Symmetry operators

Thomas Bäckdahl is a Post-Doctoral Research Assistant in the School of Mathematics at the University of Edinburgh.

Conserved quantities, for example energy and momentum, play a fundamental role in the analysis of dynamics of particles and fields. For field equations, one manifestation of conserved quantities in a broad sense is the existence of symmetry operators, i.e. linear differential operators which take solutions to solutions. A well-known example of a symmetry operator for the scalar wave equation is provided by the Lie derivative along a Killing vector field.

It is important to note that other kinds of objects

Non-CMC solutions to the constraints on AE manifolds

Caleb Meier is a postdoctoral researcher in mathematics at the University of California, San Diego.

In the n+1 formalism of general relativity, the (n+1)-dimensional space-time is decomposed into n-dimensional space-like slices that are parametrized by a time function.  This is the basis for formulating Einstein’s equation as an initial value problem.  In an effort to understand which space-times are constructible, an important question is, “What is the admissible class of initial data for this problem?”  This question is addressed by analyzing the so-called Einstein constraint equations, which are an undetermined system of equations to be solved for a metric and an extrinsic curvature tensor.

Non-CMC solutions of the Einstein constraint equations on asymptotically Euclidean manifolds

Niall ‘O Murchadha is an Editorial Board Member for Classical and Quantum Gravity and a Professor of Physics at University College Cork, Ireland

This is a very nice article which deserves to be studied carefully by anyone interested in finding solutions to the constraints. In particular, they show how to construct a solution which is far from maximal, and, at the same time, is asymptotically flat. Readers should be aware that the first theorem, Theorem 1.1, covers a much broader range of data than the second theorem, Theorem 1.2. Further, they should be aware that the titles of the theorems ‘Far-from-CMC’ (Theorem 1.1), and ‘Near-CMC’ (Theorem 1.2), especially the second one, are not particularly illuminating.

There are conditions which are surprising. No restriction is placed on Τ² (other than the AF condition), but we are asked