Comment on "Multidimensional arrow of time" (arXiv:2601.14134)

TL;DR

Rubin's MAT suggests time direction from monotonic internal-volume growth, but this work shows such growth induces a time-varying $G_N$ that clashes with Lunar Laser Ranging and BBN bounds. To resolve, it proposes a shape-dynamic extension: replace volume growth with volume-preserving shape evolution of the internal metric under volume-normalized Ricci flow, guided by Perelman's entropy $\nu$. The internal geometry is shown to undergo irreversible smoothing toward higher-entropy configurations while keeping $G_N$ fixed, thanks to normalization and the spectral interpretation of the vacuum geometry via the conjugate heat equation. This yields a geometric arrow of time grounded in differential geometry that preserves fundamental constants and connects Rubin's idea to a broader shape-dynamics framework.

Abstract

In a recent preprint [arXiv:2601.14134v1], Rubin argues that the arrow of time originates from the monotonic growth of the volume of extra dimensions. While the identification of a geometric origin for time's arrow is compelling in the case of brane-world models, we point out a possible tension between the proposed volume growth and the observational stability of the effective four-dimensional Newton's gravitational constant, G, that may arise in Kaluza-Klein (KK) theory. In standard KK approaches, such volume growth induces a time-variation of G that exceeds Big Bang Nucleosynthesis (BBN) and Lunar Laser Ranging (LLR) bounds by many orders of magnitude. To resolve this tension while preserving the author's key insight in the Kaluza-Klein case, we propose an extension: the "shape-dynamic arrow of time". By utilizing the scale-invariant monotonicity of Perelman's nu-entropy under normalized Ricci flow, we demonstrate how an arrow of time can emerge from the geometric smoothing of extra dimensions at fixed volume, thereby satisfying observational constraints on fundamental constants.