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We propose a new foliation of asymptotically Euclidean initial data sets by2-spheres of constant spacetime mean curvature (STCMC). The leaves of thefoliation have the STCMC-property regardless of the initial data set in whichthe foliation is constructed which asserts that there is a plethora of STCMC2-spheres in a neighborhood of spatial infinity of any asymptotically flatspacetime. The STCMC-foliation can be understood as a covariant relativisticgeneralization of the CMC-foliation suggested by Huisken and Yau.  We show that a unique STCMC-foliation exists near infinity of anyasymptotically Euclidean initial data set with non-vanishing energy whichallows for the definition of a new notion of total center of mass for isolatedsystems. This STCMC-center of mass transforms equivariantly under theasymptotic Poincar\'e group of the ambient spacetime and in particular evolvesunder the Einstein evolution equations like a point particle in SpecialRelativity. The new definition also remedies subtle deficiencies in theCMC-approach to defining the total center of mass suggested by Huisken and Yauwhich were described by Cederbaum and Nerz.

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