Role of anisotropy and inhomogeneity in Lemaitre-Tolman-Bondi collapse
Mena, Filipe and Nolan, Brien C. and Tavakol, Reza (2004) Role of anisotropy and inhomogeneity in Lemaitre-Tolman-Bondi collapse. Physical Review D, 70 (8). 084030-1-084030-10. ISSN 1550-7998
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We study the effects of shear and density inhomogeneities in the formation of naked singularities in spherically symmetric dust space-times. We find that in general, complete knowledge of either one of these (time-dependent) functions, along with physically motivated boundary conditions, fully determines the entire space-time and consequently uniquely specifies the end-state of the gravitational collapse. In contrast–and of more physical relevance–we show that measurements of shear or density inhomogenity at an instant of time do not uniquely determine the outcome of the collapse. We do this by (i) showing that, for open sets of initial data, the same initial shear (or initial density contrast) can give rise to both naked and covered solutions, in particular, this can happen for zero initial shear or zero initial density contrast; (ii) demonstrating that asymptotically (near the singularity) both shear and density contrast are invariant under a one parameter set of linear transformations acting on the initial data set; and (iii) showing that asymptotically one cannot in general establish a direct relationship between the rate of change of shear (or density contrast) and the nature of the singularities. However, one can uniquely determine the nature of the singularity if both the initial shear and initial density contrast are known. These results are important in understanding the effects of the initial physical state and, in particular, the role of shear in determining the end-state of the gravitational collapse.
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