# Readings of the Lichnerowicz-York equation

Research paper by **Niall O Murchadha**

Indexed on: **11 Feb '05**Published on: **11 Feb '05**Published in: **General Relativity and Quantum Cosmology**

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#### Abstract

James York, in a major extension of Andr\'e Lichnerowicz's work, showed how
to construct solutions to the constraint equations of general relativity. The
York method consists of choosing a 3-metric on a given manifold; a
divergence-free, tracefree (TT) symmetric 2-tensor wrt this metric; and a
single number, the trace of the extrinsic curvature. One then obtains a
quasi-linear elliptic equation for a scalar function, the Lichnerowicz-York
(L-Y) equation. The solution of this equation is used as a conformal factor to
transform the data into a set that satisfies the constraints. If the manifold
is compact and without boundary, one quantity that emerges is the volume of the
physical space. This article reinterprets the L-Y equation as an eigenvalue
equation so as to get a set of data with a preset physical volume. One chooses
the conformal metric, the TT tensor, and the physical volume, while regarding
the trace of the extrinsic curvature as a free parameter. The resulting
equation has extremely nice uniqueness and existence properties. A even more
radical approach would be to fix the base (conformal) metric, the physical
volume, and the trace. One also selects a TT tensor, but one is free to
multiply it by a constant(unspecified). One then solves the L-Y equation as an
eigenvalue equation for this constant. A third choice would be to fix the TT
tensor and and multiply the base metric by a constant. Each of these three
formulations has good uniqueness and existence properties.