Ngô Quốc Anh

January 18, 2010

R-G: Gauss and Codazzi equations in general relativity

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 0:01

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold $(V,g)$. This means the smooth Lorentz metric $g$ has signature $(3,1)$. We are interested in the Cauchy problem for the Einstein equations. For that reason, we will consider the initial data on a hypersurface $\Sigma$, a $3$-dimensional manifold which is a space-like.

The Einstein equations are just equations for a metric $g$ defined throughout $V$ such that the induced matric $\overline g$ onto $\Sigma$ and the second fundamental form $k$ of $\Sigma$ are identical to the initial data. Since $\Sigma$ has $1$-dimention less than that of $V$, the initial data can not chosen arbitrary, they must satisfy the Gauss and Codazzi equations.

The purpose of this entry is to derive the constraint equations for the Einstein equations. For simplicity, we consider the case when $n=3$, then the Gauss and Codazzi equations can be rewritten as the following

$\displaystyle ^{(4)}{R_{abcd}} = {R_{abcd}} + {k_{ac}}{k_{bd}} - {k_{ad}}{k_{bc}}$

and

$\displaystyle ^{(4)}{R_{\sigma bcd}}{n^\sigma } =- {\nabla _c}{k_{ab}}+{\nabla _b}{k_{ac}}$

where $n^\alpha$ the normal vector to the space-like $\Sigma$.

Further equations can be obtained from these by contraction. Indeed, from the Gauss equation, one has

$\displaystyle ^{(4)}{R_{ab}} = {R_{ab}} + {\rm trace } k{k_{ab}} - {k_{ac}}k_b^c{ - ^{(4)}}{R_{\sigma a\tau b}}{n^\sigma }{n^\tau }$

and contracting again gives

$\displaystyle ^{(4)}R + {2^{(4)}}{R_{\alpha \beta }}{n^\alpha }{n^\beta } = R - {\left( {\rm trace }k \right)^2} - {k_{ab}}{k^{ab}}$.

Similarly, contracting the Codazzi equation gives

$\displaystyle ^{(4)}{R_{\sigma a}}{n^\sigma } =- {\nabla ^a}{k_{ab}} + {\nabla _b}{\rm trace } k$.

Combining all equations with the Einstein equations gives the constrain equations

$\displaystyle\begin{gathered}R - {\left( {\rm trace }k \right)^2} - {k_{ab}}{k^{ab}} =\cdots\hfill \\{\nabla ^a}{k_{ab}} - {\nabla _b}{\rm trace } k =\cdots\hfill \\\end{gathered}$