Ngô Quốc Anh

December 10, 2012

Construction of spacetimes via solutions of the vacuum Einstein constraint equations and the propagation of the gauge condition

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 23:10

A couple of days ago, we showed that the Einstein equations are essentially hyperbolic under the harmonic gauge. To be precise, the solvability of the equations

\displaystyle {\overline {{\text{Ric}}}} - \frac{1}{2}\overline g \text{Scal}_{\overline g} +\Lambda \overline g = \kappa\overline T,

is equivalent to solving the following hyperbolic system

\displaystyle - \frac{1}{2}{\overline g ^{km}}{\overline g _{ij,km}} = \Psi_{ij}((\overline g_{pq})_{0\leqslant p,q \leqslant n},(\overline g_{pq,r})_{0\leqslant p,q,r \leqslant n}),

provided \displaystyle {\lambda ^\alpha } = 0 for all \alpha = \overline {0,n} . Later on, when we consider the Cauchy problem for the Einstein equations on some appropriate framework, the first and second fundamental forms need to verify some constraint equations, [here and here].

By tracing the above equation, we obtain

\displaystyle \text{Scal}_{\overline g} + \frac{2n}{2 - n}\Lambda = \frac{2\kappa}{2 - n}\text{trace}_{\overline g}(\overline T),

which helps us to write

\displaystyle\overline {{\text{Ric}}} = \kappa \left( {\overline T - \frac{\overline g}{{n - 2}}{\text{trace}}_{\overline g}(\overline T )} \right) + \frac{2}{{n - 2}}\Lambda \overline g.

In the vacuum case, the above equation is nothing but

\displaystyle {\overline {{\text{Ric}}}}=\frac{2}{{n - 2}}\Lambda \overline g.

(more…)

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