On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Abstract

We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a equivariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau (Invent Math 124:281–311, 1996). We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau (Invent Math 124:281–311, 1996) which were described by Cederbaum and Nerz (Ann Henri Poincaré 16:1609–1631, 2015).

Local space time constant mean curvature and constant expansion foliations

“…The non time symmetric case has been a more elusive problem with different attempts. Some of these were trying to find a generalization to the CMC foliation of Huisken and Yau like the constant expansion foliation or the STCMC foliation proposed by Cederbaum and Sakovich [4]. For a more extensive exposition of the problem see [5,6] and [3].…”

“…ii) It is important to note that in the asymptotically Euclidean case [4] the STCMC surfaces encode the information about the center of mass, however the physically relevant quantity is the center of these surfaces, in our case the center of the surfaces would be trivially p.…”

“…and Sakovich [4] proposed a different generalization of the CMC foliation and found what appears to be the surfaces that characterize the center of mass of an isolated system. They show that for an asymptotically flat initial data set with non-vanishing ADM energy, there exist an unique foliation of 2-spheres {Σ r } r>r 0 of constant space-time mean curvature (STCMC).…”

“…First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].…”

Ricci flow on surfaces along the standard lightcone in the $3+1$-Minkowski spacetime

“…and if H 2 is constant along Σ, i.e., H has constant Lorentzian length along Σ, we call Σ a surface of constant spacetime mean curvature (STCMC), cf. Cederbaum-Sakovich [7]. Finally, we define the connection one-form ζ as…”