A Weak Solution of Einstein Equations for Expanding Dust

Abstract

An expanding spherically symmetric dust cloud is considered in a framework of general relativity. Initial conditions leading to a shell-crossing singularity are chosen. 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. Junction conditions determine the motion of that thin shell. The singular part of dust stress-energy tensor is nontrivial only after the shell-crossing occurs. Before that the solution coincides with Lemaitre -TolmanBondi one. A toy model representing an underdensed region in Universe is discussed. One of the less studied properties of the LTB solution is the formation of shell-crossing singularities (SCS) for certain initial conditions. The cause for it is the intersection of initially different dust layers resulting in diverging and even negative density. The employing of frameworks other then a co-moving one merely brings the metric tensor to a regular form [5] but can't remove the singularity because it is in fact a physical but not a coordinate effect. For that reason the initial conditions leading to a SCS are usually avoided even if it seems unfortunate.The nature of SCS was investigated by different authors [6-9] and the conclusion is that it has a different ("weak" or "inessential") type from a shell-focusing singularity and therefore the solution can be extended beyond the SCS. The first example of such an extension was provided in [6] for a rather special case of spacetime. Further works [8,10] suggest the extension to be a weak solution of Einstein equations. In [11] such weak solutions are derived treating SCS as a shock wave and using Rankine -Hugoniot conditions. Unlike the classical solution the weak one is not unique to the future of the shell-crossing singularity even for well posed initial conditions. There is a weak or extended solution which has singular part in stress-energy tensor and there is still * Electronic address: tegai s f@inbox.ru a classical solution which is a special case of a weak solution with only regular distributions involved.Here we introduce another way to find a weak solution employing Israel -Darmois -Lichnerowicz junction formalism [12]. The idea is to cut out the unphysical regions with negative density and match the remaining parts of the space-time at a thin shell. Of course models with thin shells are nothing new in cosmology and were first studied in [13]. However the important difference is that the thin shell in present work arises from smooth initial conditions.Israel -Darmois -Lichnerowicz matching procedure can be viewed as a consequence of dealing with field equations in a framework of tensor distributions [14]. The same is true for relativistic Rankine -Hugoniot equations which are identical to O'Brien -Singe conditions while written in general form of energy and momentum conservation across the junction surface [15]. Thus relying on the matching scheme one can expect to get the sam...

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“…That said, there is a strategy involving thin shells which does allow us to deal with SCSs. The idea is to sur- gically excise the region containing SCSs and then identify the boundaries of the excised region using the Israel junction conditions: the problematic region is removed and replaced with a thin shell [26].…”