Peter K.F. Kuhfittig- Axially Symmetric Rotating Traversable Wormholes

8/3/2019 Peter K.F. Kuhfittig- Axially Symmetric Rotating Traversable Wormholes

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arXiv:gr-qc/0401028v18

Jan2004

AXIALLY SYMMETRIC ROTATING TRAVERSABLEWORMHOLES

PETER K. F. KUHFITTIG

Abstract. This paper generalizes the static and spherically sym-metric traversable wormhole geometry to a rotating axially sym-metric one with a time-dependent angular velocity by means of anexact solution. It was found that the violation of the weak energycondition, although unavoidable, is considerably less severe than inthe static spherically symmetric case. The radial tidal constraintis more easily met due to the rotation. Similar improvements areseen in one of the lateral tidal constraints. The magnitude of theangular velocity may have little effect on the weak energy condi-tion violation for an axially symmetric wormhole. For a sphericallysymmetric one, however, the violation becomes less severe with in-creasing angular velocity. The time rate of change of the angularvelocity, on the other hand, was found to have no effect at all.Finally, the angular velocity must depend only on the radial coor-dinate, confirming an earlier result.

PAC number(s): 04.20.Jb, 04.20.Gz

1. Introduction

It was recognized by Flamm [1] in 1916 that our universe may notbe simply connected: there may exist handles or tunnels, now calledwormholes, in the spacetime topology linking widely separated regionsof our universe or even connecting us with different universes alto-gether. That such wormholes may be traversable by humanoid travel-ers was first conjectured by Morris and Thorne [2], thereby suggestingthat interstellar travel and even time travel may some day be possible.For a detailed discussion see the book by Visser [3].

Morris-Thorne (MT) wormholes are static and spherically symmetric

and connect asymptotically flat spacetimes, here assumed to be isomet-ric. Adopting units in which c = G = 1, the metric for this wormholeis given by

(1) ds2 = e2(r)dt2 +dr2

1 b(r)/r+ r2(d2 + sin2 d2);

Date: February 7, 2008.1
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2 PETER K. F. KUHFITTIG

(r) is called the redshift function and b(r) the shape function. Theshape function describes the spatial shape of the wormhole when viewed,

for example, in an embedding diagram, described below.To hold such a wormhole open, violations of certain energy conditionsproved to be unavoidable. More precisely, all known forms of matterobey the weak energy condition (WEC) T

0 for all timelikevectors and, by continuity, all null vectors (Friedman [4]). Matter thatviolates this condition is called exotic by Morris and Thorne.

Various attempts have been made to generalize the MT wormholeby giving up spherical symmetry [3] or by including time-dependence.A particular interesting example of the latter is the inclusion of a deSitter scale factor multiplying the spatial part of the metric. The goalwas to study the possibility of enlarging a wormhole pulled out of the

spacetime foam to macroscopic size (Roman [5]). A similar scale factorwas used by Kim [6]. Yet another possibility is the use of a conformalfactor (t) [7, 8, 9, 10]:

(2) ds2 = (t)e2(r)dt2 + e2(r)dr2 + r2(d2 + sin2 d2)

.

For a metric with time-dependent functions and , see Ref. [11]. Allthese studies include a discussion of the WEC violation. Traversabilityconditions are investigated in [7] and [11].

In this paper we generalize the MT wormhole in another directionby assuming the wormhole to be rotating, not necessarily at a constant

rate, and by dropping the assumption of spherical symmetry. Instead,the wormhole is assumed to be axially symmetric, i.e., symmetric withrespect to the axis of rotation. Stationary axially symmetric wormholesare discussed by Teo [12]. It is shown that the WEC is indeed violatedbut that a traveler would not necessarily come into contact with anyof the exotic matter. These wormholes may also have an ergoregion,where a particle cannot remain stationary with respect to spatial in-finity. This is an extreme example of the well-known dragging effect ingeneral relativity.

The main purpose of this paper is to discuss both the energy viola-tion and the traversability conditions by first finding an exact solution.

Possible restrictions on the metric coefficients recently proposed byPerez Bergliaffa and Hibberd [13] are discussed in Section 7.

Proposals to search for naturally occurring wormholes, if they exist,go back at least to 1995 [14]. For a summary of these findings seeRef. [15]. While definite conclusions are still lacking, the possible exis-tence of wormholes or the existence of negative mass cannot be ruledout.


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ROTATING TRAVERSABLE WORMHOLES 3

2. The metric

The study of spacetimes that are both stationary and axially sym-metric has a long history [16, 17, 18]. A spacetime is stationary ifit possesses a time-like Killing vector field a = (/t)a generatinginvariant time translations. Axially symmetric is formally defined aspossessing a space-like Killing vector field a = (/)a generatinginvariant rotations with respect to .

Suitable metrics for stationary axially symmetric fields are discussedby Islam [19]. We will adopt the metric suggested by Teo [12], since itappears to be best-suited for obtaining an exact solution:

(3) ds2 = N2dt2 + edr2 + r2K2

d2 + sin2(d dt)2

,

where N,,K, and are all functions of r and ; is the angular

velocity d/dt. More precisely,

=d

dt=

d/d

dt/d=

u

ut,

referred to in Ref. [20] as the angular velocity relative to the asymp-totic rest frame. The connection between the metric due to a boundedrotating source and the mass and angular momentum of the source isdiscussed in Ref. [19].

To make our solution as general as possible, we will assume that = (r,,t) is time-dependent, so that the wormhole can no longerbe called stationary. The reason for proposing this model is a practical

one: if an advanced civilization were to succeed in constructing such awormhole, it is likely to be aspherical and the rate of rotation likely tobe varied. Accordingly, we will write our line element as follows:

(4) ds2 = e2(r,)dt2

+ e2(r,)dr2 + [K(r, )]2 r2

d2 sin2(d (r,,t)dt)2

.

Here K(r, ) is a positive dimensionless function of r such that Krdetermines the proper radial distance at (r, ) in the usual manner.In other words, 2(Kr)sin is the proper circumference of the circlethrough (r, ).

So far nothing has been said about the shape function. Recall thatthe wormhole geometry may be conveniently described by means ofan embedding diagram in three-dimensional Euclidean space at a fixedmoment in time and for a fixed value of , the equatorial slice =/2 [2, 21]. The resulting surface of revolution has the parametricform

f(r, ) = (r cos , r sin , z(r, 1)),


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where z = z(r, ) is some function of r and and is momentarilyheld fixed at 1. As usual, we think of the surface as connecting two

asymptotically flat universes. The radial coordinate decreases from+ in the upper universe to a minimum value r = r0 at the throat,and then increases again to + in the lower universe.

Since the embedding surface must have a vertical tangent at thethroat for any value of , we require that

limrr0+

dz

dr= +,

while limr dz/dr = 0, the meaning of asymptotic flatness. Return-ing to the line element (4), we further assume that for any fixed ,(r, ) has a vertical asymptote at r = r0: limrr0+ (r, ) = +.

Also, (r, ) is a twice differentiable function of r and (as is (r, ))and is a strictly decreasing function of r with limr (r, ) = 0.These requirements are met by z = z(r, ) for any fixed such that

dz

dr=

e2(r,) 1

(for the upper universe). Furthermore, d2z/dr2 < 0 near the throat(since d(r, 1)/dr < 0), as required by the flaring out condition inRef. [2]. The shape function is now defined by

e2(r,) =1

1 b(r,)

r

.

It follows that

b(r, ) = r

1 e2(r,)

.

Finally, at the throat itself, b must be independent of . It is readilychecked that b/ = 0 at r = r0.

3. The solution

Let us write the line element (4) in slightly more compact form:

(5) ds2 = e2dt2 + e2dr2 + K2r2 d2 + sin2(d dt)2 .

To make the analysis tractable, we choose an orthonormal basis {e}which is dual to the following 1-form basis:

(6) 0 = e dt, 1 = e dr, 2 = Krd

and

(7) 3 = Kr sin (d dt).


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(See also Ref. [23].) As a result,

(8) dt = e

0

, dr = e

1

, d =

1

Kr

2

,and

(9) d =1

Kr sin 3 + e 0.

Furthermore,

ds2 = (d0)2 + (d1)2 + (d2)2 + (d3)2.

Since the orthonormal basis is itself rotating, some information re-garding is going to be lost. We will briefly return to this topic atthe end of the last section. (In particular, it will be shown that themagnitude of must be restricted.) To do so, we need the compo-

nents of the fundamental metric tensor {g}, as well as {g}, in the(t,r,,)-coordinate system:

(10) gtt = e2 + K2r22 sin2, gt = K

2r2 sin2,

grr = e2, g = K

2r2, g = K2r2 sin2,

and

(11) gtt = e2, gt = e2, grr = e2,

g =1

K2r2, g =

1

K2r2 sin2 2e2.

The last component, g

, bears a striking similarity to d in Eq. (9).To obtain the curvature 2-forms and the components of the Riemann

curvature tensor, we use the method of differential forms (Ref. [22]).To that end we calculate the following exterior derivatives in terms ofi:

d0 =

re 1 0 +

1

Kr

2 0, d1 =

1

Kr

2 1,

d2 =

1

re +

1

K

K

re

1 2,

and

d3 = Kr r

eesin 1 0

esin 2 0

+

1

re +

1

K

K

re

1 3 +

1

Krcot +

1

K2r

K

2 3.

The connection 1-forms i k have the symmetry

0 i = i0 (i = 1, 2, 3), and

ij =

ji (i, j = 1, 2, 3, i = j)


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and are related to the basis i by

d

i

=

i

k

k

.The solution of this system is found to be

01 =1

2Kr

reesin 3 +

re 0,

02 =1

Kr

0 +

1

2

esin 3,

03 =1

2Kr

reesin 1 +

1

2

esin 2,

12 =1

Kr

1 1

r

e +1

K

K

r

e 2,13 =

1

2Kr

reesin 0

1

re +

1

K

K

re

3,

23 =1

2

esin 0

1

Krcot +

1

K2r

K

3.

The curvature 2-forms ij are calculated directly from the Cartanstructural equations

ij = dij +

ik

kj.

The results are given in the Appendix. Since the components of theRiemann curvature tensor can be read off directly using the formula

ij = 1

2R imnj

m n,

there is no need to list them explicitly. As an example, suppose we letm = 0 and n = 1 in the equation

01 = 1

2R 0mn1

m n.

Then

01 = 1

2R 0011

0 1 1

2R 0101

1 0

= R 0011 0 1 = R 0011

1 0.

Thus R 0011 = A(1, 0) in the Appendix. (As in Ref. [22], we omit thehats whenever numerical indices are used.)


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4. WEC violation

As with any traversable wormhole, we would expect a violation of

the weak energy condition near the throat: T 0 for all null

vectors. As in Morris and Thorne [2] and Roman [5] we use a radial

outgoing null vector = (t, r, 0, 0) = (1, 1, 0, 0). In our orthonormalframe we expect to have the usual stress-energy components Ttt, Trr,T, T, as well as Tt, which represents the rotation of the matterdistribution [12].

From G = R12Rg and Rab = R

cacb , we have G00 + G11 =

R00 + R11 = R0

011 + R0

022 + R0

033 R0

011 R1

122 R1

133 . A shortcalculation yields

(12) 8 (Ttt + Trr) = Rtt + Rrr

=2

re2

r+

r

+

2

K

K

re2

r+

r

+1

K2r2

cot +

1

K2r2

2

2

2

2

+1

K2r2

2

2+

4

Kr

K

re2

+ 2

K

2K

r2

e2 + 1

2

2

e2sin2 .(We have used the Einstein field equations G = 8T.)

To put this rather long expression in perspective, consider the staticspherical case [11], where only the first term survives:

(13) Ttt + Trr = =1

8

2

r

e2

d

dr+

d

dr

.

Referring to Sect. 2, recall that the throat corresponds to the valuer = r0. So if is a smooth function ofr and given that limrr0+ (r) =

+, it follows that limrr0+ d/dr = . In Ref. [11], (r) = /r, > 0, so that d/dr = /r2, making negative near the throat.That a violation of the WEC cannot be avoided regardless of the choiceof(r) can be seen geometrically. For if limrr0+ d/dr is positive andfinite, then the sum on the right side of Eq. (13) is negative near thethroat. This might be avoided if limrr0+ d/dr = +. But thenlimrr0+ (r) = and e

2(r) 0, which yields an event horizon.


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The main goal in this section is to show that for a rotating axiallysymmetric wormhole the WEC violation is, in principle, much less se-

vere than for the static spherically symmetric case by a suitable choiceof and . The functions were chosen primarily for convenience, keep-ing the analysis simple and accommodating the next section at thesame time.

Taking k,A, and B to be constants, let

(14) (r, ) = k

r

1

3

2

3+ A

, 0 <

2, k > 0,

where A is large enough to keep the expression inside the bracketspositive, and

(15) (r, ) =k

r r0

2 + B , 0 <

2, k, B > 0,

where > 0 is a small constant. On the interval [/2, ) each functionis defined to be the mirror image,i.e., and are symmetric about = /2. So the discussion may be confined to the interval (0 , /2].

The partial derivatives are listed next for easy reference:

r=

k

r2

1

3

2

3+ A

,

r=

k

(r r0)2

2

+ B

,

=

k

r

2

2,

=

k

r r0,

2

2 =

2k

r

2

,

2

2 = 0.On the right side of Eq. (12), the first two terms are similar to those

in Eq. (13). The first term is therefore negative. The fourth term isstricly positive for = /2, while the third term is positive near thethroat. In the fifth term the expression

2

2=

k2

r2

2

4

k22

(r r0)2

is similar to the first term but the quadratic factor 2 reduces the sizeof the second term, thereby making the fifth term less harmful.

To study the effect of K(r, ), we adopt a function similar to onesuggested by Teo [12],

K(r, ) = 1 +(4a sin )2

r.

Now the second and sixth terms are positive, as well. The seventhterm is close to zero near the throat, thanks to the factor e2, and sois completely overshadowed by the third and fourth terms.


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The last term is more of a problem, being strictly negative. However,for Teos choice, = 2a/r3, where a is the total angular momentum of

the wormhole, the last term is zero. Requiring to be independent ofmay be unavoidable, a conclusion also reached by Khatsymovsky [23],who states that for a macroscopic wormhole to exist, the angular ve-locity must be independent of .

With the last term eliminated, we see that a drastic reduction in theenergy condition violation is indeed possible, at least in principle.

Remark: We will see in the next section that the absolute value ofthe first term exceeds that of the second term. Otherwise the WECviolation would appear to have been eliminated completely.

5. Traversibility conditions

Another area in which improvements over the static spherically sym-metric case are possible is in the study of tidal constraints. In partic-ular, for an infalling radial observer the components of the Riemanncurvature tensor are found relative to the following orthonormal basis(from the usual Lorentz transformations):

(16)

e0 = e t v

c

er, e1 = er +

vc

et, e2 = e, e3 = e.

A traveler should not experience any tidal forces larger than those on

Earth. As outlined in Ref. [2], the radial tidal constraint is given by

|R1010| g

c2 2 m

1

(108 m)2,

assuming an observer 2 m tall. We have

(17) |R10 10| = |Rrtrt| =

e2

2

r2

r

r+

r

2

+

3

4K2r2

r 2

e2e2sin2 +1

K2r2

.

In the static spherically symmetric case only the first term survives.The second term, which involves the angular velocity, reduces the sizeof |Rrtrt|: the first term,

e2

2

r2+

r

r

+

r

2,


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is positive near the throat because the positive middle term inside thebrackets contains the factor (rr0)

2 in the denominator. For the same

reason the first term is larger than the absolute value of the second(near the throat). Since the second term is negative, the net result, sofar, is a reduction in the size of |Rrtrt|.

Concerning the last term, we need to remember that our wormhole isnot spherically symmetric. The tidal forces experienced may thereforedepend on the direction of approach. So we must ask the travelerto approach the throat in the equatorial plane = /2, and here/ = 0.

To study the first of the lateral tidal constraints, |R2 020 | (108 m)2,

we have from Eq. (16),

|R2 020 | = 2 |Rtt|+

2v

c

2|Rrr| .

As usual, this is a constraint on the velocity of the traveler. It isassumed in Ref. [2] that the spaceship decelerates until it comes to restat the throat. So only the first term needs to be examined:

(18) |Rtt| =

R 0022

=1r r e2 + 1K2r2

2

2

+ 1K2r2

2 1

K3r2K

+1

K

K

r

re2

3

4

2e2sin2

.

In the static spherically symmetric case only the positive first termsurvives. Once again requiring that the traveler approach the throatin the equatorial plane = /2, the next three terms are zero. Thelast term is also zero due to the earlier requirement / = 0. The

fourth term is negative, and for our choice of K(r, ) 1KKr r e2 0, the last term is positive. So if is large, the WEC

violation is much reduced.While the resulting reduction in the WEC violation is a welcome

surprise, some words of caution are in order. The last term in Eq.(20) suggests that if is large enough, the WEC violation can beeliminated altogether. But as explained by Teo [12], for a rapidlyrotating wormhole it may not be possible to use a radially outgoingnull vector since the gtt component of the fundamental metric tensormay no longer be negative, as can be seen from the line element, Eq.(5).

7. Additional considerations

It was pointed out in a recent paper by Perez Bergliaffa and Hibberd[13] that the metric (4) used in this paper may require further restric-tions. In particular, it is shown that a wormhole of the type studiedby Teo [12] cannot be generated by a perfect fluid or by a fluid withanisotropic stresses. In the first case the condition

(21) G12 = 0


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is violated and in the second case,

(22) G00 + G33 2 G03.

These conditions seem to be met if the metric (4) includes the func-tions and used in this paper, that is, Equations (14) and (15),respectively.

For example, to check Condition (21), a short calculation yields

G12 =1

2Kr

r

e2esin2 +

1

Kr2

+

e

+

1

K2r

K

r

+

e

1

Kr

r

e

1

Kr

2

re

1

K2r

2K

re +

1

K3r

K

r

K

e.

Each term contains e, so that G12 0 near the throat.Even more favorable is the outcome of the check on Condition (22):

five of the terms in G03 contain the factor /, which is equal tozero. The remaining terms all contain the factor e2, so that G03 0near the throat. The left side, G00 + G33, contains only two terms thatare neither strictly positive, nor zero, nor contain the factor e2. But

these two terms are strongly overshadowed by several terms that arestrictly positive. So Condition (22) is easily met in the vicinity of thethroat.

Since the vicinity of the throat is the only region that really matters,we can construct (in the usual manner) a solution with a suitable radialcut-off of the stress-energy tensor. Our solution can then be joined, atleast in principle, to an external solution also satisfying the desiredconditions. (For a discussion of the required junction conditions, seeRef. [20].)

It is indeed surprising that and , which were chosen for entirelydifferent reasons, are sufficient for satisfying Conditions (21) and (22),

thereby overcoming the objections raised to the wormhole in Teo [12].Unfortunately, Ref [13] discusses other conditions, some of which arenot so easily checked. So it may still be necessary to incorporate morerealistic (in the astrophysical sense) features, for example heat flux(Ref [13]). Future investigations of these issues could be extended toinclude stability conditions, especially for rotating wormholes, sincesuch conditions may also require more realistic features.


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8. Conclusion

In this paper the MT wormhole solution was generalized to worm-holes which are both rotating and axially symmetric, i.e., symmetricwith respect to the axis of rotation. It was concluded that the unavoid-able violation of the weak energy condition is less severe than in thespherically symmetric case. The radial tidal constraint is more easilymet due to the rotation. An improvement was also found in one of thelateral tidal constraints. Making the angular velocity time-dependentdoes not help since none of the time-derivatives in the Appendix ap-peared in the calculations. Furthermore, must be independent of, in agreement with Ref. [23]. Finally, the magnitude of the angularvelocity may have little effect on the WEC violation for an axially sym-metric wormhole. In the spherically symmetric case, however, a rapidrotation will result in a reduction in the WEC violation, as long as is not excessively large.

Appendix

This appendix lists the curvature 2-forms ij.

01 = A(1, 0) 1 0 + A(2, 0) 2 0 + A(3, 0) 3 0 + A(3, 1) 3 1

+ A(3, 2) 3 2,

where

A(1, 0) = e2

2

r2

r

r+

r

2

3

4K2r2

r

2e2e2sin2 +

1

K2r2

,

A(2, 0) =

1

Kr e 2

r

r

+

r

3

4Kr

r

e2esin2

1

Kr2

e

1

K2r

K

r

e,

A(3, 0) = 1

2Kr

t

r

e2esin ,


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16/17


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where F(1, 0) = A(3, 2), F(2, 0) = B(3, 2), F(3, 1) = E(3, 2),

F(3, 2) = 1

K2r2 +1

r2e2 +

2

Kr

K

r e2 +

1

K2K

r2

e2

1

K4r2

K

2+

1

K3r22K

2+

1

K3r2K

cot +

1

4

2e2sin2.

References

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Padmanabhan (Kluwer, The Netherlands, 1997), p. 157.[5] T. A. Roman, Phys. Rev. D 47, 1370 (1993).[6] S.-W. Kim, Phys. Rev. D 53, 6889 (1996).[7] L. A. Anchordoqui, D. F. Torres, M. L. Trobo, and S. E. Perez Bergliaffa, Phys.

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Phys. Rev. D 51, 3117 (1995).

[15] M. Safonova, D. F. Torres, and G. E. Romero, Phys. Rev. D65

, 023001 (2002).[16] A. Papapetrou, Ann. Inst. Henri Poincare, Sect. A 4, 83 (1966).[17] B. Carter, J. Math. Phys. 10, 70 (1969); Commun. Math. Phys. 17, 223 (1970).[18] B. Carter, in Gravitation and Astrophysics, edited by B. Carter and J. B.

Hartle (Plenum, New York, 1987).[19] J. N. Islam, Rotating Fields In General Relativity(Cambridge University Press,

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Department of Mathematics, Milwaukee School of Engineering, Mil-

waukee, Wisconsin 53202-3109
http://arxiv.org/abs/gr-qc/0006041http://arxiv.org/abs/gr-qc/0006041

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Peter K.F. Kuhfittig- Axially Symmetric Rotating Traversable Wormholes