Multidimensional arrow of time

TL;DR

Addresses the origin of the arrow of time by linking it to the expansion of extra dimensions in $f(R)$ gravity and the Wald entropy of the geometric background. It introduces a Postulate of Causality and analyzes a $D=4+n$ de Sitter-like bulk, showing that $S^{(D)}_{total}(t) \propto e^{(D-1)Ht}$ and $t \propto \ln S^{(D)}_{total}$, meaning the geometric entropy dominates the total budget and fixes the arrow for a 4D brane even in vacuum. The main insight is that the arrow of time is a geometric entropy gradient driven by multidimensional volume growth, not matter entropy, allowing the initial-state entropy to be inconsequential. The results connect causal structure, higher-dimensional geometry, and Wald entropy, with implications for brane-world cosmology and the universality of the thermodynamic arrow.

Abstract

This paper studies the effect of extra dimensions on the arrow of time within the framework of $f(R)$ gravity. We demonstrate that the observed irreversibility of physical processes can be explained by the monotonic growth of the extra-dimensional space. Unlike traditional cosmological approaches, our model does not link the arrow of time to the entropy of matter or radiation; rather, it identifies it with the Bekenstein-Hawking-Wald entropy of the geometric background. We establish a formal relation between the volume of the multidimensional manifold and the statistical weights of its geometric states. This leads to a fundamental relationship where the flow of time is intrinsically linked to the growth of multidimensional entropy. A key consequence of our framework is that the arrow of time remains a persistent feature for a 4D observer situated on a brane, even in the complete absence of matter or radiation. This directionality is driven by the dominant entropy production in the higher-dimensional bulk, which dominates local statistical fluctuations and ensures a stable causal direction.