How to derive the hidden symmetry behind linearized gravitation equations?

Question

I am trying to derive the "gauge-like" symmetry of linearized gravitation equation, after deriving the latter heuristically from Newton's universal of gravitation. I am roughly following Excercise 7.3 of MTW and the box that follows. (Roughly-because the derivation of linearized gravity done there,is from an action principle, but I did the same heuristically from Newton's law of gravitation,but that is of little concern here probably.)

The linearized gravitation equation is: $$\Box \bar h_{\mu\nu}+\eta_{\mu\nu}\partial^\alpha \partial^\beta \bar h_{\alpha\beta}-\partial^\alpha\partial_\mu \bar h_{\alpha\nu} -\partial^\alpha\partial_\nu \bar h_{\alpha\mu}=16\pi G T_{\mu\nu}\tag{1}$$

The symmetry equation I am concerned with is: $$\delta \bar h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu -\eta_{\mu\nu}\partial^\alpha\xi_\alpha\tag{2}$$for some arbitrary vector $\xi^\alpha$.

I know that because (2) is given in the book,and hence I can verify it by putting in (1) and doing tedious calculation involving a dozen terms!

What I want to know is how did the author come to know about the symmetry equation (2)? How to obtain the symmetry equation (2) from the main equation (1)?

I tried to follow the path in which gauge invariance of electrodynamics is obtained ,but here things are getting much complicated probably because its a two-index tensor equation and I couldn't manipulate them-So I get nothing even nearing (2). Neither could I find a book which discusses this. How can I derive the symmetry equation without taking it for granted?

Full detailed calculation,approach to the problem or references where it is done-anything will be helpful.

Answer 1

• The infinitesimal gauge symmetry (2) is not pulled out of a hat. It descends from the diffeomorphism invariance/coordinate independence $\delta g_{\mu\nu}=({\cal L}_{\xi}g)_{\mu\nu}$ in GR, which in linearized GR becomes $\delta h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$. Now use the definition $\bar{h}_{\mu\nu}~:=~h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h$ to derive eq. (2).

• Similarly, eq. (1) is the linearized EFE, and therefore inherits the corresponding symmetry.