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},

a further calculation shows that

\displaystyle\begin{array}{lcl} - ({\overline g _{\alpha i}}\lambda _{,j}^\alpha + {\overline g _{\alpha j}}\lambda _{,i}^\alpha ) &=&\displaystyle \frac{1}{2}{\overline g _{\alpha i}}{\overline g ^{km}}{\overline g ^{\alpha p}}({\overline g _{kp,mj}} + {\overline g _{mp,kj}} - {\overline g _{km,pj}}) + \hfill \\ &&\displaystyle\frac{1}{2}{\overline g _{\alpha j}}{\overline g ^{km}}{\overline g ^{\alpha q}}({\overline g _{kq,mi}} + {\overline g _{mq,ki}} - {\overline g _{km,qi}}) + \text{lower order terms}\hfill \\ &= &\displaystyle\frac{1}{2}\delta _i^p{\overline g ^{km}}({\overline g _{kp,mj}} + {\overline g _{mp,kj}} - {\overline g _{km,pj}}) + \frac{1}{2}\delta _j^q{\overline g ^{km}}({\overline g _{kq,mi}} + {\overline g _{mq,ki}} - {\overline g _{km,qi}}) + \text{lower order terms}\hfill \\ &=&\displaystyle \frac{1}{2}{\overline g ^{km}}({\overline g _{ki,mj}} + {\overline g _{mi,kj}} - {\overline g _{km,ij}}) + \frac{1}{2}{\overline g ^{km}}({\overline g _{kj,mi}} + {\overline g _{mj,ki}} - {\overline g _{km,ij}}) + \text{lower order terms}\hfill \\&=&\displaystyle {\overline g ^{km}}({\overline g _{ki,mj}} + {\overline g _{mi,kj}} - {\overline g _{km,ij}}) + \text{lower order terms}. \hfill \\ \end{array}

We now have the components of the Ricci curvature of \overline g given by

\displaystyle\overline{\text{Ric}}_{ij} = \overline \Gamma _{ij,k}^k - \overline \Gamma _{ik,j}^k + \overline \Gamma _{kl}^k\overline \Gamma _{ij}^l - \overline \Gamma _{jl}^k\overline \Gamma _{ik}^l.

A simple observation tells us that \overline \Gamma _{kl}^k\overline \Gamma _{ij}^l - \overline \Gamma _{jl}^k\overline \Gamma _{ik}^l consists of lower order terms only since there is no differentiation there. Therefore, we can write

\displaystyle {\overline {{\text{Ric}}} _{ij}} = \overline \Gamma _{ij,k}^k - \overline \Gamma _{ik,j}^k + \text{lower order terms}.

Moreover,

\displaystyle\begin{array}{lcl}\overline \Gamma _{ij,k}^k - \overline \Gamma _{ik,j}^k &=&\displaystyle \frac{1}{2}{\overline g ^{km}}({\overline g _{im,kj}} + {\overline g _{jm,ki}} - {\overline g _{ij,km}}) - \frac{1}{2}{\overline g ^{km}}({\overline g _{im,kj}} + {\overline g _{km,ij}} - {\overline g _{ik,mj}}) + \text{lower order terms}\hfill \\ &=&\displaystyle \frac{1}{2}{\overline g ^{km}}({\overline g _{jm,ki}} - {\overline g _{ij,km}} - {\overline g _{km,ij}} + {\overline g _{ik,mj}}) + \text{lower order terms}. \end{array}

In other words,

\displaystyle {\overline {{\text{Ric}}} _{ij}} = \frac{1}{2}{\overline g ^{km}}({\overline g _{jm,ki}} - {\overline g _{ij,km}} - {\overline g _{km,ij}} + {\overline g _{ik,mj}}) + \text{lower order terms}.

From this identity and our calculation for {\overline g _{\alpha i}}\lambda _{,j}^\alpha + {\overline g _{\alpha j}}\lambda _{,i}^\alpha , we find that

\displaystyle {\overline {{\text{Ric}}} _{ij}} + \frac{1}{2}({\overline g _{\alpha i}}\lambda _{,j}^\alpha + {\overline g _{\alpha j}}\lambda _{,i}^\alpha ) = - \frac{1}{2}{\overline g ^{km}}{\overline g _{ij,km}} + \text{lower order terms}.

In view of the Einstein equations which have the following form

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

we obtain

\displaystyle - \frac{1}{2}{\overline g ^{km}}{\overline g _{ij,km}} = \frac{1}{2}({\overline g _{\alpha i}}\lambda _{,j}^\alpha + {\overline g _{\alpha j}}\lambda _{,i}^\alpha ) + \underbrace {\overline T_{ij} + \frac{1}{2}\overline g {\text{Scal}}_{\overline g} + \text{lower order terms}}_{\text{lower order terms}}.

In order to get rid of high order terms on the right hand side of the preceding equation, it is usually to assume that

\displaystyle {\lambda ^\alpha } = 0, \quad \forall\alpha = \overline {0,n} .

This condition is preferred to the so-called harmonic coordinates. In the literature, this belongs to a set of a few condition that one can solve the Einstein equations.

Thus, we have shown that the Einstein equations for \overline g in the harmonic gauge are nothing but wave equations for \overline g.

I wish to conclude this topic by giving out the following two remarks:

  • On any Lorentzian manifold we can locally set up harmonic coordinates by simply solving the Cauchy problem for some linear wave equations, I leave it here;
  • The solvability of the transformed wave-like system as shown above is well-understood by the seminal work of Leray.

Reference: Justin Corvino, Introduction to General Relativity and the Einstein Constraint Equations, Lecture notes.

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