Evolution of Hawking mass under hypersurface-restricted expanding flows

TL;DR

This work investigates the robustness of Hawking mass monotonicity for non-spherical surfaces under a restricted class of uniformly expanding flows in Minkowski space. Using in-slice, time-flat flows that reduce to IMCF within a fixed spacelike hypersurface, the authors numerically evolve perturbed spheres expressed as low-order spherical-harmonic radial graphs and track the Hawking mass $m_H$. They find that monotonicity persists for single-mode perturbations across several low-order harmonics, with $m_H$ evolving monotonically (and increasing) and a discrete monitor, $\\delta m_H$, remaining zero to machine precision; however, multi-mode perturbations exhibit numerical instabilities due to the forward-backward parabolic nature of the underlying equations. These results supply concrete computational evidence that the monotonicity mechanism of uniformly expanding flows extends beyond perfectly symmetric settings and establish a practical testbed for quasi-local mass diagnostics in more general spacetime contexts. The study highlights limitations for complex mode coupling and points to future work needed to handle multi-mode interactions and fully spacetime-based UEFs, including extensions to horizons.

Abstract

We present a numerical study of the evolution of the Hawking mass for closed nonspherical surfaces evolved under a class of expanding flows in Minkowski spacetime. Although formal monotonicity of the Hawking mass under smooth inverse mean curvature flow is well established in the Riemannian setting, comparatively little is known about the robustness of this behavior in discrete numerical implementations applied to explicitly embedded surfaces away from exact symmetry. We consider surfaces defined by small spherical harmonic perturbations of a round sphere and evolve them under an in-slice, time-flat flow analogous to inverse mean curvature flow. We examine the behaviour of the Hawking mass under the flow and find that monotonicity persists for a class of nonspherical perturbations and is robust under variations in perturbation amplitude and angular frequency. We also identify regimes in which numerical instabilities arise, highlighting practical challenges associated with extending such flows beyond simple symmetry assumptions. These results provide a concrete computational testbed for future investigations of uniformly expanding flows and quasi-local mass in more general spacetime settings.

Figures (3)

• Figure 1: Hawking mass $m_H$ of a perturbed sphere in Minkowski spacetime as a function of the perturbation amplitude squared $\epsilon^2$. The linear scaling confirms the $\mathcal{O} ( \epsilon^2)$ behaviour.
• Figure 2: Deviation of the Hawking mass $\Delta m_H(t)$ for a round sphere evolved under the in-slice flow in Minkowski spacetime. Analytically, the Hawking mass of a round sphere vanishes and is preserved along the flow. The small nonzero deviation shown here is numerical and remains bounded throughout the evolution, indicating that the scheme preserves the expected invariance up to discretization error.
• Figure 3: Evolution of the Hawking mass under in-slice flow for a perturbed sphere. Shown are the results for (left) $Y_{20}$ and (right) $Y_{30}$, with $\epsilon = 0.05$ in both cases. The Hawking mass evolves monotonically for both harmonics, although higher multipole structure leads to increased curvature in the mass–time relation due to stronger angular gradients, reflecting the larger contribution of angular derivatives to the evolution in higher- $\ell$ modes. No numerical instabilities or violations of monotonic behavior are observed.