Black Hole Solutions as Topological Thermodynamic Defects

In this Letter, employing the generalized off-shell free energy, we treat black hole solutions as defects in the thermodynamic parameter space. The results show that the positive and negative winding numbers corresponding to the defects indicate the local thermodynamical stable and unstable black hole solutions, respectively. The topological number defined as the sum of the winding numbers for all the black hole branches at an arbitrary given temperature is found to be a universal number independent of the black hole parameters. Moreover, this topological number only depends on the thermodynamic asymptotic behaviors of the black hole temperature at small and large black hole limits. Different black hole systems are characterized by three classes via this topological number. This number could help us in better understanding the black hole thermodynamics and, further, shed new light on the fundamental nature of quantum gravity.

Figure 1

The red arrows represent the unit vector field $n$ on a portion of the $r_h\text{−}\mathrm{\Theta }$ plane. The zero points (ZPs) marked with black dots are at $(r_h/r_0,\mathrm{\Theta })=(1,(\pi /2))$, (1.46, $(\pi /2)$), and (2.15, $(\pi /2)$), for ${\mathrm{ZP}}_1$, ${\mathrm{ZP}}_2$, and ${\mathrm{ZP}}_3$, respectively. The blue contours $C_i$ are closed loops enclosing the zero points. (a) The unit vector field for the Schwarzschild black hole with $\tau /r_0=4\pi$. (b) The unit vector field for the RN black hole with $\tau /r_0=34.48$ and $Q/r_0=1$.

Figure 2

Zero points of the vector $\varphi$ shown in the $r_h\text{−}\tau$ plane. The blue dashed and red solid lines are for the Schwarzschild black hole (Sch BH) and RN black hole (RN BH) with $Q/r_0=1$. The black dot with ${\tau }_c=6\sqrt{3}\pi Q$ denotes the generation point for the RN black hole. At $\tau ={\tau }_1$, there is one Schwarzschild black hole, and at $\tau ={\tau }_2$, there is one Schwarzschild black hole and two RN black holes.

Figure 3

Zero points of ${\varphi }^{r_h}$ shown in the $r_h\text{−}\tau$ plane for the RN-AdS black hole with $P{r}_0^2=0.0022$ and $Q/r_0=1$. The black solid, blue dashed, and red solid lines are for the small black hole (SBH), intermediate black hole (IBH), and large black hole (LBH), respectively. Black and blue dots are the annihilation and generation points. Different color regions have different numbers of the black hole branches. However, their $W$ number is constant and equals 1.