D.E.U.S.(Dimension Embedded in Unified Symmetry)I: Five-Dimensional Manifolds and World-Lines

Adrian Sabin Popescu† §† Astronomical Institute of Romanian Academy of Science, Str. Cutitul de Argint 5, RO-040557 Bucharest,Romania

Abstract. The goal of this paper is to give a possible theoretical approach for a five-dimensional blackhole internal geometry by using self-similar minimal surfaces for the representation of the three-dimensionalprojections of the considered manifolds. Knowing that for each minimal surface it must exist a conjugatesurface and an associate family capable to describe both these surfaces, we construct the supplementarydimensions for which the partial derivatives of our conjugate manifolds (catenoid and helicoid) satisfy theCauchy-Riemann equations. After constructing the five-dimensional coordinate system andansatz for each ofthe resulting particular manifold we conclude by giving thephysical meanings hidden behind the spacetimegeometry presented in this paper.

PACS numbers: 02.40.Ky, 04.20.Gz, 04.70.-s

1. Introduction

In a classic approach, it is well known that the Einstein equations are the mathematical description of theway in which the mass-energy generates curvature and so the geometry of the spacetime. Geometric andtopological methods in which the geometry precedes physicsand area posteriori linked to the physicalreality are not something new in theory. On one side we have the study of Calabi-Yau manifolds, derivedfrom physics at energies or lengths not reached yet, which describe the string compactification withunbroken supersymmetry [14]. On the other side we have the Topological Quantum Field Theories asSeiberg-Witten theory derived from the Donaldson’s work onsmooth manifolds [18, 6] or Chern-Simonsgauge theory in which the observables are knot and link invariants [20]. The way in which the DEUS modelis constructed is by having, instead of a stress-energy tensor or an action, anab initio fundamental geometryof spacetime and a unique transformation of coordinates from the catenoid-helicoid coordinates to a systemof coordinates that proves to be orthogonal. This is possible if we consider that this geometry contains (andis contained by) the geometry of the external observer and then proving that the cosmological parameters ofthis external observer are the ones of a FRW spacetime. In (t, φ) and (tk, φk) the DEUS object exhibits, foran external observer in (r, tFRW ) coordinates, Black Hole properties as known from literature [4, 21, 27, 28],while in (r, tFRW ) the DEUS object’s helicoidis the FRW (globally) or the Minkowski (locally) spacetime.This paper will set the fundament of a five-dimensional structure (e.g., Black Hole) departing from thetrivial three-dimensional minimal surface representation of the catenoid and its conjugate surface, thehelicoid. The second section is dedicated to the construction of a five-dimensional coordinate system thatsatisfies the Cauchy-Riemann equations, while the last two sections will concentrate on resulted spacetimegeometry and its composing manifolds (ansatz, properties and correlations).

2. Catenary and Catenoid: A Short History

Galileo was the first to investigate the catenary which he mistook for parabola. Galileo’s suggestion that aheavy rope would hang in the shape of a parabola was disprovedby Jungius in 1669, but the true shape of

§ sabinp@aira.astro.ro

D.E.U.S. I 2

Figure 1. Catenoid

the ”chain-curve”, the catenary, was not found until 1690-1691, when Huygens, Leibniz and John Bernoullireplied to a challenge by James Bernoulli. The name was first used by Huygens in a letter to Leibniz in1690. In 1691 James Bernoulli obtained its true form and gavesome of its properties. David Gregory,wrote a comprehensive treatise on the ”catenarian” in 1697.

Minimal surfaces are surfaces that connect given curves in space and have the least area. This isone simple way to define minimal surfaces. The minimal surface called catenoid, first discovered by JeanBaptiste Meusnier in 1776, is the surface of the revolution of a catenary. In nature it can be seen as soapfilms, by bending wires into two parallel circles and dippingthem into soap solution. The soap film betweenthe two circles is the catenoid.

A very interesting and actual application for the catenoid is the Penning and Paul trap used for thestorage of charged particles nearly at rest in space, in the same way in which a storage ring is used for thestorage of relativistic, highly charged ions.

3. Gravitational Instantons from Minimal Surfaces

Gravitational instantons which are given by hyper-Kahlermetrics were studied intensively in the frameworkof supergravity and M-theory as well as Seiberg-Witten theory [8, 9, 25]. Generally speaking, thegravitational instantons are given by regular complete metrics with Euclidian signature and self-dualcurvature which implies that they satisfy the vacuum Einstein field equations [16, 11, 12, 26, 3]. Thesimplest gravitational instantons are obtained from the Schwarzschild-Kerr and Taub-NUT solutions byanalytically continuing them to the Euclidian sector [1].

It was observed [22] that minimal surfaces in Euclidian space can be used in the construction ofinstanton solutions, even for the case of Yang-Mills instantons [5]. For every minimal surface in three-dimensional Euclidian space there exists a gravitational instanton which is an exact solution of the Einsteinfield equations with Euclidian signature and self-dual curvature. If the surface is defined by the Mongeansatzφ = φ(x, t), then the metric establishing this correspondence is given by [1]:

ds2 = 1√

1+ φ2t + φ

2x

[(1 + φ2t )(dt2 + dy2) + (1+ φ2

x)(dx2 + dz2)+

+2φtφx(dt dx + dy dz)] .(1)

The Einstein field equations reduce to the classical equation [23]:(

1+ φ2x

)

φtt − 2φtφxφtx +(

1+ φ2t

)

φxx = 0 (2)

D.E.U.S. I 3

Figure 2. Helicoid

governing minimal surfaces inR3. This solution seems to generate many gravitational instantons, like thehelicoid (Fig. 2) defined byφ(x, t) = arctg(x/t), catenoid(Fig. 1) defined byφ(x, t) = cosh−1

(√x2 + t2

)

,or Scherk surfacegiven asφ(x, t) = log(cosx/cost). Gravitational instantons that follow from thisconstruction will admit at least two commuting Killing vectors ∂y, ∂z, which implies that they may bethe complete non-compact Ricci-flat Kahler metrics that have been considered in the context of cosmicstrings [15, 13]. The general gravitational instanton metric that results from Weierstrass’ general localsolution for minimal surfaces has been constructed in [2] using the correspondence (1) between minimalsurfaces and gravitational instantons. But an important theorem by Bernstein states thatthere are no non-trivial solutions of (2) which are defined on the whole (x, t) plane [24]. From the above examples it isalso clear that the solutions are singular at some points. These singularities are not easy to remove and,therefore, one should be careful in calling these solutionsgravitational instantons, which are supposed tobe complete non-singular solutions.

Within the framework of Dirichletp-branes Gibbons [10], considering the Born-Infeld theory BIonparticles as ends of strings intersecting the brane and ignoring gravity effects, obtains topologicallynon-trivial electrically neutral catenoidal solutions looking like two p-branes joined by a throat. Hisgeneral solution is a non-singular deformation of the catenoid if the charge is not too large and asingular deformation of the BIon solution for charges abovethat limit. Performing a duality rotationGibbons [10] obtained a monopole solution, the BPS limit being a solution of the abelian Bogolmol’nyiequations. This situation resembles that of sub and super extreme black-brane solutions of the supergravitytheories. He also shows that some specific Lagrangians submanifolds may be regarded as supersymmetricconfigurations consisting ofp-branes at angles joined by throats which are the sources of global monopoles.These catenoids are strikingly similar to the Einstein-Rosen bridges (surfaces of constant time) that oneencounters in classical super-gravity solutions representing Black Holes or blackp-branes.

4. Associate Family of Catenoid and Helicoid

Among the fundamental observations in the minimal surface theory is that every minimal surface comesin a family of minimal surfaces, the so-called associate family or Bonnet family [19]. Until now there areknown just few associate families of minimal surfaces [17]:

D.E.U.S. I 4

• the catenoid-helicoid associate family containing the catenoid and the helicoid, classic embeddedminimal surface without periodicities.

• the P-G-D associate family which contains the Schwarz P, theG (Gyroid) and and the D (Diamond)triply periodic minimal surfaces (periodic in three directions).

• the Scherk associate family containing the Scherk’s first (classic doubly periodic) and second (classicsingly-periodic) minimal surfaces.

• Scherk Torii associate family which contains the Schwarz first and second Torii minimal surfaces.

All the above families, excepting the catenoid-helicoid family, contain periodic surfaces (for example, aSchwarz surface is connected at three of its ends with similar Schwarz surfaces, only the resulting surfacebeing associate with the G or P minimal surface of the family), all the composing surfaces coexisting in theconsidered Euclidian space or in the generalRn. So, from logical point of view is more natural to considerthat, if we intend to construct a geometry of a spacetime based on minimal surfaces our first choice mustbe the simplest possible, or, in other words, to take the catenoid and helicoid as the minimal surfaces ofinterest. This choice is favorable also for the simplicity of the mathematical description of the catenoid andthe helicoid. With the catenoid given by:

C(u, v) =

cosv coshusinv coshu

u

(3)

and the helicoid by:

H(u, v) =

sinv sinhu− cosv sinhu

v

(4)

we can obtain the associate familyFϕ(u, v) of both minimal surfaces:

Fϕ(u, v) = cosϕ ·C(u, v) + sin ϕ · H(u, v) (5)

The minimal surfaces represented graphs of the functionsu : Ω ∈ Rn → R andv : Ω ∈ Rn → R [7].The parameterϕ ∈ [0, 2π] is the family parameter. Forϕ = π/2 the surface is called theconjugate

of the surface withϕ = 0, andϕ = π leads to a point mirror image. The helicoid is called theconjugatesurface of the catenoid and, in general, each pair of surfacesFϕ andFϕ+

π2 are conjugate to each other.

In (3) we may have eitheru = A cosh−1

√

t2 + φ2

A

, eitheru = θ with θ ∈ [0, 2π]. In the same way, in

(4) we may have eitherv = arctg

(

tφ

)

, eitherv = θ whereθ is the same as before.

The conjugate surfaces and the associate family display thefollowing properties [19]:

(i) The surface normals at points corresponding to an arbitrary point (u0, v0) in the domain are identical,NFϕ (u0, v0) = NC(u0, v0) = NH(u0, v0);

(ii) The partial derivatives fulfill the following correspondences:

Fϕu (u0, v0) = cosϕ ·Cu(u0, v0) − sin ϕ ·Cv(u0, v0) (6)

Fϕv (u0, v0) = sin ϕ ·Cu(u0, v0) + cosϕ ·Cv(u0, v0) (7)

in particular, the partials of catenoid and helicoid satisfy the Cauchy-Riemann equations:

Cu(u0, v0) = Hv(u0, v0)

Cv(u0, v0) = − Hu(u0, v0) ; (8)

(iii) If a minimal patch is bounded by a straight line, then its conjugate patch is bounded by a planarsymmetry line and vice versa. This can be seen in the catenoid-helicoid case, where the planarmeridians of the catenoid correspond to the straight lines of the helicoid;

D.E.U.S. I 5

(iv) Since at every point the length and the angle between thepartial derivatives are identical for the surfaceand its conjugate (both surfaces are isomeric) we have as a result, that the angles at correspondingboundary vertices of surface and conjugate surface are identical.

In a complex (C3 or Cn) analytical description of minimal surfaces the associatefamily is an easyconcept. It is a basic fact in complex analysis that every harmonic map is the real part of a complexholomorphic map. The three coordinate functions of a minimal surface in Euclidian spaceR3 are harmonicmaps. Therefore, there exist three other harmonic maps which define together with the original coordinatefunctions three holomorphic functions, ora single complex vector-valued function. The real part of thisfunction is the original minimal surface, the imaginary part is the conjugate surface, and the projectionsin-between define surfaces of the associate family.

Let now rewrite the catenoid as:

C(u1, v1) =( cosv1 coshu1

sinv1 sinhu1

)

(9)

and the helicoid as:

H(u2, v2) =(

sinv2 sinhu− cosv2 sinhu

)

(10)

The correspondence between (3) and (9) may be given troughu = A cosh−1(u1

A

)

, function which can

generate the catenoid, whereu1 =√

t2 + φ2 andv1 = θ with θ ∈ [0, 2π]. Hereφ is a spatial coordinateandt a temporal coordinate. In the same way, the correspondence between (4) and (10) is given through

v = v2 = arctg

(

tφ

)

, the function which generates the helicoid, andu2 = sinhu.

With u1 =√

t2 + φ2, v1 = θ, u2 = sinhθ, v2 = arctg

(

tφ

)

in (3) and (4):

C(u1, v1) =

u1 cosv1

u1 sinv1

A cosh−1(u1

A

)

(11)

H(u2, v2) =

u2 sinv2

− u2 cosv2

v2

(12)

In this way it will be possible to construct two distinct surfaces, the bridge between them being theassociate familyFϕ(u, v). We will have thexµ coordinates of the associate family as function ofu1, v1, u2

andv2 using (11) and (12) in (5) and an angleχ (for an object moving inside the associate family surfacefrom the catenoid to the helicoid and back, but never gettingout from it; C is the radius of this motion):

x1 = (u1 cosv1 cosϕ + u2 sin v2 sin ϕ) · cosχx2 = (u1 sin v1 cosϕ − u2 cosv2 sin ϕ) ·Cx3 =

[

A cosh−1(u1

A

)

cosϕ + v2 sin ϕ]

· sin χ(13)

5. Cauchy-Riemann Equations

We will derive a fourth and fifth coordinate for the associatefamily in the way in which to have the Cauchy-Riemann equations satisfied for the five-dimensional tensorx:

∂x∂u

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ϕ1 = 0 or 2πχ1 = 0 or 2πC = 1

=∂x∂v

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ϕ2 = −π2 or π2

χ2 = −π2 or π2

C = 1

(14)

∂x∂v

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ϕ1 = 0 or 2πχ1 = 0 or 2πC = 1

= − ∂x∂u

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ϕ2 = −π2 or π2

χ2 = −π2 or π2

C = 1

(15)

D.E.U.S. I 6

where x

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ϕ1 = 0 or 2πχ1 = 0 or 2πC = 1

is for the five-dimensional catenoid and ˆx

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ϕ2 = −π2 or π2

χ2 = −π2 or π2

C = 1

is for the five-dimensional

helicoid. (u, v) pair can be any combination fromu = A cosh−1

√

t2 + φ2

A

, u = θ, v = arctg

(

tφ

)

,

v = θ. In order to satisfy the properties of the given conjugate surfaces and of their associate family, thecatenoid and helicoid must exist in different spaces or at different times. In the associate family, if thefive-dimensional catenoid exists spacelike the five-dimensional helicoid must be timelike and the opposite.One can chosex4 andx5 as:

A.

x4 = D · u2 sin ϕ + i E · u1 cosϕx5 = F · v1 cosϕ + i G · v2 sin ϕ

(16)

B.

x4 = D · v2 sin ϕ + i E · u1 cosϕx5 = F · v1 cosϕ + i G · u2 sin ϕ

(17)

C.

x4 = D · u2 sin ϕ + i E · v1 cosϕx5 = F · u1 cosϕ + i G · v2 sin ϕ

(18)

D.

x4 = D · v2 sin ϕ + i E · v1 cosϕx5 = F · u1 cosϕ + i G · u2 sin ϕ

(19)

whereD, E, F andG are some functions that must be determined. Cauchy-Riemannconditions (14) and(15) are satisfied only forA andD cases (equations (16) and (19)). Let analyze in detail thesecases.

5.1. Case A

From (14) and (15), in the case withu = A cosh−1

√

t2 + φ2

A

andv = arctg

(

tφ

)

(catenoid-helicoid

associate hyper-surface), results thatE = G = 0. This means that (16) is reduced to:

x4 = D · u2 sin ϕ = D · sinhθ sin ϕx5 = F · v1 cosϕ = F · θ cosϕ

(20)

Whenu = A cosh−1

√

t2 + φ2

A

andv = θ and cosϕ = ±1, sinϕ = 0 we will be in the situation when

we will have only the five-dimensionalcatenoid. Because from the Cauchy-Riemann conditions (14) and(15) results that:

− 2i E = D · coshθ (21)

F = i G · 1√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t− tφ

)

(22)

the supplementary coordinates will be:

x4 = ± i E · u1 = ± i E ·√

t2 + φ2

x5 = ± F · v1 = ± F · θ (23)

D.E.U.S. I 7

or:

x4 = ∓ 12

D · coshθ u1 = ∓12

D · coshθ√

t2 + φ2

x5 = ±

i G · 1√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t− tφ

)

θ(24)

or:

x4 = ± i E · u1 = ± i E ·√

t2 + φ2

x5 = ±

i G ·1

√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t−

tφ

)

θ(25)

or:

x4 = ∓ 12

D · coshθ√

t2 + φ2

x5 = ± F · θ(26)

When cosϕ = 0 and sinϕ = ± 1, foru = θ andv = arctg

(

tφ

)

we will have a only the five-dimensional

helicoid. From (14) and (15) results that:

F = 2i G (27)

i E ·√

t2 + φ2

(

tφ− φ

t

)

= − D · coshθ , (28)

which means that:

x4 = ∓[

iE

coshθ

√

t2 + φ2

(

tφ− φ

t

)]

u2 =

= ∓[

iE

coshθ

√

t2 + φ2

(

tφ− φ

t

)]

sinhθ

x5 = ± i G · v2 = ± i G · arctg

(

tφ

)

(29)

or:

x4 = ± D · u2 = ± D · sinhθ

x5 = ± 12

F · v2 = ±12

F · arctg

(

tφ

)

(30)

or:

x4 = ∓[

iE

coshθ

√

t2 + φ2

(

tφ− φ

t

)]

u2 =

= ∓[

iE

coshθ

√

t2 + φ2

(

tφ− φ

t

)]

sinhθ

x5 = ± 12

F · v2 = ±12

F · arctg

(

tφ

)

(31)

or:

x4 = ± D · u2 = ± D · sinhθ

x5 = ± i G · v2 = ± i G · arctg

(

tφ

)

(32)

D.E.U.S. I 8

5.2. Case D

When cosϕ = ±1 and sinϕ = 0, for u = A cosh−1

√

t2 + φ2

A

andv = θ we will have only the five-

dimensionalcatenoid. From (14), (15) and (19) results that:

x4 = ± D√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t− tφ

)

v1 =

= ± D√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t− tφ

)

θ

x5 = ± F · u1 = ± F ·√

t2 + φ2

(33)

or:

x4 = ± i E · v1 = ± iE · θ

x5 = ∓ i G · 12

coshθ

sinh

√

t2 + φ2

A

cosh2

√

t2 + φ2

A

u1 =

= ∓ i G · 12

coshθ

sinh

√

t2 + φ2

A

cosh2

√

t2 + φ2

A

√

t2 + φ2

(34)

or:

x4 = ± i E · v1 = ± i E · θx5 = ± F · u1 = ± F ·

√

t2 + φ2 (35)

or:

x4 = ±D

√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t−

tφ

)

v1 =

= ± D√

t2 + φ2

cosh2

√

t2 + φ2

A

sinh

√

t2 + φ2

A

(

φ

t− tφ

)

θ

x5 = ∓ i G · 12

coshθ

sinh

√

t2 + φ2

A

cosh2

√

t2 + φ2

A

u1 =

= ∓ i G ·12

coshθ

sinh

√

t2 + φ2

A

cosh2

√

t2 + φ2

A

√

t2 + φ2

(36)

D.E.U.S. I 9

When cosϕ = 0 and sinϕ = ± 1, for u = θ andv = arctg

(

tφ

)

we will have only the five-dimensional

helicoid. From (14) and (15) results that:

i E = 2D (37)

F ·√

t2 + φ2

(

tφ− φ

t

)

= − i G · coshθ , (38)

which means that:

x4 = ± D · v2 = ± D · arctg

(

tφ

)

x5 = ∓ 1coshθ

F ·√

t2 + φ2

(

tφ− φ

t

)

u2 =

= ∓ 1coshθ

F ·√

t2 + φ2

(

tφ− φ

t

)

sinhθ

(39)

or:

x4 = ± i E12· v2 = ± i E

12· arctg

(

tφ

)

x5 = ± i G · u2 = ± i G · sinhθ(40)

or:

x4 = ± D · v2 = ± D · arctg

(

tφ

)

x5 = ± i G · u2 = ± i G · sinhθ(41)

or:

x4 = ± i E12· v2 = ± i E

12· arctg

(

tφ

)

x5 = ∓ 1coshθ

F ·√

t2 + φ2

(

tφ− φ

t

)

u2 =

= ∓1

coshθF ·

√

t2 + φ2

(

tφ−φ

t

)

sinhθ

(42)

The catenoid-helicoid associate hyper-surface (u = A cosh−1

√

t2 + φ2

A

andv = arctg

(

tφ

)

) fulfill the

Cauchy-Riemann conditions only forD = F = 0 which means that (see equation (19)):

x4 = i E · v1 cosϕ = i E · θ cosϕx5 = i G · u2 sin ϕ = i G · sinhθ sin ϕ

(43)

6. World-Lines

6.1. Ergosphere

For a better image and clarity, even that only the three dimensional projections of our five-dimensionalmanifolds can be called ”catenoids” or ”helicoids”, we willpermit ourselves to use these name conventionsalso for our multi-dimensional manifolds. Also, in order todifferentiate the helicoid coordinates from thecatenoids coordinates we will change the notation for the helicoid coordinates fromt to tk and fromφto φk keepingt andφ just for the catenoid. Having (13) and (20) (or (43)) coordinates for the associatehyper-surface we may derive all the unknown quantities withthe help the first fundamental form:

gµν ≡∂x∂xµ

∂x∂xν

by solving the system of equations formed with the help of themetric tensor components. Here it isimportant to say that no matter which choice of pairs for our unknown quantitiesD, E, F, G, (e.g., D and

D.E.U.S. I 10

G etc.) we are making in our fourth and fifth coordinates, thegµν expressions must be the same for all fourpossible choices, this being translated to only one ansatz.From this resulted that all the metric components,exceptinggtφ (or gtkφk ), gtt (or gtktk ) andgφφ (or gφkφk ), must be 0. This applies not only for the associatehyper-surface but also for the catenoid and for the helicoid. Solving the system for the associate hyper-

surface (u = A cosh−1

√

t2 + φ2

A

andv = arctg

(

tφ

)

) without any constraints onC we got some ”strong”

and some ”weak” constraints betweent, tk, φ, φk, θ andϕ coordinates. The strong constraints are:

tφ= −φk

tk

A2 cosh− 2

√

t2 + φ2

A

+ arctg2

(

tkφk

)

= 0

θ = arctg

(

tφ

)

(44)

while the weak ones are:

√

t2 + φ2 cosθ cosϕ + sinhθ sin

[

arctg

(

tkφk

)]

sin ϕ = 0

θ coshθ sin2 ϕ + sinhθ cos2 ϕ = 0

cosϕ = ± φ2

t2

(45)

For the same hyper-surface:

• In D case:

x1 =

√

t2 + φ2 cosθ cosϕ + sinhθ sin

[

arctg

(

tkφk

)]

sin ϕ

· cosχ

x2 =

√

t2 + φ2 sin θ cosϕ − sinhθ cos

[

arctg

(

tkφk

)]

sin ϕ

Cergos

x3 =

arctg

(

tkφk

)

sin ϕ ± i arctg

(

tkφk

)

cosϕ

· sin χ

x4 = ± i√

t2 + φ2 · θ cosϕx5 = ± i sinhθ sin ϕ

(46)

with Eergos = ±√

t2 + φ2 andGergos = ± 1. With Eergos andGergos determined it resulted thatCergos = 1.

• for A case:

x1 =

√

t2 + φ2 cosθ cosϕ + sinhθ sin

[

arctg

(

tkφk

)]

sin ϕ

· cosχ

x2 =

√

t2 + φ2 sin θ cosϕ − sinhθ cos

[

arctg

(

tkφk

)]

sin ϕ

Cergos

x3 =

arctg

(

tkφk

)

sin ϕ ± i arctg

(

tkφk

)

cosϕ

· sin χ

x4 = ± i sinhθ sin ϕx5 = ± i

√

t2 + φ2 · θ cosϕ

(47)

with Fergos = ± i√

t2 + φ2 and Dergos = ± i. With Fergos and Dergos determined it resulted thatCergos = 1

By using the correspondingϕ values for the catenoid and helicoid in the associate hyper-surface (andalso the above ”strong” and ”weak” constraints) we obtained(independently from thegµν = 0 equations

D.E.U.S. I 11

which gaveD, E, F andG) two possible results forC, the same forA andD cases:

C2cat = 1 and C2

cat =

sin

arctg

tφ

cosθ

cos

arctg

tφ

sinθ

C2hel = 1 and C2

hel = −cos

arctg

tkφk

cosθ

sin

arctg

ttφk

sinθ

(48)

With all the said above, the ansatz for the associate hyper-surface proved to be:

ds2ergos = Z ·

φ2k

t2k + φ2k

dt2k + Z ·t2k

t2k + φ2k

dφ2k − Z · tk φk

t2k + φ2k

dtk dφk (49)

where:

Z = sinh2 θ cos2[

arctg

(

tkφk

)]

1

t2k + φ2k

sin2 ϕ cos2 χ+

+sinh2 θ sin2

[

arctg

(

tkφk

)]

1

t2k + φ2k

sin2 ϕ · C2+

+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

cos2 ϕ sin2 χ +1

t2k + φ2k

sin2 ϕ sin2 χ

(50)

Whenθ = 0 in (50):

Ze f f ≡ Zθ=0 =

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

cos2 ϕ sin2 χ +1

t2k + φ2k

sin2 ϕ sin2 χ (51)

From now on we will entitle the associate hyper-surface of the catenoid and helicoid withC = Cergos =

Ccat = Chel = 1 with the nameergosphere. The reasons for choosing this name will be seen from thephysical interpretations of the model equations.

The strong constrainttφ= − φk

tkis having as consequence not only the fact that passing from catenoid

to helicoid the spatial coordinate transforms in a temporalcoordinate and the temporal in spatial, but alsothat the ergosphere acts like a ”mirror” for the objects passing through it (seen by an external observersituated, as we will see, on a warped external space, conformal with a Friedman-Robertson-Walker space).If the object falling through the ergosphere is seen as continuing its path to the interior of the centralobject, the external observer will see its fall reversed in time. So, in this case, the falling object will beseen like not falling at all but coming through the external observer (time reversal). Actually, becausethe external time becomes internal space the external observer will never see the object crossing the

ergosphere but ”frozen” on it. The other strong constraintθ = arctg

(

tφ

)

proved to be validonly for

C = 1 (u = ucat = uhel = v = vcat = vhel). WhenC , 1 it won’t be possible to speak anymore of existenceof a combined hyper-surface of catenoid and helicoid (u , v), they existing without their conjugate hyper-surface. Also, the hyper-surfaces for whichCcat andChel are different from 1 will not satisfy the strong

conditionθ = arctg

(

tφ

)

.

Let us consider now an uni-dimensional, rigid string-like object which rotates in time inθ plane. Itsrotation will generate a helicoidal spacetime sheet. Let denote withtk the temporal coordinate and withφk

its length. Its five-dimensional coordinates (satisfying the Cauchy-Riemann conditions and following thesame rules for their determination) will be:

D.E.U.S. I 12

• A case

x1 = x2 = x3 = x4 = 0

x5 = ± arctg

(

tkφk

)

(52)

• D case

x1 = x2 = x3 = x5 = 0

x4 = ± i arctg

(

tkφk

)

(53)

The metric for this helicoidal spacetime sheet will be:

ds2 = −φ2

k(

t2k + φ2k

)2dt2k −

t2k(

t2k + φ2k

)2dφ2

k +tk φk

(

t2k + φ2k

)2dtk dφk (54)

The first observation is that theA case (52) is spacelike, whileD case (53) is timelike. The secondremarque is that the ansatz (54) is similar to the (49) ansatz, with Z = 1, and with reversed sign. Thissimilarity will be useful when we will describe how our BlackHole will look like and how an observer willfollow different hyper-surfaces in its way to the interior. Anticipating a little bit we will name the spacetimegenerated by this uni-dimensional object ”Black Hole Internal Geometry” (BHIG).

Further on we will concentrate on the five-dimensional catenoid (cos ϕ = ± 1 ; sin ϕ = 0;

u = A cosh−1

√

t2 + φ2

A

; v = θ) and the helicoid (cosϕ = 0 ; sinϕ = ± 1 ; u = θ ; v = arctg

(

tφ

)

) from the

ergosphere (C = 1), with their correspondingx4 andx5 for A andD situations (equations (23), (24), (25),(26) for the catenoid and equations (29), (30), (31), (32) for the helicoid inA case; equations (33), (34),(35), (36) for the catenoid and equations (39), (40), (41), (42) for the helicoid in theD case). For these wederived all the relations betweenD, E, F andG. In theD case, we obtained for thecatenoid:

Fcat = ± iDcat = Ecat = 0

(55)

while for thehelicoid:

Ehel = ± 2i Dhel = 0Ghel = ±1Fhel = 0

(56)

Because the catenoid is not in the same spacetime as the helicoid it is understandable why the catenoidis not consistent anymore withD = F = 0 of theD case ergosphere. However it is consistent with theAcase ergosphere for whichE = G = 0. Analogous, in theA case we will have:

Ecat = ± 1Fcat = Gcat = 0

(57)

and:

Fhel = ± 2i Ghel

Fhel = Ghel = 0Dhel = ± i

(58)

So, for anA case ergosphere we will have aA case helicoid and aD case catenoid and for aD caseergosphere we will have aD case helicoid and aA case catenoid.

The large number of hyper-surfaces which are composing our five dimensional object and theirconsequences for the ensemble determines us to do the following development.

D.E.U.S. I 13

6.2. D Case

6.2.1. Ergosphere’s Helicoid (C = 1 ; θ = arctg

(

tφ

)

)

Parameters:

E = ± 2i D = 0G = ±1F = 0

(59)

Coordinates:

x1 = sinhθ sin

[

arctg

(

tkφk

)]

x2 = − sinhθ cos

[

arctg

(

tkφk

)]

x3 = arctg

(

tkφk

)

x4 = 0 ≡√

(

x1)2+

(

x2)2+

(

x5)2

x5 = ± i sinhθ

(60)

Ansatz:

ds2 = Z ·φ2

k(

t2k + φ2k

)2dt2k + Z ·

t2k(

t2k + φ2k

)2dφ2

k − Z · tk φk(

t2k + φ2k

)2dtk dφk (61)

where:

Z = cosh2θ −E2

4≡ cosh2θ + D2 ≡ cosh2θ (62)

If we makeθ = 0 in (61) ansatz we see thatds2θ=0 = − ds2

BHIG meaning that, in the ergosphere, not onlythat the space and time coordinates switch their roles, but also that the angleθ ”freezes” to 0. It also meansthat the object that crosses the ergosphere heading to BHIG doesn’t violate the causality for the externalobserver which sees it, as exiting from the ergosphere. We will see in the further development that, if theobject is to enter into the BHIG spacetime, it must do it from the helicoid and not from the catenoid.

Another important observation which arrises from the coordinates system is that, from the four-dimensional external observer’s perception,x4 is a ”hidden” spacetime dimension, to this contributingthex1, x2 andx5 dimensions.

6.2.2. Ergosphere’s Catenoid (C = 1 ; θ = arctg

(

tφ

)

)

Parameters:

F = ± iD = E = 0

(63)

Coordinates:

x1 =√

t2 + φ2 cosθx2 =

√

t2 + φ2 sin θ

x3 = A cosh− 1

√

t2 + φ2

A

x4 = 0 ≡√

(

x1)2+

(

x2)2+

(

x5)2

x5 = ± i√

t2 + φ2

(64)

Ansatz:

ds2 = Z · t2

t2 + φ2dt2 + Z · φ

2

t2 + φ2dφ2 + Z · t φ

t2 + φ2dt dφ (65)

D.E.U.S. I 14

where:

Z =

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

+ 1+ F2 ≡sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

(66)

By comparing theZe f f factor from (51) withZ from (62) orZ from (66) we see that, for aχ = ± π/2andϕ for catenoid or helicoid, they are equal, meaning again that, in the ergosphereθ = 0. Like in theprevious case we observe again that herex4 hidden, this happening because ofx1, x2 andx5.

6.2.3. Catenoid with C = 1 and θ , arctg

(

tφ

)

Parameters:

F = G = 0E = ±

√

t2 + φ2

D = ± i(

t2 + φ2)

sinh

√

t2 + φ2

A

cosh2

√

t2 + φ2

A

1φ

t− tφ

(67)

Coordinates:

x1 =√

t2 + φ2 cosθx2 =

√

t2 + φ2 sin θ

x3 = A cosh− 1

√

t2 + φ2

A

x4 = ± i θ√

t2 + φ2

x5 = 0

(68)

Ansatz:

ds2 = Z · t2

t2 + φ2dt2 + Z · φ

2

t2 + φ2dφ2 + Z · t φ

t2 + φ2dt dφ (69)

where:

Z = 1+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

+ F2 ≡

≡ 1+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

− 14

G2 cosh2θ ≡

≡ 1+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

(70)

Here by contrast with previous casesx5 is the one which is hidden.

D.E.U.S. I 15

6.2.4. Helicoid with C = 1 and θ , arctg

(

tφ

)

Parameters:

E = ± 2i D = ± 2√

2 iD = ±

√2

G = ±1

F = ± i cosh3θ1

√

t2k + φ2k

1tkφk− φk

tk

(71)

Coordinates:

x1 = sinhθ sin

[

arctg

(

tkφk

)]

x2 = sinhθ cos

[

arctg

(

tkφk

)]

x3 = arctg

(

tkφk

)

x4 = ± i√

2 arctg

(

tkφk

)

and x4 = ∓ i√

2 arctg

(

tkφk

)

x5 = ± i sinhθ and x5 = ∓ i sinhθ

(72)

Ansatz:

ds2 = Z ·φ2

k(

t2k + φ2k

)2dt2k + Z ·

t2k(

t2k + φ2k

)2dφ2

k − Z · tk φk(

t2k + φ2k

)2dtk dφk (73)

where:

Z = cosh2θ + D2 ≡ cosh2θ − 14

E2 ≡ cosh2θ + 2 (74)

We see that the two different forms forx4 andx5 permit the existence of four spaces all with the samemetric. These are, until now, the first real five-dimensionalhyper-surfaces. Actually these hyper-surfacesare not allowing to the external observer to see the BHIG hyper-surface (”no hair” theorem equivalent).

6.2.5. Pure Catenoidal Hyper-Surface(θ , arctg

(

tφ

)

)

Parameters:

C = ±

√

√

√

√

√

√

√

√

sin

arctg

tφ

cosθ

cos

arctg

tφ

sinθ

F = G = 0

E = ±A cosh− 1

√

t2 + φ2

A

θ

(75)

Coordinates:

x1 =√

t2 + φ2 cosθ

x2 = ±√

t2 + φ2 sin θ

√

√

√

√

√

√

√

√

sin

arctg

tφ

cosθ

cos

arctg

tφ

sinθ

x3 = A cosh− 1

√

t2 + φ2

A

x4 = ± i A cosh− 1

√

t2 + φ2

A

x5 = 0 ≡√

(

x3)2+

(

x4)2

(76)

D.E.U.S. I 16

Ansatz:

ds2 = Z · t2

t2 + φ2dt2 + Z · φ

2

t2 + φ2dφ2 + Z · t φ

t2 + φ2dt dφ (77)

where:

Z = cos2θ − sin2θ cos2θ

sin2θ −A2 cosh− 2

√

t2 + φ2

A

θ2(

t2 + φ2)

+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

+ F2 ≡

≡ cos2θ − sin2θ cos2θ

sin2θ −

A2 cosh− 2

√

t2 + φ2

A

θ2(

t2 + φ2)

+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

(78)

The hidden dimension for this case isx5.

6.2.6. Pure Helicoidal Hyper-Surface(θ , arctg

(

tφ

)

)

Parameters:

C = ± i

√

√

√

√

√

√

√

√

cos

arctg

tkφk

cosθ

sin

arctg

tkφk

sinθ

D = ± iE = ∓ 2F = G = 0

(79)

Coordinates:

x1 = sinhθ sin

[

arctg

(

tkφk

)]

x2 = ∓ i sinhθ cos

[

arctg

(

tkφk

)]

√

√

√

√

√

√

√

√

cos

arctg

tkφk

cosθ

sin

arctg

tkφk

sinθ

x3 = arctg

(

tkφk

)

x4 = ± i arctg

(

tkφk

)

x5 = 0 ≡√

(

x3)2+

(

x4)2

(80)

Ansatz:

ds2 = Z ·φ2

k(

t2k + φ2k

)2dt2k + Z ·

t2k(

t2k + φ2k

)2dφ2

k − Z · tk φk(

t2k + φ2k

)2dtk dφk (81)

D.E.U.S. I 17

where:

Z = sinh2θ

cos2[

arctg

(

tkφk

)]

− cosθsin θ

cos

[

arctg

(

tkφk

)]

sin3

[

arctg

(

tkφk

)]

sin2

[

arctg

(

tkφk

)]

+12

2

+ 1− 14

E2

≡

≡ sinh2θ

cos2[

arctg

(

tkφk

)]

− cosθsin θ

cos

[

arctg

(

tkφk

)]

sin3

[

arctg

(

tkφk

)]

sin2

[

arctg

(

tkφk

)]

+12

2

+ 1+ D2

≡

≡ sinh2θ

cos2[

arctg

(

tkφk

)]

− cosθsin θ

cos

[

arctg

(

tkφk

)]

sin3

[

arctg

(

tkφk

)]

sin2

[

arctg

(

tkφk

)]

+12

2

(82)

In this particular case, because ofx3 andx4, the hidden dimension for this case isx5.

6.3. A Case

6.3.1. Ergosphere’s Helicoid (C = 1 ; θ = arctg

(

tφ

)

)

Parameters:

F = ± 2i GF = G = 0D = ± i

(83)

Coordinates:

x1 = sinhθ sin

[

arctg

(

tkφk

)]

x2 = − sinhθ cos

[

arctg

(

tkφk

)]

x3 = arctg

(

tkφk

)

x4 = ± i sinhθ

x5 = 0 ≡√

(

x1)2+

(

x2)2+

(

x4)2

(84)

Ansatz:

ds2 = Z ·φ2

k(

t2k + φ2k

)2dt2k + Z ·

t2k(

t2k + φ2k

)2dφ2

k − Z · tk φk(

t2k + φ2k

)2dtk dφk (85)

where:

Z = cosh2θ +14

F2 ≡ cosh2θ (86)

As we saw for the same hyper-surface inD case, if we makeθ = 0 in (86) ansatz,ds2θ=0 = − ds2

BHIGmeaning again that the angleθ ”freezes” to 0 inside the ergosphere and that the object thatcrosses theergosphere heading to the BHIG spacetime doesn’t violate the causality for the external observer. Theobject can enter into the BHIG spacetime from the helicoid.

The hidden dimension here is not anymorex4 as forD case, butx5. In this way, theD case can beseen as a projection of thex4 coordinate of the five-dimensional considered hyper-surface, while theA caseas ax5 projection of the same five-dimensional hyper-surface. Theremarque thatA andD cases arex4 orx5 projections of one and the same hyper-surface (the ansatz are the same) will be valid for all the otherconsidered hyper-surfaces.

D.E.U.S. I 18

6.3.2. Ergosphere’s Catenoid (C = 1 ; θ = arctg

(

tφ

)

)

Parameters:

E = ± 1F = G = 0

(87)

Coordinates:

x1 =√

t2 + φ2 cosθx2 =

√

t2 + φ2 sin θ

x3 = A cosh− 1

√

t2 + φ2

A

x4 = ± i√

t2 + φ2

x5 = 0 ≡√

(

x1)2+

(

x2)2+

(

x4)2

(88)

Ansatz:

ds2 = Z · t2

t2 + φ2dt2 + Z · φ

2

t2 + φ2dφ2 + Z · t φ

t2 + φ2dt dφ (89)

where:

Z =

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

+ 1− E2 ≡sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

(90)

As for ourD case, by comparingZe f f factor from (51) withZ from (86) orZ from (90), forχ = ± π/2andϕ for helicoid or catenoid, we also get equality (θ = 0 in the ergosphere). Here the projection is on(x1x2x3x4) plane.

6.3.3. Catenoid with C = 1 and θ , arctg

(

tφ

)

Parameters:

D = E = 0F = ± i

√

t2 + φ2

G = ±(

t2 + φ2)

sinh

√

t2 + φ2

A

cosh2

√

t2 + φ2

A

1φ

t− tφ

(91)

Coordinates:

x1 =√

t2 + φ2 cosθx2 =

√

t2 + φ2 sin θ

x3 = A cosh− 1

√

t2 + φ2

A

x4 = 0x5 = ± i θ

√

t2 + φ2

(92)

Ansatz:

ds2 = Z ·t2

t2 + φ2dt2 + Z ·

φ2

t2 + φ2dφ2 + Z ·

t φt2 + φ2

dt dφ (93)

D.E.U.S. I 19

where:

Z = 1+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

− E2 ≡ 1+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

+14

D2 cosh2θ

≡ 1+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

(94)

Here the projection is on (x1x2x3x5) plane.

6.3.4. Helicoid with C = 1 and θ , arctg

(

tφ

)

Parameters:

F = ± 2i G = ± 2√

2 iG = ±

√2

D = ± i

E = ± cosh3θ1

√

t2k + φ2k

1tkφk− φk

tk

(95)

Coordinates:

x1 = sinhθ sin

[

arctg

(

tkφk

)]

x2 = − sinhθ cos

[

arctg

(

tkφk

)]

x3 = arctg

(

tkφk

)

x4 = ± i sinhθ and x4 = ∓ i sinhθ

x5 = ± i√

2 arctg

(

tkφk

)

and x5 = ∓ i√

2 arctg

(

tkφk

)

(96)

Ansatz:

ds2 = Z ·φ2

k(

t2k + φ2k

)2dt2k + Z ·

t2k(

t2k + φ2k

)2dφ2

k − Z · tk φk(

t2k + φ2k

)2dtk dφk (97)

where:

Z = cosh2θ −G2 ≡ cosh2θ +14

F2 ≡ cosh2θ − 2 (98)

We see that the two different forms forx4 andx5 permit the existence of four hyper-surfaces, all withthe same metric, these (as for theD case) being the five-dimensional surfaces which interdict to the externalobserver to see the BHIG hyper-surface (”no hair” theorem equivalent).

D.E.U.S. I 20

6.3.5. Pure Catenoidal Hyper-Surface(θ , arctg

(

tφ

)

)

Parameters:

C = ±

√

√

√

√

√

√

√

√

sin

arctg

tφ

cosθ

cos

arctg

tφ

sinθ

D = E = 0

F = ± i

A cosh− 1

√

t2 + φ2

A

θ

(99)

Coordinates:

x1 =√

t2 + φ2 cosθ

x2 = ±√

t2 + φ2 sin θ

√

√

√

√

√

√

√

√

sin

arctg

tφ

cosθ

cos

arctg

tφ

sinθ

x3 = A cosh− 1

√

t2 + φ2

A

x4 = 0 ≡√

(

x3)2+

(

x5)2

x5 = ± i A cosh− 1

√

t2 + φ2

A

(100)

Ansatz:

ds2 = Z · t2

t2 + φ2dt2 + Z · φ

2

t2 + φ2dφ2 + Z · t φ

t2 + φ2dt dφ (101)

where:

Z = cos2θ − sin2θ cos2θ

sin2θ −A2 cosh− 2

√

t2 + φ2

A

θ2(

t2 + φ2)

+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

− E2 ≡

≡ cos2θ − sin2θ cos2θ

sin2θ −A2 cosh− 2

√

t2 + φ2

A

θ2(

t2 + φ2)

+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

+14

D2 cosh2θ ≡

≡ cos2θ − sin2θ cos2θ

sin2θ −A2 cosh− 2

√

t2 + φ2

A

θ2(

t2 + φ2)

+

sinh2

√

t2 + φ2

A

cosh4

√

t2 + φ2

A

(102)

The projection is on (x1x2x3x5) plane.

D.E.U.S. I 21

6.3.6. Pure Helicoidal Hyper-Surface(θ , arctg

(

tφ

)

)

Parameters:

C = ± i

√

√

√

√

√

√

√

√

cos

arctg

tkφk

cosθ

sin

arctg

tkφk

sinθ

D = E = 0G = ± 1F = ± 2 i

(103)

Coordinates:

x1 = sinhθ sin

[

arctg

(

tkφk

)]

x2 = ∓ i sinhθ cos

[

arctg

(

tkφk

)]

√

√

√

√

√

√

√

√

cos

arctg

tkφk

cosθ

sin

arctg

tkφk

sinθ

x3 = arctg

(

tkφk

)

x4 = 0 ≡√

(

x3)2+

(

x5)2

x5 = ± i arctg

(

tkφk

)

(104)

Ansatz:

ds2 = Z ·φ2

k(

t2k + φ2k

)2dt2k + Z ·

t2k(

t2k + φ2k

)2dφ2

k − Z · tk φk(

t2k + φ2k

)2dtk dφk (105)

where:

Z = sinh2θ

cos2[

arctg

(

tkφk

)]

− cosθsin θ

cos

[

arctg

(

tkφk

)]

sin3

[

arctg

(

tkφk

)]

sin2

[

arctg

(

tkφk

)]

+12

2

+ 1−G2

≡

≡ sinh2θ

cos2[

arctg

(

tkφk

)]

− cosθsin θ

cos

[

arctg

(

tkφk

)]

sin3

[

arctg

(

tkφk

)]

sin2

[

arctg

(

tkφk

)]

+12

2

+ 1+14

F2

≡

≡ sinh2θ

cos2[

arctg

(

tkφk

)]

−cosθsin θ

cos

[

arctg

(

tkφk

)]

sin3

[

arctg

(

tkφk

)]

sin2

[

arctg

(

tkφk

)]

+12

2

(106)

For this hyper-surface, the projection is on (x1x2x3x5) plane.

7. Conclusions

If we pay attention to the way in which the ansatz factorsZ are looking and to theD, E, F, G parametersmodification from one ansatz to the other, we can construct a global image of our five-dimensionalspacetimes (Fig. 3) and also to see which is the path followedby an object entering/leaving into/fromthe DEUS object. Because nothing interdict us, we can consider that the five-dimensional timelike case is

D.E.U.S. I 22

Figure 3. Projection of the five-dimensional spacetime topology on (x1 x2 x3) plane. For a better clarity, forhaving both the catenoids and the helicoids on the same projection, one of them must be in ourD case while theother inA case. On the image are represented: I - Pure Helicoidal Hyper-Surface withC , 1 andθ , arctg

(

tφ

)

(blue); II - Pure Catenoidal Hyper-Surface withC , 1 andθ , arctg(

tφ

)

(magenta); III - Helicoid withC = 1

and θ , arctg(

tφ

)

(gray); IV - Ergosphere (here are living the Ergosphere’s Catenoid and the Ergosphere’s

Helicoid for whichC = 1 andθ = arctg(

tφ

)

) (green); V - Catenoid withC = 1 andθ , arctg(

tφ

)

(from red topink); VI - BHIG helicoid (pink). The Pure Catenoidal Hyper-Surface [II] will evolve, for the internal observer,together with the objects on it, from a minimal surface representation to the cylindrical geometry, through theempty space where the Catenoids for whichC = 1 andθ , arctg

(

tφ

)

[V] can exist.

D.E.U.S. I 23

the D case of the BHIG spacetime (x4 = ± i arctg

(

tkφk

)

) and the five-dimensional spacelike case is the

BHIG spacetime’sA case (x5 = ± arctg

(

tkφk

)

).

If we consider that the object enters in this space as aPure Catenoidal Hyper-Surface object in, let

sayD case, it will advance together with this hyper-surface passing through theC = 1, θ , arctg

(

tφ

)

intermediate region, in whichθ becomes 0only for the external observer which sees the object falling in.This means that the object is seen as frozen inθ angle, but still falling radially. In its proper coordinatesystem the object is actually still spinning around the black hole. When the object enters theErgosphere’s

Catenoid its θ angle becomes equal with arctg

(

tφ

)

and the coordinatesx4 andx5 switch one with the other,

meaning that, for the external observer, it becomes a spacelike object, totally frozen on the ergosphere(is ”seen” as aA case object), while, in its proper frame, the object still remains aD case object withtwo of its five coordinates rotated. The real object remains eternally frozen on the ergosphere. From thefive-dimensional external observer point of view, we may interpret intuitively (to the degree in which afive-dimensional manifold may be intuitively described) that the object (see Fig. 3) passes instantaneouslyfrom the ”equatorial” plane of our five-dimensional spacetime to its ”pole”, from where it becomes anvirtual object (spatial fluctuations) on exit path from the black hole, first as anErgosphere’s Helicoid

virtual object inA case (here becomes important the strong constrainttφ= −

φk

tk; x4 andx5 change again

their role) and, after that, to the helicoid withC = 1, θ , arctg

(

tφ

)

(A case) where the radial movement

is unfrozen, continuing to thePure Helicoidal Hyper-Surface in which also the rotation is unfrozen (alsoin A case). It is very important the fact that, even when, from five-dimensional point of view, the objectis spacelike, it is actually a four-dimensional timelike object in his system of reference for all the abovehyper-surfaces (with three spatial coordinates and one temporal coordinate), excepting the helicoid with

C = 1 andθ , arctg

(

tφ

)

which makes impossible to the four-dimensional object to see the naked four-

dimensional central singularity (”no hair” theorem), and thePure Helicoidal Hyper-Surface where we havetwo spatial and two temporal dimensions (the object is virtual for the four-dimensional internal or externalobserver).

Independently of what the external observer sees, when the object goes from theErgosphere’sCatenoid to theErgosphere’s Helicoid, it will see himself as being inside the BHIG spacetime and leavingit (ds2

θ=0 = − ds2BHIG ”mirror” effect). If the internal observer is five-dimensional (virtual) it ”will be able

to see” that its reflection on the inner horizon is real (x4 is replaced withx5 and reciprocally inA caseorthe coordinates remain unchanged but the object passes fromtheErgosphere’s Helicoid spacelikeA caseto BHIG’s D timelike case).

The external four-dimensionalD case observer ”sees” across theθ = 0 singularity anA case BHIGhelicoid rotated with 90, this happening because itsx3 world axis is becoming the BHIG’sx5 world axis,the internal phenomena seaming uncorrelated with the external ones due to the fake lack of continuity

between the BHIG helicoid and the helicoid withC = 1, θ , arctg

(

tφ

)

.

All the above story repeats itself but in reverse order if theobject enters the five-dimensional DEUSobject represented in Fig. 3 asPure Helicoidal Hyper-Surface virtual object.

The other two possible situations when a virtual object enters the five-dimensional spacetime (Fig. 3)as aPure Catenoidal Hyper-Surface or asPure Helicoidal Hyper-Surface real object are mainly the same asabove, the difference consisting into the fact that the study has to be done by replacing theA (respectivelyD) case manifolds the object crosses, withD (respectivelyA) case manifolds.

In principle the same remarques are valid also for objects (real or virtual) living inside BHIG andmoving toward exterior, as black hole ejecta.

D.E.U.S. I 24

Acknowledgments

I want to express all my gratitude to Stefan Sorescu whose computer drawing experience was truly valuablefor Fig. 3 creation.Manuscript registered at the Astronomical Institute, Romanian Academy, record No. 249 from 04/19/07.

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