[Paper Review] Thermodynamic Topology and Photon Spheres in the Hyperscaling violating black holes

J. Sadeghi, Mohammad Ali S. Afshar+2 more|arXiv (Cornell University)|Jul 24, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper investigates the thermodynamic topology and photon sphere (PS) structure in four-dimensional hyperscaling violating (HSV) black holes using topological methods. It identifies distinct topological classes—Q = -1 for black holes, Q = 0 for naked singularities, and proposes a new class Q = +1 for naked singularities when z ≥ 1. The study confirms black hole solutions only exist for 1 ≤ z < 2, with consistent topological charges across temperature and generalized Helmholtz free energy methods, revealing critical points and phase structures tied to z and θ parameters.

ABSTRACT

It was shown that a standard ring of light can be imagined outside the event horizon for stationary rotating four-dimensional black holes with axial symmetry using the topological method\cite{001,002}. Based on this concept, in this paper, we investigate the topological charge and the conditions of existence of the photon sphere (PS) for a hyperscaling violation (HSV) black hole with various values of the parameters of this model. Then, after carrying out a detailed analysis, we show the conventional topological classes viz $Q=-1$ for the mentioned black hole and $Q=0$ for the naked singularities. extit{Also, we propose a new topological class for naked singularities (Q=+1) with respect to $z\geq1$. We also determined that $z\geq2$, it either shows a naked singularity form with total topological charge $+1$ or has no solution. Therefore, we have the black hole solution only in $1\leq z&lt;2$}. Then, we will use two different methods, namely the temperature (Duan's topological current $Φ$-mapping theory) and the generalized Helmholtz free energy method, to study the topological classes of our black hole. We discuss the critical and zero points (topological charges and topological numbers) for different parameters of hyperscaling violating black holes, such as ($z, \overlineθ$) and other free parameters, and study their thermodynamic topology. We observe that for a given value of the parameters $1\leq z&lt;2$, $\overlineθ$, and other free parameters, there exist two total topological charges $(Q_{total}=-1, 0)$ with the same phase structure for the $T$ method and total topological numbers $(W=+1)$ for the generalized Helmholtz free energy method. Additionally, we summarize the results for each study as photon sphere, temperature, and generalized Helmholtz free energy in some figures and tables. Finally, we compare our findings with other related studies in the literature.

Motivation & Objective

• To investigate the existence and topological classification of photon spheres in hyperscaling violating black holes with varying z and θ parameters.
• To classify thermodynamic topological phases using two independent methods: temperature (Duan’s topological current theory) and generalized Helmholtz free energy.
• To determine the parameter regime where black hole solutions exist, particularly focusing on the dynamical exponent z and hyperscaling violation parameter θ.
• To propose a new topological class (Q = +1) for naked singularities under z ≥ 1, extending the standard classification.

Proposed method

• Applies Duan’s topological current theory (Φ-mapping method) to compute topological charges from the temperature field, identifying critical and zero points.
• Uses the generalized Helmholtz free energy method to compute topological numbers (W) and analyze phase structures in the thermodynamic state space.
• Performs detailed analysis of null geodesics and effective potential to determine photon sphere existence and location via critical point equations.
• Derives analytical expressions for critical radii and temperatures using the metric and thermodynamic relations in d-dimensional spacetime with z and θ.
• Employs vector field mapping and contour integration on (rH–θ) planes to visualize topological charges and zero-point structures.
• Validates results through numerical plots and contour analysis for z = 1, 1.3, and 1.9 across different parameter sets.

Experimental results

Research questions

• RQ1 Under what conditions does a photon sphere exist in hyperscaling violating black holes with varying z and θ parameters?
• RQ2 What are the topological classes (Q) of the thermodynamic state space for HSV black holes and naked singularities, and how do they depend on z and θ?
• RQ3 Can a new topological class (Q = +1) be consistently defined for naked singularities when z ≥ 1, and what does it imply for the solution structure?
• RQ4 Do the temperature method and generalized Helmholtz free energy method yield consistent topological classifications for the same black hole parameters?
• RQ5 What is the range of z for which black hole solutions exist, and how do critical points and phase transitions emerge in this regime?

Key findings

• Black hole solutions exist only for 1 ≤ z < 2, with z ≥ 2 yielding either naked singularities (Q = +1) or no physical solution.
• The standard topological class Q = -1 is confirmed for black holes, while Q = 0 is assigned to naked singularities, and a new class Q = +1 is proposed for naked singularities when z ≥ 1.
• For 1 ≤ z < 2, both the temperature method and generalized Helmholtz free energy method yield consistent results: total topological charge Q_total = -1 or 0, and total topological number W = +1.
• Critical points for the temperature field are analytically derived, with two distinct critical radii r_cp1 and r_cp2 depending on z, θ, d, and other parameters.
• Numerical plots confirm the presence of zero points (ZPs) and closed contours (C1, C2) that encircle or exclude ZPs, validating topological charge assignments.
• The generalized Helmholtz free energy method shows W = +1 for z = 1, 1.3, and 1.9 across various parameter sets, indicating a stable topological structure in the thermodynamic state space.