The most detailed existing proposal for the structure of spacetime singularities originates in the work of Belinskii, Khalatnikov and Lifshitz. We show rigorously the correctness of this proposal in the case of analytic solutions of the Einstein equations coupled to a scalar field or stiff fluid. More specifically, we prove the existence of a family of spacetimes depending on the same number of free functions as the general solution which have the asymptotics suggested by the Belinskii-Khalatnikov-Lifshitz proposal near their singularities. In these spacetimes a neighbourhood of the singularity can be covered by a Gaussian coordinate system in which the singularity is simultaneous and the evolution at different spatial points decouples.

We use the Fuchsian algorithm to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes. The solutions have the maximum number of arbitrary functions. Special cases correspond to polarized, or other known solutions. The method provides precise asymptotics at the singularity, which is Kasner-like. All of these solutions are asymptotically velocity-dominated. The results account for the fact that solutions with velocity parameter uniformly greater than one are not observed numerically. They also provide a justification of formal expansions proposed by Grubišić and Moncrief.

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Confirming previous heuristic analysesà la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain "subcritical" Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D−dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension dependent open neighbourhood of zero [1], while pure gravity is subcritical if D ≥ 11 [13]. Our proof relies, like the recent theorem [15] dealing with the (always subcritical [14]) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are "general" in the sense that they depend on the maximal expected number of free functions.

It is shown that for a given equation of state and a given value of the central pressure there exists a unique global solution of the Einstein equations representing a spherically symmetric static fluid body. For the proof a new theorem on singular ordinary differential equations is established which is of interest in its own right. For a given equation of state and central pressure, the fluid will either fill the entire space or be finite in extent with a vacuum exterior. Criteria are given which allow one to decide for certain equations of state which of these two cases occurs. This generalizes well known results in Newtonian theory and is proved by showing that the relativistic model inherits the property of having a finite radius from a Newtonian model. Parameter-dependent families of relativistic solutions are constructed which have a Newtonian limit in a rigorous sense. The relationship between relativistic and Newtonian equations of state is examined by looking at the example of a degenerate Fermi gas.

In the above paper the authors proved a local existence result for the spherically symmetric Vlasov-Einstein system (Theorem 3.1 ). Unfortunately, the proof contains an error: To estimate J~n in the proof of Lemma 3.3 we had in mind to differentiate the relation (3.4)with respect to t, and use the boundedness of the right-hand side of the characteristic system (3.3) and the "fact" that (X,, Vn) (S,t,x,v) is symmetric in s,t in the sense that (X~, V~) (O,t,x,v) as a function of t solves (3.3) with the signs of the righthand side reversed. This "fact" is wrong, it would be correct only if (3.3) were autonomous. In the following we indicate the main arguments which have to be added to the analysis in the above paper in order to set things right. A detailed exposition of the arguments can be obtained from the first author.As a first step we prove Lemma 3.3. By (3.26) and (3.27) we have to bound I t lips( )H~ and IlPn(t)ll~. Using the Vlasov equation to express J~ in tSn(t,x ) = f g/1 + v2j~n (t,x,v)

In numerical studies of Gowdy spacetimes evidence has been found for the development of localized features (`spikes') involving large gradients near the singularity. The rigorous mathematical results available up to now did not cover this kind of situation. In this work we show the existence of large classes of Gowdy spacetimes exhibiting features of the kind discovered numerically. These spacetimes are constructed by applying certain transformations to previously known spacetimes without spikes. It is possible to control the behaviour of the Kretschmann scalar near the singularity in detail. This curvature invariant is found to blow up in a way which is non-uniform near the spike in some cases. When this happens it demonstrates that the spike is a geometrically invariant feature and not an artefact of the choice of variables used to parametrize the metric. We also identify another class of spikes which are artefacts. The spikes produced by our method are compared with the results of numerical and heuristic analyses of the same situation.

In the above paper the authors proved a local existence result for the spherically symmetric Vlasov-Einstein system (Theorem 3.1). Unfortunately, the proof contains an error: To estimate J~n in the proof of Lemma 3.3 we had in mind to differentiate the relation (3.4) A(t,x,v) =f((x., Vn)(O,t,X,V)) with respect to t, and use the boundedness of the right-hand side of the characteristic system (3.3) and the "fact" that (X,, Vn)(S,t,x,v) is symmetric in s,t in the sense that (X~, V~)(O,t,x,v) as a function of t solves (3.3) with the signs of the right-hand side reversed. This "fact" is wrong, it would be correct only if (3.3) were autonomous. In the following we indicate the main arguments which have to be added to the analysis in the above paper in order to set things right. A detailed exposition of the arguments can be obtained from the first author. As a first step we prove Lemma 3.3. By (3.26) and (3.27) we have to bound I t lips()H~ and IlPn(t)ll~. Using the Vlasov equation to express J~ in tSn(t,x) = f g/1 + v2j~n(t,x,v)dv, integrating the term with 0~f~ by parts and using Lemma 3.2 we get l Ilpn(t)lloo, IIA(t)ll~ =< Cl(t)(1 + I[~f~(t)llo~), t c [0, T[, where C~(t) depends on the functions Z1,Z 2 introduced in Lemma 3.2. Differentiating (3.3) with respect to x and using a Gronwall argument yields the estimate t IOxy.+l(o,t,z)l + I~xVn+~(o,t,z)l ~: exp f Cl(S)(1 + II~xA(~)ll~)ds, 0