Well-posed equations of motion for charged fluids in general relativity

In general relativity, including electric charge in fluid problems results in underdetermined equations: there are only 3 dynamical equations of motion, but there are 6 dynamic degrees of freedom: 3 in the fluid and 3 in the electromagnetic (EM) current. In the literature, families of solutions are found as functions of the left-over degrees of freedom, which must be fixed to find solutions.

In this talk, additions to the EM stress-energy tensor are presented that are quadratic in the EM 4-current. With no extra fluid, there are the same number of equations as unknowns. Not all quadratic, symmetric combinations of the EM current are compatible with the EM stress-energy tensor; to satisfy both momentum and energy conservation, the scalar product of the EM current and divergence of the stress-energy tensor must be zero. The only compatible stress-energy tensor quadratic in the EM current is equivalent to that of a polytropic fluid with adiabatic exponent of 2, and negligible rest mass density (and is also equivalent to the addition in Bopp-Podolsky EM). This stress-energy tensor can be found by varying a Lagrangian  quadratic the EM current. We also present two other independent Lagrangians that are quadratic in the first derivative of the EM field tensor; one violates parity and is zero in flat space-time, and the other reduces to the first Lagrangian in flat space-time.

We find static solutions in spherical symmetry, which have infinite central density, consistent with previously published results.