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The Initial Value Problem in General Relativity

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Springer Handbook of Spacetime

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Abstract

One of the most effective ways of constructing and studying solutions of Einstein’s gravitational field equations is via the Initial Value Problem. According to this approach, one constructs spacetime solutions by choosing initial data on a spacelike manifold representing the initial state of a model universe, and one then evolves the data into a spacetime solution representing the full history of that model universe.

A set of initial data cannot be chosen freely: it must satisfy a set of partial differential equations known as the Einstein constraint equations. Not only are these constraint equations a necessary condition on initial data sets; they are as well a sufficient condition for an initial data set to admit evolution into a spacetime solution. After showing how to split the full set of Einstein’s field equations into the constraint equations and the evolution equations, we discuss the Well-Posedness Theorem, which shows that indeed all constraint-satisfying data sets can be evolved into spacetime solutions.

Our primary focus is on how to construct and parametrize initial data sets which satisfy the Einstein constraint equations. The Conformal and the Conformal Thin Sandwich Methods both provide ways of turning the constraint equations into a determined nonlinear elliptic system. These equivalent procedures are very effective for initial data sets which involve constant mean curvature or near-constant mean curvature. The challenge is to adapt these methods to more general data sets. An alternative approach for constructing and analyzing solutions of the constraints is via Gluing techniques, which we briefly outline, along with their remarkable applications.

We comment briefly on some of the main questions which arise in studying the long-time behavior of spacetime solutions of Einstein’s equations.

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Abbreviations

ADM:

Arnowitt, Deser, Misner

CMC:

constant mean curvature

CTS:

conformal thin sandwich

LCBY:

Lichnerowicz, Choquet-Bruhat, York

NUT:

Newman, Unti, Tamburino

PDE:

partial differential equation

SCC:

strong cosmic censorship

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, OR 97403, Eugene, USA

    James Isenberg Professor

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  1. James Isenberg Professor
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Correspondence to James Isenberg Professor .

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Editors and Affiliations

  1. Department of Physics, Pennsylvania State University, 16802, University Park, PA, USA

    Abhay Ashtekar

  2. Institute for Foundational Studies Hermann Minkowski, Montreal, Quebec, Canada

    Vesselin Petkov

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Isenberg, J. (2014). The Initial Value Problem in General Relativity. In: Ashtekar, A., Petkov, V. (eds) Springer Handbook of Spacetime. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41992-8_16

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  • DOI: https://doi.org/10.1007/978-3-642-41992-8_16

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