- #1
Psychosmurf
- 23
- 2
I need some help with a derivation in GR.
The linearized field equation in GR is:
[tex]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}}[/tex]
How would I use the gauge transformation
[tex]{\gamma _{ab}} \to {\gamma _{ab}} + {\partial _b}{\xi _a} + {\partial _a}{\xi _b}[/tex]
to simplify the linearized equation to:
[tex]{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} = - 16\pi {T_{ab}}[/tex]?
EDIT: I should also mention:
[tex]{{\bar \gamma }_{ab}} = {\gamma _{ab}} - \frac{1}{2}{\eta _{ab}}\gamma [/tex]
[tex]\gamma = \gamma _a^a[/tex]
and
[tex]{g_{ab}} = {\eta _{ab}} + {\gamma _{ab}}[/tex]
The linearized field equation in GR is:
[tex]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}}[/tex]
How would I use the gauge transformation
[tex]{\gamma _{ab}} \to {\gamma _{ab}} + {\partial _b}{\xi _a} + {\partial _a}{\xi _b}[/tex]
to simplify the linearized equation to:
[tex]{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} = - 16\pi {T_{ab}}[/tex]?
EDIT: I should also mention:
[tex]{{\bar \gamma }_{ab}} = {\gamma _{ab}} - \frac{1}{2}{\eta _{ab}}\gamma [/tex]
[tex]\gamma = \gamma _a^a[/tex]
and
[tex]{g_{ab}} = {\eta _{ab}} + {\gamma _{ab}}[/tex]