475 research outputs found
Initial Value Problem in General Relativity
This article, written to appear as a chapter in "The Springer Handbook of
Spacetime", is a review of the initial value problem for Einstein's
gravitational field theory in general relativity. Designed to be accessible to
graduate students who have taken a first course in general relativity, the
article first discusses how to reformulate the spacetime fields and spacetime
covariant field equations of Einstein's theory in terms of fields and field
equations compatible with a 3+1 foliation of spacetime with spacelike
hypersurfaces. It proceeds to discuss the arguments which show that the initial
value problem for Einstein's theory is well-posed, in the sense that for any
given set of initial data satisfying the Einstein constraint equations, there
is a (maximal) spacetime solution of the full set of Einstein equations,
compatible with the given set of data. The article then describes how to
generate initial data sets which satisfy the Einstein constraints, using the
conformal (and conformal thin sandwich) method, and using gluing techniques.
The article concludes with comments regarding stability and long term behavior
of solutions of Einstein's equations generated via the initial value problem.Comment: To appear as a chapter in "The Springer Handbook of Spacetime,"
edited by A. Ashtekar and V. Petkov. (Springer-Verlag, at Press
Constructing Solutions of the Einstein Constraint Equations
The first step in the building of a spacetime solution of Einstein's
gravitational field equations via the initial value formulation is finding a
solution of the Einstein constraint equations. We recall the conformal method
for constructing solutions of the constraints and we recall what it tells us
about the parameterization of the space of such solutions. One would like to
know how to construct solutions which model particular physical phenomena. One
useful step towards this goal is learning how to glue together known solutions
of the constraint equations. We discuss recent results concerning such gluing.Comment: Write-up of my GR16 talk, 18 page
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