The vacuum Einstein equations via holonomy around closed loops on characteristic surfaces

We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy H around certain closed null loops on characteristic surfaces and the light cone cut function Z, which describes the intersection of the future null...

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Bibliographic Details
Published in:Journal of geometry and physics 1996-06, Vol.19 (2), p.151-172
Main Authors: Iyer, Savitri V., Kozameh, Carlos N., Newman, Ezra T.
Format: Article
Language:English
Subjects:
Characteristic surfaces
Closed loops
Holonomy
Vacuum Einstein equations
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Summary:We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy H around certain closed null loops on characteristic surfaces and the light cone cut function Z, which describes the intersection of the future null cones from arbitrary space-times points, with future null infinity. We obtain a set of differential equations for H and Z equivalent to the vacuum Einstein equations. By finding an algebraic relation between H and Z and integrating a linear o.d.e. these equations are reduced to just two coupled equations: an integro-differential equation for Z which yields the conformal structure of the underlying space-time and a linear differential equation for the “vacuum” conformal factor. These equations, which apply to all vacuum asymptotically flat space-times are however lengthy and complicated. They nevertheless are amenable to an attractive perturbative scheme which has Minkowski space as a zeroth order solution.
ISSN:0393-0440
1879-1662
DOI:10.1016/0393-0440(95)00031-3