# Classic but controversial objects: black holes

Daniel Parrochia

► To cite this version:

Daniel Parrochia. Classic but controversial objects: black holes. 2025. hal-05168382

HAL Id: hal-05168382

https://hal.science/hal-05168382v1

Preprint submitted on 17 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Classic but controversial objects: black holes

Daniel Parrochia

University of Lyon (France)

<figure><img src="image_2.png" /><figcaption>Figure 1: Accretion disk of the black hole of the galaxy M87</figcaption></figure>

If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from light.

[Michell 1784], 50.

Il existe donc dans l'espace céleste, des corps obscurs aussi considérables, et peut-être en aussi grand nombre, que les étoiles. Un astre lumineux de même densité que la terre, et dont le diamètre seroit deux cent cinquante fois plus grand que celui du soleil, ne laisseroit en vertu de son attraction, parvenir aucun de ses rayons jusqu'à nous ; il est donc possible que les plus grands corps lumineux de l'univers, soient par cela même, invisibles.

[Laplace 1798], 347-348.
**Abstract.**

In this article, which addresses the questions raised by the existence of these objects, which are among the most mysterious in the universe – black holes – we attempt to reconstruct the genesis of the knowledge that has developed about them, before addressing crucial questions concerning their existence, nature, and form, as well as the consequences of these questions for our knowledge of the universe and its destiny. These questions are all the more crucial given that, after being anticipated, then forgotten for over a century, then rediscovered, and finally increasingly accepted, black holes are today once again contested objects, particularly due to certain advances in quantum physics.

**Keywords.** Schwarzschild radius, horizon, singularity, Kruskak-Szekeres metric, Hawking, Penrose.

# 1 Prehistory of Black Holes

As Jean-Pierre Luminet shows, "it was by combining the idea of a finite speed of light and the idea of escape velocity, inherited from Newton, that the English rev- erend John Michell (see [Michell 1784], [Montgomery et al. 09]) and Pierre Simon de Laplace (see [Laplace 1798], 347-348), at the end of the 18th century, brought to light the most fascinating consequence of gravitational attraction: the black hole" (see [Luminet 87], 22). Obviously, the word had not yet been invented. It was not until the 1960s that this was coined<fnref n="1" />. But the idea is there, even though we know nothing about the formation of these possible occluded stars. Similarly, while the philosopher Immanuel Kant did not anticipate black holes, he did, in his *General History of Nature and Theory of the Heavens* (1755), note that stars attract and condense, and thereby envisaged that "all the systems of the Universe are, by virtue of this incessant condensation, which nothing can stop, led to fall upon each other and to reunite sooner or later into a single mass" (see [Kant 86], 136), as if the

<footnote n="1">¹Science writer Marcia Bartusiak attributes the origin of the term "black hole" to physicist Robert H. Dicke, who, in the early 1960s, reportedly compared the phenomenon to the Black Hole of Calcutta, known for being a prison where one entered and never left alive. The term "black hole" was subsequently used in the magazines *Life* and *Science News* in 1963, and by science journalist Ann Ewing in a January 1964 article (see [Ewing 64]), which reported on a meeting of the American Association for the Advancement of Science held in Cleveland, Ohio (see [Bartusiak 15]). In December 1967, a student reportedly suggested the term "black hole" during a lecture by John Wheeler. Wheeler adopted it for its conciseness and "advertising interest," and it quickly caught on (see [Note of Scient. Amer. 16]), leading some to attribute the expression to him.</footnote>
universe were to become an immense black hole<fnref n="2" />.

The development of observational astronomy, the successive discoveries of white and black dwarfs, neutron stars, supernovae, and pulsars, and the increasingly proven success of the theory of general relativity gradually gave substance to these initially purely theoretical entities, long remained, as Brandon Carter wrote in a historical article (see [Carter 06] "in a state of semiclandestine gestation").

As we shall see, this happened in stages. As early as 1916, Schwarzschild wrote two articles describing exact solutions to Einstein's equations linked to what then appeared to be singularities in the metric used. The reasoning was purely theoretical, but around the 1930s, Indian physicist Subrahmanyan Chandrasekhar (1910-1995) calculated the exact limit<fnref n="3" /> beyond which a white dwarf becomes unstable and, under certain conditions, collapses into a neutron star, initiating the supernova process. This limit, now known as the Chandrasekhar limit, would earn him a heated controversy at the time with the English astronomer Sir Arthur Eddington.

Then it was Oppenheimer and Snyder, in their study of the collapse of a simplified, spherical model star with no internal pressure, who brought the subject back to the forefront. Using the equations of General Relativity, they discovered that, under certain circumstances, gravitational pressure is so great that it prevents the formation of a stable neutron star. According to them, nothing can then stop the star from contracting on itself.

Let us now delve deeper into this story and revisit its developments in more de-

<footnote n="2">²The idea that we could live in a black hole, put forward again recently (see [Schamir 25]), following the observation by the JWST telescope that all galaxies rotate in the same direction, has been often mentioned throughout history (see, for example, [Pathria 72], [Stuckey 94], [Easson-Brandenberger 01], [Tatum 18], [Chandrasekhar 31]).</footnote>

<footnote n="3">³This is a mass: the Chandrasekhar mass or Chandrasekhar limit corresponds to the limiting mass (representing 1.44 times that of the Sun) up to which a white dwarf, having exhausted its gas, exists thanks to Fermi pressure. Beyond this, it collapses to form a neutron star or, possibly, a black hole (voir [Chandrasekhar 31]). The Russian physicist Lev Landau (see [Landau 32]) made a similar discovery around the same time. The Chandrasekhar limit was observed to be true for 80 years but after 2000 it was found that there are many white dwarf whose masses are more than $1.44 M_{\odot}$ because several supernovae have been observed that are very bright, and thought to have originated from white dwarfs whose masses exceeded the Chandrasekhar limit. Stars sufficiently massive to pass the Chandrasekhar limit do not become white dwarf stars bud explode as supernovae. Then there is another limit, discovered by Toman, Oppenheimer and Volkoff. If the final mass is below the Tolman-Oppenheimer-Volkoff (or TOV) limit, then neutron degeneration pressure contributes to the balance against gravity and the result will be a neutron star; but if the total mass is above the TOV limit, the result will be a black hole. Observations currently place this limit between 2.2 and 2.7 solar masses.</footnote>
## 2 From Schwarzschild to Oppenheimer and Snyder

It all really began with Schwarzschild's two articles dating from January and February 1916, shortly after Einstein's publication of the foundations of the Theory of General Relativity. In a 1915 lecture, later published by the Royal Prussian Academy of Sciences in Berlin, Einstein presented the field equations of gravity, covariant under transformations whose determinant is equal to unity, and also showed in an addendum that each of these field equations was generally covariant when the scalar of the matter energy tensor vanished.

Recall that Einstein's equations of gravitation result from two requirements:

1. Reduce to Poisson's equation in the limit of low speeds and weak gravitational fields.

2. Satisfy Einstein's fundamental intuition that the presence of matter is equivalent to a curvature of space.

Given these constraints, the only equation that can be written takes the form:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \lambda g_{\mu\nu} = \frac{8\pi G}{a}T_{\mu\nu},$$

and constitutes the Einstein equation, with $a = 1, \lambda$ the cosmological constant, $g_{\mu\nu}$ the gravitational field, $G$ the gravitational constant, $R_{\mu\nu}$ the Riemann curvature tensor, and $T_{\mu\nu}$ the energy-momentum tensor.

An important confirmation of Einstein's theory then emerged with the secular rotation of Mercury during its orbital motion, a rotation discovered by the French astronomer Le Verrier in 1859. Indeed, in Einstein's theory, this quantity of approximately 45" per century (which constituted Mercury's perihelion shift) was now explained qualitatively and quantitatively without the specific assumptions he had formulated. Einstein also detected a stronger bending effect on light due to the gravitational field – twice as strong, in fact, as his previous study had assumed. In this last text, however, Einstein provided only approximate solutions to the field equations.

Schwarzschild, in his first article of January 1916 (see Sch1), found, to Einstein's astonishment, an exact solution to these equations.
Whatever criticisms one might level at Einstein's paper on the question of Mercury's perihelion, and whatever differences of opinion Schwarzschild and Einstein may have in their respective papers, the central point is this:

Schwarzschild's original "unique exact solution" to Einstein's field equations in vacuum for a point source is a squared world-line 4-element, which, in polar coordinates, is given by the formula:

$$ds^2 = (1 - \frac{\alpha}{R})dt^2 - (1 - \frac{\alpha}{R})^{-1}dR^2 - R^2 d\phi, \quad (1)$$

where:

* $R = (r^3 + \alpha^3)^{1/3}$;

* $\alpha$ is the Schwarzschild radius, equal to the gravitational radius $r_\alpha$ doubled, or $\alpha = 2r_g = 2GM/c_0^2$;

* $r$ is the radial distance of the coordinates, not strictly determined, but used as a measure of the distance to the source;

* $t$ is the coordinate of time ("far");

* $R$ is the approximation of $r$ to the precision of $\alpha^3$ in the original Schwarzschild metric.

We should immediately note this particularly important fact: at some point in history, the Schwarzschild metric, in the literature, took a form where $R$, an auxiliary quantity ("Hilfgrösse" in Schwarzschild's terminology) was replaced by $r$, a radial coordinate, so that the metric became divergent. On the contrary, the original metric, as mentioned, does not diverge: $R \to \alpha$ when $r \to 0$. This means that, strictly speaking, the concept of a "black hole" does not derive from the theory of general relativity. Einstein himself, as we know, resisted this idea.

When did the confusion between $R$ and $r$ arise? It is possible that Oppenheimer and Snyder were responsible<fnref n="4" /> who, in their 1939 article on the collapse of certain stars at

$$gH(r^*) = (1 - \alpha/r^*)dt^2 - (1 - \alpha/r^*)^{-1}dr^{*2} - r^{*2}d\Omega^2.$$

<footnote n="4">⁴It is also possible that an article by Hilbert (see [Hilbert 17], 24), dating from 1917, that is, a few months after Schwarzschild's articles, misled them. Hilbert, in fact, had first introduced the radial coordinate $r^* = [C(r)]^{1/2}$, then solved the vacuum field equations for the two remaining unknowns, obtaining:</footnote>

<footnote>Obviously, the use of $r^* = [C(r)]^{1/2}$ transforms the position of the point mass with coordinate $r = 0$</footnote>
the end of their lives, launched the idea of a "continuous gravitational contraction," with the line element outside the $r_b$ limit of stellar matter then taking the form:

$$ds^2 = e^{\nu}dt^2 - e^{\lambda}dr^2 - r^2(d\theta^2 + sin^2\theta d\phi^2). \qquad (2)$$

with:

$$e^{\nu} = (1 - r_0/r), \quad e^{\lambda} = (1 - r_0/r)^{-1}.$$

Here, the authors comment, "$r_0$ is the gravitational radius, related to the gravitational mass $M$ of the star by $r_0 = 2GM/c^2$, and constant. We should now expect that, since the pressure of the stellar matter is insufficient to support it against its own gravitational attraction, the star will contract, and its boundary $r_b$ will necessarily approach the gravitational radius $r_0$. Near the surface of the star, where the pressure must be low anyway, we should expect a local observer to see the matter falling inward with a speed very close to that of light; for a distant observer, this motion will be slowed by a factor $(1 - r_0/r_b))$. All the energy $e$ emitted outward from the surface of the star will be considerably reduced as it escapes, by the Doppler effect of the receding source, by the large gravitational redshift, $(1 - r_0/r_b)^{1/2}$, and by the gravitational deflection of the light which will prevent the escape of the radiation except through a cone around the outward normal of the aperture progressively narrowing as the star contracts. The star thus tends to close itself off to all communication with a distant observer; only its gravitational field persists. We will see later that although it takes, from the point of view of a distant observer, an infinite time for this asymptotic isolation to be established, for an observer co-moving with the stellar matter this time is finite and may be quite short" (see [Oppenheimer-Snyder 39], 456).

What Schwarzschild called "$\alpha$" became "$r_0$," which is not a problem. Instead, $r$ was substituted for $R$ in the denominator, and the auxiliary quantity was simply replaced by the radial coordinate, a concept Kruskal and Fronsdal would later adopt in the 1960s and which would subsequently become commonplace in all subsequent textbooks.

<footnote>into $r^* = [C(0^+)]^{1/2}$. Of course, as Schwarzschild's result shows, it is simply $\alpha$. But Hilbert, on the contrary, posited that $r^* = (x^2 + y^2 + z^2)^{1/2}$, so that the point mass represented by $gH(r^*)$ was now located at $r^* = 0$, this error not having been noticed by his contemporaries. As a result, in each spatial section, the locus $r^* = \alpha$ was no longer a point but a 2-sphere, and therefore no longer constituted a quasi-regular singularity of Hilbert's spacetime $S_H$. This error, according to Abrams, had dramatic consequences. For later, Kruskal and Fronsdal estimated that, with $r^* = \alpha$ having the character of a 2-sphere in hypersurfaces at $t = \text{constant}$, $S_H$ becomes analytically extensible to $r^* > 0$ and the spacetime thus extended contains a black hole. It follows that the theoretical basis for spherical black holes could well be based on an error by Hilbert! (see [Abrams 01], 2).</footnote>
In 1916-1917, however, thinking was different. In a second paper published in Febru-
ary 1916, dealing with the case of the gravitational field inside a ball of incompressible
fluid (see [Schwarzschild 16²]), Schwarzschild described a linear element of spherical
space with a radius of curvature that represented the geometry inside the fluid ball
(e.g., the sun). After changing the variable (χ, θ, φ replacing x₁, c₃, x₃(ix)), the linear
element in question, inside the sphere, took the form:

$$
ds^2 = \left( \frac{3\cos\chi_a - \cos\chi}{2} \right)^2 dt^2 - \frac{3}{\kappa\rho_0} [d\chi^2 + \sin^2\chi d\theta^2 + \sin^2\chi\sin^2\theta d\phi^2], \quad (3)
$$

which, for $dt = 0$, simplified to:

$$
-ds^2 = \frac{3}{\kappa \rho_0} [d\chi^2 + \sin^2\chi d\theta^2 + \sin^2\chi \sin^2\theta d\phi^2], \quad (4)
$$

which Schwarzschild immediately recognized as the geometry of a spherical space
(see [Schwarzschild 16²]).

Outside the sphere, the shape of the linear element remained the same as in the case
of the mass point.

A few months later, physicist Ludwig Flamm confirmed that the geometry inside the
sphere was not Euclidean. But he also showed that, in the case of the mass point,
the same was true, specifying in particular that, "around a central mass, each central
section has the same geometry as a surface generated by a parabola rotating around
its directrix" (see [Flamm 16]). And, connecting the two situations, he asserted that
the spherical shell that represented each central section of the fluid while retaining
its metric properties must touch the segment of the rotation paraboloid that fulfilled
the same function outside.

Clearly, if one wanted to unite the interior and the exterior, the image of the "funnel"
(as he called the surface associated with the drawings in Fig. 2) had to be connected
to a sort of small spherical cap.

3 Weyl’s Interpretation

Resuming the question of the Schwartzchild metric in his article on gravitation, Weyl focuses his attention on the cuts made in the planes $x_3 = 0$ passing through the origin, which leads to the introduction of the following spherical coordinates:

$x_1 = r\cos\theta, \quad x_2 = r\sin\theta.$
<figure><img src="image_3.png" /><figcaption>Figure 2: Flamm's Diagrams</figcaption></figure>

He then obtains:

$$d\sigma_2 = hd\sigma_2 + r^2 d\theta^2.$$

This metric, Weyl explains following Flamm, refers to the geometry of an object generated by the rotation of a parabola around an axis, whose equation is:

$$\sqrt{8a(r - 2a)},$$

according to its representation in a Euclidean space with orthogonal coordinates $x_1, x_2, z$. When we consider the projection of the object in the plane $z = 0$, it covers the exterior of the circle twice, $r \ge 2a$. But the object does not project itself into the interior of the circle, according to $r < 2a$. Each point in this domain corresponds to two points on the object. These two coverings are separated by the sphere of radius $2a$ on which the mass lies, and where the metric becomes singular. We should designate these two halves as the "exterior" and "interior" of the point mass. Weyl thus rediscovers the equation of the parabola, which is the meridian of the plane section of the hypersurface. Interestingly, however, he considers its extension along a second sheet, extending into another four-dimensional space. He explicitly specifies that there is nothing "inside" this sphere $S_2$ of radius $2a$ (the "central sphere"). The term "covering" ("revêtement" in French) is used, and particularly appropriate in
the mathematical and topological sense of the term. Thus, the spatial part of the hypersurface becomes the two-sheet covering of a manifold whose boundary is the sphere of radius $2a$.

In the remainder of the article, Weyl introduces a form of the so-called "conformal" solution, taken from the work of Adler, Bazin, and Schiffer (see [Adler et al. 75], section 6.2, 196). The authors introduce a new space coordinate that they call $\rho$. Weyl, for his part, calls his new space coordinate $r'$, and then finally replaces it with the letter $r$.

Let us introduce, Weyl explains, a new coordinate system. The transformation equations are as follows:

$$x'_{1} = \frac{r'}{r}x_{1}, \quad x'_{2} = \frac{r'}{r}x_{2}, \quad x'_{3} = \frac{r'}{r}x_{3}, \quad r = \left(r' + \frac{a}{2}\right)^{2}\frac{1}{r'}.$$

If we remove the ' after performing the transformation, we get:

$$d\sigma^2 = \left(1 + \frac{a}{2r}\right)^4 \left(dx_1^2 + (dx_2^2 + (dx_3^2))\right), \quad f' = \left|\mathrm{e}ft\left(r' + \frac{r-a/2}{r+a/2}\right)\right|^2.$$

In the new coordinates, the metric of gravitational space is:

$$(1 + \frac{a}{2r})^2.$$

$d\sigma^2$ is regular for all values of $r > 0$, $f$ is always positive and vanishes only for $r = \frac{a}{2}$.

The perimeter of the circle $x_1^2 + x_2^2 = r^2$ is:

$$2\pi r \left(1 + \frac{a}{2r}\right)^2.$$

If we now allow $r$ to traverse its range of values starting from $+\infty$, this function gradually decreases until it reaches the value of $4\pi a$ (the minimum perimeter of the central sphere), for $r = \frac{a^5}{2}$. After which, it begins to increase again as $r$ decreases further towards zero, and finally grows without limit.

"According to the above interpretation," Weyl writes, "the region $r > a/2$ would correspond to the exterior, $r < a/2$ to the interior of the mass point. Continuing the

<footnote n="5">⁵But since the potential is zero on the central sphere, the denominator of the metric there is zero. Therefore, the hypersurface is inorientable.</footnote>
analysis, $\sqrt{f} = \frac{r-a/2}{r+a/2}$ becomes negative inside, so that, for a fixed point, cosmic time $x_4$ and proper time are in opposite directions. (In nature, of course, only part of the solution can be realized, not reaching the singular sphere.)"

In these last lines, Weyl plays on the fact that:

$$ds = \frac{r - a/2}{r + a/2} dx^0.$$

The geodesics cross the central sphere. A positive $ds$ goes hand in hand with this progression along these geodesics. But the idea of a proper time inversion is, for Weyl, meaningless. What this relationship indicates is on the other side: when $r < 2$, we have $dx_0 < 0$. If an object crosses this central sphere, it travels through a portion of spacetime where the time coordinate $x_0$ has been reversed.

These last lines from Weyl should be given full attention. The physicist sees this "other side" of the central sphere as "the interior of masses." But we could also see another universe or, let's say, another side of the universe, perhaps populated by negative masses if the inversion of the time coordinate goes hand in hand with the inversion of energy and mass.

In any case, we could summarize this topological vision of Schwarzschild geometry as a "diabolo"-type surface, where the Weyl surface is a sort of extension of the "funnel" of the "Flamm surface" (see Fig. 3).

# 4 The Change of Coordinates and the Kruskal-Szekeres Metric

We have so far overlooked a section of black hole research that developed after Schwarzschild's work in 1916. In a detailed but dense article, Jean Eisenstaedt described the hesitations of scientists regarding the Schwarzschild singularity and its various interpretations in the period 1915-1923. Nothing truly conclusive emerges from all this, except that this singularity, or, beyond that, the sphere surrounding it, "is primarily thought of negatively," with the singularity being interpreted "either as the limit of the physical reality revealed by the theory, or as the limit of the validity of the theory itself," and questions and queries ("glimpsed, outlined, denied, revisited") continue to persist (see [Eisenstaedt 82], 190).

In reality, we have to go back a little in time to see the problems become clearer. In 1924, for example, Arthur Eddington (see [Eddington 24]) outlined the first coor-
<figure><img src="image_4.png" /><figcaption>Figure 3: The "diabolo" or Weyl paraboloid</figcaption></figure>

dinate system not necessarily linked to the famous Schwarzchild ray, a system that David Finkelstein would rediscover in 1958 (see [Finkelstein 58]). The Eddington-Finkelstein coordinates are obtained from the Schwarzschild coordinates by the following transformations:

$$ct' = ct + R_s \ln \left( \frac{r}{R_s} - 1 \right)$$

for the outgoing coordinates and:

$$ct' = ct - R_s \ln \left( \frac{r}{R_s} - 1 \right)$$

for the incoming coordinates, where $R_s$ is the Schwarzschild radius.

The Eddington-Finkelstein metric is non-synchronous<fnref n="6" /> but in 1938, Georges Lemaître developed a synchronous metric (since called the "Lemaître metric"), which we will not detail. Synge (see [Synge 50]) will also demonstrate that this latter metric, like that of Eddington-Finkelstein, only covers part of the geometry of Schwarzschild

<footnote n="6">⁶In general relativity, a synchronous frame of reference (or comoving frame of reference, or Gaussian normal coordinate system) is a spacetime frame of reference where the time coordinate is proper time at every point in space. In such a frame of reference, the timelines are geodesics and every point in space is in free fall. The choice of such a frame of reference is always possible. A frame of reference that is not synchronous is non-synchronous. The same is true of metrics.</footnote>
spacetime. In reality, these metrics do not allow us to consider all dynamical cases of a body in the environment of a Schwarzschild black hole. However, they showed that this radius is not a real, physical singularity, but only a singularity for the chosen metric, in this case, initially, the Schwarzschild metric. This paved the way for further advances.

These occurred in 1960, when Martin Kruskal (see [Kruskal 1960]) and George Szekeres (see [Szekeres 1960]) constructed a new metric allowing them to study all types of motion of a body outside and below the Schwarzschild radius.

The Kruskal-Szekeres metric is, in fact, the maximal analytic extension of the Schwarzschild metric. The introduction of the Kruskal-Szekeres metric provides additional solutions to those of Schwarzschild. It will also clearly show that the gravitational singularity existing at the Schwarzschild radius is not a true physical singularity but a simple coordinate singularity.

In general relativity, observable effects are described by the Riemann tensor. The singularity present in the Schwarzschild metric concerns the tidal forces experienced by an observer in free fall, and a curvature invariant such as $R_{abcd}R^{abcd}$, infinite at the Schwarzschild radius, is no longer infinite if expressed with respect to an appropriate coordinate system. Like the North Pole on Earth, which is a simple point but in certain coordinate systems becomes singular and must extend along a line, as in an equatorial projection, for example, but behaves perfectly normally in a polar projection. This type of singularity is called a coordinate singularity.

The key step in the coordinate changes used to move from the Schwarzschild metric to the Kruskal-Szekeres metric is the use of the Regge-Wheeler "turtle coordinate" $r^*$ (see [Wheeler 55], [Regge-Wheeler 57]), which takes up the Eddington-Finkelstein idea of introducing a natural logarithm into the formula. We have:

$$r^* = r + r_S \ln \left( \frac{r}{r_S} - 1 \right),$$

where $r_S$ is the Schwarzschild radius $\frac{2GM}{c^2}$ and which remains continuous at $r_S$.

From this, we can define two new coordinates:

$$u = ct + r^*;$$

$$v = ct - r^*.$$

With these coordinates, the relevant part of the Schwarzschild metric becomes:

$$ds^2 = - \left(1 - \frac{r_S}{r}\right) du dv,$$
where $r$ is considered a function of $u$ and $v$ given by:

$$r + r_S \ln \left( \frac{r}{r_S} - 1 \right) = r^* = (u - v)/2.$$

The hybrid coordinates $(u, r)$ or $(v, r)$ are called Eddington-Finkelstein coordinates.

The metric can then be rewritten as:

$$ds^2 = - \frac{r_S e^{-r/r_S}}{r} e^{v-u/2r_S} du dv.$$

By introducing the new coordinates:

$$U = -e^{-u/2r_S},$$

$$V = e^{v/2r_S},$$

the metric becomes:

$$ds^2 = \frac{4r_S^3 e^{-r/r_S}}{r} dU dV,$$

and no longer has a singularity at $r = r_S$, that is, at $U = 0$ or $V = 0$.

Schwarzschild's solution has therefore been extended to all values for which $r > 0$. As can nevertheless be verified by calculating $R_{abcd}R^{abcd}$, the singularity existing at $r = 0$ persists; it is a physical singularity and the only remaining physical singularity.

The form given by Kruskal requires a final coordinate transformation:

$$T = (V + U)/2,$$

$$X = (V - U)/2.$$

Which gives, by reintroducing the angular part:

$$ds^2 = \frac{4r_S^3 e^{-r/r_S}}{r} (dX^2 - dT^2) + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

The domain of the variables $T$ and $X$ is given by the condition $r > 0$, which leads to:

$$X^2 - T^2 > -1 :$$

Unlike the Schwarzschild metric, the Kruskal-Szekeres metric extends throughout all space, both exterior and interior to the singularity. Among its properties, we have:
1. The metric at the event horizon is non-singular;

2. $v$ remains timelike and $u$ spacelike over all spacetime;

3. The worldlines of radially moving photons satisfy $dv = \pm du$;

4. Inside the horizon, with Kruskal-Szekeres coordinates, the singularity at $r = 0$ of the Schwarzschild metric is located at $v^2 - u^2 = 1$, hence now, in fact, two singularities:

$$ \begin{cases} u = \sqrt{v^2 - 1}; \\ u = -\sqrt{v^2 - 1}. \end{cases} $$

5. The lines $r = Cte$ in Schwarzschild coordinates are the hyperbolas $u^2 - v^2 = Cte$ in Kruskal coordinates. Their asymptotes are the bisectors $u = v$ and $u = -v$;

6. The lines $t = Cte$ in Schwarzschild coordinates are the lines $v/u = Cte$ passing through the origin in Kruskal coordinates. The singularities are represented by the boundaries of the gray hyperbolic zones in the drawing in Fig. 4.

<figure><img src="image_5.png" /><figcaption>Figure 4: Representation in Kruskal-Szekeres coordinates</figcaption></figure>

7. Light-like geodesics are lines oriented at 45°. It is easy to verify that for $ds = 0$, we have $du^2 = dv^2$;
8. The Schwarzschild metric differentiates two regions of spacetime bounded by the event horizon. The region $r > 2M$ is segmented in two using the Kruskal-Szekeres metric.

9. The condition $r > R_s$ corresponds to $u^2 > v^2$, that is, to:

$$ \begin{cases} u > |v| \\ u < -|v| \end{cases}. $$

The entire Schwarzschild geometry is therefore represented by four different regions in Kruskal-Szekeres coordinates.

10. Note that time $t$ is infinite at the Schwarzschild radius $u = \pm v$. This means that an object crossing the Schwarzschild horizon and entering the black hole will be seen eternally frozen or stopped at this boundary.

# 5 The Painlevé-Gullstrand Metric and the So-Called "Flow of Space"

We also know that, just as time seems to stand still, "space flows in a black hole." This is how the press, and sometimes even some cosmologists, present the phenomenon.

In reality, this is an image used to express how space-time is extremely distorted around a black hole. It's not that space literally "flows," but rather that the gravitational field is so intense that even light cannot escape. This dynamic can be formalized very well within the framework of the theory of general relativity, but to do so, we must once again abandon the Schwarzschild metric in favor of another.

The Schwarzschild metric is perfectly valid outside the horizon ($r > r_s$), but it becomes singular at the horizon ($r = r_s$), which prevents a clear description of an observer's fall or the behavior of spacetime. To circumvent this, Paul Painlevé (see [Painlevé 21]) and then Allvar Gullstrand (see [Gullstrand 22]) proposed a form of the metric where time is measured by an observer in free fall. The coordinates are regular at the horizon, with no singularities. To construct this metric, simply start from the Schwarzschild metric. In classical units (with explicit $c$), the Schwarzschild metric is given by:

$$ ds^2 = -(1 - 2GMr)c^2dt^2 + (1 - 2GMr)dr^2 + r^2d\Omega^2, $$
where $\Omega^2 = d\theta^2 + sin^2\theta^2 d\phi^2$ is the spherical angular element. To change the time co-ordinates, we introduce a new time coordinate $t'$, corresponding to the time measured by an observer in free fall from infinity:

$$t' = t + f(r),$$

hence:

$$dt = dt' - \frac{df}{dr}dr.$$

Replacing $dt$ in the Schwarzschild metric, we obtain:

$$ds^2 = - \left(1 - \frac{2GM}{r}\right) c^2 \left(dt' - \frac{df}{dr}dr\right)^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.$$

Square expansion:

$$ds^2 = - \left(1 - \frac{2GM}{r}\right) c^2 dt'^2 + 2c^2 \left(1 - \frac{2GM}{r}\right) \frac{df}{dr} dt' dr - c^2 \left(1 - \frac{2GM}{r}\right) \left(\frac{df}{dr}\right)^2 dr^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.$$

We want the term $dr^2$ to have a coefficient equal to 1:

$$\left(1 - \frac{2GM}{r}\right)^{-1} - c^2 \left(1 - \frac{2GM}{r}\right) \left(\frac{df}{dr}\right)^2 = 1.$$

Let's solve:
$$ \begin{gathered} \frac{\frac{2GM}{r}}{1 - \frac{2GM}{r}} = c^2 \left(1 - \frac{2GM}{r}\right) \left(\frac{df}{dr}\right)^2 \\ \Rightarrow \left(\frac{df}{dr}\right)^2 = \frac{2GM}{r c^2 \left(1 - \frac{2GM}{r}\right)^2}. \end{gathered} $$

We choose the negative root for a free fall towards the center:

$$ \frac{df}{dr} = - \frac{\sqrt{\frac{2GM}{rc^2}}}{1 - \frac{2GM}{r}}. $$

Inserting this result, we obtain the mixed form:

$$ ds^2 = - \left(1 - \frac{2GM}{r}\right) c^2 dt'^2 + 2c \sqrt{\frac{2GM}{r}} dt' dr + dr^2 + r^2 d\Omega^2. $$

Or, in a more suggestive square form:

$$ ds^2 = -c^2 dt'^2 + \left( dr + \sqrt{\frac{2GM}{r}} dt' \right)^2 + r^2 d\Omega^2. $$

Thus, the metric suggests that space "falls" radially towards the center with a velocity:

$$ v(r) = \sqrt{\frac{2GM}{r}}. $$

which corresponds exactly to the speed of an object in free fall from rest to infinity. The horizon is crossed without a singularity, unlike the Schwarzschild coordinates.

It remains that there is always a singularity for $r = 0$ and, of course, a divergence. But although some dispute this possibility of entering the interior of the accretion cone and defining all its points, after Kruskal and Szekeres, or Painlevé and Gullstrand, the idea that black holes lead to a central singularity was finally accepted as an inevitable consequence of the theory of relativity.
# 6 Hawking and Penrose's Singularity Theorems

Since the singularities of spacetime have become established in general relativity, the question arose of when and why gravitation produces singularities. A number of theorems formulated by Penrose and Hawking in the years 1965-1970, known as "singularity theorems," attempted to answer this question.

## 6.1 An Attempt at a Definition

Informally, a singularity is a place that objects or light rays can reach in a finite time, where spacetime ceases to be a manifold and its curvature becomes infinite. The theory demonstrates that singularities are found in all black hole spacetimes, in the Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman metrics, as well as in all cosmological solutions lacking scalar field energy or a cosmological constant.

The notion of singularity, however, needs to be clarified. *A priori*, and by analogy with electrodynamics, one might think it reasonable to define a spacetime singularity as a point where the metric tensor remains undefined and not properly differentiable. But, as Hawking and Ellis observe (see [Hawking-Ellis 74], 256), one could then eliminate such points and consider that the remaining manifold ($M, g$), with metric $g$, still represents all of space. However, such entities cannot be considered ordinary parts of spacetime because the equations of physics do not hold at these points, and it is impossible to make any measurements there. Consequently, they must be precisely defined as points where the manifold ($M, g$) cannot be extended with the required differentiability, which means that it is incomplete. We then distinguish the following cases:

1. *Spatial* singularities: The singularity lies in the future or past of all events in a given region. The Big Bang singularity and the typical singularity inside a non-rotating, uncharged Schwarzschild black hole are of the spatial type.

2. *Time-like* singularities: These are singularities that an observer can avoid because they do not necessarily lie in the future of all events. An observer could move around a time-like singularity. They are less common in known solutions to the Einstein field equations.

3. *Null* singularities: These singularities occur on light-like or null surfaces. An example is found in certain types of black hole interiors, such as the Cauchy horizon of a charged (Reissner-Nordström) or rotating (Kerr) black hole.

A singularity can also be *strong* or *weak*:
A weak singularity is one where the tidal forces (responsible for spaghettification in black holes) are not necessarily infinite. An observer falling into a weak singularity might not be torn apart before reaching the singularity, although the laws of physics would still break down. The Cauchy horizon inside a charged or rotating black hole could be an example of a weak singularity.

A strong singularity is one where the tidal forces become infinite. In a strong singularity, any object would be destroyed by infinite tidal forces as it approaches the singularity. The singularity at the center of a Schwarzschild black hole is an example of a strong singularity.

Spacelike singularities are a characteristic of uncharged, non-rotating black holes, as described by the Schwarzschild metric, while timelike singularities are those that occur in exact solutions of charged or rotating black holes. Both exhibit the property of geodesic incompleteness, in which neither a light path nor a particle path can be extended beyond a certain proper time or affine parameter (the affine parameter being the zero analog of proper time).

Appearing in the years 1965-1970, the singularity theorems of Hawking and Penrose then specified when and how singularities could appear.

## 6.2 Informal Presentation of Theorems

Let $(M, g)$ be a Lorentzian manifold of signature $(-, +, +, +)$, $\nabla$ be the Levi-Civita connection, and $R_{ab}$ be the Ricci tensor.

**Definition 6.1** (Incomplete Inextensible Geodesic). A geodesic $\gamma : [0, a) \to M$ is called *incomplete* if it cannot be extended beyond $a < \infty$, even if its affine or proper length is finite.

**Definition 6.2** (Strong Energy Condition). Spacetime satisfies the strong energy condition if:

$$R_{ab}v^av^b \geq 0 \quad \text{for any causal vector } v^a.$$

**Definition 6.3** (Trapped Surface). A compact spacelike surface $S \subset M$ of codimension 2 whose two families of orthogonal null geodesics have a *negative expansion* everywhere.

**Definition 6.4** (Globally Hyperbolic Spacetime). A spacetime $(M, g)$ is globally hyperbolic if it admits a Cauchy hypersurface which every causal curve intersects exactly once.
**Theorem 6.1** (Penrose (1965). Singularity in Gravitational Collapse). Let (M, g) be a spacetime satisfying:

- There exists a compact trapped surface $S \subset M$,

- The zero energy condition is satisfied: $R_{ab}k^ak^b \ge 0$ for any zero vector $k^a$,

- $(M, g)$ is causally reasonable (strongly causal or globally hyperbolic).

Then, $(M, g)$ contains an inextensible and incomplete null geodesic.

**Theorem 6.2** (Hawking (1966). Cosmological singularity). Let (M, g) be a globally hyperbolic spacetime such that:

- The strong energy condition is satisfied: $R_{ab}u^au^b \ge 0$ for any timelike vector $u^a$,

- There exists a compact spacelike hypersurface $\Sigma \subset M$,

- Geodesics normal to $\Sigma$ have a strictly negative expansion everywhere.

Then any timelike geodesic normal to $\Sigma$ is incomplete toward the past.

**Theorem 6.3** (Hawking-Penrose (1970). [Unified Form of Singularity Theorems]. Let (M, g) be a spacetime satisfying:

- The strong energy condition: $R_{ab}v^av^b \ge 0$ for any causal vector $v^a$,

- Absence of closed causal curves (causality condition),

- At least one of the following conditions:

- $(M, g)$ contains a trapped surface,

- There exists a compact spacelike hypersurface with negative expansion,

- There exists a point $p \in M$ such that the zero geodesic bundle from $p$ has a finite-time focal section.

Then, $(M, g)$ contains at least one inextensible and incomplete causal geodesic.

## 6.3 Epistemological Commentary

Penrose's theorem guarantees that some form of geodesic incompleteness occurs inside any black hole as long as matter satisfies reasonable energy conditions. The energy condition required for the black hole singularity theorem is weak: it states
that light rays are always focused together by gravity, never separated, and this is
true when the energy of matter is positive.

Hawking’s singularity theorem applies to the entire universe and works in reverse: it guarantees that the (classical) Big Bang had infinite density. This theorem is very restrictive and is only valid when matter obeys a stronger energy condition, called the *strong energy condition*, in which energy is greater than pressure. All ordinary matter, except for the mathematical expectation of a scalar field, obeys this condition. During inflation, the universe violates the dominant energy condition, and it was initially suggested (e.g., by Starobinsky) that inflationary cosmologies could avoid the initial Big Bang singularity. However, it has since been shown that inflationary cosmologies are themselves still incomplete in the past and therefore require physics other than inflation to describe the past boundary of the inflationary region of spacetime. Obviously, we cannot predict what might "come out" of a Big Bang singularity in our past, nor what would happen to an observer falling "into" a black hole singularity in the future. Therefore, it is necessary to modify them.

Before Penrose, it was conceivable that singularities would only form in artificial situations. For example, during the collapse of a star to form a black hole, if the star is spinning and therefore possesses a certain angular momentum, centrifugal force could partially counteract gravity and prevent the formation of a singularity. Singularity theorems prove that this is impossible and that a singularity will always form once an event horizon forms.

In the example of a collapsing star, since all matter and energy are a source of
gravitational attraction in general relativity, the additional angular momentum only
tightens the star as it contracts: the part outside the event horizon eventually sta-
bilizes as a Kerr black hole (see the no-hair theorem). The part inside the event
horizon necessarily has a singularity somewhere. The proof is constructive: it shows
that the singularity can be found by following light rays from a surface just inside the
horizon. But this demonstration does not specify the type of singularity that occurs:
discontinuity of spatial type, of temporal type, null, orbifold<sup>7</sup> discontinuity by jump
in the metric, etc. It only guarantees that if we follow timelike geodesics into the
future, it is impossible for the boundary of the region they form to be generated by

<footnote n="7"><sup>7</sup>Let us simply say here, remaining very general, that an orbifold (sometimes also called orbivari-
ety in mathematics), is a generalization of the notion of a manifold containing possible singularities.
These spaces were explicitly introduced for the first time by the Japanese mathematician Ichiro Sa-
take (1927-2014) in 1956 under the name V-manifolds. To move from the notion of manifold (which
imposes at least an idea of continuity) to that of orbifold, we add as local models all the quotients
of open sets of R<sup>n</sup> by action of finite groups.</footnote>
null geodesics from the surface. This means that the boundary must either originate from nowhere or the entire future ends at a finite extension.

An interesting "philosophical" feature of general relativity is revealed by the sin-
gularity theorems. Since general relativity predicts the inevitable appearance of
singularities, the theory is not complete without a specification of what happens to
matter that encounters the singularity. In principle, however, one could extend gen-
eral relativity to a unified field theory, such as the Einstein-Maxwell-Dirac system,
where such singularities do not occur.<fnref n="8" />

6.4 The Background of the Theorems

From a purely mathematical point of view, there is a profound connection between
the curvature of a manifold and its topology. The Bonnet-Myers theorem states
that a complete Riemannian manifold whose Ricci curvature is everywhere greater
than a certain positive constant must be compact. The positive Ricci curvature
condition is stated as follows: for every geodesic, there exists a nearby, initially
parallel geodesic that curves towards it during its extension, and the two intersect
at a finite length.

When two nearby parallel geodesics intersect, the extension of one of them is no
longer the shortest path between their endpoints. Indeed, two parallel geodesic
paths necessarily collide after an extension of the same length, and if we follow one
of the paths to its intersection and then the other, we connect the endpoints with
a non-geodesic path of the same length. This means that for a geodesic to be a
shortest path, it must never intersect neighboring parallel geodesics.

Starting with a small sphere and extending parallel geodesics from the boundary,
assuming the manifold has a Ricci curvature bounded below by a positive constant,
none of the geodesics becomes a shortest path after a certain time, because they all
collide with a neighbor. This means that after a certain extension, all new potential
points are reached. If all points of a connected manifold are within a finite geodesic
distance of a small sphere, the manifold must be compact.

Roger Penrose reasoned similarly in relativity. If null geodesics, the paths of light
rays, are followed into the future, points in the region's future are generated. If

<footnote n="8">⁸It remains an open question, however, whether (classical) general relativity predicts spacelike singularities inside realistic, charged, or rotating black holes, or whether these are artifacts of high-symmetry solutions and transform into zero-like or timelike singularities when perturbations are added.</footnote>
a point lies on the boundary of the region's future, it can only be reached at the speed of light, no slower. Null geodesics therefore encompass the entire boundary of a region's proper future. When null geodesics intersect, they are no longer on the boundary of the future, but within it. Thus, if all null geodesics collide, there is no boundary to the future.

In relativity, the Ricci curvature, which determines the collision properties of geodesics, is determined by the energy tensor. Its projection onto light rays is equal to the null projection of the energy-momentum tensor and is always positive. This implies that the volume of a congruence of parallel null geodesics, once decreasing, will reach zero in a finite time. Once the volume reaches zero, a collapse occurs in one direction, such that each geodesic intersects a neighboring one.

Penrose concluded that whenever there is a sphere where all outgoing (and incoming) light rays initially converge, the boundary of the future of that region will terminate after a finite extension, because all null geodesics will converge.

This is of great importance because the outgoing light rays of any sphere within the horizon of a black hole solution all converge. The boundary of the future of this region is therefore either compact or originates from nowhere. The future of the interior either terminates after a finite extension or has a boundary eventually generated by new light rays that cannot be connected to the original sphere.

In this context, singularity theorems are characterized by the following elements:

### 6.4.1 Nature of a Singularity

Singularity theorems use the notion of geodesic incompleteness to replace the presence of infinite curvatures. Geodesic incompleteness is the idea that there exist geodesics, paths traveled by observers through spacetime, which can only be extended for a finite time, measured by an observer traveling along one of them. Presumably, at the end of the geodesic, the observer has fallen into a singularity or encountered some other pathology that violates the laws of general relativity.

### 6.4.2 Assumptions

In general, a singularity theorem has three fundamental elements:

1. An energy condition on matter;

2. A condition on the global structure of spacetime;
3. A condition on gravity: it must be strong enough (somewhere) to trap an entire region of spacetime.

There are different possibilities for each element, and each leads to different singularity theorems.

### 6.4.3 Specific Formulation and Proof Tool

A key tool used in the formulation and proof of singularity theorems is the Raychaudhuri equation, which describes the divergence $\theta$ of a congruence (family) of geodesics. The divergence of a congruence is defined as the derivative of the logarithm of the determinant of the congruence volume. The Raychaudhuri equation is

$$ \dot{\theta} = -\sigma_{ab}\sigma^{ab} - \frac{1}{3}\theta^2 - E[\vec{X}]_a^a, $$

where $\sigma_{ab}$ is the shear tensor of the congruence and $E[\vec{X}]_a^a = R_{mn} X^m X^n$ is also known as the Raychaudhuri scalar. The key point is that $E[\vec{X}]_a^a$ will be positive if Einstein's equations hold, if the zero-energy condition holds and geodesic congruence is zero, or if the strong-energy condition holds and geodesic congruence is timelike.

If these conditions hold, the divergence becomes infinite for a finite value of the affine parameter. Thus, all geodesics starting from a point will eventually converge after a finite time, if the appropriate energy condition holds – a result also known as the "focusing theorem".

This is relevant for singularities thanks to the following argument:

Suppose we have a globally hyperbolic spacetime and two points $p$ and $q$ that can be connected by a timelike or zero curve. Then there exists a geodesic of maximal length connecting $p$ and $q$. Let's call this geodesic $\gamma$. The geodesic $\gamma$ can be modified into a longer curve if another geodesic from $p$ inters $\gamma$ at another point, called a conjugate point. The focusing theorem shows that all geodesics from $p$ have points conjugated to finite values of the affine parameter. This is especially true for the geodesic of maximum length. But this is a contradiction: we can therefore conclude that spacetime is geodesically incomplete.

## 6.5 The Null Version

In general relativity, there are several versions of the Penrose-Hawking singularity theorem. Most of them state, roughly, that if there exists a trapped null surface and
the energy density is positive, then there exist geodesics of finite length that cannot be extended.<fnref n="9" />.

Of the many versions of the singularity theorems, we will choose to present only the null version below, which takes the following form:

Suppose that the null energy condition holds. We have a connected, non-compact Cauchy surface. We have a closed trapped null surface $T$. We then have either zero geodesic incompleteness or closed timelike curves.

*Outline of a proof by contradiction.* The future boundary of $T$, denoted $\dot{J}(T)$, is generated by null geodesic segments originating from $T$ and whose tangent vectors are orthogonal to it. Being a trapped null surface, according to the null Raychaudhuri equation, both families of null rays emanating from $T$ will encounter caustics. The null geodesics generating $\dot{J}(T)$ must, however, terminate, that is, reach their future endpoints at or before the caustics. Alternatively, one can take two null geodesic segments – changing at the caustic – and deform them slightly to obtain a time-varying curve connecting a point on the boundary to a point in $T$, which is contradictory. But since $T$ is compact, given a continuous affine parameterization of the geodesic generators, there is a lower bound on the absolute value of the expansion parameter. We therefore know that caustics will develop for each generator before a uniform bound on the affine parameter is reached. Therefore, $\dot{J}(T)$ must be compact. Either we have closed timelike curves, or we can construct a congruence by timelike curves, and each of them must intersect the non-compact Cauchy surface exactly once. Consider all these timelike curves passing through $\dot{J}(T)$ and observe their image on the Cauchy surface. Since this is a continuous map, the image must also be compact. Being a timelike congruence, the timelike curves cannot intersect, and therefore the map is injective. If the Cauchy surface were non-compact, then the image would have a boundary. We assume that spacetime is contiguous. However, $\dot{J}(T)$ is compact and without boundary, because the boundary of a boundary is empty. A continuous injective map cannot create a boundary, hence our contradiction. □

**Loopholes:** If closed timelike curves exist, then they need not intersect the partial Cauchy surface. If the Cauchy surface were compact, that is, if space were compact, the zero geodesic generators of the boundary could intersect everywhere, since they

<footnote n="9">⁹These theorems, strictly speaking, prove that there exists at least one non-spacelike geodesic that is only finitely extendable into the past, but there are cases where the conditions of these theorems are such that all spacetime paths directed toward the past end in a singularity.</footnote>
can intersect on the other side of space. Other versions of the theorem involving the
weak or strong energy condition also exist.

7 No-hair theorem and observational results

As Hawking wrote, black holes are one of the few objects in the history of science for which a theory, in the form of detailed theoretical models, was proposed before observations provided proof of their validity. They could be completely described by only three completely independent external observables: their mass, their electric charge, and their angular momentum. No more variables, "no hair", were necessary (see [Hawking 88], chap. 6).

During the 1960s, it was still unclear whether black holes really existed, but Canadian astronomer Werner Israel predicted that they must be very simple, perfectly spherical objects, their size depending only on their mass (see [Israel 67]). Moreover, and contrary to what was initially believed, they did not necessarily result from the collapse of already spherical objects, but from any star that was not rotating, since the gravitational waves predicted by relativity could, in a way, smooth it out until it became much more spherical than it could have been initially.

Israel's results only concerned the case of static black holes, but they were soon extended to Kerr black holes. In 1963, New Zealand astronomer Roy Kerr had found a set of solutions to the equations of General Relativity that described rotating black holes (see [Kerr 63]). Kerr black holes rotated at a constant speed, their size and shape depending only on their mass and rotation speed. For zero rotation, the Schwartschild solution was found. Extending the previous ideas to Kerr black holes led to the hypothesis that any rotating body would eventually settle into a stationary state described by the Kerr solution.

In the early 1970s, Brandon Carter began proving the "no-hair" hypothesis by showing that if a stationary rotating black hole had an axis of symmetry, then its size and shape would depend solely on its mass and rotational speed. In 1971, Hawking then proved that any rotating black hole must have such an axis of symmetry.

However, there is still no rigorous mathematical proof of a general no-hair theorem, and mathematicians generally prefer to refer to it as the "no-hair conjecture." Even in the case of gravity alone (i.e., zero electric fields), this conjecture has only been partially resolved by results of Stephen Hawking (see [Hawking 72]), Brandon Carter (see [Carter 71]), and David C. Robinson (see [Robinson ]), under the additional hypothesis of non-degenerate event horizons and the technical, restrictive and
difficult-to-justify assumption of real analyticity of the space-time continuum.

The no-hair theorem was originally formulated for black holes within the context of a four-dimensional spacetime, but it has since been extended to include the case where the cosmological constant is positive. Magnetic charge, if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole.

If we wanted to be exhaustive, we would have to point out some counterexamples in which the theorem fails and which concerns spacetime dimensions higher than four, in the presence of non-abelian Yang-Mills fields<fnref n="10" />, non-abelian Proca fields<fnref n="11" />, some non-minimally coupled scalar fields<fnref n="12" />, or skyrmions<fnref n="13" />, or in some theories of gravity other than Einstein’s general relativity. However, these exceptions are often unstable solutions and/or do not lead to conserved quantum numbers so that "the spirit' of the no-hair conjecture, however, seems to be maintained". It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and solitons. It as also been conjectured in 2004 that black holes could carry a finite scalar charge which might be a result of interaction with cosmological scalar fields such as the inflaton. The solution is stable and does not possess any unphysical properties; however, the existence of a scalar field with the desired properties is only speculative.

<footnote n="10">¹⁰Yang-Mills theory is the well-known quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953. It is also a generic term for the class of similar theories. The Yang,ÄiMills theory is a gauge theory based on a special unitary group SU(n), or more generally any compact Lie group. A Yang-Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups. Such a theory is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of the understanding of the Standard Model of particle physics.</footnote>

<footnote n="11">¹¹Alexandru Proca (1897-1955) extended in 1934 the Maxwell equations in quantum field theory for massive positively charged particles. He is at the origin of the theory of the massive vector bosons with spin 1. Such bosons ($W^±$ and $Z^0$ bosons) are mediators of the weak interaction. Proca,Äös equations are also used to describe spin 1 mesons, e.g. $\rho$ and $\omega$ mesons (see [Poenaru 05]).</footnote>

<footnote n="12">¹²The principle of minimal coupling has been used in the study of Higgs boson interactions. But the presence of scalar fields with non-minimal gravitational interactions have important implications for the physics of the early universe. Hence the necessity of studying non-minimally coupled scalar fields.</footnote>

<footnote n="13">¹³The skyrmion is a particle theorized in 1962 by British physicist Tony Skyrme and discovered by physicists at the Technical University of Munich in 2009. A skyrmion is a quantum superposition of baryons and resonance states. Skyrmions have been detected in Bose-Einstein condensates, superconductors, thin films with magnetic properties, and nematic chiral liquid crystals.</footnote>
As theoretical modeling of black holes developed, more and more candidates emerged in observational astronomy. For example Cygnus X-1, in our galaxy, which has six times the mass of the Sun, and two other black holes in neighboring galaxies, the Magellanic Clouds. But, since the end of the 1980s, it was known that the number of black holes was probably much higher, perhaps higher than the number of visible stars, about one hundred billion, just for our galaxy. Hawking suspected that super-massive black holes, with a mass of about one hundred million times the mass of the Sun, occupied the centers of quasars. Symmetrically, he raised the possibility of black holes with a mass much lower than that of the Sun and resulting not from gravitational collapse, but from compressions of matter at very high speeds.

# 8 Problems

Independently of the singularity issue, to which we will return, a number of problems have plagued the development of black hole theory.

## 8.1 Bekenstein Entropy

In 1971, Hawking showed that the total area of the event horizon of any classical black hole can never decrease. This property is quite similar to the second law of thermodynamics, with surface area playing the role of entropy. In theory, injecting matter into a black hole should violate this law, since the disappearance of matter from our universe would result in a decrease in its total entropy.

To avoid this consequence, physicist Jacob Bekenstein proposed that a black hole possesses entropy (without specifying its exact nature) and that it be proportional to the surface area of its horizon. The entropy of a black hole, or Bekenstein-Hawking entropy, is given by:

$$S_{BH} = \frac{k_B A c^3}{4 \hbar G},$$

where:

- $k_B$ is the Boltzmann constant;

- $A$ is the surface area of the black hole's horizon;

- $c$ is the speed of light in a vacuum;

- $\hbar$ is the reduced Planck constant;

- $G$ is Newton's gravitational constant.
The number 4 is the ratio of $8\pi$ to $2\pi$, which comes from the fact that in general relativity, the coupling constant (Einstein's constant) is $\kappa = \frac{8\pi G}{c^4}$; it appears in the denominator of the Einstein-Hilbert action and on the right-hand side of the Einstein equation.

It is also common to define the entropy of a black hole using the following equation:

$$ \frac{S}{k} = \frac{A}{4l_{\text{p}}^2}, $$

or:

$$ S = \frac{kA}{4l_{\text{p}}^2}, $$

where:

$$ l_{\text{p}} = \sqrt{\frac{\hbar G}{c^3}} $$

is the Planck length. Defined in this way, the entropy of a black hole is additive but not extensive. Its remarkable property is that it is proportional to an area, and we have:

$$ (S \propto A). $$

It would therefore be proportional to the number of Planck areas contained on the surface A, whereas in thermodynamics, entropy is proportional to volume.

At the time of this proposal, Bekenstein believed that black holes did not emit radiation and that the link with thermodynamics was merely an analogy, not a physical description of the properties of the black hole. However, Hawking soon after demonstrated through a quantum field theory calculation that it was possible to rigorously define a *temperature* associated with black hole radiation.

It turned out that black holes were, moreover, objects with maximum entropy, meaning that the maximum entropy of a region of space bounded by a given surface is equal to that of a black hole of the same surface. This observation led physicists Gerard 't Hooft and later Leonard Susskind to propose a set of ideas called the holographic principle, based on the fact that describing the surface of a region allows for the reconstruction of all the information relating to its contents, in the same way that a hologram encodes information relating to a volume on a simple surface, thus making it possible to create a relief effect from it.

The discovery of black hole entropy thus enabled the development of an extremely profound analogy between black holes and thermodynamics, with black hole ther-
modynamics potentially contributing to the understanding of a theory of quantum gravity.

## 8.2 Hawking Radiation

This second problem, also raised by S. Hawking, stems from the fact that black holes are not actually that black. More precisely, they theoretically emit "blackbody radiation," that is, radiation corresponding to their temperature. The greater a black hole's mass, the lower its temperature, and the less radiation it will emit. In fact, it's not the black hole itself that emits the radiation in question, but its immediate vicinity, and for reasons related to Quantum Mechanics. Indeed, quantum theory predicts that the vacuum continuously generates pairs of particles/antiparticles that instantly annihilate each other due to their opposite electric charges. But when they are generated near a black hole, the latter's gravitational field separates the two particles, with one of them escaping. From an observer's perspective, the particle in question then appears to be "emitted" by the black hole. Therefore, the stronger the Hawking radiation, the more mass the black hole will lose. This is the phenomenon known as "black hole evaporation."

As we saw above, the mass $M$ of a black hole can be expressed as a function of its size (actually the surface area $A$ of its horizon), and the other macroscopic parameters characterizing it, namely, for an astrophysical black hole, its electric charge $Q$ and its angular momentum $L$. Therefore, there exists a function of the form $M = M(A, Q, L)$.

The quantity $\partial M/\partial A$ can be written as:

$$ \frac{\partial M}{\partial A} = \frac{1}{8\pi G} \kappa, $$

where:

$G$ is Newton's constant and $\kappa$ is a quantity called the black hole's surface gravity, which determines how quickly a black hole's gravitational field diverges as it approaches the horizon.

Hawking's calculations on the evaporation of black holes indicate that the temperature $T$ that can be associated with them is given by:

$$ T = \frac{\hbar\kappa}{k_B 2\pi c}, $$
where $k_B$ is the Boltzmann constant, $c$ is the constant for the speed of light in a vacuum, and $\hbar$ is the reduced Planck constant, known as the Dirac constant.

This temperature is also called the Hawking temperature.

## 8.3 Information Paradox

It was Stephen Hawking who highlighted this paradox opposing general relativity and quantum mechanics in 1976. Indeed, general relativity implies that information could fundamentally disappear in a black hole following its evaporation, this loss leading to irreversibility and a non-unitary evolution of quantum states, in total contradiction with the postulates of quantum mechanics.

The Schrödinger equation, the basis of quantum mechanics, is a deterministic and time-reversible equation, so it always allows us to uniquely determine the states preceding a given state, which implies that the information they contain is always conserved.

However, the horizon of a black hole is an absolute point of no return, and if, moreover, black holes evaporate and can disappear completely, then the information they contain is irretrievably lost.

To clarify the problem, we must return to the fundamental properties of the black hole. As we have seen it is entirely characterized by three parameters: its mass ($M$), its electric charge, and its angular momentum. The black hole possesses no additional details, no "hair," to use Hawking's expression. We must therefore make the following observations:

1. During the formation of a black hole, a star, with its many properties, is transformed into an object characterized only by three numbers. The transformation emits a certain number of photons and gravitational waves, but the information carried by these emissions is very insufficient to determine the original characteristics of the star. A huge amount of information must therefore remain within, or on, the black hole's event horizon.

2. Similarly, when matter is absorbed by a black hole, it "loses its hair" by emitting gravitational waves (the "baldness" theorem), but the amount of information emitted by these waves is far from sufficient to restore the initial characteristics of the absorbed matter.

3. As long as black holes did not evaporate, and could only grow, it could always be assumed that the information was preserved within them, which remained in accord-
dance with the fundamental laws of physics. But if black holes radiate, and gradually "evaporate" as soon as their temperature rises above the ambient temperature of the cosmic microwave background radiation, the information they contain will one day disappear completely.

4. The Baldness Theorem<sup>14</sup> also implies that the Hawking radiation released by the black hole during its evaporation is thermal radiation, that is, of completely random and disordered energy, and which cannot carry any information (the geometry of the black hole beyond its horizon cannot encode information in the radiation).

The definitive loss of information therefore seems to be proven, and it calls into question at least two well-known principles in physics:

– The principle of equivalence of inertial mass and gravitational mass (in principle, there should be no distinction between a body in free fall and a motionless body, or, symmetrically, a body in accelerated motion in space free of field and a body subject to a gravitational field). However, crossing the horizon of a black hole, with the loss of information that characterizes it, would introduce a qualitative difference where there should be none.

– The principle of conservation of information: in classical physics, the laws of nature are deterministic and reversible over time. This implies total conservation of information, which is expressed in particular by Liouville’s theorem. The theorem states that the volume V of phase space is invariant with respect to time: $\frac{dV}{dt} = 0$, which is equivalent to saying that the system is reversible. Even if the information becomes illegible and scattered, it is nevertheless always present and never destroyed.

Moreover, attempting to recover the information before it is lost by means of a copy at the horizon of the black hole or elsewhere proves impossible: The Heisenberg uncertainty principle prohibits any quantum cloning. If this were possible, it would mean that two properties such as the position and momentum of certain elements of a system under study could be precisely measured. But this is precisely what the principle prohibits.

Solutions are certainly possible, but all call into question either established principles

<footnote n="14"><sup>14</sup>This is the theorem according to which every astrophysical black hole is entirely described by the Kerr-Newman metric, that is, by its position and its linear momentum, as well as by three other parameters and only three, namely: its mass *M*, its electric charge *Q* and its angular momentum *J*, and this, regardless of its mode of formation and the nature of the matter that served to form it. While Stephen Hawking played a role in its demonstration, the theorem is actually due to Werner Israel, Brandon Carter, David C. Robinson, Gary Bunting and Pawel O. Mazur.</footnote>
of current physics or aspects of known and previously accepted physical theories, whether quantum mechanics or the theory of relativity, leading to hypotheses that are difficult to justify (information stored in a Planck-sized residue, in another universe (Einstein-Cartan theory), or encoded in links between the future and the past (which leads to a counterintuitive conception of time)).

For now, the paradox therefore remains.

## 8.4 Wall of Fire

The term "wall of fire" refers to a hypothetical phenomenon believed to occur at the event horizon of a black hole. Around the hole, there must be a region of high energy density, created by the dissociation of quantum entanglements generated by Hawking radiation. This phenomenon was described in 2012 by Joseph Polchinski and his team, and is based on the fact that the vacuum exhibits quantum fluctuations. Heisenberg's inequalities demonstrate that energy can be "borrowed" from the vacuum for a very short time thanks to the existence of fluctuations. The event horizon of a black hole constantly generates particle-antiparticle pairs whose total mass is related to the energy of the fluctuation by the mass-energy relation $E = mc^2$. The type of particle generated is a function of the energy borrowed. In general, the particle-antiparticle pair annihilates immediately, unless a physical phenomenon allows one of the created particles to be separated from its antiparticle in a time less than the typical lifetime of the pair. However, at the event horizon of a black hole, the tidal forces are so intense that they can separate the particle from its antiparticle before they annihilate. One of the particles in the pair will then be absorbed by the black hole, but not the other. In this case, the energy borrowed from the vacuum will not be returned to it, and to avoid violating the principle of conservation of energy, the black hole must emit energy: this is the basic principle of Hawking radiation, which makes black holes evaporate. But breaking entanglement is an intense process that must release a large amount of energy. According to this theory, the event horizon of the black hole should resemble a ring of fire burning anything that falls into it. A region of high energy density would therefore exist around the black hole due to the enormous amount of quantum entanglement breakdown that should occur.
## 9 A recent quantum challenge?

As the authors of a recent article show (see [Carballo-Rubbio et al. 25]), despite everything we think we know about them, black holes remain, to a large extent, theoretical constructs today. Certainly, the Schwarzschild solution described the gravitational field outside a non-rotating spherical mass, which can be of finite size or a point particle. In the latter case, of course, the Schwarzschild solution presents a spacetime singularity surrounded by an event horizon, these two elements being the defining characteristics of a standard black hole, of which the Schwarzschild solution is representative.

Yet event horizons and spacetime singularities are generally considered mathematical idealizations, which, while useful for estimating reality, fail to fully capture the underlying physics. Horizons, in fact, remain completely undetectable to any finite-time experiment. Similarly, spacetime singularities are generally considered to reflect our incomplete understanding of the behavior of spacetime and matter under extreme conditions. Fundamentally, both concepts rely on the notion of infinity: event horizons imply infinite time, while spacetime singularities correspond to infinite curvature, or, equivalently, infinite energy and matter density. It is therefore to be expected, in physics, that these idealizations will be replaced by more appropriate concepts as infinities are decomposed into finite quantities. Therefore, standard black holes, in general relativity, remain idealizations. As the authors write, although this "black hole" concept has proven very successful and has played a vital role in mathematical and observational physics, "the search for a less idealized and non-singular framework for black hole physics is clearly justified." And to paraphrase Stephen Hawking, this "suggests that black holes should be redefined as metastable bound states of the gravitational field" (see [Hawking 14]). Although a complete non-singular paradigm capable of competing with the standard paradigm remains elusive, it seems that astrophysics research is moving toward a new paradigm in which singularities will be replaced by more realistic explanations.

One of the key features of a theory of quantum gravity would be the absence of singularities or event horizons (and therefore the absence of black holes (see [Kiefer 06]). More recently, in December 2018, Abhay Ashtekar, Javier Olmedo, and Parampreet Singh published a paper demonstrating that, within the framework of loop gravity theory, a black hole should not have a central singularity (see [Ashtekar et al. 18]).

The fact remains that, in this case, we do not know what happens to the matter at this point. Therefore, nothing seems, at least for the moment, to replace the classical
theory of "black holes".

# 10 Conclusion

Despite the photographs that are supposed to "prove" the existence of black holes – but which, in fact, only show not the holes themselves but accretion disks, that is, their surroundings – the fate of these objects remains uncertain even today. Certainly, there is little doubt – this has been known at least since Michell and Laplace – that the gravitational attraction of a very dense body, such as, for example, a collapsed star, can be so intense that it prevents light from escaping. This term has long been referred to as an "occluded star," and while the term was less appealing than "black hole," it had the merit of defining the situation very precisely, at least in the case of stellar black holes. But as we know, there are several types of black holes, theoretically four:

(1) the Schwartzchild black hole (zero electric charge, zero angular momentum);

(2) the Reissner-Nordström black hole (non-zero electric charge, zero angular momentum);

(3) the Kerr black hole (zero electric charge, non-zero angular momentum);

(4) finally, the Kerr-Newman black hole (non-zero electric charge, non-zero angular momentum).

Specimens (2) and (4) have a low probability of existence. But (1) and (3) raise the problem of the singularity, assumed to be point-like in (1), annular in (3), with, in this case, the possibility of a passage to another space, or even another universe.

But infinity, like speculations involving other entities that are currently only hypothetical (white holes, wormholes), leaves some questions about "black holes" themselves. Their radiation, imperfectly described since it is partly dependent on physics we do not possess (or do not yet possess), remains partly problematic. To the point that some recent physical developments (loop gravitation theory) have gone so far as to call them into question (see [Carballo-Rubbio et al. 25]). A new paradigm, in which the singularity would be "repaired" by the quantum effects that gravity must exhibit under extreme conditions, could lead to models of black holes without singularities, and even with or without event horizons. So, perhaps we could no longer define a black hole in Hawking's way, as the set of events that cannot be escaped from at great distances (see [Hawking 88], chap. 7). And, in this case, the future of the universe itself is more uncertain than we could have thought.
Time will tell. What is certain is that with the refinement of experiments, the use of increasingly sophisticated instruments, and different observation channels, we will undoubtedly learn more about these enigmatic entities. For example, in the case of mimic black holes, high-resolution imaging from the Event Horizon Telescope could eventually reveal unexpected details in the surrounding deflected light, such as more complex photon rings. Gravitational waves, likewise, could also manifest subtle anomalies consistent with non-classical space-time geometries. Finally, thermal radiation from the surface of an object without a horizon, such as a mimic black hole, could also offer other promising clues. Overall, current knowledge is not yet sufficient to precisely determine the type of disturbances to look for, nor their intensity. However, significant advances in theoretical understanding and numerical simulations could soon provide answers. This is, of course, the best that both scientists and epistemologists can hope for.

## References

[Ashtekar et al. 18] Ashtekar, A., Javier Olmedo, J. et Singh, P., "Quantum Transfiguration of Kruskal Black Holes", *Physical Review Letters*, vol. 121, no 24, December, 10th, 2018, 241301.

[Abrams 01] Abrams, L., "Black Holes: The Legacy of Hilbert's Error", arXiv:gr-qc/0102055 (or arXiv:gr-qc/0102055v1), 2001.

[Adler et al. 75] Adler, R., Bazin, M., Schiffer, M., *Introduction to General Relativity*, McGraw-Hill Book Company, New York, 1975.

[Bartusiak 15] Bartusiak, M., "Black Hole: How an Idea Abandoned by Newtonians, Hated by Einstein, and Gambled On by Hawking Became Loved", Yale University Press, New Haven CT, 2015.

[Carballo-Rubbio et al. 25] Carballo-Rubio, R., Di Filippo, F., Liberati, S., Visser, M., Arrechea, J., Barceló, C., Bonanno, A., Borissova, J., Boyanov, V., Cardoso, V., Del Porro, F., Eichhorn, A., Jampolski, D., Martin-Moruno, R., Mazza, J., McMaken, T., Panassiti, A., Pani, P., Platania, A., Luciano Rezzolla and Vania Vellucci, "Towards a non-singular paradigm of black hole physics", *Journal of Cosmology and Astroparticle Physics*, May, 6th, 2025.

[Carter 71] Carter, B., "Axisymmetric Black Hole Has Only Two Degrees of Freedom", *Phys. Rev. Lett.*, 26,6, 331-333, 1971.
[Carter 06] Carter, B., "Half century of black-hole theory: from physicists, Äô purgatory to mathematicians, Äô paradise", arXiv:gr-qc/0604064 (or arXiv:gr-qc/0604064v1

[Chakrabarty et al. 20] Chakrabarty, H., Abdujabbarov, A., Malafarina, D., Bambi, C., "A toy model for a baby universe inside a black hole", it Eur. Phys. J., C, 80, 1 (unpaginated), 2020.

[Chandrasekhar 31] Chandrasekhar, S., "The Maximum Mass of Ideal White Dwarfs", Astrophysical Journal, 74, 81-82, 1931.

[Easson-Brandenberger 01] Easson, D. A., Brandenberger R. H., "Universe generation from black hole interiors", J. High Energy Phys., 06, 1-11, 2001.

[Eddington 24] Eddington, A., "A comparison of Whitehead's and Einstein's formulae", Nature, vol. 113, no 2832, 192, february, 9th, 1924.

[Note of Scient. Amer. 16] "Pioneering Physicist John Wheeler Dies at 96", Scientific American, archived from the original on 28 November 2016, retrieved 27 November 2016.

[Eisenstaedt 82] Eisenstaedt, J., "Histoire et singularités de la solution de Schwarzschild (1915-1923)", Archive for History of Exact Sciences, vol. 27, no 2, 157-198, juin 1982.

[Ewing 64] Ewing A., " 'Black Holes' in Space", Science News Letter, vol. 85, 39-39, January 1964.

[Finkelstein 58] Finkelstein, D., "Past-future asymmetry of the gravitational field of a point particle", Phys. Rev., vol. 110, no 4, 965-967, 15 mai 1958.

[Flamm 16] Flamm, L., "Beiträge zur Einsteinschen Gravitationstheorie", Physikalische Zestschrift, XVII, 448-454, 1916.

[Gullstrand 22] A. Gullstrand, A., "Allegemeine lösung des statischen einkörper-problems in der Einsteinshen gravitations theorie", Arkiv. Mat. Astron. Fys. 16(8),1-15, 1922.

[Hawking 72] Hawking, S. W., 'ÄúBlack holes in General Relativity', Äù, Commun. Math. Phys., 25, 152-56, 1972.

[Hawking-Ellis 74] Hawking, S. W., Ellis, G. F. R., The Large Scale Structure of Space-time, Cambridge University Press, Cambridge, 1974.
[Hawking 88] Hawking, S. W., *A Brief History of Time*, Bantam Book, New York, 1988.

[Hawking 14] Hawking, S. W., "Information Preservation and Weather Forecasting for Black Holes", *arXiv:1401.5761*, 2014.

[Hilbert 17] Hilbert, D., "Die Grundlagen der Physik", *Nachr. Ges. Wiss Gottingen Math*. Phys K1, 53, 1917.

[Israel 67] Israel, W., "Event Horizons in Static Vacuum Space-Times", *Phys. Rev.* 164, 5, 1776-1779, 1967.

[Kant 86] Kant, E., *Histoire naturelle générale et théorie du ciel*, Essai sur la constitution et l'Âorigine mécanique de l'Âunivers d'Âaprès les lois de Newton, trad. C. Wolf, in *Les Hypothèses cosmogoniques*, Gauthier-Villars, Paris, 1886.

[Kerr 63] Kerr, R. P., "Gravitational field of a spinning mass as an example of algebraically special metrics", *Physical Review Letters*, vol. 11, no 5, 237-238, September 1st, 1963.

[Kiefer 06] Kiefer, C., "Quantum gravity: general introduction and recent developments", *Annalen der Physik*, vol. 15, nos 1-2, 129, 2006.

[Kruskal 1960] Kruskal, M. D., "Maximal extension of Schwarzschild metric", *Phys. Rev.*, vol. 119, no 5, 1743-1745, sept. 1960.

[Landau 32] Landau, L. D., "On the Theory of Stars", in *Collected Papers of L. D. Landau*, Gordon and Breach, New York, 1965; originally published in *Phys. Z. Sowjet.*, 1, 285, 1932.

[Laplace 1798] Laplace (P. S. de), *Exposition du Système du Monde*, 2e ed, Imprimerie du Cercle Social, Paris, 1798.

[Luminet 87] Luminet, J.-P., *Les Trous Noirs*, Belfond, Paris, 1987.

[Oppenheimer-Snyder 39] Oppenheimer, J. R., Snyder, H., "On Continued Gravitational Contraction", *Physical Review*, vol. 56, 455-459, September 1, 1939.

[Michell 1784] Michell, J., "On the Means of Discovering the Distance, Magnitude, &c. of the Fixed Stars, in Consequence of the Diminution of the Velocity of Their Light, in Case Such a Diminution Should be Found to Take Place in any of Them, and Such Other Data Should be Procured from Observations, as Would be Farther Necessary for That Purpose", *Philosophical Transactions of the Royal Society of London*, vol. 74, 35-57, 1784.
[Montgomery et al. 09] Montgomery, C., Orchiston, W., and Whittingham, I., "Michell, Laplace and the origin of the black hole concept", *Journal of Astronomical History and Heritage*, vol. 12, no 2, 90-96, juillet 2009.

[Painlevé 21] Painlevé, P., "La mécanique classique et la théorie de la relativité", *C. R. Acad. Sci.* (Paris), 173, 677-680, 1921.

[Pathria 72] Pathria, R. K., "The Universe as a Black Hole", *Nature*, vol. 240, 298-299, December 1972.

[Poenaru 05] Poenaru, D. N., "Alexandru Proca (1897-1955) the great physicist", *arXiv:physics/0508195* [physics.hist-ph], 2005.

[Regge-Wheeler 57] Regge, T. et Wheeler, J. A., "Stability of a Schwarzschild singularity", *Phys. Rev.*, vol. 108, no 4, 1063-1069, nov. 1957.

[Robinson] Robinson, D. C., 'ÄúUniqueness of the Kerr black hole,Äù, Phys, Rev. Lett., 905-906, 1975.

[Rovelli 18] Rovelli, C., "Viewpoint: Black Hole Evolution Traced Out with Loop Quantum Gravity", *Physics*, vol. 11, December, 10th, 2018.

[Schamir 25] Shamir, L., "The distribution of galaxy rotation in JWST Advanced Deep Extragalactic Survey Open Access", *Monthly Notices of the Royal Astronomical Society*, Volume 538, Issue 1, 76-91, March 2025.

[Schwarzschild 16¹] Schwarzschild, K., "Über das Gravitationsfeld Messenpunktes nach der Einsteinschen Theorie", *Sitzungsberichte der Köninglish Preussischen Akademie der Wissenschaften zu Berlin*, 1916, tr. angl. S. Antoci and A. Loinger, "On the gravitational field of a mass point according to Einstein theory", *arXiv:physics/9912033v1* [physics.hist-ph] December, 16th, 1999.

[Schwarzschild 16²] Schwarzschild, K., "Über das Gravitationsfeld einer Kugel incompressibler Flössigkeit nach der Einsteinschen Theorie", *Sitzungsberichte der Köninglish Preussischen Akademie der Wissenschaften zu Berlin, Phys-Maths Klasse 1916*, 424-434, tr. angl. "On the gravitational field of a sphere of incompressible fluid according to the Einstein,Äôs Theory", *arXiv:physics/9912033v1* [physics.hist-ph], December, 16th 1999.

[Stuckey 94] Stuckey, W. M., "The observable universe inside a black hole", *Am. J. Phys.*, 62, 788-795, 1994.

[Synge 50] Synge, J. L., "The gravitational field of a particule", *Proc. R. Irish Acad. A* 53, 83-114, 1950.
[Szekeres 1960] Szekeres, G., *On the singularities of a Riemannian manifold*, Publ. Math. (Debr.), vol. 7, 285-301, 1960.

[Tatum 18] Tatum, E. T., "Why Flat Space Cosmology Is Superior to Standard Inflationary Cosmology?", *J. Mod. Phys.*, 09, 1867-1882, 2018.

[Wheeler 55] Wheeler, J. A., "Geons", *Phys. Rev.*, vol. 97, no 2, 511-536, janv. 1955.