner-Nordström or Schwarzschild black hole, is of little astrophysical interest, given its very low probability.

Various concepts have been developed to clarify the nature and shape of black holes.

First, there is the "event horizon," which refers to the spherical zone delimiting the region from which light and matter cannot escape – a sort of surface of no return for anyone who reaches this boundary. This is accompanied not only by tidal effects, but also by changes in the perception of time: the observer inside the black hole will therefore have the impression that time is passing more quickly for their colleague
remaining outside. Conversely, the observer remaining far from the black hole will see their colleague moving increasingly slowly inside the hole, with time appearing to flow more slowly within the latter.

The black hole is also characterized by an absence of information or causal influence on the rest of the world. Of course, it is still considered to be the origin of the gravitational field exerted on the outside. However, it is not the singularity with its mass that is the source of this gravitational field, but the star just before its collapse into a black hole. This is because the effects of the collapse on the star's gravitational field take increasingly longer to be communicated to the rest of the universe, given the extreme (and even infinite) time dilation as the star's radius decreases and approaches that of the horizon. A remarkable theorem, however – Birkhoff's theorem (1923) – makes the gravitational field generated by a point region indistinguishable from that generated by a spherical mass distribution<fnref n="2" />. The same is true of the moment of inertia of the collapsing star, which is attributed to the rotating black hole, and its effects on the gravitational field.

The concept of "ergosphere" – which does not refer to a sphere – is that of a region in the vicinity of a rotating black hole in which it is impossible for an object to escape the dragging effect of this rotation.

An "innermost stable circular orbit" (ISCO) is also defined, which is the circular orbit closest to a black hole, beyond which matter inevitably falls into the black hole. For a Schwarzschild black hole, the radius of the ISCO is three times the Schwarzschild radius of the black hole, and we have:

$$R_{\text{ISCO}} = \frac{6GM}{c^2} = 3R_{\text{S}} = 6R_{\text{G}}. \qquad (2)$$

For a Kerr black hole, it varies as a function of the angular momentum of the black hole in question and is only 1.6 times the Schwarzschild radius for a black hole with maximum rotation.

We also speak of a "photon sphere" to designate a surface defined as the set of points from which a photon, an elementary particle associated with electromagnetic waves,

<footnote n="2">²In fact, in general relativity, Birkhoff's theorem states that any spherically symmetric solution to Einstein's equation must be static and asymptotically flat. But one of its consequences is this: Birkhoff's theorem generalizes Isaac Newton's two so-called "iron sphere" theorems. Given a spherical corona of negligible thickness, mass *M*, radius *R*, and center *O* at the origin, then at any exterior point (*r* > *R*) of the corona, the gravitational potential and field are equal to those of a point of mass *M* located at *O*. On the other hand, at any interior point (*r* < *R*) of the corona, the gravitational potential is constant and the gravitational field is zero. In other words, inside a spherical corona of dust, spacetime is flat.</footnote>
can be emitted and follow a closed and periodic orbit. This is an extreme case of
gravitational deflection, predicted by general relativity.

But the most important and mysterious concept in black hole theory is that of "singularity." At the center of a black hole lies a region in which the gravitational field and certain distortions of spacetime (or rather, the curvature of spacetime) diverge infinitely, regardless of the change in coordinates (see [Luminet 98]). This region is called a gravitational singularity. Describing this region is difficult within the framework of general relativity, since it can only describe regions where the curvature is finite.

Furthermore, general relativity is a theory that cannot generally incorporate quantum gravitational effects. The fact that the curvature tends to infinity is a sign that general relativity cannot fully describe reality at this point and that it is probably necessary to introduce quantum effects.

Therefore, only a theory of gravitation that incorporates all quantum effects (in this case, quantum gravity) would be capable of correctly describing gravitational singularities.

Describing a gravitational singularity is therefore problematic for the time being. It has even been recently called into question, since in December 2018, Abhay Ashtekar, Javier Olmedo, and Parampreet Singh published a scientific article demonstrating that, within the framework of the theory of loop gravitation, a black hole would not have a central singularity, without geometrically specifying the fate of matter at this point (see [Ashtekar and al. 18]; [Rovelli 18]).

## 2 Acoustic black holes

The notion of a "black hole" is not limited to astrophysics. Or rather, astrophysics has benefited from an analogy that has made it possible to find, in the field of sound waves, an acoustic equivalent of black holes affecting light waves. The question arose from a query about Hawking radiation, or Bekenstein-Hawking radiation, another name for the phenomenon of black hole evaporation. As we know, this is the phenomenon whereby an observer looking at a black hole can detect a tiny blackbody radiation emanating from the region near its event horizon.

It was Hawking who predicted this phenomenon as early as 1974, within the framework of quantum field theory in curved spacetime. This discovery, which strengthened a branch of the study of black holes called black hole thermodynamics, de-
veloped shortly before Hawking’s discovery, confirmed that it should be possible to associate a temperature with a black hole. However, at the classical level, it was demonstrated that a black hole could not emit radiation (this is, in a way, the definition of a black hole). This paradox was resolved by Stephen Hawking, who demonstrated that quantum effects could be at the origin of such a phenomenon.

The Hawking temperature of a macroscopic black hole is, however, incredibly low. For the black hole at the center of the Milky Way (Sagittarius A*), with a mass of about 4 million solar masses, it would be about $10^{-14}$ Kelvin. And for a black hole of one solar mass, this temperature would be only about 60 billionths of a Kelvin.

A direct experimental verification of such a temperature therefore seems unimaginable, at least for macroscopic black holes, since it would be drowned in the cosmic microwave background radiation, itself about 2.7 Kelvin.

Since it is impossible to manipulate gravitational black holes, it seemed particularly interesting to find near-equivalents on which experiments could be carried out safely. Such are, for example, acoustic black holes. The analogy is based on the relationship between sound waves and light waves. This was highlighted by William Unruh, who in 1981 showed the proximity of the propagation of sound waves in a moving fluid and that of light in a curved spacetime (see [Unruh 81]). Furthermore, he proposed to use this analogy to attack the problem of the very high energies at the origin of Hawking radiation: an analogue of this radiation should occur, according to him, in the most banal and ordinary of physical systems: flowing water (under particular conditions). The physical basis of his idea was disconcertingly simple: if water flows beyond a boundary faster than the speed of sound, an event horizon is formed, because any disturbance in the water, propagating at the speed of sound, will necessarily be "trapped" behind the boundary. Unruh then argued that the scattering of water wavelets at the boundary would occur with a thermalized spectrum, in perfect agreement with Hawking radiation.

In 1995, stimulated by a new wave of questions, Unruh revisited his 14-year-old idea and demonstrated its usefulness (see [Unruh 95]). Sound waves, characterized by a constant wavelength, frequency, and propagation speed, appear similar to light waves. However, as Renaud Parentani points out, "this parallelism does not generally extend to the quantum level. Indeed, in ordinary fluids, disordered movements of thermal origin prevent the appearance of quantum phenomena. However, in certain fluids cooled to such a low temperature that thermal fluctuations become negligible, all molecules are governed by quantum mechanics. In this type of fluid, sound waves "inherit" the quantum properties of the molecules among which they travel.
Everything then happens as if the sound waves were made up of particles –"grains of sound" – which we call "phonons" to emphasize the analogy with "photons". Thus, for these fluids, the description of sound waves, whether classical or quantum, is analogous to that of light" (see [Parentani 02], 42).

In a fluid at rest, phonons travel in straight lines, like light waves in a plane geometry. The same is true in the case of a uniform fluid flow, the only notable change being that the waves undergo a Doppler shift. However, when the flow is no longer uniform (more or less turbulent flow), sound waves no longer travel at a constant speed, and their frequency depends on where they are measured. Moreover, their trajectories in space-time are curved and, as William Unruh demonstrated, fall under a "curved acoustic geometry." We are therefore in a situation close to that of general relativity, except that this geometry is determined by the motion of the fluid and not by the distribution of matter.

This correspondence made it possible to find a type of effective flow that gives rise to an acoustic geometry comparable to the geometry around a black hole. In a so-called "de Laval" nozzle (a nozzle with a constriction that accelerates the fluid to supersonic speed), for example, "the velocity of the flow in the nozzle reaches and then exceeds the speed of sound in the fluid without causing a sonic explosion, as generally occurs when breaking the sound barrier" (see [Parentani 02], 43), and the acoustic geometry associated with this flow is very similar to that of a black hole. Waves propagating against the flow are inexorably pulled downstream, just like light waves in the inner region of a black hole. Once the boundary of this region is crossed, nothing can escape from the hole. Moreover, in the subsonic region, waves traveling upstream lose energy as they move away from the horizon, just like photons subject to gravitational reddening. We thus find the main characteristic properties of a black hole's horizon.

As mentioned, the analogy allows, among other things, to better understand Hawking radiation and the link between black holes and quantum mechanics. Indeed, we can compare the role of high-frequency photons in the radiation of gravitational black holes and that of high-frequency phonons in acoustic black holes. In the case of light, by virtue of the reddening effects that tend towards infinity near the horizon, the photons we receive seem to be witnesses to an unknown physics. But the comparison between phonons and photons will be enlightening here. The properties of short-wavelength phonons (therefore high-frequency and high-energy) are accessible and differ notably from those of photons. When the wavelength of a phonon approaches the intermolecular distance (of the order of some $10^{-10}$ meters), its properties change radically. It is as if the phonon becomes sensitive to the granular nature of the
matter that composes the fluid, or, if you will, to the "granulation of space-time" through which it travels. The consequence of this sensitivity is that phonons of different frequencies propagate at different speeds, and thus a dispersion phenomenon occurs.

However, when the phonons move away from the acoustic black hole, the microscopic properties of quantum radiation are erased. It is therefore reasonable to assume that this erasure also occurs for gravitational black holes, which ultimately allows us to place greater confidence in the properties of the Hawking flux, even though we know nothing about physics beyond the Planck scale. With this representation of a granular space-time and the strict analogy between light waves and sound waves, we are thus not far from rediscovering the ancient notion of aether as a vibratory medium that gives light its properties. We now realize that its abandonment means losing the possibility of distinguishing the behavior of particles at high and low energies.

Since then, analogous models of Hawking radiation have been discovered in a wide variety of fluid, solid, optical, and quantum systems. (See [Barcelo 11], [Robertson 12], [Jacobson 13], and [Faccio et al. 13] for an overview.)

The most interesting fact is that, since Unruh's arguments relied solely on the simple physical properties of escapeless boundaries and the low-energy behavior of thermalized radiation caused by field scattering off these boundaries, Unruh concluded that these so-called "mute holes" (mute because they are silent) could serve as experimentally viable proxies for testing the existence of Hawking radiation in gravitational black holes (see [Leonhardt-Philbin 08]).

In particular, the validity of analog models is defended on the grounds that the essential characteristics of Hawking radiation are due solely to a few simple formal kinematic conditions satisfied by a wide range of physical systems (see [Visser 98], [Unruh 14], [Unruth-Schützhold 05]). In particular, the manifestation of radiation-like behavior, formally analogous to true Hawking radiation from a black hole, has nothing to do with the specific dynamical characteristics of general relativity.

Therefore, the idea is that detecting an analog of Hawking radiation in one of these systems provides indirect but strong support for the confirmation of the existence of real Hawking radiation. There are, moreover, now several claims that an analogous Hawking radiation has been experimentally detected:

* [Belgiorno et al. 10] based on ultrashort laser pulse filaments, i.e., intense laser pulses in a transparent Kerr medium (those with third-order optical nonlinearity);
• [Weinfurtner et al. 11] based on obstructed supersonic fluid flow;

• [Steinhauer 14] based on a "black hole laser" composed of phonons in an Einstein-Bose condensate;

• etc. (as Erik Curiel wrote, "the list is long" (see [Curiel 09]).)

There is undoubtedly some debate as to whether Hawking radiation is experimentally confirmed by these analog experiments. But it is less, in our opinion, about the notion of an "acoustic black hole" than about the nature of the radiation, which is not necessarily clearly described and understood in the case of gravitational black holes.

Quantum field theory on curved spacetime differs from standard quantum field theory (based on the flat Minkowski spacetime of special relativity) in a profound respect, this difference ramifying into every part of the theory: a generic relativistic spacetime has no symmetry group comparable to the Poincaré group for special relativity. There is therefore no distinct vacuum state and no natural notion of a particle. This means, for example, that it is impossible to employ many familiar and useful techniques of standard quantum field theory, and that caution must be exercised in the use of most others. A worrying point is that Hawking radiation gives rise to many derivations that differ radically from one another in the mathematical rigor of the framework they adopt and the mathematical character of the structures they assume. Since almost all of them are valid in different regimes, using different types of physical systems and different approximations and idealizations, basing their arguments on different physical principles, with varying degrees of physical insight and intuition, the content of their prediction is not necessarily very convincing, and even less so since the derivations sometimes follow from contradictory assumptions. Now if A and not-A implies B, then B must be a tautology. Moreover, since the interpretation of quantum field theory itself, even in the flat spacetime of special relativity, is already contested, it can be assumed that it will encounter even more problems in curved spacetime, and a fortiori in the vicinity of a black hole. In short, one should therefore remain cautious about the astrophysical validity of acoustic experiments.

On the other hand, we have indeed succeeded in creating a sort of acoustic black hole, in the sense that sound waves remain trapped in a certain region of space-time, just like light waves in the vicinity of a highly attractive body.

This result should encourage us to look favorably on another analog transposition that we are the instigator of.
# 3 Informational black holes, informational universe and cosmology of information

## 3.1 Signal Processing

Since the medium of all information is a signal, we will begin by reviewing some methods for processing signals with a finite frequency band.

Assume we are given a signal with a finite bandwidth. According to Brillouin (see [Brillouin 88]), the analysis of signals with a finite bandwidth is as follows. Consider a signal system representing a message of total duration $\tau$ and a finite frequency band of width:

$$0 \le \Omega \le \Omega_M.$$

Such a system is said to have a finite number of "degrees of freedom," meaning that we only need a finite number of parameters $N$ to define the function $f(t)$ that corresponds to it. We prove that we have exactly:

$$N = 2\nu_M\tau \text{ (avec } \nu_M\tau \gg 1),$$

and, when the duration $\tau$ is large:

$$N \approx 2\nu_M\tau.$$

As Brillouin shows, we can then reconstruct the function from these parameters:

- either by decomposing it into a Fourier series: the degrees of freedom correspond to the amplitudes of the first $N$ terms of the series;

- or by using a sampling method due to Shannon, by which the function $f(t)$, which represents the message, is defined at $N$ equidistant points separated by:

$$\theta = \tau/N \approx 1/(2\nu_M).$$

In the latter case, we have:

$$f_m = f(m\theta),$$

And we have too:

$$f(t) = \sum f_m[\sin\pi(2\nu_M\tau - m)]/\pi(2\nu_M\tau - m).$$

But it appears that these two methods are in fact a special case of a more general method, due to Gabor, which decomposes the $(\nu, t)$-plane into rectangles or elementary information units of unit area:

$$\Delta\nu \cdot \Delta\tau = 1.$$
These cells correspond to the cells of quantum mechanics, with energy *E* and time *f* being two conjugate variables. Any cell is then defined by the relation:

$$ \Delta E \cdot \Delta t = h \quad (\text{with } E = h\nu \text{ and } \Delta\nu \cdot \Delta\tau = 1). $$

The only condition for these message functions defined with a finite bandwidth is that they exhibit no discontinuities or angular points.

## 3.2 Towards informational Lorentz relations

Let us now consider such a function $f(t)$. Since this function has *N* degrees of freedom $X_1, ..., X_N$, it can be represented by a point with coordinates $X_1, ..., X_N$ in an *N*-dimensional space. As Brillouin states, "each system of "cells" must correspond to a coordinate system in *S*, and a change of coordinates in hyperspace implies a modification of the system of cells." (see [Brillouin 88], 241). Using orthogonal functions to represent the given function, the total energy will be equal to the sum of the squares of the coordinates. With an appropriate choice of units, we can write, if we assign each sine and cosine term a certain kinetic energy *E*:

$$ E = \sum_{m=1}^{m=N} X_m^2 = R_N^2. $$

In other words, we define a space of energy components of the signal.

A change of reference frame in this space would therefore correspond to a local variation in the signal's energy along a certain component, that is, to a variation in energy within a certain period interval *τ* of the signal.

What could such a variation correspond to, in physical terms?

Let's assume the existence of thermal noise disrupting the message. The probability of a (or quantum) state with energy *E* and temperature *T* is equal to:

$$ p(E, T) = B e^{-\beta E}, $$

with:

$$ \beta = 1/KT. $$

*B* being a coefficient determined by the condition that the sum of the *p* values must be equal to unity. The total energy of the noise is given by the formula:

$$ E_n = \sum x_m^2 = r^2. $$
Thus, the message, represented by a point in the hyperspace with coordinates $X_m$, now appears augmented by the noise, defined by a point with coordinates:

$$X'_{m} = X_{m} + x_{m}.$$

Let the average energy of the noise be given and $P_n = E_n/\tau$ its average power. As Brillouin points out, if the noise power were strictly constant, it could be easily eliminated and compensated, but in fact, the disturbance comes from unpredictable fluctuations of the order of magnitude of the average power.

One possible way to make the signal locally clearer, however, would be to compress the noise locally, even if it means increasing it elsewhere. Such a method has been successfully applied to light, but it can undoubtedly be generalized to any type of signal.

Under these conditions, the modification of one of the signal components, with the exception of the others, would therefore have a well-defined physical meaning. It would be sufficient to consider a noisy channel in which the total energy $E_T$ would be equal to the sum of the signal energy $E$ and the average noise energy:

$$E_T = E + \bar{E}_n.$$

The aim would then be to locally eliminate the noise by attempting to act on the amplitude or phase, thus reducing, in a certain cell, the energy of this noise – these transformations having to satisfy a power invariance.

Let us assume:

$$R_N^2 = P^2 t^2.$$

where $P$ is the limiting power of the signal.

We say that we must necessarily assume the invariance of $P$ with respect to all hyperspace reference frames, given that the signal can only have a limited information power, and any attempt to increase this maximum power runs up against the power of the noise. In this case, it is well known that we are led to Lorentz relations. But since to this energy space (referring to signal theory) corresponds an information space (referring to the mathematical theory of communication), it is on this latter that we will go on our reasoning.

### 3.3 The universe of information

Working on the notion of information in the 1990s, as well as on certain methods for processing this information, we realized that statistics commonly used models
borrowed from classical mechanics. In J.-P. Benzecri's Data Analysis, for example (see [Benzecri 73]), where data tables were studied in the form of "clouds" of heavy points, well known theorems from classical mechanics such as Huygens' theorem were transposed into this field. This very theorem, in particular, which, via the moment of inertia, allows the decomposition of a cloud into sub-clouds, based on the expression linking its center of gravity to the center of gravity of a sub-cloud located on a parallel axis, was actually a static form of mechanics. But some clustering practitioners, such as Edwin Diday (see [Diday 71]), had imagined varying the shape of clouds by developing a method called "dynamic clouds," where information was progressively distributed and converged toward certain attractors<fnref n="3" />.

In this context, we therefore had the idea of considering the constantly expanding world of information as an *universe* in the physical sense of the term, and, under the assumption of the existence of an information constant *k* conceived as the maximum flow rate of information per unit of time, we tried to develop a theory of the relativity of information (see [Parrochia 94])<fnref n="4" />.

**Axiom 1:** The Shannon entropy formula $H = - \sum p_i \log p_i$ of an information source $S$ (see [Shannon 48]) – where the $p_i$ are statistical "weights" (i.e. probabilities) and the Log is a base 2 log – can be decomposed along the axes of an $n$-dimensional space $E$ associated with a given "informational universe" $U$.

**Axiom 2:** At each particular moment in time – say, each instant $t$ – the informational universe $U$ cannot produce more information than it produces. Therefore, there exists in $U$ an information constant $k$ such that $k = \text{Max} \frac{dH}{dt}$.

In an $n$-dimensional quadratic space $(E, Q)$, called "information space," we have thus

<footnote n="3">³Diday's method is described and commented on by Benzecri (see [Benzecri 73], 1, 293-296). But, as we wrote in our 1994 work, while it would be very useful, in order to rigorously define the field of attraction in question, to know the shape of the decrease in variance accompanying the displacement of the center of gravity of the classes at each iteration towards a new partition, unfortunately we do not have such a law (see [Parrochia 94], 155).</footnote>

<footnote n="4">⁴Actually, we were not the first to have such an idea. As early as the late 1970s, the Canadian physicist Guy Jumarie (see [Jumarie 75], [Jumarie 80], [Jumarie 83], [Jumarie 90]) attempted to rethink information theory within a relativistic framework. But our approach is quite different. Unlike this physicist, who only introduced the Lorentz relation to justify a *subjective* conception of information, related to observers, we retain the *objective* character given to it by Shannon's formula, but we decompose it along the axes of a quadratic space, which allows us to define an analogy between matter and information with comparable physical laws.</footnote>
generalized Shannon's formula $H = \sum p_i \text{Log } p_i$ by setting:

$$Q = \sum_{i=1}^{n} (x_i^2), \qquad (3)$$

with:

$$x_i = (p_i \text{Log } p_i)^{1/2}. \qquad (4)$$

which is already suggested by Brillouin when he analyzes noisy messages<fnref n="5" />. Furthermore, according to Shannon, we know that, in any communication channel, there exists a constant $C$, the maximum throughput tolerated by the channel in question, such that $H = CT$. Applying this formula to our information universe, we then posit:

$$H = \sum_{i=1}^{n} (x_i^2), \quad k = C^{1/2}, \quad t = T^{1/2},$$

hence:

$$\sum_{i=1}^{n} (x_i^2) - k^2 t^2 = 0, \qquad (5)$$

formula from which, as we know, the Lorentz relations are deduced.

Within the framework of this "flat" quadratic space, changes in the coordinate system of the space into which the information is decomposed obviously correspond to linear modifications, i.e. uniform movements of informations (without any acceleration phenomena).

Accounting for nonlinear modifications, and therefore nonuniform modifications (with acceleration phenomena), then requires the transition to general relativity, where the

$$\sum_{m=1}^{m=N} X_m^2 = R_N^2.$$

<footnote n="5">⁵Here is exactly what he writes (see [Brillouin 88], 241): "Consider a signal (or a message) defined by a certain arbitrary function, of maximum frequency $\nu_M$, and duration $\tau$. The function has $N$ degrees of freedom $X_I, X_2, \dots, X_m, \dots, X_N$, as has been pointed out (...). It can be represented by a point with coordinates $X_l, X_2, \dots, X_m, \dots, X_N$ in an $N$-dimensional space $S_N$ (*). Each system of "cells" must correspond to a coordinate system in $S_N$, and a change of coordinates in hyperspace implies a modification of the system of cells. We will use orthogonal functions to represent a given function and the total energy will be equal to the sum of the squares of the coordinates. With a choice appropriate units, we can write:</footnote>

<footnote>(*) Brillouin remarks in a note that this method had already been fully studied by Shannon as early as 1949 (see [Shannon 49], 166-167).</footnote>
previous formula generalizes in the manner of Einstein to:

$$ds^2 = \sum g_{ik} dx_i dx_k. \qquad (6)$$

The idea is that information varies in many ways, but always under the action of certain forces. As we expressed it at the time, "even if we do not specify a priori the nature of the forces at play, they will derive, more or less, from the geometry considered, because the tensor that describes the metric of the informational space-time and, consequently, its radius of curvature, will at the same time be the tensor that describes the "informational field" that curves this space-time" (see [Parrochia 94], 139).

There is no need, for this, to assume that masses attract each other like real masses subject to gravitation. It is enough to suppose the existence of an abstract force field represented by the curvature of the information space and which is held responsible for its local deformations.<fnref n="6" />.

## 3.4 Informational black holes and quantum mechanics of information

The Lorentzian reinterpretation of Shannon's formula therefore makes it possible to transpose all the formalisms of contemporary physics (theory of relativity, quantum mechanics, cosmology) to the domain of information. Several consequences follow from this:

1. Spacetime, in other words the Minkowski hypercone, with its different regions of the type "space" ($ds^2 > 0$), "time" ($ds^2 < 0$), or "light" ($ds^2 = 0$), will have as its counterpart an informational spacetime, the three regions of the previous hypercone then being respectively of the type "information," "time," or "sense" – "sense" associated with the constant $k$, being defined as the maximum information flow rate per

<footnote n="6">⁶Let us note, however, that there may be, in certain cases, real phenomena of attraction, as, for example, in the geographical space (a subspace of the information space) where centers like cities exert on their periphery an effect comparable to that of gravitation, inducing a number of displacements of the surrounding population. We have also modeled this phenomenon and proposed a generalization to geographers, who were already using the Newtonian formula $F = Gmm'/d^2$ to describe it. We have developed a theory of geographic relativity (see [Parrochia 06]), responding moreover to a request from geographers (see [Brunet-Dolfuss 90]). Note that R. Brunet himself has used the notion of "black hole" in geography. But it was a pure metaphor, without any formal sense.</footnote>
unit of time<fnref n="7" />. The interior of the cone defines the sphere of informational causality, but the exterior corresponds to an "elsewhere" where the informations are unrelated to one another.

2. We can then define, as in usual cosmology, different models of informational universes, closed or open, with an origin, a development, and, possibly, an end. In this kind of informational universes, the information, like matter in the physical universe, contributes to space curvature. Einstein equations hold and admit similar solutions. It is in this context that we can therefore encounter singularities analogous to those found in physical space, what have been called "informational black holes." Indeed, we can always assume that a sufficiently "dense" mass of information, located in a restricted region of informational space-time, can, beyond a certain limit, capture all information located in its vicinity and prevent it from escaping. The formula:

$$r_S = \frac{2gm}{k^2} \qquad (7)$$

then defines an analogue of the Schwartzschild radius, $m$ denoting a statistical weight or informational mass, $g$ a constant, and $k$ the maximum information flow per unit

<footnote n="7">⁷It may seem strange to equate "sense" with the maximum flow of information. But information serves to dissipate uncertainty about a random event, and in Shannon's theory, it is this uncertainty that is taken as the measure of information, so the more uncertain a piece of information is, the more interesting it is. We will therefore define "sense", without too much possible dispute, as the maximum relevant information (relevance being evaluated in the sense of a certain criterion). However, there is an area where maximum information and maximum relevant information tend to merge: it is when we take as reference the whole of the informational universe. This is deduced quite easily from a geometric reason. Let us admit that the whole of the information can be partitioned into the subset of messages endowed with "sense" and the complementary subset. If a message $M$ endowed with "sense" can be assimilated, from the informational point of view, to a point of a hyperspace $S^n$, and if the set of more or less noisy messages relating to $M$ is a small sphere $S$ with center $M$, then we can prove that the message $M$, whatever its deformation or its "noise", is practically on the sphere $S$ in question. Moreover, the larger the volume of the sphere $S^n$, the smaller the difference between senseful and informational noisy messages will appear, their distance becoming negligible in any case. It will be noted in fact that the volume of an $n$-dimensional hypersphere for large $n$, is practically entirely contained near its surface (on all that see [Brillouin 88], 241-244). For an immense and growing informational universe, assimilating the maximum possible quantity of information (or maximum complexity) to the quantity of "sense" then becomes an almost acceptable approximation. This is what led us, in the following, to name such an approximation "sense" (in quotation marks). On these bases, we apply a theory of information, expanded by means of a certain number of invariance theorems. It will be noted that the growth of the sphere prohibits a purely mathematical resolution of the problem of absolute coding, the question of the optimum stacking of spheres, moreover not resolved in the case of a well-defined sphere, having no meaning in such a case.</footnote>
of time. But nothing prevents, in this context, the existence of non-point singularities, of the annular type, which would then correspond to what one could call informational Kerr black holes.<fnref n="8" />

3. Now assuming that we can isolate particles of information (infons) or sense (semions), we can consider transposing elements of quantum mechanics into the domain of information. In the 1994 book, we gave indications for defining, in the manner of Mandelbrot, an "informational temperature", the analogue of radiation, the existence of an informational Doppler effect, step by step, de Broglie relations, an uncertainty principle, a Schrödinger equation, possible perturbations, collisions and fissions of certain informational "nuclei". Hence a physical interpretation of many particular situations (informational vacuum, initial informational singularity, etc.)<fnref n="9" />.

4. If a precise parallel could be defined between gravitational black holes and informational black holes, the latter would therefore inherit the properties and problems associated with the former, properties and problems that we have studied elsewhere (see [Parrochia 25], namely:

- Informational black holes, which can be static or rotating (like de Kerr black holes) in the multidimensional space derived from raw data, can be defined by their mass (or "weight" in the sense of probability theory), their angular momentum (defined in Diday's dynamic clouds method), and their "semantic charge," associated with the semions. The question arises as to whether we can find here the equivalent of a "no-hair" conjecture.

- Informational black holes, like their gravitational model, could present the same kind of questioning: Hawking radiation (here from the semions), Bekenstein entropy (here expressed in a Shannonian (and not Boltzmannian) form), the information paradox (where does the information go in an informational black hole and is it irretrievably lost?), wall of fire (as long as we admit radiation made of semions, corresponding to more or less intense sense effects.)

As in the case of acoustic black holes, we can also read things backwards: gravita-

<footnote n="8">⁸In a science fiction short story inspired by our work, Claude Baltz showed what would happen if some observer found himself drifting towards a singularity of this kind in informational space (see [Baltz 94]).</footnote>

<footnote n="9">⁹This pursuit of the analogy has sometimes been criticized. But this is to forget that great semanticists such as A. J. Greimas had once attempted to explore the semantic structure of language with notions borrowed from physics (every atom of sense decomposes into a semic nucleus (analogous to atomic nuclei) and contextual semes (analogous to electronic structures). See [Greimas 66].</footnote>
tional black holes are also, in a certain way, informational black holes, and all of apparently materialistic physics actually speaks to us, from start to finish, about information: physicists like Wheeler or von Weizäcker, who inspired him (see [Parrochia 21], were close to thinking this. Some recent physicists are going even further today by showing the problematic aspect of a purely materialist description of the universe (which is also called into question by the quantum conception) (see [Kakushadze 17]), by basing this quantum conception on quantum computation (see [Lloyd 18]), or even by describing the entire universe, including its early inflationary period, in terms of quantum information (see [Zizzi 00], [Zizzi 03]). All this physics of information is not without excesses. We previously opposed the total identification of physics with mathematics, a thesis defended by Max Tegmark (see [Tegmark 15]), and consider that the assimilation of the universe to a computer and of the reality to which we have access to a pure simulation (see [Vopson 23]) are baseless assertions. It remains that we can maintain the existence of two planes of reality that are in some way parallel: that of matter and that of information, which was, moreover, Landauer's way of seeing things<fnref n="10" />.

5. Among other connections, the link between the theory of relativity of information and informational quantum mechanics can be manifested locally using a formula comparable to Dirac's. Indeed, in classification theory, Shannon's formula can be associated with the information contained in the classes c of weight $p_c$ of a partition Π, such that we state:

$$H_{\Pi} = -\sum_{c} p_c \log p_c, \quad \text{for all } c \in \Pi. \qquad (8)$$

Applying substitution (4), we obtain:

$$H_{\Pi} = -\sum X_{c}^{2}. \qquad (9)$$

And for the set C of all partitions Π of a set E:

$$H_{C} = \sum_{\Pi \in C} H_{\Pi} = (Y_{\Pi})^{2} = \sum_{\Pi} \left( \sum_{C} (X_{\Pi})^{2} \right).$$

Setting:

$$Y_{\Pi} = (H_{\Pi})^{1/2},$$

<footnote n="10">¹⁰Philosophically speaking, this was already Spinoza's conception in the past.</footnote>
we obtain:

$$ (Y_{\Pi})^2 = \sum_c (X_c)^2. \qquad (10) $$

This formula, which, as in the Dirac equation for the relativistic Hamiltonian of the wave function, equates a quadratic form and the square of a linear form, generalizes here to all levels of information complexity. It brings the idea of complex information admitting more degrees of freedom (or "informational spins") the more levels of integration or internal organization it has. Indeed, we demonstrate that there exist non-zero coefficients $\alpha_i$ such that:

$$ (\sum_i \alpha_i h_i^2)^{(2\nu)} = (\sum_i \alpha_i h_i^{2\nu})^2, \qquad (11) $$

These coefficients are found in a non-commutative algebra (Hermitian or pseudo-Hermitian matrix algebra) and must verify, in the algebra considered:

$$ \alpha_1^2 = -\alpha_2^2 = -\alpha_3^2 = \dots = -\alpha_n^2 = 1, \qquad (12) $$

$$ \alpha_1\alpha_2 + \alpha_2\alpha_1 = \alpha_1\alpha_3 + \alpha_3\alpha_1 = \dots = 0. \qquad (13) $$

We therefore finally obtain equation (11) which, taking into account (12) and (13), gives:

$$ (\sum_i \alpha_i h_i^2)^{(2\nu)} = \sum_i h_i^{2\nu}, \qquad (14) $$

And since:

$$ (\sum_i \alpha_i h_i^2)^{(2\nu)} = (\sum_i \alpha_i h_i^{2\nu})^2, \qquad (15) $$

if we laid:

$$ h_i^{2\nu} = h_i'^2, \qquad (16) $$

we obtain:

$$ (\sum_i \alpha_i h_i^2)^{(2\nu)} = \sum_i \pm (h_i')^2. \qquad (17) $$

Generally speaking, given a quadratic space $(E, Q)$, $Q$ being the quadratic form of the information, we can define a mapping of $(E, Q)$ into an algebra $A$ over a field $K$, associative and possessing a unit $1_A$, which reveals the form $Q$ as the square of a linear form. The algebra in question is called a "Clifford algebra".

Two final remarks are worth mentioning:
1. Note that the solutions to equations (15) and (16) reveal what, in relation to matter/antimatter symmetry, could be called "anti-information," which remains to be interpreted. This sign-changed information is a kind of counterpart to physical negentropy. Positing an exact parallel between matter and information (as does the algorithmic theory of Bennett, Landauer, and Zurek), we reject, with Popper (see [Popper 57] and [Popper 74]), the theory of communicating vessels of Szilard and Brillouin, which claims to establish an equivalence between negentropy and information. It has been proven that it is not the acquisition of information that increases the negentropy of the system from which it is extracted. It is the destruction of information that is irreversible and requires, for the lost information to be recovered when possible, a negentropic expenditure (see the first part of [Parrochia 94]). On the algorithmic theory of information and the entire issue, see [Leff-Rex 1990], which brings together all the fundamental texts on the subject.

2. Similarly, It would be appropriate to give meaning to the notion of "informational spin." One possible interpretation suggested by the preceding construction is that the spinor groups of Clifford algebras can be realized in groups of permutations of classes, partitions, or classifications associated with the decomposition of heavy point clouds in data analysis. However, it is true that we currently lack a real algebra of classifications and an authentic general theory of the domain (see Par4).

# 4 Conclusion

In summary, we must conclude that the notion of a gravitational "black hole" has, in recent history, been extended into fields other than astrophysics. In the 1980s, an acoustic analogue of gravitational black holes was discovered, which subsequently proved useful for simulating a number of their properties. But in the midst of In the 1990s, as part of a general reflection on a theory of information, transposing into a field linked to Shannon's work the formalisms from recent physics (theory of relativity and quantum mechanics), the idea arose that there could also be informational black holes, leading to the existence of a region of space (meaning the abstract space of the different components of information) from which no information can escape. Under the hypothesis that there is an informational constant (maximum information flow of the universe per unit of time) comparable in this to the speed of light, a theory of special and general relativity, as well as the transposition of certain elements from quantum mechanics lead to an almost identical formalism that we described in an old book, published in French. We hope that this brief presentation will make the reader want to go further and learn more about this perspective.
To end on a more poetic note, we would like to respond to the reader who might ask us to provide an example or an intuitive view of this famous "information black hole" we have introduced. If they were to ask us this question, we would simply tell them to rewatch the famous Otto Preminger's film "River of No Return", with Marilyn Monroe and Robert Mitchum, and listen carefully to the lyrics of the song by Ken Darby and Lionel Newman:

"There is a river called the river of no return.
Sometimes it's peaceful and sometimes wild and free!
Love is a traveler on the river of no return.(...)

This reference is not entirely irrelevant, if we remember that, in the case of acoustic black holes, it is water flowing beyond a boundary faster than the speed of sound, which generates the black hole (knowing that the sound will necessarily be "trapped" behind the boundary). In the case of information black holes, beyond a certain horizon, a flow of informational events with particular weights takes on the same unidirectional character as entropy, the subject who undergoes this situation being caught in an irreversible process (thus the gambler who is going to ruin himself, the drug addict condemned to always increase the doses, the patient suffering from a fatal disease and whose condition deteriorates every day, etc.<fnref n="11" />).

So, as in astrophysics with gravitational black holes, there are also journeys in life without return. There are also journeys of no return in life. And perhaps love is one of them. But more surely life itself.

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