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Вестник Челябинского государственного университета. 2009. № 8 (146). Физика. Вып. 4. С. 54-61.
АСТРОФИЗИКА
1.1. Nigmatzyanov, N. G. Migranov, K. K. Nandi
ENERGETICS IN MAzuR-MoTTOLA GRAviSTAR AND TRuNCATED WoRMHoLES
It is known that the total gravitational energy in localized sources having static spherical symmetry and satisfying energy conditions, is negative (attractive gravity). A natural query is how the gravitational energy behaves under circumstances where energy conditions are violated. To answer this, the known expression for the gravitational energy is suitably modified to account for situations like the ones occurring in wormhole space-time. It is then exemplified that in many cases the modified expression yields desirable answers. The implications are discussed.
Key words: gravitational energy, wormhole, Ellis wormhole, Mazur-Mottola star, Lobo phantom wormhole, Lemos-Lobo-Oliveira wormhole, attractive gravity, repulsive gravity.
i. introduction. Classical wormholes, just as black holes, represent self consistent solutions of Einstein's theory of general relativity. Topologically they are like handles connecting two distant regions of space-time. Wormhole solutions were conceived as particle models by Einstein himself (Einstein-Rosen bridge [1]) in 1935. (A 1916 predecessor of wormholes is Flamm [2] paraboloid). The seminal theoretical framework laid in 1988 by Morris, Thorne and Yurtsever [3; 4] has since led to serious investigations into the topic of wormhole physics. In addition to the traditional method of solving Einstein's equations, there exists what we call Morris — Thorne — Yurtsever reverse method in which one first fixes the space-time geometry and then computes, via field equations, the stress components needed to support such geometry. The resulting stress components automatically satisfy local conservation laws in virtue of Bianchi identities. Either method has led to several wormhole solutions in well known theories such as in Brans-Dicke theory [5-8], scalar field theory with potential [9], low energy string theory [10-13], braneworld model [15-17], phantom model [18-20], Chaplygin gas model [21], Thin-Shell model [22-24] and in cosmology [25-27]. Configurations resulting from these theories could be potential candidates to occur in a natural way and are of great astrophysical interest [28-32]. A largely unnoticed but important work was carried out in 1948 by Fisher [33] who discovered formal solutions to minimally coupled scalar field Einstein equations with a positive sign kinetic term. Thereafter, in 1973, Ellis [34] and independently, Bronnikov [35] found wormhole solutions of the Einstein minimally coupled theory with a negative sign kinetic term. All wormhole solutions require exotic material for their construction. However,
to our knowledge the gravitational energy content in the interior of exotic matter distribution has not yet been studied. An initiative along this direction can be taken by employing the Lynden-Bell — Katz — Bicak (LKB) formulation [36] of gravitational energy. A conformal factor interpretation of gravitational energy density, which is new, is also given in [36].
The energy formulation by Katz [37], or that by Lynden-Bell, Katz and Bicak, is supposedly intended for isolating and calculating the attractive gravitational energy Eglkb in perfect fluid (ordinary star). In our view, their formulation did not require any compelling restriction on the energy conditions of the source matter. In fact, the equation for Lorentz boosted energy density is valid equally well for exotic matter [3; 4] defined by p + Pr - 0 where p is the matter energy density and pr is the radial pressure. (Transverse pressures p± are not considered as they refer strictly to ordinary matter.) This is the definition for the violation of Null Energy Condition which is a minimal requirement to have defocusing of light trajectories (repulsive gravity) passing across the wormhole throat. The necessity of Null Energy Condition violation in wormholes is provided by the Topological Censorship Theorem [38] or by dynamical circumstances [39].
In this paper, we first modify EGfB and denote the modified version by EG. Then we investigate the behavior of Eg in certain static spherically symmetric model solutions that violate the Null Energy Condition either marginally or fully. The examples we consider, one star and three wormholes, are somewhat critical in nature. Explicit calculations in the Ellis III wormhole show that the definition of EG is robust enough. In the case of de Sitter star, one has to calculate Eglkb which yields the right quantity of gravitational
energy, though not the right sign. In the case of phantom wormhole, we find that EG is repulsive irrespective of the velocity of the observer. Such a behavior of Eg may serve as a constraint with regard to the practical feasibility of localized wormholes.
The paper is organized as follows: In Sec.II, we modify the equation for gravitational energy so as to account for the wormhole geometry. In Sec. III, we apply the energy budget to the Mazur-Mottola star [40] and show that it quantitatively yields the special relativistic mass-energy relation for un-binding gravity. In Sec.IV, we calculate contribution to EG coming from the thin shell. In Sections V and VI. we investigate wormholes which are localized by space-time cut-off at any radii. Sec.VII summarizes the results. Units are chosen so that G = c = 1, unless specifically restored.
II. Gravitational energy. We shall consider spherically symmetric static space-time with the metric expressed in “standard” coordinates as
ds2 = -e2° dt2 + grr (r )dr2 + +r 2(d 02 + sin2 0d y2).
(1)
Suppose that the fluid is moving radially (v ^ 0) orthogonal to a spacelike hypersurface t = const. Then a static observer at (r, 0, y) constant sees the energy density T0 as
'T'0 ____
T0 _
P + Pr 1 - v2
Pr ,
(2)
in which 0 < v < 1, p is the matter energy density and pr is the radial pressure of the fluid in its rest frame. The gravitational energy EG appropriate for stars is given by Lynden-Bell, Katz and Bicak as [36]
EGK = Mc2 - EM =
= 21 [1 - (gj FoVdr, (3)
2 0
where the total mass-energy within the standard coordinate radius r is provided by Einstein's equations as
Mc2 = - T0r2 dr
(4)
and the sum of other forms of energy like rest energy, kinetic energy, internal energy etc is defined by
1 r
Em = 2 f T0( g„)* r 2 dr.
The factor 1 comes from -jjn The EM is similar to the geometric definition given by Wald [41].
The Misner-Thorne-Wheeler formulation is the special case of Eglkb when v = 0 [42]. Since (grr)1 >1 by definition (proper radial length larger than the Euclidean length), one immediately deduces the criteria that EG < 0 (attractive) if T00 > 0 and that Eg > 0 (repulsive) if T00 < 0.
For wormhole space-time, consider the spherically symmetric space-time metric in the generic Morris -Thorne — Yurtsever form
ds2 = -e2 °(r} dt2 + ■
dr2
1 -
b (r) r
+r (d0 + sin 0dy ),
(6)
where O(r) and b(r) are redshift and shape functions respectively. The wormhole geometry has no center, but has instead a hole and so we shall change the lower limit of integration in eq. (3) to the minimum allowed radius or throat r0 defined by b(r0) = r0. The radius r has the significance that it is the embedding space radial coordinate; it decreases from +<x> to r = r0 in the lower side and again increases to +<x in the upper side. This requires us to change the integrals (4) and (5) to
Mc2 = T00 r2 dr + r°;
9 J 0 9
E = — Em 2
1 r
2 f T00( grr ) ^ r 2 dr,
(7)
(8)
where grr =(l - ^) , the entire space-time geometry being assumed to be free of singularities. The
in eq. (7), coming from the integration
dM 1 T 0 2
constant 2 .
of Einstein's equation = 2 T^ rz, is fixed in the
rest frame of the fluid (v = 0) in such a way as to recover the known values of M in a given wormhole configuration. When T00 = 0, we should fix r0 = 0 in order to recover M = 0. In geometries with a regular center, one has r0 = 0, which then reproduces Eqs. (4) and (5) respectively. The difference between the above integrals, viz.,
Eg = Mc - EM =
= 2 2 - (grr) *] T00 r2 dr
(9)
is what we call the modified EG appropriate for wormholes. One immediately notices that due to the presence of the positive nonzero last term, the sign (5) of T00 does not necessarily determine the sign of EG,
as would be the case otherwise.
Lynden-Bell, Katz and Bicak show that the gravitational energy density can be written in remarkable
analogy with electrical energy density of Maxwell electrodynamics: The total gravitational energy EG can be written as an integral of a perfect square F2. In case of wormhole space-time, we should modify the relevant expression for EG to
EG =-f ^ 1 -g )"’
G I 2
r
'0
dV +
+= -f F 2dV + r0, 9 J 9
(10)
where
F = ±1 r
= ±(g" )
1 -(g" )
-1 d¥
dr
,dV = (grr)2r drd0dy. (11)
The ± sign corresponds respectively to repulsive and attractive nature of gravitational potential ¥ and the choice is generally open unless we are dealing with perfect fluid for which only the -ve sign is provided by independent physical observations. Lynden-Bell, Katz and Bicak have shown that the function ¥ can be interpreted as the conformal factor of the transformation that makes the spatial metric (1) flat.
A typical wormhole solution may be derived from sources (T^) that have an overall wrong sign (negative) so that all energy conditions are violated. A well known example is the Ellis-Bronnikov wormhole. Then the Lynden-Bell-Katz-Bicak equation for ¥ [their eq. (9)] should be rephrased as
1
1
V 2¥ =— T00 +-(V¥ )2,
2
2
(12)
which shows that a positive gravitational energy density is acting alongside negative exotic matter density (- -2 T00) as a source of ¥, now a repulsive potential.
iii. Energetics in the Mazur-Mottola star.
Consider the static spherically symmetric vacuum condensate star devised by Mazur and Mottola [40]. The star has an isotropic de-Sitter vacuum (p + pr= 0) in the interior, the matter marginally violating the Null Energy Condition and strictly violating the Strong Energy Condition p + pr> °. From eq. (2) we have T00 = p which is independent of v, implying that both static and moving observers see the same density p. The boundary r = r0 of the star is matched to the Schwarzschild exterior (p = 0, p = 0) of mass M across a thin shell of stiff matter (p = +p). We thus
have a spherically localized source with the spacetime having a nonsingular center and asymptotic flatness. Hence we should use EJlkb .
The self-consistent interior de Sitter metric for a constant density vacuum p = pvac = const > 0 is given by
d t2 =-
,„2 A
1 -
v
,,2 V1
dt +
1 -
R2
dr2 + r2 (d02 + sin2 0dy2), (13)
I
where R2 = -
3
. The transverse pressures in the
8nPv
thin shell serve to act more like a Roman arch supporting the star than making any substantial contribution to mass-energy. The contribution has been shown [40] to be negligible, since Mshell ~ sM where 0 < s << 1. Israel-Darmois junction conditions then imply a negative surface tension at the inner interface of the shell which balances the outward force exerted by the repulsive vacuum within. Likewise, the positive surface tension at the outer interface balances the inward force from without. Using the thin shell approach, Visser and Wiltshire [42] studied dynamic stability of similar type of configurations. The mass-energy contained within the radius r = rb is given by
, 4n 3
M =— 'bPv
> 0.
(14)
The physical volume element of the spacelike hypersurface orthogonal to the time-like Killing vector field is dt
(
dV =
1 -— v R2 y
r2dr sin 0d0dy. (15)
Thus dV is larger than 4nr2dr by a factor
. The total physical volume is
'b
V=fdV
4n 3
= — r:
and hence the average density <p> is given by
/ . M (p) = y = pv
(16)
(17)
Clearly, if pvac were constant physical density, it would have equaled <p> which is not the case. Hence we get a mass defect AM given by
AM - Vpa -M = Vpv
which, to leading order, yields 16 15
AM =^2 rfpL >0.
(18)
(19)
Following eq. (5), we can identify Vpvac = EM, so that we have from eq. (3)
ELKB = Mc2 - EM = -c2AM < 0. (20)
Since here Tg° = pvac>0, the original Lynden-Bell-Katz-Bicak equation [ - V 2¥ = 1T00 - ^(V¥)2 ] indicates attractive potential ¥ and the result EGLKB < 0 implies that gravitational energy in the de Sitter star is attractive. These would be wrong conclusions. The independent physical information is that the de Sitter space has repulsive gravity. As argued after eq. (11), the Lynden-Bell-Katz-Bicak formulation does allow repulsive potential ¥ which, in turn, allows us to quantitatively estimate the gravitational energy content, as can be seen from the following argument.
Due to repulsion, a vacuum fluid element has to exert an inward directed force to stay at any fixed r. This fact can be interpreted in a Newtonian way as follows. Imagine an interior concentric sphere of radius r which has on its surface a repulsive Newtonian potential
¥(r) = +
4n P vacr3 4n
3
3
P r .
vac
(21)
Increasing the radius by dr, an amount of exotic matter 4npvacr2dr is moved away from the center by the force of repulsion. The work done by vacuum results in a loss of energy (as opposed to gain in gravitational energy in the case of attractive gravity) of amount
dE (r) = which integrates to
rb
E = f dE
16n pvac r4dr,
_16 2 5 2
= 15 ^ 'b p-.
(22)
(23)
The similarity between eqs. (19) and (20) prompts us to identify E with
EGkb = c2 AM. (24)
This is a curious result but not altogether new, only the circumstances are different. Note that, for a ‘constant’ density ordinary
star described by Schwarzschild interior metric, Adler, Bazin and Schiffer [43] calculated that E = -c2AM, their equation being a special case (p = const, v = 0) of the Lynden-Bell-Katz-Bicak eq. (3). In the next sections, we shall consider truncated wormholes.
iv. Thin shell contribution. Several asymptotically flat wormholes are known in the literature with matter threading the wormhole all the way to infinity with radial fall-offs in the stress quantities. Such wormholes might be existing in nature as an end result of some past astrophysical phenomena. To apply the energy budget, however, we should look for sources in the form of localized exotic matter just like the de Sitter star. Suitable candidates are truncated wormholes. For completeness, we shall calculate the thin shell contribution to EG although, practically, the contribution is negligible.
The idea of a truncated wormhole is the following. One wants to artificially create a wormhole by localizing the exotic matter within a finite radius around the throat r = r0 of a given solution. This can be achieved by taking a cut-off at any finite radius away from r = r0, say, at r = a > r0 and matching the surface at r = a to an exterior Schwarzschild vacuum. The matching brings into play junction conditions as follows: The induced metric on the spacelike junction interface E is given by
ds2 = -dt2 + a2 (d02 + sin2 0dy2); (25)
where t is the proper time on the surface. On this surface the matter energy density c and transverse pressures are calculated from the jump in the extrinsic curvature [Kp]- = Kj - Kj as r ^ a±. The result is [44; 45]
a = —
1
4n a
P = P =----------
0 ^ o
8n a
b(a)
(26)
(27)
where Z = 1 + a^r | . When c = 0, P0 =P = 0, the
^ dr lr =a 0 y 7
surface r=a is called the boundary. However, a more interesting possibility is to consider an arbitrarily thin shell of quasi-normal matter (that is, matter satisfying both Week Energy Condition and Null Energy Condition) at r = a. Then the total mass-energy is given by [18]
M = b(a) + M„
1 -
b(a) Msl
2a
, (28)
r
3
2
a
where MsheI1 = 4na2c is the shell mass. If b(a) = = 2M, then c = 0. To have an idea of how c ^ 0 contributes to the gravitational energy, we should fix the shape function b(a) to a value slightly away from 2M. For instance, we can fix b(a) = 2M - sM where 0 < s << 1 is a parameter related to the infinitesimally thin thickness of the shell. In this case we get, to leading order in s,
eM 2 .
M
(29)
Up to a factor (-2 ), this is exactly the same result as that obtained in ref. [40]. To get an idea of the measure of EM in the shell, we can regard the density to be approximately the constant c throughout the shell in which the space-time can be approximately described by a Schwarzschild metric for mass M. Then
77 shell _____
1 -
2M
"2dr :
e M
-[3a2 + aM ] = O(e2). (30)
2 4n a
The total gravitational energy of the truncated wormhole therefore becomes
Eg = M - Em =
1 f [1 - (gr) ’] T0' '-dr
+ -° +
+M
shell
1 -
b(a) M,
shell
a
2a
- E
shell ‘M ,
(31)
where T00is given by eq. (2). The term M2shell as well may be neglected as being of order s2. To
no 17 shell as EM
calculate MshelI for a given shape function b(r), we express it in terms of b(a) as follows
M shell = 2
b(a) 2 - e
eb(a) 4 :
(32)
r0 e b(a)
1 -
b(a)
(33)
Let us apply this formula to analyze a couple of known truncated solutions.
v. Lobo phantom wormhole. The metric is given
by
ds2 =-
dr2
+------------:—
(\1-a
r)
I r0 ) 1-a
1 - 0
( r I
1+aro
1-a
dt +
+ r
(d02 + sin2 0dy2), (34)
where a and ra are constant parameters and 0 < a < 1. The phantom equation of state further demands that p/p = ra < -1. The shape function and the redshift function respectively are
b(r) = ra rJ-“,
O(r) = 1 | ln
1 1 - a
1 -I "0
(35)
The density and radial pressure for this wormhole
are
P r3 ( r
Pr =■
aror„
so that, from eq. (2), we have
r0a (1 + o>v2) |'
rrt0 ___
= '
1 - v2
(36)
(37)
to first order in s. So the only contribution MshJ,l 1 - ] to mass-energy coming from the
thin shell reduces to -b-!. [^ 1 -] which is always positive. This may be added to the right hand side of eq. (9).
The thin shell contribution to energy is more of academic interest than anything substantial because of the limit s ^ 0. See for instance [45]. Nonetheless, in all, we can write
Eg = M - EM =
To have the space-time free of singularities, we must impose a constraint 1 + ara = 0. We thus obtain the Null Energy Condition violating:
ar
p + Pr =-tH - | (----)<0
(38)
a
satisfied for all r.
As discussed by Lobo [18], this wormhole can be truncated at some finite radius at r = a away from the throat r = r0 to match to an exterior Schwarzschild space-time. Neglecting the thin shell contribution O(s), we can numerically integrate to get EG using eq. (9). Using the metric (34), and taking for example, a = 1, so that ra = -3, we obtain
a
1
3 •
r
0 _ (1 - 3v2) 11
10 ~
1
(39)
3(1 - v ) ( r.
Its sign depends on v. Even though p > 0, some observers might see T00 < 0. This is recognized in [3; 4] as a deeply troublesome aspect of Null Energy Condition violation. In the present case, p > 0 but if -jj < v < 1, then T00 > 0 and if 0 < v < , then
T00 < 0. For a quantitative estimate, we shall explicitly take a finite cut-off, say at a = 9r0 and choose mass units in which r0 = 1. The interesting result is that the velocity dependence of T00 does not alter the sign of Eg which is always positive:
1
1 11 - 3v2) 9 , / - 2 \1
r If 1 - (1 - r 3 )
2 I 1 -v2 Jf V /
11 - 3v2) ( 1 - v' y
P r dr +----+ O(e) =
x (-0.74) + - + O(e ) > 0. (40)
That is, the expression for EG consistently shows repulsion to observers irrespective of his/her state of motion.
VI. Lemos-Lobo-Oliveira wormhole (LLO).
The metric inside r0 < r < a is given by [46]
ds2 =-
1 -
dt +
+ -
dr'
1 -(')
which gives
r + r2 (d02 + sin2 0dy2), (41)
b(r) = ^Fro,
O =1 ln
2
1 - r. 1i"
( a y
= const.
(42)
(43)
The exterior vacuum metric a < r < <x> is (r0a) 1
ds =
1-
dt +
+-
dr2
■ + r
( '0a )2
(d02 + sin2 0dy2). (44)
The energy density and radial pressure are
p = j. ±=JL > 0,
r dr 2r
(45)
Pr =
„5/2
< 0,
Rr. = -2. P
(46)
(47)
We have a phantom equation of state (ra = -2) here although the metric properties are different from the earlier example. Null Energy Condition is violated everywhere, including at the throat, since
nr
p + Pr = - --'5°- . The throat appears at r = r0, the space-time is perfectly regular there and we see from eq. (2) that
T o __
T0 =
11 - 2v2) 1 - v2
p > 0.
(48)
To get an estimate, we shall again take the cutoff at a = 9r0 and choose mass units in which r0 = 1. Using the metric (39), we can express EG of eq. (33) in dimensionless form as follows
eg = -G 2
1 - 2v 1 - v2
pr2dr +
1 e b(9)
2 4
1 -
b(9)
r1 - 2v2) v y
x (-1.59) +1 + 0.61e. (49)
2
When v is small, we see that E„ < 0, which conG
veys the impression that the space-time is attractive. To offset this attraction, one might add the thin shell contribution, but then the shell can not be as thin as possible. So this is out of question. Higher values of v however give Eg > 0.
The physical situation in any wormhole is that the cross-sectional area of a bundle of light rays entering one mouth must decrease and then increase while emerging at the other mouth. This can be produced only by the gravitational repulsion of matter. It was argued that, even though p > 0, some observers with v ^ 0 must see the energy density T00 < 0. In the case under discussion, observers with v > -rr do find
Eg > 0.
VII. Summary. The original derivation of the formula for EGLKB for a static spherically symmetric asymptotically flat space-time, as given in [42; 36; 37], does not require any restriction on the nature of the fluid whether it is perfect or exotic. On the other hand, Einstein field equations do admit exotic matter sources that automatically satisfy local conservation
G
r
laws. There is a statement in [37] to the effect that Egk < 0 for localized sources satisfying energy conditions. The statement is certainly true for perfect fluids. However, the converse question, namely, whether Eglkb > 0 in case of non-perfect fluids violating energy conditions such as in wormholes, remained essentially open.
To handle wormhole configurations which require repulsion, we straightforwardly modified EGLKB which correctly produced the gravitational energy picture in the Lobo phantom wormhole. We obtained that the LLO wormhole provides a velocity range for which there is the desired repulsion. The Mazur-Mottola star does not have wormhole topology but it still produced a quantitatively correct result.
What are the possible implications of these results? We recall that p > 0 wormholes are not ruled out [3; 4] but as mentioned before, some observers need to see defocusing of light rays, hence repulsion, at or in the vicinity of the throat. Looking at eq. (9) we realize that the integral can result in values having either signs depending on the wormhole model chosen. If it so happens that the integral is large and negative for all v overcoming the additive factor "2, then we end up with Eg < 0 or lack of repulsion everywhere. We might rule out such wormhole configurations as physically unrealistic or unrealizable, though they might be technically valid solutions. A future task is to extend the concept of gravitational energy to rotating systems. Rotation effects provide a more complex and enriching scenario in astrophysical circumstances [47].
References
1. Einstein, A. The Particle Problem in the General Theory of Relativity / A. Einstein, N. Rosen // Physical Rev. 1935. Vol. 48. P. 73-77.
2. Flamm, L. // Z. Physik. 1916. Vol. 17. P. 448.
3. Morris, M. S. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity / M. S. Morris, K. S. Thorne // American J. of Physics. 1988. Vol. 56. P. 395-412.
4. Morris, M.S. Wormholes, Time Machines, and the Weak Energy Condition / M. S. Morris, K. S. Thorne, U. Yurtsever // Physical Rev. Letters. 1988. Vol. 61. P. 1446-1449.
5. Agnese, A. G. Wormholes in the Brans-Dicke theory of gravitation / A. G. Agnese, M. La Camera // Physical Rev. D. 1995. Vol. 51. P. 2011-2013.
6. Nandi, K. K. Brans wormholes / K. K. Nandi, A. Islam, J. Evans // Physical Rev. D. 1997. Vol. 55. P. 2497-2500.
7. Anchordoqui, L. A. Brans-Dicke wormholes in nonvacuum spacetime / L. A. Anchordoqui, S. P. Berg-liaffa, D.F. Torres // Physical Review D. 1997. Vol. 55. P. 5226 -5229.
8. Nandi, K. K. Brans-Dicke wormholes in the Jordan and Einstein frames / K. K. Nandi, B. Bhat-tacharjee, S. M. K. Alam, J. Evans // Physical Rev. D. 1998. Vol. 57. P. 823-828.
9. Bronnikov, K. A. Regular Phantom Black Holes / K. A. Bronnikov, J. C. Fabris // Physical Rev. Letters. 2006. Vol. 96. P. 251101-251105.
10. Cadoni, M. Cosmological and wormhole solutions in low-energy effective string theory / M. Ca-doni, M. Cavaglia // Physical Rev. D. 1994. Vol. 50. P. 6435-6443.
11. Nandi, K. K. Stringy Wormholes / K. K. Nandi,
S. M. K. Alam // General Relativity and Gravitation. 1998. Vol. 30. P. 1331-1340.
12. Vollick, D. N. Wormholes in string theory / D. N. Vollick // Classical and quantum gravity. 1999. Vol. 16. P. 1599-1604.
13. Nandi, K. K. Traversable Lorentzian wormholes in the vacuum low energy effective string theory in Einstein and Jordan frames / K. K. Nandi, Y. Z. Zhang // Physical Rev. D. 2004. Vol. 70. P. 044040-044050.
14. K. A. Bronnikov // Gravitation and cosmology. 1998. Vol. 4. P. 49.
15. Barcelo, C. Brane surgery: energy conditions, traversable wormholes, and voids / C. Barcelo, M. Vis-ser // Nuclear Physics B. 2000. Vol. 584. P. 415-435.
16. Bronnikov, K. A. Possible wormholes in a brane world / K.A. Bronnikov, S.-W. Kim // Physical Rev. D. 2003. Vol. 67. P. 064027-064034.
17. Rodrigo, E. Higher-dimensional bulk wormholes and their manifestations in brane worlds // Physical Rev. D. 2006. Vol. 74. P. 104025-104036.
18. Lobo, F. S. N. Phantom energy traversable wormholes // Physical Rev. D. 2005. Vol. 71. P. 084011-084019.
19. Sushkov, S. V. Wormholes supported by a phantom energy // Physical Rev. D. 2005. Vol. 71. P. 043520-043525.
20. Zaslavskii, O. B. Exactly solvable model of a wormhole supported by phantom energy // Physical Rev. D. 2005. Vol. 72. P. 061303-061306.
21. Lobo, F. S. N. Chaplygin traversable wormholes // Physical Rev. D. 2006. Vol. 73. P. 064028-064037.
22. Rahaman, F. Thin shell wormholes in higher dimensional Einstein-Maxwell theory / F. Rahaman, M. Kalam, S. Chakraborty // General Relativity and Gravitation. 2006. Vol. 38. P. 1687-1695.
23. Richarte, M. Thin-shell wormholes supported by ordinary matter in Einstein-Gauss-Bonnet gravity / M. Richarte, C. Simeone // Physical Rev. D. 2007. Vol. 76. P. 087502-087506.
24. Eiroa, E. F. Stability of Chaplygin gas thin-shell wormholes / E. F. Eiroa, C. Simeone // Physical Rev. D. 2007. Vol. 76. P. 024021-024029.
25. Hochberg, D. Wormhole cosmology and the horizon problem / D. Hochberg, T. W. Kephart // Physical Rev. Letters. 1993. Vol. 70. P. 2665-2668.
26. Barcelo, C. Wormholes in spacetimes with cosmological horizons // International J. of Modern Physics D. 1999. Vol. 8. P. 325-335.
27. Sushkov, S. V. Scalar wormholes in a cosmological setting and their instability / S. V. Sush-kov, Y. Z. Zhang // Physical Rev. D. 2008. Vol. 77. P. 024042-024050.
28. Cramer, J. G. Natural wormholes as gravitational lenses / J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, G. A. Landis // Physical Rev. D. 1995. Vol. 51. P. 3117-3120.
29. Virbhadra, K. S. Schwarzschild black hole lensing / K. S. Virbhadra, G. F. R. Ellis // Physical Rev. D. 2002. Vol. 62. P. 084003-084011.
30. Bozza, V. Gravitational lensing in the strong field limit // Physical Rev. D. 2002. Vol. 66. P. 103001103011.
31. Kardashev, N. S. Astrophysics of wormholes / N. S. Kardashev, I. D. Novikov, A. A. Shatskiy // International J. of Modern Physics D. 2007. Vol. 16. P. 909-926.
32. Nandi, K. K. Gravitational lensing by wormholes / K. K. Nandi, Y. Z. Zhang, A. V. Zakharov // Physical Rev. D. 2006. Vol. 74. P. 024020-024033.
33. Fisher, I. Z. Scalar mesostatic field with regard for gravitational effects // Zh. Eksp. Teor. Fiz. 1948. Vol. 18. P. 636-640.
34. Ellis, H. G. Ether flow through a drainhole: A particle model in general relativity // J. of Mathematical Physics. 1973. Vol. 15. P. 104.
35. Bronnikov, K. A. // Acta Phys. Polon. B. 1973. Vol. 4. P. 251.
36. Lynden-Bell, D. Energy and angular momentum densities of stationary gravitational fields / D. Lynden-Bell, J. Katz, J. Bicak // Physical Rev. D. 2007. Vol. 75. P. 024040-024049.
37. Katz, J. Gravitational energy // Classical Quantum Gravity. 2005. Vol. 22. P. 5169-5179.
38. Friedman, J. L. Topological censorship / J. L. Friedman, K. Schleich, D. M. Witt // Physical Rev. Letters. 1993. Vol. 71. P. 1486-1489.
39. Hochberg, D. Null Energy Condition in Dynamic Wormholes / D. Hochberg, M. Visser // Physical Rev. Letters. 1998. Vol. 81. P. 746-749.
40. Mazur, P. O. Gravitational vacuum condensate stars / P. O. Mazur, E. Mottola // Proc. Nat. Acad. Sc. USA. 2004. Vol. 101. P. 9545-9550.
41. Wald, R. M. General Relativity. Chicago : University of Chicago Press, 1984.
42. Visser, M. Stable gravastars — an alternative to black holes? / M. Visser, D.L. Wiltshire // Classical Quantum Gravity. 2004. Vol. 21. P. 1135-1151.
43. Adler, R. Introduction to General Relativity / R. Adler, M. Bazin, M. Schiffer. N. Y. : McGraw Hill, 1965.
44. Visser, M. Lorentzian Wormholes-From Einstein to Hawking. N. Y. : AIP Press, 1995.
45. Visser, M. Traversable Wormholes with Arbitrarily Small Energy Condition Violations / M. Visser, S. Kar, N. Dadhich // Physical Rev. Letters. 2003. Vol. 90. P. 201102-201106.
46. Lemos, J. P. S. Morris-Thorne wormholes with a cosmological constant / J. P. S. Lemos, F. S. N. Lobo, S. Q. de Oliveira // Physical Rev. D. 2003. Vol. 68. P. 064004-064019.
47. Dudorov, A. E. Evolution of Collapse Nonuniformity for Rotating Magnetic Interstellar Clouds / A. E. Dudorov, A. G. Zhilkin, N. Y. Zhilkina // Astronomy Letters. 2006. Vol. 32. P. 622-632.
