How is the effective Lagrangian for the Graviton coupling to matter derived?

[Karatechop250]

Could someone point to me where they derive the follow effective Lagrangian for the Graviton coupling to matter

$$L = \frac{1}{M_{pl{}}}h^{\mu \nu}T_{\mu \nu}$$

 

[Bill_K]

A great many books discuss the coupling of gravity to matter in the framework of the classical weak field approximation. One ref that does it QM-ically is Quantum Field Theory by Zee, Sect VIII, "Gravity and Beyond".

The fundamental definition T^μν = - (2/√-g) δS_M/δg_μν tells us that coupling of the graviton to matter (in the weak field limit) can be included by adding the term - ∫d⁴x ½h_μν T^μν to the action...

 

[Karatechop250]

Thanks, however this just seems as circular logic to me and confuses me cause we defined the stress energy tensor that way. I will read this book though.

 

[Vanadium 50]

Since physics is an experimental science, ultimately the answer is "so that our calculations match what we see". This may sound circular, but it's how (and why) science works.

 

[Bill_K]

It's not circular logic, it's just two ways of writing the same thing. Do you want to say T = δS/δg, or do you want to say δS = T δg?

More explicitly, the first form is T^μν = (2/√-g) δS_M/δg_μν

while the second form is δS_M = ½∫d⁴x √-g T^μν δg_μν

They both say that when you subject g_μν to an infinitesimal variation, the change in the action is linear in δg_μν, and the coefficient is defined to be T^μν.

 

[dextercioby]

You can rigorously prove the Th coupling at first order through deformation of the master equation in the BRST antibracket-antifield formalism: http://arxiv.org/abs/0704.2321v1