Asymmetric thin-shell wormholes
Eur. Phys. J. C
Asymmetric thin-shell wormholes
S. Danial Forghani 0
S. Habib Mazharimousavi 0
Mustafa Halilsoy 0
0 Department of Physics, Faculty of arts and sciences, Eastern Mediterranean University , Famagusta, North Cyprus, via Mersin 10 , Turkey
Spacetime wormholes in isotropic spacetimes are represented traditionally by embedding diagrams which were symmetric paraboloids. This mirror symmetry, however, can be broken by considering different sources on different sides of the throat. This gives rise to an asymmetric thin-shell wormhole, whose stability is studied here in the framework of the linear stability analysis. Having constructed a general formulation, using a variable equation of state and related junction conditions, the results are tested for some examples of diverse geometries such as the cosmic string, Schwarzschild, Reissner-Nordström and Minkowski spacetimes. Based on our chosen spacetimes as examples, our finding suggests that symmetry is an important factor to make a wormhole more stable. Furthermore, the parameter γ , which corresponds to the radius dependency of the pressure on the wormholes's throat, can affect the stability in a great extent.
1 Introduction
The history of wormholes goes back to the embedding
diagrams of Ludwig Flamm [1] in the newly discovered
Schwarzschild metric in 1916. Later on, in 1935, Einstein
and Rosen [2] in search of a geometric model for elementary
particles rediscovered a wormhole as a tunnel connecting two
asymptotically flat spacetimes. The minimum radius of the
tunnel, now known as the throat connecting two geometries,
was interpreted as the radius of an elementary particle. The
idea of wormhole did not go in much popularity until
Morris and Thorne [3,4] gave a detailed analysis and in certain
sense initiated the modern age of wormholes as tunnels
connecting two spacetimes. It was already stated by Morris and
Thorne that the energy density of such an object, if it ever
exists, must be negative; a notorious concept in the realm of
classical physics. In quantum theory, however, rooms exist
to manipulate and live along peacefully with negative energy
densities. Being a classical theory, general relativity must
find the remedy within its classical regime without
resorting to any quantum. At this stage, an important contribution
came from Visser, who found a way to confine the
negative energy density zone to a very narrow band of
spacetime known as the thin-shell [5,6]. The idea of thin-shell
wormholes (TSWs) became as popular and interesting as the
standard wormholes, verified by the large literature in that
context [7–9]. For some more recent works we refer to [10–
17,46]. Let us also remark that there have been attempts to
construct TSWs with total positive energy against the
negative local energy density [18–25]. This has been possible only
by changing the geometrical structure of the throat, namely
from spherical/circular to non-spherical/non-circular
geometry, depending on the dimensionality. Stability of TSW is
another important issue that deserves mentioning and
investigation for the survival of a wormhole (Fig. 1) [26–37].
In this paper, we introduce TSWs, that are constructed
from asymmetric spacetimes in the bulk [38–40]. So far, the
two spacetimes on different sides of the throat, are made from
the same bulk material. Our intention is to consider different
spacetimes, or at least different sources in common types of
spacetimes in order to create a difference between the two
sides. Naturally, the reflection symmetry about the throat in
the upper and lower halves will be broken and in
consequence new features are expected to arise which is the basic
motivation for the present study. Note that for non-isotropic
bulks, asymmetric TSWs emerge naturally. For example, we
consider Reissner–Nordström (RN) spacetimes on both sides
with different masses and charges in two sides of the throat;
Or two cosmic string (CS) spacetimes with different deficit
angles to be joined at the throat. This type of TSW, which
we dub as asymmetric TSW (ATSW), has not been
investigated so far. For this reason, we will be focusing on such
wormholes. One may anticipate that the asymmetry of the
wormhole will have an impact on particle geodesics, light
lensing, and other matters. Asymmetry may act instrumental
in the identification of TSWs in nature, if there exists such
structures. Our next concern will be to study the stability
of such ATSW and novelties that will give rise, if there are
any at all. As in the previous studies, an equation of state
(EoS) is introduced at the throat with pressure and density
to be used as the surface energy-momentum tensor. Then,
the Israel junction conditions [41–45] relate these variables
within an energy equation (see Eq. 16), in which Veff (a) is an
effective potential. Taking derivative of the energy equation
(
16
) will naturally yield the equation of motion. Expansion of
Veff (a) about an equilibrium radius of the throat, say (...truncated)