Table Of Content∆ string - a hybrid between wormhole and string
−
V. Dzhunushaliev ∗
Dept. Phys. and Microel. Engineer., KRSU, Bishkek,
Kievskaya Str. 44, 720000, Kyrgyz Republic
Abstract
The fluxtubesolutions in 5D Kaluza-Klein theory can be considered as a string-like object - ∆−string.
The initial 5D metric can be reduced to some inner degrees of freedom living on the ∆−string. The
3 propagation of electromagnetic waves through the ∆−string is considered. It is shown that the difference
0 between∆andordinarystringsareconnectedwiththefactthatforthe∆−stringsuchlimitationsascritical
0 dimensions are missing.
2
n
1 Introduction
a
J
4 Thedifferencebetweenpoint-likeparticlesandstringsontheonehand,andEinstein’spointofviewonaninner
1 structure of matter on the other hand is that according to Einstein everything must have an inner structure.
Even more : at the origin of matter should be vacuum. In string theory a string is a vibrating 1-dimensional
1
objectandthestringhasmanydifferentharmonicsofvibration,andinthiscontextdifferentelementaryparticles
v
are interpreted as different harmonics of the string. The string degrees of freedom are the coordinates of string
6
4 points in an ambient space.
0 Inthispaperwewouldliketoconsiderthe situationwhenthestringhasaninnerstructure. Thequestionin
1 this situationis : what willbe changedinthis situation? Definitely wecansaythatin this situationthe string
0
isanobjectwhicheffectivelyarisefromafieldtheoryandsuchobjecthasinnerdegreesoffreedomwhicharenot
3
connected with an external space. As a model of such kind of string-like object we will consider gravitational
0
/ flux tubes. These tubes are the solutions in 5D Kaluza-Klein gravity [1] filled with electric E and magnetic H
c
fields. If E = H we have an infinite tube, if E H (but E > H) the length of the tube can be arbitrary long
q ≈
- and the cross section can be lPl (lPl is the Planck length) [2]. Such flux tube can be considered as a string
r attached to two Universes or≈to remote parts of a single Universe. For the observer in the outer Universe the
g
: attachment points looks like to point-like electric and magnetic charges (see Fig. 1).
v
We have to note that similar construction was presented in Ref. [3] : the matching of two remote regions
i
X was done using 4D infinite flux tube which is the Levi-Civita - Bertotti - Robinson solution [4], [5] filled with
r the electric and magnetic fields.
a
2
q -q
D - string
Q
-Q
1
spacetime foam
Figure 1: The gravitational flux tube attached to Figure 2: ∆ string is the gravitational flux tube
−
two Universes (1) or to remote parts of a single surroundedwith the handles ofspacetime foamat
Universe (2). The cross section is in order of the the ends.
Plancklengthandthetubelengthcanbearbitrary
long. Such construction is similar to the string
attached to D-brane(s).
∗E-mail: [email protected]
2 Gravitational flux tube
In this section we will describe the gravitationalflux tube and why it can be considered as a ∆ string. Let us
−
consider the following 5D metric
ds2 =e2ν(r)dt2 e2ψ(r)−2ν(r)[dχ+ω(r)dt+Qcosθdϕ]2
− (1)
dr2 a(r)(dθ2+sin2θdϕ2),
− −
where χ is the 5th extra coordinate; r,θ,ϕ are 3D spherical-polar coordinates; Q is the magnetic charge. The
solution of 5D Kaluza-Klein equations depends on the relation δ = 1 H/E between electric E and magnetic
−
H fields. If δ =0 we have an infinite flux tube filled with electric and magnetic fields [1]
q2
a = =const, (2)
0
2
r√2
eψ =eν =cosh , (3)
q
r√2
ω =√2sinh (4)
q
here q is the electric charge. If 0 < δ 1 the part of spacetime located between two hypersurfaces ds2(r =
≪
r ) = 0 1 is a finite flux tube filled with electric and magnetic fields. In both cases the cross sectional sizes
H
±
can be arbitrary but we choose its in the Planck region ( l2 ). This condition is very important for the
≈ Pl
idea presented here : the gravitational flux tube can be considered as the ∆ string attached to two different
−
Universes (or to remote parts of a single Universe). From the physical point of view this is 1D object as the
Planck length is a minimal length in the physical world.
Now our strategy is to consider small perturbations of 5D metric. Generally speaking they are 5D gravita-
tional waves on the gravitationalflux tube (or on the string language - vibrations of ∆ string). In the general
−
case 5D metric is
ds2 =g dxµdxν φ2(dχ+A dxµ)2 (5)
µν µ
−
here g is the 4D metric; µ,ν = 0,1,2,3; A is the electromagnetic potential; φ is the scalar field. The
µν µ
corresponding 5D Kaluza-Klein’s equations are (for the reference see, for example, [7])
1 φ2 1
R g R = T [ (∂ φ) g (cid:3)φ], (6)
µν µν µν µ ν µν
− 2 − 2 − φ ∇ −
∂ φ
Fµν = 3 ν Fµν, (7)
ν
∇ − φ
φ3
(cid:3)φ = FαβF (8)
αβ
− 4
whereR isthe4DRiccitensor;F =∂ A ∂ A isthe4DMaxwelltensorandT istheenergy-momentum
µν µν µ ν ν µ µν
−
tensor for the electromagnetic field.
WewillconsideronlyδA perturbations,δg andδφdegreesoffreedomarefrozen. Forthisapproximation
µ µν
we have equation
δFµν =0. (9)
ν
∇
We introduce only one small perturbation in the electromagnetic potential
δA =f(t,r) (10)
θ
sinceforthebackgroundmetricφ=const. GenerallyspeakingA -componentshouldhavesomedependenceon
θ
theθ-angle. Butthecrosssectionofthe∆ stringisinPlanckregionandconsequentlythepointswithdifferent
−
θ andϕ(r =const)physicallyarenotdistinguishable. Thereforeallphysicalquantitiesonthe∆ stringshould
−
be averagedover polar angles θ and ϕ. It means that the δA in Eq. (10) is averagedquantity.
θ
After this (very essential) remark we have the following wave equation for the function f(t,r)
∂ f(t,x) coshx∂ coshx∂ f(t,x) =0 (11)
tt x x
− (cid:18) (cid:19)
here we have introduced the dimensionless variables t/√a0 t and r/√a0 x. The solution is
→ →
f(t,x)=f F (t 2arctanex)+f F (t+2arctanex) (12)
0 1
−
here f are some constants and F and H are arbitrary functions. This solution has more suitable form if we
0,1
introduce new coordinate y =2arctanex. Then
f(t,y)=f F(t y)+f H(t+y). (13)
0 1
−
1atthesepointsexp(−2ν(r=±rH))=0
The metric is
a dy2
ds2 = 0 dt2 a dθ2+sin2θdϕ2 (dχ+ωdt+Qcosθdϕ)2. (14)
sin2y − sin2y − 0 −
(cid:0) (cid:1)
Thus the simplest solution is electromagnetic waves moving in both directions along the ∆ string.
−
3 The comparison with string theory
Now we want to compare this situation with the situation in string theory. How is the difference between the
result presented here (Eq. (11), (12)) and the string oscillation in the ordinary string theory ? The action for
bosonic string is
T
S = d2σ√hhab∂ µ∂ (15)
a b µ
−2 Z X X
here σa = σ,τ is the coordinates on the world sheet of string; µ are the string coordinates in the ambient
{ } X
spacetime; h is the metric on the world sheet. The variation with respect to µ give us the usual 2D wave
ab
X
equation
∂2 ∂2
(cid:3) µ µ =0 (16)
X ≡(cid:18)∂σ2 − ∂τ2(cid:19)X
which is similar to Eq. (11). The difference is that the variation of the action (15) with respect to the metric
hab gives us some constraints equations in string theory
1
T =∂ µ∂ h hcd∂ µ∂ =0 (17)
ab a b µ ab c d µ
X X − 2 X X
but for ∆ string analogous variation gives the dynamical equation for 2D metric h .
ab
−
The moredetaileddescriptionisthe following. Thetopologyof5DKaluza-Kleinspacetime is M2 S2 S1
× ×
whereM2 isthe2Dspace-timespannedonthetimeandlongitudinalcoordinater; S2 isthecrosssectionofthe
flux tube solution and it is spanned on the ordinary spherical coordinates θ and ϕ; S1 = U(1) is the Abelian
gauge group which in this consideration is the 5th dimension. The initial 5D action for ∆ string is
−
(5)(5)
S = g R (18)
Z q−
(5) (5)
where g is the 5D metric (5); R are 5D Ricci scalars. We want to reduce the initial 5D Lagrangian(18) to a
2D Lagrangian (here we will follow to Ref. [6]). Our basic assumption is that the sizes of 5th and dimensions
spanned on polar angles θ and ϕ is approximately l . At first we have usual 5D 4D Kaluza-Klein
Pl
≈ →
dimensional reduction. Following, for example, to review [7] we have
1 (5) 1 (4) 1 2∂ φ∂αφ
(5) R= 16πG R −4φ2FµνFµν + 3 αφ2 (19)
16π G
(5) (4)
where G=G dx5 is 5Dgravitationalconstant;Gis 4Dgravitationalconstant; R isthe 4DRicciscalars. The
R (5) (4)
determinants of 5D and 4D metrics are connected as g=g φ. One of the basis paradigm of quantum gravity
is that a minimal length in the Nature is the Planck scale. Physically it means that not any physical fields
depend on the 5th, θ and ϕ coordinates.
The next step is reduction from 4D to 2D. The 4D metric can be expressed as
(4)
d s2 = g dxµdxν =g (xc)dxadxb+
µν ab
χ(xc) ω¯iidyi+B¯ai(xc)dxa ωj¯jdyj +B¯ia(xc)dxa η¯i¯j (20)
(cid:16) (cid:17)(cid:16) (cid:17)
where a,b,c=0,1; xa are the time and longitudinal coordinates; −(cid:16)ω¯iidyi(cid:17)(cid:16)ωj¯jdyj(cid:17)η¯i¯j =dl2 is the metric on
the 2DsphereS2; yi arethe coordinatesonthe 2Dsphere S2; allphysicalquantities gab,χ andB¯ia candepend
only on the physical coordinates xa. Accordingly to Ref. [8] we have the following dimensional reduction to 2
dimensions
(4) (2) 1
R = R +R(S2)− 4φ¯iabφ¯aib−
1
hijhkl D h Dah +D h Dahkl a hijD h (21)
a ik jl a ij a ij
2 −∇
(cid:0) (cid:1) (cid:0) (cid:1)
(2)
where R is the Ricci scalar of 2D spacetime; Dµ and φ¯iab are, respectively, the covariant derivative and the
curvature of the principal connection B¯i and R(S2) is the Ricci scalar of the sphere S2 = SU(2)/U(1) with
a
linear sizes l ; h is the metric on
Pl ij
≈ L
su(2) = Lie(SU(2))=u(1) , (22)
⊕L
u(1) = Lie(U(1)) (23)
here is the orthogonal complement of the u(1) algebra in the su(2) algebra; the index i . The metric h
ij
L ∈L
is proportional to the scalar χ in Eq. (17).
The 2D action (which is our goal) is
(2)
S =Z d2xaφ(detgab)(cid:16)detω¯ii(cid:17)16RπG − 41φ2(cid:0)FtθFtθ+FrθFrθ(cid:1)+other terms. (24)
The second term in the [...] brackets give us the wave equation (11) and the most important is that the first
term is not total derivative in contrast with the situation in ordinary string theory in the consequence of the
factorsφand detω¯i . Thereforethevariationwithrespectto2Dmetricg giveussomedynamicalequations
i ab
(cid:16) (cid:17)
contrary to string theory where this variation leads to the constraint equations (17).
This remarkallows us to say that the ∆ string do not have such peculiarities as criticaldimensions (D=26
−
for bosonic string). The reason for this is very simple : the comprehending space for ∆ string is so small that
−
itcoincideswith∆ string. Thuswecansupposethatthecriticaldimensionsinstringtheoryisconnectedwith
−
the fact that the string curves the external space but the back reaction of curved space on the metric of the
string world sheet is not taken into account.
4 Discussion and conclusions
Finally : the most important difference between ∆ and ordinary strings is that in the first case the dynamical
equation for 2D metric is replaced by the constraint equation for the second case. Physically it means that
in the second case we ignore the back reaction of string on the metric of world sheet. It is supposed that in
ordinary string theory this back reaction is zero but for the ∆ string this reaction is very big since the string
−
coincides with the ambient space.
One can say that the ∆ string is a hybrid between the wormhole and the string as it is the wormhole-like
−
solution in 5D Kaluza-Klein gravity on the one hand and it is approximately 1D object on the other hand.
For the outer observer the attachment point of ∆ string to outer Universe looks like a distributed electric
−
and magnetic charges as this point is spread in the consequence of the appearance of spacetime foam handles
between ∆ string and the outer Universe. The ∆ string is a bridge between two Universes (or remote parts
− −
of a single Universe) like to wormhole on the one hand and has an arbitrarylong throat like to the string. The
∆ string can be considered as a model of Wheeler’s “mass without mass” and “charge without charge”. We
−
have shown that such object can transfer the electromagnetic waves from one Universe to another one or from
one part of Universe to another one.
The discussion of electromagnetic waves propagating through the ∆ string is not full as we have frozen
−
δg andδφ perturbations. EvidentlyδA perturbations willinitiate δg and δφwavesandcertainlythe back
µν µ µν
reaction takes place.
5 Acknowledgment
I am very grateful to the ISTC grant KR-677 for the financial support.
References
[1] V. Dzhunushaliev and D. Singleton, Phys. Rev. D59, 064018(1999).
[2] V. Dzhunushaliev, Class. Quant. Grav., 19, 4817 (2002).
[3] E. I. Guendelman, Gen. Relat. Grav., 23, 1415 (1991).
[4] Levi-Civita, T. (1917). Atti Acad. Naz. Lincei 26,519
[5] Bertotti, B. (1959). Phys. Rev. 116, 1331; Robinson, I. (1959). Bull. Akad. Pol. 7, 351
[6] V. Dzhunushaliev, “Strings from Flux Tube Solutions in Kaluza-Klein Theory”, gr-qc/0208023, to be
published in Phys. Lett. B.
[7] J. M. Overduin and P. S. Wesson, Phys. Rept., 283, 303 (1997).
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