# Higher Derivative Gravity and Conformal Gravity from Bimetric and Partially Massless Bimetric Theory

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## Abstract

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## 1. Introduction

#### 1.1. A Review of the Different Theories Considered

#### 1.1.1. Higher Derivative Gravity

#### 1.1.2. Conformal Gravity

#### 1.1.3. Ghost-Free Bimetric Theory

#### 1.1.4. Linear Partially Massless Theory and Beyond

#### 1.1.5. Partially Massless Bimetric Theory

#### 1.2. Summary of Results

## 2. Higher Curvature Gravity from Bimetric Theory

#### 2.1. Outline of Obtaining Higher Derivative Gravity from Bimetric Theory

#### 2.2. Review of Ghost-Free Bimetric Gravity

#### 2.3. The Algebraic Solutions for S and f

#### 2.3.1. Exact Solution in the ${\beta}_{1}$ Model:

#### 2.3.2. Perturbative Solution for General ${\beta}_{n}$:

#### 2.4. Higher Derivative Gravity from Bimetric Theory

## 3. The Ghost Issue and Relevance to New Massive Gravity

## 4. Conformal Gravity from Partially Massless Bimetric Theory

#### 4.1. The Correspondence in $d=4$

#### 4.2. A Step Further: Equivalence between CG and PM Bimetric Theory

## 5. Discussions

- (1)
- The analysis in this paper shows that the HR bimetric theory captures the essential features of higher derivative gravity action Equation (1), while at the same time avoiding the spin-2 ghost problem. The correspondence between the two theories found here is not a complete equivalence of equations of motion, but can still be used to generate higher derivative completions of the four-derivative gravity actions.
- (2)
- The equation of motion in the candidate PM bimetric theory at the four-derivative level was shown to coincide with the Bach equation of conformal gravity. While this result was motivated by the general correspondence between bimetric and HD gravity actions, it turns out to bypass the general correspondence and, in fact, was an equivalence at the level of equations of motion. As a result, it has genuine consequences for the bimetric PM proposal, as discussed in the paper.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## A. Higher Derivative Treatment of Free Massive Spin-0 and Spin-2 Fields

#### A.1. Higher Derivative Treatment of Massive Scalars

#### A.2. First Approach: The Equivalent Higher Derivative Equations

#### A.3. Second Approach: A More General Higher Derivative Action

#### A.4. Truncation to a Four Derivative Theory

#### A.5. Higher Derivative Treatment of Linearized Bimetric Theory

## B. The General Perturbative Solution of the ${g}_{\mu \nu}$ Equation for ${f}_{\mu \nu}$

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**MDPI and ACS Style**

Hassan, S.F.; Schmidt-May, A.; Von Strauss, M.
Higher Derivative Gravity and Conformal Gravity from Bimetric and Partially Massless Bimetric Theory. *Universe* **2015**, *1*, 92-122.
https://doi.org/10.3390/universe1020092

**AMA Style**

Hassan SF, Schmidt-May A, Von Strauss M.
Higher Derivative Gravity and Conformal Gravity from Bimetric and Partially Massless Bimetric Theory. *Universe*. 2015; 1(2):92-122.
https://doi.org/10.3390/universe1020092

**Chicago/Turabian Style**

Hassan, Sayed Fawad, Angnis Schmidt-May, and Mikael Von Strauss.
2015. "Higher Derivative Gravity and Conformal Gravity from Bimetric and Partially Massless Bimetric Theory" *Universe* 1, no. 2: 92-122.
https://doi.org/10.3390/universe1020092