Gumede Sfundo C, Govinder Keshlan S, Maharaj Sunil D
Astrophysics Research Centre, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa.
Department of Mathematical Sciences, Mangosuthu University of Technology, P.O. Box 12363, Jacobs 4026, South Africa.
Entropy (Basel). 2022 May 4;24(5):645. doi: 10.3390/e24050645.
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.
方程(y_{xx}=f(x)y^{2}+g(x)y^{3})是埃姆登 - 福勒方程的带电推广形式，在球对称无剪切时空的研究中至关重要。此版本源于针对带电无剪切物质分布的爱因斯坦 - 麦克斯韦方程组。我们对方程进行积分并得到一个新的第一积分。为使该解存在，作为可积性条件出现了两个积分方程。可积性条件可转化为非线性微分方程，其根据初等函数和特殊函数给出了(f(x))和(g(x))的显式形式。显式形式(f(x)\sim\frac{1}{x^{5}}(1 - \frac{1}{x})^{-\frac{11}{5}})和(g(x)\sim\frac{1}{x^{6}}(1 - \frac{1}{x})^{-\frac{12}{5}})作为一个四次多项式的重根出现。这是爱因斯坦 - 麦克斯韦方程组的一个新解。我们的结果补充了早期关于中性和带电物质的工作，表明带电自引力流体的复杂性与第一积分的存在有关。
