Jean-Pierre Petit, Gilles d’Agostini, Jean-Marc Pechinot
Manaty Research Group, Glanon, France.
DOI: 10.4236/jmp.2026.178045   PDF    HTML   XML   0 Downloads   30 Views   Citations

Abstract

We present the solution to Einstein’s stationary, spherically symmetric field equation, expressed using variables initially chosen by Schwarzschild. We show that the geodesics of the Flamm surface are not the geodesics of the hypersurface solution and how the Flamm’s 2D cross-section allows us to understand the topology of the 4D object.

Keywords

Schwarzschild Metric, Schwarzschild Geometry, Throat Sphere

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1. Introduction

This article will be particularly long. We will begin by revisiting how Karl Schwarzschild constructed the first exact solution to Einstein’s equation in 1916—a stationary, spherically symmetric solution [1].

Very few people have followed his calculations in detail, as they require close attention. Indeed, Schwarzschild performs several successive changes of variable, making it quite difficult to follow his work. We shall therefore go through this again in detail, specifying his assumptions and variables at each step. This yields the line element shown in Equation (3), which is what Schwarzschild should have written when expressing his solution metric in terms of his coordinates $\left(t,r,\theta ,\phi \right)$ .

The reason why he did not do so is then explained. Quite simply, his primary goal was to recover the approximate solution published a few months earlier by Einstein, which accounted for the advance of Mercury’s perihelion [2]. To achieve this, he needed to calculate the geodesic trajectories of a test mass (Mercury) around the mass (the Sun). He chose to do this not using the radial coordinate r, but rather using what he termed an intermediate quantity R (Hilfsgröße), a choice he explained in a letter sent to Einstein as early as December 1915 [3] (see Figure 8), even before the article had been published by the journal.

It is absolutely essential to understand that the real formulation of this metric solution, whose construction is presented in details in the present paper, is nothing other than the true form of his original solution—expressed in the coordinates $\left(t,r,\theta ,\phi \right)$ he had chosen, and that this calculation involves no extraneous input or additional geometric or physical assumptions. The invariant remains the length s. What is truly extraordinary is that no one has revisited his calculation in the way we are about to do together.

Naturally, this analysis is inextricably linked to the black hole model. The present work stems from an article published in 1989 by the Canadian mathematician L.S. Abrams [4]—a paper that went entirely unnoticed at the time. Abrams pointed out what he regarded as an error in D. Hilbert’s work [5]. However, the issue is more complex; it is not, strictly speaking, an error. In fact, Hilbert based his geometry not on length measurement but on a bilinear form—an approach that appeared more general to him—which entails the extension of geodesics along autoparallel curves. This aspect will be analyzed in a future article, as will the reinterpretation of the solution by Kruskal and matters relating to Birkhoff’s theorem. The present article is confined to a precise examination of the solution constructed by Karl Schwarzschild in February 1916 [1] and a similarly precise description of Flamm’s surface [6].

2. Detailed Re-Examination of Karl Schwarzschild’s Calculation

Figure 1 shows how Schwarzschild presents his solution [1]. To convince the reader of the importance of this examination, we deemed it appropriate to also provide the German versions of the passages from the articles.

We felt it important to present an excerpt from the original article, published in German, noting that the English translation of this document was only made available in 1999, 83 years after its publication. The corresponding excerpt, in English translation, follows on Figure 1.

Figure 1. The way in which Karl Schwarzschild presented his exterior metric solution in January 1916 [1].

It seems essential to draw attention to the fact that this form of the Schwarzschild metric, described in the literature as “standard”, is expressed using a variable R, auxiliary quantity (Hilfsgröbe), and not the variable r [1], explicitly designated by Schwarzschild as a radial variable. See Figure 2.

Figure 2. The true Schwarzschild’s radial coordinate [1].

In his article, Schwarzschild presents the above expression as, in his view, the most general form of the solution to Einstein’s spherically symmetric and time-translation invariant equation. He then performs a series of changes of variable, for the sake of computational convenience, which means that, in this complex process, we lose the true final expression of his solution, expressed in its initial coordinate system, with its variable r and not this intermediate quantity R. We will reconstruct it precisely. He goes from its coordinates $\left(t,r,\theta ,\varphi \right)$ to a set $\left(x_4,x_1,x_2,x_3\right)$ according to Figure 3.

Figure 3. The first Schwarzschild’s coordinate change [1].

Which gives, see Figure 4:

Figure 4. Schwarzschild’s line element with his new space coordinates [1].

In the following pages, he calculates the Christoffel symbols, introduces all of this into the field equation, solves the equations, and obtains the expressions of his functions $f_1,f_2,f_3,f_4$ . See Figure 5.

By exploiting this result, and taking into account de $f_3=f_2,x_4=t$ , we obtain:

$\begin{array}{c}\text{d}{s}^2=\left[1-\alpha {\left(3x_1+\rho \right)}^{-1/3}\right]\text{d}{t}^2-\frac{{\left(3x_1+\rho \right)}^{-4/3}}{1-\alpha {\left(3x_1+\rho \right)}^{-1/3}}\text{d}{x}_1^2\\ -{\left(3x_1+\rho \right)}^{2/3}\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\varphi }^2\right)\end{array}$ (1)

Using the change mentioned on Figure 6:

Figure 5. The f functions [1].

Figure 6. From original document [1].

We give for the first time, presented rigorously in a peer-reviewed scientific journal, 110 years after the publication of the original article, using its variables.

$\left(t,r=\sqrt{x^2+y^2+z^2}\ge 0,\theta ,\varphi \right)$ (2)

the true solution, given in 1916 by Karl Schwarzschild, of Einstein’s homogeneous equation, in spherical symmetry and invariant under time translation.

$\left(\begin{array}{c}\text{d}{s}^2=\frac{{\left(r^3+{\alpha }^3\right)}^{1/3}-\alpha }{{\left(r^3+{\alpha }^3\right)}^{1/3}}{c}^2\text{d}{t}^2-\frac{r^4\text{d}{r}^2}{\left(r^3+{\alpha }^3\right)\left[{\left(r^3+{\alpha }^3\right)}^{1/3}-\alpha \right]}\\ -{\left(r^3+{\alpha }^3\right)}^{2/3}\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)\end{array}\right)$ (3)

Following Schwarzschild, Einstein’s homogeneous equation—assuming spherical symmetry and invariance under time translation—is presented here in its author’s original coordinates. This is the first time in 110 years that this form—despite its essential nature—has been presented in an article.

The invariant remains the length s. It is simply a reformulation of the exact solution provided by Schwarzschild, expressed not in terms of his intermediate quantity R, but in terms of his true radial coordinate:

$r=\sqrt{x^2+y^2+z^2}\ge 0$ . (4)

We have represented and highlighted it in red in the hope that the many specialists will notice this expression, which is being presented to them for the first time. At this point, a question immediately arises: Why didn’t Schwarzschild present it in this form, with its variable $r=\sqrt{x^2+y^2+z^2}$ , in his article, instead of using this intermediate variable R? The answer is found in a letter Schwarzschild sent to Einstein on December 22, 1915 [3]. He wrote, see Figure 7:

Figure 7. Letter from Schwarzschild to Einstein, December 2015 [3].

Einstein and Schwarzschild were in contact and exchanged letters regularly. In his letter of December 22, 1915, Schwarzschild responded to the article Einstein had just published on November 25, in which he calculated the perihelion advance of Mercury using an approximate solution [2]. Schwarzschild was preparing to send what was then the exact solution. But as he specified in his article, he knew that the radius, for the Sun, was limited to 3 km. He therefore decided to present his result using an intermediate variable R by performing a series expansion, as indicated in the article. We thus find here the origin of what would later be called the “standard form of the Schwarzschild solution”.

3. December 1916: David Hilbert’s Impact on Geometric Cosmology [5]

During 1915, for seven months, through numerous exchanges—first in person, then through copious correspondence—a very close relationship developed between Einstein and Hilbert, in an atmosphere of great mutual respect. Through Einstein’s words, Hilbert discovered the link between advanced mathematics and the development of a new physics, toward which he had always been skeptical. Einstein spoke of the project in which he had become completely invested: a geometric translation, within a framework of covariance of the force of gravity. Intrigued, Hilbert had access to the most advanced mathematical tools of his time. It was a world in which Einstein, first and foremost a fantastic physicist endowed with exceptional intuition, always admitted to feeling uneasy. In this context, Hilbert conceived the project of integrating the two forces known at the time—gravitation and electromagnetism—into a single formalism. Today we would call this a “theory of everything”. For him, cosmic geometry is that of a four-dimensional space, designated by the letters $\left(w_1,w_2,w_3,w_4\right)$ , where the indices 1 to 3 refer to space and where $w_4$ is the time coordinate. Space is quasi-Euclidean and stationary (invariant under time translation): the phenomenon of cosmic expansion had not yet been discovered by Edwin Hubble. The geometry of the cosmos is determined by a number of functions, the set forming a tensor $g_{\mu \nu }$ , from which one can calculate the components of the Ricci tensor, which Hilbert denotes by $K_{\mu \nu }$ . The contraction of this tensor gives the Ricci scalar, designated by the letter K.

Hilbert then published a long article on November 20, 1915 [7] entitled “Foundations of Physics”. While Einstein struggled as best he could with tensors, aided by his friend the mathematician Grassmann, Hilbert took a further step into abstraction by deriving a field equation from a variational calculus, centered on an action constructed around a “world function” H:

${\displaystyle \int H\sqrt{\left|g\right|}\text{d}\omega }$ (5)

$\left|g\right|=\left|\det g_{\mu \nu }\right|$ (6)

See Figure 8.

Figure 8. The Hilbert’s world function [7].

His stroke of genius was to identify this world function as the sum of two terms, the first being the Ricci scalar and the second the Lagrangian of matter, defining a field where, in principle, gravitation and electromagnetism are ambitiously intertwined. The publication of this article was a real shock for Einstein, who, when he sent his own five days later, on November 25, 1915 [8], presenting the goal of his quest—the famous field equation of this general relativity, which completes his special relativity—discovered that Hilbert had beaten him to it. See Figure 9.

If we replace the letter K with the letter R (the Ricci tensor and scalar r), and setting:

$\frac{1}{\sqrt{g}}\frac{\partial \sqrt{g}L}{\partial g^{\mu \nu }}=\chi T_{\mu \nu }$ (7)

The reader will recognize the field equation of general relativity with which they are familiar. Einstein then wrote Hilbert a letter of protest, and the following weeks reflected a certain amount of confusion. However, by mutual agreement, to preserve the feelings of esteem that bound them, the decision was made that this equation would bear only Einstein’s name. In the literature, the reader will now find references to the Einstein-Hilbert collaboration.

Figure 9. The Hilbert’s field equation [7].

History makes its choices. It is, indeed essential to note that at no point in Hilbert’s article does the word “metric” appear, nor expression beginning with “ds² =”.

Schwarzschild then published, in quick succession, in January and February 1916, his two solutions to this field equation [1] [9]. Hilbert felt compelled to urgently incorporate this into a second version of his paper, bearing the same title, which he published on December 23, 1916 [5]. Only the solution to the homogeneous equation [1] held his attention. He made no mention of the second.

For Hilbert, a solution to the field equation was not a metric but a bilinear form which, in his view, determined the geometry. This is a crucial point that must be emphasized. Hilbert is, first and foremost, a mathematician. It therefore seems natural to him to consider the most general formulation from a purely mathematical perspective, specifically, a geometry derived from a bilinear form that could have any sign. He operates, in other words, like someone living in the world of mathematics. Einstein, by contrast, inhabits the world of physics. That is where the entire difference lies. See Figure 10.

Figure 10. Hilbert’s Quadratic form [5].

The tedious presentation of these numerous plates is intended to show how Hilbert’s highly personal conception of this “geometric cosmology”, very different from that of Einstein [8], Schwarzschild [1], Droste [10], Flamm [6], Weyl [11], and others, profoundly influenced the development of this discipline. We will show later how the main proponents of these ideas echoed, without citing their source, the choices made by Hilbert.

Thus, in the plate above, we find the origin of the change in signature. Among the authors cited, this signature was ubiquitous:

The « old » signature: $\left(+---\right)$ (8)

In the sentences underlined in red in the excerpt from Hilbert’s December 1916 article, he explicitly introduces what will become the “modern” signature:

$\left(+++-\right)\text{or}\left(-+++\right)$ (9)

This change will occur gradually and will eventually become widespread in the postwar period, even though, at least in cosmology, there is no article that provides an argument in favor of such a change.

But what does Hilbert mean by pseudo-geometry? The answer is in the following excerpt. See Figure 11.

Figure 11. Hilbert’s pseudo (Euclidean) geometry [5].

In equation (37) of the extract, it is clear that for Hilbert the universe is not only stationary (independent of its time coordinate $w_4$ ) but practically Euclidean. All this relativistic physics introduced by Einstein manifests itself only as a tiny perturbation, through the presence of a term $g_{44}=\epsilon h_{44}$ [5]. The conception of spacetime introduced by Hermann Minkowski [12] in 1909¹, with its metric, is completely neglected. See Minkowski’s statement on Figure 12.

These elements were very well assimilated by people like Schwarzschild [1]. See Figure 13.

In Schwarzschild’s work, as with Schwarzschild and all others at that time, the assumptions are unambiguous:

• The coordinates are defined in ${ℝ}^4$ .

• The element of length ds, associated with the hypersurface, is fundamentally real.

• We are looking for curves that minimize the length.

Figure 12. Minkowski’s axiomatic definition [12] of spacetime (1909).

Figure 13. Schwarzschild foreword [1].

In short, it’s very clear and simple:

When the element ds ceases to be real, we are outside the hypersurface.

In doing so, he is merely conforming to the prestigious geometrization of physics associated with the lecture Minkowski gave in 1909 [12], just months before his death.

Since Hilbert defines his geometry not in terms of an elementary length ds, but rather based on a bilinear form, he arrives at what appears to him to be the most general geometry. It therefore seems logical to him to define two lengths rather than one, a step he takes by introducing an arbitrary sign change. See Figure 14.

For Hilbert, it is possible to define two real lengths from different expressions, depending on the sign of the quadratic form G. When this form is negative, this length becomes the proper time τ and a real value, measured along timelines, time-like trajectories, and is then obtained by inserting a negative sign in front of it. In the domain of definition, when this form is positive, the scalar obtained is then called the length of a segment, along curves described today as being “space-like”. Hilbert is therefore the first to achieve an extension of the solution, now called an “extension to the whole of spacetime”. These are thus two different interpretations of the solution. In Schwarzschild’s work, the result of the variational calculus is limited to the determination of authentic geodesics where the length is minimized:

$\delta {\displaystyle \int \text{d}s}=\delta {\displaystyle \int \sqrt{g_{\mu \nu }\text{d}{x}^{\mu }\text{d}{x}^{\nu }}}=0$ (10)

Figure 14. Hilbert’s two different lengths [5].

Hilbert constructs the set of curves directly from the quadratic form, without worrying about whether it is positive or negative.

$\delta {\displaystyle \int \left(g_{\mu \nu }\text{d}{x}^{\mu }\text{d}{x}^{\nu }\right)}=0$ (11)

Hilbert was also the first to introduce the concept of the light cone, which henceforth has an interior and an exterior, where curves of a spatial nature then unfold.

4. Hilbert’s Construction of the Solution to the Field Equation, Stationary and Spherically Symmetric [5]

For Hilbert, a solution to the field equation is a quadratic form, which can then have any sign. Using polar coordinates, he writes, see Figure 15.

Figure 15. Hilbert’s coordinate choice [5].

He denotes his time coordinate by the letter $l$ . This coordinate does not intervene at any point in his calculation, since the solution was assumed to be independent of it. For Hilbert, what determines the geometry is a quadratic form. It is therefore in this form that he presents his result (the typographical errors of the original manuscript, reproduced in the English translation, have been corrected):

$G\left(\text{d}r,\text{d}\theta ,\text{d}\vartheta ,\text{d}l\right)=\frac{r}{r-\alpha }\text{d}{r}^2+r^2\text{d}{\theta }^2+r^2{\sin }^2\theta \text{d}{\vartheta }^2+\frac{r-\alpha }{r}\text{d}{l}^2$ (12)

And it is only at this stage that Hilbert adopts the interpretation of relativity suggested in 1902 by the mathematician Henri Poincaré [13]. For Poincaré, as or Hilbert, the time coordinate is purely imaginary ($l=it$ ). See Figure 16.

Figure 16. When Hilbert identifies [5] its own result to the Schwarzschild’s one.

Like Hilbert, Schwarzschild adopts what appears to him to be the most general form of the solution to Einstein’s equation without second member, stationary and spherically symmetric. See Figure 17.

Figure 17. Schwarzschild’s line element [1].

As can be seen, in Schwarzschild coordinates $\left(t,r,\theta ,\vartheta \right)$ , this corresponds to

$G{r}^2={\left(r^3+{\alpha }^3\right)}^{2/3}\ne r^2$ (13)

In his calculation, Hilbert introduces a simplification concerning the coefficient of $\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\vartheta }^2\right)$ which seemed to him likely to converge more rapidly toward what he identified as Schwarzschild’s result in $\left(t,r,\theta ,\vartheta \right)$ . See Figure 18.

Figure 18. Hilbert’s approximation [5].

He then identifies his variable $r^{\ast }$ with $r$ . At a time when no one could have imagined for a single moment the existence of astronomical objects whose characteristic scale would be anything other than an infinitesimal perturbation, this error had little importance. But in reality, this amounts to identifying $r=\sqrt{x^2+y^2+z^2\ge 0}$ with $R={\left(r^3+{\alpha }^3\right)}^{1/3}\ge \alpha$ (the intermediate Schwarzschild variable). This point was only pointed out in 1989 by L.S. Hence, the expression of Hilbert solution, as presented in its Equation (45) is not strictly identical to the Schwarzschild’s one since it is expressed with the intermediary variable R, as noticed in 1989 by the Canadian mathematician Abrams [4], 73 years after the publication of Hilbert’s paper.

5. Impact on Physicists

The first person to widely publicize these different works on this cosmology was Arthur Eddington [14], as early as 1919. His work testifies to his extraordinary openness to the science of his time. Representing a considerable undertaking, the book presents the tools of differential geometry upon which the progress represented by general relativity is based. However, on page 139, in Chapter III, he reproduces the manner in which Hilbert treats this solution of Einstein’s equation. See Figure 19.

Figure 19. From Eddington’s book [14], 1919.

One should note the old signature $\left(+---\right)$ as well as the use of exponentials ${\text{e}}^{\lambda }$ and ${\text{e}}^{\nu }$ to ensure invariance when coordinates and functions belong to the real domain.

The same formulation appears identically in the book published in 1934 by the mathematician Richard Tolman [15], see on this book, Equations (94.2) and (95.1).

6. Geometric Interpretation of the Schwarzschild Solution

The young German mathematician Ludwig Flamm provided [6], immediately after publication of Schwarzschild’s papers, the geometric interpretation of the two solutions [1] [9]. Since these are invariant under time translation, the study reduces to three-dimensional geometric structures undergoing translation in time. The 3D structure results from the joining of two manifolds along their common boundary, which is the spherical surface of the star. The author then considers the section of these two objects at constant θ, which yields 3D objects. See Figure 20.

Figure 20. L.Flamm’s choice for the external metric representation [6].

Let us note in passing that Flamm adopts the signature $\left(+---\right)$ and identifies the trajectories of test masses with geodesics minimizing the length s, identified with the proper time $\tau$ . See Figure 21.

Figure 21. Flamm’s variational procedure [6].

He then embeds the object into ${ℝ}^3$ . See Figure 22.

Figure 22. L. Flamm’s embedment operation [6].

He then obtains the meridian $\left(r,z\right)$ of this manifold with boundary in the form of a sideways parabola. See Figure 23.

Figure 23. Flamm’s medidian [6].

This Schwarzschild geometry is thus perfectly defined. The points G and G’ represent the ends of the segment corresponding to the diameter of the sphere along which the two 3D surface elements are joined. The circular arc GG’ completes the parabolic portions Gp and G’p’.

The Flamm surface is often described in the literature as a pedagogical artifact intended to illustrate how masses curve the surrounding space. When the spacetime geodesics arising from the 4D hypersurface corresponding to the exterior metric are projected, one obtains curves suggestive of an attractive effect, with the classical perihelion precession, as shown on Figure 24.

Figure 24. 2D $\left(R,\phi \right)$ projection of 4D Schwarzschild geodesic [16].

The system of geodesics of the Flamm surface was first studied only very recently [17], 113 years after its first presentation by its author. Later (2019!) this study was extended [18]. In this last paper, the geodesics are grouped into two families, depending on whether or not they cross the throat circle. The authors used a color code to distinguish portions of curves suggesting attraction, and at greater distance, repulsion. Consequently, they cannot constitute a pedagogical image intended to illustrate the way masses modify the geometry of the vacuum surrounding them. See Figure 25.

Figure 25. Flamm’s surface geodesics. first family [18].

In Figure 15 of the cited reference, the authors represented in green the portions of geodesics that do not cross the throat sphere, whose concavity is directed toward the symmetry axis (suggesting attraction). At larger distances, this concavity reverses (red color: repulsion).

This shows that the purely spatial geodesics of the Flamm surface must not be identified with the back-projections onto that surface of the 2D projections, in $\left(R,\phi \right)$ , of spacetime geodesics. Under these conditions, what information does the Flamm surface provide? It is of a topological nature. Since Flamm’s analysis shows that the 4D geodesics necessarily project onto this surface, conversely there cannot exist real geodesics crossing the throat sphere that correspond to the region $0<R<\alpha$ in the representation $\left(t,R,\theta ,\phi \right)$ .

Schwarzschild’s approach is perfectly consistent from the astrophysical point of view. He sought to construct two stationary, spherically symmetric solutions of Einstein’s equation. He constructed the interior metric solution [9] in the coordinate system $\left(t,\chi ,\theta ,\phi \right)$ .

$\text{d}{s}^2={\left(\frac{3\cos {\chi }_a-\cos \chi }{2}\right)}^2{c}^2\text{d}{t}^2-{\widehat{R}}^2\left(\text{d}{\chi }^2+{\sin }^2\chi \text{d}{\theta }^2+{\sin }^2\chi {\sin }^2\theta \text{d}{\phi }^2\right)$ (14)

With:

$R=R_a\sin \chi$ (15)

$R_a$ being the ray of the star. With:

${\widehat{R}}^2=\frac{3c^2}{8\pi G\rho }$ (16)

By going back to the system $\left(t,R,\theta ,\phi \right)$ it comes:

$\begin{array}{c}\text{d}{s}^2={\left[\frac{3}{2}\sqrt{1+\frac{8\pi G\rho R_a^2}{3c^2}}-\frac{1}{2}\sqrt{1+\frac{8\pi G\rho R^2}{3c^2}}\right]}^2{c}^2\text{d}{t}^2\\ +\frac{\text{d}{R}^2}{1-\frac{8\pi G\rho R^2}{3c^2}}-R^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)\end{array}$ (17)

The geodesics of the two solutions join along the sphere representing the boundary of the star, assimilated to a sphere of radius $R_a$ , filled with an incompressible material of constant density $\rho$ . The geometric solution presented by Schwarzschild is illustrated in the following figure, expressing the union of a portion of the Flamm surface with a spherical cap. The global result is a contractible object, as shown on Figure 26.

Figure 26. The global geometry of the Schwarzschild’s solution.

This constitutes the true geometry of “Schwarzschild spacetime”.

7. The Flamm Surface as a Whole

However, some authors considered the isolated exterior metric as a possible geometric description of masses: first H. Weyl in 1917² [11], then Einstein and Rosen in 1935 [19]. In the postwar period, the isolated exterior metric solution was considered either as describing a “frozen snapshot” of an object undergoing rapid implosion [20], or as a wormhole [21]. See the Kruskal’s wormhole on Figure 27.

Figure 27. Kruskal wormhole [21].

In both cases, this leads us to consider the Flamm surface in its entirety. Below are its geodesics when they cross the throat circle, as shown on Figure 28.

Figure 28. Flamm’s surface geodesics. Second family [18].

The same color coding is found again: green when these geodesics direct their concavity toward the axis (attraction), and red when this concavity reverses (repulsion). This is also visible when these Flamm surface geodesics are projected. See Figure 29.

Figure 29. Geodesic projection of Flamm surface [17].

In these figures the purely spatial geodesics of the Flamm surface must not be identified with the back-projections onto that surface of the 2D projections, in $\left(R,\phi \right)$ , of spacetime geodesics. This shows that since the 4D geodesics necessarily project onto this surface, there cannot exist real geodesics crossing the throat sphere that correspond to the region $R<\alpha$ in the representation $\left(t,R,\theta ,\phi \right)$ with $\left(0<R<\alpha \right)$ .

When one constructs the curves solving the Lagrange equations, one finds in the projection $\left(R,\phi \right)$ real curve segments that give the illusion that test masses penetrate into the region $0<R<\alpha$ . This illusion disappears when one constructs 3D images of these geodesics in a coordinate system $\left(t,R,\phi \right)$ , as shown on Figure 30.

In Figure 30(A), the trajectory of a test mass converging toward the boundary and then emerging from it has been represented. Both operations occur, with this coordinate t, over an infinite time interval. In Figure 30(B), one has the impression that the test mass “bounces” off this boundary. In Figure 30(C), this projection Figure 30(B) of the geodesic has been back-projected onto the Flamm surface along the green curve, which therefore appears distorted. It should be kept in mind that this is not a geodesic of the Flamm surface.

If one considers separating the two sheets of the hypersurface associated with this isolated exterior metric solution, it then becomes appropriate to introduce a new variable [17]:

$R=\propto \left(1+\ln ch\rho \right)$ (18)

$\text{d}{s}^2=\frac{\ln ch\rho }{1+\ln ch\rho }{c}^2\text{d}{t}^2-{\alpha }^2\left[\frac{1+\ln ch\rho }{\ln ch\rho }t{h}^2\rho \text{d}{\rho }^2+{\left(1+\ln ch\rho \right)}^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)\right]$ (19)

The throat sphere corresponds to the value $\rho =0$ . The two sheets are distinct and correspond to $\rho >0$ and to $\rho <0$ . At the throat:

$G_{tt}\propto \frac{{\rho }^2}{2}\to +0g_{\rho \rho }\to 2$ (20)

Figure 30. 4D geodesics and projections.

If one restricts oneself to the spatial part described by the metric:

$\text{d}{\sigma }^2={\alpha }^2\left[\frac{1+\ln ch\rho }{\ln ch\rho }t{h}^2\rho \text{d}{\rho }^2+{\left(1+\ln ch\rho \right)}^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)\right]$ (21)

it is regular everywhere. Let us now consider the projections in $\left(\rho ,\phi \right)$ of spacetime geodesics with $\theta =\pi /2$ . In a representation $\left(t,R,\theta ,\phi \right)$ , the projections $\left(R,\phi \right)$ are solutions of the classical equation:

$\text{d}\phi =\pm \frac{\text{d}R}{R^2\sqrt{\frac{c^2{l}^2}{h^2}+\frac{\alpha }{h^{2}R}-\frac{1}{R^2}+\frac{\alpha }{R^3}}}$ (22)

where $l$ and $h$ are parameters of the solution curve. Let $\varphi$ denote the angle formed by the tangent to the curve and the radius vector. At the throat:

$tg\varphi =\alpha {\left|\frac{\text{d}\varphi }{\text{d}R}\right|}_{R=\alpha }=\frac{h}{\alpha cl}$ (23)

This quantity is finite. This is what gives the impression, when projecting the geodesic lying on the other sheet, of a discontinuity. It seems to bounce on the throat sphere. But this is only an artifact resulting from the embedding into ${ℝ}^3$ . If one instead adopts a representation $\left(t,\rho ,\theta ,\phi \right)$ and projects the geodesics with $\theta =\pi /2$ onto the plane $\left(\rho ,\phi \right)$ , this angular discontinuity disappears. Let $\beta$ denote the angle formed by the tangent to the curve and the radius vector in this representation.

${\left(tg\beta \right)}_{\rho \to 0}=\frac{\rho \text{d}\phi }{\text{d}\rho }=\frac{R\text{d}\phi }{\text{d}R}\frac{\rho }{R}\frac{\text{d}R}{\text{d}\rho }=\frac{h}{{\alpha }^{2}cl}{\rho }^2\to 0$ (24)

One can therefore draw the corresponding projection of the geodesic. See Figure 31.

Figure 31. Projection of spatiotemporal geodesics in the representation $\left( \rho ,\phi \right)$ [17].

This illustrates the interpretation of the Schwarzschild exterior metric solution as a two-sheeted covering of a manifold. Such a situation could be visualized by drawing the geodesics $\left(\rho >0\right)$ on the front side of a sheet of paper and extending them with the curves $\left(\rho <0\right)$ on the back side (dashed curve). The throat sphere is then reduced to a point. What opposes our intuition is that this sphere reduced to a point nevertheless possesses a nonzero area $4\pi {\alpha }^2$ , so that the manifold is locally non-contractible even though it visually appears contractible. A similar representation would be obtained by exploiting the true expression of the Schwarzschild exterior metric solution in coordinates $\left(t,r,\theta ,\phi \right)$ . See Figure 10. In a projection $\left(r,\phi \right)$ , using the true variable $r=\sqrt{x^2+y^2+z^2}\ge 0$ the interior of the sphere of area $4\pi {\alpha }^2$ would likewise reduce to a point.

These elements show that artifacts may arise when one attributes physical reality to a representation obtained through embedding in ${ℝ}^3$ . That being said, can the Schwarzschild exterior metric solution be considered in isolation? The first response was given in 1939 by Oppenheimer and Snyder [20]. In article [22], a second interpretation is proposed, outlining a model for a neutron star reaching Schwarzschild criticality [9], leading to the establishment of a passage toward a second sheet. Whatever physical meaning one assigns to this model, the following figure illustrates the P-symmetry of the two sheets. In three-dimensional space, four points form a tetrahedron whose volumetric orientation is linked to the orientation of its faces. When this tetrahedron crosses the throat sphere, its orientation becomes inverted, showing that the two sheets are related by an enantiomorphic relationship, as shown on Figure 32.

Figure 32. Crossing the throat sphere produces space-inversion [23].

In general, a choice of coordinates leads to a representation of the solution-object that is both geometric and physical. Each such choice corresponds to a particular way of conceiving spacetime, both geometrically and physically. When the stationary, spherically symmetric exterior solution to Einstein’s equations is viewed in isolation as a physical object, a black hole, this choice of a geometry that allows for contraction implies a swapping of the roles of space and time coordinates within the object; this issue does not arise in interpretations derived from Flamm’s, in which the object is non-contractile and possesses no interior.

8. Conclusion

We must now draw conclusions from this review of the various works in question. This could be summarized by saying that “everything depends on what one calls spacetime”. As defined in 1909 by H. Minkowski, it is a hypersurface defined by a real elementary line element: $\text{d}{s}^2\ge 0$ . This was also the position held by Schwarzschild, Einstein, Droste, Weyl, Flamm, and many others at the time. In December 1916, the mathematician D. Hilbert proposed that the solution hypersurface be defined directly from the bilinear form—representing what is today referred to as the “extension of the solution to the whole of spacetime”. Consequently, the signature ceases to act as a rigid constraint and can take either the form $\left(+---\right)$ or $\left(-+++\right)$ . This corresponds to the standard view of spacetime expressed through the geometric interpretation of the Schwarzschild solution known as the black hole model. This article demonstrates that this interpretation differs from the reading of the geometry provided by Flamm. Achieving this “extension to the whole of spacetime” requires that geodesics penetrating “into the interior”—beyond the Schwarzschild sphere—connect differently. That is, instead of extending onto the second sheet of Flamm’s surface (organized around a throat sphere and corresponding to a wormhole), they extend into a region of spacetime “where time and space coordinates exchange roles”. These correspond to two different topological options. Mathematically, Flamm’s physical interpretation corresponds to a non-contractile geometric object, as can be seen directly from the line element in Figure 10, whereas the black hole model reflects the preservation of the object’s contractibility, at the cost of swapping the roles of the coordinates. This article represents only the beginning of a broader line of reasoning linked to the black hole model. The ultimate goal is to clarify the nature of the geometric extension associated with this model. A future article will address the Kruskal extension and aspects related to Birkhoff’s theorem.

Author Contributions

J.-P. Petit is the author of the article. G. D’Agostini verified the calculations, and J.-M. Pechinot contributed to the production of some of the figures.

Supplementary Information

Figure S1. German version of Figure 1 (the line element expressed with the intermediate quantity R).

Figure S2. German version of Figure 2 (the true Schwarzschild’s radial coordinate r).

Figure S3. German version of Figure 3 (Schwarzschild’s first coordinate change).

Figure S4. German version of Figure 4 (the three functions f).

Figure S5. German version of Figure 4 (expressions of the three functions f) [1].

Figure S6. German version of Figure 8 (the letter to Einstein).

Figure S7. German version of Figure 9 (the Hilbert’s world function).

Figure S8. German version of Figure 10 (The Hilbert’s Field Equation).

Figure S9. German version of Figure 11 (Hilbert’s quadratic form).

Figure S10. German version of Figure 12 (Hilbert’s pseudo Euclidean space).

Figure S11. German version of Figure 13 (Minkowski’s axiomatic definition of spacetime).

Figure S12. German version of Figure 14 (foreword of the Schwarzschild’s paper).

Figure S13. German version of Figure 15 (Hilbert’s definition of lengths).

Figure S14. German version of Figure 16 (Hilbert’s coordinates’ choice).

Figure S15. German version of Figure 17-German version of Figure 17 (Hilbert’s identification of his solution to Schwarzschild’s).

Figure S16. German version of Figure 18 (Schwarzschild’s line element).

Figure S17. German version of Figure 19 (Hilbert’s approximation).

Figure S18. German version of Figure 21 (Flamm’s choice for the metric representation).

Figure S19. German version of Figure 22 (Flamm’s variational procedure).

Figure S20. German version of Figure 23 (Flamm’s embedment).

NOTES

¹Minkowski died a few months later from appendicitis that degenerated into a generalized infection. Antibiotics had not yet been invented. A detail of crucial importance.

²Using the isotropic coordinate system he introduced, and noting the possible inversion of the time factor on a second sheet, Weyl was the first to suggest a two-sheeted covering structure involving an inversion of the time coordinate x⁴. This will be analyzed in a future article; detailing it here would make the exposition overly cumbersome.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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