# Field Equation of Einstein’s General Relativity

Gauge freedom is the freedom of choosing a coordinate system. Fixing a gauge means choosing a particular coordinate system.

# Perturbation Theory of General Relativity

Gauge tranformation is Lie derivative along some arbitary vector here.

Line element

\begin{eqnarray} \tilde g _ {00} &=& -a^2(1+2 A Y) \newline \tilde g _ {0j} &=& -a^2 B Y _ j \newline \tilde g _ {ij} &=& a^2(\gamma _ {ij} +2 H _ L Y \gamma _ {ij} +2 H _ T Y _ {ij} ) \end{eqnarray}

Energy momentum tensor is

\begin{eqnarray} \tilde T^0 _ {\phantom{0}0} = -\rho (1+\delta Y) \newline \tilde T^0 _ {\phantom{0} j} = (\rho + p)(v - B) Y \newline \tilde T^j _ {\phantom{j}0 } = -(\rho + p)v Y^{j} \end{eqnarray}

For a infinitesimal gauge transformation along some vector (X=T\partial _ t + L^i \partial _ i), gauge variables are

Symbol Physics Gauge Transformation Note
$\tilde A$

Through that we can find out gauge invariant variables