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New formalism for perturbations of massive gravity theories around arbitrary background spacetimes

Kieran Wood

TL;DR

This work tackles perturbations in dRGT-type massive gravity on arbitrary backgrounds by formulating perturbations in the vielbein language and then relating them to metric perturbations, thereby avoiding the problematic matrix square roots of the metric formulation. The method illuminates the ghost-free structure through explicit W- and B-tensor constructions and yields linearised equations that generalise to any number of interacting metrics and to higher orders. The authors apply the framework to proportional, cosmological, and black-hole backgrounds, obtaining new linear results (including non-proportional rotating black-hole perturbations) and derive the cubic-order multi-gravity potential around a generic background, aligning with known results in special cases. The approach provides a powerful, generalizable toolkit for analyzing perturbations in massive gravity theories with broad applicability to cosmology and black-hole physics, with clear paths to higher-order studies and richer interaction structures.

Abstract

We develop a new technique for studying the perturbations of dRGT-type massive gravity theories around arbitrary background spacetimes. Built initially from the vielbein formulation of the theory, but switching back to the metric formulation afterwards, our approach bypasses many of the complications that arise in previous metric formulation approaches to linearising massive gravity around generic backgrounds, naturally elucidates the ghost-free structure of the interactions, and readily generalises to higher orders in perturbation theory, as well as to multiple interacting metric tensor fields. To demonstrate the power of our technique, we apply our formalism to a number of commonly occurring example backgrounds - proportional, cosmological, and black hole - recovering and extending many known results from the literature at linear order. Lastly, we provide, for the first time, the cubic order multi-gravity potential around a generic background spacetime.

New formalism for perturbations of massive gravity theories around arbitrary background spacetimes

TL;DR

This work tackles perturbations in dRGT-type massive gravity on arbitrary backgrounds by formulating perturbations in the vielbein language and then relating them to metric perturbations, thereby avoiding the problematic matrix square roots of the metric formulation. The method illuminates the ghost-free structure through explicit W- and B-tensor constructions and yields linearised equations that generalise to any number of interacting metrics and to higher orders. The authors apply the framework to proportional, cosmological, and black-hole backgrounds, obtaining new linear results (including non-proportional rotating black-hole perturbations) and derive the cubic-order multi-gravity potential around a generic background, aligning with known results in special cases. The approach provides a powerful, generalizable toolkit for analyzing perturbations in massive gravity theories with broad applicability to cosmology and black-hole physics, with clear paths to higher-order studies and richer interaction structures.

Abstract

We develop a new technique for studying the perturbations of dRGT-type massive gravity theories around arbitrary background spacetimes. Built initially from the vielbein formulation of the theory, but switching back to the metric formulation afterwards, our approach bypasses many of the complications that arise in previous metric formulation approaches to linearising massive gravity around generic backgrounds, naturally elucidates the ghost-free structure of the interactions, and readily generalises to higher orders in perturbation theory, as well as to multiple interacting metric tensor fields. To demonstrate the power of our technique, we apply our formalism to a number of commonly occurring example backgrounds - proportional, cosmological, and black hole - recovering and extending many known results from the literature at linear order. Lastly, we provide, for the first time, the cubic order multi-gravity potential around a generic background spacetime.

Paper Structure

This paper contains 17 sections, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of multi-gravity
  3. Metric formalism
  4. Vielbein formalism
  5. Multi-gravity perturbations: general formalism
  6. Quadratic action and linearised field equations
  7. Multi-gravity perturbations: examples
  8. Proportional backgrounds
  9. Cosmological backgrounds
  10. Black hole backgrounds
  11. Cubic potential
  12. Conclusion
  13. Details of the previous approaches to linear multi-gravity perturbations
  14. Matrix equation approach
  15. Redefined fluctuation variables approach
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Directed theory graph representing some generic multi-metric theory. The circular nodes represent different metrics, the edges indicate interactions and the arrows point in the direction of positive interaction orientation. Each metric generically has a number of interactions of either orientation, and each edge contributes a term to the field equations of the two metrics it connects; these terms are orientation-dependent.