Canonical derivation of Landau-Lifshitz pseudotensor

[QZQZ]

A normal gravitation-less energy-momentum tensor can be canonically derived from a Lagrangian as Talpha beta = (1/sqrt(g)) d (L sqrt(g)) / d galpha beta, where sqrt(g) is the square root of the absolute value of the determinant of the metric g, L is the Lagrangian density, and "d" means functional derivative.

Is there a similar approach by which one can derive the Landau-Lifshitz pseudotensor from a Lagrangian that includes gravity? In particular, if I have an alternate theory of gravitation, which has a Lagrangian different from that of general relativity, how can I find a Landau-Lifshitz-like pseudotensor for energy-momentum?

[Bill_K]

I don't know what kind of alternative theories you have in mind, but presumably your field equations will be of the form Tμν = Gμν where Gμν = δLgrav/δgμν is your analog of the Einstein tensor. Landau and Lifshitz do it by splitting Gμν into two parts. They put the first derivatives into the pseudotensor tμν, and then show the second derivatives can be written in terms of a potential hμνσ which is antisymmetric on ν and σ:

(-g) (Tμν + tμν) = hμνσ,σ

If you can do that, then the divergence of the right hand side vanishes identically and so the divergence of the left hand side must vanish also.

So it all depends on whether you can find an hμνσ for your theory.