_{1}

This paper discusses the effect that conformal symmetry can have on a charged wormhole. The analysis yields a physical interpretation of the conformal factor in terms of the electric charge. The rate of change of the conformal factor determines much of the outcome, which ranges from having no solution to wormholes having either one or two throats.

Wormholes are handles or tunnels in spacetime connecting different regions of our Universe or different universes altogether. That wormholes could be actual physical structures suitable for interstellar travel was first proposed by Morris and Thorne [

using units in which

In this paper, we study the effect of conformal symmetry on wormholes that have an electric charge. More precisely, we assume the existence of a conformal Killing vector

where

In addition to studying its effect on a charged wormhole, we obtain a physical interpretation of the conformal factor in terms of the electric charge. The combination of electric charge and conformal symmetry results in a wormhole model that may actually have two throats.

As noted in the Introduction, we assume that our static spherically symmetric spacetime admits a one-parameter group of conformal motions, i.e., motions along which the metric tensor remains invariant up to a scale factor. Equivalently, there exist conformal Killing vectors such that

where the left-hand side is the Lie derivative of the metric tensor and

To study the effect of conformal symmetry, it is convenient to use an alternate form of the metric [

Using this form, the Einstein field equations become

and

To keep the analysis tractable, we follow Herrera and Ponce de León [

and

From Equations (8) and (9), we then obtain

where

Substituting in Equation (10) and using

Solving for

where

The Einstein field equations can be rewritten as follows:

and

It now becomes apparent that

so that

where

It was proposed by Kim and Lee [

Given that the usual form is

Kim and Lee go on to note that with

The effective shape function also has the usual properties, to be discussed later.

In this section we return to the assumption of conformal symmetry mentioned in Section 2. Let us first consider the Kim-Lee model, Equation (19). Then by Equation (11),

which is impossible. So this model is not compatible with the assumption of conformal symmetry. This difficulty can be overcome, however, by introducing a new differentiable function

Evidently,

by Equation (17). Since

As already noted, given the effective shape function

The major objective in this section is to obtain a physical interpretation of the conformal factor

Next, from Equations (4), (12), and (21) (and recalling that

Since

and

The condition

Also, since

and

So Equations (26) and (27) give us a physical interpretation for the conformal factor

Having just learned that

From the inequality

Solving, we obtain the inequality

Similarly, since

Solving, we obtain the second inequality:

Our final task is to check the violation of the null energy condition (NEC) required to hold the wormhole open. Recall that the NEC states that given the stress-energy tensor

for a generic shape function. For

holds whenever

associated with

since

For

showing that

to require that

(Since

In thia section we study the various conditions under which the NEC is violated. So let us restate Inequality (29) and the modified Inequality (30):

and

From

we obtain the following half-open intervals:

Interval I:

Interval II:

and

Interval III:

We now observe that whenever

Now recall that the event horizon of a black hole is often viewed as the analogue of the throat of the wormhole. In fact, according to Hayward [

Remark: The existence of two throats invites the following speculation: a variation on the Kim-Lee model is

Unlike our earlier model, this metric can lead to an event horizon. In particular, suppose

where

This paper discusses the effect that conformal symmetry can have on a charged wormhole. Conversely, the physical requirements are seen to place severe constraints on the wormhole geometry.

The analysis yields a physical interpretation of the conformal factor

of the charge Q. Moreover, the outcome is heavily dependent on

and ranges from having no solution to wormholes having two throats. The latter case can be viewed as the analogue of the Kerr-Newman black hole.

Kuhfittig, P.K.F. (2016) The Effect of Conformal Symmetry on Charged Wormholes. Journal of Applied Mathematics and Physics, 4, 2117-2125. http://dx.doi.org/10.4236/jamp.2016.412209