Tawfik A.N., Alshehri A.A. «Reissner–Nordström black hole: curvature and singularity with quantized fundamental tensor» Нелинейный мир, 22, № 3, с. 49-61 (2024)

To reveal the nature of curvatures and singularitis which are emerged with the proposed quantization imposed on the fundamental tensor, the timelike geodesic congruence of the Reissner–Nordström metric shall be derived, analytically, and analyzed, numerically. The evolution of the geodesic congruence expansion is found nonvanishing everywhere. Furthermore, as the radial distance from the singularity decreases, an extremely large geodesic congruence expansion evolution occurs. The proposed quantization seems to largely enhance and apparently enrich the profile of the geodesic congruence expansion evolution. That the Kretschmann scalar for both versions of the fundamental tensor is found finite everywhere allows for an unambiguous assessment that the curvatures and singularities are likely real and essential (not artifact in some coordinate systems). We conclude that the proposed quantization seems to locally sharpen the curvatures and hence the singularities of the charged, non-rotating, spherically symmetric, and massive Reissner–Nordström black hole. This finding would alter the Schwarzschild radius and even the entire black hole geometry, especially at relativistic quantum scales. We also conclude that the additional curvatures even with their approximate qualitative estimation point to a rich spacetime structure which is apparently overseen in the classical limit.

Tong Zhicheng, Li Yong «Quantitative uniform exponential acceleration of averages along decaying waves» Известия Российской академии наук. Серия математическая, 89, № 6, с. 105-130 (2025)

In this study, utilizing a specific exponential weighting function, we investigate the uniform exponential convergence of weighted Birkhoff averages along decaying waves and delve into several related variants. A key distinction from traditional scenarios is evident here: despite reduced regularity in observables, our method still maintains exponential convergence. In particular, we develop new techniques that yield very precise rates of exponential convergence, as evidenced by numerical simulations. Furthermore, this innovative approach extends to quantitative analyses involving different weighting functions employed by others, surpassing the limitations inherent in prior research. It also enhances the exponential convergence rates of weighted Birkhoff averages along quasi-periodic orbits via analytic observables. To the best of our knowledge, this is the first result on the uniform exponential acceleration beyond averages along quasi-periodic or almost periodic orbits, particularly from a quantitative perspective.