content_image/407297.jpg content_image/407302.jpg content_image/407307.jpg content_image/407309.jpg content_image/407300.jpg content_image/407301.jpg content_image/407303.jpg content_image/407312.jpg content_image/407304.jpg
# Some remarks on exact wormhole solutions 

Department of Mathematics Milwaukee School of Engineering Milwaukee, Wisconsin 53202-3109 USA 

Peter K. F. Kuhfittig 

Exact wormhole solutions, while eagerly saught after, often have the appearance of being overly specialized or highly artificial. A case for the possible existence of traversable wormholes would be more compelling if an abundance of solutions could be found. It is shown in this note that for many of the wormhole geometries in the literature, the exact solutions obtained imply the existence of large sets of additional solutions. 

One of the fundamental problems in the general theory of relativity is finding exact solutions of the Einstein field equations [1]. An example is the Schwarzschild solution, which proved to be a major milestone. Similarly, find- ing exact solutions of wormhole geometries has occupied many researchers. The difference is that in this setting exact solutions are hardly a virtue if they have the ap- pearance of being artificial or, at best, overly special- ized. It is shown in this note that exact solutions may be used to show the existence of entire families of solutions, which in turn helps strengthen the case for the possible existence of traversable wormholes. 

A wormhole may be defined as a handle or tunnel in the spacetime topology linking different universes or widely separated regions of our own Universe [9]. A general line element is the following: 

<img src='content_image/407297.jpg'>

where Φ( r ) → 0 and Λ( r ) → 0 as r → ∞ , usually re- ferred to as asymptotic flatness . We also assume that Φ and Λ have continuous derivatives in their respective domains. The function Φ = Φ( r ) is called the redshift function , which must be everywhere finite to prevent an event horizon. The function Λ = Λ( r ) is related to the shape function b ( r ) = r (1 − e − 2Λ( r $^{)}$), that is, e 2Λ( r ) = 1 / [1 − b ( r ) /r ]. The minimum radius r = r$_{0}$ is the throat of the wormhole, where b ( r$_{0}$ ) = r$_{0}$ . Also needed is b $^{′}$( r$_{0}$ ) < 1, usually referred to as the flare-out condition . As a consequence, Λ( r ) has a vertical asymp- tote at r = r$_{0}$ : lim$_{r}$ → r 0 $_{+}$Λ( r ) = + ∞ . For the redshift function we also have the additional requirement that Φ $^{′}$( r$_{0}$ ) be finite. Finally, to hold a wormhole open, the null energy condition must be violated [9]. 

The components of the Einstein tensor in the orthonor- 

PAC numbers: 04.20.Jb, 04.20.Gz Key words: wormholes, exact solutions 

## I. INTRODUCTION 

## II. WORMHOLE SOLUTIONS 

mal frame are listed next [4]. 

<img src='content_image/407302.jpg'>

The case that will be examined in some detail is a wormhole supported by phantom energy, whose equation of state (EoS) is p = − Kρ , K > 1. For this case, ρ + p = ρ (1 − K ) < 0, in violation of the null energy condition. 

From the Einstein field equations G$_{ˆ}$ α ˆ β = 8 πT$_{ˆ}$ α ˆ β and Eqs. (2) and (3), we now get 

<img src='content_image/407307.jpg'>

Solving for Λ $^{′}$( r ), we have 

<img src='content_image/407309.jpg'>

Observe that lim$_{r}$ → r 0 $_{+}$Λ $^{′}$( r ) = −∞ . Otherwise Λ $^{′}$( r ) is continuous in the interval ( r$_{0}$, ∞ ). 

The continuity assumptions play a critical role: thus for a > r$_{0}$ , the integral ∫ b a e Λ( r $^{)}$dr exists since Λ( r ) is continuous. Now from the line element (1), the proper radial distance ℓ ( r ) from the throat r = r$_{0}$ to any point away from the throat is given by 

<img src='content_image/407300.jpg'>

<img src='content_image/407301.jpg'>

<img src='content_image/407303.jpg'>

<img src='content_image/407312.jpg'>

<img src='content_image/407304.jpg'>

