Well, it makes sense for the effective field theory form of GR, for light at least.
The key to remembering how to derive the Euler-Lagrange equation (for me) is to remember that the variation in L vanishes at the boundary. This is what’s going to let you do an integration by parts and throw away the constant term. Actually, once you have an intuitive grasp of what’s going on, it’s kind of fun to derive generalized EL equations for Lagrangians with more complicated stuff in them.
Well, it makes sense for the effective field theory form of GR, for light at least.
The key to remembering how to derive the Euler-Lagrange equation (for me) is to remember that the variation in L vanishes at the boundary. This is what’s going to let you do an integration by parts and throw away the constant term. Actually, once you have an intuitive grasp of what’s going on, it’s kind of fun to derive generalized EL equations for Lagrangians with more complicated stuff in them.