[Paper Review] Thermodynamic Topological Classifications of Well-Known Black Holes

Aqsa Mehmood, M. Umair Shahzad|arXiv (Cornell University)|Oct 15, 2023

Black Holes and Theoretical Physics39 references4 citations

TL;DR

This paper introduces a thermodynamic topological classification of black holes in dRGT massive gravity, 5D Yang-Mills massive gravity, and D-dimensional RN-AdS black holes with quintessence and string clouds. By computing topological charges and winding numbers from critical points in phase diagrams, it reveals that all models exhibit total topological charges of 0 or 1, indicating trivial or simple topology, with phase transitions linked to changes in winding numbers.

ABSTRACT

In this work, we investigate the thermodynamic properties of black holes (BHs) that have non-trivial topological features in their phase diagrams. We consider three different models of BHs: (1) a class of BHs in dRGT massive gravity, which adds a mass term to general relativity; (2) a class of BHs in 5D Yang-Mills massive gravity, which combines dRGT massive gravity with a non-Abelian gauge field; and (3) a D-dimensional RN-AdS BH surrounded by Quintessence and a cloud of strings, which are strange forms of matter that change the thermodynamics of the BH. Our goal is to find the critical points of these BHs, which provide the location of first-order phase transitions and figure out their corresponding topological charges. Topological charges are numbers that show how complicated the BH topology is. Then, we look at these BHs as topological defects in the thermodynamic domain, which is the space of thermodynamic variables like pressure and temperature. We calculate winding numbers to analyze topology on a global and local scale at these defects, which are integers that indicate how many times a curve encircling the defect wraps around the origin. Our analysis reveals that the total topological charge is either equal to 0 or 1 for all models, meaning that the BHs have either a trivial or simple topology. In some cases, we see that the BH's topology belongs to a different thermodynamic topological class. This means that the BHs can go through topological phase transitions.

Motivation & Objective

• To classify black holes based on their thermodynamic and topological properties using advanced topological invariants.
• To identify critical points in black hole phase diagrams where first-order phase transitions occur.
• To compute topological charges and winding numbers as indicators of non-trivial topology in black hole spacetimes.
• To analyze the stability and phase structure of black holes in modified gravity models with exotic matter fields.
• To explore the physical implications of topological phase transitions in black hole thermodynamics.

Proposed method

• Identify critical points in black hole phase diagrams using the condition $(\partial T/\partial S)_P = 0$, marking phase transitions.
• Define a thermodynamic potential $\Phi = \frac{1}{\sin\theta} \tilde{T}(S, \dots)$ to simplify the topology of critical points.
• Construct a vector field $\phi^a = (\partial \Phi / \partial S, \partial \Phi / \partial \theta)$ from gradients of the potential.
• Compute the topological current $j^\mu = \frac{1}{2\pi} \epsilon^{\mu\nu\lambda} \epsilon_{ab} \partial_\nu n^a \partial_\lambda n^b$ using normalized vector field $n^a = \phi^a / ||\phi||$.
• Calculate the topological charge $Q = \frac{1}{2\pi} \int_\Sigma j^\mu d^2\Sigma_\mu = \sum_i w_i$ by integrating over a closed surface enclosing critical points.
• Determine winding numbers $w_i$ from the number of times the vector field wraps around the origin, with conventional critical points having $w = -1$ and novel unstable points $w = +1$.

Experimental results

Research questions

• RQ1 What are the critical points in the thermodynamic phase diagrams of black holes in dRGT massive gravity, 5D Yang-Mills massive gravity, and D-dimensional RN-AdS black holes with quintessence and string clouds?
• RQ2 How do topological charges and winding numbers classify the topological structure of black hole phase transitions?
• RQ3 What is the total topological charge of each black hole model, and what does it imply about the global topology of the system?
• RQ4 How do changes in winding numbers correlate with phase transitions and stability in black hole thermodynamics?
• RQ5 Can black holes be classified as topological defects in thermodynamic space, and what are the implications for their phase behavior?

Key findings

• The dRGT massive gravity black hole with $\alpha=2$, $\beta=10$, $c=1$ has a total topological charge of 1, while the case with $\alpha=-15$, $\beta=10$, $c=10$ yields a total charge of 0.
• The 5D Yang-Mills massive gravity black hole exhibits a total topological charge of 1 across all studied parameter sets, with winding numbers $-1, +1, -1$ at critical points.
• The D-dimensional RN-AdS black hole with quintessence and string cloud has a total topological charge of 1, with winding numbers $+1, -1, +1$ at three distinct zero points.
• The phase diagrams of all models show three branches—small, intermediate, and large black holes—where small and large branches are stable, and intermediate is unstable, resembling the van der Waals liquid-gas system.
• Specific heat curves show divergences at extremal points, marking boundaries between stable and unstable regions, with black and purple dashed lines indicating these critical transitions.
• The interchange of winding numbers across critical points signals topological phase transitions, suggesting that changes in the order parameter space topology can induce physical changes in black hole properties such as mass and spin.