In the hunt for new physics beyond the Standard Model, scientists are now looking closely at an unusual class of particle interactions called neutral triple gauge couplings (nTGCs). These interactions do not appear in the present Standard Model (SM) and only show up when we consider more subtle, higher-order effects. Even more intriguingly, some of these interactions could break CP symmetry—a tiny imbalance that might help explain why the universe contains more matter than anti-matter.
Black holes are usually pictured as pure gravitational objects, where spacetime is so strongly curved that even light cannot escape. Yet since the pioneering work of Bekenstein and Hawking, physicists have also understood black holes as thermodynamic systems: they have temperature, entropy, and phase transitions. A new invited review, published in Science China Physics, Mechanics & Astronomy, highlights recent developments in applying topological methods to black hole thermodynamics.
Topology is the branch of mathematics concerned with properties that remain unchanged under smooth deformation, and it has become a robust tool for probing the intrinsic properties of black holes. In black hole thermodynamics, the idea is more subtle. Researchers construct mathematical vector fields from thermodynamic quantities and look for special points where those fields vanish, following Duan's topological current theory. These zero points act like "defects" in an abstract thermodynamic landscape. By counting how the vector field winds around them, researchers can assign each one a topological charge. The total charge then gives a global topological number that characterizes the system as a whole.
The review explains how this framework has been explored for several aspects of black hole thermodynamics. These include critical points, where first-order small-large black hole phase transitions end; Davies points, which are associated with divergences in heat capacity; Hawking-Page phase transitions, which describe transitions between thermal radiation and stable large black holes in anti-de Sitter space; and black hole solutions themselves.
Among these approaches, the topology of black hole solutions has attracted particular attention because of its clear physical interpretation and broad applicability. When black hole solutions are treated as topological defects, the sign of the winding number indicates whether a branch is locally stable or unstable. Moreover, although the local winding characteristics of zero points, or the number of these local defects, may vary with black hole parameters such as charge and pressure, the global topological number can remain unchanged, indicating a universal characteristic of the topology. In this framework, Schwarzschild, Reissner-Nordstrom, and charged Reissner-Nordstrom anti-de Sitter black holes fall into different topological classes.
The review also surveys how the topological number behaves across different black hole settings, including rotating black holes in various dimensions, C-metric and NUT spacetimes, negative or positive cosmological constants, regular black holes without singularities, different thermodynamic ensembles, statistical mechanics with modified entropy formalisms, multiple defect curves, and temperature-dependent cases. These results reveal the distinctive topological attributes of diverse black hole solutions.
Building on these insights, the review summarizes recent work toward universal topological classifications of black holes. These classifications organize black hole solutions according to their global topological number and the signs of the innermost and outermost winding numbers. The universal thermodynamic behaviors of different topological classes are investigated in limits such as innermost and outermost states, as well as low- and high-temperature regimes.
The review concludes that topological methods have become a robust testing ground in black hole physics. Similar ideas have already been used to study photon spheres/light rings, light deflection, timelike circular orbits, and Hawking temperature. Exploring the interplay between topology, gravity, and thermodynamics may help uncover the nature of black holes and spacetime from a thermodynamic perspective, offering valuable insights for developing a comprehensive quantum gravity framework.
Science China Physics Mechanics and Astronomy
Systematic review