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Thermodynamic topology of black holes and an invariant of spacetime

Cao H. Nam

Abstract

We represent a new approach to exploring the thermodynamic topology of black holes, without introducing the nonphysical variable $Θ\in[0,π]$ considered in previous studies, where black holes can exchange both energy and matter with the environment, leading to a thermal and chemical equilibrium. We construct a conserved topological tensor based on the gradient flow of the off-shell grand free energy corresponding to a two-dimensional or higher-dimensional vector field whose zeros are black hole solutions. We obtain a topological charge that is the sum of the index of all zeros. We find that black holes that share the same background geometry would have the same topological charge, hence they belong to the same kind of solutions. This point implies that the topological charge characterizing the black hole thermodynamics is also an invariant of spacetime, leading to valuable insight into the observed cosmological constant and the AdS distance conjecture.

Thermodynamic topology of black holes and an invariant of spacetime

Abstract

We represent a new approach to exploring the thermodynamic topology of black holes, without introducing the nonphysical variable considered in previous studies, where black holes can exchange both energy and matter with the environment, leading to a thermal and chemical equilibrium. We construct a conserved topological tensor based on the gradient flow of the off-shell grand free energy corresponding to a two-dimensional or higher-dimensional vector field whose zeros are black hole solutions. We obtain a topological charge that is the sum of the index of all zeros. We find that black holes that share the same background geometry would have the same topological charge, hence they belong to the same kind of solutions. This point implies that the topological charge characterizing the black hole thermodynamics is also an invariant of spacetime, leading to valuable insight into the observed cosmological constant and the AdS distance conjecture.

Paper Structure

This paper contains 24 equations, 3 figures, 1 table.

Figures (3)

  • Figure 1: The left and right panels represent the unit vector field $\Vec{\phi}/||\Vec{\phi}||$ for the RN and RN-AdS black holes, respectively, with $\tau_1=1$, $\tau_2=5$, and $l=0.3$. Red dots refer to the zeros of $\Vec{\phi}$. The blue closed curves denote the curves $\mathcal{C}_i$, enclosing solely each zero, through which we compute $\text{index}_{x_i}(\Vec{\phi})=\frac{1}{2\pi}\int_{\mathcal{C}_i}\epsilon_{ab}n^adn^b$.
  • Figure 2: The behavior of the RN-AdS black hole temperature in the entropy $S$ for two regions of the time parameter $\tau_2$ with the AdS radius $l$ kept fixed.
  • Figure 3: The behavior of $\Vec{\phi}/||\Vec{\phi}||$ in the $(S,Q,J)$ space with $\tau_1=1$, $\tau_2=4$, and $\tau_3=5$. The green dot refers to a zero of $\Vec{\phi}$. The light orange closed surface refers to the surface $\mathcal{S}_i$, enclosing the zero, by which we compute $\text{index}_{x_i}(\Vec{\phi})=\frac{1}{\omega_2}\int_{\mathcal{S}_i}\epsilon_{abc}n^{a}dn^{b}\wedge dn^{c}$ with $\wedge$ being the wedge product.