Bondi Mass, Memory Effect and Balance Law of Polyhomogeneous Spacetime

TL;DR

This work addresses mass and memory in spacetimes with polyhomogeneous asymptotics, where metric expansions include both powers of $1/r$ and logarithmic terms. The authors apply the Iyer–Wald formalism directly in the physical spacetime to derive NP asymptotics, asymptotic symmetries, and mass-like charges, revealing that logarithmic terms do not modify the leading memory/balance structure. They construct a canonical energy $\mathcal{E}$ that reduces to the Newman–Unti mass when logs vanish and introduce a reduced energy $\hat{\mathcal{E}}$ that obeys a standard mass-loss formula $\frac{d\hat{\mathcal{E}}}{du}=-\frac{1}{4\pi}\int_{S^2}|\dot{\sigma}_{2,0}|^2\,dS$, generalizing Bondi mass loss to polyhomogeneous cases. The results extend gravitational scattering and memory analyses to slower decays without conformal compactification and recover known results in special limits (e.g., $\Psi_0^3=0$), enabling a broader understanding of observable memory effects in such spacetimes.

Abstract

Spacetimes with metrics admitting an expansion in terms of a combination of powers of 1/r and ln r are known as polyhomogeneous spacetimes. The asymptotic behaviour of the Newman-Penrose quantities for the vacuum polyhomogeneous spacetimes is presented under certain gauges. The Bondi mass is revisited via the Iyer-Wald formalism. The memory effect of the gravitational radiation in the vacuum polyhomogeneous spacetimes is also discussed. It is found that the appearance of the logarithmic terms does not affect the balance law and it remains identical to that of spacetimes with metrics admitting an expansion in terms of powers of 1/r.