Noether and Lie symmetries for charged perfect fluids. (English) Zbl 1217.83015
Summary: We study the underlying nonlinear partial differential equation that governs the behaviour of spherically symmetric charged fluids in general relativity. We investigate the conditions for the equation to admit a first integral or be reduced to quadratures using symmetry methods for differential equations. A general Noether first integral is found. We also undertake a comprehensive group analysis of the underlying equation using Lie point symmetries. The existence of a Lie symmetry is subject to solving an integro-differential equation in general; we investigate the conditions under which it can be reduced to quadratures. Earlier results for uncharged fluids and particular first integrals for charged matter are regained as special cases of our treatment.
MSC:
83C15 | Exact solutions to problems in general relativity and gravitational theory |
83C40 | Gravitational energy and conservation laws; groups of motions |
83C50 | Electromagnetic fields in general relativity and gravitational theory |
83C55 | Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.) |
35L15 | Initial value problems for second-order hyperbolic equations |
45K05 | Integro-partial differential equations |
35K55 | Nonlinear parabolic equations |
83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |