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Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨ur Physik, Universit¨at Oldenburg, D-2
Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨ur Physik, Universit¨at Oldenburg, D-2
Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨ur Physik, Universit¨at Oldenburg, D-2
Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨ur Physik, Universit¨at Oldenburg, D-2
Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨ur Physik, Universit¨at Oldenburg, D-2
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Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨ur Physik, Universit¨at Oldenburg, D-2

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Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory …

Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory
Panagiota Kanti
Division of Theoretical Physics, Department of Physics,
University of Ioannina, Ioannina GR-45110, Greece
Burkhard Kleihaus, Jutta Kunz
Institut f¨ur Physik, Universit¨at Oldenburg, D-26111 Oldenburg, Germany
(Dated: August 16, 2011)

We construct traversable wormholes in dilatonic Einstein-Gauss-Bonnet theory in four spacetime
dimensions, without needing any form of exotic matter. We determine their domain of existence,
and show that these wormholes satisfy a generalised Smarr relation. We demonstrate linear stability
with respect to radial perturbations for a subset of these wormholes.
PACS numbers: 04.70.-s, 04.70.Bw, 04.50.-h
Introduction.– When the first wormhole, the
‘Einstein-Rosen bridge’, was discovered in 1935 [1] as
a feature of Schwarzschild geometry, it was considered
a mere mathematical curiosity of the theory. In the
50’s, Wheeler showed [2] that a wormhole can connect
not only two different universes but also two distant
regions of our own universe. However, the dream
of interstellar travel short-cuts was shattered by the
following findings: (i) the Schwarzschild wormhole
is dynamic - its ‘throat’ expands to a maximum radius
and then contracts again to zero circumference so
quickly that not even a particle moving at the speed of
light can pass through [3], (ii) the past horizon of the
Schwarzschild geometry is unstable against small perturbations
- the mere approching of a traveller would
change it to a proper, and thus impenetrable, one [4].
However, in 1988Morris and Thorne [5] found a new
class of wormhole solutions which possess no horizon,
and thus could be traversable. The throat of these
wormholes is kept open by a type of matter whose
energy-momentum tensor violates the energy conditions.
A phantom field, a scalar field with a reversed
sign in front of its kinetic term, was shown to be a
suitable candidate for the exotic type of matter necessary
to support traversable wormholes [6].

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  • 1. Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory Panagiota Kanti Division of Theoretical Physics, Department of Physics, University of Ioannina, Ioannina GR-45110, Greece Burkhard Kleihaus, Jutta Kunz Institut f¨r Physik, Universit¨t Oldenburg, D-26111 Oldenburg, Germany u a (Dated: August 16, 2011) We construct traversable wormholes in dilatonic Einstein-Gauss-Bonnet theory in four spacetime dimensions, without needing any form of exotic matter. We determine their domain of existence,arXiv:1108.3003v1 [gr-qc] 15 Aug 2011 and show that these wormholes satisfy a generalised Smarr relation. We demonstrate linear stability with respect to radial perturbations for a subset of these wormholes. PACS numbers: 04.70.-s, 04.70.Bw, 04.50.-h Introduction.– When the first wormhole, the term, the Gauss-Bonnet (GB) term, and a scalar field ‘Einstein-Rosen bridge’, was discovered in 1935 [1] as (the dilaton) coupling exponentially to the GB term, a feature of Schwarzschild geometry, it was considered so that the latter has a non-trivial contribution to the a mere mathematical curiosity of the theory. In the four-dimensional field equations. 50’s, Wheeler showed [2] that a wormhole can con- Here we investigate the existence of wormhole solu- nect not only two different universes but also two dis- tions in the context of the EGBd theory. No phantom tant regions of our own universe. However, the dream scalar fields or other exotic forms of matter are intro- of interstellar travel short-cuts was shattered by the duced. Instead, we rely solely on the existence of the following findings: (i) the Schwarzschild wormhole higher-curvature GB term that follows naturally from is dynamic - its ‘throat’ expands to a maximum ra- the compactification of the 10-dimensional heterotic dius and then contracts again to zero circumference so superstring theory down to four dimensions. quickly that not even a particle moving at the speed of EGBd Theory.– We consider the following ef- light can pass through [3], (ii) the past horizon of the fective action [9–12] motivated by the low-energy het- Schwarzschild geometry is unstable against small per- erotic string theory [7, 8] turbations - the mere approching of a traveller would change it to a proper, and thus impenetrable, one [4]. 1 √ 1 S= d4 x −g R − ∂µ φ ∂ µ φ + αe−γφ RGB ,(1) 2 However, in 1988 Morris and Thorne [5] found a new 16π 2 class of wormhole solutions which possess no horizon, and thus could be traversable. The throat of these where φ is the dilaton field with coupling constant wormholes is kept open by a type of matter whose γ, α is a numerical coefficient given in terms of the energy-momentum tensor violates the energy condi- Regge slope parameter, and RGB = Rµνρσ Rµνρσ − 2 µν 2 tions. A phantom field, a scalar field with a reversed 4Rµν R + R is the GB correction. sign in front of its kinetic term, was shown to be a Here we consider only static, spherically-symmetric suitable candidate for the exotic type of matter nec- solutions of the field equations. Hence we may write essary to support traversable wormholes [6]. the spacetime line-element in the form [9] In order to circumvent the use of exotic matter to obtain traversable wormholes, one is led to consider ds2 = −eΓ(r) dt2 + eΛ(r) dr2 + r2 dθ2 + sin2 θdϕ2 . generalized theories of gravity. Higher curvature the- (2) ories of gravity are suitable candidates to allow for the In [9] it was demonstrated that EGBd theory admits existence of stable traversable wormholes. In partic- static black hole solutions, based on this line-element. ular, the low-energy heterotic string effective theory But it was also observed that, besides the black hole [7, 8] has provided the framework for such a general- solutions, the theory admits other classes of solutions. ized gravitational theory in four dimensions where the One of the examples presented showed a pathological curvature term R of Einstein’s theory is supplemented behavior for the grr metric component and the dila- by the presence of additional fields as well as higher- ton field at a finite radius r = r0 but had no proper curvature gravitational terms. The dilatonic Einstein- horizon with gtt being regular over the whole radial Gauss-Bonnet (EGBd) theory offers a simple version regime. Since the solution did not exhibit any singu- that contains, in addition to R, a quadratic curvature lar behaviour of the curvature invariants at r0 , it was
  • 2. 2concluded that the pathological behaviour was due to Wormhole Properties.– For a wormhole solutionthe choice of the coordinate system. 2 the radius r = l2 + r0 has to be minimal at l = 0. Here we argue that this class of asymptotically flat To cast this condition in a coordinate independentsolutions is indeed regular and represents a class of way, we define the proper distance from the throat by l√ lwormholes with r0 being the radius of the throat. In- ξ = 0 gll dl′ = 0 f (l′ )dl′ . Then the conditionsdeed, the coordinate transformation r2 = l2 + r0 leads 2 2to a metric without any pathology, for a mininal radius dr d r dξ l=0 = 0, dξ 2 l=0 > 0 follow from the substitution of the expansion at the throat.ds2 = −e2ν(l) dt2 +f (l)dl2 +(l2 +r0 ) dθ2 + sin2 θdϕ2 . 2 In order to examine the geometry of the space man- (3) ifold, we consider the isometric embedding of a planeIn terms of the new coordinate, the expansion at the passing through the wormhole. Choosing the θ = π/2-throat l = 0, yields f (l) = f0 + f1 l + · · · , e2ν(l) = plane, we set f (l)dl2 +(l2 +r0 )dϕ2 = dz 2 +dη 2 +η 2 dϕ2 , 2e2ν0 (1+ν1 l)+· · · , φ(l) = φ0 +φ1 l+· · · , where fi , νi and where {z, η, ϕ} are a set of cylindrical coordinates inφi are constant coefficients. All curvature invariants, the three-dimensional Euclidean space R3 . Regardingincluding the GB term, remain finite for l → 0. z and η as functions of l, we find η(l) and z(l). We The expansion coefficients f0 , ν0 and φ0 are free pa- note that the curvature radius of the curve {η(l), z(l)}rameters, as well as the radius of the throat r0 and the at l = 0 is given by R0 = r0 f0 . From this equation wevalue of α – the value of the constant γ is set to 1 in obtain an independent meaning for the parameter f0the calculations. The set of equations remains invari- as the ratio of the curvature radius and the radius ofant under the simultaneous changes φ → φ + φ∗ and the throat, f0 = R0 /r0 .(r, l) → (r, l)e−φ∗ /2 . The same holds for the changes Essential for the existence of the wormhole so-α → kα and φ → φ + ln k. As a result, out of the lution is the violation of the null energy conditionparameter set (α, r0 , φ0 ) only one is independent: we Tµν nµ nν ≥0, for any null vector field nµ . For spher-thus fix the value of φ0 , in order to have a zero value of ically symmetric solutions, this condition can be ex-the dilaton field at infinity, and create a dimensionless pressed as −G0 + Gl ≥ 0 and −G0 + Gθ ≥ 0, where 0 l 2 0 θparameter α/r0 out of the remaining two. Also, since the Einstein equations have been employed. The nullonly the derivatives of the metric function ν appear energy condition is violated in some region if one ofin the equations of motion, we fix the value of ν0 to these conditions does not hold. By using the expan-ensure asymptotic flatness at radial infinity. sion of the fields near the throat, we find that there Of particular interest is the constraint on the valueof the first derivative of the dilaton field at the throat 2 −G0 + Gl 0 l l=0 =− 2 <0 , (5)which, in terms of the expansions translates into a f0 r0constraint on the value of the parameter φ1 , i.e. provided e2ν(0) = 0, i.e. in the absence of a horizon. f0 (f0 − 1) The wormhole solutions satisfy a Smarr-like mass φ2 1 = . (4) α formula 2αγ 2 e−γφ0 f0 − 2(f0 − 1) r2 e−γφ0 0 κ D 1 √ dφ ˜Since the left-hand-side of the above equation is M = 2Sth − + −gg ll 1 + 2αe−γφ R dΩ2positive-definite, we must impose the constraint f0 ≥ 2π 2γ 8πγ dl1. This constraint introduces a boundary in the phase where κ denotes the surface gravity at the throat andspace of the wormhole solutions. The expression in-side the square brackets in the denominator remains 1 √ Sth = ˜ −h 1 + 2αe−γφ R dΩ2 .positive and has no roots if a/r0 < eφ0 /2 - this con- 2 4straint is automatically satisfied for the set of solutionspresented. ˜ Here hµν is the induced metric on the throat, R the For l → ∞ we demand asymptotic flatness for the scalar curvature of h, and the integral is evaluated attwo metric functions and a vanishing dilaton field. l = 0. Thus the known EGBd mass formula for blackThen, the corresponding asymptotic expansion yields holes [12] is augmented by a contribution which mayν → − M + · · · , f → 1 + 2M + · · · , φ → − D + · · · l l l be interpreted as a modified throat dilaton charge,where M and D are identified with the mass and dila- where the GB modification is of the same type as theton charge of the wormhole, respectively. Unlike the GB modification of the area (or entropy, in case ofcase of the black hole solutions [9], the parameters M black holes).and D characterizing the wormholes at radial infinity Numerical Results.– For the numerical calcula-are not related, in agreement with the classification of tions we use the line element (3). At l = 0 regularitythis group of solutions as two-parameter solutions. requires relation (4) to hold with f (0) = f0 or φ(0) =
  • 3. 3 2 indicated by asterisks corresponds to the limit f0 = 1 and coincides with the black hole curve since −g00 (r0 ) 1 f(l) tends to zero in this limit. The boundary indicated by crosses corresponds to the limit f0 → ∞. In this 0 φ(l) limit, the mass M and the dilaton charge D assume -1 finite values. The same holds for the redshift func- tion, the dilaton field and all curvature invariants at -2 2 α/r0=0.02, f0=1.1 the throat. The third boundary, indicated by dots, -3 2 α/r0=0.02, f0=1.5 is characterized by curvature singularities; it emerges 2 α/r0=0.02, f0=2.0 2 ν(l) when branches of solutions with fixed α/r0 terminate -4 at singular configurations with the derivatives of the 0.001 0.01 0.1 l 1 10 100 functions developing a discontinuity at some point lcrit outside the throat. Stability.– A crucial requirement for traversableFIG. 1: The metric and dilaton functions for f0 = 1.1 wormhole solutions is their stability. We therefore 2(red), 1.5 (green), 2.0 (blue), and α/r0 = 0.02, versus l. assess the stability of the EGBd wormhole solutions with respect to radial perturbations. We allow the metric and dilaton functions to depend on both l andφ0 in order to obtain a unique solution. Note that t, and we decompose them into an unperturbed partf → 1 for l → ∞ is always satisfied. Thus the asymp- and the perturbations: ν (l, t) = ν(l) + λδν(l)eiσt , ˜totic boundary conditions read ν → 0 and φ → 0. ˜ ˜ f (l, t) = f (l) + λδf (l)eiσt , φ(l, t) = φ(l) + λδφ(l)eiσt , The field equations, reducing to a system of ODE’s, where λ is considered as small. Substituting the abovewere solved for 1.0001 ≤ f0 ≤ 20.0 fixing γ = 1. in the (time-dependent) Einstein and dilaton equa- 2Wormhole solutions were found for every value of α/r0 tions and linearizing in λ, we obtain a system of linearconsidered up to 0.13 - this translates into a lower ODEs for the functions δν(l), δf (l) and δφ(l).bound on the radius of the throat r0 that can be arbi- Rearranging appropriately the system of ODEs, wetrarily large. In Fig. 1, we show the metric and dilaton derive a decoupled second-order equation for δφ,functions for an indicative set of wormhole solutions.In Fig. 2, we present the isometric embedding of the (δφ)′′ + q1 (δφ)′ + (q0 + qσ σ 2 ) δφ = 0 , (6) 2solution with α/r0 = 0.02 and f0 = 1.1. 1.8 2 1.04 α/r0 0.02 1.6 1.02 0.05 0.06 1.0 0.08 -0.02 0.0 0.02 1.4 A/16πM2 0.09 0.10 r0 R0 0.12 1.2 0.128 2 σ =0 1 0.8 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D/MFIG. 2: Isometric embedding for a wormhole solution for 2α/r0 = 0.02 and f0 = 1.1. FIG. 3: The scaled area of the throat versus the scaled In Fig. 3 we exhibit the domain of existence of the 2 dilaton charge for several values of α/r0 . The domain ofwormhole solutions. Here we show the scaled area of existence is bounded by the black hole solutions (aster-the throat A/16πM 2 versus the scaled dilaton charge isks), the f0 → ∞ solutions (crosses), and curvature sin- 2 gularity solutions (dots). The shaded areas indicate linearD/M for several values of α/r0 . We observe that thedomain of existence is bounded by three curves indi- stability (lilac), instability (red), undecided yet (white)cated by dots, crosses and asterisks. The boundary w.r.t. radial perturbations.
  • 4. 4 0 2 Junction Conditions.– When we extend the α/r0 -0.001 0.020 wormhole solutions to the second asymptotically flat -0.002 0.050 part of the manifold (l → −∞) in a symmetric way, -0.003 0.060 jumps appear in the derivatives of the metric and dila- 0.080 -0.004 0.090 ton functions at l = 0. These can be attributed to the presence of matter at the throat of the wormhole. 2 -0.005 σ √ -0.006 ˜ Introducing the action (λ1 + λ0 2αe−φ0 R) −h d3 x, -0.007 ˜ 2 where R = 2/r0 , the junction conditions take the form -0.008 -0.009 4αe−φ0 8αe−φ0 φ′ -0.01 ρ − λ0 2 − λ1 = 2 √0 , (8) 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 r0 r0 f0 f0 ′ 2ν0 p + λ1 = √ , (9) f0 4αe−φ0 ρdil 8αe−φ0 ′ 1FIG. 4: The eigenvalue σ 2 versus f0 for several values of λ0 + = φ′ + 0 ν0 √ .(10) 2 r0 2 2 r0 2α/r0 . f0 We have also assumed that the matter at the throatwhere q1 , q0 and qσ depend on the unperturbed so- takes the form of a perfect fluid and a dilaton chargelution. All coefficients take up constant values as density ρdil . A simple numerical analysis shows thatl → ∞ and, in order to ensure normalizability, δφ the junction conditions can be easily satisfied for nor-is demanded to vanish in that limit. However, while mal matter with positive energy density and pressure,qσ is bounded at l = 0, q1 and q0 diverge as 1/l. To for appropriately chosen constants λ1 and λ0 . Theavoid this singularity, we consider the transformation presence of ρdil ensures that the stability considera-δφ = F (l)ψ(l), where F (l) satisfies F ′ /F = −q1 (l)/2. tions are not affected by the matter on the throat.This yields Conclusions.– Traversable wormholes do not ex- ist in General Relativity, unless some exotic matter ψ ′′ + Q0 ψ ′ + σ 2 qσ ψ = 0 (7) is introduced. However, if string theory corrections are taken into account the situation changes dramat- ′ 2 ically. Here we investigated wormhole solutions inwhere Q0 = −q1 /2 − q1 /4 + q0 is bounded at l = 0.Then, for regular, normalizable solutions, ψ has to EGBd theory without introducing exotic matter. Wevanish both asymptotically and at l = 0 . determined the domain of existence and showed lin- We solved the ODEs together with the normaliza- ear stability with respect to radial perturbations for 2 a subset of solutions. Since the radius of the throattion constraint for several values of α/r0 and f0 . Asolution exists only for certain values of the eigen- is bounded from below only, the wormholes can bevalue σ 2 . In Fig. 4, we exhibit the negative modes arbitrarily large. Astrophysical consequences will beof families of wormhole solutions, which are thus un- addressed in a forthcoming paper as well as the exis-stable. However, there is also a region in parameter tence of stationary rotating wormhole solutions in thespace, where no negative mode exists, i.e. a region EGBd theory.where the EGBd wormhole solutions are linearly sta- Acknowledgements.– We gratefully acknowl-ble w.r.t. radial perturbations. The different solutions edge disscussions with Eugen Radu. B.K. acknowl-are also marked in the domain of existence (Fig. 3). edges support by the DFG. [1] A. Einstein and N. Rosen, Phys. Rev. 48 (1935) 73. 2736. [2] J.A. Wheeler, Geometrodynamics (Academic, New [5] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 York, 1962). (1988). [3] M. D. Kruskal, Phys. Rev. 119, 1743 (1960); [6] H. G. Ellis, J. Math. Phys. 14, 104 (1973); R. W. Fuller and J. A. Wheeler, Phys. Rev. 128, 919 K. A. Bronnikov, Acta Phys. Polon. B 4, 251 (1962). (1973); T. Kodama, Phys. Rev. D 18, 3529 (1978); [4] I.H. Redmount, Prog. Theor. Phys. 73 (1985) 1401; C. Armendariz-Picon, Phys. Rev. D 65, 104010 D.M. Eardley, Phys. Rev. Lett. 33 (1974) 442; R. (2002). Wald and S. Ramaswamy, Phys. Rev. D21 (1980) [7] D. J. Gross, J. H. Sloan, Nucl. Phys. B291 (1987)
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