Cosmology Questionable black hole.
Jean-Pierre Petit Observatoire de Marseille, France Pierre Midy CRI Orsay, France For correspondence:
Abstract
Starting from the so-called black hole model, considered as a physical interpretation of Schwarzschild geometry, we reconsider the problem of the fate of a neutron star when it exceeds its stability limit. We first present a new geometric tool: hypertoric geometry, through 2D and 3D examples (section 2). We show that pathologies associated with metrics, arising from their line element expressed in a given coordinate system, can be cured by a more suitable choice phrased in terms of "local topology." For example, we show that in the two given examples, the 2D surface and 3D hypersurface, whose isometry groups are O2 and O3, are not simply connected.
We extend the method to Schwarzschild geometry. We show that singular features can be fully eliminated by considering a non-simply connected spacetime hypersurface. We give Schwarzschild geometry a different physical significance: a bridge connecting two universes, ours and a twin universe.
We show that the "freezing of time," cornerstone of the black hole model, is simply a consequence of an arbitrary choice of a particular time marker. Using another marker, inspired by Eddington's work (1924), we derive a completely different scenario, involving radial frame dragging (similar to the azimuthal frame dragging of the Kerr metric). We show that the Schwarzschild solution can be interpreted as a "spatial bridge," linking two universes, two spacetimes, acting as a one-way bridge. We show that the transit time of a test particle is finite and short, which immediately makes the classical black hole model questionable.
By extending the isometry group of the Schwarzschild metric, we show that the two universes are enantiomorphic (P-symmetric) and possess opposite time markers (t* = -t). Using a group-theory tool: the coadjoint action of a group on its momentum space, we give physical meaning to this "time inversion," through the spherical throat surface, the Schwarzschild sphere: when a positive-mass particle passes through the spatial bridge, its contribution to the gravitational field is inverted: m* = -m (as shown by J.M. Souriau in 1974, time marker inversion is equivalent to mass and energy inversion).
Since the question of the fate of an unstable neutron star remains an open problem, we present a project for an alternative model: the hyperspatial transfer of part of its matter through a spatial bridge, with this matter flowing toward the twin universe at relativistic speed.
By the way, we recall some well-known defects of the Kruskal model, particularly the fact that it is not asymptotically Lorentzian at infinity.
We suggest considering Schwarzschild geometry as a hypersurface embedded in a ten-dimensional space. By linking this work to earlier ones based on group theory, we construct a CPT-symmetric model. The matter-antimatter duality is preserved in both folds. When matter is transferred toward the twin universe, it undergoes CPT symmetry and its mass (its contribution to the gravitational field) is reversed. But it remains matter. Similarly, antimatter flowing through the spatial bridge remains antimatter, with opposite mass, since time marker inversion, as shown by Souriau, implies mass inversion.
- The black hole model.
Neutron stars cannot exceed a critical mass, close to 2.5 solar masses. For higher masses, their material can no longer withstand the enormous internal pressure due to gravitational force. Then gravitational collapse occurs. For a long time, theoreticians tried to describe the fate of such an object. Examining the Schwarzschild metric, expressed below in terms of
coordinates, where Rs is the so-called Schwarzschild radius (1),
people imagined that this solution of Einstein's equation:
(2) S = 0
with a zero right-hand side could solve the problem. Indeed, if t is chosen as "cosmic time for an external observer," the free-fall time of a test particle following a "radial geodesic" from any distant point outside the Schwarzschild sphere r = Rs is found to be infinite, while this free-fall time Ds, expressed in proper time, remains finite. Then the "physical description" is as follows:
-
The object (a neutron star exceeding its stability limit) undergoes gravitational collapse. Its mass rapidly falls toward "the geometric center of the system," described as a "central singularity." This phenomenon lasts for a finite duration Ds, in terms of proper time s.
-
But, for an "external observer" located at some distance from the object, this process appears "frozen in time." Furthermore, the Schwarzschild sphere is an infinite redshift surface (due to the vanishing of the gtt term of the metric at r = Rs).
This is the model of a spherically symmetric black hole.
r is identified as a "radial distance," which means one can think about "what lies inside the Schwarzschild sphere." Roughly speaking, this means assuming that the "local topology" is "spherical": inside the Schwarzschild sphere, a "smaller sphere is assumed to exist," and so on, up to the "geometric center" of the object.
Later, the model was extended to axially symmetric geometry (Kerr metric). But this extension brings no fundamental conceptual change. That's why we will focus in the following on spherically symmetric systems (we believe this study could later be extended to the Kerr metric).
It is somewhat strange that such a dense object can be described by a solution of equations (2), which a priori refers to an empty region of the Universe where there is no matter-energy.
If one keeps the description (a particular choice of coordinates), many difficulties arise. For example, as r approaches Rs, the grr term tends to infinity.
The signature of the metric, expressed with this particular coordinate choice, is: (+ - - -) for r > Rs, (- + - -) for r < Rs.
When a test particle penetrates inside the Schwarzschild sphere, its mass becomes imaginary and its velocity exceeds that of light: it becomes a tachyon.
Considering the change in signature, some people said:
- No problem: inside the Schwarzschild sphere, r simply becomes time and t becomes radial distance.
A French cosmologist, Jean Heidmann, is known to say: "When thinking about black holes, one must abandon all common sense."
By the way, there are very few candidates for black holes, which is the most troubling aspect. Indeed, supernovae, white dwarfs, and neutron stars were predicted before being observed. For example, Fritz Zwicky presented the supernova model in a famous lecture at Caltech in 1931, long before anyone had observed one. But years later, the model was confirmed, and we now know of hundreds of such objects. The same applies to rotating neutron stars, identified as pulsars. Why so few observed black holes?
Regardless, astrophysicists believe black holes exist, even though there is so little observational data about them. They "use" models of "giant black holes," assumed to reside at the centers of galaxies or galaxy clusters, to "explain" some of their enigmatic dynamical features.
In the following, we wish to suggest a different fate for neutron stars that have exceeded their stability limit. Let us begin by introducing new geometric tools.
- Hypertoric geometry.
Consider the Riemannian metric g, in two dimensions, whose line element, written in terms of two coordinates [r, θ], is:
(3)
where:
θ is defined on ℝ, modulo 2π.
Rs is a constant.
This metric becomes asymptotically Euclidean as r approaches infinity:
(4)
In this particular coordinate system, the signature is: (+, +) for r > Rs, (-, +) for r < Rs.
The determinant:
(5)
becomes infinite at r = Rs. We show that this is due to this particular choice of coordinates. Introduce the following coordinate transformation:
(6)
The line element becomes (7)
whose associated determinant is:
(8)
It no longer vanishes for any value (which further shows that, in a metric, the vanishing of the determinant of the line element depends on the choice of coordinate system, as Eddington showed in 1924 [ref. [10]] for the Schwarzschild metric). As θ tends to zero (which corresponds to
the determinant tends to:
θ varies from -∞ to +∞, which is equivalent to r ≥ Rs.
The metric g, regardless of the chosen coordinate system, describes a surface, a two-dimensional object. This surface has its own system of geodesics, fundamentally invariant with respect to coordinates. Let us study this system in a coordinate system via Lagrange equations. Introduce the following function F:
(9)
The corresponding Lagrange equations are:
(10)
(11)
Equation (11) gives:
(12)
h being positive, negative, or zero. Furthermore, if in (3) we divide both sides by r², we obtain, classically:
(13)
from which we can derive the differential equation describing planar geodesics in this coordinate system:
(14)
The condition |h| ≤ r, according to (12), means that the absolute value of the cosine of the angle between the tangent to the geodesic and the radial vector is ≤ 1.
Now, embed the surface in ℝ³ by adding an extra immersion coordinate z. We choose cylindrical coordinates
The surface is axially symmetric around the z-axis.
The geodesics (θ = const) are the meridians of this surface, where:
(15)
which immediately gives the equation of the meridional curve of this surface embedded in ℝ³. It is the parabola:
(16)
Figure 1 shows a 3D view of this surface embedded in ℝ³, accompanied by a geodesic and its projection onto a plane with polar coordinates.
This surface is not simply connected. Among the orbits of the isometry group O2, there is a circle of minimal perimeter: the throat circle (p = 2Rs).
Fig. 1: The surface embedded in ℝ³ and its representation in a coordinate system.
In Figure 2, several geodesics are shown in this representation.
Fig. 2: Representation of some geodesics. Fig. 3: A particular geodesic crossing the throat circle.
Note that this representation of geodesics on a plane is not isometric. If we measure length on this plane, it does not correspond to the length measured on the surface.
If we impose that the length dS be real, we see that it determines what we might call the local topology. Let us call such a geometric structure a toroidal bridge. We can also say that this surface has a local toroidal topology. It has a single fold, which can be viewed as the union of two bounded half-folds, glued along their circular boundaries along the throat circle, whose perimeter is 2Rs. These circles are not geodesics (except for this very special geodesic, the throat circle itself, the only closed one). On each half-fold, as the distance from the "toroidal bridge" tends to infinity, the metric tends toward the Euclidean metric (2). In Figure 2, corresponding to a representation [r, θ], the upper portions of geodesics crossing the throat circle are shown as solid lines, while the portions corresponding to the other half-fold are shown as dashed lines. Note that one half-fold corresponds to (θ ∈ [0, π]), hence the other corresponds to (θ ∈ [π, 2π]). The throat circle corresponds to θ = 0.
The metric g, regardless of the chosen coordinate system, describes a surface, a two-dimensional object. This object has its own system of geodesics, fundamentally invariant with respect to coordinates. Let us study this system in a coordinate system via Lagrange equations. Introduce the following function F:
(9)
The corresponding Lagrange equations are:
(10)
(11)
Equation (11) gives:
(12)
where h is positive, negative, or zero. Furthermore, if in (3) we divide both sides by , we obtain, classically:
(13)
from which we can derive the differential equation describing plane geodesics in the coordinate system:
(14)
The condition |h| ≤ r, according to (12), means that the absolute value of the cosine of the angle between the tangent to the geodesic and the radial vector is ≤ 1.
Now, let us embed the surface in R³ by adding an extra immersion coordinate z. We choose cylindrical coordinates
The surface is axially symmetric with respect to the z-axis.
The geodesics ( = constant) are the meridional lines of this surface, where:
(15)
which immediately gives the equation of the meridional curve of this surface embedded in R³. It is the parabola:
(16)
Figure 1 shows a 3D view of this surface embedded in R³, together with a geodesic and its projection onto a plane using polar coordinates.
This surface is not simply connected. Among the orbits of the isometry group O₂, we find a circle of minimal perimeter: the throat circle (p = 2 Rs).
Fig. 1: The surface, embedded in R³
and its representation in a coordinate system.
In Figure 2, several geodesics are shown in this representation system.
Fig. 2: Representation of some geodesics. Fig. 3: A particular geodesic crossing the throat circle.
Note that this planar representation of geodesics is not isometric. If we measure length on this plane, it does not correspond to the length measured on the surface.
If we impose that the length dS be real, we see that it determines what we might call the local topology. Let us refer to such a geometric structure as a toroidal bridge. We can also say that this surface possesses a local toroidal topology. It has a single fold, which can be regarded as a pair of bounded half-folds joined along their circular edges along the throat circle, whose perimeter is 2 Rs. These circles are not geodesic lines (except for this very special geodesic, the throat circle itself, the only closed one). On each half-fold, as the distance from the "toroidal bridge" tends to infinity, the metric tends toward the Euclidean metric (2). In Figure 2, corresponding to a [r, ] representation, the upper parts of the geodesics crossing the throat circle are shown as solid lines, while the parts corresponding to the other half-fold are shown as dashed lines. Note that one half-fold corresponds to ( ), and the other to ( ). The throat circle corresponds to = 0. Summary Next page