Globally defined Carroll symmetry of gravitational waves

The local Carroll symmetry of a gravitational wave found in Baldwin-Jeffery-Rosen coordinates is extended to a globally defined one by switching to Brinkmann coordinates. Two independent globally defined solutions of a Sturm-Liouville equation allow us to describe both the symmetries (translations and Carroll boosts) and the geodesic motions. One of them satisfies particular initial conditions which imply zero initial momentum, while the other does not. Pure displacement arises when the latter is turned off by requiring the momentum to vanish and when the wave parameters take, in addition, some particular values which correspond to having an integer half-wave number. The relation to the Schwarzian derivative is highlighted. We illustrate our general statements by the Pöschl-Teller profile.

💡 Research Summary

The paper addresses a long‑standing limitation of the Carroll symmetry description of plane gravitational waves: in Baldwin‑Jeffery‑Rosen (BJR) coordinates the symmetry is only locally defined because the coordinate transformation matrix P(u) becomes singular at isolated values of the affine parameter u. By moving to globally defined Brinkmann coordinates (X,U,V) the authors construct a fully global Carroll symmetry. The key technical step is to solve the matrix Sturm‑Liouville equation

  P″(U)=A(U) P(U),  (Pᵀ P′)=(Pᵀ)′ P,

where A(U) encodes the wave profile. Two independent solutions of this equation, denoted P(U) and Q(U), play distinct roles. P(U) is fixed by the asymptotic condition P(−∞)=𝟙, P′(−∞)=0, guaranteeing that particles are initially at rest. Q(U) is defined as Q=P S, where S(U) is the Souriau matrix S(U)=∫⁽ᵘ⁾(PᵀP)⁻¹ dt. Although S diverges at the zeros of P, the product Q remains regular for all U, thereby removing the singularities that plague the BJR description.

In Brinkmann coordinates the infinitesimal Carroll generators acquire a transparent form:

 Θ_B = h ∂_V + c·