Spherically symmetric gravitational collapse of general fluids

We express Einstein’s field equations for a spherically symmetric ball of general fluid such that they are conducive to an initial value problem. We show how the equations reduce to the Vaidya spacetime in a non-null coordinate frame, simply by designating specific equations of state. Furthermore, this reduces to the Schwarzschild spacetime when all matter variables vanish. We then describe the formulation of an initial value problem, whereby a general fluid ball with vacuum exterior is established on an initial spacelike slice. As the system evolves, the fluid ball collapses and emanates null radiation such that a region of Vaidya spacetime develops. Therefore, on any subsequent spacelike slice there exists three regions; general fluid, Vaidya and Schwarzschild, all expressed in a single coordinate patch with two free-boundaries determined by the equations. This implies complicated matching schemes are not required at the interfaces between the regions, instead, one simply requires the matter variables tend to the appropriate equations of state. We also show the reduction of the system of equations to the static cases, and show staticity necessarily implies zero “heat flux.” Furthermore, the static equations include a generalization of the Tolman-Oppenheimer-Volkoff equations for hydrostatic equilibrium to include anisotropic stresses in general coordinates.

Figure 1

Compactification diagram for the collapse of a ball of fluid: $i$ represents spacelike infinity, ${\mathcal{I}}^{+}$ is future null infinity, ${\mathcal{I}}^{+}$ is future timelike infinity and ${\Sigma }_0$ is the initial spacelike hypersurface.

Figure 2

Compactification diagram showing the evolution of spacelike hypersurfaces.

Figure 3

Compactification diagram for static scenarios.