Multidimensional arrow of time

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.