Regge-Teitelboim analysis of the symmetries of electromagnetic and gravitational fields on asymptotically null spacelike surfaces

TL;DR

This work extends the Regge-Teitelboim framework to asymptotically null, spacelike hourglass surfaces to study radiation and asymptotic symmetries in electromagnetism and gravity. It develops a duality-invariant Hamiltonian formulation with surface charges (electric and magnetic BMS charges) and a two-potential description, yielding a consistent Lorentz generator and a conserved, duality-covariant angular momentum in the presence of radiation. The paper reveals a gravity–electromagnetism correspondence for BMS structures, identifies a Taub-NUT magnetic pole in gravity, and derives memory effects (fiber/charge and supertranslations) together with emission/absorption rates, all tied to explicit boundary conditions and a robust hourglass formalism. It provides explicit boundary-condition analysis, a dictionary to standard light-cone variables, and checks against Taub–NUT and Kerr solutions, highlighting the practical impact for understanding radiation and asymptotic symmetries in GR and EM. Overall, the hourglass approach yields a coherent, duality-aware framework for radiative spacetimes with well-defined conserved charges and memory effects.

Abstract

We present a new application of the Regge-Teitelboim method for treating symmetries which are defined asymptotically. It may be regarded as complementary to the one in their original 1974 paper. The formulation is based on replacing an asymptotic plane by the two--sheeted ``hourglass" shaped surface obtained by joining smoothly an incoming hyperboloid with an outgoing one. The hyperboloids have a fixed radius, and as one moves the center of the hourglass along the time axis one covers the whole of spacetime. The motivation is to study radiation, and the hourglass is well suited to the task because it is asymptotically null, and thus is able to register the details of the process. A simple parity condition for the fields on the hyperboloid is given. It specifies that as much radiation as is coming in as it is going out. With it, a Hamiltonian formulation of the symmetry of Bondi, van der Burg, Metzner and Sachs is developed fir both electromagnetism and gravitation. It is indispensable for the construction to have electric--magnetic duality asymptotically. For gravitation, a formulation for the linearized theory on the hourglass has not been explicitly constructed; but enough rudiments of it are given so that the main results can be established. A definition for angular momentum wish is conserved (for which the ``magnetic sector'' is essential) is given. It incorporates an interrelationship between spin and charge. For the gravitational field, Taub-NUT space appears as the analog of a magnetic pole.

Figures (3)

• Figure 1: The hyperbolic hourglass. The figure shows a two dimensional cut of an incoming hyperboloid and an outgoing one which are joint smoothly at at $r=0$. The arrows show the direction of increasing $r$, which coincide asymptotically with the direction of propagation of a wave that comes in, goes through itself, and then comes out. If the incoming wave is not spherically symmetric, the spacetime point at which the wavefront goes through itself will be different for different $(\vartheta,\varphi)$. For the analysis of the asymptotic region the details of what happens inside are irrelevant. The key aspects are the asymptotic hyperbolic shape and its orientation inherited from that of an incoming wave that goes through itself and becomes outgoing.
• Figure 2: Slicing by hyperbolic hourglasses. A given spacetime point is labeled by two set of coordinates. In the case of the point $P$ shown in the figure, these are $(t=0, r, \vartheta,\varphi)$ and $(t=3\tau_0, -r, \pi-\vartheta,\varphi+\pi)$.
• Figure 3: Limits $\tau_{0}\rightarrow0$ and $\tau_{0}\rightarrow \infty$ for Minkowski space. The succession of conformal diagrams shows from left to right how the surfaces of the hourglass foliation are deformed from nearly light cones to nearly planes as $\tau_{0}$ increases from a very small value to a very large one. To better illustrate the effect, different members of the foliation are shown in the different figures of the sequence; but, to keep track of the deformation, the surface at $t=0$ (shown with a heavy line) in all cases. The Penrose diagram has been doubled to admit negative values of $r$ in the left triangular area. This doubling shows how the curves of constant $t,\vartheta,\varphi$ are smooth spacelike curves that connect asymptotically past and future null infinities. The scale of the lenght $\tau_0$ is irrelevant for the effect described in the figure, which only depends on the ratios between the different $\tau_0$'s shown.