# Derivation of gauge condition in linearized GR

by WannabeNewton
Tags: condition, derivation, gauge, linearized
 C. Spirit Sci Advisor Thanks P: 5,637 Hey there guys! So we know that in linearized GR we work with small perturbations $\gamma _{ab}$ of the background flat minkowski metric. In deriving the linearized field equations the quantity $\bar{\gamma _{ab}} = \gamma _{ab} - \frac{1}{2}\eta _{ab}\gamma$ is usually defined, where $\gamma = \gamma ^{a}_{a}$. Under the action of an infinitesimal diffeomorphism (generator of flow), $\gamma _{ab}$ transforms as $\gamma' _{ab} = \gamma _{ab} + \partial _{b}\xi _{a} + \partial _{a}\xi _{b}$ (this comes out of the lie derivative of the minkowski metric with respect to the flow generated by this vector field). This implies that $\bar{\gamma' _{ab}} = \bar{\gamma _{ab}} + \partial _{b}\xi _{a} + \partial _{a}\xi _{b} - \eta _{ab}\partial^{c}\xi _{c}$. Since we have the freedom to then fix the gauge by choosing some $\xi ^{a}$, we can take one satisfying $\partial ^{b}\partial _{b}\xi _{a} = -\partial ^{b}\bar{\gamma _{ab}}$ which, after differentiating the expression for $\bar{\gamma' _{ab}}$, gives $\partial^{b}\bar{\gamma' _{ab}} = 0$. Apparently we can then conclude from this that $\partial^{b}\bar{\gamma _{ab}} = 0$ but why is that? Is it because in a background flat space - time we can regard $\partial^{b}\bar{\gamma' _{ab}} = 0$ as a covariant equation due to being able to treat $\triangledown _{a}$ as $\partial _{a}$ therefore, since $\bar{\gamma '_{ab}}, \bar{\gamma _{ab}}$ are related by a diffeomorphism, the equation must remain invariant under the transformation $\bar{\gamma '_{ab}}\rightarrow \bar{\gamma _{ab}}$ (in the context of GR)?
 Sci Advisor Thanks P: 4,160 I don't see the difference. You use the gauge transformation to put γab into the Hilbert gauge and then just drop the prime.
 C. Spirit Sci Advisor Thanks P: 5,637 Hi Bill! Thanks for responding. My question is why are we allowed to drop the prime? Thanks again mate.
 Sci Advisor P: 8,658 Derivation of gauge condition in linearized GR Seems to be a discussion after Eq 20, 21 of http://www.tapir.caltech.edu/~chirata/ph236/lec08.pdf. Something about non-uniqueness of the gauge.