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A number of experiments show that the decay lifetimes of muons bound to atomic nuclei are longer than the decay lifetimes of free muons. In this paper, a scheme of extending quantum mechanics (EQM) is proposed to resolve this problem. The Schr$\ddot{\text{o}}$dinger's equation is obtained to prove the validation of this attempt. The decay ratio of bound muons is also calculated in EQM, and the result is in good agreement with the experimental data., Comment: 5 pages, 1 figure, 2 tables

In this paper, we study the thermodynamics of Schwarzschild-anti-de Sitter black holes within the framework of non-commutative geometry. By solving the Einstein's equations, we derive the corrected Schwarzschild-AdS black hole with Lorentzian distribution and analyze the thermodynamics. Our results confirm that if the energy-momentum tensor outside the event horizon is related to the mass of the black hole, the conventional first law of thermodynamics will be violated. The study of criticality reveals that the black hole undergoes a small black hole-large black hole phase transition similar to that of the Van der Waals system, with a critical point and a critical ratio slightly smaller than that of the Van der Waals fluid. As the non-commutative parameter increases, the phase transition process shortens, leading to a critical point, and ultimately to the disappearance of the phase transition. The violation of the conventional first law results in a discontinuity of the Gibbs free energy during the phase transition, indicating the occurrence of zeroth-order phase transition. Moreover, we investigate the Joule-Thomson expansion, obtaining the minimum inversion temperature and the minimum inversion mass., Comment: 37pages, 11figures

In this article, we firstly studied wave function in static spherically symmetric spacetime and obtained effective potential of perturbed fields with spin. Then we applied $6^{\rm{th}}$ order WKB approximation to analyze quasinormal modes of Schwarzschild black hole in the Cold Dark Matter halo in perturbed fields with different spins and derived quasinormal frequencies. Further, to study the relation between quasinormal frequencies and optics, we compare the results of WKB method with eikonal limit formula. At last, we discussed greybody factor in different perturbed fields under this spacetime., Comment: 22 pages, 5 figures, 4 tables

We obtained the metric of the Schwarzschild-like black hole with loop quantum gravity (LQG) corrections in anti-de Sitter (AdS) space-time, under the assumption that the cosmological constant is decoupled in LQG. We investigated its thermodynamics, including the equation of state, criticality, heat capacity, and Gibbs free energy. The $P-v$ graph was plotted, and the critical behavior was calculated. It was found that, due to the LQG effect, the quantum-corrected Schwarzschild-AdS black hole exhibits a critical point and a critical ratio of $7/18$, which differs from the Reissner-Nordstr$\ddot{\mathrm{o}}$m-AdS black hole's ratio of $3/8$ (the same as that of the Van der Waals system) slightly. However, there are still some similarities compared to the Van der Waals system, such as the same critical exponents and a similar $P-v$ graph. Moreover, it is concluded that the energy-momentum tensor related to the black hole's mass could violate the conventional first law of thermodynamics. This modified first law may violate the conservation of Gibbs free energy during the small black hole-large black hole phase transitions, potentially indicating the occurrence of the zeroth-order phase transition. The Joule-Thomson expansion was also studied. Interestingly, compared to the Schwarzschild-AdS black hole, the LQG effect leads to inversion points. The inversion curve divides the $\left(P,T\right)$ coordinate system into two regions: a heating region and a cooling region, as shown in detail by the inversion curves and isenthalpic curves. The results indicated that there is a minimum inversion mass, below which any black hole will not possess an inversion point., Comment: 38 pages, 8 figures, 1 table

The optical properties of Euler-Heisenberg (EH) black hole (BH) surrounded by Cold Dark Matter (CDM) halo are investigated. By changing BH's parameters, we found that the radius of horizon r_{h} and radius of photon sphere r_{ph} will transparently increase as CDM halo parameters R and \rho increase. To show the influence of CDM halo on the BH's optical characteristics, we took two sets of R and \rho with prominent differences and plot the first four orders of images for thin accretion disk with different angle of inclination \theta of observer. The images with light intensity distributions using Novikov-Thorne (N-T) model are also derived, as well as the effective potential, photon orbits. Especially, analysis of intersection behaviors between photon trajectories with different impact parameters and circular time-like orbits in accretion disk will help better understand the image of thin accretion disk. Our results showed that CDM halo will make BH become more larger and dimmer distinctly., Comment: 42 pages,16 figures,4 tables

A generation method of new metric in the case of static spherically symmetric space-time is derived. Using this approach, we construct a metric which describes Euler-Heisenberg black hole surrounded by perfect fluid dark matter and investigate its optical and thermodynamic properties. We found that radius of shadow will increase with the increase of dark matter effect, and more strong dark matter will diminish the light intensity of accretion disk generally. Moreover, in thermodynamics, when quantum electrodynamic parameter is positive, there will be a critical value of dark matter parameter, which determine the number of black hole's critical points., Comment: 34 pages, 11 figures

The Schwarzschild black hole in the Cold Dark Matter (CDM) halo is studied, and the radiation laws of the thin accretion disk near the black hole are discussed and summarized. The orbits of light around the black hole are also calculated. Additionally, using the Novikov-Thorne model's light intensity function of the thin accretion disk, it is possible to solve for the shadow created by the thin accretion disk near the Schwarzschild black hole as well as the observed luminosity of the disk., Comment: 23pages