Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-06-09
Class.Quant.Grav.23:6683-6708,2006
Physics
High Energy Physics
High Energy Physics - Theory
31 pages, 14 color figures. v2: Discussion of the metric on the space of metrics corrected and expanded, references added
Scientific paper
10.1088/0264-9381/23/23/006
Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to General Relativity, where gradient flow is Ricci flow, and focus on the example of 4-dimensional Euclidean gravity with boundary S^1 x S^2, representing the canonical ensemble for gravity in a box. At high temperature the action has three saddle points: hot flat space and a large and small black hole. Adding a time direction, these also give static 5-dimensional Kaluza-Klein solutions, whose potential energy equals the 4-dimensional action. The small black hole has a Gross-Perry-Yaffe-type negative mode, and is therefore unstable under Ricci flow. We numerically simulate the two flows seeded by this mode, finding that they lead to the large black hole and to hot flat space respectively, in the latter case via a topology-changing singularity. In the context of string theory these flows are world-sheet renormalization group trajectories. We also use them to construct a novel free energy diagram for the canonical ensemble.
Headrick Matthew
Wiseman Toby
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