[Paper Review] General Backgrounds for higher spin massive particles

Ignacio Cortese, Manuela Kulaxizi|arXiv (Cornell University)|Nov 30, 2017

Black Holes and Theoretical Physics27 references3 citations

TL;DR

This paper proposes a consistent system of dynamical equations and constraints for massive higher-spin (s ≥ 3) bosonic fields in generic curved spacetimes by introducing additional background fields beyond the metric. Using a deformation procedure based on closure of the algebra generated by d'Alembertian, divergence, and trace operators, the authors show that consistent propagation is possible in asymptotically de Sitter, anti-de Sitter, and flat black hole geometries, as well as in domain-wall spacetimes like FRW, provided specific relations between curvature invariants and auxiliary background fields are satisfied. The key result is the extension of consistent higher-spin propagation beyond constant-curvature spacetimes.

ABSTRACT

We consider the propagation of totally symmetric bosonic fields on generic background spacetimes. The mutual compatibility of the dynamical equations and constraints severely constrains the set of geometries where consistent propagation is possible. To enlarge this set in this article we allow several background fields to be turned on. We were able to show that massive fields of spin s greater than or equal to three may consistently propagate in a large set of non-trivial spacetimes, such as asymptotically de-Sitter, flat and anti-de-Sitter black holes geometries, as long as certain conditions between the various background fields are met. For the special case of massive spin-2 fields the set of allowed spacetimes is larger and includes domain-wall-type geometries, such as the Freedman-Robertson-Walker metric. We comment on the assumptions underlying our study and on possible applications of our results.

Motivation & Objective

• To extend the consistent propagation of massive higher-spin fields beyond constant-curvature spacetimes, where prior formulations fail due to incompatibility of equations of motion and constraints.
• To address the breakdown of Fierz-Pauli systems in nontrivial backgrounds by systematically deforming the dynamical equations and constraints using curvature and auxiliary fields.
• To generalize previous results by allowing additional background fields to be turned on, thereby enlarging the set of allowed geometries for consistent propagation.
• To derive necessary conditions on background fields (curvature invariants and auxiliary fields) ensuring closure of the operator algebra and mutual compatibility of equations and constraints.
• To demonstrate that massive spin-2 fields can propagate consistently in domain-wall-type geometries such as the Friedmann-Robertson-Walker (FRW) metric, which are not covered by standard formulations.

Proposed method

• A systematic deformation of the Fierz-Pauli equations using the formalism of Kaparulin et al. to preserve mutual compatibility of equations of motion and constraints in curved spacetimes.
• Introduction of additional background fields (beyond the metric) to relax constraints on spacetime geometry, enabling consistent propagation in non-constant curvature backgrounds.
• Derivation of closure conditions for the algebra generated by the deformed d'Alembertian, divergence, and trace operators, leading to constraints on curvature invariants (Y, Z, X) and auxiliary fields.
• Explicit construction of deformed equations of motion for symmetric, traceless rank-s tensors, including non-minimal couplings to Ricci curvature, Ricci scalar, and Weyl tensor.
• Use of irreducible Lorentz tensor components (Yμνρ, Zμνρ, Xμνρσλ) to express curvature constraints, ensuring the system remains involutive and free of inconsistencies.
• Computation of O(R²) anomaly terms to enforce second-order cancellation, leading to a system of equations that must be satisfied by the background fields for exact consistency.

Experimental results

Research questions

• RQ1 Can massive higher-spin fields (s ≥ 3) propagate consistently in non-constant curvature spacetimes such as de Sitter, anti-de Sitter, and black hole geometries?
• RQ2 What additional background fields must be introduced to restore consistency between equations of motion and constraints in generic curved spacetimes?
• RQ3 How does the inclusion of auxiliary fields modify the conditions on curvature invariants (Y, Z, X) for consistent propagation?
• RQ4 Can the formalism be extended to include domain-wall-type geometries like the Friedmann-Robertson-Walker metric, particularly for spin-2 fields?
• RQ5 What are the necessary and sufficient conditions on background fields to ensure cancellation of anomalies at second order in curvature?

Key findings

• Massive higher-spin fields of spin s ≥ 3 can consistently propagate in asymptotically de Sitter, anti-de Sitter, and flat black hole spacetimes when specific relations between curvature invariants and auxiliary background fields are satisfied.
• For spin-2 fields, the allowed class of geometries is enlarged to include domain-wall-type spacetimes such as the Friedmann-Robertson-Walker (FRW) metric, which are not covered by standard formulations.
• The consistency of the system is achieved by introducing auxiliary background fields that modify the equations of motion and constraints, ensuring closure of the operator algebra.
• The necessary conditions for consistency are encoded in the vanishing of irreducible curvature tensors Yμνρ, Zμνρ, and Xμνρσλ, which constrain the background geometry.
• For spin s ≥ 3, the system requires b1Sλ(−Rρλ + a2/s Uρλ) = 0 and additional tensor equations involving Ricci and Weyl curvature to hold, leading to constraints like a2 = 0 or S² = 0.
• Second-order anomaly cancellation in curvature leads to a system of equations involving R, W, and auxiliary fields, which must be solved to achieve exact consistency across all orders in curvature.