DIFFERENTIAL GEOMETRY/PDE SEMINAR

Wednesday, November 8 (Note: Change of Date) Padelford C-36 3:45 PM

Gluing and wormholes for the Einstein constraint equations

Daniel Pollack (UW)

Initial data for Einstein’s general relativistic field equations for the gravitational field consist of a Riemannian metric γ and a symmetric tensor K specified on a three-manifold Σ^{3}, with γ and K satisfying, the vacuum constraint equations

div_{γ}K − ∇trK = 0
R_{γ} − |K|_{γ}^{2} + (trK)^{2} = 0

Here ∇ is the covariant derivative corresponding to γ, div_{γ} is the divergence, R is the scalar curvature, |K|_{γ}^{2} is the squared norm of K and trK is the trace of K with respect to γ. In 1952, Choquet-Bruhat proved that Einstein’s vacuum field equations G_{μν} ≡ R_{μν} − \frac{1}{2}R g_{μν} = 0, for μ, ν ∈ \{0, 1, 2, 3\}, form a locally well posed hyperbolic system. Thus for any choice of (Σ^{3}, γ, K) satisfying the vacuum constraint equations, there exists an ϵ > 0 and a Lorentz metric g defined on the spacetime manifold Σ^{3} × (−ϵ, +ϵ), with g satisfying the Einstein equations, with γ being the induced intrinsic metric of the hypersurface Σ^{3} × {0}, and with K being the induced second fundamental form. Using this result, one can seek to understand solutions to the field equations by understanding solutions to the vacuum constraint equations.

We will present a general gluing construction for solutions to the vacuum constraint equations with trK = τ constant. Using the “conformal method” of Choquet-Bruhat, Lichnerowicz and York, this involves solving a semi-coupled system consisting of a linear elliptic system for a vector field and a scalar semilinear elliptic equation for the induced metric. Applications of the construction to connected sums of space-times and the existence of wormholes will be given.

This is joint work with Jim Isenberg and Rafe Mazzeo.