# Self-interactions in linearized gravity

## Main Question or Discussion Point

self-interactions in "linearized" gravity

This question is really about the gauge invariance seen in linearized gravity. I'm trying to derive the 3-point self-interaction term in the GR lagrangian (in the weak field limit), and the algebra is just a nightmare. I finally gave up and determined that I could find it by just writing down the most general terms, and requiring gauge invariance. By inspecting the Ricci tensor, it is clear that all of these terms are of the form [TEX]h\partial{h}\partial{h}[/TEX]. The most general lagrangian has 10 terms, but under the infinitesimal gauge transformation [TEX]h_{\mu\nu}\rightarrow h_{\mu\nu} + \partial_\mu\alpha_\nu + \partial_\nu\alpha_\mu[/TEX], it can not be made invariant. In fact, even the 4 point term seems to break gauge invariance. Is this gauge invariance purely a feature of the linear terms or am I doing something wrong?

*edit* and if anyone knows this lowest order self-interaction term that would be incredibly helpful.

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Bill_K

I'm interested in this question, although I don't have an immediate answer. Isn't gauge invariance an automatic consequence of general covariance?

Expanding on what you said, the first few terms of the Lagrangian must be of the form hμν,στAμνστ + hμν,σhαβ,γBμνσαβγ + hδεhμν,σhαβ,γCδεμνσαβγ + ... where A, B and C are symmetrized products of kronecker deltas.

Is the problem that the gauge transformation is nonlinear? It's not really hμν → ξμ,ν + ξν,μ, it's hμν → hμαξα + hανξα

fzero
Homework Helper
Gold Member

For the question of gauge symmetries, it might be helpful to read the 't Hooft and Veltman paper, which you can get free from http://www.numdam.org/numdam-bin/recherche?h=nc&id=AIHPA_1974__20_1_69_0&format=complete [Broken]

I dug through some old notes and found an expression that I'd worked out years ago for the cubic interaction from the Einstein-Hilbert action. The attached pdf has the expressions before any gauge fixing.

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