Ngô Quốc Anh

December 5, 2012

Why do the Einsteins equations describe the propagation of wavelike phenomena?

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 6:20

In order to formulate the initial value problem for the Einstein equations as nonlinear wave equations, we express the Einstein equations in terms of a partial di erential equation along with a gauge condition.

We suppose that (V,\overline g) is a Lorentzian manifold of the dimension n+1. The dummy indices will be from 0 up to n. In a coordinate system that will be fixed from now on, we have

\displaystyle\overline \Gamma _{ij}^k = \frac{1}{2}{\overline g ^{km}}({\overline g _{im,j}} + {\overline g _{jm,i}} - {\overline g _{ij,m}})

as Christoffel symbols for the metric \overline g.

By lower order terms we mean terms consisting of either no derivative or first order derivative of the metric \overline g. As such, terms consisting of derivatives of order higher than two will be called high order terms.

Let us now introduce the following notation

\displaystyle {\lambda ^\alpha } = {\square _{\overline g }}{x^\alpha } = {\overline g ^{ij}}x_{.ij}^\alpha = {\overline g ^{ij}}( - \overline \Gamma _{ij}^kx_{.k}^\alpha ) = - {\overline g ^{ij}}\overline \Gamma _{ij}^\alpha .

It is obvious to see that

\displaystyle - ({\overline g _{\alpha i}}\lambda _{,j}^\alpha + {\overline g _{\alpha j}}\lambda _{,i}^\alpha ) = {\overline g _{\alpha i}}{\overline g ^{km}}\overline \Gamma _{km,j}^\alpha + {\overline g _{\alpha j}}{\overline g ^{km}}\overline \Gamma _{km,i}^\alpha .

Since

\displaystyle\overline \Gamma _{km,j}^\alpha = \frac{1}{2}{\overline g ^{\alpha p}}({\overline g _{kp,mj}} + {\overline g _{mp,kj}} - {\overline g _{km,pj}}) + \text{lower order terms}

and

\displaystyle\overline \Gamma _{km,i}^\alpha = \frac{1}{2}{\overline g ^{\alpha q}}({\overline g _{kq,mi}} + {\overline g _{mq,ki}} - {\overline g _{km,qi}}) + \text{lower order terms},

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