Charged Shear-Free Fluids and Complexity in First Integrals

The equation yxx=f(x)y2+g(x)y3 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)∼1x51-1x-11/5 and g(x)∼1x61-1x-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.