Scalar fields as a fundamental interaction in physics are one of the main predictions of the Kaluza-Klein and the Superstring theories. Scalar fields are also a fundamental component of the Brans-Dicke theory and of the inflationary models. Furthermore, in the standard model of Weinberg, Glashow and Salam scalar fields are needed as a primordial component for given mass to the particles. More recently, using an exact solution of the scalar-tensor field equations of gravity, it has been able to show that scalar fields are a very good candidate to be the dark matter in spiral galaxies \cite{fco,GMV}.However, it is fair to say that there are still some open questions in this issue, which should be solved in order to give a more solid evidence of their presence, and thus establish a proof for the existence of scalar fields in nature.

The possibility of existance of scalar fields together with the spontaneous scalarization in compact stars \cite{dam2} implies that astrophysical objects could contain scalar fields inherent in them. In other words, if scalar fields exist, the way it was established in \cite{fco,GMV}, a compact star will prefer to have one, in order to save energy. Even when these fundamental scalar fields have not been observed, they are one of the main ingredients of modern physics. Of course the question arise, why being them so important in physics have we never seen one? The answer to this question could be because they interact very weakly with matter. It can be shown that many of the theories containing scalar fields are in concordance with measurements in weak gravitational fields \cite{dam,brena}. We expect that scalar fields are important in strong gravitational fields like at the origin of the universe or in pulsars or black holes. A great effort has been done in order to detect scalar fields in strong gravitational fields \cite{esp}. In some sense, inflation at the origin of the universe could be a proof of such an interaction. Nevertheless, this effort has been done using perturbative methods \cite{dam} or using static exact solutions \cite{MNQ,ma,brena}. The problem with these two approaches is that for the first, perturbative methods are very efficient only in weak fields and for the second, astrophysical objects are in general non-static, thus the approach is not realistic. Even for binary pulsars, perturbative methods have shown a great success because the distance between the pulsars is so that the gravitational field is too strong to be understood with Newtonian mechanics but enough weak to be described by pertubative methods in general relativity \cite{esp}. Nevertheless the gravitational interaction is too weak for deciding which theory containing scalar fields could be the right one. It has been possible to discard a series of theories which did not agree with measurements or to bound some parameters of some other theories using perturbative methods or static exact solutions \cite{brena,dam}. But the most interesting effects of scalar fields are expected to be very near of a black hole or of a pulsar and are expected to be non-perturbative. If we want to understand scalar fields in a strong regime one way to follow is to find rotating exact solutions of the theory containing scalar fields and comparing them with observations. The problem then is that the field equations are very complicated to be solved in an exact manner. In a past work \cite{MNQ} we gave a very powerful method for finding exact static solutions of the Einstein-Maxwell-Dilaton field equations using harmonic maps, we found classes of solutions with arbitrary electromagnetic fields and gravitational arbitrary multipole momentums. In the present work we want: i) to give the details of the calculations made in \cite{MNQ}; ii) to complete the schema of that work, and iii) to present a way to derive exact rotating dilatons with arbitrary coupling between the scalar field and the electromagnetic one. In order to do so, we start from the Lagrangian

L = \sqrt{-g}\,(R-2 (\nabla \phi )^{2}-e^{-2 \alpha \phi }F^{2}), {lag0}

where $g$ is the determinant of the metric tensor, $R$ is the scalar curvature, $\phi $ the dilaton field and $F$ the Maxwell one. The constant $\alpha $ is a free parameter which governs the strength of the coupling of the dilaton to the Maxwell field. When $\alpha =0$, the action reduces to the Einstein-Maxwell scalar theory. When $\alpha =1$, the action is part of the low-energy action of string theory. For $\alpha =\sqrt{3}$, the Lagrangian (\ref{lag0}) leads to the Kaluza-Klein field equations obtained from the dimensional reduction of the five-dimensional Einstein vacuum equations. However, we will consider this theory for all values of $\alpha $.

On the other hand, the harmonic maps ansatz has probed to be an excellent tool for finding exact solutions of systems of non-linear partial differential equations \cite{ma24}, in particular, this method has been very useful in solving the chiral equations derived from a non-linear $\sigma $ model \cite{ma29}. Einstein equations in vacuum can be reduced to a non-linear $\sigma $ model with structural group $SL(2,R)$ in the space-time and to a structural group $SU(1,1)$ in the potential spaces, $i.e.$, in terms of the Ernst potentials. The electro-vacuum case can be also reduced to a non-linear $\sigma $ model with structural group $SU(2,1)$ in terms of the extended Ernst potentials \cite{neu}, \cite{kra}. The Kaluza-Klein field equations can be cast into a $SL(3,R)$ non-linear $\sigma $ model in the space-time as well as in the potential space \cite{ma1}, \cite{ma24}. This is possible because the corresponding potential space, defined bellow, is a symmetric Riemannian space only for $\alpha =0$ and $\alpha =\sqrt{3}$, but this is not the case for the low energy limit in super strings theory, where $\alpha =1$. In this work we will extend the techniques of the harmonic maps ansatz \cite{neu1}, \cite{ma24}, \cite{mis}, even for non-symmetric Riemannian spaces, maintaining $\alpha $ as an arbitrary constant. In the present work, the reduction of the field equations to a non-linear $\sigma $ model is not needed.

 

References
{fco}T. Matos and F. S. Guzmán, Class. Quant. Grav., 17, (2000), L9-L16. Available at: gr-qc/9810028. T. Matos and F. S. Guzmán. Ann. Phys. (Leipzig) 9, (2000), S133-S136. Available at: astro-ph/0002126

{GMV} T. Matos and F. S. Guzmán. Rev. Mex. A.A 37(2001)63-72. Avalable at: astro-ph/9811143.

{dam2}T. Damour and G. Esposito-Farese. Phys. Rev. Lett. 70, (1993), 2220.

{dam}T. Damour and G. Esposito-Farese. Class. Quant. Grav., 9, (1992), 2093.

{brena}Tonatiuh Matos and Hugo Villegas. Class. Quant. Grav., 17, (2000), 1455-1466. Tonatiuh Matos, Octavio Obregon and Hugo Villegas. Mod. Phys. Lett. A13, (1998), 3161.

{esp}G. Esposito-Farese, gr-qc/9612039.

{MNQ}T. Matos, D. Nuñez and H. Quevedo. Phys. Rev. D51, (1995), R310.

{ma}T. Matos. Gen. Rel. Grav., 30, (1998), 5.

{ma24}T. Matos. J. Math. Phys. 35, (1994), 1302.

{ma29}T. Matos. `Exact Solutions of G-invariant Chiral Equations' Math. Notes, 58, (1995), 1178-1182. Available here.

{neu}G. Neugebauer. Habilitationsschrift} (Universitat Jena Press, Jena, Germany) (1969).

{kra}D. Kramer, H. Stephani, M. MacCallum and E. Held. Exact Solutions of Einstein Field Equations. (1980), DVW, Berlin.

{ma1}T. Matos. Astron. Nachr. 307, (1986), 317.

{neu1}G. Neugebauer and D. Kramer. Ann. Phys. (Leipzig), 24}, (1969), 62.

{mis}Tonatiuh Matos, Guadalupe Rodriguez and Ricardo Becerril. J. Math. Phys. 33, (1992), 3521.