Relativistic model of a spherical star emitting neutrinos

We present a simple but complete relativistic model of a spherical star emitting neutrinos, with its basis in the coupled Einstein-Dirac equations. The interior of the star is assumed to be a perfect fluid—described by its energy-matter density, pressure $p$, and baryon number density $n$—bounded in space. Matter is considered transparent for neutrinos and the exterior region contains only neutrinos and the gravitational field. The question of compatibility of neutrinos with spherically symmetric gravitational fields is discussed and a redefinition proposed for the physical energy-momentum tensor of neutrinos, which enters the right-hand side of Einstein equations. The analytical solutions are shown to correspond to a description of emission of neutrinos with cooling and contraction of the configuration. The local conservation laws and the junction and boundary conditions of the exterior and interior solutions in the surface of the fluid are studied and allowed to characterize two classes of solutions. In one case the solution describes the stage of neutrino emission with consequent contraction of the configuration of the star immediately before the fluid is totally contained inside its Schwarzschild radius, when the emission of neutrinos ceases. The other possibility can correspond to a quasistatic configuration emitting neutrinos; the relativistic equation of radiative equilibrium for neutrinos is derived and permits us to define the equivalent of a "radiation pressure" for neutrinos, which has an additive contribution to the gravitational pressure and is not a purely relativistic effect.