# 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.