We use Ising-like models to probe the thermal nature of Euclidean spacetime backgrounds. We determine which properties of the background—curvature, the presence of a horizon, or temperature—play a role in phase transitions. The geometries we use are Euclidean Schwarzschild, Rindler, extremal Reissner-Nördstrom (ERN), anti–de Sitter (AdS), and de Sitter. Among these, Rindler is flat, AdS does not have a horizon, and both AdS and ERN have zero temperatures. We find second-order phase transitions as the metric parameter is varied in all cases except for Rindler. Specifically, we find that the transition from order to disorder occurs as the curvature—or Euclidean gravity—increases. This supports our conjecture that Euclidean gravity is an essential ingredient for these phase transitions, as opposed to the presence of a horizon or temperature. Separately, since the selected geometries are position-dependent, the Ising-like models constructed on them are inhomogeneous, whereby they generalize the standard Ising model. We find that a consequence of this is that criticality does not correspond to maximal correlation lengths and scale invariance.

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• Classical black holes
• General relativity
• Phase transitions
• Quantum field theory
• Quantum fields in curved spacetime
• Monte Carlo methods