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Cancellation of the central singularity of the Schwarzschild

solution with natural mass inversion process.

Published inModern Physics Letters A, Vol. 30, No. 9 (2015) 1550051

J.P. Petit∗G. D’Agostini†

March 25, 2021

Abstract

We reconsider the classical Schwarzschild solution in the context of a Janus cosmological

model. We show that the central singularity can be eliminated through a simple coordinate

change and that the subsequent transit from one fold to the other is accompanied by mass

inversion. In such scenario matter swallowed by black holes could be ejected as invisible

negative mass and dispersed in space.

Keywords: Black hole; space bridge; Schwarzschild metric; Kerr metric; central singularity; Janus

cosmological model; Gaussian coordinates; mass inversion process.

PACS Nos.: 98.80.Bp, 98.80.Qc

1 Introduction

When Schwarzschild published his solution1of Einstein equation, in 1916, the basic hypothesis

was just time-independence and spherical symmetry. Nobody suspected that there was an addi-

tional one: null-homotopy. The consequence of this last one was that, in the proposed solution,

rwas a radial coordinate:

ds2=1−Rs

rc2dt2−dr2

1−Rs

r−r2dθ2+ sin2θdϕ2.(1)

A ﬁrst problem at r=Rs, considered as coordinate singularity, was eliminated by various

coordinates changes. See Fig. 1.

But r= 0 was considered as a true singularity, a point of space where the geodesics end.

The goal of this paper is to show, through a very simple coordinate change, that this so-called

“central singularity” is due to a wrong choice of local topology.

We must introduce the concept of representation space which is the space in which we build

a mental image of an object. As an example we believe we live in R3 and that space and time

are separated. This old belief is useful and enough for today’s life. But if the speed of light

was much smaller we should shift to Minkowski’s spacetime. Similarly, when we send a probe

to Jupiter or predict an eclipse we neglect space curvature. If this last was not neglectible, we

should use Einstein’s equation instead of Newton’s. If we intend to describe an object through

a line element it is easy to show, through 2D examples that inadequate coordinate choice may

induce wrong image of a geometric object. For example, consider the following metric:

ds2=dr2

1−r2

R2

s

+r2dϕ, (2)

∗CNRS, BP 55, 84122 Pertuis, France, e-mail: jppetit1937@yahoo.fr

†dagostini.gilles@laposte.net

1

arXiv:2103.12845v1 [gr-qc] 23 Mar 2021

Coordinates Line element

Eddington-Finkelstein

(ingoing) 1−rs

sdv2−2 dvdr−r2dΩ2regular at horizon,

extends across

future horizon

Eddington-Finkelstein

(outgoing) 1−rs

sdu2−2 dudr−r2dΩ2regular at horizon,

extends across

past horizon

Gullstrand-Painlevé 1−rs

sdT2−2rτs

rdTdr−dr2−r2dΩ2regular at horizon

Isotropic 1−rs

4R

1 + τs

4R2dt2−1 + rs

4R4dx2+ dy2+ dz2isotropic lightcones

on constant

time slices

Fig. 1. Schwarzschild’s geometry. Alternative coordinates.

whose signature (+,+) changes into (−,+) when r > Rs. But, through the simple following

coordinate change

r=Rssin θ, (3)

this object becomes a sphere:

ds2=R2

sdθ2+ sin2θdϕ2.(4)

Another example. Consider the metric:

ds2=dr2

−r2+ 2rR +r2

0−R2+r2dϕ2(5)

The signature is (+,+) if

R−r0< r < R +r0,(6)

but through the new coordinate change

r=R+r0cos θ, (7)

we get the well-known metric of the torus:

ds2=r0dθ2+ (R+r0cos θ) dϕ2.(8)

Now, let us focus on the space part of Schwarzschild’s line element, limited to {r, ϕ}coordi-

nates:

dΣ2=dr2

1−Rs

r

+r2dϕ2(9)

For r < Rs, the signature (+,+) is changed into (−,+). Let us make the change of variable:

r=Rs(1 + Log ch ρ),(10)

which gives:

dΣ2=R2

s(1 + Log ch ρ)

Log ch ρth2ρdρ2+ (1 + Log ch ρ)2dϕ2.(11)

2

All singularities disappear, r=Rscorresponds to ρ= 0. In this point the determinant of

the metric

det g=R4

s

(1 + Log ch ρ)2

Log ch ρth2ρ,

is no longer zero. The metric is well deﬁned for all values of ρ. If we embed the surface in a

3D-Euclidean space we can deﬁne the meridians, corresponding to

dΣ2=dr2

1−Rs

r

+ dz2(12)

And we immediately get the meridians as∗

z=±2Rsrr

Rs

−1, r2=Rs+z2

4Rs

.(13)

The surface is a space bridge, a “2D diabolo” linking two 2D-Euclidean surfaces.

Fig. 2. The 2D diabolo embedded in R3.

The problem of the signature has disappeared. From Lagrange equations we can calculate

the geodesics in the [ρ, ϕ]coordinate system. If embedded, the surface owns a throat circle whose

perimeter is 2πRs. We can shape the surface as a twofold F(+) and F(−)cover of a M2manifold

with a 1D common circular border, and create induced mapping between adjacent points M(+)

and M(−).

Figure 4shows the projection αβγ of a geodesic ABC (see Fig. 2) tangent to the common

boundary, which ensures the continuity of the geodesics of the fold F(+) with the ones of the fold

F(−)(dotted lines). Radial line λµν is the image of a geodesic meridian curve LMN of Fig. 2.

This will be important for the following.

Figure 5shows how vicinities of adjacent points are linked by enantiomorphy relationship.

Figures 3–5correspond to a reduction of our representation space to a 2D Euclidean rep-

resentation space, through a non-isometric imbedding which artiﬁcially transforms the throat

into a pleat and couples points through an adjacent and enantiomorphic relationship. “The circle

owns no center and there is nothing inside”, because we are out of the considered 2D surface,

the 2D diabolo.

∗Erratum page 11

3

Fig. 3. Twofold cover of a manifold with a circle as common boundary.

Fig. 4. Projection of geodesics.

Now introduce the 3D metric

dΣ2=dr2

1−Rs

r

+r2dθ2+ sin2θdϕ2,(14)

which is Euclidean at inﬁnity. When r < Rsthe signature (+ + +) is changed into (−+ +).

Applying (10) we get

ds2=R2

s(1 + Log ch ρ)

Log ch ρth2ρdρ2+ (1 + Log ch ρ)2dθ2+ sin2θdϕ2.(15)

Its determinant never vanishes. The metric is well deﬁned for all values of ρand is Euclidean

at inﬁnite. It is a 3D space bridge linking to 3D Euclidean spaces. Its throat, corresponding to

ρ= 0, is a S2 sphere.

Unfortunately we do not own a 4D Euclidean representation space to operate an isometric

imbedding. So that, similarly as in Fig. 3, we will use a non-isometric imbedding in 3D Euclidean

space. Then the throat is converted into a “pleat” along the projected S2 sphere. Similarly, some

geodesics, continuous in this 3D hypersurface, seem to turn back on the sphere (see Fig. 6).

To show 3D space orientation we will use a tetrahedron (see Fig. 7). To illustrate the enan-

tiomorphy relationship we need to project this object through the S2 throat sphere, each vertice

following a geodesic, as showed in Fig. 8.

4

Fig. 5. When the triangle crosses the common boundary, the orientation is reversed.

Fig. 6. When geodesics of the 3D hypersurface cross the throat sphere they appear tangent to it in this 3D

non-isometric imbedding in a 3D Euclidean space.

Fig. 7. Oriented tetrahedron.

5

Fig. 8. By crossing the throat sphere, the tetrahedron is inverted.

In the 2D {ρ, ϕ}representation the adjacent points are deﬁned by the relation:

M: (ρ, θ)→M0: (−ρ, θ).

In the 3D {ρ, θ, ϕ}representation the adjacent points in 2D and 3D are deﬁned by the

relation:

M: (ρ, θ, ϕ)→M0: (−ρ, θ, ϕ).

The association of points Mand M0goes hand in hand with an enantiomorphic relation

between their corresponding neighborhoods.

Now let us go back to (2) and apply (10). We get

ds2=Log ch ρ

1 + Log ch ρc2dt2−R2

s(1 + Log ch ρ)

Log ch ρth2ρdρ2+ (1 + Log ch ρ)2dθ2+ sin2θdϕ2.

(16)

When ρtends to ±∞,Log ch ρ→ρand th ρ→1. The metric tends to Lorentz metric. Space

is extended to (ρ > 0; ρ < 0) domain. The hypersurface becomes a spacetime bridge, linking two

Lorentz spaces through a throat surface S2. When we calculate the geodesics in the plane θ=π

2

in the {t, r, θ, ϕ}representation we ﬁnd the following (Eq. (6.90) in Ref. 8):

dϕ=±1

r2

dr

qc2l2−1

h2+Rs

h2r−1

r2+Rs

f3

,(17)

where land hare the classical parameters of the quasi-Keplerian trajectory (Eqs. (6.80) and

(6.81) in Ref. 8). On the Schwarzschild’s sphere (r=Rs)we get:

tg α=Rs

dϕ

dr

r=Rs

=h

Rscl .(18)

In a {t, ρ, θ, ϕ}representation, with θ=π

2, we obtain curves ρ=f(ϕ)that will be inscribed

in two (adjacent) folds F(+) and F(−).

With

(tg β)ρ→0=ρdϕ

dρ=rdϕ

dr

ρ

r

dr

dρ=h

R2

scl ρ2→0.(19)

The throat sphere r=Rsis reduced to a point. But the geodesics of the fold F(+) can be

prolonged continuously in the adjacent fold F(−).The central singularity disappears. Now let us

deal with spacetime structures.

Referring to Introduction to General Relativity, Sec. 2.6 of Ref. 8:

6

Fig. 9. Geodesical path in a [ρ, ϕ]representation (plane, θ=π/2).

By letting a family of geodesics play a particular role among the coordi-nates lines

Gauss introduced a useful coordinate system. Consider a 4D space with hyperbolic

metric with signature (1,−1,−1,−1). Assume we can imbed a 3D hypersurface Σ3,

imbedded in the 4D space Σ4. Assume, in some place, we can deﬁne a vector n

normal to Σ3, which satisﬁes:

n0n0+n1n1+n1n1+n1n1>0

which, in the familiar language of special relativity theory, implies that Σ3is “oriented

in space” (whereas the vector n, normal to s, is “oriented in time”).

We introduce in the surface Σ3three coordinates x∗1,x∗2,x∗3which serve to charac-

terise the point P∗∈Σ3. Through each point P∗of the 3D surface Σ3we draw the

geodesic which is orthogonal to Σ3at P∗. These geodesic will form a non-intersecting

curves in some neighborhood Mof Σ3such that, through each point Pof Mthere

will be exactly one of the geodesics constructed. We introduce now, in the entire

4D domain M, coordinates as follows: Given P, we consider the geodesic passing

through Pand its original point P∗∈Σ3. We deﬁne the coordinate xiof Pin terms

of the arc length P∗Pof the geodesic and of the coordinate x∗iof P∗.

In this manner, the three coordinates x1,x2,x3remain constant along any geodesic

perpendicular to Σ3. It follows that, along such a geodesic,

ds2= (d ˙x)2, g00 = 1.

This is the classical way Gaussian coordinates are deﬁned. This is possible if and only if

g00 6= 0, if the term of the line element related to time-marker is nonzero. Note that, in Fig. 1,

Fig. 10

7

Fig. 11. Building a 3D spacetime.

Fig. 12. In turning the throat circle, the arrow of time is inverted.

through various choices of coordinate systems, in Schwarzschild solution this term g00 (called

gtt) vanishes on the so-called horizon (r=Rs), or throat surface (ρ= 0). It means that on this

peculiar portion of the hypersurface, normal vector and space orientation cannot be deﬁned.

Let us build a 3D spacetime on a 2D surface, as the twofold cover of a manifold M2, as shown

in Fig. 11.

As shown in Fig. 12 we will ﬁnd some problem at the common boundary, where it is impossible

to deﬁne space orientation and arrow of time (identiﬁed to normal vector). But, for adjacent

regions we have imbricated PT-symmetrical spacetime structures. Such coupling concept, called

at that time “twin universe theory”, was ﬁrst presented by Sakharov in 196712–15 and later in

Ref. 11. In addition, this goes with recent works9,10 (Janus Cosmological Model).

We have to deal with 4D spacetime, not 3D spacetime, which is just a didactic image of such

geometric structure.

Back to the expression of the Schwarzschild metric in the new coordinates (16) we see that,

on the throat surface (ρ= 0), gtt vanishes (as well as in expressions shown in Fig. 1). According

to Sec. 2.6 of Ref. 8, Gaussian coordinates cannot be deﬁned, which means that on the throat

surface, time and space cannot be oriented. This ﬁts time and space inversion from fold F(+) to

fold F(−), so that we must write joint metrics:

ds(+)2 =Log ch ρ

1 + Log ch ρc2dt(+)2

−R2

s(1 + Log ch ρ)

Log ch ρth2ρdρ2+ (1 + Log ch ρ)2dθ2+ sin2θdϕ2,(20a)

ds(−)2 =Log ch ρ

1 + Log ch ρc2dt(−)2

−R2

s(1 + Log ch ρ)

Log ch ρth2ρdρ2+ (1 + Log ch ρ)2dθ2+ sin2θdϕ2,(20b)

8

with

t(−)=−t(+).(21)

In the neighborhood of ρ= 0 it is possible to write the nearby expressions:

ds(+)2=ρ2

2dt(+)2−2 dρ2−R2

sdϕ2,(22a)

ds(−)2=ρ2

2dt(−)2−2 dρ2−R2

sdϕ2.(22b)

The inversion of the time variable does not imply a change in the sign of the proper time, and

from one fold to the other ds(−)takes the reins of ds(+). It is not possible to have an inversion

in the length measure along a geodesic. In other words we have ds(+)ds(−)>0. But t(+)

and t(−)are nothing more than “time markers”, simple coordinates, and thus one will have:

dt(−)=−dt(+). So here one ﬁnds again the central idea of diﬀerential geometry: the length

element has an only intrinsic reality. Lagrange equations give always in the vicinity of ρ= 0 the

relations:

¨ϕ= ¨ρ= 0.(23)

These functions are linear and monotonic as a function of the proper times (lengths s(+) and

s(−)) that enchain themselves in passing from one fold to the other without inversion.

dt(+)

ds(+) =C

ρ2,dt(−)

ds(−)=−C

ρ2, C =Cst .(24)

The sign of the constant Cdepends on the sense adopted for the passage from one fold to the

other. The object is thus a “black hole” and a “white fountain” at the same time. But, if measured

with variable t(+) the passage is achievable only in an inﬁnite time. If the object results from

the implosion of a neutron star, its mass would be transferred to the negative energy region. But

for the observers located in one of the folds such phenomena of implosion–explosion will appear

to be “freezed in time”.

As shown in Ref. 16 time-inversion goes with mass and energy inversion. In Refs. 11 and 12,

according to Janus Cosmological Model the universe is composed by positive and negative energy

(and mass if they own) particles, respectively described by metrics g(+)

µν and g(−)

µν , solutions of a

coupled ﬁeld equation system. Spacetime bridges operate mass inversion. From Ref. 11 we know

that masses with opposite signs repel each other. On another hand negative matter does not

interact with positive matter by electromagnetic, strong or weak interaction.

2 Conclusion and Discussion

If black holes exist and swallow matter this last, instead to be crushed in a central singularity,

would be discretely rejected as an invisible negative matter.

The main feature of the theory presented here is the mass (and space) inversion process. The

advantage is to avoid the puzzling problem of a “central singularity” and to explain the fate of

matter swallowed by black holes. But it implies injection of negative energy (and mass if the

own) particles in spacetime, considered as a manifold plus two metrics g(+)

µν and g(−)

µν .

As precised in Ref. 9we assumed that particles of opposite masses do not interact neither

by electromagnetic forces nor strong or weak forces, they could not enter into a collision. Some

colleagues have criticized this idea arguing that “the particles are on the same spacetime”. The

answer to this question is: if one considers the problem on purely geometrical grounds, those

encounters would be “geometrically impossible” because the two subsets move along disjoint

families of geodesics.

9

A second criticism may rely on the immediate instability of a quantum vacuum which could

create pairs (+m, −m). But it is based on the theoretical framework of quantum gravity that,

however, still today remains purely hypothetical. The creation and annihilation of pairs of par-

ticles of opposite mass has not been de-scribed till today.

A third criticism may issue from Quantum Field Theory, which excludes straight away states

of negative energy “because a particle could not have an energy less than that of the vacuum”

(p. 76 of Ref. 17). We quote:

If we suppose that Tis linear and unitary then we should face the disastrous conclu-

sion that for any state Ψof energy Ethere is another state T−1Ψof energy −E. To

avoid this we are forced here to conclude that Tis anti-linear and anti-unitary.

In order to refute this statement we would say that it is bootstrap talk and that the conclusion

is contained in the hypothesis, as occurs with the “CPT theorem”.

In his book Weinberg,17 let us quote his sentence on p. 104:

No examples are known of particles that furnish unconventional representation of

inversions, so these possibilities will not be pursued here. From now on, the inversions

will be assumed to have the conventional action assumed in Sec. 2.6.

This sentence refers of course to the hypothesis expressed on p. 76 of Ref. 17 about the

anti-unitary and anti-linear character of the Toperator. However, cosmic acceleration implies

the action of a negative pressure and hence of negative energy (pressure is an energy density by

unit volume). The discovery of such quite un-foreseen phenomenon18–27 makes it compelling for

Quantum Field Theory to be extended in order to include negative energy states.

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Erratum

Equation (13) page 3has a typo (extra square) and should be written:

r=Rs+z2

4Rs

11