Properties of Dual Supertranslation Charges in Asymptotically Flat Spacetimes

TL;DR

The paper investigates dual supertranslation charges in asymptotically flat spacetimes by introducing dyonic boundary conditions that activate both real and imaginary components of the boundary graviton. It demonstrates that the global dual charge is topological and cannot be generated by diffeomorphisms, with Taub-NUT furnishing a concrete topological example. The work resolves a transformation puzzle for superrotations by decomposing the soft sector and clarifies the phase-space action of dual charges, showing a separation between memory effects: standard memory for supertranslations exists, while dual supertranslations yield no integrated memory. A vacuum scattering solution of impulsive gravitational waves illustrates a spacetime transition between vacua with different dual charges, offering a physical interpretation in terms of gravitomagnetic cosmic strings. Overall, the results position dual supertranslations as a topological, Abelian symmetry distinct from BMS superrotations and with potential connections to monopole-like spacetime topology.

Abstract

We study several properties of some new charges of asymptotically flat spacetimes. These dual supertranslation charges are akin to the magnetic large $U(1)$ charges in QED. In this paper we find the symmetries associated with these charges and show that the global dual supertranslation charge is topological because it is invariant under globally defined, smooth variations of the asymptotic metric. We also exhibit spacetimes where the charge does not vanish and we find dynamical processes that interpolate between regions with different values of these charges.

Figures (6)

• Figure 1: The analogy between our work and the theory of magnetic monopoles. For brevity, we refer to dual supertranslation charged objects simply as gravitational monopoles, which are analogous to magnetic monopoles in electrodynamics. In the same way that the magnetic monopole charge partitions the space of gauge fields into distinct topological sectors, the gravitational monopole charge partitions the space of spacetime metrics into topologically distinct sectors.
• Figure 2: The Penrose diagram of the scattering process of two impulsive gravitational plane waves. The two waves propagate along $u=0$ and $v=0$, respectively and travel freely until the instant of collision. The interaction region is locally isomorphic to the Taub region of the Taub-NUT metric and can be extended beyond the even horizon which is described by the dashed line.
• Figure 3: Left: the interaction region projected onto the $T,X,\theta$ subspace. The two waves propagate inside the light cones of the two spacetime points $U=V=0, \, \theta =0 ,\pi$. Right: the interaction region projected onto the $T,X$ subspace.
• Figure 4: The NUT region of the Taub-NUT metric in the $x_3$-$\rho$ plane. The two string segments are stretched along the $x_3$-axis in the regions $|x_3| \geq \sigma$. The horizon at $r=r_+$, or equivalently $\rho=0$, lies in the domain $|x_3| \leq \sigma$ and is depicted by the dashed line.
• Figure 5: The plane waves solution in the $x_3$-$\rho$ plane. The shockwaves at $u=0$ and $v=0$ correspond to $x_3-\rho-\sigma=0$ and $-x_3-\rho-\sigma=0$, respectively. The null lines at $u=1$ and $v=1$ correspond to $x_3-\rho+\sigma =0$ and $-x_3-\rho+\sigma =0$, respectively. On these lines there is a conical singularity and therefore the metric cannot be extended beyond them. The gray area is the interaction region IV (before the extension across the horizon), which is isomorphic to the part $m<r<r_+$ of the Taub space, and it can be extended beyond the horizon (depicted by the dashed line). Across the horizon the metric describes the NUT space, which includes two string segments (depicted by the zigzag lines) stretching from the horizon along the $x_3$-axis. In the NUT region $(\rho>0)$ the coordinate $\rho$ is spacelike, while in the region $\rho<0$ it is timelike. The two null lines $x_3-\rho+\sigma =0$ and $-x_3-\rho+\sigma =0$ are naturally interpreted as the endpoints of the two string segments moving along their axis towards each other at the speed of light until they reach the horizon, after which spacetime is described by the static NUT metric.