# Lorentz-Violating Gravity Models and the Linearized Limit

## Abstract

**:**

## 1. Introduction

## 2. Linearized Equations for Lorentz-Violating Gravity

- The kinetic terms can vary in two ways when we change the metric. The more obvious way is that the full contraction of the tensor $\nabla \mathsf{\Psi}$ with itself may require some raising and lowering of indices, which requires an implicit use of the metric or the inverse metric; these terms will change when the metric is varied. A more subtle point (but in some ways a more important one) is that the covariant derivative operator ${\nabla}_{a}$ also varies when the metric is varied. One could also consider Palatini-type theories, where the connection is viewed as a variable independent from the metric. As the correspondence between the “metric” and “Palatini” versions of a generalized gravity theory is not straightforward [11], I focus on the “metric” versions here for simplicity.
- The coupling terms explicitly depend on the Riemann tensor, which varies with the metric. As with the kinetic terms, they can also contain implicit factors of the metric or inverse metric that arise from index raising and lowering.
- The potential term must be a spacetime scalar constructed out of ${\mathsf{\Psi}}^{\cdots}$; this will again usually require the raising and lowering of $\mathsf{\Psi}$’s indices, and so, implicitly depends on the metric.

- We can obtain a linearized second-order equation for ${h}_{ab}$ that is manifestly gauge-invariant. Since the linearized Riemann tensor contains all of the “gauge-invariant information” encoded in the linearized metric, this means that the resulting equation is simply of the form$$\delta {G}_{ab}+\xi {\overline{\mathsf{\Psi}}}^{\cdots}\delta {R}_{\cdots}=0.$$The process for obtaining this equation may (and, in the examples below, will) require boundary conditions to be set, but should not rely on any particular choice of gauge. Equations of this type are sometimes taken as the starting point for the study of Lorentz violation in gravitational physics [12,13], while remaining agnostic about the underlying model of Lorentz violation. However, it has also been pointed out that not all LV gravity models lead to such an equation [14].
- We cannot obtain a linearized second-order equation for ${h}_{ab}$ alone that is gauge-invariant, but we can find some particular gauge that allows us to eliminate $\tilde{\psi}$ from the Equation (11b). Again, the imposition of some appropriate set of boundary conditions may be necessary.
- The Equations (11a) and (11b) are inextricably coupled. This means that we cannot write down a second-order equation for the metric alone without knowing the behavior of the LV field as well. In such a case, it could, in principle, still be possible to solve Equation (11b) for $\tilde{\psi}$ in terms of h and whatever other fields are present by Green’s functions; one could then insert this solution for $\tilde{\psi}$ into (11a). However, the resulting equation would include some kind of integral over spacetime and would not result in a local second-order partial differential equation for ${h}_{ab}$.

## 3. n-Form Fields

## 4. Tensor Klein–Gordon Fields

## 5. Discussion

## Funding

## Acknowledgments

## References

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Seifert, M.
Lorentz-Violating Gravity Models and the Linearized Limit. *Symmetry* **2018**, *10*, 490.
https://doi.org/10.3390/sym10100490

**AMA Style**

Seifert M.
Lorentz-Violating Gravity Models and the Linearized Limit. *Symmetry*. 2018; 10(10):490.
https://doi.org/10.3390/sym10100490

**Chicago/Turabian Style**

Seifert, Michael.
2018. "Lorentz-Violating Gravity Models and the Linearized Limit" *Symmetry* 10, no. 10: 490.
https://doi.org/10.3390/sym10100490