Rotating Black Hole Solutions with Axion Dilaton and Two Vector Fields and Solutions with Metric and Fields of the Same Form

We present two rotating black hole solutions with axion $\xi$, dilaton $\phi$ and two U(1) vector fields. By applying the "Newman-Janis trick" to a metric with 3 arbitrary parameters we find a rotating metric $g_{\mu\nu}$ with 4 such parameters $(M, a, Q_E, Q_M)$, and then a solution with this $g_{\mu\nu}$ as metric. Our solution is asymptotically flat and has angular momentum $J=M a$, gyromagnetic ratio $g=2$, two horizons, the singularities of Kerr's solution, axion and dilaton singular only for $r=a\cos\theta=0$. Applying to the solution we have found the $S-$duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metric, each vector field and the $\lambda=\xi+ie^{-2\phi}$ of our solutions and the solution of : Sen for $Q_E$, Sen for $Q_E$ and $Q_M$, Kerr-Newman for $Q_E$ and $Q_M$, Kerr, Ref. 9, STW, GM-GHS, Reissner-Nordstr\"{o}m,Schwarzschild are the same function of $a$, and two functions $\rho^2=r(r+b)+a^2\cos^2\theta$ and $\Delta=\rho^2-2Mr+c$, of $a$, $b$ and two functions, and of $a$, $b$ and $d$ respectively, where $a$, $b$, $c$ and $d$ are constants. It is shown that from our solutions a number of known solutions can be obtained, which together with our solutions are listed in an Appendix. Also it is shown that all solutions which are mentioned in the paper satisfy all energy conditions, and mass formulae are obtained for them.