The Arrow of Time and the Growing Causal Sphere: Reading Penrose’s Weyl Curvature Hypothesis as the Growth of the Light Sphere’s Entropy — The Close of the Thermodynamic Arc.
The second law requires a low-entropy past — the “past hypothesis” — and gravitationally this is subtle: a smooth, uniform distribution is the LOW gravitational-entropy state, and clumping into black holes is the HIGH one. Penrose’s Weyl Curvature Hypothesis encodes this: the initial singularity had vanishing Weyl curvature (smooth, low), final singularities have diverging Weyl (clumped, high); the arrow of time is the growth of gravitational entropy from a smooth beginning. This paper reads that arrow through the system’s own geometry. The causal (light) sphere r=ct of Paper 61 grows with time; its area A=4π(ct)² grows; and its Bekenstein–Hawking entropy S=A/(4l_P²) (Papers 65, 73) grows accordingly, S(t)=πc²t²/l_P². This monotone growth is a geometric face of the horizon-entropy term of the generalised second law (Bekenstein; Gibbons–Hawking), and the expanding-horizon COMPLEMENT of Penrose’s clumping Weyl growth; both share the smooth low-entropy origin. This closes the thermodynamic arc of Papers 62, 72, 73, 74: under the emergent-gravity umbrella, the arrow of time is gravity-as-thermodynamics read in time. IMPORTANT: this does NOT solve the arrow-of-time problem and does NOT explain the low-entropy past or the vanishing initial Weyl (Penrose’s own open conjecture); it is a placement/reading, not a derivation.
A note on the aim of this work. This is a structural interpretation — a geometric reading of the gravitational arrow placed within Penrose’s hypothesis and the generalised second law — not a solution to the arrow-of-time problem and not a new physical law. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Arrow of Time and the Growing Causal Sphere: Reading Penrose’s Weyl Curvature Hypothesis as the Growth of the Light Sphere’s Entropy (The Close of the Thermodynamic Arc) [Paper 75]. Zenodo.
DOI: 10.5281/zenodo.21670183
ORCID: 0009-0006-8587-3450