Topological and optical signatures of modified black-hole entropies

Thermodynamic topology and photon spheres in the hyperscaling violating black holes

In this article, we investigate the thermodynamic topology of some black holes, namely AdS Reissner Nordstrom (R–N), AdS Einstein–Gauss–Bonnet (EGB), and AdS Einstein-power-Yang–Mills (EPYM), from different frameworks: bulk-boundary (BB) and restricted phase space (RPS). Using the generalized off-shell Helmholtz free energy method, we calculate the thermodynamic topology of the selected black holes in each space separately and determine their topological classifications. We show that the addition of GB terms, dimensions, and other factors do not affect the topological classes of black holes in both spaces. The calculations and plots indicate that the AdS R–N and AdS EGB black holes show similar behavior and their topological numbers sets in both spaces, i.e., BB and RPS, are similar and equal to (W=+1). However, AdS EPYM black holes show an interesting behavior. In addition to BBT and RPS, we also consider the extended phase space thermodynamics (EPST) and evaluate the thermodynamic topology for AdS EPYM black hole. The changing (r−τ) in both spaces shows similar behavior. Also, the topological number and the total topological numbers for this black hole in the BB, RPS and EPS thermodynamics are completely same, i,e., (ωBBT=ωRPS=ωEPST=+1,−1) or WBBT=WRPS=WEPST=0. An important point is that the Einstein–Yang–Mills black hole has thermodynamic topology equivalence in three spaces. The present result may be due to the non-linear YM charge parameter and the difference between the gauge and gravity corrections in the above black holes.

In this study, we explore the thermodynamic properties of rotating regular black holes (RBH) in the context of conformal massive gravity (CMG). Our methodology involves calculating the specific heat, Gibbs free energy, and winding numbers (w) in canonical and grand canonical ensembles. These winding numbers play a crucial role in determining the thermodynamic stability of the system via topology. We derive the topological numbers of rotating regular BHs in conformal massive gravity by varying the spin (a), charge (Q), and hair parameters ([Formula: see text]). Our observations reveal that lower values of the hair parameter result in a stable topological classification, while higher values lead to an unstable topological classification, which indicates the presence of a phase transition in both ensembles. This shows that these parameters play a crucial role in stability and topological classification.

In this work, we present a new spherically symmetric black hole solution surrounded by a King-type dark matter halo. The construction begins by formulating the spacetime metric corresponding to a pure dark matter distribution, utilizing the established relationship between the tangential velocity in the equatorial plane and the metric coefficients of a spherically symmetric geometry. By incorporating the King dark matter profile into the energy-momentum tensor within Einstein’s field equations, we obtain an exact black hole solution describing the coupled system. To investigate the influence of the halo parameters-specifically, the core density and core radius-on the optical properties of the black hole, we analyze null geodesics within the Lagrangian framework. The resulting effective potential, photon sphere, and shadow structures are systematically examined. Notably, the shadow radius is found to exceed the corresponding Schwarzschild value, displaying a significant increase as the core radius of the King profile grows. Finally, we analyze the thermodynamic properties of the obtained black hole solution and found that the dark matter halo parameters strongly influence its thermodynamic behavior. These parameters significantly affect the Hawking temperature, Gibbs free energy, and specific heat capacity, thereby shaping the black hole’s stability and phase structure. Furthermore, within the framework of thermodynamic topology, our analysis of the generalized free energy shows a topological charge $$W = -1$$ W = - 1 , indicating that the black hole shares the same topological class as the Schwarzschild case and possesses a single thermodynamically stable phase without multiple phase transitions in the explored regime.

In this paper, we study the thermodynamic topology of AdS Einstein–power–Yang–Mills black holes, examining them through both the bulk-boundary and restricted phase space (RPS) frameworks. We consider various non-extensive entropy models, including Barrow (δ), Rényi (λ), Sharma–Mittal (β, α), Kaniadakis (K), and Tsallis-Cirto entropy (Δ). Initially, we analyze the thermodynamic topology within the bulk-boundary framework. Our findings highlight the influence of free parameters on topological charges. We observe two topological charges (ω=+1,-1) with respect to the non-extensive Barrow parameter and also with (δ=0) in Bekenstein–Hawking entropy. For Rényi entropy, different topological charges are observed depending on the value of the λ with a notable transition from three topological charges (ω=+1,-1,+1) to a single topological charge (ω=+1) as λ increases. Also, by setting λ to zero results in two topological charges (ω=+1,-1). Sharma–Mittal entropy exhibits three distinct ranges of topological charges influenced by the α and β with different classifications viz, if β exceeds α, we will have (ω=+1,-1,+1); if β=α, we have (ω=+1,-1); and if α exceeds β, we obtain (ω=-1). Also, Kaniadakis entropy shows variations in topological charges; viz., we observe (ω=+1,-1) for any acceptable value of K, except when K=0, where a single topological charge (ω=-1) appears. In the case of Tsallis-Cirto entropy, for small parameter Δ values, we have (ω=+1) and when Δ increases to 0.9, we will have (ω=+1,-1). A particularly intriguing aspect of this research is its application to the RPS framework. When we extend our analysis to this space using the specified entropies, we find that the topological charge consistently remains (ω=+1) independent of the specific values of the free parameters for Rényi, Sharma–Mittal, and Tsallis–Cirto. Additionally, for Barrow entropy in RPS, when δ increases from 0 to 0.8, the number of topological charges rises. Finally for Kaniadakis entropy, at small values of K, we observe (ω=+1). However, as the non-extensive parameter K increases, we encounter different topological charges and classifications with (ω=+1,-1).

In 2005 Zakharov et al. predicted an opportunity to reconstruct a shadow in Sgr A* with ground based or space—ground interferometer acting in mm or sub-mm band (the Millimetron was mentioned for such needs). The prediction was confirmed in May 2022 since the Event Horizon Telescope (EHT) Collaboration presented results of a shadow reconstruction for our Galactic Center (the shadow around the supermassive black hole in M87 was reconstructed in 2019). These reconstructions were based on EHT observations done in 2017. In 2005 Zakharov et al. also derived analytical expressions for shadow size as a function of charge for Reissner–Nordström metric and later these results were generalized for a tidal charge case. We discuss opportunities to evaluate parameters of alternative theories of gravity with shadow size estimates done by the EHT Collaboration, in particular, a tidal charge could be estimated from these observations. We also discuss opportunities to use Millimetron facilities for shadow reconstructions in M87* and Sgr A*. In our recent studies we discuss shadow formations for cases where naked singularities, wormholes or more exotic models substitute conventional black holes in galactic centers.

The classification of the static magnetic domain wall structures of tube- and envelope-type is made in an unified way using the homotopy theory. The sets of topological classes for such two kinds of magnetic domain walls, GWn and GWn, are corresponding respectively one-by-one to the sets of homotopy classes relative to n + l base points for the S2→S2 and S3→S4 continuous maps. Either GW(n) ro GW(n), therefore, can be constructed into group isomorphic to Z, the additive group of integers. (Then we call them the tube-wall group and the envelope-wall group of type n, respectively). The ‘winding number' introduced by Slon-czewski et al. is considered anew. The sufficient and necessary conditions under which the ‘winding number' is allowed to be taken as the index of tube-wall class are obtained. Finally, the topological classification of the magnetization states with M tube-walls and N envelope-walls coexisting is discussed. It is shown that the set of the corresponding topological classes, GW(M,N), can be constructed into group isomorphic to ZM+N, the M + N dimensional lattice vector group. (It is then referred to as the mix-wall group of type [M, N] ).

Dynamical quantum phase transitions (DQPTs), which serve as a theoretical framework for understanding far-from-equilibrium physics in quantum many-body systems, have recently been observed experimentally. Their topological properties are typically characterized by the winding number, which acts as an order parameter. While DQPTs exhibiting both integer and half-integer jumps in the winding number have been reported, the underlying mechanisms behind these distinct topological behaviors, as well as the potential existence of other topological classes, remain open questions. To address this, we investigate DQPTs in the one-dimensional XY model under a quench protocol. We show that the observed topological diversity originates from the nature of the critical modes, which we classify into two categories: boundary modes and interior modes. Specifically, critical interior modes always lead to DQPTs with an integer winding number, while critical boundary modes always result in DQPTs characterized by a half-integer winding number. By analyzing the number and classification of critical modes, we provide a classification of the topological properties of DQPTs in the one-dimensional XY model. According to their distinct topological features, we categorize DQPTs into six types, three of which have not been previously identified in the literature. We discuss in detail the conditions associated with each type and present the corresponding dynamical phase diagrams. Our framework is not restricted to the XY model; it is applicable to other two-band models in one-dimensional systems, including the SSH model, Kitaev chain, Rice-Mele model, and Creutz model.

Consideration of extra spatial dimensions is motivated by the unification of gravity with other interactions, the achievement of the ultimate framework of quantum gravity, and fundamental problems in particle physics and cosmology. Much attention has been focused on the effect of these extra dimensions on the modified theories of gravity. Analytically examining astrophysical phenomena like black hole shadows is one approach to understand how extra dimensions would affect the modified gravitational theories. The purpose of this study is to derive a higher-dimensional metric for a dark compact object in STVG theory and then examine the behaviour of the shadow shapes for this solution in STVG theory in higher dimensions. We apply the Carter method to formulate the geodesic equations and the Hamilton–Jacobi method to find photon orbits around this higher-dimensional MOG dark compact object. We investigate the effects of extra dimensions and the STVG parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on the black hole’s shadow size. Next, we compare the shadow radius of this higher-dimensional MOG dark compact object to the shadow size of the supermassive black hole M87*, which has been realized by the Event Horizon Telescope (EHT) collaborations, in order to restrict these parameters. We find that extra dimensions in the STVG theory typically lead to a reduction in the shadow size of the higher-dimensional MOG dark compact object, whereas the effect of parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on this black hole’s shadow is suppressible. Remarkably, given the constraints from EHT observations, we find that the shadow size of the four-dimensional MOG dark compact object lies in the confidence levels of the EHT data. Finally, we investigate the issue of acceleration bounds in higher-dimensional MOG dark compact object in confrontation with EHT data of M87*.

We establish a number of results about smooth and topological concordance of knots in S 1 × S 2 . The winding number of a knot in S 1 × S 2 is defined to be its class in H 1 ( S 1 × S 2 ; Z ) ≅ Z . We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl–Nagel–Orson–Powell in the topological category. We say a knot in S 1 × S 2 is slice (respectively, topologically slice) if it bounds a smooth (respectively, locally flat) disk in D 2 × S 2 . We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than ± 1 , there are infinitely many topological concordance classes even within the collection of slice knots. Additionally, we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are pairwise topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and are pairwise topologically, but not smoothly, concordant.

We classify topological phases of non-Hermitian systems in the Altland-Zirnbauer classes with an additional reflection symmetry in all dimensions. By mapping the non-Hermitian system into an enlarged Hermitian Hamiltonian with an enforced chiral symmetry, our topological classification is thus equivalent to classifying Hermitian systems with both chiral and reflection symmetries, which effectively change the classifying space and shift the periodical table of topological phases. According to our classification tables, we provide concrete examples for all topologically nontrivial non-Hermitian classes in one dimension and also give explicitly the topological invariant for each nontrivial example. Our results show that there exist two kinds of topological invariants composed of either winding numbers or $\mathbb{Z}_2$ numbers. By studying the corresponding lattice models under the open boundary condition, we unveil the existence of bulk-edge correspondence for the one-dimensional topological non-Hermitian systems characterized by winding numbers, however we did not observe the bulk-edge correspondence for the $\mathbb{Z}_2$ topological number in our studied $\mathbb{Z}_2$-type model.

Topology is a promising approach toward to the light ring in a generic black hole background, and equatorial timelike circular orbit in a stationary black hole background. In this paper, we consider the distinct topological configurations of the timelike circular orbits in static, spherically symmetric, and asymptotic flat black holes. By making use of the equation of motion of the massive particles, we construct a vector with its zero points exactly relating with the timelike circular orbits. Since each zero point of the vector can be endowed with a winding number, the topology of the timelike circular orbits is well established. Stable and unstable timelike circular orbits respectively have winding number +1 and -1. In particular, for given angular momentum, the topological number of the timelike circular orbits also vanishes whether they are rotating or not. Moreover, we apply the study to the Schwarzschild, scalarized Einstein-Maxwell, and dyonic black holes, which have three distinct topological configurations, representations of the radius and angular momentum relationship, with one or two pairs timelike circular orbits at most. It is shown that although the existence of scalar hair and quasi-topological term leads to richer topological configurations of the timelike circular orbits, they have no influence on the total topological number. These results indicate that the topological approach indeed provides us a novel way to understand the timelike circular orbits. Significantly, different topological configurations can share the same topology number, and hence belong to the same topological class. More information is expected to be disclosed when other different topological configurations are present.

In the recent proposal [S. W. Wei et al., Black Hole Solutions as Topological Thermodynamic Defects, Phys. Rev. Lett. 129, 191101 (2022).], the black holes were viewed as topological thermodynamic defects by using the generalized off shell free energy. In this paper, we follow such proposal to study the local and global topological natures of the Gauss-Bonnet black holes in anti--de Sitter (AdS) space. The local topological natures are reflected by the winding numbers, where the positive and negative winding numbers correspond to the stable and unstable black hole branches. The global topological natures are reflected by the topological numbers, which are defined as the sum of the winding numbers for all black hole branches and can be used to classify the black holes into different classes. When the charge is present, we find that the topological number is independent on the values of the parameters, and the charged Gauss-Bonnet AdS black holes can be divided into the same class of the Reissner-Nordstr\"om anti-de Sitter black hole black holes with the same topological number 1. However, when the charge is absent, we find that the topological number has certain dimensional dependence. This is different from the previous studies, where the topological number is found to be a universal number independent of the black hole parameters. Furthermore, the asymptotic behaviors of curve $\ensuremath{\tau}({r}_{h})$ in small and large radii limit can be a simple criterion to distinguish the different topological number. We find a new asymptotic behavior as $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}0)=0$ and $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}\ensuremath{\infty})=0$ in the black hole system, which shows topological equivalency with the asymptotic behaviors $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}0)=\ensuremath{\infty}$ and $\ensuremath{\tau}({r}_{h}\ensuremath{\rightarrow}\ensuremath{\infty})=\ensuremath{\infty}$. We also give an intuitional proof of why there are only three topological classes in the black hole system under the condition $({\ensuremath{\partial}}_{{r}_{h}}S{)}_{P}>0$.

More than two years ago the Event Horizon Telescope collaboration presented the first image reconstruction around the shadow for the supermassive black hole in M87*. It gives an opportunity to evaluate the shadow size. Recently, the Event Horizon Telescope collaboration constrained parameters (“charges”) of spherical symmetrical metrics of black holes from an estimated allowed interval for shadow radius from observations of M87* in 2017. Earlier, analytical expressions for the shadow radius as a function of charge (including a tidal one) in the case of Reissner–Nordström metric have been obtained. Some time ago, Bin-Nun proposed to apply a Reissner–Nordström metric with a tidal charge as an alternative to the Schwarzschild metric in Sgr A*. If we assume that a Reissner–Nordström black hole with a tidal charge exists in M87*, therefore, based on results of the shadow size evaluation for M87* done by the Event Horizon Telescope collaboration we constrain a tidal charge. Similarly, we evaluate a tidal charge from shadow size estimates for Sgr A*.

One of the possible ways to test gravity theories and get constraints on parameters of a gravity theory and a black hole is based on studies of black hole shadow applying Event Horizon Telescope (EHT) data from the shadow sizes of M87* and Sgr A*. In this sense, we study the shadow of rotating charged black holes in Einstein–Maxwell scalar (EMS) theory. First, we obtain a rotating EMS black hole solution and analyze the horizon properties. We derive the effective potential for the circular motion of photons along null geodesics around the rotating black hole and obtain the black hole shadow using celestial coordinates. The effects of the black charge and spin and EMS theory parameters on the shape of the black hole shadow, its radius, and distortion parameters are analyzed in detail. We have obtained upper and lower limits for spin and black hole charges of Sgr A* and M87* using their shadow size for various values of EMS parameters. Lastly, we computed and examined the standard shadow radius, equatorial, and polar quasinormal modes using the geometric-optic relationship between the parameters of the quasinormal mode and the conserved values along the geodesics.