A Quasigroup Approach for Conservation Laws in Asymptotically Flat Spacetimes
Alfonso Zack Robles, Alexander I. Nesterov, Claudia Moreno
TL;DR
The paper addresses the challenge of defining conserved quantities such as energy–momentum and angular momentum at future null infinity in asymptotically flat spacetimes, where the infinite-dimensional Newman–Unti group induces supertranslation ambiguities. It introduces a smooth quasigroup approach that reduces the NU symmetry to a finite-dimensional Poincaré quasigroup and constructs a complex Noether charge $Q_{\xi_c}$ whose real part yields gauge-invariant, supertranslation-free conserved quantities ${\cal L}_{\xi}$. Explicit surface integrals are derived for momentum $P_a=-\frac{1}{4\pi}\Re\oint l_a(\Psi^0_2+\sigma^0\lambda^0)d\Omega$ and angular momentum $M_i=-\frac{1}{4\pi}\Re\oint (\xi_i(\Psi^0_2+\sigma^0\lambda^0)+\bar{L}_i\Psi^0_1)d\Omega$, with a flux law for energy–momentum $\dot P_a$ and a CM-frame intrinsic angular momentum given by $J_i=-\frac{1}{4\pi}\Re\oint \bar{L}_i(\Psi^0_1+\sigma^0\eth\bar{\sigma}^0)d\Omega$. The center-of-mass frame is defined by a Dixon-like condition $K_i=0$, leading to an intrinsic angular momentum in the Komar form, and the framework connects to nice sections of ${\mathscr I}^{+}$ where supermomentum components vanish. Overall, the work provides a geometric, gauge-invariant method to extract physically meaningful conserved quantities in radiating spacetimes and offers a pathway to applications in gravitational-wave physics and memory phenomena.
Abstract
In the framework of the quasigroup approach to conservation laws in general relativity, we show how the infinite-parametric Newman-Unti group of asymptotic symmetries can be reduced to the Poincare quasigroup. We compute Noether's charges associated with any element of the Poincare quasialgebra. The integral conserved quantities of energy momentum and angular momentum, being linear on generators of the Poincare quasigroup, are identically equal to zero in Minkowski spacetime. We present a definition of the angular momentum free of the supertranslation ambiguity. We provide an appropriate notion of intrinsic angular momentum and a description of the mass reference frame's center at future null infinity. Finally, in the center of mass reference frame, the momentum and angular momentum are defined by the Komar expression.