The
wormhole solutions to the Einstein equations started with Einstein himself
where he has interested in giving a field representation of particles \cite{ER}.
The idea was further developed by Ellis, \cite{Ellis}, where instead of particles
people try to model them as "bridges" between two regions of thespace-time.
The idea of considering such solutions as actual connections between two separated
regions of the Universe, has attracted a lot of attention since the seminal
Morris and Thorne's work\cite{MT}. These solutions have evolved from a science-fiction
type of solutions, to a solid scientific topic, even to consider their actual
possibility of existence. Indeed, Carl Sagan, worried for the impossibility
of star travel imposed by the fundamental laws of Physics, being the most
serious problems the huge distances together with the finiteness of the speed
of light, and the Lorentz time contraction, asked for help to physicist and
turned out that the solution proposed by Ellis \cite{Ellis}, actually could
be interpreted as the identification, the union, of two different regions,
no matter how far they where or even if those two regions were in the same
space-time, you could just identify two regions, and obtain a solution which
allows to go from one region to another, by means of this gluing.

Such solutions need a type of matter called "exotic", as an understatement of matter which violated the energy conditions which we are used to see fulfilled everywhere, see \cite{Viser1} for a detailed review on this subject. Thus, the solutions exist but to be generated they need matter which apparently does not exist. Actually you do need something very peculiar to warp the space-time or to make holes on it. This feature was a serious backdrop to consider their actual existence in Nature, so they remained in the realm of fiction. However, as often happens with these can-not-be laws or conjectures, more and more evidence was building towards the presence in our Universe of unknown types of matter and energy which do not necessarily obey the energy conditions. Indeed, the Universe is know to be formed by $73 \%$ of dark energy. This new, for us, type of matter composes the most overwhelming majority in the Universe and happens to be everywhere \cite{EO}. Works have also been made discussing the plausibility of energy conditions violations at a quantum level, see for example \cite{Rom}. There is now an agreement among the scientific community that matter which violates some of the energy conditions is very plausible to exit. Thus, the issue that the wormholes can be rejected due to the type of matter that they need is, at least, diminished.

Another mayor
problem faced by the wormhole solutions is their stability. By construction,
wormhole solutions are transversable, that is, a test particle can go from
one side of the throat to the other in a finite time, measured by the observer
at the test particle and by the one far away from it, and without facing
large tidal forces. The stability problem of the "bridges" has
been studied since the 60's by Penrose \cite{Pen}, in connection to the
stability of the Cauchy horizons. However, the stability of the throat of
a wormhole was just recently studied numerically by Shinkay and Hayward
\cite{shin}, where they show that the wormhole proposed by Thorne \cite{MT}
when perturbed by a scalar field with

stress-energy tensor defined with the usual sign, the wormhole collapses
possibly towards a black hole, the throat closes. And when the perturbation
is due to a scalar field of the same type as that making the wormhole, the
throat grows exponentially, thus

showing that the solution is highly unstable.

Intuitively is clear that a rotating solution would have more possibilities of being stable, as well as more general static spherically symmetric solutions than the one proposed by Thorne. Some studies have been made on rotating wormhole solutions \cite{rot}, but non have put forward an exact solution to the Einstein equations describing such a wormhole. In the present project we do so.

{ER}A. Einstein, and N. Rosen, Phys. Rev. {\bf 48}, 73, (1935).

{Ellis}H. G. Ellis, J. Math. Phys., {\bf 14}, 395, (1973).

{MT}M. S. Morris, K. S. Thorne, Am. J. of Physics, {\bf 56}, N. 5, 365, (1988).

{Viser1}M.
Visser,

{\it Lorentzian wormholes: form Einstein to Hawking}, I. E. P. Press, Woodsbury,
N. Y. 1995.

{EO}F. S. N. Lobo, Phys.Rev. D71 (2005) 084011, e-print: gr-qc/0502099.

{Rom}T. Roman, e-print: gr-qc/0409090.

{Pen}R. Penrose,
Battele Rencontres,

ed. by B. S. de Witt and J. A. Wheeler, Benjamin, New York, (1968).

{shin}H. Shinkay and S. A. Hayward, Phys. Rev. {\bf D} , (2002), e-print: gr-qc/0205041.

{rot}E. Teo,
e-print: gr-qc/9803098, P. K. F. Kuhfitting,

e-print: gr-qc/0401023, e-print: gr-qc/0401028, e-print: gr-qc/0401048.