Wormholes Supported by a Combination of Normal and Quintessential Matter in Einstein and Einstein-Maxwell Gravity ()
1. Introduction
Traversable wormholes, first conjectured by Morris and Thorne [1], are handles or tunnels in the spacetime topology connecting different regions of our Universe or of different universes altogether. Interest in traversable wormholes has increased in recent years due to an unexpected development, the discovery that our Universe is undergoing an accelerated expansion [2,3]. This acceleration is due to the presence of dark energy, a kind of negative pressure, implying that 
in the Friedmann equation
. In the equation of state
, the range of values 
results in
. This range is referred to as quintessence dark energy. Smaller values of 
are also of interest. Thus 
corresponds to Einstein’s cosmological constant [4]. The case 
is referred to as phantom energy [5-10]. Here we have
, in violation of the null energy condition. As a result, phantom energy could, in principle, support wormholes and thereby cause them to occur naturally.
Sections 2-4 discuss a combined model of quintessence matter and ordinary matter that could support a wormhole in Einstein-Maxwell gravity, once again suggesting that such wormholes could occur naturally. The theoretical construction by an advanced civilization is also an inviting prospect since the model allows the assumption of zero tidal forces. Section 4 considers the effect of eliminating the electric field. A wormhole solution can still be obtained but only by introducing a redshift function that results in enormous radial tidal forces, suggesting that some black holes may actually be wormholes fitting the conditions discussed in this paper and so may be capable of transmitting signals, a possibility that can in principle be tested.
2. The Model
Our starting point for a static spherically symmetric wormhole is the line element
(1)
where
. Here 
is the shape function and 
is the redshift function, which must be everywhere finite to prevent an event horizon. For the shape function, 
, where 
is the radius of the throat of the wormhole. Another requirement is the flare-out condition, 
(in conjunction with
), since it indicates a violation of the weak energy condition, a primary prerequisite for the existence of wormholes [1].
In this paper the model proposed for supporting the wormhole consists of a quintessence field and a second field with (possibly) anisotropic pressure representing normal matter. Here the Einstein field equations take on the following form (assuming
):
(2)
where 
is the energy momentum tensor of the quintessence-like field, which is characterized by a free parameter 
such that
. Following Kiselev [11], the components of this tensor satisfy the following conditions:
(3)
(4)
Furthermore, the most general energy momentum tensor compatible with spherically symmetry is
(5)
with
. The Einstein-Maxwell field equations for the above metric corresponding to a field consisting of a combined model comprising ordinary and quintessential matter are stated next [12,13]. Here 
is the electric field strength, 
the electric charge density, and 
the electric charge.
(6)
(7)
(8)
(9)
Equation (9) can also be expressed in the form
(10)
where 
is the total charge on the sphere of radius
.
3. Solutions
We assume that for the normal-matter field we have the following equation of state for the radial pressure [14]:
(11)
For the lateral pressure we assume the equation of state
(12)
Generally, 
is not equal to
, unless, of course,
.
Following Ref. [14], the factor 
is assumed to have the form
, where 
is an arbitrary constant and 
is the charge density at
. As a result,
(13)
(14)
and
(15)
The next step is to obtain the shape function 
by deriving a differential equation that can be solved for
. The easiest way to accomplish this is to solve Equation (6) for 
and substituting the resulting expression in Equation (7), which, in turn, is solved for
. After substituting this expression in Equation (8) and making use of Equations (11) and (12), we obtain the simplified form
(16)
Here
, 
, and 
are dimensionless quantities given by the following:
(17)
where
(18)
and
(19)
Equation (16) is linear and would readily yield an exact solution provided that 
and 
are constants. This can only happen if 
for some constant
. In the first part of this paper we will assume that
, leading to the zero-tidal-force solution [1]. Whether occurring naturally or constructed by an advanced civilization, such a wormhole would be suitable for humanoid travelers.
Returning to Equation (16) and using Equation (14), the integrating factor 
yields the solution
(20)
where 
is an integration constant. From 
in Section 2, we obtain the shape function
(21)
4. Wormhole Structure
In Equation (20), 
is an integration constant. So mathematically, 
is a solution for every
, leading to 
in Equation (21). Physically, however, 
is going to satisfy the requirements of a shape function only for a range of values of
. This problem can best be approached graphically by assigning some typical values to the various parameters and adjusting the value of
, as exemplified by Figure 1. First observe that if
, then
. For the given values
, 
, 
, 
, and
, a suitable value for 
is
, as we will see. Substituting in Equation (21), we obtain
(22)
To locate the throat 
of the wormhole, we define the function 
and determine where 
intersects the 
-axis, as shown in Figure 2. Observe that Figure 2 indicates that for
, 
, so that 
for
, an essential requirement for a shape function. Furthermore, 
is a decreasing function near
; so
, which implies that
, the flare-out condition. With the flare-out condition now satisfied, the shape function has produced the desired wormhole structure. For completeness let us
note that 
and
. (Suitable choices for 
corresponding to other parameters will be discussed at the end of the section).
To the right of
, 
keeps rising, but at
, 
is still less than unity. So at
, the interior shape function, Equation (22), can be joined smoothly to the exterior function
To check this statement, observe that
while
To the right of
, 
as
, so that after adjusting the constant redshift function, the wormhole spacetime is asymptotically flat. (The components 
and 
are already continuous for the exterior and interior components, respectively [15-17]).
Returning to Equation (21), an example of an anisotropic case is
, 
, 
, 
, and
; a suitable choice for 
is
. The result is
Here 
and
.
An example of a value of 
closer to −1, the lower end of the quintessence range, is the following:
, 
, 
, and
. Letting
, the shape function is
This time 
and
.
5. Could the Electric Field Be Eliminated?
The purpose of this section is to study conditions under which a combined model of quintessential and ordinary matter may be sufficient without the electric field
.
If 
is eliminated, then the assumption of zero tidal forces becomes too restrictive. So we assume that 
for some nonzero constant
. This, in turn, means that
(23)
Now Equation (16) yields
(24)
Both 
and 
are positive integration constants. (The reason that 
has to be positive is that 
is close to zero whenever 
is close to
); 
and 
now become (for
)
(25)
and
(26)
The last two equations are similar to those in Ref. [12], which deals with galactic rotation curves.
As noted in Section 2, the shape function 
is obtained from
, so that
(27)
To meet the condition
, we must have
Solving for
, we obtain the radius of the throat:
(28)
Since
, 
and 
must have opposite signs. From
, we have
and, after substituting Equation (28),
which simplifies to
. It follows immediately that if
, then
, so that the flare-out condition cannot be met. To get a value for 
between 0 and 1, the exponent 
in the redshift function, Equation (23), has to be negative and sufficiently large in absolute value. Such a value will cause 
to be negative, which can best be seen from a simple numerical example: for convenience, let us choose
, the lower end of the quintessence range, and
. Then we must have
. The result is a large positive numerator in Equation (25) because the last term is positive and 
is large. So 
and 
have opposite signs, as expected. (Observe that for the isotropic case, if
, then the values of 
and 
are independent of 
and
).
Continuing the numerical example, if we let 
and
, then
, 
, and
From Equation (28), 
, while
.
As we have seen, 
is independent of
. So we are free to choose a smaller value in Equation (28) to obtain a larger throat size.
We conclude that we can readily find an interior wormhole solution around 
without
, provided that we are willing to choose a sufficiently large (and negative) value for
, resulting in what may be called an unpalatable shape function:
. At the throat, 
, which indicates the presence of an enormous radial tidal force, even for large throat sizes. (Recall that from Ref. [1], to meet the tidal constraint, we must have roughly
. Such a wormhole would not be suitable for a humanoid traveler, but it may still be useful for sending probes or for transmitting signals.
The enormous tidal force is actually comparable to that of a solar-mass black hole of radius 2.9 km near the event horizon, making the solution physically plausible: since we have complete control over 
and
, we are not only able to satisfy the flare-out condition but we can place the throat wherever we wish. Moreover, the assumption 
is equivalent to Einstein’s cosmological constant, the best model for dark energy [18]. Also physically desirable is the assumption of isotropic pressure, i.e., 
in the respective equations of state. As we have seen, in the isotropic case our conclusions are independent of 
and
. So by placing the throat just outside the event horizon of a suitable black hole, it is possible in principle to construct a “transmission station” for transmitting signals to a distant advanced civilization and, conversely, receiving them. If such a wormhole were to exist, it would be indistinguishable from a black hole at a distance. This suggests a possibility in the opposite direction: A black hole could conceivably be a wormhole fitting our description. The easiest way to test this hypothesis is to listen for signals, artificial or natural, emanating from a (presumptive) black hole.
6. Conclusions
This paper discusses a class of wormholes supported by a combined model consisting of quintessential matter and ordinary matter, first in Einstein-Maxwell gravity and then in Einstein gravity, that is, in the absence of an electric field. To obtain an exact solution, it was necessary to assume that the redshift function has the form 
for some constant
. In the Einstein-Maxwell case, this constant could be taken as zero, thereby producing a zero-tidal-force solution, which, in turn, would make the wormhole traversable for humanoid travelers. Without the electric field
, the exponent 
has to be nonzero and leads to a less desirable solution with large tidal forces. Concerning the exact solution, it is shown in Ref. [19] that the existence of an exact solution implies the existence of a large set of additional solutions, suggesting that wormholes of the type discussed in this paper could occur naturally.
It is argued briefly in the Einstein case with a quintessential-dark-energy background that some black holes may actually be wormholes with enormous tidal forces, a hypothesis that may be testable.