v.m.khatsymovsky- rotating vacuum wormhole

8/3/2019 V.M.Khatsymovsky- Rotating vacuum wormhole

1/10

arXiv:gr-qc/9803027v16

Mar1998

Rotating vacuum wormhole

V.M.KhatsymovskyBudker Institute of Nuclear Physics

Novosibirsk, 630090, Russia

E-mail address: [email protected]

Abstract

We investigate whether self-maintained vacuum traversible wormhole can exist

described by stationary but nonstatic metric. We consider metric being the sumof static spherically symmetric one and a small nondiagonal component whichdescribes rotation sufficiently slow to be taken into account in the linear approx-imation. We study semiclassical Einstein equations for this metric with vacuumexpectation value of stress-energy of physical fields as the source. In suggestionthat the static traversible wormhole solution exists we reveal possible azimuthalangle dependence of angular velocity of the rotation (angular velocity of the localinertial frame) that solves semiclassical Einstein equations. We find that in themacroscopic (in the Plank scale) wormhole case a rotational solution exists butonly such that, first, angular velocity depends on radial coordinate only and, sec-ond, the wormhole connects the two asymptotically flat spacetimes rotating with

angular velocities different in asymptotic regions.

1
http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1http://arxiv.org/abs/gr-qc/9803027v1


2/10

1.Introduction. The possibility of existence of static spherically-symmetricaltraversible wormhole as topology-nontrivial solution to the Einstein equations has beenfirst studied by Morris and Thorne in 1988 [1]. Since that time much activity has beendeveloped in studying the wormhole subject (see, e.g., review by Visser [2]). Rather inter-

esting is the possibility of existence of self-consistent wormhole solutions to semiclassicalEinstein equations. Checking this possibility requires finding vacuum expectation valueof the stress-energy tensor as functional of geometry and solving the Einstein equationswith quantum backreaction, i.e. with such the induced stress-energy as a source.

Recently some arguments in favour of this possibility has been given. In Refs. [3, 4, 5]the gravity induced vacuum stress-energy tensor in the wormhole background has beenfound to violate energy conditions just as it is required for this tensor itself be thesource for such the wormhole metric [1, 6]. (We consider physical vacuum of spin 1 and1/2 massless fields in these papers). In Ref. [7] self-consistent spherically-symmetricalwormhole solution has been found numerically for the quantised scalar field vacuumplaying the role of a source for gravitation.

The problem of existence of self-consistent static wormhole is a particular case of themore fundamental problem of self-consistent solutions to semiclassical Einstein equationswith vacuum expectations of stress-energy tensor of physical fields as a source. In Ref.[8] it has been found that such the problem linearised over metric perturbations offMinkowski spacetime gives solutions (apart from some unphysical ones) coinciding withthose for classical gravity wave problem. If, however, topology is not Minkowski one asin the wormhole case at hand, new solutions can appear such as static wormhole itself.The natural next step may be the search for stationary but nonstatic topology nontrivialsolution, namely rotating vacuum wormhole. In the case of slow rotation one simply addssmall nondiagonal polar-angle-time component of metric to the self-maintained staticwormhole metric:

ds2 = exp (2)dt2 d2 r2[d2 + sin2 (d2 + 2hddt)] (1)where h(, ) has the sense of angular velocity of the local inertial frame and will becalled the angular velocity of rotation in what follows. The static metric (), r() isassumed to exist as a self-consistent solution of static semiclassical Einstein equations,and solution for h is to be found. One can take h arbitrarily small in order to limitoneself to the theory linearised in h. In practice, in order that the quadratic in hterms in the Riemann tensor could be disregarded, the space derivatives of h shouldbe negligible as compared to the derivatives of the static part of metric in the scale oftypical wormhole size. Note that in the linear approximation the only component of

Einstein equations being new compared to the static case is the t one, as it followsfrom symmetry considerations. The role of the source is played there by the inducedvacuum energy flow (angle-time stress-energy component).

In the given note we show that the rotational solution exists at least in the case whenthe coefficient at the Weyl term in the induced stress-energy is large and if one can neglectother terms. As noted in Refs. [4, 5], this corresponds to the wormhole of macroscopicsize (that is, large in Plank units). The rotational solution which we argue should exist

2


3/10

is such that h depends on radial distance only and has different asymptotic limitsin the two asymptotically flat spacetimes connected by wormhole. This, in particular,means that these two spacetimes cannot be glued together in asymptotic region, sothat this wormhole cannot be considered as that connecting the two regions ofthe same

asymptotically flat spacetime.As for the general case when we do not assume the Weyl term to dominate, wefind the two kinds of azimuthal angle dependence of angular velocity h for which theradial and angular variables , are separated and the Einstein equation with quantumbackreaction for h reduces to that for the function of purely . One possibility is theabove mentioned angle-independent angle velocity; another one is proportional to cos velocity.

2.Classical rotation. By classical we mean rotation considered without takinginto account corresponding quantum backreaction, i.e. induced vacuum energy flow.However, it is implied that the static wormhole problem is already solved (the static

metric is found) with taking into account corresponding backreaction. We show (in thelinearised in h theory used throughout the paper) that only solution for h not dependingon is physically acceptable such that it has different finite limits at + and at .

The t Einstein equation of interest can be conveniently written using the tetradcomponents introduced like the following basic 1-forms a = eadx

[9]:

0 = exp ()dt, 1 = (d + hdt)r sin , 2 = d, 3 = rd. (2)

Taking the expressions for the Riemann tensor presented in Ref. [9] we find for theequation of interest:

R10

= exp ()r sin

Rt

=

1

2r3

exp()r4h

sin

+1

2rexp() 1

sin2

sin3

h

= 0. (3)

Remind that we neglect the induced vacuum energy flow Tt in this section. Separatingthe variables, h = f()Y(), we find hypergeometric function for the angle dependence:

Y F(a, 3 a; 2; 1 cos 2

), (4)

a = const. This function diverges at = unless a = k, k = 0, 1, 2,...; in the lattercase it reduces to the Gegenbauer polynomial C(3/2)k (cos ). Then the radial part satisfiesthe equation

(r4 exp()f) = k(k + 3)r2 exp()f. (5)By asymptotical flatness 0, r at and thus f k3 at largedistances. Another solution, f k, should be discarded at k 1 as unphysical one.Therefore if we choose some large L > 0, we shall have solutions at || > L parametrised

3


4/10

by two constants C (f Ck3 at ), whereas in the intermediate region|| < L the Eq. (5) is regular in the wormhole geometry and has a solution parametrisedby two constants C1, C2. The four constants C1, C2, C+, C are subject to four uniformequations which are matching conditions for f and its derivative f at = +L and at

= L. Requiring for this system to have nonzero solution, i.e. zero determinant, weget some constraint on the already known metric functions r(), (). Therefore theset of possible solutions for f() at k 1 has zero measure as compared to the set ofpossible static solutions r, . At k = 0 the Eq. (5) can be easily integrated and leadsto physically admissible h not depending on and being monotonic function of whichhas finite but different limits at : h() = h(+). Because of the lattercircumstance the two asymptotically flat spacetimes connected by the wormhole channelcannot be glued together in asymptotic region (glueing at a shorter distance wouldspoil spherical symmetry of the background static solution described by two functions r,) so we cannot derive from such the rotating wormhole the wormhole connecting thetwo distinct regions of the same asymptotically flat spacetime.

3.Quantum backreaction and angle dependence of rotation. Now considerpossible dependence of the angle velocity h on the azimuthal angle that could solvethe semiclassical Einstein equation (with backreaction). Here we show that the only twoversions of the angle dependence for this equation to be solved by separation of variablesare the following ones: h = f() or h = f()cos .

Evidently, the problem reduces to studying the angle dependence of T10 for a givenangle dependence ofh. Given any physical field, we should solve equations of motion forthis field in curved spacetime and sum vacuum contributions into stress-energy from allthe eigenmodes. The resulting expression can be regularised by, e.g., covariant geodesicpoint separation and renormalised by subtracting the divergent parts known for physical

fields [10]. Choosing such separation in the radial direction we avoid discussing therenormalisation issue as far as the angle dependence is concerned.

Let us consider general structure of the equations of motion and stress-energy forarbitrary field in the metric (1) and illustrate this by the case of massless fields of spin1 (electromagnetic) and 1/2 (neutrino). Most natural to display effect of rotation inthe stationary axisymmetrical nonstatic metric is to use the complex Newman-Penroseformalism in analogy with that applied to Kerr metric [9]. In particular, up to the linearorder in h, we can choose isotropic tetrad of real l, n and complex m, m (astericsmeans complex conjugation) of which l, n are tangential to some geodesics and m

(and thus m) are orthogonal to l, n. Besides that, normalisation can be chosen such

that ln

= 1, m

m

= 1. These are the distinctive properties of the Newman-Penrosetetrad which can be written with the help of the derivatives over directions as

l = exp(2)[t + exp () h],n =

1

2[t exp () h], (6)

m = (r

2)1[ + i(sin )1],

4


5/10

m = (r

2)1[ i(sin )1].Calculation gives the following values for the standard 12 complex spin-connection coef-ficients , , , , , , , , , , , [9]:

= = 0, = 2 exp (2) = 2 =i

2 exp(2)h sin , = 2 = r

rexp(), = = i

2

2exp()hr sin , (7)

= = 12

2

cot

r i

4

2exp()hr sin , = 1

2 exp () i

8h sin

where subscript on h means corresponding derivative; prime on r, means derivativeover . The main feature of the covariant equations of motion for an arbitrary physicalfield is therefore occurence of h in the form h sin , h sin and h there. In thediagrammatic language, h-field-field vertex is combination of these expressions. Besides

that, in the Newman-Penrose formalism operators acting on the angle variables appearin the form of spin raising and lowering operators

Ls = isin

+ s cot , L+s = +i

sin + s cot , (8)

0 s s0, s0 being spin of the field. These operators simplify in the basis of spinspherical harmonics proportional to the elements of rotation matrix Dlsm(,, 0) [11]: Lsand L+s transform Dlsm to Dls+1,m and Dls1,m, respectively. In this basis the expressionfor T10 turns out to be a value of spin weight s = 1 (and m = 0), that is, combinationof Dl10 (or D

l1,0) for different l. Quantum contribution to T10 can be viewed as some

loop diagram with external T10- and h-legs. Since h can be expanded in the Legendre

polynomials Pk Dk

00, we can take Dk

00 as probe function for the angle dependenceof h. Then the expressions h sin and h sin appearing in the vertex on h-line arecombinations of the values Dk+110 and D

k110 . This corresponds to the angular momenta

k + 1 and k 1 flowing through the h-line. It is less evident but shown at the end ofthis section that the vertex h corresponds to the combination of angular momentak + 1, k 1, k 3, ... . By conservation of angular momentum the T10 also shouldbe combination of Dk+12n10 , n = 0, 1, 2, ... . In particular, at k = 0, 1 the only term(D110 or D

210) remains and T10 factorises into the functions of and of. Moreover, since

Dk+110 C3/2k just for k = 0, 1, the same -dependence also factors out in the LHS ofEinstein equation. Therefore we conclude: h = f() or h = f()cos solves for the-dependence of semiclassical Einstein equation.

Finally, let us illustrate the above said by the examples of electromagnetic and neu-trino fields; for more detail on the Newman-Penrose description of these fields see Ref.[9]. Electromagnetic field is described by three complex functions f0, f1, f2; for our choiceof complex tetrad (6) these are related to the electromagnetic field strength tensor Fas follows:

2f0 = Ft + hF + exp ()F + [Ft + exp ()F]i

sin ,

5


6/10

2f1 = r2 exp()(Ft + hF) + F isin

, (9)

2f2 = (Ft + hF) + exp ()F + [Ft exp ()F] isin

.

The eight real Maxwell equations can be recast into the following four complex ones:

f1 L1f0 = hf1,+f1 + L+1 f0 = hf1, (10)+f0 +

exp (2)

r2L+0 f1 = i

f0 f22

h sin ih exp () sin f1 hf0,

f2 exp (2)r2

L0f1 = if0 f22

h sin + ih exp()sin f1 + hf2,

where exp () t, it = being energy. Each mode should be normalised sothat its full energy

Ttt r

2 exp ()d sin dd =

sin dd exp()d

f0 f0 + f2 f2

+2exp (2)

r2f1 f1 2h sin Im[(f0 f2)f1]

(11)

be equal to the vacuum value /2 and then substituted into the expression for thestress-energy component studied,

T10

=

exp ()

r sin Tt

= 2exp()

r3Im[(f0 f2)f1]. (12)

Analogously, massless fermion field is described by two complex values g1, g2 obeyingthe field equations

g1 +exp ()

rL1/2g2 = i

4sin (g1h rhg2) + hg1,

exp ()

rL+1/2g1 +g2 =

i

4sin (rhg1 g2h) + hg2. (13)

Expression for the energy of each mode takes the form (on the field equations)

T

tt r

2

exp ()d sin dd =

sin dd exp()d 4(g1g1 + g

2g2) (14)

while the stress-energy component of interest is

T10 =exp()

r3

g1L+1/2g1 + (L+1/2g1)g1 g2L1/2g2 (L1/2g2)g2 (g1g1 g2g2)

+(r r)(g2g1 + g1g2) + 2r exp()[g2(t h)g1 g1(t h)g2]} .(15)

6


7/10

Solutions to the above equations of motion can be constructed iteratively. In zeroorder one takes

f0f1f2

(0)

=

RD

l1,m

l(l + 1)RDl0m

+RDl

+1,m

(16)

which are the well-known TE-modes for the electromagnetic field (TM-modes follow bymultiplying this by i =

1) and

g1g2

(0)=

Z1D

l+1/2,m

Z2Dl1/2,m

(17)

for the neutrino field. The R and Z1, Z2 are some radial functions. Substituting Eqs.(16) and (17) into the RHS of the equations of motion we find for the first O(h) correctionan expression of the type

f0f1f2

(1)

=j

i(1)m . . . Dj1,m. . . Dj0m

. . . Dj+1,m

l k 1 jm 0 m

+

. . . Dj1,m. . . Dj0m

. . . Dj+1,m

l k + 1 jm 0 m

+

. . . Dj1,m. . . Dj0m

. . . Dj+1,m

m l k j

m 0 m (18)

and quite analogous one for the fermion field. The dots mean (real) factors which donot depend on , and m. The linear order in h of interest comes from interplay inthe bilinear T10 between zero order (16) and first correction (18). By properties of 3j-symbols and rotation matrix elements Dlsm summation over m just yields combination

of the harmonics D

k+p

10 , p = 1, 1, 3, . . .. Important is that the last term in (18) whichstems from h operator in the equations of motion is representable as combination of3j-symbols as

m

l k jm 0 m

=n0

An

l k + 1 2n jm 0 m

(19)

where An does not depend on m. In T10 this and other terms enter multiplied byDl0m(D

j1,m)

and summed over m thus giving just combination of Dk+12n10 , n 0.4.Macroscopic wormhole and radial dependence of rotation. Usually, if one

does not assume existence of fundamental scales in the theory other than the Plankscale one expects the typical wormhole size be of the Plank scale too. However, a new

scale can exist connected with coefficient of the Weyl term in the effective action. Thiscoefficient is subject to renormalisation in both infrared (if massless fields are presentin the theory) and ultraviolet regions. Possible large value of this coefficient can enableexistence of the wormhole of macroscopic size.

Here we argue that if the Weyl term coefficient is large and one can disregard otherterms in the effective action then the conclusion concerning the existence of rotatingwormholes resembles that for the case of classical rotation in Sect. 2.

7


8/10

The effective gravity Lagrangian density with taking into account the Weyl term canbe written as proportional to R + (22)1CC

[12]. Up to the full derivative, theWeyl term CC

is equivalent to 2(RR 1

3R2). We calculate the latter up to

the second order in h (required to get the first order in the equations of motion) using

Riemann tensor given in Ref. [9] in the tetrad components. Varying in h gives the desiredt-component of the Einstein equations. Consider both versions of -dependence of hfound in Sect. 3, h 1 and h cos , and introduce new variable z via dz = exp ()dand the function r = r exp(). For h not depending on , h = f() = f((z)), theresult reads

(r4fz exp (2))z =1

2

r4

1

r4(r4fz)z

z

+

10

3(r2z 1) +

2

3rzz r

r2fz

z

(20)

(subscript z means differentiation over z). For h = f()cos we find

(r4fz exp (2))z

4 exp (2)r2f =1

2 r4

1

r4

(r4fz)zz

+

10

3r2z +

2

3rzz r 34

3

r2fz

z

+8

3(rzz r + r2z + 8)f

. (21)

The infrared contribution to the coefficient 2 goes from the massless fields. In theconsidered case of 2 large in the Plank scale the typical wormhole size r20 is defined

just by 2 as r20 = (32)1 [4, 5]. For example, in the vacuum of N1 spin 1 and N1/2

spin 1/2 massless fields we have

(32)1 = r20 =G

120(4N1 + N1/2) ln

120

G

2

4N1 + N1/2

, (22)

being infrared cut off.Consider first Eq. (20) which upon integrating both parts and denoting fz g

reduces to the second order one:

2 exp (2)r4g + C = r4

1

r4(r4g)z

z

+

10

3(r2z 1) +

2

3rzz r

r2g (23)

where C = const. Asymptotical form of this equation readsd

d

1

4d

d4g 2g = C

4. (24)

The general solution is the sum of a particular one which behaves as g 4 + O(6)at and arbitrary combination of the two independent solutions to the uniformequation. Of the latter two one exponentially grows at + or at andshould be omitted as unphysical solution while another one proportional to exp ()(at +) or exp() (at ) should be kept. Therefore, if we choose, asin Sect. 2, some large L > 0 we shall have physically acceptable solutions at || > Lparametrised by three constants, one of which is C. Meanwhile, in the intermediate

8


9/10

region || < L the equation (23) is regular in the wormhole geometry and has solutionparametrised by maximal set of three constants, one of which is C. The overall set offive constants is subject to four uniform equations which are matching conditions for gand for its derivative at = +L and at = L. This defines all five constants up

to an overall factor. Note that imposing additional condition+ gdz = 0 (that is,h() = h(+)) is, generally speaking, contradictory since it would be condition not

on a freely chosen constant, but on the already defined static metric (), r().Next consider Eq. (21) which has asymptotic form (at = 0, r = )

4d

d

1

4d

d4f =

1

24

d

d

1

4d

d42

f. (25)

Assuming this form at || > L we findd

d

1

4d

d4f 2f = C1 +

C+44

(26)

where C+1, C++4 (C

1, C

+4) are some constants which parametrise the solution at > L

(at < L). To get physical solution we put C1 = 0. Of the two solutions of uniformequation we discard exponentially growing one in each region > L or < L andretain exponentially falling off another solution. Thus, in each region > L or < Lthe solution is specified by two constants. At the same time, the regular fourth orderdifferential equation has solution parametrised by four constants at || < L. The overallnumber of constants is eight. These should ensure validity of eight matching conditionsfor f, f, f and f at = L. The determinant of this uniform system should be zero.This imposes a constraint on the already known static metric , r. Therefore the subsetof rotating wormhole solutions with h = f()cos should have zero measure w.r.t. the

set of spherically symmetrical static wormhole solutions (), r().

5.Conclusion. We have shown that if the coefficient at the Weyl term is large(infrared cut off is large) and one can discard other terms in the effective action then therotation existing for any static wormhole background (), r() is that which proceedswith the angular velocity h not depending on the azimuthal angle and having thedifferent finite limits h(+) and h() in the asymptotic region . We see thatdespite that the structure of the equations is drastically changed because of enhancingthe maximal order of derivatives upon taking into account vacuum polarisation, theresult practically does not differ from that for the classical case of Sect. 2.

Note that the sign of infrared divergent coefficient 2 at the Weyl tensor squared in

the effective action is crucial for existence (or, rather, nonexistence) of more rotationalsolutions in our case. Were 2 substituted by negative value, the equations above wouldhave oscillating instead of monotonic exponential solutions, and we would not have toomit some of them as unphysical ones. Then the general solution of interest would beparametrised by more constants, and the set of such solutions would be larger.

Also we can say that if we denote 2 x and extend the equations to arbitrary realx, the problem will be singular at x = 0.

9


10/10

In the case if the infrared logarithm is not large we do not have macroscopic vacuumwormhole, and the following interesting question arises: whether microscopic wormholecan rotate so that it would have the macroscopic tail of rotation (when h falls off inpower law in asymptotic region). Answering this question implies rather complicated

problem of calculation and analysis of the terms in stress-energy other than the Weylterm.

This work was supported in part by the President Council for Grants through grantNo. 96-15-96317.

References

[1] M.S.Morris and K.S.Thorne, Amer.J.Phys. 56 (1988) 395.

[2] M.Visser, Lorentzian Wormholes: from Einstein to Hawking(American Institute of

Physics, Woodbury, 1995).

[3] V.M.Khatsymovsky, Phys.Lett. B320 (1994) 234.



[6] M.S.Morris, K.S.Thorne and U.Yurtsever, Phys.Rev.Lett. 61 (1988) 1446.

[7] D.Hochberg, A.Popov and S.V.Sushkov, Self-consistent Wormhole Solutions ofSemiclassical Gravity (Preprint LAEFF 96/25, KSPU-96-03, gr-qc/9701064),

Phys.Rev.Lett. 78 (1997) 2050.

[8] G.T.Horowitz, Phys.Rev. D21 (1980) 1445.

[9] S.Chandrasekhar, The mathematical theory of black holes (Clarendon Press, Oxford,1983).

[10] S.M.Christensen, Phys.Rev. D17 (1978) 946.

[11] J.N.Goldberg, A.J.Macfarlane, E.T.Newman, F.Rohrlich, E.C.G.Sudarshan,J.Math.Phys. 8 (1967) 2155.

[12] B.S.DeWitt, Phys.Rep. C19 (1975) 295.

10
http://arxiv.org/abs/gr-qc/9701064http://arxiv.org/abs/gr-qc/9701064

v.m.khatsymovsky- rotating vacuum wormhole

Documents