Master Degree Project Presentation: On the notion of center of mass in General Relativity

Abstrakt: In this presentation, we review and compare different notions of the center of mass of isolated gravitating systems in General Relativity. Such systems are modelled using asymptotically Euclidean initial data for the Einstein equations consisting of a Riemannian manifold (M,g) equipped with a symmetric 2-tensor K such that g approaches the Euclidean metric and K approaches zero near infinity. In this context, we focus on three notions of center of mass: 1) Hamiltonian definition of Beig and Ó Murchadha, 2) definition of Huisken and Yau using constant mean curvature surfaces, and 3) definition of Sakovich and Cederbaum using spacetime constant mean curvature surfaces. After reviewing these definitions we discuss their advantages and disadvantages focusing on properties such as convergence (well-definiteness) and coordinate equivariance. For this, we use an example, originally due to Cederbaum and Nerz, of asymptotically Euclidean data of critical order induced on a graph in Schwarzschild spacetime. In this example we can see that the definitions 1) and 2) that both only depend on (M,g) but not on K give a divergent center of mass, while the definition 3) that also takes K into account yields a convergent center of mass. In particular, we can see that the term depending on K in definition of Sakovich and Cederbaum compensates for terms that yield divergent flux integrals in the definition of Beig and Ó Murchadha. In conclusion, it appears that extrinsic geometry of initial data for Einstein equations represented by tensor K should not be neglected when aiming at an adequate definition of center of mass for isolated gravitational systems.