Quasi-local masses in General relativity and their positivity: Spinor approach

We study the quasi-local masses arising in general relativity using spinors and prove their positivity property. This leads to the question of a pure quasi-local proof of the positivity of the Wang-Yau \cite{yau} quasi-local mass. More precisely we prove that the gravitational mass bounded by a spacelike topological $2-$sphere is non-negative in a generic spacetime verifying dominant energy condition and vanishes only if the surface is embedded in the Minkowski space. This construction is purely quasi-local in nature and in particular does not rely on Bartanik's gluing and asymptotic extension construction \cite{bartnik1993quasi} and subsequent application of the positive mass theorem \cite{schoen1979proof,schoen1981proof} to prove the positivity of quasi-local mass. The result involves solving Dirac equation on a compact Riemannian manifold with boudary using MIT Bag and APS boundary condition.