Massive gravity - Corrections to the Pauli Fierz mass term

[Chris Harrison]

Homework Statement

I'm looking at solving a simple theory of massive gravity for Scwarzchild type solutions. I've attatched the paper that I'm working with. I've tried to add the 3 possible cubic terms to L_mass parametrized by constants. It doesn't seem possible to solve for B as a function of r only by giving the constants certain values, as is done in (3.6) in the paper. Am i missing something important?

 

[fzero]

Homework Statement

I'm looking at solving a simple theory of massive gravity for Scwarzchild type solutions. I've attatched the paper that I'm working with. I've tried to add the 3 possible cubic terms to L_mass parametrized by constants. It doesn't seem possible to solve for B as a function of r only by giving the constants certain values, as is done in (3.6) in the paper. Am i missing something important?

You might want to provide some more details (at least your version of (3.4) and (3.5)). It's clear that your modification makes the equations even more nonlinear than they already are, so there's no a priori reason to expect the solution to be as simple (or even explicit).

 

[Chris Harrison]

I'm using the same ansatz for f as they do in the paper. Does this look right so far? I can post the non vanishing components of the EMT soon if necessary. Note i achieved the EMT by varying the mass term with respect to the inverse of f.
$$L_{Mass}=\frac{-M^2\sqrt{-\eta}}{4k_{f}^2} \left ( (f^{ k\lambda}-\alpha\eta^{k\lambda})(f^{\sigma\rho}-\beta\eta^{\sigma\rho} \right )\left ( \eta_{k\sigma}\eta_{\lambda\rho}-\eta_{k\lambda}\eta_{\sigma\rho} \right ) +\gamma_{1}(f^{k\lambda}f_{k\lambda}f{_{\sigma}}^{\sigma})+\gamma_{2}(f{_{\lambda}}^{\lambda})^3+\gamma_{3}(f{_{\lambda}}^{\sigma}f{_{\sigma}}^{\rho}f{_{\rho}}^{\lambda}))$$
I found the corresponding EMT to be
$$T_{\mu\nu}=\frac{M^2}{4k_{f}^2}\frac{\sqrt{-\eta}}{\sqrt{-f}}((2f^{k\lambda}-(\alpha+\beta)\eta^{k\lambda})(\eta_{k\mu}\eta_{\lambda\nu}-\eta_{k\lambda}\eta_{\mu\nu})+\gamma_{1}(2f_{\mu\nu}f{_{\lambda}}^{\lambda}+\eta_{\mu\nu}f^{\sigma\rho}f_{\sigma\rho})+3\gamma_{2}(\eta_{\mu\nu}(f{_{\lambda}}^{\lambda})^2)+3\gamma_{3}(f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}))$$

 

[fzero]

That looks ok so far, but the expressions involving $A,B,\ldots$ are what you actually need to discuss solutions. In the meantime, I would suggest that you might try perturbative solutions of your equations. Introduce a parameter $\epsilon$ that controls the size of the cubic terms, so $\gamma_i = \epsilon g_i$, for some new numbers $g_i$. Then you will look for solutions of the form $B=2r^2/3 + \epsilon b(r)$, etc. To start you will solve the equations at first order in $\epsilon$, but it might be possible to consider higher orders as well.