Singularity-free static fluid spheres in general relativity

The relations between the metric coefficients, the energy density, the pressure, and the speed of sound at the center of a perfect fluid sphere which are necessary for singularity-free static solutions are considered. In order to find exact singularity-free static solutions and to construct reasonable star models it is suggested that we modify the known methods of obtaining exact spherically symmetric solutions using these relations. A known class and the new class of such solutions are obtained and the extreme static star model is constructed. At the center of the sphere the equation of state is ultrarelativistic and the sound speed is equal to the speed of light in a vacuum.