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Gravitational wave polarization modes and stability analysis in Weyl geometry gravity

Yu-Zhi Fan, Xiao-Bin Lai, Yu-Qi Dong, Yu-Xiao Liu

Abstract

We investigate the gravitational wave polarization modes and stability in Weyl geometry gravity within a Minkowski background. Our results indicate that the tensor sector consists of two standard modes propagating at the speed of light. Although the vector sector possesses a dynamical degree of freedom, it generates no polarization modes. The scalar sector, in contrast, features a mixture mode of breathing and longitudinal modes associated with a single scalar degree of freedom. This degree of freedom exhibits superluminal propagation and intrinsic amplitude decay, both driven by the background Weyl gauge field. We further discuss the observational detectability of this scalar mode. Our stability analysis reveals that, while the tensor and vector sectors are free from ghost and Laplacian instabilities, the scalar sector suffers from an Ostrogradsky ghost instability. These findings clarify the unique gravitational wave propagation characteristics in Weyl geometry gravity and provide theoretical foundations for testing the theory through future multi-messenger observations.

Gravitational wave polarization modes and stability analysis in Weyl geometry gravity

Abstract

We investigate the gravitational wave polarization modes and stability in Weyl geometry gravity within a Minkowski background. Our results indicate that the tensor sector consists of two standard modes propagating at the speed of light. Although the vector sector possesses a dynamical degree of freedom, it generates no polarization modes. The scalar sector, in contrast, features a mixture mode of breathing and longitudinal modes associated with a single scalar degree of freedom. This degree of freedom exhibits superluminal propagation and intrinsic amplitude decay, both driven by the background Weyl gauge field. We further discuss the observational detectability of this scalar mode. Our stability analysis reveals that, while the tensor and vector sectors are free from ghost and Laplacian instabilities, the scalar sector suffers from an Ostrogradsky ghost instability. These findings clarify the unique gravitational wave propagation characteristics in Weyl geometry gravity and provide theoretical foundations for testing the theory through future multi-messenger observations.

Paper Structure

This paper contains 22 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An overview of Weyl geometry gravity
  3. Weyl geometry
  4. Action in Weyl geometry gravity
  5. Effective matter Lagrangian and self-consistency condition
  6. Field equations in Weyl geometry gravity
  7. Linear perturbations and gauge invariants
  8. Background equations and linear perturbation equations
  9. Gauge invariants
  10. Polarization modes of GWs
  11. Geodesic deviation equation and polarization modes
  12. Gauge-invariant perturbation equations and polarization modes
  13. Discussion on scalar modes
  14. Detectability of scalar modes
  15. Stability analysis for tensor and vector perturbations
  16. ...and 7 more sections

Figures (1)

  • Figure 1: The relative motions of freely falling test particles induced by the six polarization modes of GWs Eardley:1973zuo. Initially, the test particles are positioned on a spherical shell. The GWs travel in the $+z$ direction, as denoted by the icons in the upper-right corner of each subfigure. The solid and dashed curves both illustrate the relative positions of the test particles at their peak deviations, separated by a phase shift of half a cycle.