Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3 , m 1, are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition, S 15 has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon and Karcher [BK78]. In 1982 Ziller proved that these are the only homogeneous Einstein metrics on spheres [Zil82]. ...
This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in the ﬁrst paper. ...
Institute for Theoretical Physics University of California Santa Barbara, CA 93106 firstname.lastname@example.org December 1997
Abstract These notes represent approximately one semester’s worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: gravitational radiation, black holes, and cosmology. Individual chapters, and potentially updated versions, can be found at http://itp.ucsb.edu/~carroll/notes/.
Table of Contents
This paper considers a trapped characteristic initial value problem for the spherically symmetric Einstein-Maxwell-scalar ﬁeld equations. For an open set of initial data whose closure contains in particular Reissner-Nordstr¨m data, o the future boundary of the maximal domain of development is found to be a light-like surface along which the curvature blows up, and yet the metric can be continuously extended beyond it. This result is related to the strong cosmic censorship conjecture of Roger Penrose.