Solutions of Einstein-Maxwell field equations for a static charged perfect fluid sphere (Q760820)

For the static line element in spherical symmetry \(ds^ 2=e^{\nu}dt^ 2-e^{\lambda}dr^ 2-r^ 2(d\theta^ 2+\sin^ 2 \theta d\phi^ 2)\) Einstein-Maxwell field equations for charged perfect fluid are: \(r^{- 2}-e^{-\lambda}(r^{-2}-\lambda '/r)=8\pi \delta +q^ 2/r^ 4\), (*) \(d((e^{-\lambda}-1)/r^ 2)/dr+d((e^{-\lambda}\nu '/2r))/dr+e^{- \lambda -\nu}d((e^{\nu}\nu '/2r))/dr=4q^ 2/r^ 5\), \(q=4\pi \int^{r}_{0}\rho e^{\lambda /2}r^ 2dr\), \(\mu =4\pi \int^{r}_{0}\delta r^ 2dr\); here \(\nu\), \(\lambda\), \(\delta\), \(\rho\), q, \(\mu\) are all functions of r; q and \(\mu\) are ''charge'' and ''mass'' inside the sphere of radius r. Introducing (**) \(X=\nu '/2r\), from (*) one gets the linear differential equation for \(e^{-\lambda}\) with known general solution. Moreover, introducing \(\phi\) (r) as follows: (***) \(r^ 2(X^ 1+X^ 2r-r^{-3})=(X\phi -r^{-1})(1+Xr^ 2)\), one gets from (*), (**), (***) the equation for X(r) of Bernoulli type, again with known general solution. Using the obtained solutions, the authors show that some previously known results are particular cases of the general solutions.