# Gauge Transformations in GR

1. Nov 14, 2013

### Psychosmurf

I need some help with a derivation in GR.

The linearized field equation in GR is:

$$G_{ab}^{(1)} = - \frac{1}{2}{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} + {\partial ^c}{\partial _{(b}}{{\bar \gamma }_{a)c}} - \frac{1}{2}{\eta _{ab}}{\partial ^c}{\partial ^d}{{\bar \gamma }_{cd}} = 8\pi {T_{ab}}$$

How would I use the gauge transformation

$${\gamma _{ab}} \to {\gamma _{ab}} + {\partial _b}{\xi _a} + {\partial _a}{\xi _b}$$

to simplify the linearized equation to:

$${\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} = - 16\pi {T_{ab}}$$?

EDIT: I should also mention:

$${{\bar \gamma }_{ab}} = {\gamma _{ab}} - \frac{1}{2}{\eta _{ab}}\gamma$$

$$\gamma = \gamma _a^a$$

and

$${g_{ab}} = {\eta _{ab}} + {\gamma _{ab}}$$

2. Nov 16, 2013

### Psychosmurf

Nvm. I figured it out.