Geometrization of Hawking Radiation via Ricci Flow

Winona, Minnesota

In 1982, Richard S. Hamilton formulated Ricci flow along manifolds of three dimensions of positive Ricci curvature as an attempt to resolve Poincaré’s Conjecture. However, it took until 2006 by Grigori Perelman to resolve the conjecture with Ricci flow. Since then, research in pure mathematics on Ricci Flow increased exponentially, and people began to apply it towards physics. For example, Ricci flow has been found to be the Renormalization Group flow of the bosonic string and sigma model. However, Ricci flow’s analogous counterpart being the heat equation, makes it appear to have more applications. For this reason, we have studied the implication of Ricci flow as being the geometrization of Hawking Radiation, or black hole evaporation. Ricci flow behaves as the evolution of the metric along a parameter λ. It is equal to negative the Ricci tensor (if not normalized) or stated as ∂λ g = -Ric(g). We choose λ to be t for time and see how the metric evolves under time. In this presentation, we will explore the difficulties with this approach, including the consequences of Birkhoff's theorem, a conformal transformation of the metric, Ruppeiner’s methods of relationship between curvature and thermodynamics, and direct manipulation of the flow parameter λ.