Anisotropic fluid in a spherical spacetime: I. Radiation from a compact star

Generalizing work of Misner and Sharp, we develop the general relativistic theory of any spherically symmetric spacetime, allowing for shearing stresses (hence unequal principal pressures) in the matter. Assumptions specific to the system considered are required to determine the time-evolution of the principal pressures. For a gas of noninteracting massless particles, the necessary equations follow from relativistic kinetic theory. The equations for null radiation are solved in the steady-state case, neglecting both: (a) the mass loss of the central body, and (b) the effect of the radiation on the background (Schwarzschild) geometry. Once the velocity-field of the radiation is specified, its energy-momentum tensor follows by integration. The form of the velocity-field cannot be uniquely fixed by the theory because it varies from case to case. Asymptotically, at large distances R, the energy density and principal pressures of null radiation usually fall off as R^?4 while its proper speed increases as R. A one-parameter family of model stars emitting null radiation in flat spacetime is constructed. These differ in how narrow a cone the radiation is beamed into. Radiation emitted isotropically differs markedly from that which is emitted with sharp directionality.