Electromagnetic wormholes via handlebody constructions
Electromagnetic wormholes via
handlebody constructions
Allan Greenleaf, Yaroslav Kurylev,
Matti Lassas and Gunther Uhlmann ∗
Abstract
Cloaking devices are prescriptions of electrostatic, optical or elec-
tromagnetic parameter fields (conductivity σ(x), index of refraction
n(x), or electric permittivity ǫ(x) and magnetic permeability µ(x))
which are piecewise smooth on R3 and singular on a hypersurface Σ,
and such that objects in the region enclosed by Σ are not detectable to
external observation by waves. Here, we give related constructions of
invisible tunnels, which allow electromagnetic waves to pass between
possibly distant points, but with only the ends of the tunnels visible
to electromagnetic imaging. Effectively, these change the topology of
space with respect to solutions of Maxwell’s equations, corresponding
to attaching a handlebody to R3. The resulting devices thus function
as electromagnetic wormholes.
∗AG and GU are supported by US NSF, ML by CoE-programm 213476 of Academy of
Finland.
http://arxiv.org/abs/0704.0914v1
1 Introduction
There has recently been considerable interest, both theoretical [16, 18, 19, 3,
13] and experimental [20], in invisibility (or “cloaking”) from observation by
electromagnetic (EM) waves. (See also [17] for a treatment of cloaking in the
context of elasticity.) Theoretically, cloaking devices are given by specifying
the conductivity σ(x) (in the case of electrostatics), the index of refraction
n(x) (for optics in the absence of polarization, where one uses the Helmholtz
equation), or the electric permittivity ǫ(x) and magnetic permeability µ(x)
(for the full system of Maxwell’s equations.) In the constructions to date,
the EM parameter fields ( σ;n; ǫ and µ ) have been piecewise smooth and
anisotropic. (See, however, [5, Sec.4] for an example that can be interpreted
as cloaking with respect to Helmholtz by an isotropic negative index of re-
fraction material.) Furthermore, the EM parameters have singularities, with
one or more eigenvalues of the tensors going to zero or infinity as one ap-
proaches from on or both sides the cloaking surface Σ, which encloses the
region within which objects may be hidden from external observation. Such
constructions might have remained theoretical curiosities, but the advent of
metamaterials[1] allows one, within the constraints of current technology, to
construct media with fairly arbitrary ǫ(x) and µ(x).
It thus becomes an interesting mathematical problem with practical signif-
icance to understand what other new phenomena of wave propagation can
be produced by prescribing other arrangements of ǫ and µ. Geometrically,
cloaking can be viewed as arising from a singular transformation of R3. In-
tuitively, for a spherical cloak [6, 7, 18], it is as if an infinitesimally small
hole in space has been stretched to a ball D; an object can be inserted in-
side the hole so created and is then invisible to external observations. On
the level of the EM parameters, homogeneous, isotropic parameters ǫ, µ are
pushed forward to become inhomogeneous, anisotropic and singular as one
approaches Σ = ∂D from the exterior. There are then two ways, referred to
as the single and double coating in [3], of continuing ǫ, µ to within D so as to
rigorously obtain invisibility with respect to locally finite energy waves. We
refer to either process as blowing up a point. As observed in [3], one can use
the double coating to produce a manifold with a different topology, but with
the change in topology invisible to external measurements.
To define the solutions of Maxwell’s equations rigorously in the single coating
case, one has to add boundary conditions on Σ. Physically, this corresponds
to the lining of the interior of the single coating material, e.g., in the case
of blowing up a point, with a perfectly conducting layer, see [3]. We point
out here that in the recent preprint [21], the single coating construction is
supplemented with selfadjoint extensions of Maxwell operators in the interior
of the cloaked regions; these implicitly impose interior boundary conditions
on the boundary of the cloaked region, similar to the PEC boundary condi-
tion suggested in [3]. For the case of an infinite cylinder the Soft-and-Hard
(SH) interior boundary condition is used in [3] to guarantee cloaking of active
objects, and is needed even for passive ones.
In this paper, we show how more elaborate geometric constructions, cor-
responding to blowing up a curve, enable the description of tunnels which
allow the passage of waves between distant points, while only the ends of
the tunnels are visible to external observation. These devices function as
electromagnetic wormholes, essentially changing the topology of space with
respect to solutions of Maxwell’s equations.
We form the wormhole device around an obstacle K ⊂ R3 as follows. First,
one surrounds K with metamaterials, corresponding to a specification of EM
parameters ε̃ and µ̃. Secondly, one lines the surface of K with material im-
plementing the Soft-and-Hard (SH) boundary condition from antenna theory
[8, 10, 11]; this condition arose previously [3] in the context of cloaking an
infinite cylinder. The EM parameters, which become singular as one ap-
proaches K, are given as the pushforwards of nonsingular parameters ε and
µ on an abstract three-manifold M , described in Sec. 2. For a curve γ ⊂ M ,
we construct the diffeomorphism F fromM \γ to the wormhole device in Sec.
3. For the resulting EM parameters ε̃ and µ̃, we have singular coefficients of
Maxwell’s equations at K, and so it is necessary to formulate an appropriate
notion of locally finite energy solutions (see Def. 4.1). In Theorem 4.2, we
then show that there is a perfect correspondence between the external mea-
surements of EM waves propagating through the wormhole device and those
propagating on the wormhole manifold.
It was shown in [3] that the cloaking constructions are mathematically valid
at all frequencies k. However, both cloaking and the wormhole effect stud-
ied here should be considered as essentially monochromatic, or at least very
narrow-band, using current technology, since, from a practical point of view
the metamaterials needed to implement the constructions have to be fabri-
cated and assembled with a particular wavelength in mind, and theoretically
are subject to significant dispersion [18]. Thus, as for cloaking in [16, 18, 3],
here we describe the wormhole construction relative to electromagnetic waves
at a fixed positive frequency k. We point out that the metamaterials used in
the experimental verification of cloaking [20] should be readily adaptable to
yield a physical implementation, at microwave frequencies, of the wormhole
device described here. See Remark 1 in Sec. 4.2 for further discussion.
The results proved here were announced in [4].
2 The wormhole manifold M
First we explain, somewhat informally, what we mean by a wormhole. The
concept of a wormhole is familiar from general relativity [9, 22], but here
we define a wormhole as an object obtained by gluing together pieces of
Euclidian space equipped with certain anisotropic EM parameter fields. We
start by describing this process heuristically; later, we explain more precisely
how this can be effectively realized vis-a-vis EM wave propagation using
metamaterials.
We first describe the wormhole as an abstract manifold M , see Fig. 1; in
the next section we will show how to realize this concretely in R3, as a
wormhole device N . Start by making two holes in the Euclidian space R3 =
{(x, y, z)|x, y, z ∈ R}, say by removing the open ball B1 = B(O , 1) with
center at the origin O and of radius 1, and also the open ball B2 = B(P, 1),
where P = (0, 0, L) is a point on the z-axis having the distance L > 3 to
the origin. We denote by M1 the region so obtained, M1 = R
3 \ (B1 ∪ B2),
which is the first component we need to construct a wormhole. Note that
M1 is a 3-dimensional manifold with boundary, the boundary of M1 being
∂M1 = ∂B1∪∂B2, the union of two 2-spheres. Thus, ∂M1 can be considered
as a disjoint union S2 ∪ S2, where we will use S2 to denote various copies of
the two-dimensional unit sphere.
The second component needed is a 3−dimensional cylinder, M2 = S2× [0, 1].
This cylinder can be constructed by taking the closed unit cube [0, 1]3 in R3
and, for each value of 0 < s < 1, gluing together, i.e., identifying, all of the
points on the boundary of the cube with z = s. Note that we do not identify
points at the top of the boundary, at z = 1, or at the bottom, at z = 0. We
then glue together the boundary ∂B(O , 1) ∼ S2 of the ball B(O , 1) with the
lower end (boundary component) S2×{0} of M2, and the boundary ∂B(P, 1)
with the upper end, S2 × {1}. In doing so we identify the point (0, 0, 1) ∈
∂B(O , 1) with the point NP × {0} and the point (0, 0, L − 1) ∈ ∂B(P, 1)
with the point NP × {1}, where NP is the north pole on S2.
The resulting manifold M no longer lies in R3, but rather is the connected
sum of the components M1 and M2 and has the topology of R
3 with a
3−dimensional handle attached. Note that adding this handle makes it pos-
sible to travel from one point in M1 to another point in M1, not only along
curves lying in M1 but also those in M2.
To consider Maxwell’s equations on M , let us start with Maxwell’s equations
on R3 at frequency k ∈ R, given by
∇× E = ikB, ∇×H = −ikD, D(x) = ε(x)E(x), B(x) = µ(x)H(x).
Here E and H are the electric and magnetic fields, D and B are the electric
displacement field and the magnetic flux density, ε and µ are matrices corre-
sponding to permittivity and permeability. As the wormhole is topologically
different from the Euclidian space R3, we use a formulation of Maxwell’s
equations on a manifold, and as in [3], do this in the setting of a general Rie-
mannian manifold, (M, g). For our purposes, as in [14, 3] it suffices to use
ε, µ which are conformal, i.e., proportional by scalar fields, to the metric g.
In this case, Maxwell’s equations can be written, in the coordinate invariant
form, as
dE = ikB, dH = −ikD, D = ǫE, B = µH in M,
where E,H are 1-forms, D,B are 2-forms, d is the exterior derivative, and ǫ
and µ are scalar functions times the Hodge operator of (M, g), which maps
1-forms to the corresponding 2-forms [2]. In local coordinates these equations
are written in the same form as Maxwell’s equations in Euclidian space with
matrix valued ε and µ. Although not necessary, for simplicity one can choose
a metric on the wormhole manifold M which is Euclidian on M1, and on M2
is the product of a given metric g0 on S
2 and the standard metric of [0, 1].
More generally, can also choose the metric on M2 to be a warped product.
Even the simple choice of the product of the standard metric of S2 and the
metric δ2ds2, where δ is the “length” of the wormhole, gives rise to interesting
ray-tracing effects for rays passing through the wormhole tunnel. For δ << 1,
the image through one end of the wormhole (of the region beyond the other
end) would resemble the image in a a fisheye lens; for δ & 1, multiple images
and greater distortion occur. (See [4, Fig.2].)
The proof of the wormhole effect that we actually give is for yet another
variation, where the balls that form the ends have their boundary spheres
flattened; this may be useful for applications, since it allows for there to be a
vacuum (or air) in a neighborhood of the axis of the wormhole, so that, e.g.,
instruments may be passed through the wormhole. We next show how to
construct, using metamaterials, a device N in R3 that effectively realizes the
geometry and topology of M , relative to solutions of Maxwell’s equations at
frequency k, and hence functions as an electromagnetic wormhole.
3 The wormhole device N in R3
We now explain how to construct a “device” N in R3, i.e., a specification
of permittivity ε and permeability µ, which affects the propagation of elec-
tromagnetic waves in the same way as the presence of the handle M2 in the
wormhole manifold M . What this means is that we prescribe a configuration
of metamaterials which make the waves behave as if there were an invisible
tube attached to R3, analogous to the handle M2 in the wormhole manifold
M . In the other words, as far as external EM observations of the wormhole
device are concerned, it appears as if the topology of space has been changed.
We use cylindrical coordinates (θ, r, z) corresponding to a point (r cos θ, r sin θ, z)
in R3. The wormhole device is built around an obstacle K ⊂ R3. To de-
fine K, let S be the two-dimensional finite cylinder {θ ∈ [0, 2π], r = 2, 0 ≤
z ≤ L} ⊂ R3. The open region K consists of all points in R3 that have
distance less than one to S and has the shape of a long, thick-walled tube
with smoothed corners.
Let us first introduce a deformation map F from M to N = R3 \K or, more
precisely, from M \γ to N \Σ, where γ is a closed curve in M to be described
shortly and Σ = ∂K. We will define F separately on M1 and M2 denoting
the corresponding parts by F1 and F2.
To describe F1, let γ1 be the line segment on the z−axis connecting ∂B(O , 1)
and ∂B(P, 1) in M1, namely, γ1 = {r = 0, z ∈ [1, L − 1]}. Let F1(r, z) =
(θ, R(r, z), Z(r, z)) be such that (R(r, z), Z(r, z)), shown in Fig. 2,
PSfrag replacements
Figure 1: Schematic figure: a wormhole manifold is glued from two com-
ponents, the “handle” and space with two holes. Note that in the actual
construction, the components are three dimensional.
PSfrag replacements
A B C D A
Figure 2: The map (R(r, z), Z(r, z)) in cylindrical coordinates (z, r).
transforms in the (r, z) coordinates the semicircles AB and CD in the left
picture to the vertical line segments A′B′ = {r ∈ [0, 1], z = 0} and C ′D′ =
{r ∈ [0, 1], z = L} in the right picture and the cut γ1 on the left picture to the
curve B′C ′ on the right picture. This gives us a map F1 : M1 \ γ1 → N1 \Σ,
where the closed region N1 in R
3 is obtained by rotation of the region exterior
to the curve A′B′C ′D′ around the z−axis. We can choose F1 so that it is
the identity map in the domain U = R3 \ {−2 ≤ z ≤ L+ 2, 0 ≤ r ≤ 4}.
To describe F2, consider the line segment, γ2 = {NP} × [0, 1] on M2 . The
sphere without the north pole can be ”flattened” and stretched to an open
disc with radius one which, together with stretching [0, 1] to [0, L], gives us a
map F2 from M2 \γ2 to N2\Σ. The region N2 is the 3−dimensional cylinder,
N2 = {θ ∈ [0, 2π], r ∈ [0, 1], z ∈ [0, L]}. When flattening S2 \ NP , we do
it in such a way that F1 on ∂B(O , 1) and ∂B(P, 1) coincides with F2 on
(S2 \NP )× {0} and (S2 \NP )× {1}, respectively.
Thus, F maps M \ γ, where γ = γ1 ∪ γ2 is a closed curve in M , onto N \ Σ;
in addition, F is the identity on the region U .
Now we are ready to define the electromagnetic material parameter tensors
on N . We define the permittivity to be
ε̃ = F∗ε(y) =
(DF )(x)· ε(x)· (DF (x))t
det(DF )
x=F−1(y)
where DF is the derivative matrix of F , and similarly the permeability to
be µ̃ = F∗µ. These deformation rules are based on the fact that permittivity
and permeability are conductivity type tensors, see [14].
Maxwell’s equations are invariant under smooth changes of coordinates. This
means that, by the chain rule, any solution to Maxwell’s equations in M \ γ,
endowed with material parameters ε, µ becomes, after transformation by F ,
a solution to Maxwell’s equations in N \Σ with material parameters ε̃ and µ̃,
and vice versa. However, when considering the fields on the entire spaces M
and N , these observations are not enough, due to the singularities of ε̃ and µ̃
near Σ; the significance of this for cloaking was observed and analyzed in [3].
In the following, we will show that the physically relevant class of solutions to
Maxwell’s equations, namely the (locally) finite energy solutions, remains the
same, with respect to the transformation F , in (M ; ε, µ) and (N ; ε̃, µ̃). One
can analyze the rays in M and N endowed with the electromagnetic wave
propagation metrics g =
εµ and g̃ =
ε̃µ̃, respectively. Then the rays on
M are transformed by F into the rays in N . As almost all the rays on M do
not intersect with γ, therefore, almost all the rays on N do not approach Σ.
This was the basis for [16, 18] and was analyzed further in [19]; see also [17]
for a similar analysis in the context of elasticity. Thus, heuristically one is
led to conclude that the electromagnetic waves on (M ; ε, µ) do not feel the
presence of γ, while those on (N ; ε̃, µ̃) do not feel the presence of K, and
these waves can be transformed into each other by the map F .
Although the above considerations are mathematically rigorous, on the level
both of the chain rule and of high frequency limits, i.e., ray tracing, in the
exteriors M \ γ and N \Σ, they do not suffice to fully describe the behavior
of physically meaningful solution fields on M and N . However, by carefully
examining the class of the finite-energy waves in M and N and analyzing
their behavior near γ and Σ, respectively, we can give a complete analysis,
justifying the conclusions above. Let us briefly explain the main steps of the
analysis using methods developed for theory of invisibility (or cloaking) at
frequency k > 0 [3] and at frequency k = 0 in [6, 7]. The details will follow.
First, to guarantee that the fields in N are finite energy solutions and do not
blow up near Σ, we have to impose at Σ the appropriate boundary condition,
namely, the Soft-and-Hard (SH) condition, see [8, 11],
eθ ·E|Σ = 0, eθ ·H|Σ = 0,
where eθ is the angular direction. Secondly, the map F can be considered
as a smooth coordinate transformation on M \ γ; thus, the finite energy
solutions on M \ γ transform under F into the finite energy solutions on
N \ Σ, and vice versa. Thirdly, the curve γ in M has Hausdorff dimension
equal to one. This implies that the possible singularities of the finite energy
electromagnetic fields near γ are removable [12], that is, the finite energy
fields in M \ γ are exactly the restriction to M \ γ of the fields defined on all
of M .
Combining these steps we can see that measurements of the electromagnetic
fields on (M ; ε, µ) and on (R3 \K; ε̃, µ̃) coincide in U . In the other words, if
we apply any current on U and measure the radiating electromagnetic fields
it generates, then the fields on U in the wormhole manifold (M ; ε, µ) coincide
with the fields on U in (R3 \K; ε̃, µ̃), 3-dimensional space equipped with the
wormhole device construction.
Summarizing our construction, the wormhole device consists of the metama-
terial coating of the obstacle K. This coating should have the permittivity
ε̃ and permeability µ̃. In addition, we need to impose the SH boundary
condition on Σ, which may be realized by fabricating the obstacle K from a
perfectly conducting material with parallel corrugations on its surface [8, 11].
In the next section, the permittivity ε̃ and and permeability µ̃ are described
in a rather simple form. (As mentioned earlier, in order to allow for a tube
around the axis of the wormhole to be a vacuum or air, we deal with a
slightly different construction than was described above, starting with flat-
tened spheres). It should be possible to physically implement an approxima-
tion to this mathematical idealization of the material parameters needed for
the wormhole device, using concentric rings of split ring resonators as in the
experimental verification of cloaking obtained in [20].
4 Rigorous construction of the wormhole
Here we present a rigorous model of a typical wormhole device and justify
the claims above concerning the behavior of the electromagnetic fields in the
wormhole device in R3 in terms of as the fields on the wormhole manifold
(M, g).
4.1 The wormhole manifold (M, g) and the wormhole
device N
Here we prove the wormhole effect for a variant of the wormhole device
described in the previous sections. Instead of using a round sphere S2 as
before, we present a construction that uses a deformed sphere S2flat that is
flat the near the south and north poles, SP and NP . This makes it possible
to have constant isotropic material parameters near the z-axis located inside
the wormhole. For possible applications, see [4].
We use following notations. Let (θ, r, z) ∈ [0, 2π]×R+×R be the cylindrical
coordinates of R3, that is the map
X : (θ, r, z) → (r cos θ, r sin θ, z)
that maps X : [0, 2π] × R+ × R → R3. In the following, we identify [0, 2π]
and the unit circle S1.
Let us start by removing from R3 two “deformed” balls which have flat
portions near the south and north poles. More precisely, let M1 = R
3 \ (P1∪
P2), where in the cylindrical coordinates
P1 = {X(θ, r, z) : −1 ≤ z ≤ 1, 0 ≤ r ≤ 1}
∪{X(θ, r, z) : (r − 1)2 + z2 ≤ 1},
P2 = {X(θ, r, z) : 1 ≤ z − L ≤ 1, 0 ≤ r ≤ 1}
∪{X(θ, r, z) : (r − 1)2 + (z − L)2 ≤ 1}.
We say that the boundary ∂P1 of P1 is a deformed sphere with flat portions,
and denote it by S2flat. We say that the intersection points of S
flat with the
z-axis are the north pole, NP , and the south pole, SP .
Let g1 be the metric on M1 inherited from R
3, and let γ1 be the path
γ1 = {X(0, 0, z) : 1 < z < L− 1} ⊂ M1.
A1 = M1 \ V1/4,
Vt = {X(θ, r, z) : 0 ≤ r ≤ t, 1 < z < L− 1}, 0 < t < 1,
and consider a map G0 : M1 \γ1 → A1; see Fig. 3. G0 defined as the identity
map on M1 \ V1/2 and, in cylindrical coordinates, as
G0(X(θ, r, z)) = X(θ,
, z), (θ, r, z) ∈ V1/2.
Clearly, G0 is C
0,1−smooth.
Let U(x) ∈ R3×3, x = X(θ, r, z), be the orthogonal matrix that maps the
standard unit vectors e1, e2, e3 of R
3 to the Euclidian unit vectors correspond-
ing to the θ, r, and z directions, that is,
U(x)e1 = (− sin θ, cos θ, 0), U(x)e2 = (cos θ, sin θ, 0), U(x)e3 = (0, 0, 1).
Then the differential of G0 in the Euclidian coordinates at the point x ∈ V1/2
is the matrix
DG0(x)U(y)
) 0 0
0 0 1
U(x)−1, x = X(θ, r, z), y = G0(x). (1)
Later we impose on part of the boundary, Σ0 = ∂A1 ∩ {1 < z < L− 1}, the
soft-and-hard boundary condition (marked red in the figures).
Next, let (θ, z, τ) = (θ(x), z(x), τ(x)) be the Euclidian boundary normal
coordinates associated to Σ0, that is, τ(x) = distR3(x,Σ0) and (θ(x), z(x))
are the θ and z-coordinates of the closest point of Σ0 to x.
Denote by (G0)∗g1 the push forward of the metric g1 in G0, that is, the
metric obtained from g1 using the change of coordinates G0, see [2]. The
metric (G0)∗g1 coincides with g1 in A1 \ V1/2, and in the Euclidian boundary
normal coordinates of Σ0, on A1 ∩ V1/2, the metric (G0)∗g1, has the length
element
ds2 = 4τ 2 dθ2 + dz2 + 4dτ 2.
PSfrag replacements
Figure 3: A schematic figure on the map G0, considered in the (r, z) coor-
dinates. Later, we impose the SH boundary condition on the portion of the
boundary coloured red.
Next, let
q3 = conv
{(r, z) : (r − 2)2 + (z − (−2))2 ≤ 1}
∪{(r, z) : (r − 2)2 + (z − (L+ 2))2 ≤ 1}
q4 = {(r, z) : 0 ≤ r ≤ 1, −1 ≤ z ≤ L+ 1},
where conv(q) denotes the convex hull of the set q.
N1 = R
3 \ (P3 ∪ P4),
P3 = {X(θ, r, z) : (r, z) ∈ q3},
P4 = {X(θ, r, z) : (r, z) ∈ q4},
Σ1 = ∂N1 \ ∂P4.
We can find a Lipschitz smooth map G1 : A1 → N1, see Fig. 4, of the form
G1(X(θ, r, z)) = X(θ, R(r, z), Z(r, z))
such that it maps Σ0 to Σ1, and in A1 near Σ0 it is given by
G1(x+ tν0) = G1(x) + tν1. (2)
Here, x ∈ Σ0, ν0 is the Euclidian unit normal vector of Σ0, ν1 is the Euclidian
unit normal vector of Σ1, and 0 < t <
. Moreover, we can find a G1 so that
it is the identity map near the z-axis, that is,
G1(x) = x, x ∈ A1 ∩ {0 ≤ r <
} (3)
and such that G1 is also the identity map in the set of points with the
Euclidian distance 4 or more from P1 ∪ P2. Note that we can find such a
G1 such that both G1 and its inverse G
1 are Lipschitz smooth up to the
boundary. Thus the differentialDG1 ofG1 at x ∈ A1 in Euclidian coordinates
DG1(x) = U(y)
a11(r, z) 0
0 A(r, z)
U(x)−1, x = X(θ, r, z), y = G1(x),
where c0 ≤ a11(r, z) ≤ c1 and A(r, z) is a symmetric (2×2)-matrix satisfying
c0I ≤ A(r, z) ≤ c1I
with some c0, c1 > 0.
The map F1(x) = G1(G0(x)) then maps F1 : M1\γ1 → N1. Let g̃1 = (F1)∗g1
be metric on N1. From the above considerations, we see that the differential
DF1 of F1 at x ∈ M1 \ γ1 near Σ0, in Euclidian coordinates, is given by
DF1(x) = U(y)
b11(θ, r, z) 0
0 B(r, z)
U(x)−1, (4)
b11(θ, r, z) =
c11(r, z)
distR3(X(θ, r, z),Σ0)
x = X(θ, r, z), y = F1(x)
where c0 ≤ c11(r, z) ≤ c1, and B(r, z) is a symmetric (2×2)-matrix satisfying
c0I ≤ B(r, z) ≤ c1I,
for some c0, c1 > 0.
Note that ∂P4∩{r < 1} consists of two two-dimensional discs, B2(0, 1)×{−1}
and B2(0, 1)× {L+ 1}. Below, we will use the map
f2 = F1|∂P1\NP : ∂P1 \NP → B2(0, 1)× {−1} ⊂ ∂N1.
The map f2 can be considered as the deformation that “flattens” S
flat \NP
to a two dimensional unit disc.
PSfrag replacements
Figure 4: Map G1 in (r, z)-coordinates.
To describe f2, consider S
flat as a surface in Euclidian space and define on it
the θ coordinate corresponding to the θ coordinate of R3 \ {z = 0}. Let then
s(y) be the intrinsic distance of y ∈ S2flat to the south pole SP . Then (θ, s)
define coordinates in S2flat\{SP,NP}. We denote by y(θ, s) ∈ S2flat\{SP,NP}
the point corresponding to the coordinates (θ, s).
By the above construction, the map f2 has the form, with respect to the
coordinates used above,
f2(y(θ, s)) = X(θ, R(s),−1) ∈ B2(0, 1)× {−1}, where (5)
R(s) = s, for 0 < s <
R(s) = 1− 1
[(π + 4)− s], for (π + 4)− 1
< s < (π + 4),
cf. formulae (2) and (3). In the following we identify B2(0, 1) × {−1} with
the disc B2(0, 1).
Let h1 be the metric on ∂P1 \NP inherited from (M1, g1). Let h2 = (f2)∗h1
be the metric on B2(0, 1). We observe that the metric h2 makes the disc
B2(0, 1) isometric to S
flat \NP , endowed with the metric inherited from R3.
Thus, let
M2 = S
flat × [−1, L+ 1].
OnM2, let the metric g2 be the product of the metric of S
flat inherited from R
and the metric α2(z)dz
2, α2 > 0 on [−1, L+1]. Let γ2 = {NP}× [−1, L+1]
be a path on M2.
Define N2 = P4 = {X(θ, r, z) : 0 ≤ r < 1,−1 ≤ z ≤ L + 1} ⊂ R3,
Σ2 = ∂N2 ∩ {r = 1}, and let F2 : M2 \ γ2 → N2 be the map of the form
F2(y, z) = (f2(y), z) ∈ R3, (y, z) ∈ (S2flat \NP )× [−1, L+ 1]. (6)
Let g̃2 = (F2)∗g2 be the resulting metric on N2.
Figure 5: The set N2 in the (r, z) coordinates. Later, we impose the SH
boundary condition on the portion of the boundary colored red.
Denote byM 1 = M1∪∂M1 the closure ofM1 and let (M, g) = (M 1, g1)#(M2, g2)
be the connected sum of M 1 and M2, that is, we glue the boundaries ∂M1
and ∂M2. The set N = N1 ∪ N2 ⊂ R3 is open, and its boundary ∂N is
Σ = Σ1 ∪ Σ2.
Let F be the map F : M \ γ → N defined by the maps F1 : M1 \ γ1 → N1
and F2 : M2 \ γ2 → N2, and finally, let γ = γ1 ∪ γ2 and g̃ = F∗g.
Figure 6: The set N = N1 ∪ N2 ⊂ R3 having the complement K, presented
in the (r, z) coordinates. Later, the SH boundary condition is imposed on
Let K = R3 \N . On the surface Σ = ∂K we can use local coordinates (t̃, θ̃),
where θ̃ is the θ-coordinate of the ambient space R3 and t̃ is either the r or
z -coordinate of the ambient space R3 restricted to Σ. Denote also
τ̃ = τ̃ (x) = distR3(x, ∂K).
Then by formula (2) we see that in N1, in the Euclidian boundary normal
coordinates (θ̃, t̃, τ̃) associated to the surface Σ1, the metric g̃ has the length
element
ds2 = 4dτ̃ 2 + α1(t̃) dt̃
2 + 4τ̃ 2 dθ̃2, 0 < τ̃ <
, c−10 ≤ α1(t̃) ≤ c0, c0 ≥ 1.
The construction of F2 yields that in N2 , in the Euclidian boundary normal
coordinates (θ̃, t̃, τ̃) with t̃ = z, associated to the surface Σ2 = ∂K ∩ ∂N2,
the metric g̃ has the length element, near Σ2,
ds2 = 4dτ̃ 2 + α2(t̃)dt̃
2 + 4τ̃ 2 dθ̃2, 0 < τ̃ <
Here, near ∂N1 ∩ ∂N2, we use t̃ = z on Σ1. Choosing the map G1 in the
construction of the map F1 appropriately, we have α2(−1) = α1(−1), α2(L+
1) = α1(L+ 1), and the resulting map is Lipschitz.
On M1, N1, and N2 that are subsets of R
3 we have the well defined cylin-
drical coordinates (θ, r, z). Similarly, M2 = S
flat × [−1, L + 1] we define
the coordinates (θ, s, z), where (θ, s) are the above defined coordinates on
flat \ {SP,NP}.
We can also consider on N ⊂ R3 also the Euclidian metric, denoted by
ge. In Euclidean coordinates, (ge)ij = δjk. Consider next the above defined
Euclidian boundary normal coordinates (θ̃, t̃, τ̃) associated to ∂K. They are
well defined in a neighborhood of ∂K. We define the vector fields
ξ̃ = ∂eτ , η̃ = ∂eθ, ζ̃ = ∂et
on N near ∂K. These vector fields are orthogonal with respect to the metric
g̃ and to the metric ge.
On M near γ, we use coordinates (θ, t, τ). On M1, near γ1 they in the terms
of the cylindrical coordinates are (θ, t, τ) = (θ, z, r). On M2, they are the
coordinates (θ, t, τ) = (θ, z, s), where s is the intrinsic distance to the north
pole NP . We define also the vector fields
ξ = ∂τ , η = ∂θ, ζ = ∂t
on M \ γ near γ. These vector fields are orthogonal with respect to the
metric g.
In the sequel, we consider the differential of F as the linear map
DF : (TxM, g) → (TyN, ge), y = F (x), x ∈ M \ γ.
Using formula (4) in M1 and formulas (5), (6) in M2, we see that DF
−1(x)
at x ∈ N near ∂N is a bounded linear map that satisfies
|(η,DF−1(x)η̃)g| ≤ C τ̃(x), (ζ,DF−1(x)η̃)g = 0, (ξ,DF−1(x)η̃)g = 0,
(η,DF−1(x)ζ̃)g = 0, |(ζ,DF−1(x)ζ̃)g| ≤ C, |(ξ,DF−1(x)ζ̃)g| ≤ C,
(η,DF−1(x)ξ̃)g = 0, |(ζ,DF−1(x)ξ̃)g| ≤ C, |(ξ,DF−1(x)ξ̃)g| ≤ C,
where C > 0 and (· , · )g is the inner product defined by the metric g. More-
over, we obtain similar estimates for DF in terms of the Euclidian metric
|(η̃, DF (y)η)ge| ≤ C τ(y)−1, (ζ̃ , DF (y)η)ge = 0, (ξ̃, DF (y)η)ge = 0,
(η̃, DF (y)ζ)ge = 0, |(ζ̃ , DF (y)ζ)ge| ≤ C, |(ξ̃, DF (y)ζ)ge| ≤ C,
(η̃, DF (y)ξ)ge = 0, |(ζ̃ , DF (y)ξ)ge| ≤ C, |(ξ̃, DF (y)ξ)ge| ≤ C
for y ∈ M \ γ near γ with C > 0.
Next, consider DF (y) at y ∈ M \γ. Recall that the singular values sj(y), j =
1, 2, 3 of DF (y) are the square roots of the eigenvalues of (DF (y))tDF (y),
where (DF )t is the transpose of DF . By (7), the singular values sj = sj(y),
j = 1, 2, 3, of DF (y), numbered in increasing order, satisfy
c1 ≤ s1(y) ≤ c2,
c1 ≤ s2(y) ≤ c2,
≤ s3(y) ≤
where c1, c2 > 0.
The determinant of the matrix DF (y) can be computed in terms of its sin-
gular values by det(DF ) = s1s2s3. Later, we need the norm of the matrix
det(DF (y))−1DF (y). It satisfies by formula (8)
‖det(DF (y))−1DF (y)‖ = ‖(
s−1k )diag (s1, s2, s3)‖ = max
1≤j≤3
k 6=j
s−1k ≤ c−21 . (9)
4.2 Maxwell’s equations on the wormhole with SH coat-
Let dV0(x) denote the Euclidian volume element on N ⊂ R3. Recall that
N ⊂ R3 is open set with boundary ∂N = Σ. Let dVg be the Riemannian
volume on (M, g). We consider below the map F : M\γ → N as a coordinate
deformation. The map F induces for any differential form Ẽ on N a form
E = F ∗Ẽ in M \ γ called the pull back of Ẽ in F , see [2].
Next, we consider Maxwell equations with degenerate material parameters ε̃
and µ̃ on N with SH boundary conditions on Σ. On M and N we define the
permittivity and permeability by setting
εjk = µjk = det(g)1/2gjk, on M, (10)
ε̃jk = µ̃jk = det(g̃)1/2g̃jk, on N.
Here, and below, the matrix [gjk(x)] is the representation of the metric g in
local coordinates, [gjk(x)] is the inverse of the matrix [gjk(x)], and det(g) is
the determinant of [gjk(x)]. We note that the metric g̃ is degenerate near
Σ, and thus ε̃ and µ̃, represented as matrices in the Euclidian coordinates,
have elements that tend to infinity at Σ, that is, the matrices ε̃ and µ̃ have
a singularity near Σ.
Remark 1. Modifying the above construction by replacing M2 with M2 =
flat× [l1, l2] for appropriate l1, l2 ∈ R and choosing F1 in an appropriate way,
we can use local coordinates (θ̃, t̃) on Σ such that the Euclidian distance
along Σ of points (θ̃, t̃1) and (θ̃, t̃2) is proportional to |t̃1− t̃2|, and the metric
g̃ in the Euclidian boundary normal coordinates (θ̃, t̃, τ̃) associated to ∂K
has the form
ds2 = 4dτ̃ 2 + dt̃2 + 4τ̃ 2 dθ̃2, 0 < τ̃ <
The metric corresponding to the metamaterials used in the physical exper-
iment in [20] has the same form in Euclidian boundary normal coordinates
associated to an infinitely long cylinder B2(0, 1)×R. Thus it seems likely that
metamaterials similar to those used in the experimental verification of cloak-
ing could be used to create physical wormhole devices working at microwave
frequencies.
4.3 Finite energy solutions of Maxwell’s equations
and the equivalence theorem
In the following, we consider 1-forms Ẽ =
j Ẽjdx̃
j and H̃ =
j H̃jdx̃
the Euclidian coordinates (x̃1, x̃2, x̃3) of N ⊂ R3. In the sequel, we use Ein-
stein’s summation convention and omit the sum signs. We use the Euclidian
coordinates as we want to consider N with the differential structure inherited
from the Euclidian space. We say that Ẽj and H̃j are the (Euclidian) coeffi-
cients of the forms Ẽ and H̃, correspondingly. We say that these coefficients
are in L
loc(N, dV0), 1 ≤ p < ∞, if∫
|Ej(x)|p dV0(x) < ∞, for all bounded measurable sets W ⊂ N.
Definition 4.1 We say that the 1-forms Ẽ and H̃ are finite energy solutions
of Maxwell’s equations in N with the soft-and-hard (SH) boundary conditions
on Σ and the frequency k 6= 0,
∇× Ẽ = ikµ̃(x)H̃, ∇× H̃ = −ikε̃(x)Ẽ + J̃ on N,
η̃ · Ẽ|Σ = 0, η̃ · H̃|Σ = 0,
if 1-forms Ẽ and H̃ and 2-forms D̃ = ε̃Ẽ and B̃ = µ̃H̃ in N have coefficients
in L1loc(N, dV0) and satisfy
‖Ẽ‖2
L2(W,|eg|1/2dV0)
ε̃jk Ẽj Ẽk dV0(x) < ∞,
‖H̃‖2
L2(W,|eg|1/2dV0))
µ̃jk H̃j H̃k dV0(x) < ∞
for all bounded measurable sets W ⊂ N , and finally,
((∇× h̃) · Ẽ − ikh̃ · µ̃(x)H̃) dV0(x) = 0,
((∇× ẽ) · H̃ + ẽ · (ikε̃(x)Ẽ − J̃)) dV0(x) = 0,
for all 1-forms ẽ and h̃ with coefficients in C∞0 (N) that satisfy
η̃ · ẽ|Σ = 0, η̃ · h̃|Σ = 0, (11)
where η̃ = ∂θ is the angular vector field that is tangential to Σ.
Below, we use for 1-forms E = Ejdx
j and H = Hjdx
j , given in local coordi-
nates (x1, x2, x3) on M , the notations
∇× E = dH, ∇· (εE) = d ∗ E, ∇· (µH) = d ∗H,
where d is the exterior derivative and ∗ is the Hodge operator on (M, g), cf.
formula (10).
We have the following “equivalent behavior of electromagnetic fields on N
and M” result, analogous to the results of [3] for cloaking.
Theorem 4.2 Let E and H be 1-forms on M \ γ and Ẽ and H̃ be 1-forms
with coefficients in L1loc(N, dV0) such that E = F
∗Ẽ, H = F ∗H̃. Let J̃
and J = F ∗J̃ be 2-forms with smooth coefficients in N and M \ γ that are
supported away from Σ and γ.
Then the following are equivalent:
1. On N , the 1-forms Ẽ and H̃ satisfy Maxwell’s equations with SH bound-
ary conditions in the sense of Definition 4.1.
2. On M , the forms E and H can be extended on M so that they are
classical solutions E and H of Maxwell’s equations,
∇× E = ikµH, in M,
∇×H = −ikεE + J, in M.
Proof. Assume first that E and H satisfy Maxwell’s equations on M with
source J supported away from γ. Then E and H are C∞ smooth near γ.
Using F−1 : N → M \ γ we define the 1-forms Ẽ, H̃ and 2-form J̃ on N
by Ẽ = (F−1)∗E, H̃ = (F−1)∗H , and J̃ = (F−1)∗J. These fields satisfy
Maxwell’s equations in N ,
∇× Ẽ = ikµ̃(x)H̃, ∇× H̃ = −ikε̃(x)Ẽ + J̃ in N. (12)
Now, writing E = Ej(x)dx
j on M near γ, we see using the transformation
rule for differential 1-forms that the form Ẽ = (F−1)∗E is in local coordinates
Ẽ = Ẽj(x̃)dx̃
j , Ẽj(x̃) = (DF
−1)kj (x̃)Ek(F
−1(x̃)), x̃ ∈ N. (13)
Using the smoothness of E and H near γ on M and formulae (7), we see
that Ẽ, H̃ are forms on N with L1loc(N, dV0) coefficients. Moreover,
ε̃(x)Ẽ(x) = det(DF (y))−1DF (y)ε(y)DF (y)t(DF (y)t)−1E(y)
= det(DF (y))−1DF (y)ε(y)E(y)
where x ∈ N , y = F−1(x) ∈ M \ γ. Formula (9) shows that D̃ = ε̃Ẽ, and
B̃ = µ̃H̃ are 2-forms on N with L1loc(N, dV0) coefficients.
Let Σ(t) ⊂ N be the t-neighbourhood of Σ in the g̃-metric. Note that for
small t > 0 the set Σ(t) is the Euclidian (t/2)-neighborhood of ∂K. Denote
by ν be the unit exterior Euclidian normal vector of ∂Σ(t) and the Euclidian
inner product by (η̃, Ẽ)ge = η̃ · Ẽ.
Formulas (7) and (13) imply that the angular components satisfy
|η̃ · Ẽ| ≤ Ct, x ∈ ∂Σ(t),
|ζ̃ · Ẽ| ≤ C, x ∈ ∂Σ(t)
with some C > 0. Thus denoting by dS the Euclidian surface area on ∂Σ(t),
Stokes’ formula, formula (12), and the identity ν × ξ̃ = ±η̃ yield
((∇× h̃) · Ẽ − ikh̃ · µ̃H̃) dV0(x)
= lim
N\Σ(t)
((∇× h̃) · Ẽ − ikh̃ · µ̃H̃) dV0(x)
= − lim
∂Σ(t)
(ν × Ẽ) · h̃ dS(x)
= − lim
∂Σ(t)
ν × ((η̃ · Ẽ)η̃ + (ζ̃ · Ẽ)ζ̃) · h̃ dS(x)
for a test function h̃ satisfying formula (11).
Similar analysis for H̃ shows that 1-forms Ẽ and H̃ satisfy Maxwell’s equa-
tions with SH boundary conditions in the sense of Definition 4.1.
Next, assume that Ẽ and H̃ form a finite energy solution of Maxwell’s equa-
tions on (N, g) with a source J̃ supported away from Σ, implying in particular
ε̃jkẼjẼk ∈ L1(W, dV0), µ̃jkH̃jH̃k ∈ L1(W, dV0)
where W = F (U \ γ) ⊂ N and U ⊂ M is a relatively compact open neigh-
bourhood of γ, supp (J̃)∩W = ∅. Define E = F ∗Ẽ, H = F ∗H̃ , and J = F ∗J̃
on M \ γ. Therefore we conclude that
∇×E = ikµ(x)H, ∇×H = −ikε(x)E + J, in M \ γ
εjkEjEk ∈ L1(U \ γ, dVg), µjkHjHk ∈ L1(U \ γ, dVg).
As representations of ε and µ, in local coordinates of M , are matrices that
are bounded from above and below, these imply that
∇× E ∈ L2(U \ γ, dVg), ∇×H ∈ L2(U \ γ, dVg),
∇· (εE) = 0, ∇· (µH) = 0, in U \ γ.
Let Ee, He ∈ L2(U, dVg) be measurable extensions of E and H to γ. Then
∇× Ee − ikµ(x)He = 0, in U \ γ,
∇× Ee − ikµ(x)He ∈ H−1(U, dVg),
∇×He + ikε(x)Ee = 0, in U \ γ,
∇×He + ikε(x)Ee ∈ H−1(U, dVg),
where H−1(U, dVg) is the Sobolev space with smoothness (−1) on (U, g).
Since γ is a subset with (Hausdorff) dimension 1 of the 3-dimensional domain
U , it has zero capacitance. Thus, the Lipschitz functions on U that vanish on
γ are dense in H1(U), see [12]. Therefore, there are no non-zero distributions
in H−1(U) supported on γ. Hence we see that
∇× Ee − ikµ(x)He = 0, ∇×He + ikε(x)Ee = 0 in U.
This also implies that
∇· (εEe) = 0, ∇· (µHe) = 0 in U,
which, by elliptic regularity, imply that Ee and He are C∞ smooth in U .
In summary, E and H have unique continuous extensions to γ, and the
extensions are classical solutions to Maxwell’s equations. ✷
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Department of Mathematics
University of Rochester
Rochester, NY 14627, USA, allan@math.rochester.edu
Department of Mathematical Sciences
University of Loughborough
Loughborough, LE11 3TU, UK, Y.V.Kurylev@lboro.ac.uk
Institute of Mathematics
Helsinki University of Technology
Espoo, FIN-02015, Finland, Matti.Lassas@tkk.fi
Department of Mathematics
University of Washington
Seattle, WA 98195, USA, gunther@math.washington.edu
	Introduction
	The wormhole manifold M
	The wormhole device N in R3
	Rigorous construction of the wormhole
	The wormhole manifold (M,g) and the wormhole device N
	Maxwell's equations on the wormhole with SH coating 
	Finite energy solutions of Maxwell's equations  and the equivalence theorem
