Spherically symmetric free fall collapse

Abstract

The general dynamical equations for spherical gravitational collapse are derived by introducing the eigenvalue of the conformal Weyl tensor in the 2-2 component of the Einstein tensor and assuming the material content of the models to be a perfect fluid. Since this eigenvalue is coupled always with the material energy density, it has been interpreted as theenergy density of the free gravitational field whose presence is related with anisotropy and inhomogeneity. As a particular case, the collapse of a spherically symmetric dust (zero pressure) with vanishing radial acceleration (free fall collapse) is discussed. It is shown that the model is inhomogeneous with non-vanishing shear of the congruence of world lines of the dust particles. The model contains gravitational radiation by Szekere’s criterion since both shear invariant and the spatial gradient of density are non-vanishing. This is in contrast to the Oppenheimer-Synder model for which both the above mentioned characteristics are absent. A particular solution which is anisotropic and inhomogeneous has been given to prove the emission of gravitational radiation by the freely falling dust and in this case the energy density of the free gravitational field contains a typeN term superposed on the coulombian field.