CC BY
11
Chelyabinsk Physical and Mathematical Journal. 2017. Vol. 2, iss. 2. P. 245-252.
УДК 530.12
THIN SHELL SCHWARZSCHILD-PHANTOM WORMHOLE STABILITY
R. Kh. Karimova, R. N. Izmailovb
M. Akmullah Bashkir State Pedagogical University, Ufa, Russia akarimov_ramis_92@mail.ru; bizmailov.ramil@gmail.com
It has been suggested that a possible candidate for the present accelerated expansion of the Universe is "phantom energy". The latter possesses an equation of state of the form ш = p/p < —1, consequently violating the null energy condition. As this is the fundamental ingredient to sustain traversable wormholes, this cosmic fluid presents us with a natural scenario for the existence of these exotic geometries. Recently, it has been shown by Lobo that phantom energy with ш = pr/p < —1 could support phantom wormholes. Several classes of such solutions have been derived by him. While the inner spacetime is represented by asymptotically flat phantom wormhole that have repulsive gravity, it is most likely to be unstable to perturbations. Hence, we consider a situation, where a phantom wormhole is somehow trapped inside a Schwarzschild sphere across a thin shell. Applying the method developed by Garcia, Lobo and Visser (GLV), we shall exemplify that the shell can possess zones of stability depending on certain constraints.
Keywords: phantom wormhole, thin-shell technique, stability.
Introduction
Wormholes are topological tunnels that connect two universes or two distant regions of spacetime, that has received some attention after the influential work by Morris and Thorne [1]. Wormholes have not been observed in any experiment but have not yet been ruled out by observations either. There could nonetheless be a modest interest in the topic of wormholes so long as the exterior does not deviate from the Schwarzschild vacuum but allows other matter like phantom in the interior. We then call it phantom wormhole. Since most wormhole are solutions unstable [2], we look for stable solutions, and hence study the stability of phantom wormholes.
An alternative way to analise wormhole stability using Tangherlini's approachh [3] was applied for Ellis and phantom wormhole solutions. It was shown that depending on location, an observer might get stability or instability.
Phantom wormholes could be interesting in the sense that they are built out of phantom energy defined by equation of state ш = pr/p < —1, which is speculated to be a possible cause driving late-time cosmic acceleration [4]. Both wormholes or phantom equation of state have really not been confirmed by observations. However, mathematical solutions exist. It has been recently shown by Lobo [5] that wormholes with phantom energy could actually be found as exact solutions of Einstein's equations. However, since the phantom wormholes are built of the Null Energy Condition (NEC) violating exotic matter (since pr + p < 0), they are likely to be unstable though an exact formulation of such instability is still unavailable. We try to imagine a situation, where some kind of stability involving phantom wormholes would still be possible.
Renewed interest in the subject is due in part to the discovery that our universe is undergoing an accelerated expansion [6; 7], that is, a > 0 in the Friedmann equation
a/a = —4n/3(p + 3p). The acceleration is caused by a negative pressure dark energy with equation of state p = up, u < —1/3 and p > 0. A value of u < —1/3 is required for accelerated expansion; u =1 corresponds to a cosmological constant [8]. Of particular interest is the case u = —1, referred to as phantom energy. For this case, p + p < 0, in violation of the null energy condition.
The strategy in this paper is to start with a general line element, together with the above equation of state, and to find zones of stability of obtained solutions.
To that end, we thought it useful to consider that a phantom wormhole is somehow trapped inside a Schwarzschild sphere across a thin shell. We shall employ a novel and recent formalism to study the stability of such thin shells developed by Garcia, Lobo and Visser (GLV) [9]. The formalism was primarily used in [10; 11] for the stability of the thin-shell wormholes. We shall consider asymptotically flat phantom wormhole solution derived by Lobo [12], gluing them to a vacuum Schwarzschild exterior. The configuration would resemble a gravastar of certain radius having in its interior repulsive phantom matter and a Schwarzschild vacuum in the exterior. For an outside observer, there would therefore be no distinction between a true Schwarzschild mass and a gravastar of this kind.
1. The GLV method
We shall only cite their results that will be used here and take the liberty to restate the new concepts they have developed. For details of their ingenious arguments and calculations, the reader is asked to read the original paper [9]. They take the spacetimes on two sides of the thin shell as
ds2 = - e2*±
1
b±(r±) r±
dt\ +
1
b±(r±) r±
dr\ + r\dQi.
The method allows any two arbitrary spherically symmetric spacetimes to be glued together by cut and paste procedure. Thus, for the static and spherically symmetric spacetime, the single manifold M is obtained by gluing two bulk spherically symmetric spacetimes M+ and M_ at a timelike junction surface £, i.e., at f (r, r) = r — a(r) = 0. The surface stress-energy tensor may be written in terms of the surface energy density a and the surface pressure P as Sj = diag(—a, P,P). GLV work out the general conservation law
d(aA) _dA
dr
+ dr
iAa,
where a = da/dr, the shell surface area A = 4na2 and there is an entirely new term
1
4na
K(a)\l i — + a2 + (ah/1 — ^^ + a2
b-(a)
The first term in Equation (2) represents the variation of the internal energy of the shell, the second term is the work done by the internal force of the shell. The right hand side is the net discontinuity in the conservation law of the surface stresses of the bulk momentum flux and is physically interpreted as the work done by external forces on the thin shell. In short it is the "external force" term occurring due to = 0. This is a new concept not noted so far in the literature. Only when $'±(a) = 0, we have S = 0, and then one recovers the familiar conservation law on the shell.
Assuming integrability of a [9], which allows a = a(a), it is possible to define the mass of the thin shell of exotic matter residing on wormhole throat as ms(a) = 4na(a)a2.
GLV derived the workable master inequalities about stability around a given radius a0 after long calculations, which are the constraint from the "mass"
m"(ao) > if
[Map) - aob/+(ao)]2 + [Map) - ^-(ap)]2
L [1 - 6+(ao)/ao]3/2 + [1 - b_(ao)/ap]3/2 J
+
1
+ 2
b+ (ao)
+
b'L (ao)
A/1 - b+(ao)/ao A/1 - b_(ao)/ao
and when $'±(a0) < 0, the constraint from the «external force»
[4nS(a)a]" |O0 > - b+(a)/a + (a)^1 - b_(a)/a
«0
1
4
$+ (a)
(b+(a)/a)/ A/1 - 6+(a)/a
+ $// (a)
(b_(a)/a)/ a/1 - b_(a)/a
«0
*+M, [<b+Ca?<a.>/]3/2 + (a) l(6_(a)/a)/'2
[1 - b+(a)/a]3/2
$+(a)
(b+(a)/a)// - b+(a)/a
+ $/_ (a)
[1 - b_(a)/a]3/2 (b_(a)/a)//
a/1 - b_(a)/a
«0
«0
(4)
Similar, but not the same, force constraint appears for $'±(a0) > 0 as well. We shall not quote it here as our example has $'±(a0) < 0. Such a force constraint however disappears, if $'±(a0) = 0.
Inequalities (3) and (4) are the ones that will be used in the case of the Lobo phantom wormhole.
2. Lobo phantom wormhole
We shall consider a spherically simmetric and static wormhole
ds2 = -
1 -I ^
ro
a—1
1 + g^ 1 — a
dt2 +
dr2
1 - ( ^
r 0
a3T + r2 (d02 + sin2 0d02),
where 0 < a < 1 is a constant and ro is the minimum radius.
The throat appears at rth = ro and wro, but we ignore the second radius because it is negative. Thus the wormhole spacetime is defined for ro < r < x>. The shape function and the redshift function respectively are
b_ = ro ( —
Jo
1 /1 + aw
2 1 1 - a
ln
1-
1a
(6)
Using the Einstein field equation, G= 8nTMV, the (orthonormal) stress-energy tensor components in the bulk are given by
P
a\ —
r0
a1
8nr2
Pr
aw —
r0
a1
8nr2
a
r
Pt
a (ro)2(a - 1)u + (au2 - (a - 3)u + 1) (£) ^
1—a
32nr2 ( ( ^ ) - 1
Here p is the energy density, pr is the radial pressure, and pt is the lateral pressure measured in the orthogonal direction to the radial direction. We thus obtain the NEC violating condition
a-1
a(1 + u) I £
K1 + u)( r0)C
p + pr=-snT2- Vr'
The total gravitational energy EG for wormholes is
1 fr
Eg = - (1 - y/grr)pr2dr-2 Jro
From Equation (5) we have
1
a-1
1 - (
ro
Signature protection in the metric (5) requires that 1 — /gTr which together with the information that p > 0 should yield a negative value of the integral in (7).
We glue the horizonless Lobo phantom wormhole with Schwarzschild exterior at some given radius r = a0 > 2M, r0 (throat radius of phantom wormhole), that is, we just join the two spacetimes, Lobo phantom wormhole and Schwarzschild, at a radius above the Schwarzschild horizon, i.e., at r > rh0r = 2M. The interior regions r < 2M, ro are surgically excised out from respective spacetimes because we don't want the presence of any horizon in the resultant wormhole. This ao is the radius about which linear spherical perturbations are assumed to take place. Casting the Schwarzschild metric in the GLV form (1), we get b+ = 2M, = 0 and similarly casting phantom wormhole metric (5) in the form of GLV metric (1), we get Equations (6).
Since 0 < a < 1, we have $'± < 0 for a0 > r0, and there will occur the effect of «external force» influencing the thin shell motion. Putting the above functions in the inequalities (3), (4), and defining x = M/a0, y = r0/M, we find, respectively
,, 2 », W „ 1 2x — 3x2 , (a — 2)(a — 1)(xy)1-a , (a — 1)2x(xy)2<'-a>
4a°m"(a0) - f(x'y) = (1——WT* + 2^1 — (xy)-a ' + (1 — (xy)1-a)3/2 •
8a3 [4n5(a)af |ao - g(x,y),
where
. . a [1+ u(39 — 28a + 5a2)(xy)2<1-a> — 2(1 + u)(a — 2)(5a — 21)(xy)a-1l
g(x,y) =--
8(1 - (xy)1—a)3/2(1 - (xy)a—1)3
a [1 + u(39 - 28a + 5a2)(xy)2(1—a) - 2(1 + u)(a - 2)(5a - 21)(xy)a—1] 8(1 - (xy)1—a)3/2(1 - (xy)a—1 )3
a [-111 + 72a - 9a2 + 2(1 + u)(33 - 19a + 2a2)(xy)1—a] + 8(1 - (xy)1—a)3/2(1 - (xy)a—1)3 '
g
Fig. 1. Stability zone for the Schwarzschild — Lobo phantom wormhole thin shell. The ranges are defined by x = M/a0, y = r0/M, where x e [0,0.3], y e [2,5] and ce = 1/10, cj = -10
Fig. 2. Stability zone for the Schwarzschild — Lobo phantom wormhole thin shell. The ranges are defined by x = M/a0, y = r0/M, where x e [0, 0.3], y e [2, 5] and a = 1/2, u = -2
Fig. 3. Stability zone for the Schwarzschild — Lobo phantom wormhole thin shell. The ranges are defined by x = M/ao, y = ro/M, where x e [0, 0.3], y e [2, 5] and a = 9/10, u = —10/9
3. Results and discussion
A possible candidate for the accelerated expansion of the Universe is speculated to be phantom energy, a cosmic fluid governed by an equation of state of the form w = pr/p < —1, which consequently violates the null energy condition. We have analyzed the physical properties and characteristics of thin-shell phantom wormholes gluing with exterior Schwarzschild spacetime. The reason for choosing Schwarzschild exterior is two fold: first, this exterior spacetime is the simplest and most well discussed, especially in the context of the existence of a black hole in the galactic center. Second, the speculation of phantom energy curling up by some high energy process into a Schwarzschild-like star is by itself of interest.
We have analyzed the stability regions of such configurations by including the effects of "external forces", in addition to that of the "mass" term. The "force constraint" is a momentum flux across the shell and is included here only for completeness. We enumerated the interior energy content by using the gravitational energy integral by Lynden — Bell, Katz and Bicak [10; 11]. It turns out that, even though the interior mass in positive, the integral is positive implying repulsive energy in the interior. This is consistent with the fact that the mass is phantom in nature. To make stability analysis physically meaningful, one should be able to delineate the possible parameter ranges for which stability can be achieved. The general and unified GLV stability analysis provides an excellent way to achieve this goal.
The stability condition can be expected as an explicit inequality involving the second derivative of the "mass" of the throat, m!£(a). In the absence of "external forces", this inequality is given by (3), and this is the only stability constraint one requires. However, once one has external forces (a0) = 0), there arises additional constraints in the
form of inequality (4), which is imposed from the external force over those from the linearized perturbations around static solutions of the Einstein field equations, as worked out above.
Given three graphs describe wormholes with phantom energy, e.g., for which equation of state is u < -1. We consider three following cases.
Case (1): The interior is a phantom energy wormhole with a = 1/10 and u = -10. In this case we see that thin shell resulted from gluing phantom wormhole and Schwarcshield spacetime is the most stable when x goes to 0 independently of y. We also see that, then x goes to 0, 5 the stability region goes to 0. However, with increasing y the decreasing of stability region has an exponential character.
Case (2): The interior is a phantom energy wormhole with a =1/2 and u = -2. The stability region has similar structure with the first case. However, the stability zone is bigger then the first case.
Case (3): The interior is a phantom energy wormhole with a = 9/10 and u = -10/9. In this case the stability region is the largest.
Note for all three considered cases the thin shell is the most stable then v goes to 0 and y goes to 2. Analysing the graphs we can conclude that stability zone for then shell increases with increasing a. This informative picture has been available thanks only to the effect of the newly developed "external force" constraint by GLV.
References
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2. Bronnikov K.A., Fabris J.C., Zhidenko A. On the stability of scalar-vacuum spacetimes. European Physics Journal C, 2011, vol. 71, p. 1791.
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4. Watanabe M., Kanno S., Soda J. Inflationary Universe with anisotropic hair. Physical Review Letters, 2009, vol. 102, p. 191302.
5. Lobo F.S.N., Parsaei F., Riazi N. New asymptotically flat phantom wormhole solutions. Physical Reviews D, 2013, vol. 87, p. 084030.
6. Riess A.G., et al. Observational evidence from supernovae for an accelerating Universe and a cosmological constant. Astrophysics Journal, 1998, vol. 116, p. 1009.
7. Perlmutter S. et al. Measurements of omega and lambda from 42 high-redshift supernovae. Astrophysics Journal, 1999, vol. 517, pp. 565.
8. Peebles P.J.E., Ratra B. The cosmological constant and dark energy. Reviews of Modern Physics, 2003, vol. 75, p. 559.
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10. Khaybullina A., Akhtaryanova G., Mingazova R., Saha D., Izmailov R.
Stability of the thin-shell Schwarzschild-Ellis wormhole. Physical Reviews D, 2016, vol. 93, p. 104044.
11. Lukmanova R., Khaibullina A., Izmailov R., Yanbekov A., Karimov R., Potapov A. Note on the Schwarzschild-phantom wormhole. Indian Journal of Physics, 2016, vol. 90, pp. 1319-1323.
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Accepted article received 23.04-2017 Corrections received 17.06.2017
Челябинский физико-математический журнал. 2017. Т. 2, вып. 2. С. 245-252. УДК 530.12
СТАБИЛЬНОСТЬ ТОНКОЙ ОБОЛОЧКИ ШВАРЦШИЛЬД-ФАНТОМНОЙ КРОТОВОЙ НОРЫ
Р. Х. Каримов", Р. Н. Измаилов6
Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия
"karimov_ramis_92@mail.ru; bizmailov.ramil@gmail.com
Предполагается, что возможной причиной, объясняющей ускоренное расширение Вселенной, является «фантомная энергия». Фантомная энергия обладает уравнением состояния вида ш = p/p < -1, следовательно, нарушает нулевое энергетическое условие. Поскольку это является фундаментальным условием существования проходимых кротовых нор, то этот вид материи представляет собой естественный сценарий существования подобных экзотических геометрий. Недавно Лобо было показано, что фантомная энергия с ш = pr/р < —1 может поддерживать фантомные кротовые норы. Им было получено несколько классов таких решений. В то время как внутреннее пространство-время представлено асимптотически плоской фантомной кротовой норой, которая обладает отталкивающей гравитацией, она, скорее всего, неустойчива к возмущениям. Рассмотрена ситуация, когда фантомная кротовая нора заключена внутри сферы, описываемой пространством-временем Шварцшильда, вдоль тонкой оболочки. Применяя метод, разработанный Гарсией, Лобо и Виссером (GLV), мы продемонстрировали, что оболочка может обладать зонами устойчивости в зависимости от определённых ограничений.
Ключевые слова: фантомные кротовые норы, техника тонкой оболочки, стабильность.
Поступила в редакцию 23.04.2017 После переработки 17.06.2017
Сведения об авторах
Каримов Рамис Хамитович, аспирант физико-математического факультета, Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия; e-mail: karimov_ramis_92@mail.ru.
Измаилов Рамиль Наильевич, кандидат физико-математических наук, доцент кафедры прикладной физики и нанотехнологий, Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия; e-mail: izmailov.ramil@gmail.com.
