Symmetry operators

Thomas Bäckdahl

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 Continue reading

Non-CMC solutions to the constraints on AE manifolds

Caleb Meier

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

Niall O'Murchadha

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 Continue reading