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## Homework Statement

We will treat the graviton as a symmetric ##2##-index tensor field. It couples to a current ##T_{\mu\nu}## also symmetric in its two indices, which satisfies the conservation law ##\partial_{\mu}T_{\mu\nu}=0##.

(a) Assume the Lagrangian is ##\mathcal{L}=-\frac{1}{2}h_{\mu\nu}\Box h_{\mu\nu} + \frac{1}{M_{\text{Pl}}}h_{\mu\nu}T_{\mu\nu}.## Solve ##h_{\mu\nu}##'s equations of motion, and substitute back to find an interaction like ##T_{\mu\nu}\frac{1}{k^{2}}T_{\mu\nu}##.

(b) Write out the ##10## terms in the interaction ##T_{\mu\nu}\frac{1}{k^{2}}T_{\mu\nu}## explicitly in terms of ##T_{00}, T_{01},## etc.

(c) Use current conservation to solve for ##T_{\mu 1}## in terms of ##T_{\mu 0}, \omega## and ##\kappa##. Substitute in to simplify the interaction. How many causally propagating degrees of freedom are there?

(d) Add to the interaction another term of the form ##cT_{\mu\mu}\frac{1}{k^{2}}T_{\nu\nu}##. What value of ##c## can reduce the number of propagating modes? How many are there now?

## Homework Equations

## The Attempt at a Solution

(a) ##\frac{\partial \mathcal{L}}{\partial h_{\mu\nu}}=-\frac{1}{2}\Box h_{\mu\nu}+\frac{1}{M_{\text{Pl}}}T_{\mu\nu}##

I'm having trouble finding ##\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})}##:

##\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})} = \frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}\Big( -\frac{1}{2}h_{\mu\nu}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu} \Big) = \frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}\Big( -\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}\partial_{\sigma}\partial_{\rho}h_{\mu\nu} \Big)=-\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}\partial_{\sigma}\frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}(\partial_{\rho}h_{\mu\nu})##