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Applications of Vesture of Harmonic Maps to General Relativity and Cosmology

11 December 2013

Time: 4:30pm
Venue: Maths 103
The London Relativity and Cosmology Seminar
Shabnam Behesti (Rutgers)

How customizable are solutions to the Einstein Equations, i.e., is it possible to construct spacetimes with a prescribed set of asymptotic observables (e.g., total mass and angular momentum) and a prescribed causal and/or singular structure (e.g. number of ring singularities), given a specified set of initial parameters?  In this talk, we provide the first steps in answer to such a question.

First, we review two well-known Killing field reductions of Einstein’s equations, unifying them using the framework of harmonic maps.  Two general theorems on integrability and vesture (or dressing) of harmonic maps into symmetric spaces are presented.  We then place the gravitational field equations in this context by combining the dressing technique with a control-theory approach. 

An example of the technique is carried out explicitly for the Einstein vacuum equations, where a novel asymptotic expansion demonstrates that given any real numbers M > 0 and J, it is indeed possible to treat the arbitrary constants of the dressing procedure as control parameters and produce a 1-solitonic harmonic map that, when viewed as a stationary axisymmetric spacetime metric, has total ADM mass equal to M and total ADM angular momentum equal to J, and is otherwise free of unwanted pathologies. 

We shall discuss how this approach may be modified to address cosmological models and other geometric field theories of physical interest.

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