CC BY
35
UDC 530.1; 539.1
Wormholes in 5D models
L. N. Lipatova1, N. R. Khusnutdinov2
Department of Physics, Kazan Federal University, Kazan, 420008, Russia.
E-mail: il.nAipaQgrnail.com, 2 7nail7@gmail.com
We consider our world as a brane embedded in the 5D space-time which is a solution of the 5D Einstein equations with A-term. We do not solve the modified Einstein equations on the brane, instead of this we use exact solution of 5D Einstein equations. The energy-momentum tensor appears as the Israel jump condition on the brane. This tensor at the infinity gives positive tension in one side of the wormhole and negative tension in the second part of the wormhole space-time. In the case of the AdS5 model without Randall-Sundrum asymptotic we found the wormhole metric which satisfies the energy conditions.
Keywords: wormhole, Randall-Sundrum model, multidimensional theory, energy conditions.
1 Introduction
The central problem of wormhole physics is the fact that the wormhole’s source violates the energy conditions. The exotic matter is a necessary condition for the existence of wormholes [1]. In multidimensional models the 4D space-time is understood as a surface embedded in space-time of large dimensions. The
4D
spacetime maybe embedded in the 5D anti de Sitter space-time (AdS5). As was shown in Ref. [4] the grav-
4D
different from Einstein’s equations. Additional terms contain terms quadratic in the stress-energy tensor of matter, as well as a specific term that depends on the
5D
Weyl tensor and the sign of this term maybe arbitrary. From this point of view, we can expect that wormholes can be the solution of the modified Einstein equations
5D
tensor. In this paper we use the exact solutions of 5dimensional Einstein equations and consider different hypersurfaces with wormhole’s metric. In framework of this approach the modified Einstein equations are satisfied automatically and the main problem is the choice of the hypersurface. We consider the simplest 5D
and black string model metric [5]. We also consider the general case of a static spherically symmetric metric. In all these cases the condition, that the wormhole’s hypersurface goes into Randall-Sundrum model’s hypersurface far from the wormhole’s throat, leads to a violation of the energy conditions. However, if we assume that this condition is violated, in the case of fivedimensional space-time of the Randall-Sundrum model the wormhole’s metric can be presented without violation the energy conditions.
2 Randall-Sundrum model
In framework of the Randall-Sundrum model [2,3] the matter is localized in 4D hypersurface in AdS5. The metric,
dsRS = e-2|y|/1 (dp2 + p2dQ2 2) — dt2) + dy2, (1)
is a solution of a 5D vacuum Einstein equations Gab = k^Tab = K5( A5qab + SabS(y)) (2)
(A5 = — J2) with stress-energy tensor of brane S^v = —Aguv (3)
and positive tension A: k5 A = +6/1 > 0.
3 Effective 4D Einstein equations
Following the approach suggested in the work [4] the effective 4D gravitational field equations in the vacuum Randall-Sundrum brane read
GHv = —A4qnv + — , (4)
where
= — 4TM«Tv“+112+8qv— 24%vт‘2, (5)
tmv is the stress-energy tensor of brane, EMV is the projection of the 5D Weyl tensor on the brane and A4 = 2 k5 (A5 + 6 k5A2) , K2 = 6 k5A are constants.
The sign of projection of the Weyl tensor projection maybe arbitrary and we can assume that wormhole is the solution of modified Einstein equations without violation the energy conditions.
-3 -2 -1 \ V 2 3
-10
-20
-30
-40
Figure 1: The stress-energy tensor components SX, S$, St (thick, medium, thin) for embedding functions of the form u = Vx2 + a2, v = 2 (1 + tanh2 ) (l = 4, a = 0.5, b =
0.6, c = 0.8) as function of the radial coordinate x.
Figure 2: The stress-energy tensor components SX, S$, Sf (thick, medium, thin) for embedding functions of the form
u = Vl-X/fe, v = 1+e-x/b (1 = 10, a = 1,b = 0.51) as
x
4 Stress-energy tensor of the brane
The brane is delta-like distribution of matter and tension in the bulk. The Israel matching conditions
[KMV - gMVK] = 8nG5Smv ,
(6)
connect the jump of the extrinsic curvature of the hypersurface with stress-energy tensor of the brane matter: SMV = -\gMV + tMV, where tMV is the stress-energy tensor. Thus the geometry of the brane produces the stress-energy tensor of the matter.
We consider a metric in the form of RandallSundrum solution:
/2
ds5 = (dp2 + p2d^2) - dt2 + dz2),
(7)
where z > 0, p > 0, and choose the simplest section z = v(x), p = u(x) with x ^ 0. In this case the metric of the section is
/2 r 'i
ds2 = |(u/2 + v/2)dx2 + u2d^2) - dt2j . (8)
Stress-energy tensor of the brane reads
SM = -2(K£ - 5MK),
or in manifest form
Sx 2 3uu' + 2vv'
= 2 luVu'2 + v'2 ,
(9)
^0 rfvv' (u,2 + v,2)+u(3ui (u'2 + v'2)+v(u' v''-v'u"))
lu(u'2 +v'2)3/2 ,
t rj 2vv' (u'2 +v'2 )+u(3u' (u/2 +v'2 )+v(u'v''-v' u''))
_ lu(u'2 +v'2 )3/2
St = -2
We impose the asymptotic conditions:
1) RS brane at infinity: limx^±TO v(x) = c± =0,
2) Flatness at infinity: limx^±TO sgn(u/) = ±1. Then we have the asymptotic value of the stress-
energy tensor
Su = -sgn(u/).
(11)
The signs of components of the stress-energy tensor are different on each side of the wormhole’s throat in the asymptotic region (Fig. 1). Therefore, if on one side of the wormhole space-time the energy conditions are satisfied then on other side they are violated.
Similarly, we consider section r = u(x), z = v(x) of the Black string space-time [5]
/2
ds2 = (-U (r)dt2 + U (r)
z2
-1
dr2 + r2d^22) + dz2), (12)
(10)
where U (r) = 1 - 2^ ? and spherically symmetric metric of the general form
/2
ds2 = — (ef dr2 + r2ehd^2) - epdt2 + eqdz2), (13)
where f = f (r,z),h = h(r,z),p = p(r,z),q = q(r,z). In both cases requirements in the asymptotic region limx^±TO v(x) = c± = 0, limx^±TO sgn(u/) = ±1 lead to violation of energy conditions, as in the case of the RS metric, limx^±TO SM = Tf 5M-
In the case of the RS model, the components of the stress-energy tensor of the brane can have the same signs when x ^ and x ^ -to, if we allow the violation of condition limx^±TO v(x) = c± =0. For example this is the case (see Fig. 2) for
u=
Vx2 + a2
1 + e-x/b ,
/
v=
1 + e-x/b
(14)
Simple analysis of the geodesics gives some general conclusion. Without loss of generality we choose the plain of motion 9 = n/2. When 0 = const the first integral of geodesic equations is t = c4 . For a massless
particle from geodesic equations we obtain the square
Figure 3: The trajectory of the massive particle in the plain 6 = n/2. Thick line shows the trajectory in region, where x(s) > 0, thin line shows the trajectory in re gion, where x(s) < 0, point indicates the location oft he throat, xo is the initial location of the particle.
of velocity
x2 =
.4 „2„,2
l4u2 (u'2 + v'2 ) ’
1
V • = ( I) =
For a massive particle these equations read
u'2 + v
2
x2 =
-v2 (-v2 c2 + 1)
l2 (u'2 + v'2) 2
(15)
(16)
(17)
(18)
V2 =( dx\ = -l2 ( — JT 4 +1)
\ dt ) c4v2 (u'2 + v'2 ) ’
Because the square of the velocity should be positive we obtain the condition
x > —bln(|c41 — 1).
(19)
This means, that the particle can not penetrate into specific region of space-time, corresponding to negative values of the x coordinate. When 0 = const, the trajectory of motion for massive particle is represented on Fig. 3.
5 Conclusion
In this paper we studied the section of the 5D Einstein spacetime with the geometry of a 4D wormhole. It is shown, that the considered sections can not simultaneously satisfy the energy conditions and at the same time coincide with the brane metric in the Randall-Sundmm model in the asymptotic region. We have presented the space-time of the wormhole which corresponds to a brane embedded in the Randall-Sundmm space-time. In this case the matter of the wormhole preserves the energy conditions blit the metric asymptotically does not coincide with the brane metric in the Randall-Sundmm model.
Acknowledgements
This work was supported by the Russian Foundation for Basic Research grant no. 11-02-01162-a.
References
[1] M. Visser, M. Lorentzian Wormholes: from Einstein to Hawking, AIP, (1995).
[2] L. Randall, R. Sundrum, Phys. Rev. Lett., 83, 3370 (1999).
[3] L. Randall, R. Sundrum, Phys. Rev. Lett., 83, 4690 (1999).
[4] T. Shiromizu, Kei-ichi Maeda, M. Sasaki, Phys. Rev. D62, 024012 (2000).
[5] A. Chamblin, S. W. Hawking, H. S. Rea 11, Phys. Rev. D61, 065007 (2000).
[6] W. Israel, Nuovo Cim. 44, 1 (1966).
4
Received 01.10.2012
Л. H. Липатова, H. P. Хуснутдинов КРОТОВЫЕ ПОРЫ В 5D МОДЕЛЯХ
Мы рассматриваем нашу Вселенную как брану, погруженную в пятимерное пространство-время, которое является решением 5D уравнений Эйнштейна с Л членом. Мы не решаем модифицированных уравнений Эйнштейна на бране, вместо этого мы используем известные решения 5D уравнений Эйнштейна. Тензор энергии-импульса вычисляем с помощью условий сшивки Израэля. Этот тензор на бесконечности дает положительное натяжение с одной стороны от горловины и отрицательное натяжение во второй части пространства-времени кротовой норы. В случае моделаА^£б (без асимптотики Рандалл-Сундрума) мы нашли метрику кротовой норы, которая удовлетворяет энергетическим условиям.
Ключевые слова: кротовые норы, многомерные теории, модель Рандалл-Сундрума, браны, энергетические условия.
Хуснутдинов Н.Р., доктор физико-математических наук, профессор.
Казанский государственный университет.
Ул. Кремлевская, 18, Казань, Россия, 420008.
E-mail: 7nail7@gmail.com
Липатова Л. Н., аспирант.
Казанский государственный университет.
Ул. Кремлевская, 18, Казань, Россия, 420008.
E-mail: l.n.lipa@gmail.com
