# Massive gravity - Corrections to the Pauli Fierz mass term

• Chris Harrison
In summary, the author is trying to solve a theory of massive gravity for Scwarzchild type solutions, but he has difficulty getting started because of the complexity of the equations. He has modified the equations to include possible cubic terms, but he is still having difficulty solving for B. He suggests perturbative solutions as a starting point.
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?

#### Attachments

• Class of solutions for the strong gravity equations - PhysRevD.16.2668-1 (1).pdf
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Chris Harrison said:

## 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).

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
$$T_{\mu\nu}=\frac{M^2}{4k_{f}^2}\frac{\sqrt{-\eta}}{\sqrt{-f}}((2f^{k\lambda}-(\alpha+\beta)\eta^{k\lambda})(\eta_{k\mu}\eta_{\lambda\nu}-\eta_{k\lambda}\eta_{\mu\nu})+\gamma_{1}(2f_{\mu\nu}f{_{\lambda}}^{\lambda}+\eta_{\mu\nu}f^{\sigma\rho}f_{\sigma\rho})+3\gamma_{2}(\eta_{\mu\nu}(f{_{\lambda}}^{\lambda})^2)+3\gamma_{3}(f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}))$$

Last edited:
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.

## 1. What is massive gravity?

Massive gravity is a theory in physics that attempts to explain the nature of gravity by introducing a non-zero mass for the graviton, the hypothetical particle that is believed to mediate the force of gravity.

## 2. What are corrections to the Pauli Fierz mass term?

Corrections to the Pauli Fierz mass term refer to modifications made to the original mass term proposed by Pauli and Fierz in their theory of massive gravity. These corrections are necessary in order to make the theory consistent with experimental observations and to avoid any potential inconsistencies or problems.

## 3. What is the significance of these corrections?

The corrections to the Pauli Fierz mass term are crucial for making the theory of massive gravity physically viable. Without these corrections, the theory would not be able to accurately describe and predict the behavior of gravity at large distances, such as in the case of galaxies and clusters of galaxies.

## 4. How do these corrections affect our understanding of gravity?

The corrections to the Pauli Fierz mass term provide a more complete and accurate understanding of the nature of gravity. They allow for the theory of massive gravity to better align with experimental data and observations, helping us to further our understanding of the fundamental forces of the universe.

## 5. Are there any current experiments or observations testing the validity of these corrections?

Yes, there are ongoing experiments and observations aimed at testing the validity of these corrections. Some examples include the use of gravitational wave detectors and observations of the large-scale structure of the universe. These experiments and observations will help to further refine our understanding of massive gravity and its corrections to the Pauli Fierz mass term.

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