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2015 J. Phys.: Conf. Ser. 600 012038(http://iopscience.iop.org/1742-6596/600/1/012038)
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Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution
IOP Publishing
Journal of Physics: Conference Series 600 (2015) 012038
doi:10.1088/1742-6596/600/1/012038

Wormhole in higher-dimensional space-time
Hisa-aki Shinkai1,2 , Takashi Torii 3
1

Faculty of Information Science and Technology, Osaka Institute of Technology, Hirakata,Osaka 573-0196, Japan
2
Computational Astrophysics Laboratory, Institute of Physical & Chemical Research(RIKEN), Hirosawa, Wako, Saitama, 351-0198 Japan
3
Faculty of Engineering, Osaka Institute of Technology, Osaka, Osaka 535-8585, Japan
E-mail: hisaaki.shinkai@oit.ac.jp
E-mail: takashi.torii@oit.ac.jp
Abstract. We introduce our recent studies on wormhole, especially its stability aspect inhigher-dimensional space-time both in general relativity and in Gauss-Bonnet gravity. We
derived the Ellis-type wormhole solution in n-dimensional general relativity, and found existenceof an unstable mode in its linear perturbation analysis. We also evolved it numerically in dualnull coordinate system, and confirmed its instability. The wormhole throat will change into
black hole horizons for the input of the (relatively) positive energy, while it will change into
inflationary expansion for the (relatively) negative energy input. If we add Gauss-Bonnet terms(higher curvature correction terms in gravity), then wormhole tends to expand (or change toblack hole) if the coupling constant α is positive (negative), and such bifurcation of the throat
horizon is observed earlier in higher dimension.

1. Introduction
Wormhole is a popular tool for interstellar travel in science fiction. It connects two diﬀerent
space-time points directly, which does not contradict with the Einstein’s field theory. Althoughsuch an astrophysical object is not yet observed, studying wormhole structure gives us many
chances to understand unknown aspects of the gravitational field.
The most famous and the common starting model of wormhole is, we think, the one announced
by Morris and Thorne as a “traversable wormhole”[1]. They considered “traversable conditions”
for human travel through wormholes responding to Carl Sagan’s idea for his novel Contact, andconcluded that such a wormhole solution is available if we allow “exotic matter” (negative-energymatter).
The introduction of exotic matter sounds to be unusual for the first time, but such matter
appears in quantum field theory and in alternative gravitational theories such as scalar-tensor
theories. The Morris-Thorne solution is constructed with a massless Klein-Gordon field whose
gravitational coupling takes the opposite sign to normal, which is found in Ellis’s earlier work[2], so that we call it Ellis wormhole, hereafter. (See a review e.g. by Visser [3] for earlier works;
See also e.g. Lobo [4] for recent works).
Ellis wormhole solution was studied in many contexts. Among them, we focus on its
dynamical features in higher-dimensional space-time.
The first numerical simulation on Ellis wormhole in 4-dimensional space-time was reported
by one of the authors [5]. They use a dual-null formulation for spherically symmetric space-timeContent from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd1Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution
IOP Publishing
Journal of Physics: Conference Series 600 (2015) 012038
doi:10.1088/1742-6596/600/1/012038

integration, and observed that the wormhole is unstable against Gaussian pulses in either exotic
or normal massless Klein-Gordon fields. The wormhole throat suﬀers a bifurcation of horizons
and either explodes to form an inflationary universe or collapses to a black hole, if the total input
energy is negative or positive, respectively. These basic behaviors were repeatedly confirmed byother groups [6, 7].
Black HoleorInflationary
expansion

x minusthroat"throat"Figure 1. Partial Penrose diagram of the evolved
space-time. Suppose we live in the right-side
region and input a pulse to an Ellis-wormhole inthe middle of the diagram. The wormhole throat
suﬀers a bifurcation of horizons and either explodes
to form an inflationary universe or collapses to a
black hole, if the total input energy is negative or
positive, respectively [5].

pulseinput

x plus

The changes of wormhole either to a black hole or an expanding throat support a unified
understanding of black holes and traversable wormholes proposed by Hayward [8]. His
proposal is that the two are dynamically interconvertible, and that traversable wormholes areunderstandable as black holes under the existence of negative energy density.
In this article, we introduce our recent studies on the stability in higher-dimensional spacetime. The higher-dimensional theories such as string/M theories are applied for various unsolved
problems in gravitational phenomena and cosmology, and gain new insights into them. We
believe that wormholes will also give us new fundamental physical landscapes. We therefore
demonstrate wormhole dynamics also in Gauss-Bonnet (GB) gravity, which is one of the modified
gravity theory including higher-order corrections of curvatures, one of the string-motivatedgravity theories.
The main four contents in this article are: (a) constructing Ellis solutions in higherdimensional general relativity, (b) stability analysis using a linear perturbation method in ndimensional general relativity (GR) [9], (c) stability analysis using a numerical evolution method
in 5, 6, 7-dimensional GR, and (d) the similar numerical analysis with the GB coupling.
2. Wormhole solution and its linear perturbation in higher-dimensional GR
2.1. Field equations
We start from the n-dimensional Einstein-Klein-Gordon system∫√d x −gn

S=M

[]

11
R − ϵ(∇ϕ)2 − V (ϕ) ,22κn2(1)where κ2n is the n-dimensional gravitational constant. The scalar field ϕ is the normal (or ghost)field if ϵ = 1 (−1).
The metric of the space-time is assumed to be
ds2 = −f (t, r)e−2δ(t,r) dt2 + f (t, r)−1 dr2 + R(t, r)2 hij dxi dxj ,(2)where hij dxi dxj represents the line element of a unit (n − 2)-dimensional constant curvature
space with curvature k = ±1, 0 and volume Σk . In order to construct a static wormhole solution,
we restrict the metric function as f = f (r), R = R(r), ϕ = ϕ(r), and δ = 0.
2

Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution
IOP Publishing
Journal of Physics: Conference Series 600 (2015) 012038
doi:10.1088/1742-6596/600/1/012038

Figure 2. The n-dimensional wormhole solutions; (a) The circumference radius R and (b) thescalar field ϕ are plotted as a function of the radial coordinate r.
For massless ghost scalar field (V (ϕ) = 0 and ϵ = −1), the Klein-Gordon equation is
integrated as ϕ′ = C/(f Rn−2 ), where C is an integration constant, and ′ = d/dr. By locating
the throat of the wormhole at r = 0, and imposing the reflection symmetry at the throat, thesolution of the field equations is obtained only for the spherically symmetric case k = 1, as√f ≡ 1,′

R =1−( a )2(n−3)

R√,

ϕ=(n − 2)(n − 3) n−3aκn∫1dr,
R(r)n−2(3)where a is the radius of the throat, i.e. R(0) = a.
The throat of the wormhole has large curvature and the scalar field ϕ becomes steeper as ngoes higher. We plotted these behaviors in Figure 2. For n → ∞, the functions have the limiting
solution, R = r + a and ϕ = π/2 (r > 0).
2.2. Existence of unstable mode in linear perturbation analysis
We study linear stability of the solution obtained above. We focus on the “spherical” modes,
where the (n − 2)-dimensional constant curvature space is not perturbed. For metric variablesf, δ, R and for scalar field ϕ, we assume the perturbed functions in the form of x(t, r) =x0 (r) + x1 (r)eiωt , where x0 denotes the static solution above. Following [7], we obtain themaster equation for gauge-invariant variable. After some transformations of variable, we reachthe single Schrödinger-type master equation, which potential does not diverge at the throat,−Ψ′′1 + W (r)Ψ1 = ω 2 Ψ1 ,whereW (r) = −]1 [ 3(n − 2)2−(n−4)(n−6).
4R02 R2(n−3)(4)

0

We search eigenfunctions Ψ1 (r) of eq. (4), and find that in any dimension n there exists one
negative eigenvalue for ω 2 , which implies that the solution is unstable. We find large negative
ω 2 for higher n, which indicates the timescale of instability becomes shorter [9].
3. Numerical evolutions of wormhole solutions
3.1. Dual-null scheme
We implemented our evolution code [5] for higher-dimensional space-time, and with GB gravity
terms. The system we consider is spherical symmetry, and expressed using dual-null coordinate.

3

Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution
IOP Publishing
Journal of Physics: Conference Series 600 (2015) 012038
doi:10.1088/1742-6596/600/1/012038

The use of dual-null coordinate simplifies the treatment of horizon dynamics and radiation
propagation clearly.
The Einstein-Gauss-Bonnet (EGB) action in n-dimensional space-time is described as∫√dn X −g[]1{αGR (R − 2Λ) + αGB LGB } + Lmatter ,2κ2M
where LGB = R2 − 4Rµν Rµν + Rµνρσ Rµνρσ .

S=(5)
(6)

The additional term LGB forms ghost-free combinations [10] and does not give higher derivativeequations but an ordinary set of equations with up to the second derivative in spite of the higher
curvature combinations.
We adopt the line element−ds2 = −2ef (x ,x ) dx+ dx− + r2 (x+ , x− )hij dz i dz j ,+(7)where the coordinate (x+ , x− ) are along to null propagation directions.
For writing down the Einstein equations, we introduce the variables
1Ω= ,r

ϑ± = (n − 2)∂± r,

ν± = ∂± f,p± = ∂± ϕ,(8)which are conformal factor, expansions, in-aﬃnities, and scalar momentum, respectively.

Evolution

x+x−

Σ0

Figure 3. Numerical grid structure. Initial data
are given on null hypersurfaces Σ± (x∓ = 0, x± >0) and their intersection S.

We prepare our numerical integration range as drawn in Figure 3. The grid will cover bothuniverses connected by the wormhole throat x+ = x− . We give initial data on a surface S andthe two null hypersurfaces Σ± generated from it. Generally the initial data have to be given as(Ω, f, ϑ± , ϕ) on S: x+ = x− = 0(ν± , p± ) on Σ± : x∓ = 0, x± > 0.

(9)
(10)

We then evolve the data u = (Ω, ϑ± , f, ν± , ϕ, p± ) on a constant-x− slice to the next.
As a virtue of the dual-null scheme, we can follow the wormhole throat or black hole horizons
easily. They are both trapping horizons, hypersurfaces where ϑ+ = 0 or ϑ− = 0. In order toevaluate the energy, we apply the Misner-Sharp mass in n-dimensional EGB gravity [11].
3.2. Evolutions of 4, 5, and 6-dimensional wormhole solutions in GR
We checked our numerical code whether it reproduces the static wormhole solution obtainedin §2. We next add a perturbation to the scalar momentum on the static wormhole solution
in the form of Gaussian pulses from the right-hand universe, δp+ = c1 exp(−c2 (l − c3 )2 ),where4

Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolutionIOP Publishing
Journal of Physics: Conference Series 600 (2015) 012038
doi:10.1088/1742-6596/600/1/012038

Figure 4. Location of the expansion ϑ+ (red lines) and ϑ− (blue lines) for evolutions of asolution in 4, 5, and 6-dimensional General Relativity (αGB = 0) as a function of (x+ , x− ). The
throat begins expanding if we input negative energy scalar flux (left panel), while the throat
turns to be a black hole if we input positive energy scalar flux (right panel).
c1 , c2 , c3 are parameters. In Figure 4, we plotted the cases with small amplitude c1 = ±0.01 andwidth c2 = 3, and the initial location c3 = 1. That is, the pulse will hit the wormhole throat at
x+ = x− = 1. Positive (or negative) c1 corresponds to enhancing (or reducing) the supporting
ghost field.
Figure 4 shows the results of n = 4, 5, and 6 dimensional wormhole solution with above
perturbation. The plot shows locations of vanishing expansions, ϑ± = 0, in (x+ , x− ) plane. We
see first the wormhole throat locates where ϑ± = 0, but after a small pulse hit it, then the throat(or horizon) split into two (ϑ+ = 0 and ϑ− = 0), depending on the signature of the energy ofpulse.
If the location of ϑ+ is outer (in x+ -direction) than that of ϑ− , then the region ϑ− < x < ϑ+is judged as a black hole. Otherwise the region ϑ+ < x < ϑ− can be judged as an expanding
throat. The throat begins expanding if we input negative energy scalar flux (left panel in Figure4), while the throat turns to be a black hole if we input positive energy scalar flux (right panel).
3.3. Evolutions of wormhole solution in EGB gravity
We also construct the initial data with GB terms αGB ̸= 0, and study their eﬀects to theevolutions. We confirm 4-dimensional EGB equation reproduces the equivalent results withthose of GR.
Figure 5 shows the case of 5 and 6-dimensional EGB gravity. The initial data of wormhole on
Σ± = 0 are obtained numerically, solving the set of equations. The evolutions with αGB ̸= 0 arequite unstable, and we are hard to keep the static configurations long enough. We see if αGB < 0
the throat turns to be a black hole (left panel in Figure 5). On the contrary, if αGB > 0, then
the throat begins to expand (right two panels).
4. Conclusions and Discussions
We studied the simplest wormhole solutions and their stability in higher-dimensional space-time
both in GR and EGB gravity. The space-time is assumed to be static and spherically symmetric,
has ghost scalar field, and has reflection symmetry at the throat.
5

Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution
IOP PublishingJournal of Physics: Conference Series 600 (2015) 012038
doi:10.1088/1742-6596/600/1/012038

Figure 5. The same with Figure 4, but for Einstein-Gauss-Bonnet gravity. When αGB < 0(left panel), the throat turns to be a black hole, while for αGB > 0 the throat begins expanding(middle/left panel).
Using the linear perturbation technique, we showed that the solutions have at least one
negative mode, which concludes that all wormholes are linearly unstable. The time scale ofinstability becomes shorter as n becomes large. This prediction is confirmed using numericalevolutions.
Numerical evolutions show that if we put a scalar pulse with small amplitude to the throat,then the wormhole throat bifurcates, its horizon structure changes into a black hole or an
expanding throat depending the pulse energy is positive or negative, respectively. We also
investigated the eﬀect of the higher-order curvature corrections using Gauss-Bonnet terms, andfound that such corrections do not work for stabilization of wormholes.
All the behavior of wormholes may be explained simply with energy balance. In [5], forsmall perturbations, an existence of critical solution is suggested. The similar behaviors are also
observed in our investigations, which will be reported elsewhere.
We guess a wormhole with exotic matter is a disguise to avoid public notice, and does prefer
to appear as a black hole or an expanding universe.
Acknowledgments
This work was supported in part by the Grant-in-Aid for Scientific Research Fund of the JSPS
(C) No. 25400277. Numerical computations were carried out on SR16000 at YITP in Kyoto
University, and on the RIKEN Integrated Cluster of Clusters (RICC).
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[2] H. G. Ellis, J. Math. Phys. 14, 395 (1973).
[3] M. Visser, Lorentzian Wormholes (AIP Press, 1995).
[4] F. S. N. Lobo, in Classical and Quantum Gravity Research (Nova Sci. Pub., 2008). [arXiv:0710.4474][5] H. Shinkai & S.A. Hayward, Phys. Rev. D 66, 044005 (2002).
[6] A. Doroshkevich, J. Hansen, I. Novikov, & A. Shatskiy, Int. J. Mod. Phys. D 18, 1665 (2009)[7] J. A. Gonzalez, F. S. Guzman & O. Sarbach, Class. Quant. Grav. 26, 015010, 015011 (2009); Phys. Rev. D80, 024023 (2009). O. Sarbach & T. Zannias, Phys. Rev. D 81, 047502 (2010).
[8] S A Hayward, Int. J. Mod. Phys. D 8, 373 (1999).
[9] T. Torii & H. Shinkai, Phys. Rev. D 88, 064027 (2013).
[10] B. Zwiebach, Phys. Lett. B 156, 315 (1985).
[11] H. Maeda & M. Nozawa, Phys. Rev. D 78, 024005 (2008).

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