# Matter Graviton Coupling.

1. Jun 5, 2013

### Karatechop250

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

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

2. Jun 6, 2013

### 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".

3. Jun 6, 2013

### 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.

Last edited: Jun 6, 2013
4. Jun 6, 2013

Staff Emeritus
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.

5. Jun 6, 2013

### 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 Tμν = (2/√-g) δSM/δgμν

while the second form is δSM = ½∫d4x √-g Tμν δgμν

They both say that when you subject gμν to an infinitesimal variation, the change in the action is linear in δgμν, and the coefficient is defined to be Tμν.

6. Jun 7, 2013

### 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