Canonical derivation of Landau-Lifshitz pseudotensor

[QZQZ]

A normal gravitation-less energy-momentum tensor can be canonically derived from a Lagrangian as T^{alpha beta} = (1/sqrt(g)) d (L sqrt(g)) / d g_{alpha 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-Lifgarbagez 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-Lifgarbagez-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 T^μν = G^μν where G^μν = δL_grav/δg_μν is your analog of the Einstein tensor. Landau and Lifgarbagez do it by splitting G^μν into two parts. They put the first derivatives into the pseudotensor t^μν, and then show the second derivatives can be written in terms of a potential h^μνσ which is antisymmetric on ν and σ:

(-g) (T^μν + t^μν) = h^μνσ_,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 h^μνσ for your theory.