In this talk, I will introduce a new proof of the stability of Minkowski spacetime in exterior regions first obtained in the seminal monograph by Klainerman and Nicolò.

In 1993, Chirstodoulou and Klainerman proved the global stability of Minkowski spacetime by maximal foliation. In 2002, Klainerman and Nicolò gave a second proof of the stability of Minkowski not in the whole region, but restricting outside of a future cone by double null foliation. Klainerman and Nicolò estimated third order derivatives of Ricci coefficients \(\Gamma\) and second order derivatives of curvature components \(R\). In this new proof, we only consider their first order derivatives. Since we have the same order of derivatives of \(\Gamma\) and \(R\), we use geodesic foliation instead of canonical foliation on the last slice. One essential step in the proof of Klainerman and Nicolò is using the vectorfield method to estimate the curvature components \(R\). For this step, we apply the \(r^p\)-weighted estimate, a new method introduced by Dafermos and Rodnianski in 2010.

I will start with an introduction of the Einstein-vacuum equations and the Minkowski spacetime. Then I will introduce the wave coordinates introduced by Lindblad and Rodnianski, relative to which the Einstein-vacuum equations are equivalent to a reduced system comprising quasilinear wave equations. Next, I will introduce the method of Klainerman invariant vectorfield and its applications to the wave equations. Afterward, I will talk about the double null foliations of spacetime, which induce the null structure equations and Bianchi equations. Finally, I will present my main result and try to introduce the main ideas of its proof.