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Polarisation Singularities of Gravitational Waves

Claire Rigouzzo, Sebastian Golat, Alex J. Vernon, Kyan Louisia, Eugene Lim, Francisco J. Rodriguez-Fortuno

Abstract

Departure from idealised plane waves gives rise to intricate geometric structures in wave fields. One such structure is the polarisation singularity, which emerges when multiple monochromatic waves interfere (such as would be the case for stochastic backgrounds), producing loci of purely circular or linear polarisation. In this work, we extend the theory of polarisation singularities to gravitational waves and higher spin fields. Building on the electromagnetic description, we formulate the gravitational analogue of polarisation singularities and show that they are generic features of gravitational waves. Their dimension, however, depends on the spin of the field. We illustrate these results with simulations of plane-wave interference and analyse the resulting singularity densities.

Polarisation Singularities of Gravitational Waves

Abstract

Departure from idealised plane waves gives rise to intricate geometric structures in wave fields. One such structure is the polarisation singularity, which emerges when multiple monochromatic waves interfere (such as would be the case for stochastic backgrounds), producing loci of purely circular or linear polarisation. In this work, we extend the theory of polarisation singularities to gravitational waves and higher spin fields. Building on the electromagnetic description, we formulate the gravitational analogue of polarisation singularities and show that they are generic features of gravitational waves. Their dimension, however, depends on the spin of the field. We illustrate these results with simulations of plane-wave interference and analyse the resulting singularity densities.
Paper Structure (9 sections, 32 equations, 3 figures)

This paper contains 9 sections, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Polarisation singularities for electromagnetic waves
  3. Polarisation singularities for gravitational waves
  4. Simulations of polarisation singularities
  5. Discussion and conclusions
  6. Codimension of polarisation singularities
  7. Polarisation singularities for spin-s fields
  8. Visualising polarisation singularities
  9. Plane wave polarisation basis for a general propagation direction

Figures (3)

  • Figure 1: Generic electric field polarisation. (a) left: a polarisation ellipse in the $xy$ plane described by $\mathbf{E}=\mathbf{P}+i\mathbf{Q}$, and its semi-major and semi-minor axes aligned with the real vectors $\mathbf{a}$ and $\mathbf{b}$. (a) right: polarisation ellipses with arbitrary 3D orientation determined by the normal to the plane of the ellipse, $\mathbf{n}_\mathbf{E}$. (b): generic depiction of red C lines and green L lines in a volume of space containing an electromagnetic field (e.g., random 3D plane wave interference).
  • Figure 2: Generic gravitational field polarisation at a single point $\mathbf{r}$. A gravitational wave displaces the local orthonormal basis vectors $\hat{\mathbf e}_i$ by $\boldsymbol{\mathcal{h}}_i(\mathbf r,t)=\tfrac{1}{2}\mathcal{h}^a{}_i(\mathbf r,t)\hat{\mathbf e}_a$, which traces an ellipse in time. The displacement vectors $\boldsymbol{\mathcal{h}}_i$ are shown in red, and the normal to each ellipse, $\mathbf n_i$, is shown in green.
  • Figure 3: Polarisation singularities for the interference of $N=3,4,100$ random plane waves. Left: electromagnetic C (red) and L (green) lines. Right: gravitational C lines (red) and L points (green). The gradient of green and red is to help show depth. The code is available online https://github.com/KZL358/Polarisation-Singularitiesfigshare.