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arXiv · 2510.24209

Thermodynamic topology of black holes and an invariant of spacetime

Cao H. Nam
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Abstract

We present a formalism for exploring the thermodynamic topology of black holes, without introducing the auxiliary variable $\Theta\in[0,\pi]$ employed in previous studies. The formalism is based on an off-shell grand free energy appropriate to thermodynamic systems that can exchange both energy and matter with their environment, thereby allowing thermal and chemical equilibrium. We define a vector field as the gradient of the off-shell grand free energy, whose zeros correspond to equilibrium configurations and hence to black hole solutions, and then we construct a conserved topological tensor from its normalized components. The associated topological flux is given by the sum of local topological indices of all zeros contained within a hypersurface at fixed ensemble parameters. We further define the asymptotic topological flux as the limiting value of the conserved topological flux through a family of finite hypersurfaces as they approach the asymptotic boundary of the ensemble parameter space. We find that the asymptotic topological flux is determined by the asymptotic/background spacetime geometry. This motivates the conjecture that the asymptotic topological flux is an invariant associated with the asymptotic/background spacetime geometry. Within the spacetime classification based on the asymptotic topological flux, the flat-space limit of AdS is found to be nontrivial, providing a potentially complementary perspective on the AdS distance conjecture.

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Cao H. Nam. 2025-10-28. Thermodynamic topology of black holes and an invariant of spacetime. https://arxiv.org/abs/2510.24209

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