Nonexistence of apparent horizons for elongated configurations of matter having small mass

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Abstract
The authors considers a time-symmetric, conformally flat and axisymmetric Cauchy hypersurface Sigma . It is assumed that the distribution of matter present in Sigma is strongly elongated in the direction of the symmetry axis and has compact support. Let L denote the length of this configuration as measured in the flat background space, and M the ADM mass associated with Sigma . It is shown that if the ratio M/L is sufficiently small, then such a configuration of matter cannot be entirely surrounded by an apparent horizon. The proof makes use of the second variation formula. Moreover, if the Penrose inequality is true, one can obtain more quantitative estimation of this ratio: if M<or=L/96, the an apparent horizon, entirely enclosing the configuration of matter, cannot exist.