Hey there guys! So we know that in linearized GR we work with small perturbations [itex]\gamma _{ab}[/itex] of the background flat minkowski metric. In deriving the linearized field equations the quantity [itex]\bar{\gamma _{ab}} = \gamma _{ab} - \frac{1}{2}\eta _{ab}\gamma [/itex] is usually defined, where [itex]\gamma = \gamma ^{a}_{a}[/itex]. Under the action of an infinitesimal diffeomorphism (generator of flow), [itex]\gamma _{ab}[/itex] transforms as [itex]\gamma' _{ab} = \gamma _{ab} + \partial _{b}\xi _{a} + \partial _{a}\xi _{b}[/itex] (this comes out of the lie derivative of the minkowski metric with respect to the flow generated by this vector field). This implies that [itex]\bar{\gamma' _{ab}} = \bar{\gamma _{ab}} + \partial _{b}\xi _{a} + \partial _{a}\xi _{b} - \eta _{ab}\partial^{c}\xi _{c}[/itex]. Since we have the freedom to then fix the gauge by choosing some [itex]\xi ^{a}[/itex], we can take one satisfying [itex]\partial ^{b}\partial _{b}\xi _{a} = -\partial ^{b}\bar{\gamma _{ab}}[/itex] which, after differentiating the expression for [itex]\bar{\gamma' _{ab}}[/itex], gives [itex]\partial^{b}\bar{\gamma' _{ab}} = 0[/itex]. Apparently we can then conclude from this that [itex]\partial^{b}\bar{\gamma _{ab}} = 0[/itex] but why is that? Is it because in a background flat space - time we can regard [itex]\partial^{b}\bar{\gamma' _{ab}} = 0[/itex] as a covariant equation due to being able to treat [itex]\triangledown _{a}[/itex] as [itex]\partial _{a}[/itex] therefore, since [itex]\bar{\gamma '_{ab}}, \bar{\gamma _{ab}}[/itex] are related by a diffeomorphism, the equation must remain invariant under the transformation [itex]\bar{\gamma '_{ab}}\rightarrow \bar{\gamma _{ab}}[/itex] (in the context of GR)?(adsbygoogle = window.adsbygoogle || []).push({});

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# Derivation of gauge condition in linearized GR

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