THIN-SHELL WORMHOLES AND GRAVASTARS Are wormholes or ’gravastars’ mimicking gravitational-wave signals from black holes?

Dr. Ali Övgün Ph.D in Physics, Eastern Mediterranean University, Northern Cyprus with Prof. Mustafa Halilsoy 21 September 2017 Now FONDECYT Postdoc. Researcher at Instituto de Física, Pontificia Universidad Católica de Valparaíso with Prof. Joel Saavedra

My Research Areas 1. COSMOLOGY: Inflation and acceleration of universe with Nonlinear Electrodynamics 1 PAPER 2. WORMHOLES: Thinshell Wormholes and Wormholes 9 PAPER 3. GRAVITATIONAL LENSING 6 PAPER 4. COMPACT STARS, BLACK HOLES, GRAVASTARS 2 PAPERS 5. WAVE PROPERTIES OF BHs or WHs: Hawking Radiation, Greybody factors, Quasinormal Modes of Black Holes, Quantum Singularities 20 PAPER 6. PARTICLE PROPERTIES OF BHs or WHs: Geodesics, particle collision near BHs and BSW effect of BHs 2 PAPER

Table of contents

1. INTRODUCTION 2. WHAT IS A WORMHOLE? 3. MOTIVATION 4. THIN-SHELL WORMHOLES 5. HAYWARD THIN-SHELL WH IN 3+1-D 6. ROTATING THIN-SHELL WORMHOLE 7. THIN-SHELL GRAVASTARS

INTRODUCTION

Figure 1: General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915

Figure 2: Gravity

Figure 3: General relativity explains gravity as the curvature of spacetime

Figure 4: The inter-changeable nature of gravity and acceleration is known as the principle of equivalence.

WHAT IS A WORMHOLE?

Figure 5: Hypothetical shortcut between distant points

Figure 6: Can we make journeys to farther stars?

Figure 7: How can we open gate into space-time?

• How can we connect two regions of space-time? • Can we make stable and traversable wormholes?

MOTIVATION

Figure 8: Wormhole

-We do not know how to open the throat without exotic matter. -Thin-shell methods with Israel junction conditions can be used to minimize the exotic matter needed. However, the stability must be saved.

Figure 9: How to realize Wormholes in real life

History of Wormholes

Figure 10: Firstly , Flamm’s work on the WH physics using the Schwarzschild solution (1916).

Figure 11: Einstein and Rosen (ER) (1935), ER bridges connecting two identicalsheets.

Figure 12: J.Wheeler used ”geons” (self-gravitating bundles of electromagnetic fields) by giving the first diagram of a doubly-connected-space (1955).

Wheeler added the term ”wormhole” to the physics literature at the quantum scale.

Figure 13: First traversable WH, Morris-Thorne (1988).

Figure 14: Then Morris, Thorne, and Yurtsever investigated the requirements of the energy condition for traversable WHs.

Figure 15: A technical way to make thin-shell WHs by Visser (1988).

Figure 16: Published in 1995

Traversable Wormhole Construction Criteria • Obey the Einstein field equations. • Must have a throat that connects two asymptotically flat regions of spacetime. • No horizon, since a horizon will prevent two-way travel through the wormhole. • Tidal gravitational forces experienced by a traveler must be bearably small. • Traveler must be able to cross through the wormhole in a finite and reasonably small proper time. • Physically reasonable stress-energy tensor generated by the matter and fields. • Solution must be stable under small perturbation. • Should be possible to assemble the wormhole. ie. assembly should require much less than the age of the universe.

Figure 17: ER=EPR

Traversable Lorentzian Wormholes The first defined traversable WH is Morris Thorne WH with a the red-shift function f(r) and a shape function b(r) : ds2 = −e2f(r) dt2 +

1 1−

b(r) r

dr2 + r2 (dθ2 + sin2 θdϕ2 ).

(1)

• Spherically symmetric and static • Radial coordinate r such that circumference of circle centered around throat given by 2πr • r decreases from +∞ to b = b0 (minimum radius) at throat, then increases from b0 to +∞ • At throat exists coordinate singularity where r component diverges • Proper radial distance l(r) runs from −∞ to +∞ and vice versa

Figure 18: This hypothesis was originally put forward by Mazur and Mottola in 2004.

• • • •

Gravastar literally means Gravitational Vacuum Condensate Star Black holes havea very large entropy value. Gravastars, on the other hand, have quite a low entropy. As a star collapses further the particles fall into a Bose-Einstein state where the entire star nears absolute zero and get very compact. • Acts as a giant atom composed of bosons. • The interior of Gravastars is a de Sitter Spacetime, a positive vacuum energy, an internal negative pressure.

• Mazur and Mottola think that event horizon is actually the outer shell of the Bose-Einstein matter, similar to neutron star

THIN-SHELL WORMHOLES

• Constructing WHs with non-exotic (normal matter) source is a difficult issue in General Relativity. • First, Visser use the thin-shell method to construct WHs by minimizing the exotic matter on the throat of the WHs.

Input: Two space-times -Use the Darmois –Israel formalism and match an interior spacetime to an exterior spacetime -Use the Lanczos equations to find a surface energy density σ and a surface pressure p. -Use the energy conservation to find the equation of motion of particle on the throat of the thin-shell wormhole -Check the stability by using different EoS equations. Outputs Thin-shell wormhole σ < 0 with extrinsic curvature K > 0 of the throat means it required exotic matter.

HAYWARD THIN-SHELL WH IN 3+1-D

A. Ovgun et.al Eur. Phys. J. C 74 (2014): 2796 • The metric of the Hayward BH is given by ds2 = −f(r)dt2 + f(r)−1 dr2 + r2 dΩ2 . with the metric function

( f (r) = 1 −

2mr2 r3 + 2ml2

(2)

) (3)

and dΩ2 = dθ2 + sin2 θdϕ2 .

(4)

• It is noted that m and l are free parameters. • At large r, the metric function behaves as a Schwarzchild BH ( ) 2m 1 lim f (r) → 1 − +O 4 , r→∞ r r

(5)

whereas at small r becomes a de Sitter BH lim f (r) → 1 −

r→0

r2 2

( ) + O r5 .

(6)

• It is noted that the singularity located at r = 0. • Thin-shell is located at r = a . • The throat must be outside of the horizon (a > rh ). • Then we paste two copies of it at the point of r = a. • For this reason the thin-shell metric is taken as ) 2( ds2 = −dτ 2 + a (τ ) dθ2 + sin2 θdϕ2

(7)

where τ is the proper time on the shell. • The Einstein equations on the shell are [ ] Kji − [K] δij = −Sji

(8)

− where [Kij ] = K+ ij − Kij .

• It is noted that the extrinsic curvature tensor is Kji . • Moreover, K stands for its trace. • Setting coordinates ξ i = (τ, θ), the extrinsic curvature formula connecting the two sides of the shell is simply given by ( 2 γ ) α β ∂ x γ ∂x ∂x ± K± = −n + Γ , γ ij αβ ∂ξ i ∂ξ j ∂ξ i ∂ξ j

(9)

where the unit normals (nγ nγ = 1) are αβ ∂H ∂H −1/2 ∂H ± , nγ = ± g ∂xα ∂xβ ∂xγ

(10)

with H(r) = r − a(τ ). • The surface stresses, i.e., surface energy density σ and surface pressures Sθθ = p = Sϕϕ , are determined by the surface stress-energy tensor Sji . • The energy and pressure densities are obtained as √ 4 σ=− f (a) + a˙ 2 (11) a (√ ) ¨a + f′ (a) /2 f (a) + a˙ 2 p=2 +√ . (12) a f (a) + a˙ 2 • Then they reduce to simple form in a static configuration (a = a0 ) 4√ σ0 = − f (a0 ) (13) a0 and

(√ ) f (a0 ) f′ (a0 ) /2 p0 = 2 + √ . a0 f (a0 )

(14)

Once σ ≥ 0 and σ + p ≥ 0 hold, then WEC is satisfied. • It is obvioust hat negative energy density violates the WEC, and consequently we are in need of the exotic matter for constructing thin-shell WH. • Stability of such a WH is investigated by applying a linear perturbation with the following EoS p = ψ (σ)

(15)

• Moreover the energy conservation is Sij;j = 0

(16)

which in closed form it equals to ik j Sij,j + Skj Γiµ kj + S Γkj = 0

(17)

after the line element in Eq.(7) is used, it opens to ∂ (

) ∂ ( 2) σa2 + p a = 0.

(18)

• The 1-D equation of motion is a˙ 2 + V (a) = 0,

(19)

in which V (a) is the potential, V (a) = f −

( aσ )4 4

.

(20)

• The equilibrium point at a = a0 means V′ (a0 ) = 0 and V′′ (a0 ) ≥ 0. • Then it is considered that f1 (a0 ) = f2 (a0 ), one finds V0 = V′0 = 0. • To obtain V′′ (a0 ) ≥ 0 we use the given p = ψ (σ) and it is found as follows ) ( dσ 2 = − (σ + ψ) (21) σ′ = da a and σ ′′ = where ψ ′ =

dψ dσ .

V′′ (a0 ) = f′′0 −

2 (σ + ψ) (3 + 2ψ ′ ) , a2

(22)

After we use ψ0 = p0 , finally it is found that ] 1[ 2 (σ0 + 2p0 ) + 2σ0 (σ0 + p0 ) (1 + 2ψ ′ (σ0 )) (23)

• The equation of motion of the throat, for a small perturbation becomes a˙ 2 +

V′′ (a0 ) (a − a0 )2 = 0. 2

• Note that for the condition of V′′ (a0 ) ≥ 0 , TSW is stable where the motion √ of the throat is oscillatory with angular frequency ω=

V′′ (a0 ) . 2

Some models of EoS Now we use some models of matter to analyze the effect of the parameter of Hayward in the stability of the constructed thin-shell WH. Linear gas For a LG, EoS is choosen as ψ = η0 (σ − σ0 ) + p0 in which η0 is a constant and ψ ′ (σ0 ) = η0 .

(24)

Figure 19: Stability of Thin-Shell WH supported by LG.

Fig. shows the stability regions in terms of η0 and a0 with different Hayward’s parameter. It is noted that the S shows the stable regions.

Chaplygin gas For CG, we choose the EoS as follows

( ψ = η0

1 1 − σ σ0

)

where η0 is a constant and ψ ′ (σ0 ) = − ση02 . 0

+ p0

(25)

Figure 20: Stability of Thin-Shell WH supported by CG.

In Fig., the stability regions are shown in terms of η0 and a0 for different values of ℓ. The effect of Hayward’s constant is to increase the stability of the Thin-Shell WH.

Generalized Chaplygin gas The EoS of the GCG is taken as ( ψ (σ) = η0

1 1 − ν σν σ0

) + p0

(26)

where ν and η0 are constants. We check the effect of parameter ν in the stability and ψ becomes

ψ (σ) = p0

( σ )ν 0

σ

.

(27)

Figure 21: Stability of Thin-Shell WH supported by GCG.

We find ψ ′ (σ0 ) = − σp00 ν. In Fig., the stability regions are shown in terms of ν and a0 with various values of ℓ.

Modified Generalized Chaplygin gas In this case, the MGCG is

( ψ (σ) = ξ0 (σ − σ0 ) − η0

1 1 − ν σν σ0

) + p0

(28)

in which ξ0 , η0 and ν are free parameters. Therefore,

ψ ′ (σ0 ) = ξ0 + η0

η0 ν . σ0ν+1

(29)

Figure 22: Stability of Thin-Shell WH supported by MGCG.

To go further we set ξ0 = 1 and ν = 1. In Fig., the stability regions are plotted in terms of η0 and a0 with various values of ℓ. The effect of Hayward’s constant is to increase the stability of the Thin-Shell WH.

Logarithmic gas Lastly LogG is choosen by follows

σ ψ (σ) = η0 ln + p0 σ0

(30)

in which η0 is a constant. For LogG, we find that

ψ ′ (σ0 ) =

η0 . σ0

(31)

Figure 23: Stability of Thin-Shell WH supported by LogG.

In Fig., the stability regions are plotted to show the effect of Hayward’s parameter clearly. The effect of Hayward’s constant is to increase the stability of the Thin-Shell WH.

Conclusions • On the thin-shell we use the different type of EoS with the form p = ψ (σ) and plot possible stable regions. • We show the stable and unstable regions on the plots. • Stability simply depends on the condition of V′′ (a0 ) > 0. • We show that the parameter ℓ, which is known as Hayward parameter has a important role. • Moreover, for higher ℓ value the stable regions are increased. • It is checked the small velocity perturbations for the throat. • It is found that throat of the thin-shell WH is not stable against such kind of perturbations. • Hence, energy density of the WH is found negative so that we need exotic matter.

ROTATING THIN-SHELL WORMHOLE

A. Ovgun, Eur.Phys.J.Plus 131 (2016) no.11, 389 Constructing The Rotating Thin-Shell Wormhole The 5-d rotating Myers-Perry (5DRMP) black hole solution, which is the generalization of the Kerr solution to higher dimensions, is given by the following space-time metric : 2

ds2 = −F(r)2 dt2 +G(r)2 dr2 +r2b gab dxa dxb + H(r)2 [dψ + Ba dxa − K(r)dt] , (32) in which ( )−1 r2 2MΞ 2Ma2 1+ 2 − 2 + 4 G(r)2 = , (33) ℓ r r ( ) 2Ma2 2Ma H(r)2 = r2 1 + 4 , K(r) = 2 , (34) r r H(r)2 F(r)

=

r , G(r)H(r)

Ξ=1−

a2 , ℓ2

(35)

where B = Ba dxa and b gab dxa dxb =

1(

) 1 dθ2 + sin2 θ dϕ2 , B = cos θ dϕ .

(36)

Note that taking the limit of the Anti-de-Sitter (AdS) length ℓ → ∞ , the asymptotically flat case can be recovered. One writes the mass M and angular momentum J of the spacetime as ) ( πM a2 M= J = πMa . (37) 3+ 2 , 4 ℓ For convenience we move to a comoving frame to eliminate cross terms in the induced metrics by introducing dψ −→ dψ ′ + K± (R(t))dt .

(38)

We choose a radius R(t), which is the throat of the wormhole, and take ˜ ± for the interior and exterior regions with two copies of this manifold M r ≥ R to paste them at an identical hypersurface Σ = {xµ : t = T (τ ), r = R(τ )}, which is parameterized by coordinates yi = {τ, ψ, θ, ϕ} on the 4-d surface. The line element in the interior and the exterior sides become ds2±

=

−F± (r)2 dt2 + G± (r)2 dr2 + r2 dΩ + H± (r)2 {dψ ′ + Ba dxa + [K± (R(t)) −

For simplicity in the comoving frame, we drop the prime on ψ ′ . The ˜ =M ˜+ U M ˜ − . We use geodesically complete manifold is satisfied as M

the Darmois-Israel formalism to construct the rotating thin-shell wormhole. Using the Israel junction conditions, the Einstein’s equations for the wormhole produce

β(R2 H)′ J (RH)′ , φ = − , 4π R3 2π 2 R4 H [ ]′ β H H [ 2 ]′ R β , ∆P = , 4πR3 4π R

ρ

= −

(40)

P

=

(41)

where primes stand for d/dR and √ β ≡ F(R) 1 + G(R)2 R˙ 2 .

(42)

Without rotation or in the case of a corotating frame, the momentum φ and the anisotropic pressure term ∆P are equal to zero. Consequently, the static energy and pressure densities at the throat of wormhole R =

R0 are given by ρ0

=

P0

=

F(R20 H)′ J (R0 H)′ , φ = − , 0 2π 2 R40 H 4π R30 [ ]′ F H H [ 2 ]′ R F , ∆P = . 0 0 4π R0 4πR30

−

(43) (44)

To check the stability of the wormhole, we use the linear equation of state (EoS): P = ωρ. (45) The stability of the wormhole solution depends upon the conditions of V′′eff (R0 ) > 0 and V′eff (R0 ) = Veff (R0 ) = 0 as 1 ′′ 2 V (R0 ) (R − R0 ) . (46) 2 eff Let us then introduce x = R − R0 and write the equation of motion again: 1 x˙ 2 + V′′eff (R0 ) x2 = 0 (47) 2 which after a derivative with respect to time reduces to Veff ∼

1 ¨x + V′′eff (R0 ) x = 0.

(48)

We show the conditions for a positive stability value. Our main aim is to discover the behavior of V′′eff (R0 ) as √ V′′eff

1 = [−40 (R0 R0 10

( ) R0 4 + 2 a2 −2 ω −4 ω 2 ) R0 a + 2 R0 10 + 12 a2 − 12 R0 6 4 R0 +40 a2 R0 4 ].

Note that a = ω = 0 corresponds to a non-rotating case. It is easy to see that a has a crucial role in this stability analysis; the stability regions are shown in the following Figs.:

猀

猀

Figure 24: Stability of wormhole supported by linear gas in terms of ω and R0 for a = 0.1.

猀

猀 Figure 25: Stability of wormhole supported by linear gas in terms of ω and R0 for a = 0.4.

猀

猀 Figure 26: Stability of wormhole supported by linear gas in terms of ω and R0 for a = 1.

Discussions • We have studied a TSWH with rotation in 5-D constructed using a Myers-Perry black hole with cosmological constants using a cut-and-paste procedure. • The standard stability approach has been applied by considering a linear gas model at the TSWH throat. • A key feature of the current analysis is the inclusion of rotation in the form of non-zero values of angular momentum. • Another key aspect of the current analysis is the focus on different values of parameter (a), which plays a crucial role in making the TSWH more stable in five dimensions. • We observe that the stability of the wormhole is fundamentally linked to the behavior of the constant (a). • The amount of exotic matter required to support the TSWH is always a crucial issue; unfortunately, we are not able to completely eliminate exotic matter during the constructing of the stable rotating TSWH.

THIN-SHELL GRAVASTARS

A. Ovgun et. al Eur. Phys. J. C (2017) 77:566 Exterior Of Gravastars: Noncommutative geometry inspired Charged BHs • The metric of a noncommutative charged black hole is described by the metric given in S. Ansoldi et al.Phys.Lett.B645,261(2007): ds2 = −f(r)dt2 + f(r)−1 dr2 + r2 dΩ2 , ) ( Q2 with f(r) = 1 − 2Mr θ + r2θ .

(51)

STRUCTURE EQUATIONS OF CHARGED GRAVASTARS • The metrics of interior is the nonsingular de Sitter spacetimes: r2− 2 r2− −1 2 )dt + (1 − ) dr− + r2− dΩ2− (52) − α2 α2 and exterior of noncommutative geometry inspired charged spacetimes: ds2 = −(1 −

2 2 2 ds2 = −f(r)+ dt2+ + f(r)−1 + dr+ + r+ dΩ+ .

(53)

• Then using the relation of Si j = diag(−σ, P, P), one can find the surface energy density, σ, and the surface pressure, P, as follows: √ σ

P

= −

=

1 2Mθ+ 1− + 4πa a

Q2θ+ a2

√( + a˙ 2 − (

1−

a2 α2

)

, + a˙ 2 (54)

2a2 α2

Mθ+ 1 + a¨a + a − ˙2 1 √1 + a + a¨a − a √( − ) Q2 8πa 2 ˙2 1 − 2Maθ+ + aθ+ 1 − αa 2 + a˙ 2 2 + a

˙2

) . (55)

• To calculate the surface mass of the thin-shell, one can use this equation Ms (a) = 4πa2 σ. To find stable solution, we consider a static case [a0 ∈ (r− , r+ )].

Then the surface charge and pressure at static case reduce to

σ(a0 ) =

P(a0 ) =

√ √( ) Q2θ+ a20 1 2Mθ+ − 1− + 2 − 1− 2 , (56) 4πa0 a0 α a0 ( ) 2a20 Mθ+ 1 − α2 1 − a0 1 √ √ − ( ) . 8πa0 Q2θ+ a20 Mθ+ 1 − α2 1 − a0 + a2 0

(57)

Figure 27: We plot σ0 + p0 as a function of Mθ and a0 . We choose Qθ = 1 and α = 0.4. Note that in this region the NEC is satisfied.

Stability of the Charged Thin-shell Gravastars in Noncommutative Geometry • We use the surface energy density σ(a) on the thin-shell of the gravastars to check stability: 1 2 a˙ + V(a) = 0, 2

(58)

with the potential, 1 V(a) = 2

{

[ ]2 [ ]2 } B(a) Ms (a) D(a) 1− − − . a 2a Ms (a)

(59)

It is noted that B(a) and D(a) are

[( B(a) =

2Mθ+ −

Q2θ+ a

2

)

] 3 + ( αa 2 )

,

[( 2Mθ+ − D(a) =

Q2θ+ a

2

)

3

− ( αa 2 )

]

. (60

• One can also easily obtain the surface mass as a function of the potential: √ √ 2 2 Q 2M a θ+ Ms (a) = −a 1 − + θ+ − 2V(a) − (1 − 2 ) − 2V(a) . a a2 α (61) • Then the surface charge and the pressure are rewritten in terms of potential as follows: √ √ Q2θ+ 1 2Mθ+ a2 σ=− 1− + 2 − 2V − (1 − 2 ) − 2V , 4πa a a α ( ) M 1 − 2V − aV′ − α2a2 1 1 − 2V − aV′ − aθ+ . √ − √ P= Q2θ+ 8πa a2 2Mθ+ (1 − α2 ) − 2V 1 − a + a2 − 2V

(62)

(63)

• To find the stable solution, we linearize it using the Taylor expansion around the a0 to second order as follows: V(a) =

1

V′′ (a0 )(a − a0 )2 + O[(a − a0 )3 ] .

(64)

• Note that for stability, the conditions are V(a0 ) = V′ (a0 ) = 0, a˙ 0 = ¨a0 = 0 and V′′ (a0 ) > 0. • Using the relation Ms (a) = 4πσ(a)a2 , we use the M′′s (a0 ) instead of V′′ (a0 ) ≥ 0 as following: 2 −a30 2 [ 2(Mθ+ +Qθ+ ) ]2 [ ] 1 2 a 0 α − M′′s (a0 ) ≥ 3 2 2 a 4a0 [1 − 2Mθ+ + Qθ+ [(1 − α02 )]3/2 ]3/2 a0 a20 2Q2θ+ 4a0 1 a30 2 √ + −√ α 2 , 2 a0 Q2θ+ 2M θ+ 1− (1 − α2 ) a0 + a2 0

• so for the stable solution V′′ (a0 ) = −

3M2s (a0 ) [ M′s (a0 ) M′′s (a0 ) ] + − Ms (a0 ) 4a40 4a20 a30

−

B′′ (a0 ) B′ (a0 ) B(a0 ) M′s (a0 )2 + − − 2a0 a2 4a2 a3

(65)

−

D′2 (a0 ) + D(a0 )D′′ (a0 ) 4D(a0 )D′ (a0 )M′s (a0 ) + D2 (a0 )M′′s (a0 ) + M2s (a0 ) M3s (a0 ) 3D2 (a0 )(M′s )2 (a0 ) − , (66) M4s (a0 )

where M′s (a0 ) = 8πa0 σ0 − 8πa0 (σ0 + p0 ),

(67)

and M′′s (a0 )

=

8πσ0 − 32π(σ0 + p0 )

+

4π [2(σ0 + p0 ) + 4(σ0 + p0 )(1 + η)] .

(68)

• Moreover, we have also introduced η(a) = P ′ (a)/σ ′ (a)|a0 , as a parameter which will play a fundamental role in determining the stability regions of the respective solutions. • Generally, η interpreted as the speed of sound, so that one would expect the range of 0 < η ≤ 1, that the speed of sound should not exceed the speed of light.

• But the range of η may be lying outside the range of 0 < η ≤ 1, on the surface layer. • Therefore, in this work the range of η will be relaxed and we use graphical reputation to determine the stability regions given by the Eq. (34), due to the complexity of the expression.

Figure 28: Stability regions of the charged gravastar in terms of η = P′ /σ ′ as a function of a0 . We choose Mθ = 2, Qθ = 1.5, α = 0.4.

Figure 29: Stability regions of the charged gravastar in terms of η = P′ /σ ′ as a function of a0 . We choose Mθ = 1.5, Qθ = 1, α = 0.2.

Figure 30: Stability regions of the charged gravastar in terms of η = P′ /σ ′ as a function of a0 . We choose Mθ = 3, Qθ = 2.5, α = 0.5.

Conclusions • We have studied the stability of a particular class of thin-shell gravastar solutions, in the context of charged noncommutative geometry. • We have considered the de Sitter geometry in the interior of the gravastar by matching an exterior charged noncommutative solution at a junction interface situated outside the event horizon. • We have showed that gravastar’s shell satisfies the null energy conditions. • We further explored the gravastar solution by the dynamical stability of the transition layer, which is sufficient close to the event horizon. • We have found that for specific choices of mass Mθ , charge Qθ and the values of α, the stable configurations of the surface layer do exists which is sufficiently close to where the event horizon is expected to form.

Figure 31: Tae-shack-cure-lair (Pronunciation in Turkish)

Dr. Ali Övgün Ph.D in Physics, Eastern Mediterranean University, Northern Cyprus with Prof. Mustafa Halilsoy 21 September 2017 Now FONDECYT Postdoc. Researcher at Instituto de Física, Pontificia Universidad Católica de Valparaíso with Prof. Joel Saavedra

My Research Areas 1. COSMOLOGY: Inflation and acceleration of universe with Nonlinear Electrodynamics 1 PAPER 2. WORMHOLES: Thinshell Wormholes and Wormholes 9 PAPER 3. GRAVITATIONAL LENSING 6 PAPER 4. COMPACT STARS, BLACK HOLES, GRAVASTARS 2 PAPERS 5. WAVE PROPERTIES OF BHs or WHs: Hawking Radiation, Greybody factors, Quasinormal Modes of Black Holes, Quantum Singularities 20 PAPER 6. PARTICLE PROPERTIES OF BHs or WHs: Geodesics, particle collision near BHs and BSW effect of BHs 2 PAPER

Table of contents

1. INTRODUCTION 2. WHAT IS A WORMHOLE? 3. MOTIVATION 4. THIN-SHELL WORMHOLES 5. HAYWARD THIN-SHELL WH IN 3+1-D 6. ROTATING THIN-SHELL WORMHOLE 7. THIN-SHELL GRAVASTARS

INTRODUCTION

Figure 1: General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915

Figure 2: Gravity

Figure 3: General relativity explains gravity as the curvature of spacetime

Figure 4: The inter-changeable nature of gravity and acceleration is known as the principle of equivalence.

WHAT IS A WORMHOLE?

Figure 5: Hypothetical shortcut between distant points

Figure 6: Can we make journeys to farther stars?

Figure 7: How can we open gate into space-time?

• How can we connect two regions of space-time? • Can we make stable and traversable wormholes?

MOTIVATION

Figure 8: Wormhole

-We do not know how to open the throat without exotic matter. -Thin-shell methods with Israel junction conditions can be used to minimize the exotic matter needed. However, the stability must be saved.

Figure 9: How to realize Wormholes in real life

History of Wormholes

Figure 10: Firstly , Flamm’s work on the WH physics using the Schwarzschild solution (1916).

Figure 11: Einstein and Rosen (ER) (1935), ER bridges connecting two identicalsheets.

Figure 12: J.Wheeler used ”geons” (self-gravitating bundles of electromagnetic fields) by giving the first diagram of a doubly-connected-space (1955).

Wheeler added the term ”wormhole” to the physics literature at the quantum scale.

Figure 13: First traversable WH, Morris-Thorne (1988).

Figure 14: Then Morris, Thorne, and Yurtsever investigated the requirements of the energy condition for traversable WHs.

Figure 15: A technical way to make thin-shell WHs by Visser (1988).

Figure 16: Published in 1995

Traversable Wormhole Construction Criteria • Obey the Einstein field equations. • Must have a throat that connects two asymptotically flat regions of spacetime. • No horizon, since a horizon will prevent two-way travel through the wormhole. • Tidal gravitational forces experienced by a traveler must be bearably small. • Traveler must be able to cross through the wormhole in a finite and reasonably small proper time. • Physically reasonable stress-energy tensor generated by the matter and fields. • Solution must be stable under small perturbation. • Should be possible to assemble the wormhole. ie. assembly should require much less than the age of the universe.

Figure 17: ER=EPR

Traversable Lorentzian Wormholes The first defined traversable WH is Morris Thorne WH with a the red-shift function f(r) and a shape function b(r) : ds2 = −e2f(r) dt2 +

1 1−

b(r) r

dr2 + r2 (dθ2 + sin2 θdϕ2 ).

(1)

• Spherically symmetric and static • Radial coordinate r such that circumference of circle centered around throat given by 2πr • r decreases from +∞ to b = b0 (minimum radius) at throat, then increases from b0 to +∞ • At throat exists coordinate singularity where r component diverges • Proper radial distance l(r) runs from −∞ to +∞ and vice versa

Figure 18: This hypothesis was originally put forward by Mazur and Mottola in 2004.

• • • •

Gravastar literally means Gravitational Vacuum Condensate Star Black holes havea very large entropy value. Gravastars, on the other hand, have quite a low entropy. As a star collapses further the particles fall into a Bose-Einstein state where the entire star nears absolute zero and get very compact. • Acts as a giant atom composed of bosons. • The interior of Gravastars is a de Sitter Spacetime, a positive vacuum energy, an internal negative pressure.

• Mazur and Mottola think that event horizon is actually the outer shell of the Bose-Einstein matter, similar to neutron star

THIN-SHELL WORMHOLES

• Constructing WHs with non-exotic (normal matter) source is a difficult issue in General Relativity. • First, Visser use the thin-shell method to construct WHs by minimizing the exotic matter on the throat of the WHs.

Input: Two space-times -Use the Darmois –Israel formalism and match an interior spacetime to an exterior spacetime -Use the Lanczos equations to find a surface energy density σ and a surface pressure p. -Use the energy conservation to find the equation of motion of particle on the throat of the thin-shell wormhole -Check the stability by using different EoS equations. Outputs Thin-shell wormhole σ < 0 with extrinsic curvature K > 0 of the throat means it required exotic matter.

HAYWARD THIN-SHELL WH IN 3+1-D

A. Ovgun et.al Eur. Phys. J. C 74 (2014): 2796 • The metric of the Hayward BH is given by ds2 = −f(r)dt2 + f(r)−1 dr2 + r2 dΩ2 . with the metric function

( f (r) = 1 −

2mr2 r3 + 2ml2

(2)

) (3)

and dΩ2 = dθ2 + sin2 θdϕ2 .

(4)

• It is noted that m and l are free parameters. • At large r, the metric function behaves as a Schwarzchild BH ( ) 2m 1 lim f (r) → 1 − +O 4 , r→∞ r r

(5)

whereas at small r becomes a de Sitter BH lim f (r) → 1 −

r→0

r2 2

( ) + O r5 .

(6)

• It is noted that the singularity located at r = 0. • Thin-shell is located at r = a . • The throat must be outside of the horizon (a > rh ). • Then we paste two copies of it at the point of r = a. • For this reason the thin-shell metric is taken as ) 2( ds2 = −dτ 2 + a (τ ) dθ2 + sin2 θdϕ2

(7)

where τ is the proper time on the shell. • The Einstein equations on the shell are [ ] Kji − [K] δij = −Sji

(8)

− where [Kij ] = K+ ij − Kij .

• It is noted that the extrinsic curvature tensor is Kji . • Moreover, K stands for its trace. • Setting coordinates ξ i = (τ, θ), the extrinsic curvature formula connecting the two sides of the shell is simply given by ( 2 γ ) α β ∂ x γ ∂x ∂x ± K± = −n + Γ , γ ij αβ ∂ξ i ∂ξ j ∂ξ i ∂ξ j

(9)

where the unit normals (nγ nγ = 1) are αβ ∂H ∂H −1/2 ∂H ± , nγ = ± g ∂xα ∂xβ ∂xγ

(10)

with H(r) = r − a(τ ). • The surface stresses, i.e., surface energy density σ and surface pressures Sθθ = p = Sϕϕ , are determined by the surface stress-energy tensor Sji . • The energy and pressure densities are obtained as √ 4 σ=− f (a) + a˙ 2 (11) a (√ ) ¨a + f′ (a) /2 f (a) + a˙ 2 p=2 +√ . (12) a f (a) + a˙ 2 • Then they reduce to simple form in a static configuration (a = a0 ) 4√ σ0 = − f (a0 ) (13) a0 and

(√ ) f (a0 ) f′ (a0 ) /2 p0 = 2 + √ . a0 f (a0 )

(14)

Once σ ≥ 0 and σ + p ≥ 0 hold, then WEC is satisfied. • It is obvioust hat negative energy density violates the WEC, and consequently we are in need of the exotic matter for constructing thin-shell WH. • Stability of such a WH is investigated by applying a linear perturbation with the following EoS p = ψ (σ)

(15)

• Moreover the energy conservation is Sij;j = 0

(16)

which in closed form it equals to ik j Sij,j + Skj Γiµ kj + S Γkj = 0

(17)

after the line element in Eq.(7) is used, it opens to ∂ (

) ∂ ( 2) σa2 + p a = 0.

(18)

• The 1-D equation of motion is a˙ 2 + V (a) = 0,

(19)

in which V (a) is the potential, V (a) = f −

( aσ )4 4

.

(20)

• The equilibrium point at a = a0 means V′ (a0 ) = 0 and V′′ (a0 ) ≥ 0. • Then it is considered that f1 (a0 ) = f2 (a0 ), one finds V0 = V′0 = 0. • To obtain V′′ (a0 ) ≥ 0 we use the given p = ψ (σ) and it is found as follows ) ( dσ 2 = − (σ + ψ) (21) σ′ = da a and σ ′′ = where ψ ′ =

dψ dσ .

V′′ (a0 ) = f′′0 −

2 (σ + ψ) (3 + 2ψ ′ ) , a2

(22)

After we use ψ0 = p0 , finally it is found that ] 1[ 2 (σ0 + 2p0 ) + 2σ0 (σ0 + p0 ) (1 + 2ψ ′ (σ0 )) (23)

• The equation of motion of the throat, for a small perturbation becomes a˙ 2 +

V′′ (a0 ) (a − a0 )2 = 0. 2

• Note that for the condition of V′′ (a0 ) ≥ 0 , TSW is stable where the motion √ of the throat is oscillatory with angular frequency ω=

V′′ (a0 ) . 2

Some models of EoS Now we use some models of matter to analyze the effect of the parameter of Hayward in the stability of the constructed thin-shell WH. Linear gas For a LG, EoS is choosen as ψ = η0 (σ − σ0 ) + p0 in which η0 is a constant and ψ ′ (σ0 ) = η0 .

(24)

Figure 19: Stability of Thin-Shell WH supported by LG.

Fig. shows the stability regions in terms of η0 and a0 with different Hayward’s parameter. It is noted that the S shows the stable regions.

Chaplygin gas For CG, we choose the EoS as follows

( ψ = η0

1 1 − σ σ0

)

where η0 is a constant and ψ ′ (σ0 ) = − ση02 . 0

+ p0

(25)

Figure 20: Stability of Thin-Shell WH supported by CG.

In Fig., the stability regions are shown in terms of η0 and a0 for different values of ℓ. The effect of Hayward’s constant is to increase the stability of the Thin-Shell WH.

Generalized Chaplygin gas The EoS of the GCG is taken as ( ψ (σ) = η0

1 1 − ν σν σ0

) + p0

(26)

where ν and η0 are constants. We check the effect of parameter ν in the stability and ψ becomes

ψ (σ) = p0

( σ )ν 0

σ

.

(27)

Figure 21: Stability of Thin-Shell WH supported by GCG.

We find ψ ′ (σ0 ) = − σp00 ν. In Fig., the stability regions are shown in terms of ν and a0 with various values of ℓ.

Modified Generalized Chaplygin gas In this case, the MGCG is

( ψ (σ) = ξ0 (σ − σ0 ) − η0

1 1 − ν σν σ0

) + p0

(28)

in which ξ0 , η0 and ν are free parameters. Therefore,

ψ ′ (σ0 ) = ξ0 + η0

η0 ν . σ0ν+1

(29)

Figure 22: Stability of Thin-Shell WH supported by MGCG.

To go further we set ξ0 = 1 and ν = 1. In Fig., the stability regions are plotted in terms of η0 and a0 with various values of ℓ. The effect of Hayward’s constant is to increase the stability of the Thin-Shell WH.

Logarithmic gas Lastly LogG is choosen by follows

σ ψ (σ) = η0 ln + p0 σ0

(30)

in which η0 is a constant. For LogG, we find that

ψ ′ (σ0 ) =

η0 . σ0

(31)

Figure 23: Stability of Thin-Shell WH supported by LogG.

In Fig., the stability regions are plotted to show the effect of Hayward’s parameter clearly. The effect of Hayward’s constant is to increase the stability of the Thin-Shell WH.

Conclusions • On the thin-shell we use the different type of EoS with the form p = ψ (σ) and plot possible stable regions. • We show the stable and unstable regions on the plots. • Stability simply depends on the condition of V′′ (a0 ) > 0. • We show that the parameter ℓ, which is known as Hayward parameter has a important role. • Moreover, for higher ℓ value the stable regions are increased. • It is checked the small velocity perturbations for the throat. • It is found that throat of the thin-shell WH is not stable against such kind of perturbations. • Hence, energy density of the WH is found negative so that we need exotic matter.

ROTATING THIN-SHELL WORMHOLE

A. Ovgun, Eur.Phys.J.Plus 131 (2016) no.11, 389 Constructing The Rotating Thin-Shell Wormhole The 5-d rotating Myers-Perry (5DRMP) black hole solution, which is the generalization of the Kerr solution to higher dimensions, is given by the following space-time metric : 2

ds2 = −F(r)2 dt2 +G(r)2 dr2 +r2b gab dxa dxb + H(r)2 [dψ + Ba dxa − K(r)dt] , (32) in which ( )−1 r2 2MΞ 2Ma2 1+ 2 − 2 + 4 G(r)2 = , (33) ℓ r r ( ) 2Ma2 2Ma H(r)2 = r2 1 + 4 , K(r) = 2 , (34) r r H(r)2 F(r)

=

r , G(r)H(r)

Ξ=1−

a2 , ℓ2

(35)

where B = Ba dxa and b gab dxa dxb =

1(

) 1 dθ2 + sin2 θ dϕ2 , B = cos θ dϕ .

(36)

Note that taking the limit of the Anti-de-Sitter (AdS) length ℓ → ∞ , the asymptotically flat case can be recovered. One writes the mass M and angular momentum J of the spacetime as ) ( πM a2 M= J = πMa . (37) 3+ 2 , 4 ℓ For convenience we move to a comoving frame to eliminate cross terms in the induced metrics by introducing dψ −→ dψ ′ + K± (R(t))dt .

(38)

We choose a radius R(t), which is the throat of the wormhole, and take ˜ ± for the interior and exterior regions with two copies of this manifold M r ≥ R to paste them at an identical hypersurface Σ = {xµ : t = T (τ ), r = R(τ )}, which is parameterized by coordinates yi = {τ, ψ, θ, ϕ} on the 4-d surface. The line element in the interior and the exterior sides become ds2±

=

−F± (r)2 dt2 + G± (r)2 dr2 + r2 dΩ + H± (r)2 {dψ ′ + Ba dxa + [K± (R(t)) −

For simplicity in the comoving frame, we drop the prime on ψ ′ . The ˜ =M ˜+ U M ˜ − . We use geodesically complete manifold is satisfied as M

the Darmois-Israel formalism to construct the rotating thin-shell wormhole. Using the Israel junction conditions, the Einstein’s equations for the wormhole produce

β(R2 H)′ J (RH)′ , φ = − , 4π R3 2π 2 R4 H [ ]′ β H H [ 2 ]′ R β , ∆P = , 4πR3 4π R

ρ

= −

(40)

P

=

(41)

where primes stand for d/dR and √ β ≡ F(R) 1 + G(R)2 R˙ 2 .

(42)

Without rotation or in the case of a corotating frame, the momentum φ and the anisotropic pressure term ∆P are equal to zero. Consequently, the static energy and pressure densities at the throat of wormhole R =

R0 are given by ρ0

=

P0

=

F(R20 H)′ J (R0 H)′ , φ = − , 0 2π 2 R40 H 4π R30 [ ]′ F H H [ 2 ]′ R F , ∆P = . 0 0 4π R0 4πR30

−

(43) (44)

To check the stability of the wormhole, we use the linear equation of state (EoS): P = ωρ. (45) The stability of the wormhole solution depends upon the conditions of V′′eff (R0 ) > 0 and V′eff (R0 ) = Veff (R0 ) = 0 as 1 ′′ 2 V (R0 ) (R − R0 ) . (46) 2 eff Let us then introduce x = R − R0 and write the equation of motion again: 1 x˙ 2 + V′′eff (R0 ) x2 = 0 (47) 2 which after a derivative with respect to time reduces to Veff ∼

1 ¨x + V′′eff (R0 ) x = 0.

(48)

We show the conditions for a positive stability value. Our main aim is to discover the behavior of V′′eff (R0 ) as √ V′′eff

1 = [−40 (R0 R0 10

( ) R0 4 + 2 a2 −2 ω −4 ω 2 ) R0 a + 2 R0 10 + 12 a2 − 12 R0 6 4 R0 +40 a2 R0 4 ].

Note that a = ω = 0 corresponds to a non-rotating case. It is easy to see that a has a crucial role in this stability analysis; the stability regions are shown in the following Figs.:

猀

猀

Figure 24: Stability of wormhole supported by linear gas in terms of ω and R0 for a = 0.1.

猀

猀 Figure 25: Stability of wormhole supported by linear gas in terms of ω and R0 for a = 0.4.

猀

猀 Figure 26: Stability of wormhole supported by linear gas in terms of ω and R0 for a = 1.

Discussions • We have studied a TSWH with rotation in 5-D constructed using a Myers-Perry black hole with cosmological constants using a cut-and-paste procedure. • The standard stability approach has been applied by considering a linear gas model at the TSWH throat. • A key feature of the current analysis is the inclusion of rotation in the form of non-zero values of angular momentum. • Another key aspect of the current analysis is the focus on different values of parameter (a), which plays a crucial role in making the TSWH more stable in five dimensions. • We observe that the stability of the wormhole is fundamentally linked to the behavior of the constant (a). • The amount of exotic matter required to support the TSWH is always a crucial issue; unfortunately, we are not able to completely eliminate exotic matter during the constructing of the stable rotating TSWH.

THIN-SHELL GRAVASTARS

A. Ovgun et. al Eur. Phys. J. C (2017) 77:566 Exterior Of Gravastars: Noncommutative geometry inspired Charged BHs • The metric of a noncommutative charged black hole is described by the metric given in S. Ansoldi et al.Phys.Lett.B645,261(2007): ds2 = −f(r)dt2 + f(r)−1 dr2 + r2 dΩ2 , ) ( Q2 with f(r) = 1 − 2Mr θ + r2θ .

(51)

STRUCTURE EQUATIONS OF CHARGED GRAVASTARS • The metrics of interior is the nonsingular de Sitter spacetimes: r2− 2 r2− −1 2 )dt + (1 − ) dr− + r2− dΩ2− (52) − α2 α2 and exterior of noncommutative geometry inspired charged spacetimes: ds2 = −(1 −

2 2 2 ds2 = −f(r)+ dt2+ + f(r)−1 + dr+ + r+ dΩ+ .

(53)

• Then using the relation of Si j = diag(−σ, P, P), one can find the surface energy density, σ, and the surface pressure, P, as follows: √ σ

P

= −

=

1 2Mθ+ 1− + 4πa a

Q2θ+ a2

√( + a˙ 2 − (

1−

a2 α2

)

, + a˙ 2 (54)

2a2 α2

Mθ+ 1 + a¨a + a − ˙2 1 √1 + a + a¨a − a √( − ) Q2 8πa 2 ˙2 1 − 2Maθ+ + aθ+ 1 − αa 2 + a˙ 2 2 + a

˙2

) . (55)

• To calculate the surface mass of the thin-shell, one can use this equation Ms (a) = 4πa2 σ. To find stable solution, we consider a static case [a0 ∈ (r− , r+ )].

Then the surface charge and pressure at static case reduce to

σ(a0 ) =

P(a0 ) =

√ √( ) Q2θ+ a20 1 2Mθ+ − 1− + 2 − 1− 2 , (56) 4πa0 a0 α a0 ( ) 2a20 Mθ+ 1 − α2 1 − a0 1 √ √ − ( ) . 8πa0 Q2θ+ a20 Mθ+ 1 − α2 1 − a0 + a2 0

(57)

Figure 27: We plot σ0 + p0 as a function of Mθ and a0 . We choose Qθ = 1 and α = 0.4. Note that in this region the NEC is satisfied.

Stability of the Charged Thin-shell Gravastars in Noncommutative Geometry • We use the surface energy density σ(a) on the thin-shell of the gravastars to check stability: 1 2 a˙ + V(a) = 0, 2

(58)

with the potential, 1 V(a) = 2

{

[ ]2 [ ]2 } B(a) Ms (a) D(a) 1− − − . a 2a Ms (a)

(59)

It is noted that B(a) and D(a) are

[( B(a) =

2Mθ+ −

Q2θ+ a

2

)

] 3 + ( αa 2 )

,

[( 2Mθ+ − D(a) =

Q2θ+ a

2

)

3

− ( αa 2 )

]

. (60

• One can also easily obtain the surface mass as a function of the potential: √ √ 2 2 Q 2M a θ+ Ms (a) = −a 1 − + θ+ − 2V(a) − (1 − 2 ) − 2V(a) . a a2 α (61) • Then the surface charge and the pressure are rewritten in terms of potential as follows: √ √ Q2θ+ 1 2Mθ+ a2 σ=− 1− + 2 − 2V − (1 − 2 ) − 2V , 4πa a a α ( ) M 1 − 2V − aV′ − α2a2 1 1 − 2V − aV′ − aθ+ . √ − √ P= Q2θ+ 8πa a2 2Mθ+ (1 − α2 ) − 2V 1 − a + a2 − 2V

(62)

(63)

• To find the stable solution, we linearize it using the Taylor expansion around the a0 to second order as follows: V(a) =

1

V′′ (a0 )(a − a0 )2 + O[(a − a0 )3 ] .

(64)

• Note that for stability, the conditions are V(a0 ) = V′ (a0 ) = 0, a˙ 0 = ¨a0 = 0 and V′′ (a0 ) > 0. • Using the relation Ms (a) = 4πσ(a)a2 , we use the M′′s (a0 ) instead of V′′ (a0 ) ≥ 0 as following: 2 −a30 2 [ 2(Mθ+ +Qθ+ ) ]2 [ ] 1 2 a 0 α − M′′s (a0 ) ≥ 3 2 2 a 4a0 [1 − 2Mθ+ + Qθ+ [(1 − α02 )]3/2 ]3/2 a0 a20 2Q2θ+ 4a0 1 a30 2 √ + −√ α 2 , 2 a0 Q2θ+ 2M θ+ 1− (1 − α2 ) a0 + a2 0

• so for the stable solution V′′ (a0 ) = −

3M2s (a0 ) [ M′s (a0 ) M′′s (a0 ) ] + − Ms (a0 ) 4a40 4a20 a30

−

B′′ (a0 ) B′ (a0 ) B(a0 ) M′s (a0 )2 + − − 2a0 a2 4a2 a3

(65)

−

D′2 (a0 ) + D(a0 )D′′ (a0 ) 4D(a0 )D′ (a0 )M′s (a0 ) + D2 (a0 )M′′s (a0 ) + M2s (a0 ) M3s (a0 ) 3D2 (a0 )(M′s )2 (a0 ) − , (66) M4s (a0 )

where M′s (a0 ) = 8πa0 σ0 − 8πa0 (σ0 + p0 ),

(67)

and M′′s (a0 )

=

8πσ0 − 32π(σ0 + p0 )

+

4π [2(σ0 + p0 ) + 4(σ0 + p0 )(1 + η)] .

(68)

• Moreover, we have also introduced η(a) = P ′ (a)/σ ′ (a)|a0 , as a parameter which will play a fundamental role in determining the stability regions of the respective solutions. • Generally, η interpreted as the speed of sound, so that one would expect the range of 0 < η ≤ 1, that the speed of sound should not exceed the speed of light.

• But the range of η may be lying outside the range of 0 < η ≤ 1, on the surface layer. • Therefore, in this work the range of η will be relaxed and we use graphical reputation to determine the stability regions given by the Eq. (34), due to the complexity of the expression.

Figure 28: Stability regions of the charged gravastar in terms of η = P′ /σ ′ as a function of a0 . We choose Mθ = 2, Qθ = 1.5, α = 0.4.

Figure 29: Stability regions of the charged gravastar in terms of η = P′ /σ ′ as a function of a0 . We choose Mθ = 1.5, Qθ = 1, α = 0.2.

Figure 30: Stability regions of the charged gravastar in terms of η = P′ /σ ′ as a function of a0 . We choose Mθ = 3, Qθ = 2.5, α = 0.5.

Conclusions • We have studied the stability of a particular class of thin-shell gravastar solutions, in the context of charged noncommutative geometry. • We have considered the de Sitter geometry in the interior of the gravastar by matching an exterior charged noncommutative solution at a junction interface situated outside the event horizon. • We have showed that gravastar’s shell satisfies the null energy conditions. • We further explored the gravastar solution by the dynamical stability of the transition layer, which is sufficient close to the event horizon. • We have found that for specific choices of mass Mθ , charge Qθ and the values of α, the stable configurations of the surface layer do exists which is sufficiently close to where the event horizon is expected to form.

Figure 31: Tae-shack-cure-lair (Pronunciation in Turkish)