Charged Shear-Free Fluids and Complexity in First Integrals

The equation y(xx) = f (x)y(2) + g(x)y(3) is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the Einstein-Maxwell system for a charged shear-free matter distribution. We integrate this equation and find a new first integral. For this solution to exist, two integral equations arise as integrability conditions. The integrability conditions can be transformed to nonlinear differential equations, which give explicit forms for f (x) and g(x) in terms of elementary and special functions. The explicit forms f (x) similar to 1/x(5) (1 - 1/x)(-11/5) and g(x) similar to 1/x(6) (1 - 1/x)(-12/5) arise as repeated roots of a fourth order polynomial. This is a new solution to the Einstein-Maxwell equations. Our result complements earlier work in neutral and charged matter showing that the complexity of a charged self-gravitating fluid is connected to the existence of a first integral.