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A weak solution of Einstein equations for expanding dust

An expansion of a spherically symmetric dust cloud is considered with initial conditions leading to a shell-crossing singularity. The way to construct a weak solution for such a case is proposed. Suggested method consists in cutting off the region containing the shell-crossing and matching the remaining parts of space-time at a thin shell. The motion of that thin shell is then found from junction conditions. The stress-energy tensor of the extended solution has singular part which is nontrivial only after the shell-crossing occurs. Before that the solution coinsides with Lemaitre – Tolman - Bondi one.

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