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The Einstein equations in wave coordinates are an example of a systemÌýwhich does not obey the "null condition". This leads to manyÌýdifficulties, most famously when attempting to prove global existence,Ìýotherwise known as the "nonlinear stability of Minkowski space".ÌýPrevious approaches to overcoming these problems suffer from a lack ofÌý generalisability - among other things, they make the a priori assumptionÌýthat the space is approximately scale-invariant. Given the currentÌýinterest in studying the stability of black holes and other relatedÌýproblems, removing this assumption is of great importance.

The p-weighted energy method of Dafermos and Rodnianski promises toÌýovercome this difficulty by providing a flexible and robust tool toÌýprove decay. However, so far it has mainly been used to treat linearÌýequations. In this talk I will explain how to modify this method so thatÌýit can be applied to nonlinear systems which only obey the "weak null condition" - a large class of systems that includes the EinsteinÌýequations. This involves combining the p-weighted energy method withÌýmany of the geometric methods originally used by Christodoulou andÌýKlainerman. Among other things, this allows us to enlarge the class ofÌýwave equations which are known to admit small-data global solutions, it gives a new proof of the stability of Minkowski space, and it also yields a detailed description of null infinity. In particular, in some situations we can understand the geometric origin of the slow decay towards null infinity exhibited by some of these systems: it is due to the formation of "shocks at infinity".