The momentum constraints of general relativity and spatial conformal isometries

Transverse-tracefree (TT-) tensors on (R ³,g _ab), withg _ab an asymptotically flat metric of fast decay at infinity, are studied. When the source tensor from which these TT tensors are constructed has fast fall-off at infinity, TT tensors allow a multipole-type expansion. Wheng _ab has no conformal Killing vectors (CKV's) it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable TT tensor. When CKV's exist there are obstructions — certain (combinations of) moments have to vanish — which we study.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P9376-PHY.Partially supported by Forbairt Grant SC/94/225.     

Gravitational potentials: A constructive approach to general relativity

Recent investigations of the initial-value problem of general relativity have shown that the initial-value constraints can be formulated in all cases as a system of elliptic equations with well-defined physical and mathematical properties. The solutions of these equations can be regarded as generalized gravitational potentials. These potentials are interrelated and depend on their sources quasilinearly. They are particularly useful in analyzing asymptotically flat solutions of Einstein's equations. We have found from these results (1) a technique for constructing physically meaningful initial data in the integration of Einstein's equations, and (2) a method for characterization and analysis of the spacelike mass, momentum, angular momemtum, and multipole moments of gravitational fields.This essay received the third award from the Gravity Research Foundation for the year 1975. (Editor.)The research for this paper was partially supported by National Science Foundation grant GP43909 awarded to the University of North Carolina, Chapel Hill, North Carolina.     

Morphological Transition of Schooling in 19th-Century France

The transformation over time of level sets—the sets of states for which a given function takes a given value—has no reason to be regular, especially in social science, where the structure of these level sets of a given variable may be altered by a fluctuating demography or disrupted by unexpected events, and where the actors contribute to shaping their space through investment policies. The pioneering construction of graphic derivatives of set-valued maps renews the understanding of dynamic level sets in social science. The exemplary case of schooling in 19th-century France shows the set derivatives of level sets and illustrates how the Franco-Prussian War of 1870–1871 disturbed the directions taken in 1867. A comparison of the observed changes of level sets with those observed under rational theoretical policies leads to the conclusion that the actors behaved with the ambition of optimizing the overall schooling incidence while reducing the gap between boys and girls.