To understand what a wormhole is, it is better to follow the history of wormholes first. Interestingly, an insight into a wormhole shares the same year of discovery as the first black hole. In 1916, Schwarzschild found his famous black hole solution of the Einstein equations, the Schwarzschild black hole. In 1916, Flamm found that the Schwarzschild metric has a hidden tunnel structure which connects two asymptotically flat spaces; he developed what is now called the embedding diagram [
2]. Almost twenty years after Flamm’s work, Einstein and Rosen published their famous paper about the “Einstein–Rosen bridge” which is the Schwarzschild spacetime that can be interpreted as a solution joining the two same Schwarzschild geometries at their horizons [
3]. The bridge acts as a spacetime tunnel since it connects two asymptotically flat regions.
The outline of this article is following:
Section 1 is the introduction, where wormhole properties are reviewed. In
Section 2, we show the construction and linear stability against perturbations preserving symmetries of thin-shell wormholes. In
Section 3, we mention that there are many generalizations of the thin-shell wormhole by Poisson and Visser. In this section, we review some of its generalizations. In
Section 4, we concentrate on pure tension wormholes in Einstein gravity. This part is based on the work of [
4]. We introduce wormholes with a negative-tension brane and we analyze the existence of static solutions, stability and horizon avoidance in spherical, planar and hyperbolic symmetries. In
Section 5, we treat pure tension wormholes in Einstein–Gauss–Bonnet gravity. This part is on the work of [
5]. At first, assuming that the shell is made of tension together with a perfect fluid, we derive the equation of motion for the shell and review the basic properties of the static shell After that, we show that the shell has a negative energy density and hence the weak energy condition is violated. (However, the negative-tension brane still satisfies the null energy condition.) Next, we study the existence and instability of static spherically symmetric thin-shell wormholes. The signature of the metric is taken as diag
, and Greek indices run over all spacetime indices. In this section, the
d-dimensional gravitational constant
is retained.
Section 6 is dedicated to the conclusion.
1.1. Einstein–Rosen Bridge
The Einstein–Rosen bridge is unstable since the throat pinches off quickly. To understand this mechanism, let us see the dynamics of the bridge to understand the reason of the pinch off.
We show here how this tunnel structure is recognized. The Schwarzschild metric is written in the spherical coordinates
as
where
M is a constant mass parameter. Suppose we take a constant time slice,
const. Since the spacetime is spherically symmetric, we can take the equator slice,
, without loss of generality. Then, the metric reduces to
Equation (
2) has an axial symmetry:
const. The metric or the line element can be expressed as a two-dimensional surface in a three-dimensional flat space;
at
const and
is embedded into the three dimensional Euclidean space
. In
, the infinitesimal distance
with the cylindrical coordinates (
) (
: distance from the
z axis,
: angle around the
z axis) is given by
A surface in a flat space can be described as
in the cylindrical coordinates. Since the coordinates
r and
are functions of
and
, we must have relation between (
) and (
) to identify the equation of the surface, i.e.,
. Since the metric (
2) is axially symmetric, which suggests
and
, the surface function becomes the function of
r,
. Summarizing the above, we find
Substituting (
4) into (
3), then comparing this metric and (
2), we get the relations
The simultaneous equations reduces to a single differential equation and is easy to integrate;
A plot for (
6) is shown in
Figure 1 below, in the assumption
. One finds that two asymptotically flat regions (
as
) are connected by the neck
. Due to the shape of a neck, we call it a
throat.
At this stage, one may ask a question such as “if we live in an asymptotically flat region, namely, in the upper space of
Figure 1, what does the other region correspond to? Can we pass through the throat and go to this other region?”. A clear answer to this question, is produced in the paper by Fuller and Wheeler [
6]. They revealed the dynamical properties of the throat and also showed that no traveller can safely pass through the throat to go to the other region. We will show the dynamics of the throat by the following this argument.
The Kruskal diagram of the Schwarzschild spacetime is given by
Figure 2. Here,
v is a timelike coordinate while
u is a spacelike coordinate. Regions I and III represent the outside of the black hole which corresponds to the region of
in the Schwarzschild coordinates. Region II is the inside of the black hole,
, while Region IV is the white hole, that is considered as a time-reversal of the black hole. The straight lines
correspond to
, the event horizon of the spacetime. The dashed bold lines are the curvature singularity
. Straight lines between
and
are
const., while hyperbolas are
const. surfaces.
Here, we take a particular spacelike slice for the diagram as
which becomes
, as
. We draw
const. surfaces of (
7) in
Figure 3 as gray curves ((a) to (h)). As one can see,
plays the role of time here; when
increases, the surface (
7) moves in the direction of increasing
v. Since the surface (
7) is spacelike, it moves in the time direction. In this figure, the red dotted straight line describes a null geodesic
released from Region IV, while the blue one is a null geodesic
released from Region III. The throat cannot stay static and its dynamics is as described in
Figure 4. The process occurs in the order of (a) to (h):
- (a)
Photons and initially are in the lower sheet. They go to the center . The values and correspond to and , respectively. The vertical bold line is the curvature singularity . At this moment, the singularity is in between two quasi-Euclidean spaces.
- (b)
Both photons go to the center. A throat is going to appear.
- (c)
The throat just opened. The circumference of the throat is smaller than .
- (d)
The maximal throat, . The photon has passed though the throat.
- (e)
The throat is shrinking. Both photons have passed though the throat.
- (f)
The moment of the throat closing. In this stage, both photons are still in the upper sheet, while the photon approaches the central singularity.
- (g)
Photon is just caught. Then, disappears in the singularity and stops existing.
- (h)
Photon keeps escaping to the null infinity of the upper sheet.
Although we have used a specific spacelike slice (
7) to show the dynamical feature of the bridge, this dynamics does not change as long as the slice is spacelike.
From the above, we conclude that, although a timelike traveller might go to the upper space in just a finite time, it cannot come back to the lower space. Hence, the Schwarzschild solution provides one-way travel. One can speculate that a two-way travel needs a Penrose diagram such as
Figure 5. Apparently, this diagram shows that a timelike worldline can cross the throat again and again without hitting any singularities. The introduction of such a two-way tunnel spacetime is explained in the next subsection.
1.2. Wormhole Properties in Brief
It was Morris and Thorne who established modern wormhole physics. In this subsection, we follow their approach to understand what properties wormholes should have. They pioneered qualitative study for static and spherically symmetric spacetimes which have “two-way“
traversable wormholes [
7]. Since they knew what kinds of geometries describe tunnel structures, they deduced the metric which has such a geometry. Then, substituting the metric into the Einstein equation, they recovered the matter properties and its distribution. Here, we begin with a brief overview of their discussion.
A convenient choice of coordinates to describe static and spherically symmetric wormhole spacetimes is
where
and
b are both functions of
r. To simplify calculations, we introduce an orthonormal basis of reference frame of the static observers:
In this basis, the metric takes the Minkowskian form:
. Then, the non-zero components of the Einstein tensor yield
where
. Since the geometry is both static and spherically symmetric, the vacuum equation must be the Schwarzschild black hole (Birkhoff’s theorem), a non-traversable wormhole. Thus, if we want to build a wormhole spacetime, we must handle spacetimes with a specific form of stress–energy tensors. As the Einstein tensor takes a diagonal form, the corresponding non-zero stress–energy tensor must also be diagonal. In the orthonormal basis, we can then give each component of the stress–energy tensor the physical interpretation as
, where
is the energy density, that static observers measure,
is the radial tension that they measure in the radial direction (negative of the radial pressure) and
p is the pressure that they measure in the lateral direction.
The Einstein equations
give the following non-trivial equations for
and
p:
One may solve the above equations to get the form of
b and
by imposing a specific component choice for
, i.e., a specific form of
and
p. An alternative way to solve them is that one imposes an equation of state as
and
, and then one solves for (
12).
1.2.1. Embedding Wormholes and Asymptotic Flatness
The surface
actually describes the throat. The reason for this is obvious from the embedding operation. We can play the same game in
Section 1.1 to get the embedding of the metric (
8). Going through the same process in
Section 1.1, we obtain a differential equation for
z;
This differential equation can now be integrated if
is determined. Thus,
b is called the shape function. Obviously, (
13) diverges when
. Since the schematic picture of (
13) is similar to
Figure 1, one finds that the sphere with radius of
describes the throat (
Figure 6). We denote the throat,
, as the minimum surface. As mentioned, at the throat,
. Morris and Thorne further imposed the asymptotically flatness condition which means
as
.
1.2.2. Flaring-Out Condition
In
Section 1.2.1, we show that (
13) diverges at the throat. In other words, the inverse function of
, i.e.,
satisfies
For a spacetime to be a wormhole, there must be a throat that flares out. The flaring-out condition states
at and near the throat.
1.2.3. Absence of the Horizon
For the wormhole to be traversable, there must be no horizons in a spacetime. By using the function
in the metric (
8), it states
1.2.4. Magnitude of the Tension at the Throat
The shape function
b gives restrictions on
and
p through Equation (
12). A critical restriction is at the throat,
. Then,
. Reviving
c and
G, this yields the huge tension;
if
is not too large.
1.2.5. Exotic Matter
Besides some of the peculiar features about wormholes noted above, the most difficult thing to digest in wormhole physics is the necessity of exotic matters that can violate energy conditions. In general relativity, Morris–Thorne’s static spherically symmetric traversable wormholes need stress–energy tensors that violate energy conditions at and near the throat. To see what happens to the relation between the tension
and energy density
near the throat, we introduce a dimensionless function
as
Since (
15) and
is satisfied around the throat,
reduces to
at and near the throat. We call the matter which has property
as an
exotic matter because the conventional matter satisfies the null energy condition,
.
1.2.6. Other Properties
Morris and Thorne required some additional conditions on traversable wormholes, tidal forces and a time to pass through wormholes. We do not explain these conditions here since we think that the conditions mentioned above (
Section 1.2.2,
Section 1.2.3,
Section 1.2.4 and
Section 1.2.5) are the primary problems for wormholes. We refer the reader to the work of [
7] for the details of tidal forces and time to pass through wormholes.
1.3. Simple Exact Solutions and Their Stability
If we somehow solved all of the difficulties about wormhole properties discussed above, there is still an important problem, i.e., the stability of wormhole spacetimes. Given a spacetime, its stability analysis against gravitational/matter perturbations is a problem with critical importance in the sense that only stable spacetimes may exist in the “real world”.
During few decades, after the paper by Einstein and Rosen, several exact solutions to the Einstein equations have been found and they have tunnel structures as described above [
8,
9,
10]. These types of spacetimes are assumed to have a massless scalar field with the opposite sign of its kinetic term to the sign of the Einstein-Hilbert term in the Lagrangian. We often call such a scalar field a
ghost or
phantom scalar field. We refer to the simplest exact wormhole solution as the Ellis solution (or the Ellis–Bronnikov solution).
Although wormhole solutions have been known for decades, their stability analysis had not been conducted until quite recently. The first stability analysis was by [
11]. In this paper, it is shown that the Ellis wormhole is stable against gravitational perturbations in a restricted class which do not change the throat radius. Subsequently, Shinkai and Hayward showed that the Ellis wormhole is unstable against either a normal and a phantom gaussian pulse of massless scalar field [
12]. When a normal (ghost) pulse is injected into the throat, the throat must shrink (inflate). Gonzalez et al. also proved that the Ellis wormhole is unstable against linear and non-linear spherically symmetric perturbations in which the throat is not fixed [
13,
14]. They showed that a charged generalization of the wormhole is also unstable [
15]. This unstable feature is invariant in the higher dimensional generalization of the Ellis spacetime [
16].