Globally defined Carroll symmetry of gravitational waves

TL;DR

This work addresses extending the Carroll symmetry of gravitational waves from a locally defined to a globally defined framework by recasting plane wave spacetimes in Brinkmann coordinates and solving a matrix Sturm-Liouville problem for $P(U)$ with profile ${\mathcal A}(U)$. The authors derive global Carroll generators $\Theta^{Brink}$ using two independent SL solutions $P$ and $Q$ (with $Q = P S$) and show that the memory effects—Displacement Memory (DM) and Velocity Memory (VM)—are governed by these solutions, with DM occurring when the wave parameters yield pure geodesic focusing. A Schwarzian-derivative perspective links the profile ${\mathcal A}$ to the quotient of SL solutions, $f = \varphi_1/\varphi_2$, via ${\mathcal A} = -\tfrac{1}{2} \mathfrak{S}(f)$, and reveals a reciprocal symmetry between $P$ and $Q$. The Pöschl–Teller example concretely demonstrates caustics, global symmetry, and the unification of memory effects, while the Brinkmann formulation cures local-singularity issues in BJR coordinates and suggests generalizations to higher dimensions and related profiles.

Abstract

The locally defined Carroll symmetry of a gravitational wave is extended to a globally defined one by switching to Brinkmann coordinates. Translations and Carroll boosts associated with two independent globally defined solutions of a Sturm-Liouville equation allow us to describe the motions. The Displacement Memory Effect arises for particular choices of the parameters which yield trajectories with zero momentum. The relation to the Schwarzian derivative is highlighted. We illustrate our general statements by the Pöschl-Teller profile.

Figures (10)

• Figure 1: For integer $m$ the Sturm-Liouville solutions $\bf P$ in \ref{['SL+cond']} are DM trajectories, written in Brinkmann coordinates with wave numbers ${\bf m=1}$ and ${\bf m=2}$, respectively. The Souriau matrix S diverges where $\bf P=0$, but the pull-back to Brinkmann $\textcolor{rgb(0,128,0)}{\bf Q = PS}$ is regular for all $U$.
• Figure 2: For a DM wave in $D=1$ transverse dimension Brinkmann trajectories focus to the caustic points determined by $P_m(U_k)=0$, $k=1,\dots,m$. For $m=1$ there is one focal point at the origin, and for $m=2$, we have two of them, determined by $\tanh U_{\pm}=\pm{1}/{\sqrt{3}}$.
• Figure 3: For wave number ${\bf m=1}$, the (a) translation generator ${\bf P}_1$ [which is also a trajectory] is odd. (b) the generator of a Carroll boost, ${\bf Q}_1 = {\bf P_1S_1}$, is even. The only zero of ${\bf P}_1$ is at $U=0$.
• Figure 4: Infinitesmal Carroll generators \ref{['CarrollPTm2']} with wave number ${\bf m=2}$. (a) For a translation, the transverse component ${\bf P}_2$ is even, and has two zeros, at $U_{\mp}$. (b) For a boost, ${\bf Q}_2 = {\bf P_2S_2}$ is odd.
• Figure 5: