Revisiting thermodynamic topologies of black holes

Abstract

In the generalized off-shell free energy landscape, black holes can be treated as thermodynamic topological defects. The local topological properties of the spacetime can be reflected by the winding numbers at the defects, while the global topological nature can be classified by the topological number which is the sum of all local winding numbers. We propose that the winding numbers can be calculated via the residues of isolated one-order pole points of characterized functions constructed from the off-shell free energy. Using the residue method, we show that the topologies of black holes can be divided into three classes with the topological numbers being -1, 0, and 1, respectively, being consistent with the results obtained in [Phys. Rev. Lett. 129, 191101 (2022)] by using the topological current method. Moreover, we point out that standard defect points, generation and annihilation points, and critical points can be distinguished by coefficients of the Laurent series of the off-shell characterized function at those singular points.

Classifying topology of consistent thermodynamics of four-dimensional Lorentzian NUT-charged spacetimes

“…Note that the topological number W still stays the same if the pressure takes another arbitrary value. Additionally, similar to the results given above for the RN-NUT spacetime and KN-NUT spacetime, it also gives a counterexample that significantly differs from the conclusions reached in all prior works [52][53][54][55][56][57][58][59][60][61], which state that a black hole solution only has one certain topological number, independent of the values of the solution parameters.…”

“…Thus, the whole class of four-dimensional NUTcharged spacetimes should be viewed as generic black holes from the viewpoint of the thermodynamic topological approach. Furthermore, the RN-NUT spacetime, KN-NUT spacetime, and RN-NUT-AdS spacetime provide three counterexamples that significantly differs from the conclusions reached in all prior works [52][53][54][55][56][57][58][59][60][61], which conclude that a black hole solution only has one definite topological number, independent of the values of the solution parameters.…”

“…Based on the local property of the zero points, for the RN-NUT spacetime, we have the topological number W = −1 when the parameter values n > q, and W = 1 − 1 = 0 when n ≤ q. Thus, it provides the first example violates the consensus in all previous works [52][53][54][55][56][57][58][59][60][61] that one black hole solution has only one definite topological number, regardless of the values of the solution parameters. In addition, since the topological number of the Taub-NUT spacetime is −1, while that of RN-NUT spacetimes is −1 or 0, it implies that the electric charge parameter has an important effect on the topological number for the neutral static Taub-NUT spacetime.…”

“…Because of its simplicity and easy maneuverability of the procedure, the thermodynamic topological approach proposed in Ref. [52] soon attracted a great deal of attention and was then successfully applied to investigate the topological numbers of some known black hole solutions [53][54][55][56][57][58][59][60][61]. It is natural to study the topological numbers of NUT-charged spacetimes to determine whether they are black holes.…”

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“…The topological number of a black hole is obtained by summing over the individual winding numbers of all the black hole branches. Following [43], study of black holes as topological defects has been extended to a number of black holes in [44][45][46][47][48][49][50][51][52][53][54].…”