| We discuss center of mass for asymptotically flat manifolds satisfying the Regge-Teitelboim condition from two different points of view: the Hamiltonian formulation as flux integrals at infinity, and the geometric description using foliations by surfaces with constant mean curvature. The equivalence of those different notions of center of mass is also proven. Physicists have proposed a notion of center of mass as a flux integral at infinity from the centroid for the distribution of energy. We propose another flux integral for center of mass involving the three dimensional Einstein tensor. This notion is more intrinsic because it has a coordinate-free expression and natural properties. Moreover, it is equivalent to the previous one. The main tool is a new density theorem for data satisfying the Regge-Teitelboim condition.;The new density theorem says that the solutions with harmonic asymptotics to the constraint equations are dense among solutions which satisfies the Regge-Teitelboim condition in some weighted Sobolev spaces. Moreover, mass, linear momentum, center of mass, and angular momentum converge to the ones from the original initial data.;Using this density theorem as one ingredient, we relate some integrals involving mean curvature of Euclidean spheres to center of mass. Then we prove the existence and uniqueness of foliations by surfaces with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition. We also show that the foliation is asymptotically concentric, and its geometric center is equal to the other two notions of center of mass. In addition, the unique constant mean curvature foliation may provide a polar coordinate system for such manifolds, and potentially enable us to understand their geometric structure. |
