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Abstract
We start with a recently introduced spherically symmetric geodesic fluid model (arXiv: 1601.07030) whose energy-momentum tensor in the comoving frame is dust-like with nontrivial energy flux. In the non-comoving energy frame (vanishing energy flux) the same EMT contains besides dust only radial pressure. We present Einstein's equations together with the matter equations in static spherically symmetric coordinates. These equations are self-contained (four equations for four unknowns). We solve them analytically except for a resulting nonlinear ordinary differential equation (ODE) for the gravitational potential. This ODE can be rewritten as a Lienard differential equation which, however, may be transformed into a rational Abel differential equation of the first kind. Finally we list some open mathematical problems and outline possible physical applications (galactic halos, dark energy stars) and related open problems.
Highlights
Analytic solutions of the coupled Einstein-matter equations for the stationary anisotropic and spherically symmetric case, without supplying any external input, are rather rare
We start with a recently introduced spherically symmetric geodesic fluid model whose energy-momentum tensor (EMT) in the comoving frame is dust-like with nontrivial energy flux
In the non-comoving energy frame, the same EMT contains besides dust only radial pressure
Summary
Analytic solutions of the coupled Einstein-matter equations for the stationary anisotropic and spherically symmetric case, without supplying any external input, are rather rare. We describe another model for which we derive an analytic solution except for one remaining ordinary differential equation. We consider the resulting Einstein’s field equations together with the matter equations in static spherically symmetric coordinates These equations are self-contained (four equations for four unknowns). We solve them analytically except for a resulting nonlinear ordinary differential equation (ODE) for the gravitational potential. This ODE turns out to be the general relativistic generalization of a corresponding ODE derived in [5] for the nonrelativistic darkon fluid model.
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